diff --git a/thys/Bernoulli/Bernoulli_Zeta.thy b/thys/Bernoulli/Bernoulli_Zeta.thy
--- a/thys/Bernoulli/Bernoulli_Zeta.thy
+++ b/thys/Bernoulli/Bernoulli_Zeta.thy
@@ -1,502 +1,502 @@
 section \<open>Bernoulli numbers and the zeta function at positive integers\<close>
 theory Bernoulli_Zeta
 imports 
-  "HOL-Analysis.Analysis"
+  "HOL-Complex_Analysis.Complex_Analysis"
   Bernoulli_FPS
 begin
 
 (* TODO: Move *)
 lemma joinpaths_cong: "f = f' \<Longrightarrow> g = g' \<Longrightarrow> f +++ g = f' +++ g'"
   by simp
 
 lemma linepath_cong: "a = a' \<Longrightarrow> b = b' \<Longrightarrow> linepath a b = linepath a' b'"
   by simp
 
 text \<open>
   The analytic continuation of the exponential generating function of the Bernoulli numbers
   is $\frac{z}{e^z - 1}$, which has simple poles at all $2ki\pi$ for $k\in\mathbb{Z}\setminus\{0\}$.
   We will need the residue at these poles:
 \<close>
 lemma residue_bernoulli:
   assumes "n \<noteq> 0"
   shows   "residue (\<lambda>z. 1 / (z ^ m * (exp z - 1))) (2 * pi * real_of_int n * \<i>) = 
              1 / (2 * pi * real_of_int n * \<i>) ^ m"
 proof -
   have "residue (\<lambda>z. (1 / z ^ m) / (exp z - 1)) (2 * pi * real_of_int n * \<i>) =
           1 / (2 * pi * real_of_int n * \<i>) ^ m / 1"
     using exp_integer_2pi[of "real_of_int n"] and assms
     by (rule_tac residue_simple_pole_deriv[where s="-{0}"])
        (auto intro!: holomorphic_intros derivative_eq_intros connected_open_delete_finite 
              simp add: mult_ac connected_punctured_universe)
   thus ?thesis by (simp add: divide_simps)
 qed
 
 text \<open>
   At positive integers greater than 1, the Riemann zeta function is simply the infinite
   sum $\zeta(n) = \sum_{k=1}^\infty k^{-n}$. For even $n$, this quantity can also be
   expressed in terms of Bernoulli numbers.
 
   To show this, we employ a similar strategy as in the meromorphic asymptotics approach:
   We apply the Residue Theorem to the exponential generating function of the Bernoulli numbers:
   \[\sum_{n=0}^\infty \frac{B_n}{n!} z^n = \frac{z}{e^z - 1}\]
   Recall that this function has poles at $2ki\pi$ for $k\in\mathbb{Z}\setminus\{0\}$.
   In the meromorphic asymptotics case, we integrated along a circle of radius $3i\pi$ in order
   to get the dominant singularities $2i\pi$ and $-2i\pi$. Now, however, we will not use a 
   fixed integration path, but we let the integration path become bigger and bigger. 
   Because the integrand decays relatively quickly if $n > 1$, the integral vanishes in the limit 
   and we obtain not just an asymptotic formula, but an exact representation of $B_n$ as an 
   infinite sum.
 
   For odd $n$, we have $B_n = 0$, but for even $n$, the residues at $2ki\pi$ and $-2ki\pi$ 
   combine nicely to $2\cdot(-2k\pi)^{-n}$, and after some simplification we get the formula
   for $B_n$.
 
   Another difference to the meromorphic asymptotics is that we now use a rectangle instead
   of a circle as the integration path. For the asymptotics, only a big-oh bound was needed
   for the integral over one fixed integration path, and the circular path was very convenient.
   However, now we need to explicitly bound the integral for a whole sequence of integration paths
   that grow in size, and bounding $e^z - 1$ for $z$ on a circle is very tedious. On a rectangle,
   this term can be bounded much more easily. Still, we have to do this separately for all four
   edges of the rectangle, which will be a bit tedious.
 \<close>
 theorem nat_even_power_sums_complex:
   assumes n': "n' > 0"
   shows   "(\<lambda>k. 1 / of_nat (Suc k) ^ (2*n') :: complex) sums
              of_real ((-1) ^ Suc n' * bernoulli (2*n') * (2 * pi) ^ (2 * n') / (2 * fact (2*n')))"
 proof -
   define n where "n = 2 * n'"
   from n' have n: "n \<ge> 2" "even n" by (auto simp: n_def)
   define zeta :: complex where "zeta = (\<Sum>k. 1 / of_nat (Suc k) ^ n)"
 
   have "summable (\<lambda>k. 1 / of_nat (Suc k) ^ n :: complex)"
     using inverse_power_summable[of n] n
     by (subst summable_Suc_iff) (simp add: divide_simps)
   hence "(\<lambda>k. \<Sum>i<k. 1 / of_nat (Suc i) ^ n) \<longlonglongrightarrow> zeta"
     by (subst (asm) summable_sums_iff) (simp add: sums_def zeta_def)
   also have "(\<lambda>k. \<Sum>i<k. 1 / of_nat (Suc i) ^ n) = (\<lambda>k. \<Sum>i\<in>{0<..k}. 1 / of_nat i ^ n)"
     by (intro ext sum.reindex_bij_witness[of _ "\<lambda>n. n - 1" Suc]) auto
   finally have zeta_limit: "(\<lambda>k. \<Sum>i\<in>{0<..k}. 1 / of_nat i ^ n) \<longlonglongrightarrow> zeta" .
 
   \<comment> \<open>This is the exponential generating function of the Bernoulli numbers.\<close>
   define f where "f = (\<lambda>z::complex. if z = 0 then 1 else z / (exp z - 1))"
 
   \<comment> \<open>We will integrate over this function, since its residue at the origin
       is the $n$-th coefficient of @{term f}. Note that it has singularities
       at all points $2ik\pi$ for $k\in\mathbb{Z}$.\<close>
   define g where "g = (\<lambda>z::complex. 1 / (z ^ n * (exp z - 1)))"
 
   \<comment> \<open>We integrate along a rectangle of width $2m$ and height $2(2m+1)\pi$
       with its centre at the origin. The benefit of the rectangular path is that
       it is easier to bound the value of the exponential appearing in the integrand.
       The horizontal lines of the rectangle are always right in the middle between 
       two adjacent singularities.\<close>
   define \<gamma> :: "nat \<Rightarrow> real \<Rightarrow> complex" 
     where "\<gamma> = (\<lambda>m. rectpath (-real m - real (2*m+1)*pi*\<i>) (real m + real (2*m+1)*pi*\<i>))"
 
   \<comment> \<open>This set is a convex open enclosing set the contains our path.\<close>
   define A where "A = (\<lambda>m::nat. box (-(real m+1) - (2*m+2)*pi*\<i>) (real m+1 + (2*m+2)*pi*\<i>))"
 
   \<comment> \<open>These are all the singularities in the enclosing inside the path
       (and also inside @{term A}).\<close>
   define S where "S = (\<lambda>m::nat. (\<lambda>n. 2 * pi * of_int n * \<i>) ` {-m..m})"
 
   \<comment> \<open>Any singularity in @{term A} is of the form $2ki\pi$ where $|k| \leq m$.\<close>
   have int_bound: "k \<in> {-int m..int m}" if "2 * pi * k * \<i> \<in> A m" for k m
   proof -
     from that have "(-real (Suc m)) * (2 * pi) < real_of_int k * (2 * pi) \<and> 
                         real (Suc m) * (2 * pi) > real_of_int k * (2 * pi)"
       by (auto simp: A_def in_box_complex_iff algebra_simps)
     hence "-real (Suc m) < real_of_int k \<and> real_of_int k < real (Suc m)"
       by simp
     also have "-real (Suc m) = real_of_int (-int (Suc m))" by simp
     also have "real (Suc m) = real_of_int (int (Suc m))" by simp
     also have "real_of_int (- int (Suc m)) < real_of_int k \<and> 
                  real_of_int k < real_of_int (int (Suc m)) \<longleftrightarrow> k \<in> {-int m..int m}" 
       by (subst of_int_less_iff) auto
     finally show "k \<in> {-int m..int m}" .
   qed
 
   have zeros: "\<exists>k\<in>{-int m..int m}. z = 2 * pi * of_int k * \<i>" if "z \<in> A m" "exp z = 1" for z m
   proof -
     from that(2) obtain k where z_eq: "z = 2 * pi * of_int k * \<i>"
       unfolding exp_eq_1 by (auto simp: complex_eq_iff)
     with int_bound[of k] and that(1) show ?thesis by auto
   qed
   have zeros': "z ^ n * (exp z - 1) \<noteq> 0" if "z \<in> A m - S m" for z m
     using zeros[of z] that by (auto simp: S_def)
 
   \<comment> \<open>The singularities all lie strictly inside the integration path.\<close>
   have subset: "S m \<subseteq> box (-real m - real(2*m+1)*pi*\<i>) (real m + real(2*m+1)*pi*\<i>)" if "m > 0" for m
   proof (rule, goal_cases)
     case (1 z)
     then obtain k :: int where k: "k \<in> {-int m..int m}" "z = 2 * pi * k * \<i>"
       unfolding S_def by blast
     have "2 * pi * -m + -pi < 2 * pi * k + 0"
       using k by (intro add_le_less_mono mult_left_mono) auto
     moreover have "2 * pi * k + 0 < 2 * pi * m + pi"
       using k by (intro add_le_less_mono mult_left_mono) auto
     ultimately show ?case using k \<open>m > 0\<close>
       by (auto simp: A_def in_box_complex_iff algebra_simps)
   qed
   from n and zeros' have holo: "g holomorphic_on A m - S m" for m
     unfolding g_def by (intro holomorphic_intros) auto
 
   \<comment> \<open>The integration path lies completely inside $A$ and does not cross
       any singularities.\<close>
   have path_subset: "path_image (\<gamma> m) \<subseteq> A m - S m" if "m > 0" for m
   proof -
     have "path_image (\<gamma> m) \<subseteq> cbox (-real m - (2 * m + 1) * pi * \<i>) (real m + (2 * m + 1) * pi * \<i>)"
       unfolding \<gamma>_def by (rule path_image_rectpath_subset_cbox) auto
     also have "\<dots> \<subseteq> A m" unfolding A_def
       by (subst subset_box_complex) auto
     finally have "path_image (\<gamma> m) \<subseteq> A m" .
     moreover have "path_image (\<gamma> m) \<inter> S m = {}"
     proof safe
       fix z assume z: "z \<in> path_image (\<gamma> m)" "z \<in> S m"
       from this(2) obtain k :: int where k: "z = 2 * pi * k * \<i>"
         by (auto simp: S_def)
       hence [simp]: "Re z = 0" by simp
       from z(1) have "\<bar>Im z\<bar> = of_int (2*m+1) * pi"
         using \<open>m > 0\<close> by (auto simp: \<gamma>_def path_image_rectpath)
       also have "\<bar>Im z\<bar> = of_int (2 * \<bar>k\<bar>) * pi"
         by (simp add: k abs_mult)
       finally have "2 * \<bar>k\<bar> = 2 * m + 1"
         by (subst (asm) mult_cancel_right, subst (asm) of_int_eq_iff) simp
       hence False by presburger
       thus "z \<in> {}" ..
     qed
     ultimately show "path_image (\<gamma> m) \<subseteq> A m - S m" by blast
   qed
 
   \<comment> \<open>We now obtain a closed form for the Bernoulli numbers using the integral.\<close>
   have eq: "(\<Sum>x\<in>{0<..m}. 1 / of_nat x ^ n) =
               contour_integral (\<gamma> m) g * (2 * pi * \<i>) ^ n / (4 * pi * \<i>) -
               complex_of_real (bernoulli n / fact n) * (2 * pi * \<i>) ^ n / 2" 
     if m: "m > 0" for m
   proof -
     \<comment> \<open>We relate the formal power series of the Bernoulli numbers to the
         corresponding complex function.\<close>
     have "subdegree (fps_exp 1 - 1 :: complex fps) = 1"
       by (intro subdegreeI) auto
     hence expansion: "f has_fps_expansion bernoulli_fps" 
       unfolding f_def bernoulli_fps_def by (auto intro!: fps_expansion_intros)
 
     \<comment> \<open>We use the Residue Theorem to explicitly compute the integral.\<close>
     have "contour_integral (\<gamma> m) g =
              2 * pi * \<i> * (\<Sum>z\<in>S m. winding_number (\<gamma> m) z * residue g z)"
     proof (rule Residue_theorem)
       have "cbox (-real m - (2 * m + 1) * pi * \<i>) (real m + (2 * m + 1) * pi * \<i>) \<subseteq> A m"
         unfolding A_def by (subst subset_box_complex) simp_all
       thus "\<forall>z. z \<notin> A m \<longrightarrow> winding_number (\<gamma> m) z = 0" unfolding \<gamma>_def
         by (intro winding_number_rectpath_outside allI impI) auto
     qed (insert holo path_subset m, auto simp: \<gamma>_def A_def S_def intro: convex_connected)
     \<comment> \<open>Clearly, all the winding numbers are 1\<close>
     also have "winding_number (\<gamma> m) z = 1" if "z \<in> S m" for z
       unfolding \<gamma>_def using subset[of m] that m by (subst winding_number_rectpath) blast+
     hence "(\<Sum>z\<in>S m. winding_number (\<gamma> m) z * residue g z) = (\<Sum>z\<in>S m. residue g z)"
       by (intro sum.cong) simp_all
     also have "\<dots> = (\<Sum>k=-int m..int m. residue g (2 * pi * of_int k * \<i>))"
       unfolding S_def by (subst sum.reindex) (auto simp: inj_on_def o_def)
     also have "{-int m..int m} = insert 0 ({-int m..int m}-{0})"
       by auto
     also have "(\<Sum>k\<in>\<dots>. residue g (2 * pi * of_int k * \<i>)) = 
                  residue g 0 + (\<Sum>k\<in>{-int m..m}-{0}. residue g (2 * pi * of_int k * \<i>))"
       by (subst sum.insert) auto
     \<comment> \<open>The residue at the origin is just the $n$-th coefficient of $f$.\<close>
     also have "residue g 0 = residue (\<lambda>z. f z / z ^ Suc n) 0" unfolding f_def g_def
       by (intro residue_cong eventually_mono[OF eventually_at_ball[of 1]]) auto
     also have "\<dots> = fps_nth bernoulli_fps n"
       by (rule residue_fps_expansion_over_power_at_0 [OF expansion])
     also have "\<dots> = of_real (bernoulli n / fact n)" 
       by simp
     also have "(\<Sum>k\<in>{-int m..m}-{0}. residue g (2 * pi * of_int k * \<i>)) = 
                  (\<Sum>k\<in>{-int m..m}-{0}. 1 / of_int k ^ n) / (2 * pi * \<i>) ^ n"
     proof (subst sum_divide_distrib, intro refl sum.cong, goal_cases)
       case (1 k)
       hence *: "residue g (2 * pi * of_int k * \<i>) = 1 / (2 * complex_of_real pi * of_int k * \<i>) ^ n"
         unfolding g_def by (subst residue_bernoulli) auto
       thus ?case using 1 by (subst *) (simp add: divide_simps power_mult_distrib)
     qed
     also have "(\<Sum>k\<in>{-int m..m}-{0}. 1 / of_int k ^ n) =
                  (\<Sum>(a,b)\<in>{0<..m}\<times>{-1,1::int}. 1 / of_int (int a) ^ n :: complex)" using n
       by (intro sum.reindex_bij_witness[of _ "\<lambda>k. snd k * int (fst k)" "\<lambda>k. (nat \<bar>k\<bar>,sgn k)"])
          (auto split: if_splits simp: abs_if)
     also have "\<dots> = (\<Sum>x\<in>{0<..m}. 2 / of_nat x ^ n)"
       using n by (subst sum.Sigma [symmetric]) auto
     also have "\<dots> = (\<Sum>x\<in>{0<..m}. 1 / of_nat x ^ n) * 2"
       by (simp add: sum_distrib_right)
     finally show ?thesis 
       by (simp add: field_simps)
   qed
 
   \<comment> \<open>The ugly part: We have to prove a bound on the integral by splitting
       it into four integrals over lines and bounding each part separately.\<close>
   have "eventually (\<lambda>m. norm (contour_integral (\<gamma> m) g) \<le> 
           ((4 + 12 * pi) + 6 * pi / m) / real m ^ (n - 1)) sequentially"
     using eventually_gt_at_top[of "1::nat"]
   proof eventually_elim
     case (elim m)
     let ?c = "(2*m+1) * pi * \<i>"
     define I where "I = (\<lambda>p1 p2. contour_integral (linepath p1 p2) g)"
     define p1 p2 p3 p4 where "p1 = -real m - ?c" and "p2 = real m - ?c" 
                          and "p3 = real m + ?c" and "p4 = -real m + ?c"
     have eq: "\<gamma> m = linepath p1 p2 +++ linepath p2 p3 +++ linepath p3 p4 +++ linepath p4 p1"
       (is "\<gamma> m = ?\<gamma>'") unfolding \<gamma>_def rectpath_def Let_def
       by (intro joinpaths_cong linepath_cong) 
          (simp_all add: p1_def p2_def p3_def p4_def complex_eq_iff)
     have integrable: "g contour_integrable_on \<gamma> m" using elim
       by (intro contour_integrable_holomorphic_simple[OF holo _ _ path_subset])
          (auto simp: \<gamma>_def A_def S_def intro!: finite_imp_closed)
     have "norm (contour_integral (\<gamma> m) g) = norm (I p1 p2 + I p2 p3 + I p3 p4 + I p4 p1)"
       unfolding I_def by (insert integrable, unfold eq)
                          (subst contour_integral_join; (force simp: add_ac)?)+
     also have "\<dots> \<le> norm (I p1 p2) + norm (I p2 p3) + norm (I p3 p4) + norm (I p4 p1)"
       by (intro norm_triangle_mono order.refl)
 
     also have "norm (I p1 p2) \<le> 1 / real m ^ n * norm (p2 - p1)" (is "_ \<le> ?B1 * _")
       unfolding I_def
     proof (intro contour_integral_bound_linepath)
       fix z assume z: "z \<in> closed_segment p1 p2"
       define a where "a = Re z"
       from z have z: "z = a - (2*m+1) * pi * \<i>"
         by (subst (asm) closed_segment_same_Im)
            (auto simp: p1_def p2_def complex_eq_iff a_def)
       have "real m * 1 \<le> (2*m+1) * pi"
         using pi_ge_two by (intro mult_mono) auto
       also have "(2*m+1) * pi = \<bar>Im z\<bar>" by (simp add: z)
       also have "\<bar>Im z\<bar> \<le> norm z" by (rule abs_Im_le_cmod)
       finally have "norm z \<ge> m" by simp
       moreover {
         have "exp z - 1 = -of_real (exp a + 1)" using exp_integer_2pi_plus1[of m]
           by (simp add: z exp_diff algebra_simps exp_of_real)
         also have "norm \<dots> \<ge> 1"
           unfolding norm_minus_cancel norm_of_real by simp
         finally have "norm (exp z - 1) \<ge> 1" .
       }
       ultimately have "norm z ^ n * norm (exp z - 1) \<ge> real m ^ n * 1"
         by (intro mult_mono power_mono) auto
       thus "norm (g z) \<le> 1 / real m ^ n" using elim
         by (simp add: g_def divide_simps norm_divide norm_mult norm_power mult_less_0_iff)
     qed (insert integrable, auto simp: eq)
     also have "norm (p2 - p1) = 2 * m" by (simp add: p2_def p1_def)
 
     also have "norm (I p3 p4) \<le> 1 / real m ^ n * norm (p4 - p3)" (is "_ \<le> ?B3 * _")
       unfolding I_def
     proof (intro contour_integral_bound_linepath)
       fix z assume z: "z \<in> closed_segment p3 p4"
       define a where "a = Re z"
       from z have z: "z = a + (2*m+1) * pi * \<i>"
         by (subst (asm) closed_segment_same_Im)
            (auto simp: p3_def p4_def complex_eq_iff a_def)
       have "real m * 1 \<le> (2*m+1) * pi"
         using pi_ge_two by (intro mult_mono) auto
       also have "(2*m+1) * pi = \<bar>Im z\<bar>" by (simp add: z)
       also have "\<bar>Im z\<bar> \<le> norm z" by (rule abs_Im_le_cmod)
       finally have "norm z \<ge> m" by simp
       moreover {
         have "exp z - 1 = -of_real (exp a + 1)" using exp_integer_2pi_plus1[of m]
           by (simp add: z exp_add algebra_simps exp_of_real)
         also have "norm \<dots> \<ge> 1"
           unfolding norm_minus_cancel norm_of_real by simp
         finally have "norm (exp z - 1) \<ge> 1" .
       }
       ultimately have "norm z ^ n * norm (exp z - 1) \<ge> real m ^ n * 1"
         by (intro mult_mono power_mono) auto
       thus "norm (g z) \<le> 1 / real m ^ n" using elim
         by (simp add: g_def divide_simps norm_divide norm_mult norm_power mult_less_0_iff)
     qed (insert integrable, auto simp: eq)
     also have "norm (p4 - p3) = 2 * m" by (simp add: p4_def p3_def)
 
     also have "norm (I p2 p3) \<le> (1 / real m ^ n) * norm (p3 - p2)" (is "_ \<le> ?B2 * _")
       unfolding I_def
     proof (rule contour_integral_bound_linepath)
       fix z assume z: "z \<in> closed_segment p2 p3"
       define b where "b = Im z"
       from z have z: "z = m + b * \<i>"
         by (subst (asm) closed_segment_same_Re)
            (auto simp: p2_def p3_def algebra_simps complex_eq_iff b_def)
       from elim have "2 \<le> 1 + real m" by simp
       also have "\<dots> \<le> exp (real m)" by (rule exp_ge_add_one_self)
       also have "exp (real m) - 1 = norm (exp z) - norm (1::complex)"
         by (simp add: z)
       also have "\<dots> \<le> norm (exp z - 1)"
         by (rule norm_triangle_ineq2)
       finally have "norm (exp z - 1) \<ge> 1" by simp
       moreover have "norm z \<ge> m"
         using z and abs_Re_le_cmod[of z] by simp
       ultimately have "norm z ^ n * norm (exp z - 1) \<ge> real m ^ n * 1" using elim
         by (intro mult_mono power_mono) (auto simp: z)
       thus "norm (g z) \<le> 1 / real m ^ n" using n and elim
         by (simp add: g_def norm_mult norm_divide norm_power divide_simps mult_less_0_iff)
     qed (insert integrable, auto simp: eq)
     also have "p3 - p2 = of_real (2*(2*real m+1)*pi) * \<i>" by (simp add: p2_def p3_def)
     also have "norm \<dots> = 2 * (2 * real m + 1) * pi"
       unfolding norm_mult norm_of_real by simp
 
     also have "norm (I p4 p1) \<le> (2 / real m ^ n) * norm (p1 - p4)" (is "_ \<le> ?B4 * _")
       unfolding I_def
     proof (rule contour_integral_bound_linepath)
       fix z assume z: "z \<in> closed_segment p4 p1"
       define b where "b = Im z"
       from z have z: "z = -real m + b * \<i>"
         by (subst (asm) closed_segment_same_Re)
            (auto simp: p1_def p4_def algebra_simps b_def complex_eq_iff)
       from elim have "2 \<le> 1 + real m" by simp
       also have "\<dots> \<le> exp (real m)" by (rule exp_ge_add_one_self)
       finally have "1 / 2 \<le> 1 - exp (-real m)"
         by (subst exp_minus) (simp add: field_simps)
       also have "1 - exp (-real m) = norm (1::complex) - norm (exp z)"
         by (simp add: z)
       also have "\<dots> \<le> norm (exp z - 1)"
         by (subst norm_minus_commute, rule norm_triangle_ineq2)
       finally have "norm (exp z - 1) \<ge> 1 / 2" by simp
       moreover have "norm z \<ge> m"
         using z and abs_Re_le_cmod[of z] by simp
       ultimately have "norm z ^ n * norm (exp z - 1) \<ge> real m ^ n * (1 / 2)" using elim
         by (intro mult_mono power_mono) (auto simp: z)
       thus "norm (g z) \<le> 2 / real m ^ n" using n and elim
         by (simp add: g_def norm_mult norm_divide norm_power divide_simps mult_less_0_iff)
     qed (insert integrable, auto simp: eq)
     also have "p1 - p4 = -of_real (2*(2*real m+1)*pi) * \<i>" 
       by (simp add: p1_def p4_def algebra_simps)
     also have "norm \<dots> = 2 * (2 * real m + 1) * pi"
       unfolding norm_mult norm_of_real norm_minus_cancel by simp
 
     also have "?B1 * (2*m) + ?B2 * (2*(2*real m+1)*pi) + ?B3 * (2*m) + ?B4 * (2*(2*real m+1)*pi) =
                  (4 * m + 6 * (2 * m + 1) * pi) / real m ^ n"
       by (simp add: divide_simps)
     also have "(4 * m + 6 * (2 * m + 1) * pi) = (4 + 12 * pi) * m + 6 * pi"
       by (simp add: algebra_simps)
     also have "\<dots> / real m ^ n = ((4 + 12 * pi) + 6 * pi / m) / real m ^ (n - 1)"
       using n by (cases n) (simp_all add: divide_simps)
     finally show "cmod (contour_integral (\<gamma> m) g) \<le> \<dots>" by simp
   qed
 
   \<comment> \<open>It is clear that this bound goes to 0 since @{prop "n \<ge> 2"}.\<close>
   moreover have "(\<lambda>m. (4 + 12 * pi + 6 * pi / real m) / real m ^ (n - 1)) \<longlonglongrightarrow> 0"
     by (rule real_tendsto_divide_at_top tendsto_add tendsto_const 
           filterlim_real_sequentially filterlim_pow_at_top | use n in simp)+
   ultimately have *: "(\<lambda>m. contour_integral (\<gamma> m) g) \<longlonglongrightarrow> 0"
     by (rule Lim_null_comparison)
 
   \<comment> \<open>Since the infinite sum over the residues can expressed using the
       zeta function, we have now related the Bernoulli numbers at even
       positive integers to the zeta function.\<close>
 
   have "(\<lambda>m. contour_integral (\<gamma> m) g * (2 * pi * \<i>) ^ n / (4 * pi * \<i>) -
              of_real (bernoulli n / fact n) * (2 * pi * \<i>) ^ n / 2) \<longlonglongrightarrow>
            0 * (2 * pi * \<i>) ^ n / (4 * pi * \<i>) - 
            of_real (bernoulli n / fact n) * (2 * pi * \<i>) ^ n / 2"
     using n by (intro tendsto_intros * zeta_limit) auto
   also have "?this \<longleftrightarrow> (\<lambda>m. \<Sum>k\<in>{0<..m}. 1 / of_nat k ^ n) \<longlonglongrightarrow> 
                - of_real (bernoulli n / fact n) * (2 * pi * \<i>) ^ n / 2"
     by (intro filterlim_cong eventually_mono [OF eventually_gt_at_top[of "0::nat"]])
        (use eq in simp_all)
   finally have "(\<lambda>m. \<Sum>k\<in>{0<..m}. 1 / of_nat k ^ n)
                    \<longlonglongrightarrow> - of_real (bernoulli n / fact n) * (of_real (2 * pi) * \<i>) ^ n / 2" 
     (is "_ \<longlonglongrightarrow> ?L") .
   also have "(\<lambda>m. \<Sum>k\<in>{0<..m}. 1 / of_nat k ^ n) = (\<lambda>m. \<Sum>k\<in>{..<m}. 1 / of_nat (Suc k) ^ n)"
     by (intro ext sum.reindex_bij_witness[of _ Suc "\<lambda>n. n - 1"]) auto
   also have "\<dots> \<longlonglongrightarrow> ?L \<longleftrightarrow> (\<lambda>k. 1 / of_nat (Suc k) ^ n) sums ?L"
     by (simp add: sums_def)
   also have "(2 * pi * \<i>) ^ n = (2 * pi) ^ n * (-1) ^ n'"
     by (simp add: n_def divide_simps power_mult_distrib power_mult power_minus')
   also have "- of_real (bernoulli n / fact n) * \<dots> / 2 =
                of_real ((-1) ^ Suc n' * bernoulli (2*n') * (2*pi)^(2*n') / (2 * fact (2*n')))"
     by (simp add: n_def divide_simps)
   finally show ?thesis unfolding n_def .
 qed
 
 corollary nat_even_power_sums_real:
   assumes n': "n' > 0"
   shows   "(\<lambda>k. 1 / real (Suc k) ^ (2*n')) sums
              ((-1) ^ Suc n' * bernoulli (2*n') * (2 * pi) ^ (2 * n') / (2 * fact (2*n')))"
     (is "?f sums ?L")
 proof -
   have "(\<lambda>k. complex_of_real (?f k)) sums complex_of_real ?L"
     using nat_even_power_sums_complex[OF assms] by simp
   thus ?thesis by (simp only: sums_of_real_iff)
 qed
 
 text \<open>
   We can now also easily determine the signs of Bernoulli numbers: the above formula 
   clearly shows that the signs of $B_{2n}$ alternate as $n$ increases, and we already know
   that $B_{2n+1} = 0$ for any positive $n$. A lot of other facts about the signs of
   Bernoulli numbers follow.
 \<close>
 corollary sgn_bernoulli_even:
   assumes "n > 0"
   shows   "sgn (bernoulli (2 * n)) = (-1) ^ Suc n"
 proof -
   have *: "(\<lambda>k. 1 / real (Suc k) ^ (2 * n)) sums
              ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))"
     using assms by (rule nat_even_power_sums_real)
   from * have "0 < (\<Sum>k. 1 / real (Suc k) ^ (2*n))"
     by (intro suminf_pos) (auto simp: sums_iff)
   hence "sgn (\<Sum>k. 1 / real (Suc k) ^ (2*n)) = 1"
     by simp
   also have "(\<Sum>k. 1 / real (Suc k) ^ (2*n)) = 
                (- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n))"
     using * by (simp add: sums_iff)
   also have "sgn \<dots> = (-1) ^ Suc n * sgn (bernoulli (2 * n))"
     by (simp add: sgn_mult)
   finally show ?thesis
     by (simp add: minus_one_power_iff split: if_splits)
 qed
 
 corollary bernoulli_even_nonzero: "even n \<Longrightarrow> bernoulli n \<noteq> 0"
   using sgn_bernoulli_even[of "n div 2"] by (cases "n = 0") (auto elim!: evenE)
 
 corollary sgn_bernoulli: 
   "sgn (bernoulli n) = 
      (if n = 0 then 1 else if n = 1 then -1 else if odd n then 0 else (-1) ^ Suc (n div 2))"
   using sgn_bernoulli_even [of "n div 2"] by (auto simp: bernoulli_odd_eq_0)
 
 corollary bernoulli_zero_iff: "bernoulli n = 0 \<longleftrightarrow> odd n \<and> n \<noteq> 1"
   by (auto simp: bernoulli_even_nonzero bernoulli_odd_eq_0)
 
 corollary bernoulli'_zero_iff: "(bernoulli' n = 0) \<longleftrightarrow> (n \<noteq> 1 \<and> odd n)"
   by (auto simp: bernoulli'_def bernoulli_zero_iff)
 
 corollary bernoulli_pos_iff: "bernoulli n > 0 \<longleftrightarrow> n = 0 \<or> n mod 4 = 2"
 proof -
   have "bernoulli n > 0 \<longleftrightarrow> sgn (bernoulli n) = 1"
     by (simp add: sgn_if)
   also have "\<dots> \<longleftrightarrow> n = 0 \<or> even n \<and> odd (n div 2)"
     by (subst sgn_bernoulli) auto
   also have "even n \<and> odd (n div 2) \<longleftrightarrow> n mod 4 = 2"
     by presburger
   finally show ?thesis .
 qed
 
 corollary bernoulli_neg_iff: "bernoulli n < 0 \<longleftrightarrow> n = 1 \<or> n > 0 \<and> 4 dvd n"
 proof -
   have "bernoulli n < 0 \<longleftrightarrow> sgn (bernoulli n) = -1"
     by (simp add: sgn_if)
   also have "\<dots> \<longleftrightarrow> n = 1 \<or> n > 0 \<and> even n \<and> even (n div 2)"
     by (subst sgn_bernoulli) (auto simp: minus_one_power_iff)
   also have "even n \<and> even (n div 2) \<longleftrightarrow> 4 dvd n"
     by presburger
   finally show ?thesis .
 qed
 
 
 text \<open>
   We also get the solution of the Basel problem (the sum over all squares of positive
   integers) and any `Basel-like' problem with even exponent. The case of odd exponents
   is much more complicated and no similarly nice closed form is known for these.
 \<close>
 
 corollary nat_squares_sums: "(\<lambda>n. 1 / (n+1) ^ 2) sums (pi ^ 2 / 6)"
   using nat_even_power_sums_real[of 1] by (simp add: fact_numeral)
 
 corollary nat_power4_sums: "(\<lambda>n. 1 / (n+1) ^ 4) sums (pi ^ 4 / 90)"
   using nat_even_power_sums_real[of 2] by (simp add: fact_numeral)
 
 corollary nat_power6_sums: "(\<lambda>n. 1 / (n+1) ^ 6) sums (pi ^ 6 / 945)"
   using nat_even_power_sums_real[of 3] by (simp add: fact_numeral)
 
 corollary nat_power8_sums: "(\<lambda>n. 1 / (n+1) ^ 8) sums (pi ^ 8 / 9450)"
   using nat_even_power_sums_real[of 4] by (simp add: fact_numeral)
 
 end
diff --git a/thys/Bernoulli/ROOT b/thys/Bernoulli/ROOT
--- a/thys/Bernoulli/ROOT
+++ b/thys/Bernoulli/ROOT
@@ -1,16 +1,16 @@
 chapter AFP
 
-session Bernoulli (AFP) = "HOL-Analysis" +
+session Bernoulli (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 600]
   sessions
     "HOL-Analysis"
     "HOL-Number_Theory"
     "HOL-Computational_Algebra"
   theories
     Bernoulli
     Periodic_Bernpoly
     Bernoulli_FPS
     Bernoulli_Zeta
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/Cartan_FP/Cartan.thy b/thys/Cartan_FP/Cartan.thy
--- a/thys/Cartan_FP/Cartan.thy
+++ b/thys/Cartan_FP/Cartan.thy
@@ -1,534 +1,534 @@
 theory Cartan
-imports "HOL-Analysis.Analysis"
+imports "HOL-Complex_Analysis.Complex_Analysis"
 
 begin
 
 section\<open>First Cartan Theorem\<close>
 
 text\<open>Ported from HOL Light. See
       Gianni Ciolli, Graziano Gentili, Marco Maggesi.
       A Certified Proof of the Cartan Fixed Point Theorems.
       J Automated Reasoning (2011) 47:319--336    DOI 10.1007/s10817-010-9198-6\<close>
 
 lemma deriv_left_inverse:
   assumes "f holomorphic_on S" and "g holomorphic_on T"
       and "open S" and "open T"
       and "f ` S \<subseteq> T"
       and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
       and "w \<in> S"
     shows "deriv f w * deriv g (f w) = 1"
 proof -
   have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
     by (simp add: algebra_simps)
   also have "... = deriv (g o f) w"
     using assms
-    by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
+    by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_chain image_subset_iff)
   also have "... = deriv id w"
     apply (rule complex_derivative_transform_within_open [where s=S])
     apply (rule assms holomorphic_on_compose_gen holomorphic_intros)+
     apply simp
     done
   also have "... = 1"
     using higher_deriv_id [of 1] by simp
   finally show ?thesis .
 qed
 
 lemma Cauchy_higher_deriv_bound:
     assumes holf: "f holomorphic_on (ball z r)"
         and contf: "continuous_on (cball z r) f"
         and "0 < r" and "0 < n"
         and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
       shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
 proof -
   have "0 < B0" using \<open>0 < r\<close> fin [of z]
     by (metis ball_eq_empty ex_in_conv fin not_less)
   have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
     apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
     apply (auto simp: \<open>0 < r\<close>  dist_norm norm_minus_commute)
     apply (rule continuous_intros contf)+
     using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
     done
   have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
     using \<open>0 < n\<close> by simp
   also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
     by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close> holomorphic_on_const)
   finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
   have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
     by (rule contf continuous_intros)+
   have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
     by (simp add: holf holomorphic_on_diff holomorphic_on_const)
   define a where "a = (2 * pi)/(fact n)"
   have "0 < a"  by (simp add: a_def)
   have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
     using \<open>0 < r\<close> by (simp add: a_def divide_simps)
   have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
     using \<open>0 < r\<close> \<open>0 < n\<close>
     by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
   have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
         \<le> (B0/r^(Suc n)) * (2 * pi * r)"
     apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
     using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
     using \<open>0 < B0\<close> \<open>0 < r\<close>
     apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
     done
   then show ?thesis
     using \<open>0 < r\<close>
     by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
 qed
 
 lemma higher_deriv_comp_lemma:
     assumes s: "open s" and holf: "f holomorphic_on s"
         and "z \<in> s"
         and t: "open t" and holg: "g holomorphic_on t"
         and fst: "f ` s \<subseteq> t"
         and n: "i \<le> n"
         and dfz: "deriv f z = 1" and zero: "\<And>i. \<lbrakk>1 < i; i \<le> n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0"
       shows "(deriv ^^ i) (g o f) z = (deriv ^^ i) g (f z)"
 using n holg
 proof (induction i arbitrary: g)
   case 0 then show ?case by simp
 next
   case (Suc i)
   have "g \<circ> f holomorphic_on s" using "Suc.prems" holf
     using fst  by (simp add: holomorphic_on_compose_gen image_subset_iff)
   then have 1: "deriv (g \<circ> f) holomorphic_on s"
     by (simp add: holomorphic_deriv s)
   have dg: "deriv g holomorphic_on t"
     using Suc.prems by (simp add: Suc.prems(2) holomorphic_deriv t)
   then have "deriv g holomorphic_on f ` s"
     using fst  by (simp add: holomorphic_on_subset image_subset_iff)
   then have dgf: "(deriv g o f) holomorphic_on s"
     by (simp add: holf holomorphic_on_compose)
   then have 2: "(\<lambda>w. (deriv g o f) w * deriv f w) holomorphic_on s"
     by (blast intro: holomorphic_intros holomorphic_on_compose holf s)
   have "(deriv ^^ i) (deriv (g o f)) z = (deriv ^^ i) (\<lambda>w. deriv g (f w) * deriv f w) z"
     apply (rule higher_deriv_transform_within_open [OF 1 2 [unfolded o_def] s \<open>z \<in> s\<close>])
-    apply (rule complex_derivative_chain)
+    apply (rule deriv_chain)
     using holf Suc.prems fst apply (auto simp: holomorphic_on_imp_differentiable_at s t)
     done
   also have "... = (\<Sum>j=0..i. of_nat(i choose j) * (deriv ^^ j) (\<lambda>w. deriv g (f w)) z * (deriv ^^ (i - j)) (deriv f) z)"
     apply (rule higher_deriv_mult [OF dgf [unfolded o_def] _ s \<open>z \<in> s\<close>])
     by (simp add: holf holomorphic_deriv s)
   also have "... = (\<Sum>j=i..i. of_nat(i choose j) * (deriv ^^ j) (\<lambda>w. deriv g (f w)) z * (deriv ^^ Suc (i - j)) f z)"
   proof -
     have *: "(deriv ^^ j) (\<lambda>w. deriv g (f w)) z = 0"  if "j < i" and nz: "(deriv ^^ (i - j)) (deriv f) z \<noteq> 0" for j
     proof -
       have "1 < Suc (i - j)" "Suc (i - j) \<le> n"
         using \<open>j < i\<close> \<open>Suc i \<le> n\<close> by auto
       then show ?thesis  by (metis comp_def funpow.simps(2) funpow_swap1 zero nz)
     qed
     then show ?thesis
       apply (simp only: funpow_Suc_right o_def)
       apply (rule comm_monoid_add_class.sum.mono_neutral_right, auto)
       done
   qed
   also have "... = (deriv ^^ i) (deriv g) (f z)"
     using Suc.IH [OF _ dg] Suc.prems by (simp add: dfz)
   finally show ?case
     by (simp only: funpow_Suc_right o_def)
 qed
 
 
 lemma higher_deriv_comp_iter_lemma:
     assumes s: "open s" and holf: "f holomorphic_on s"
         and fss: "f ` s \<subseteq> s"
         and "z \<in> s" and [simp]: "f z = z"
         and n: "i \<le> n"
         and dfz: "deriv f z = 1" and zero: "\<And>i. \<lbrakk>1 < i; i \<le> n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0"
       shows "(deriv ^^ i) (f^^m) z = (deriv ^^ i) f z"
 proof -
   have holfm: "(f^^m) holomorphic_on s" for m
     apply (induction m, simp add: holomorphic_on_ident)
     apply (simp only: funpow_Suc_right holomorphic_on_compose_gen [OF holf _ fss])
     done
   show ?thesis using n
   proof (induction m)
     case 0 with dfz show ?case
       by (auto simp: zero)
   next
     case (Suc m)
     have "(deriv ^^ i) (f ^^ m \<circ> f) z = (deriv ^^ i) (f ^^ m) (f z)"
       using Suc.prems holfm \<open>z \<in> s\<close> dfz fss higher_deriv_comp_lemma holf s zero by blast
     also have "... = (deriv ^^ i) f z"
       by (simp add: Suc)
     finally show ?case
       by (simp only: funpow_Suc_right)
   qed
 qed
 
 lemma higher_deriv_iter_top_lemma:
     assumes s: "open s" and holf: "f holomorphic_on s"
         and fss: "f ` s \<subseteq> s"
         and "z \<in> s" and [simp]: "f z = z"
         and dfz [simp]: "deriv f z = 1"
         and n: "1 < n" "\<And>i. \<lbrakk>1 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0"
       shows "(deriv ^^ n) (f ^^ m) z = m * (deriv ^^ n) f z"
 using n
 proof (induction n arbitrary: m)
   case 0 then show ?case by simp
 next
   case (Suc n)
   have [simp]: "(f^^m) z = z" for m
     by (induction m) auto
   have fms_sb: "(f^^m) ` s \<subseteq> s" for m
     apply (induction m)
     using fss
     apply force+
     done
   have holfm: "(f^^m) holomorphic_on s" for m
     apply (induction m, simp add: holomorphic_on_ident)
     apply (simp only: funpow_Suc_right holomorphic_on_compose_gen [OF holf _ fss])
     done
   then have holdfm: "deriv (f ^^ m) holomorphic_on s" for m
     by (simp add: holomorphic_deriv s)
   have holdffm: "(\<lambda>z. deriv f ((f ^^ m) z)) holomorphic_on s" for m
     apply (rule holomorphic_on_compose_gen [where g="deriv f" and t=s, unfolded o_def])
     using s \<open>z \<in> s\<close> holfm holf fms_sb by (auto intro: holomorphic_intros)
   have f_cd_w: "\<And>w. w \<in> s \<Longrightarrow> f field_differentiable at w"
     using holf holomorphic_on_imp_differentiable_at s by blast
   have f_cd_mw: "\<And>m w. w \<in> s \<Longrightarrow> (f^^m) field_differentiable at w"
     using holfm holomorphic_on_imp_differentiable_at s by auto
   have der_fm [simp]: "deriv (f ^^ m) z = 1" for m
     apply (induction m, simp add: deriv_ident)
     apply (subst funpow_Suc_right)
-    apply (subst complex_derivative_chain)
+    apply (subst deriv_chain)
     using \<open>z \<in> s\<close> holfm holomorphic_on_imp_differentiable_at s f_cd_w apply auto
     done
   note Suc(3) [simp]
   note n_Suc = Suc
   show ?case
   proof (induction m)
     case 0 with n_Suc show ?case
       by (metis Zero_not_Suc funpow_simps_right(1) higher_deriv_id lambda_zero nat_neq_iff of_nat_0)
   next
     case (Suc m)
     have deriv_nffm: "(deriv ^^ n) (deriv f o (f ^^ m)) z = (deriv ^^ n) (deriv f) ((f ^^ m) z)"
       apply (rule higher_deriv_comp_lemma [OF s holfm \<open>z \<in> s\<close> s _ fms_sb order_refl])
       using \<open>z \<in> s\<close> fss higher_deriv_comp_iter_lemma holf holf holomorphic_deriv s
         apply auto
       done
     have "deriv (f ^^ m \<circ> f) holomorphic_on s"
       by (metis funpow_Suc_right holdfm)
     moreover have "(\<lambda>w. deriv f ((f ^^ m) w) * deriv (f ^^ m) w) holomorphic_on s"
       by (rule holomorphic_on_mult [OF holdffm holdfm])
     ultimately have "(deriv ^^ n) (deriv (f ^^ m \<circ> f)) z = (deriv ^^ n) (\<lambda>w. deriv f ((f ^^ m) w) * deriv (f ^^ m) w) z"
       apply (rule higher_deriv_transform_within_open [OF _ _ s \<open>z \<in> s\<close>])
-      by (metis comp_funpow complex_derivative_chain f_cd_mw f_cd_w fms_sb funpow_swap1 image_subset_iff o_id)
+      by (metis comp_funpow deriv_chain f_cd_mw f_cd_w fms_sb funpow_swap1 image_subset_iff o_id)
     also have "... =
           (\<Sum>i=0..n. of_nat(n choose i) * (deriv ^^ i) (\<lambda>w. deriv f ((f ^^ m) w)) z *
                      (deriv ^^ (n - i)) (deriv (f ^^ m)) z)"
       by (rule higher_deriv_mult [OF holdffm holdfm s \<open>z \<in> s\<close>])
     also have "... = (\<Sum>i \<in> {0,n}. of_nat(n choose i) * (deriv ^^ i) (\<lambda>w. deriv f ((f ^^ m) w)) z *
                      (deriv ^^ (n - i)) (deriv (f ^^ m)) z)"
     proof -
       have *: "(deriv ^^ i) (\<lambda>w. deriv f ((f ^^ m) w)) z = 0"  if "i \<le> n" "0 < i" "i \<noteq> n" and nz: "(deriv ^^ (n - i)) (deriv (f ^^ m)) z \<noteq> 0" for i
       proof -
         have less: "1 < Suc (n-i)" and le: "Suc (n-i) \<le> n"
           using that by auto
         have "(deriv ^^ (Suc (n - i))) (f ^^ m) z = (deriv ^^(Suc (n - i))) f z"
           apply (rule higher_deriv_comp_iter_lemma [OF s holf fss \<open>z \<in> s\<close> \<open>f z = z\<close> le dfz])
           by simp
         also have "... = 0"
           using n_Suc(3) less le le_imp_less_Suc by blast
         finally have "(deriv ^^ (Suc (n - i))) (f ^^ m) z = 0" .
         then show ?thesis by (simp add: funpow_swap1 nz)
       qed
       show ?thesis
         by (rule comm_monoid_add_class.sum.mono_neutral_right) (auto simp: *)
     qed
     also have "... = of_nat (Suc m) * (deriv ^^ n) (deriv f) z"
       apply (subst Groups_Big.comm_monoid_add_class.sum.insert)
       apply (simp_all add: deriv_nffm [unfolded o_def] of_nat_Suc [of 0] del: of_nat_Suc)
       using n_Suc(2) Suc
       apply (auto simp del: funpow.simps simp: algebra_simps funpow_simps_right)
       done
     finally have "(deriv ^^ n) (deriv (f ^^ m \<circ> f)) z = of_nat (Suc m) * (deriv ^^ n) (deriv f) z" .
     then show ?case
       apply (simp only: funpow_Suc_right)
       apply (simp add: o_def del: of_nat_Suc)
       done
   qed
 qed
 
 
 text\<open>Should be proved for n-dimensional vectors of complex numbers\<close>
 theorem first_Cartan_dim_1:
     assumes holf: "f holomorphic_on s"
         and "open s" "connected s" "bounded s"
         and fss: "f ` s \<subseteq> s"
         and "z \<in> s" and [simp]: "f z = z"
         and dfz [simp]: "deriv f z = 1"
         and "w \<in> s"
       shows "f w = w"
 proof -
   obtain c where "0 < c" and c: "s \<subseteq> ball z c"
     using \<open>bounded s\<close> bounded_subset_ballD by blast
   obtain r where "0 < r" and r: "cball z r \<subseteq> s"
     using \<open>z \<in> s\<close> open_contains_cball \<open>open s\<close> by blast
   then have bzr: "ball z r \<subseteq> s" using ball_subset_cball by blast
   have fms_sb: "(f^^m) ` s \<subseteq> s" for m
     apply (induction m)
     using fss apply force+
     done
   have holfm: "(f^^m) holomorphic_on s" for m
     apply (induction m, simp add: holomorphic_on_ident)
     apply (simp only: funpow_Suc_right holomorphic_on_compose_gen [OF holf _ fss])
     done
   have *: "(deriv ^^ n) f z = (deriv ^^ n) id z" for n
   proof -
     consider "n = 0" | "n = 1" | "1 < n" by arith
     then show ?thesis
     proof cases
       assume "n = 0" then show ?thesis by force
     next
       assume "n = 1" then show ?thesis by force
     next
       assume n1: "n > 1"
       then have "(deriv ^^ n) f z = 0"
       proof (induction n rule: less_induct)
         case (less n)
         have le: "real m * cmod ((deriv ^^ n) f z) \<le> fact n * c / r ^ n" if "m\<noteq>0" for m
         proof -
           have holfm': "(f ^^ m) holomorphic_on ball z r"
             using holfm bzr holomorphic_on_subset by blast
           then have contfm': "continuous_on (cball z r) (f ^^ m)"
             using \<open>cball z r \<subseteq> s\<close> holfm holomorphic_on_imp_continuous_on holomorphic_on_subset by blast
           have "real m * cmod ((deriv ^^ n) f z) = cmod (real m * (deriv ^^ n) f z)"
             by (simp add: norm_mult)
           also have "... = cmod ((deriv ^^ n) (f ^^ m) z)"
             apply (subst higher_deriv_iter_top_lemma [OF \<open>open s\<close> holf fss \<open>z \<in> s\<close> \<open>f z = z\<close> dfz])
             using less apply auto
             done
           also have "... \<le> fact n * c / r ^ n"
             apply (rule Cauchy_higher_deriv_bound [OF holfm' contfm' \<open>0 < r\<close>, where y=z])
             using less.prems apply linarith
             using fms_sb c r ball_subset_cball
             apply blast
             done
           finally show ?thesis .
         qed
         have "cmod ((deriv ^^ n) f z) = 0"
           apply (rule real_archimedian_rdiv_eq_0 [where c = "(fact n) * c / r ^ n"])
           apply simp
           using \<open>0 < r\<close> \<open>0 < c\<close>
           apply (simp add: divide_simps)
           apply (blast intro: le)
           done
         then show ?case by simp
       qed
       with n1 show ?thesis by simp
     qed
   qed
   have "f w = id w"
     by (rule holomorphic_fun_eq_on_connected
                  [OF holf holomorphic_on_id \<open>open s\<close> \<open>connected s\<close> * \<open>z \<in> s\<close> \<open>w \<in> s\<close>])
   also have "... = w" by simp
   finally show ?thesis .
 qed
 
 
 text\<open>Second Cartan Theorem.\<close>
 
 lemma Cartan_is_linear:
   assumes holf: "f holomorphic_on s"
       and "open s" and "connected s"
       and "0 \<in> s"
       and ins: "\<And>u z. \<lbrakk>norm u = 1; z \<in> s\<rbrakk> \<Longrightarrow> u * z \<in> s"
       and feq: "\<And>u z. \<lbrakk>norm u = 1; z \<in> s\<rbrakk> \<Longrightarrow> f (u * z) = u * f z"
     shows "\<exists>c. \<forall>z \<in> s. f z = c * z"
 proof -
   have [simp]: "f 0 = 0"
     using feq [of "-1" 0] assms by simp
   have uneq: "u^n * (deriv ^^ n) f (u * z) = u * (deriv ^^ n) f z"
        if "norm u = 1" "z \<in> s" for n u z
   proof -
     have holfuw: "(\<lambda>w. f (u * w)) holomorphic_on s"
       apply (rule holomorphic_on_compose_gen [OF _ holf, unfolded o_def])
       using that ins by (auto simp: holomorphic_on_linear)
     have hol_d_fuw: "(deriv ^^ n) (\<lambda>w. u * f w) holomorphic_on s" for n
       by (rule holomorphic_higher_deriv holomorphic_intros holf assms)+
     have *: "(deriv ^^ n) (\<lambda>w. u * f w) z = u * (deriv ^^ n) f z" if "z \<in> s" for z
     using that
     proof (induction n arbitrary: z)
       case 0 then show ?case by simp
     next
       case (Suc n)
       have "deriv ((deriv ^^ n) (\<lambda>w. u * f w)) z = deriv (\<lambda>w. u * (deriv ^^ n) f w) z"
         apply (rule complex_derivative_transform_within_open [OF hol_d_fuw])
         apply (auto intro!: holomorphic_higher_deriv holomorphic_intros assms Suc)
         done
       also have "... = u * deriv ((deriv ^^ n) f) z"
         apply (rule deriv_cmult)
         using Suc \<open>open s\<close> holf holomorphic_higher_deriv holomorphic_on_imp_differentiable_at by blast
       finally show ?case by simp
     qed
     have "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u ^ n * (deriv ^^ n) f (u * z)"
       apply (rule higher_deriv_compose_linear [OF holf \<open>open s\<close> \<open>open s\<close>])
       apply (simp add: that)
       apply (simp add: ins that)
       done
     moreover have "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u * (deriv ^^ n) f z"
       apply (subst higher_deriv_transform_within_open [OF holfuw, of "\<lambda>w. u * f w"])
       apply (rule holomorphic_intros holf assms that)+
       apply blast
       using * \<open>z \<in> s\<close> apply blast
       done
     ultimately show ?thesis by metis
   qed
   have dnf0: "(deriv ^^ n) f 0 = 0" if len: "2 \<le> n" for n
   proof -
     have **: "z = 0" if "\<And>u::complex. norm u = 1 \<Longrightarrow> u ^ n * z = u * z" for z
     proof -
       have "\<exists>u::complex. norm u = 1 \<and> u ^ n \<noteq> u"
         using complex_not_root_unity [of "n-1"] len
         apply (simp add: algebra_simps le_diff_conv2, clarify)
         apply (rule_tac x=u in exI)
         apply (subst (asm) power_diff)
         apply auto
         done
       with that show ?thesis
         by auto
     qed
     show ?thesis
       apply (rule **)
       using uneq [OF _ \<open>0 \<in> s\<close>]
       by force
   qed
   show ?thesis
     apply (rule_tac x = "deriv f 0" in exI, clarify)
     apply (rule holomorphic_fun_eq_on_connected [OF holf _ \<open>open s\<close> \<open>connected s\<close> _ \<open>0 \<in> s\<close>])
     using dnf0 apply (auto simp: holomorphic_on_linear)
     done
 qed
 
 text\<open>Should be proved for n-dimensional vectors of complex numbers\<close>
 theorem second_Cartan_dim_1:
   assumes holf: "f holomorphic_on ball 0 r"
       and holg: "g holomorphic_on ball 0 r"
       and [simp]: "f 0 = 0" and [simp]: "g 0 = 0"
       and ballf: "\<And>z. z \<in> ball 0 r \<Longrightarrow> f z \<in> ball 0 r"
       and ballg: "\<And>z. z \<in> ball 0 r \<Longrightarrow> g z \<in> ball 0 r"
       and fg: "\<And>z. z \<in> ball 0 r \<Longrightarrow> f (g z) = z"
       and gf: "\<And>z. z \<in> ball 0 r \<Longrightarrow> g (f z) = z"
       and "0 < r"
     shows "\<exists>t. \<forall>z \<in> ball 0 r. g z = exp(\<i> * of_real t) * z"
 proof -
   have c_le_1: "c \<le> 1"
     if "0 \<le> c" "\<And>x. 0 \<le> x \<Longrightarrow> x < r \<Longrightarrow> c * x < r" for c
   proof -
     have rst: "\<And>r s t::real. 0 = r \<or> s/r < t \<or> r < 0 \<or> \<not> s < r * t"
       by (metis (no_types) mult_less_cancel_left_disj nonzero_mult_div_cancel_left times_divide_eq_right)
     { assume "\<not> r < c \<and> c * (c * (c * (c * r))) < 1"
      then have "1 \<le> c \<Longrightarrow> (\<exists>r. \<not> 1 < r \<and> \<not> r < c)"
           using \<open>0 \<le> c\<close> by (metis (full_types) less_eq_real_def mult.right_neutral mult_left_mono not_less)
       then have "\<not> 1 < c \<or> \<not> 1 \<le> c"
         by linarith }
     moreover
     { have "\<not> 0 \<le> r / c \<Longrightarrow> \<not> 1 \<le> c"
           using \<open>0 < r\<close> by force
       then have "1 < c \<Longrightarrow> \<not> 1 \<le> c"
         using rst \<open>0 < r\<close> that
         by (metis div_by_1 frac_less2 less_le_trans mult.commute not_le order_refl pos_divide_le_eq zero_less_one) }
     ultimately show ?thesis
       by (metis (no_types) linear not_less)
   qed
   have ugeq: "u * g z = g (u * z)" if nou: "norm u = 1" and z: "z \<in> ball 0 r" for u z
   proof -
     have [simp]: "u \<noteq> 0" using that by auto
     have hol1: "(\<lambda>a. f (u * g a) / u) holomorphic_on ball 0 r"
       apply (rule holomorphic_intros)
       apply (rule holomorphic_on_compose_gen [OF _ holf, unfolded o_def])
       apply (rule holomorphic_intros holg)+
       using nou ballg
       apply (auto simp: dist_norm norm_mult holomorphic_on_const)
       done
     have cdf: "f field_differentiable at 0"
       using \<open>0 < r\<close> holf holomorphic_on_imp_differentiable_at by auto
     have cdg: "g field_differentiable at 0"
       using \<open>0 < r\<close> holg holomorphic_on_imp_differentiable_at by auto
     have cd_fug: "(\<lambda>a. f (u * g a)) field_differentiable at 0"
       apply (rule field_differentiable_compose [where g=f and f = "\<lambda>a. (u * g a)", unfolded o_def])
       apply (rule derivative_intros)+
       using cdf cdg
       apply auto
       done
     have "deriv g 0 = deriv g (f 0)"
       by simp
     then have "deriv f 0 * deriv g 0 = 1"
       by (metis open_ball \<open>0 < r\<close> ballf centre_in_ball deriv_left_inverse gf holf holg image_subsetI)
     then have equ: "deriv f 0 * deriv (\<lambda>a. u * g a) 0 = u"
       by (simp add: cdg deriv_cmult)
     have der1: "deriv (\<lambda>a. f (u * g a) / u) 0 = 1"
       apply (simp add: field_class.field_divide_inverse deriv_cmult_right [OF cd_fug])
-      apply (subst complex_derivative_chain [where g=f and f = "\<lambda>a. (u * g a)", unfolded o_def])
+      apply (subst deriv_chain [where g=f and f = "\<lambda>a. (u * g a)", unfolded o_def])
       apply (rule derivative_intros cdf cdg | simp add: equ)+
       done
     have fugeq: "\<And>w. w \<in> ball 0 r \<Longrightarrow> f (u * g w) / u = w"
       apply (rule first_Cartan_dim_1 [OF hol1, where z=0])
       apply (simp_all add: \<open>0 < r\<close>)
       apply (auto simp: der1)
       using nou ballf ballg
       apply (simp add: dist_norm norm_mult norm_divide)
       done
     have "f(u * g z) = u * z"
       by (metis \<open>u \<noteq> 0\<close> fugeq nonzero_mult_div_cancel_left z times_divide_eq_right)
     also have "... = f (g (u * z))"
       by (metis (no_types, lifting) fg mem_ball_0 mult_cancel_right2 norm_mult nou z)
     finally have "f(u * g z) = f (g (u * z))" .
     then have "g (f (u * g z)) = g (f (g (u * z)))"
       by simp
     then show ?thesis
       apply (subst (asm) gf)
       apply (simp add: dist_norm norm_mult nou)
       using ballg mem_ball_0 z apply blast
       apply (subst (asm) gf)
       apply (simp add: dist_norm norm_mult nou)
       apply (metis ballg mem_ball_0 mult.left_neutral norm_mult nou z, simp)
       done
   qed
   obtain c where c: "\<And>z. z \<in> ball 0 r \<Longrightarrow> g z = c * z"
     apply (rule exE [OF Cartan_is_linear [OF holg]])
     apply (simp_all add: \<open>0 < r\<close> ugeq)
     apply (auto simp: dist_norm norm_mult)
     done
   have gr2: "g (f (r/2)) = c * f(r/2)"
     apply (rule c) using \<open>0 < r\<close> ballf mem_ball_0 by force
   then have "norm c > 0"
     using \<open>0 < r\<close>
     by simp (metis \<open>f 0 = 0\<close> c dist_commute fg mem_ball mult_zero_left perfect_choose_dist)
   then have [simp]: "c \<noteq> 0" by auto
   have xless: "x < r * cmod c" if "0 \<le> x" "x < r" for x
   proof -
     have "x = norm (g (f (of_real x)))"
     proof -
       have "r > cmod (of_real x)"
         by (simp add: that)
       then have "complex_of_real x \<in> ball 0 r"
         using mem_ball_0 by blast
       then show ?thesis
         using gf \<open>0 \<le> x\<close> by force
     qed
     then show ?thesis
       apply (rule ssubst)
       apply (subst c)
       apply (rule ballf)
       using ballf [of x] that
       apply (auto simp: norm_mult dist_0_norm)
       done
   qed
   have 11: "1 / norm c \<le> 1"
     apply (rule c_le_1)
     using xless apply (auto simp: divide_simps)
     done
   have "\<lbrakk>0 \<le> x; x < r\<rbrakk> \<Longrightarrow> cmod c * x < r" for x
     using c [of x] ballg [of x] by (auto simp: norm_mult dist_0_norm)
     then have "norm c \<le> 1"
     by (force intro: c_le_1)
   moreover have "1 \<le> norm c"
     using 11 by simp
   ultimately have "norm c = 1"  by (rule antisym)
   with complex_norm_eq_1_exp c show ?thesis
     by metis
 qed
 
 end
diff --git a/thys/Cartan_FP/ROOT b/thys/Cartan_FP/ROOT
--- a/thys/Cartan_FP/ROOT
+++ b/thys/Cartan_FP/ROOT
@@ -1,10 +1,10 @@
 chapter AFP
 
-session Cartan_FP (AFP) = "HOL-Analysis" +
+session Cartan_FP (AFP) = "HOL-Complex_Analysis" +
   description "The Cartan Fixed Point Theorems"
   options [timeout = 300]
   theories
     Cartan
   document_files
     "root.bib"
     "root.tex"
diff --git a/thys/Count_Complex_Roots/ROOT b/thys/Count_Complex_Roots/ROOT
--- a/thys/Count_Complex_Roots/ROOT
+++ b/thys/Count_Complex_Roots/ROOT
@@ -1,13 +1,13 @@
 chapter AFP
 
-session "Count_Complex_Roots" (AFP) = "HOL-Analysis" +
+session "Count_Complex_Roots" (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 600]
   sessions
     Sturm_Tarski
     Winding_Number_Eval
   theories
     Count_Complex_Roots
     Count_Complex_Roots_Examples
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/Dirichlet_Series/Dirichlet_Series_Analysis.thy b/thys/Dirichlet_Series/Dirichlet_Series_Analysis.thy
--- a/thys/Dirichlet_Series/Dirichlet_Series_Analysis.thy
+++ b/thys/Dirichlet_Series/Dirichlet_Series_Analysis.thy
@@ -1,3087 +1,3087 @@
 (*
     File:      Dirichlet_Series_Analysis.thy
     Author:    Manuel Eberl, TU München
 *)
 section \<open>Analytic properties of Dirichlet series\<close>
 theory Dirichlet_Series_Analysis
 imports
-  "HOL-Analysis.Analysis"
+  "HOL-Complex_Analysis.Complex_Analysis"
   "HOL-Library.Going_To_Filter"
   "HOL-Real_Asymp.Real_Asymp"
   Dirichlet_Series
   Moebius_Mu
   Partial_Summation
   Euler_Products
 begin
 
 (* TODO Move *)
 lemma frequently_eventually_frequently:
   "frequently P F \<Longrightarrow> eventually Q F \<Longrightarrow> frequently (\<lambda>x. P x \<and> Q x) F"
   by (erule frequently_rev_mp, erule eventually_mono) auto
 
 text \<open>
   The following illustrates a concept we will need later on: A property holds for
   \<open>f\<close> going to \<open>F\<close> if we can find e.\,g.\ a sequence that tends to \<open>F\<close> and whose elements
   eventually satisfy \<open>P\<close>.
 \<close>
 lemma frequently_going_toI:
   assumes "filterlim (\<lambda>n. f (g n)) F G"
   assumes "eventually (\<lambda>n. P (g n)) G"
   assumes "eventually (\<lambda>n. g n \<in> A) G"
   assumes "G \<noteq> bot"
   shows   "frequently P (f going_to F within A)"
   unfolding frequently_def
 proof
   assume "eventually (\<lambda>x. \<not>P x) (f going_to F within A)"
   hence "eventually (\<lambda>x. \<not>P x) (inf (filtercomap f F) (principal A))"
     by (simp add: going_to_within_def)
   moreover have "filterlim (\<lambda>n. g n) (inf (filtercomap f F) (principal A)) G"
     using assms unfolding filterlim_inf filterlim_principal
     by (auto simp add: filterlim_iff_le_filtercomap filtercomap_filtercomap)
   ultimately have "eventually (\<lambda>n. \<not>P (g n)) G"
     by (rule eventually_compose_filterlim)
   with assms(2) have "eventually (\<lambda>_. False) G" by eventually_elim auto
   with assms(4) show False by simp
 qed
 
 lemma frequently_filtercomapI:
   assumes "filterlim (\<lambda>n. f (g n)) F G"
   assumes "eventually (\<lambda>n. P (g n)) G"
   assumes "G \<noteq> bot"
   shows   "frequently P (filtercomap f F)"
   using frequently_going_toI[of f g F G P UNIV] assms by (simp add: going_to_def)
 
 lemma frequently_going_to_at_topE:
   fixes f :: "'a \<Rightarrow> real"
   assumes "frequently P (f going_to at_top)"
   obtains g where "\<And>n. P (g n)" and "filterlim (\<lambda>n. f (g n)) at_top sequentially"
 proof -
   from assms have "\<forall>k. \<exists>x. f x \<ge> real k \<and> P x"
     by (auto simp: frequently_def eventually_going_to_at_top_linorder)
   hence "\<exists>g. \<forall>k. f (g k) \<ge> real k \<and> P (g k)"
     by metis
   then obtain g where g: "\<And>k. f (g k) \<ge> real k" "\<And>k. P (g k)"
     by blast
   have "filterlim (\<lambda>n. f (g n)) at_top sequentially"
     by (rule filterlim_at_top_mono[OF filterlim_real_sequentially]) (use g in auto)
   from g(2) and this show ?thesis using that[of g] by blast
 qed
 
 text \<open>
   Apostol often uses statements like `$P(s_k)$ for all $k$ in an infinite sequence
   $s_k$ such that $\mathfrak{R}(s_k)\longrightarrow\infty$ as $k\to\infty$'.
 
   Instead, we write @{prop "frequently P (Re going_to at_top)"}. This lemma shows that
   our statement is equivalent to his.
 \<close>
 lemma frequently_going_to_at_top_iff:
   "frequently P (f going_to (at_top :: real filter)) \<longleftrightarrow>
      (\<exists>g. \<forall>n. P (g n) \<and> filterlim (\<lambda>n. f (g n)) at_top sequentially)"
   by (auto intro: frequently_going_toI elim!: frequently_going_to_at_topE)
 (* END TODO *)
 
 lemma surj_bullet_1: "surj (\<lambda>s::'a::{real_normed_algebra_1, real_inner}. s \<bullet> 1)"
 proof (rule surjI)
   fix x :: real show "(x *\<^sub>R 1) \<bullet> (1 :: 'a) = x"
     by (simp add: dot_square_norm)
 qed  
 
 lemma bullet_1_going_to_at_top_neq_bot [simp]:
   "((\<lambda>s::'a::{real_normed_algebra_1, real_inner}. s \<bullet> 1) going_to at_top) \<noteq> bot"
   unfolding going_to_def by (rule filtercomap_neq_bot_surj[OF _ surj_bullet_1]) auto
 
 
 
 lemma fds_abs_converges_altdef:
   "fds_abs_converges f s \<longleftrightarrow> (\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on {1..}"
   by (auto simp add: fds_abs_converges_def abs_summable_on_nat_iff
            intro!: summable_cong eventually_mono[OF eventually_gt_at_top[of 0]])
 
 lemma fds_abs_converges_altdef':
   "fds_abs_converges f s \<longleftrightarrow> (\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on UNIV"
   by (subst fds_abs_converges_altdef, rule abs_summable_on_cong_neutral) (auto simp: Suc_le_eq)
 
 lemma eval_fds_altdef:
   assumes "fds_abs_converges f s"
   shows   "eval_fds f s = (\<Sum>\<^sub>an. fds_nth f n / nat_power n s)"
 proof -
   have "fds_abs_converges f s \<longleftrightarrow> (\<lambda>n. fds_nth f n / nat_power n s) abs_summable_on UNIV"
     unfolding fds_abs_converges_altdef 
     by (intro abs_summable_on_cong_neutral) (auto simp: Suc_le_eq)
   with assms show ?thesis unfolding eval_fds_def fds_abs_converges_altdef
     by (intro infsetsum_nat' [symmetric]) simp_all
 qed
 
 lemma multiplicative_function_divide_nat_power:
   fixes f :: "nat \<Rightarrow> 'a :: {nat_power, field}"
   assumes "multiplicative_function f"
   shows   "multiplicative_function (\<lambda>n. f n / nat_power n s)"
 proof
   interpret f: multiplicative_function f by fact
   show "f 0 / nat_power 0 s = 0" "f 1 / nat_power 1 s = 1"
     by simp_all
   fix a b :: nat assume "a > 1" "b > 1" "coprime a b"
   thus "f (a * b) / nat_power (a * b) s = f a / nat_power a s * (f b / nat_power b s)"
     by (simp_all add: f.mult_coprime nat_power_mult_distrib)
 qed
 
 lemma completely_multiplicative_function_divide_nat_power:
   fixes f :: "nat \<Rightarrow> 'a :: {nat_power, field}"
   assumes "completely_multiplicative_function f"
   shows   "completely_multiplicative_function (\<lambda>n. f n / nat_power n s)"
 proof
   interpret f: completely_multiplicative_function f by fact
   show "f 0 / nat_power 0 s = 0" "f (Suc 0) / nat_power (Suc 0) s = 1"
     by simp_all
   fix a b :: nat assume "a > 1" "b > 1"
   thus "f (a * b) / nat_power (a * b) s = f a / nat_power a s * (f b / nat_power b s)"
     by (simp_all add: f.mult nat_power_mult_distrib)
 qed
 
 subsection \<open>Convergence and absolute convergence\<close>
 
 class nat_power_normed_field = nat_power_field + real_normed_field + real_inner + real_algebra_1 +
   fixes real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"
   assumes real_power_nat_power: "n > 0 \<Longrightarrow> real_power (real n) c = nat_power n c"
   assumes real_power_1_right_aux: "d > 0 \<Longrightarrow> real_power d 1 = d *\<^sub>R 1"
   assumes real_power_add: "d > 0 \<Longrightarrow> real_power d (a + b) = real_power d a * real_power d b"
   assumes real_power_nonzero [simp]: "d > 0 \<Longrightarrow> real_power d a \<noteq> 0"
   assumes norm_real_power: "x > 0 \<Longrightarrow> norm (real_power x c) = x powr (c \<bullet> 1)"
   assumes nat_power_of_real_aux: "nat_power n (x *\<^sub>R 1) = ((real n powr x) *\<^sub>R 1)"
   assumes has_field_derivative_nat_power_aux:
             "\<And>x::'a. n > 0 \<Longrightarrow> LIM y inf_class.inf
               (Inf (principal ` {S. open S \<and> x \<in> S})) (principal (UNIV - {x})).
                 (nat_power n y - nat_power n x - ln (real n) *\<^sub>R nat_power n x * (y - x)) /\<^sub>R
                 norm (y - x) :> Inf (principal ` {S. open S \<and> 0 \<in> S})"
   assumes has_vector_derivative_real_power_aux:
             "x > 0 \<Longrightarrow> filterlim (\<lambda>y. (real_power y c - real_power x (c :: 'a) -
                (y - x) *\<^sub>R (c * real_power x (c - 1))) /\<^sub>R
                norm (y - x)) (INF S\<in>{S. open S \<and> 0 \<in> S}. principal S) (at x)"
   assumes norm_nat_power: "n > 0 \<Longrightarrow> norm (nat_power n y) = real n powr (y \<bullet> 1)"
 begin
 
 lemma real_power_diff: "d > 0 \<Longrightarrow> real_power d (a - b) = real_power d a / real_power d b"
   using real_power_add[of d b "a - b"] by (simp add: field_simps)
 
 end
 
 lemma real_power_1_right [simp]: "d > 0 \<Longrightarrow> real_power d 1 = of_real d"
   using real_power_1_right_aux[of d] by (simp add: scaleR_conv_of_real)
 
 
 lemma has_vector_derivative_real_power [derivative_intros]:
   "x > 0 \<Longrightarrow> ((\<lambda>y. real_power y c) has_vector_derivative c * real_power x (c - 1)) (at x within A)"
   by (rule has_vector_derivative_at_within)
      (insert has_vector_derivative_real_power_aux[of x c],
       simp add: has_vector_derivative_def has_derivative_def
                 nhds_def bounded_linear_scaleR_left)
               
 lemma has_field_derivative_nat_power [derivative_intros]:
   "n > 0 \<Longrightarrow> ((\<lambda>y. nat_power n y) has_field_derivative ln (real n) *\<^sub>R nat_power n x)
      (at (x :: 'a :: nat_power_normed_field) within A)"
   by (rule has_field_derivative_at_within)
      (insert has_field_derivative_nat_power_aux[of n x],
       simp only: has_field_derivative_def has_derivative_def netlimit_at,
       simp add: nhds_def at_within_def bounded_linear_mult_right)
               
 lemma continuous_on_real_power [continuous_intros]: 
   "A \<subseteq> {0<..} \<Longrightarrow> continuous_on A (\<lambda>x. real_power x s)"
   by (rule continuous_on_vector_derivative has_vector_derivative_real_power)+ auto
 
 instantiation real :: nat_power_normed_field
 begin
 
 definition real_power_real :: "real \<Rightarrow> real \<Rightarrow> real" where
   [simp]: "real_power_real = (powr)"
   
 instance proof (standard, goal_cases)
   case (7 n x)
   hence "((\<lambda>x. nat_power n x) has_field_derivative ln (real n) *\<^sub>R nat_power n x) (at x)"
     by (auto intro!: derivative_eq_intros simp: powr_def)
   thus ?case unfolding has_field_derivative_def netlimit_at has_derivative_def
     by (simp add: nhds_def at_within_def)
 next
   case (8 x c)
   hence "((\<lambda>y. real_power y c) has_vector_derivative c * real_power x (c - 1)) (at x)"
     by (auto intro!: derivative_eq_intros
              simp: has_field_derivative_iff_has_vector_derivative [symmetric])
   thus ?case by (simp add: has_vector_derivative_def has_derivative_def nhds_def)
 qed (simp_all add: powr_add)
 
 end
 
 
 instantiation complex :: nat_power_normed_field
 begin
   
 definition nat_power_complex :: "nat \<Rightarrow> complex \<Rightarrow> complex" where
    [simp]: "nat_power_complex n z = of_nat n powr z"
 
 definition real_power_complex :: "real \<Rightarrow> complex \<Rightarrow> complex" where
   [simp]: "real_power_complex = (\<lambda>x y. of_real x powr y)"
   
 instance proof
   fix m n :: nat and z :: complex
   assume "m > 0" "n > 0"
   thus "nat_power (m * n) z = nat_power m z * nat_power n z"
     unfolding nat_power_complex_def of_nat_mult by (subst powr_times_real) simp_all
 next
   fix n :: nat and z :: complex
   assume "n > 0"
   show "norm (nat_power n z) = real n powr (z \<bullet> 1)" unfolding nat_power_complex_def
     using norm_powr_real_powr[of "of_nat n" z] by simp
 next
   fix n :: nat and x :: complex assume n: "n > 0"
   hence "((\<lambda>x. nat_power n x) has_field_derivative ln (real n) *\<^sub>R nat_power n x) (at x)"
     by (auto intro!: derivative_eq_intros simp: powr_def scaleR_conv_of_real mult_ac)
   thus "LIM y inf_class.inf (Inf (principal ` {S. open S \<and> x \<in> S})) (principal (UNIV - {x})).
            (nat_power n y - nat_power n x - ln (real n) *\<^sub>R nat_power n x * (y - x)) /\<^sub>R
            cmod (y - x) :> (Inf (principal ` {S. open S \<and> 0 \<in> S}))"
     unfolding has_field_derivative_def netlimit_at has_derivative_def
     by (simp add: nhds_def at_within_def)
 next
   fix x :: real and c :: complex assume "x > 0"
   hence "((\<lambda>y. real_power y c) has_vector_derivative c * real_power x (c - 1)) (at x)"
     by (auto intro!: derivative_eq_intros has_vector_derivative_real_field)
   thus "LIM y at x. (real_power y c - real_power x c - (y - x) *\<^sub>R (c * real_power x (c - 1))) /\<^sub>R
            norm (y - x) :> INF S\<in>{S. open S \<and> 0 \<in> S}. principal S"
     by (simp add: has_vector_derivative_def has_derivative_def nhds_def)
 next
   fix n :: nat and x :: real
   show "nat_power n (x *\<^sub>R 1 :: complex) = (real n powr x) *\<^sub>R 1"
     by (simp add: powr_Reals_eq scaleR_conv_of_real)
 qed (auto simp: powr_def exp_add exp_of_nat_mult [symmetric] algebra_simps scaleR_conv_of_real
           simp del: Ln_of_nat)
 
 end
 
 lemma nat_power_of_real [simp]:
   "nat_power n (of_real x :: 'a :: nat_power_normed_field) = of_real (real n powr x)"
   using nat_power_of_real_aux[of n x] by (simp add: scaleR_conv_of_real)
 
 lemma fds_abs_converges_of_real [simp]: 
   "fds_abs_converges (fds_of_real f) 
      (of_real s :: 'a :: {nat_power_normed_field,banach}) \<longleftrightarrow> fds_abs_converges f s"
   unfolding fds_abs_converges_def
   by (subst (1 2) summable_Suc_iff [symmetric]) (simp add: norm_divide norm_nat_power)
 
 lemma eval_fds_of_real [simp]:
   assumes "fds_converges f s"
   shows   "eval_fds (fds_of_real f) (of_real s :: 'a :: {nat_power_normed_field,banach}) = 
              of_real (eval_fds f s)"
   using assms unfolding eval_fds_def by (auto simp: fds_converges_def suminf_of_real)
 
 lemma fds_abs_summable_zeta_iff [simp]:
   fixes s :: "'a :: {banach, nat_power_normed_field}"
   shows "fds_abs_converges fds_zeta s \<longleftrightarrow> s \<bullet> 1 > (1 :: real)"
 proof -
   have "fds_abs_converges fds_zeta s \<longleftrightarrow> summable (\<lambda>n. real n powr -(s \<bullet> 1))"
     unfolding fds_abs_converges_def
     by (intro summable_cong always_eventually)
        (auto simp: norm_divide fds_nth_zeta powr_minus norm_nat_power divide_simps)
   also have "\<dots> \<longleftrightarrow> s \<bullet> 1 > 1" by (simp add: summable_real_powr_iff)
   finally show ?thesis .
 qed
 
 lemma fds_abs_summable_zeta: 
   "(s :: 'a :: {banach, nat_power_normed_field}) \<bullet> 1 > 1 \<Longrightarrow> fds_abs_converges fds_zeta s"
   by simp
 
 
 lemma fds_abs_converges_moebius_mu:
   fixes s :: "'a :: {banach,nat_power_normed_field}"
   assumes "s \<bullet> 1 > 1"
   shows   "fds_abs_converges (fds moebius_mu) s"
   unfolding fds_abs_converges_def
 proof (rule summable_comparison_test, intro exI allI impI)
   fix n :: nat
   show "norm (norm (fds_nth (fds moebius_mu) n / nat_power n s)) \<le> real n powr (-s \<bullet> 1)"
     by (auto simp: powr_minus divide_simps abs_moebius_mu_le norm_nat_power norm_divide
                    moebius_mu_def norm_power)
 next
   from assms show "summable (\<lambda>n. real n powr (-s \<bullet> 1))" by (simp add: summable_real_powr_iff)
 qed
 
 
 definition conv_abscissa
     :: "'a :: {nat_power,banach,real_normed_field, real_inner} fds \<Rightarrow> ereal" where
   "conv_abscissa f = (INF s\<in>{s. fds_converges f s}. ereal (s \<bullet> 1))"
 
 definition abs_conv_abscissa
     :: "'a :: {nat_power,banach,real_normed_field, real_inner} fds \<Rightarrow> ereal" where
   "abs_conv_abscissa f = (INF s\<in>{s. fds_abs_converges f s}. ereal (s \<bullet> 1))"
 
 lemma conv_abscissa_mono:
   assumes "\<And>s. fds_converges g s \<Longrightarrow> fds_converges f s"
   shows   "conv_abscissa f \<le> conv_abscissa g"
   unfolding conv_abscissa_def by (rule INF_mono) (use assms in auto)
 
 lemma abs_conv_abscissa_mono:
   assumes "\<And>s. fds_abs_converges g s \<Longrightarrow> fds_abs_converges f s"
   shows   "abs_conv_abscissa f \<le> abs_conv_abscissa g"
   unfolding abs_conv_abscissa_def by (rule INF_mono) (use assms in auto)
 
 
 class dirichlet_series = euclidean_space + real_normed_field + nat_power_normed_field +
   assumes one_in_Basis: "1 \<in> Basis"
     
 instance real :: dirichlet_series by standard simp_all
 instance complex :: dirichlet_series by standard (simp_all add: Basis_complex_def)
 
 context
   assumes "SORT_CONSTRAINT('a :: dirichlet_series)"
 begin
 
 lemma fds_abs_converges_Re_le:
   fixes f :: "'a fds"
   assumes "fds_abs_converges f z" "z \<bullet> 1 \<le> z' \<bullet> 1"
   shows   "fds_abs_converges f z'"
   unfolding fds_abs_converges_def
 proof (rule summable_comparison_test, intro exI allI impI)
   fix n :: nat assume n: "n \<ge> 1"
   thus "norm (norm (fds_nth f n / nat_power n z')) \<le> norm (fds_nth f n / nat_power n z)"
     using assms(2) by (simp add: norm_divide norm_nat_power divide_simps powr_mono mult_left_mono)
 qed (insert assms(1), simp add: fds_abs_converges_def)
 
 lemma fds_abs_converges:
   assumes "s \<bullet> 1 > abs_conv_abscissa (f :: 'a fds)"
   shows   "fds_abs_converges f s"
 proof -
   from assms obtain s0 where "fds_abs_converges f s0" "s0 \<bullet> 1 < s \<bullet> 1"
     by (auto simp: INF_less_iff abs_conv_abscissa_def)
   with fds_abs_converges_Re_le[OF this(1), of s] this(2) show ?thesis by simp
 qed
 
 lemma fds_abs_diverges:
   assumes "s \<bullet> 1 < abs_conv_abscissa (f :: 'a fds)"
   shows   "\<not>fds_abs_converges f s"
 proof
   assume "fds_abs_converges f s"
   hence "abs_conv_abscissa f \<le> s \<bullet> 1" unfolding abs_conv_abscissa_def
     by (intro INF_lower) auto
   with assms show False by simp
 qed
 
 
 lemma uniformly_Cauchy_eval_fds_aux:
   fixes s0 :: "'a :: dirichlet_series"
   assumes bounded: "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"
   assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > s0 \<bullet> 1"
   shows   "uniformly_Cauchy_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
 proof (cases "B = {}")
   case False
   show ?thesis
   proof (rule uniformly_Cauchy_onI', goal_cases)
     case (1 \<epsilon>)
     define \<sigma> where "\<sigma> = Inf ((\<lambda>s. s \<bullet> 1) ` B)"
     have \<sigma>_le: "s \<bullet> 1 \<ge> \<sigma>" if "s \<in> B" for s
       unfolding \<sigma>_def using that 
       by (intro cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded B) auto
     have "\<sigma> \<in> ((\<lambda>s. s \<bullet> 1) ` B)"
       unfolding \<sigma>_def using B \<open>B \<noteq> {}\<close>
       by (intro closed_contains_Inf bounded_inner_imp_bdd_below compact_imp_bounded B 
             compact_imp_closed compact_continuous_image continuous_intros) auto
     with B(2) have \<sigma>_gt: "\<sigma> > s0 \<bullet> 1" by auto
     define \<delta> where "\<delta> = \<sigma> - s0 \<bullet> 1"
       
     have "bounded B" by (rule compact_imp_bounded) fact
     then obtain norm_B_aux where norm_B_aux: "\<And>s. s \<in> B \<Longrightarrow> norm s \<le> norm_B_aux" 
       by (auto simp: bounded_iff)
     define norm_B where "norm_B = norm_B_aux + norm s0"
     from norm_B_aux have norm_B: "norm (s - s0) \<le> norm_B" if "s \<in> B" for s
       using norm_triangle_ineq4[of s s0] norm_B_aux[OF that] by (simp add: norm_B_def)
       
     define A where "A = sum_upto (\<lambda>k. fds_nth f k / nat_power k s0)"
     from bounded obtain C_aux where C_aux: "\<And>n. norm (\<Sum>k\<le>n. fds_nth f k / nat_power k s0) \<le> C_aux"
       by (auto simp: Bseq_def)
     define C where "C = max C_aux 1"
     have C_pos: "C > 0" by (simp add: C_def)
     have C: "norm (A x) \<le> C" for x
     proof -
       have "A x = (\<Sum>k\<le>nat \<lfloor>x\<rfloor>. fds_nth f k / nat_power k s0)"
         unfolding A_def sum_upto_altdef by (intro sum.mono_neutral_left) auto
       also have "norm \<dots> \<le> C_aux" by (rule C_aux)
       also have "\<dots> \<le> C" by (simp add: C_def)
       finally show ?thesis .
     qed
     
     have "(\<lambda>m. 2 * C * (1 + norm_B / \<delta>) * real m powr (-\<delta>)) \<longlonglongrightarrow> 0" unfolding \<delta>_def using \<sigma>_gt
       by (intro tendsto_mult_right_zero tendsto_neg_powr filterlim_real_sequentially) simp_all
     from order_tendstoD(2)[OF this \<open>\<epsilon> > 0\<close>] obtain M where
       M: "\<And>m. m \<ge> M \<Longrightarrow> 2 * C * (1 + norm_B / \<delta>) * real m powr - \<delta> < \<epsilon>" 
       by (auto simp: eventually_at_top_linorder)
       
     show ?case
     proof (intro exI[of _ "max M 1"] ballI allI impI, goal_cases)
       case (1 s m n)
       from 1 have s: "s \<bullet> 1 > s0 \<bullet> 1" using B(2)[of s] by simp
       have mn: "m \<ge> M" "m < n" "m > 0" "n > 0" using 1 by (simp_all add: )
       have "dist (\<Sum>n\<le>m. fds_nth f n / nat_power n s) (\<Sum>n\<le>n. fds_nth f n / nat_power n s) =
               dist (\<Sum>n\<le>n. fds_nth f n / nat_power n s) (\<Sum>n\<le>m. fds_nth f n / nat_power n s)"
         by (simp add: dist_commute)
       also from 1 have "\<dots> = norm (\<Sum>k\<in>{..n}-{..m}. fds_nth f k / nat_power k s)"
         by (subst Groups_Big.sum_diff) (simp_all add: dist_norm)
       also from 1 have "{..n} - {..m} = real -` {real m<..real n}" by auto
       also have "(\<Sum>k\<in>\<dots>. fds_nth f k / nat_power k s) = 
                    (\<Sum>k\<in>\<dots>. fds_nth f k / nat_power k s0 * real_power (real k) (s0 - s))"
         (is "_ = ?S") by (intro sum.cong refl) (simp_all add: nat_power_diff real_power_nat_power)
       also have *: "((\<lambda>t. A t * ((s0 - s) * real_power t (s0 - s - 1))) has_integral
                     (A (real n) * real_power n (s0 - s) - A (real m) * real_power m (s0 - s) - ?S))
                     {real m..real n}" (is "(?h has_integral _) _") unfolding A_def using mn
         by (intro partial_summation_strong[of "{}"]) 
            (auto intro!: derivative_eq_intros continuous_intros)
       hence "?S = A (real n) * nat_power n (s0 - s) - A (real m) * nat_power m (s0 - s) -
                     integral {real m..real n} ?h"
         using mn by (simp add: has_integral_iff real_power_nat_power)
       also have "norm \<dots> \<le> norm (A (real n) * nat_power n (s0 - s)) +
                     norm (A (real m) * nat_power m (s0 - s)) + norm (integral {real m..real n} ?h)"
         by (intro order.trans[OF norm_triangle_ineq4] add_right_mono order.refl)
       also have "norm (A (real n) * nat_power n (s0 - s)) \<le> C * nat_power m ((s0 - s) \<bullet> 1)"
         using mn \<open>s \<in> B\<close> C_pos s
         by (auto simp: norm_mult norm_nat_power algebra_simps intro!: mult_mono C powr_mono2')
       also have "norm (A (real m) * nat_power m (s0 - s)) \<le> C * nat_power m ((s0 - s) \<bullet> 1)"
         using mn by (auto simp: norm_mult norm_nat_power intro!: mult_mono C)
       also have "norm (integral {real m..real n} ?h) \<le> 
                    integral {real m..real n} (\<lambda>t. C * (norm (s0 - s) * t powr ((s0 - s) \<bullet> 1 - 1)))"
       proof (intro integral_norm_bound_integral ballI, goal_cases)
         case 1
         with * show ?case by (simp add: has_integral_iff)
       next
         case 2
         from mn show ?case by (auto intro!: integrable_continuous_real continuous_intros)
       next
         case (3 t)
         thus ?case unfolding norm_mult using C_pos mn 
           by (intro mult_mono C) (auto simp: norm_real_power dot_square_norm algebra_simps)
       qed
       also have "\<dots> = C * norm (s0 - s) * integral {real m..real n} (\<lambda>t. t powr ((s0 - s) \<bullet> 1 - 1))"
         by (simp add: algebra_simps dot_square_norm)
       also {
         have "((\<lambda>t. t powr ((s0 - s) \<bullet> 1 - 1)) has_integral 
                      (real n powr ((s0 - s) \<bullet> 1) / ((s0 - s) \<bullet> 1) - 
                       real m powr ((s0 - s) \<bullet> 1) / ((s0 - s) \<bullet> 1))) {m..n}" 
           (is "(?l has_integral ?I) _") using mn s
           by (intro fundamental_theorem_of_calculus)
              (auto intro!: derivative_eq_intros 
                    simp: has_field_derivative_iff_has_vector_derivative [symmetric] inner_diff_left)
         hence "integral {real m..real n} ?l = ?I" by (simp add: has_integral_iff)
         also have "\<dots> \<le> -(real m powr ((s0 - s) \<bullet> 1) / ((s0 - s) \<bullet> 1))" using s mn
           by (simp add: divide_simps inner_diff_left)
         also have "\<dots> = 1 * (real m powr ((s0 - s) \<bullet> 1) / ((s - s0) \<bullet> 1))"
           using s by (simp add: field_simps inner_diff_left)
         also have "\<dots> \<le> 2 * (real m powr ((s0 - s) \<bullet> 1) / ((s - s0) \<bullet> 1))" using mn s
           by (intro mult_right_mono divide_nonneg_pos) (simp_all add: inner_diff_left)
         finally have "integral {m..n} ?l \<le> \<dots>" .
       }
       hence "C * norm (s0 - s) * integral {real m..real n} (\<lambda>t. t powr ((s0 - s) \<bullet> 1 - 1)) \<le>
                C * norm (s0 - s) * (2 * (real m powr ((s0 - s) \<bullet> 1) / ((s - s0) \<bullet> 1)))" 
         using C_pos mn
         by (intro mult_mono mult_nonneg_nonneg integral_nonneg
               integrable_continuous_real continuous_intros) auto
       also have "C * nat_power m ((s0 - s) \<bullet> 1) + C * nat_power m ((s0 - s) \<bullet> 1) + \<dots> =
                    2 * C * nat_power m ((s0 - s) \<bullet> 1) * (1 + norm (s - s0) / ((s - s0) \<bullet> 1))"
         by (simp add: algebra_simps norm_minus_commute)
       also have "\<dots> \<le> 2 * C * nat_power m (-\<delta>) * (1 + norm_B / \<delta>)" 
         using C_pos s mn \<sigma>_le[of s] \<open>s \<in> B\<close> \<sigma>_gt unfolding nat_power_real_def
         by (intro mult_mono mult_nonneg_nonneg powr_mono frac_le add_mono norm_B)
            (simp_all add: inner_diff_left \<delta>_def)
       also have "\<dots> = 2 * C * (1 + norm_B / \<delta>) * real m powr (-\<delta>)" by simp
       also from \<open>m \<ge> M\<close> have "\<dots> < \<epsilon>" by (rule M)
       finally show ?case by - simp_all
     qed
   qed
 qed (auto simp: uniformly_Cauchy_on_def)
 
 lemma uniformly_convergent_eval_fds_aux:
   assumes "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k (s0 :: 'a))"
   assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > s0 \<bullet> 1"
   shows   "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
   by (rule Cauchy_uniformly_convergent uniformly_Cauchy_eval_fds_aux assms)+
 
 lemma uniformly_convergent_eval_fds_aux':
   assumes conv: "fds_converges f (s0 :: 'a)"
   assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > s0 \<bullet> 1"
   shows   "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
 proof (rule uniformly_convergent_eval_fds_aux)
   from conv have "convergent (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"
     by (simp add: fds_converges_def summable_iff_convergent')
   thus "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)" by (rule convergent_imp_Bseq)
 qed (insert assms, auto)
 
 lemma bounded_partial_sums_imp_fps_converges:
   fixes s0 :: "'a :: dirichlet_series"
   assumes "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)" and "s \<bullet> 1 > s0 \<bullet> 1"
   shows   "fds_converges f s"
 proof -
   have "uniformly_convergent_on {s} (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)" using assms(2)
     by (intro uniformly_convergent_eval_fds_aux[OF assms(1)]) auto
   thus ?thesis
     by (auto simp: fds_converges_def summable_iff_convergent'
              dest: uniformly_convergent_imp_convergent)
 qed
 
 theorem fds_converges_Re_le:
   assumes "fds_converges f (s0 :: 'a)" "s \<bullet> 1 > s0 \<bullet> 1"
   shows   "fds_converges f s"
 proof -
   have "uniformly_convergent_on {s} (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
     by (rule uniformly_convergent_eval_fds_aux' assms)+ (insert assms(2), auto)
   then obtain l where "uniform_limit {s} (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z) l at_top"
     by (auto simp: uniformly_convergent_on_def)
   from tendsto_uniform_limitI[OF this, of s]
   have "(\<lambda>n. fds_nth f n / nat_power n s) sums l s" unfolding sums_def'
     by (simp add: atLeast0AtMost)
   thus ?thesis by (simp add: fds_converges_def sums_iff)
 qed
 
 lemma fds_converges:
   assumes "s \<bullet> 1 > conv_abscissa (f :: 'a fds)"
   shows   "fds_converges f s"
 proof -
   from assms obtain s0 where "fds_converges f s0" "s0 \<bullet> 1 < s \<bullet> 1"
     by (auto simp: INF_less_iff conv_abscissa_def)
   with fds_converges_Re_le[OF this(1), of s] this(2) show ?thesis by simp
 qed
 
 lemma fds_diverges:
   assumes "s \<bullet> 1 < conv_abscissa (f :: 'a fds)"
   shows   "\<not>fds_converges f s"
 proof
   assume "fds_converges f s"
   hence "conv_abscissa f \<le> s \<bullet> 1" unfolding conv_abscissa_def
     by (intro INF_lower) auto
   with assms show False by simp
 qed
 
 theorem fds_converges_imp_abs_converges:
   assumes "fds_converges (f :: 'a fds) s" "s' \<bullet> 1 > s \<bullet> 1 + 1"
   shows   "fds_abs_converges f s'"
   unfolding fds_abs_converges_def
 proof (rule summable_comparison_test_ev)
   from assms(2) show "summable (\<lambda>n. real n powr ((s - s') \<bullet> 1))"
     by (subst summable_real_powr_iff) (simp_all add: inner_diff_left)
 next
   from assms(1) have "(\<lambda>n. fds_nth f n / nat_power n s) \<longlonglongrightarrow> 0"
     unfolding fds_converges_def by (rule summable_LIMSEQ_zero)
   from tendsto_norm[OF this] have "(\<lambda>n. norm (fds_nth f n / nat_power n s)) \<longlonglongrightarrow> 0" by simp
   hence "eventually (\<lambda>n. norm (fds_nth f n / nat_power n s) < 1) at_top"
     by (rule order_tendstoD) simp_all
   thus "eventually (\<lambda>n. norm (norm (fds_nth f n / nat_power n s')) \<le> 
           real n powr ((s - s') \<bullet> 1)) at_top"
   proof eventually_elim
     case (elim n)
     thus ?case
     proof (cases "n = 0")
       case False
       have "norm (fds_nth f n / nat_power n s') =
               norm (fds_nth f n) / real n powr (s' \<bullet> 1)" using False
         by (simp add: norm_divide norm_nat_power)
       also have "\<dots> = norm (fds_nth f n / nat_power n s) / real n powr ((s' - s) \<bullet> 1)" using False
         by (simp add: norm_divide norm_nat_power inner_diff_left powr_diff)
       also have "\<dots> \<le> 1 / real n powr ((s' - s) \<bullet> 1)" using elim
         by (intro divide_right_mono elim) simp_all
       also have "\<dots> = real n powr ((s - s') \<bullet> 1)" using False
         by (simp add: field_simps inner_diff_left powr_diff) 
       finally show ?thesis by simp
     qed simp_all
   qed
 qed
 
 lemma conv_le_abs_conv_abscissa: "conv_abscissa f \<le> abs_conv_abscissa f"
   unfolding conv_abscissa_def abs_conv_abscissa_def
   by (intro INF_superset_mono) auto
 
 lemma conv_abscissa_PInf_iff: "conv_abscissa f = \<infinity> \<longleftrightarrow> (\<forall>s. \<not>fds_converges f s)"
   unfolding conv_abscissa_def by (subst Inf_eq_PInfty) auto
 
 lemma conv_abscissa_PInfI [intro]: "(\<And>s. \<not>fds_converges f s) \<Longrightarrow> conv_abscissa f = \<infinity>"
   by (subst conv_abscissa_PInf_iff) auto
     
 lemma conv_abscissa_MInf_iff: "conv_abscissa (f :: 'a fds) = -\<infinity> \<longleftrightarrow> (\<forall>s. fds_converges f s)"
 proof safe
   assume *: "\<forall>s. fds_converges f s"
   have "conv_abscissa f \<le> B" for B :: real
     using spec[OF *, of "of_real B"] fds_diverges[of "of_real B" f]
     by (cases "conv_abscissa f \<le> B") simp_all
   thus "conv_abscissa f = -\<infinity>" by (rule ereal_bot)    
 qed (auto intro: fds_converges)
 
 lemma conv_abscissa_MInfI [intro]: "(\<And>s. fds_converges (f::'a fds) s) \<Longrightarrow> conv_abscissa f = -\<infinity>"
   by (subst conv_abscissa_MInf_iff) auto
 
 lemma abs_conv_abscissa_PInf_iff: "abs_conv_abscissa f = \<infinity> \<longleftrightarrow> (\<forall>s. \<not>fds_abs_converges f s)"
   unfolding abs_conv_abscissa_def by (subst Inf_eq_PInfty) auto
 
 lemma abs_conv_abscissa_PInfI [intro]: "(\<And>s. \<not>fds_converges f s) \<Longrightarrow> abs_conv_abscissa f = \<infinity>"
   by (subst abs_conv_abscissa_PInf_iff) auto
 
 lemma abs_conv_abscissa_MInf_iff: 
   "abs_conv_abscissa (f :: 'a fds) = -\<infinity> \<longleftrightarrow> (\<forall>s. fds_abs_converges f s)"
 proof safe
   assume *: "\<forall>s. fds_abs_converges f s"
   have "abs_conv_abscissa f \<le> B" for B :: real
     using spec[OF *, of "of_real B"] fds_abs_diverges[of "of_real B" f]
     by (cases "abs_conv_abscissa f \<le> B") simp_all
   thus "abs_conv_abscissa f = -\<infinity>" by (rule ereal_bot)    
 qed (auto intro: fds_abs_converges)
 
 lemma abs_conv_abscissa_MInfI [intro]: 
   "(\<And>s. fds_abs_converges (f::'a fds) s) \<Longrightarrow> abs_conv_abscissa f = -\<infinity>"
   by (subst abs_conv_abscissa_MInf_iff) auto
 
 lemma conv_abscissa_geI:
   assumes "\<And>c'. ereal c' < c \<Longrightarrow> \<exists>s. s \<bullet> 1 = c' \<and> \<not>fds_converges f s"
   shows   "conv_abscissa (f :: 'a fds) \<ge> c"
 proof (rule ccontr)
   assume "\<not>conv_abscissa f \<ge> c"
   hence "c > conv_abscissa f" by simp
   from ereal_dense2[OF this] obtain c' where "c > ereal c'" "c' > conv_abscissa f" by auto
   moreover from assms[OF this(1)] obtain s where "s \<bullet> 1 = c'" "\<not>fds_converges f s" by blast
   ultimately show False using fds_converges[of f s] by auto
 qed
 
 lemma conv_abscissa_leI:
   assumes "\<And>c'. ereal c' > c \<Longrightarrow> \<exists>s. s \<bullet> 1 = c' \<and> fds_converges f s"
   shows   "conv_abscissa (f :: 'a fds) \<le> c"
 proof (rule ccontr)
   assume "\<not>conv_abscissa f \<le> c"
   hence "c < conv_abscissa f" by simp
   from ereal_dense2[OF this] obtain c' where "c < ereal c'" "c' < conv_abscissa f" by auto
   moreover from assms[OF this(1)] obtain s where "s \<bullet> 1 = c'" "fds_converges f s" by blast
   ultimately show False using fds_diverges[of s f] by auto
 qed
 
 lemma abs_conv_abscissa_geI:
   assumes "\<And>c'. ereal c' < c \<Longrightarrow> \<exists>s. s \<bullet> 1 = c' \<and> \<not>fds_abs_converges f s"
   shows   "abs_conv_abscissa (f :: 'a fds) \<ge> c"
 proof (rule ccontr)
   assume "\<not>abs_conv_abscissa f \<ge> c"
   hence "c > abs_conv_abscissa f" by simp
   from ereal_dense2[OF this] obtain c' where "c > ereal c'" "c' > abs_conv_abscissa f" by auto
   moreover from assms[OF this(1)] obtain s where "s \<bullet> 1 = c'" "\<not>fds_abs_converges f s" by blast
   ultimately show False using fds_abs_converges[of f s] by auto
 qed  
 
 lemma abs_conv_abscissa_leI:
   assumes "\<And>c'. ereal c' > c \<Longrightarrow> \<exists>s. s \<bullet> 1 = c' \<and> fds_abs_converges f s"
   shows   "abs_conv_abscissa (f :: 'a fds) \<le> c"
 proof (rule ccontr)
   assume "\<not>abs_conv_abscissa f \<le> c"
   hence "c < abs_conv_abscissa f" by simp
   from ereal_dense2[OF this] obtain c' where "c < ereal c'" "c' < abs_conv_abscissa f" by auto
   moreover from assms[OF this(1)] obtain s where "s \<bullet> 1 = c'" "fds_abs_converges f s" by blast
   ultimately show False using fds_abs_diverges[of s f] by auto
 qed
 
 lemma conv_abscissa_leI_weak:
   assumes "\<And>x. ereal x > d \<Longrightarrow> fds_converges f (of_real x)"
   shows   "conv_abscissa (f :: 'a fds) \<le> d"
 proof (rule conv_abscissa_leI)
   fix x assume "d < ereal x"
   from assms[OF this] show "\<exists>s. s \<bullet> 1 = x \<and> fds_converges f s"
     by (intro exI[of _ "of_real x"]) auto
 qed
 
 lemma abs_conv_abscissa_leI_weak:
   assumes "\<And>x. ereal x > d \<Longrightarrow> fds_abs_converges f (of_real x)"
   shows   "abs_conv_abscissa (f :: 'a fds) \<le> d"
 proof (rule abs_conv_abscissa_leI)
   fix x assume "d < ereal x"
   from assms[OF this] show "\<exists>s. s \<bullet> 1 = x \<and> fds_abs_converges f s"
     by (intro exI[of _ "of_real x"]) auto
 qed
 
 lemma conv_abscissa_truncate [simp]: 
   "conv_abscissa (fds_truncate m (f :: 'a fds)) = -\<infinity>"
   by (auto simp: conv_abscissa_MInf_iff)
 
 lemma abs_conv_abscissa_truncate [simp]: 
   "abs_conv_abscissa (fds_truncate m (f :: 'a fds)) = -\<infinity>"
   by (auto simp: abs_conv_abscissa_MInf_iff)
 
   
 theorem abs_conv_le_conv_abscissa_plus_1: "abs_conv_abscissa (f :: 'a fds) \<le> conv_abscissa f + 1"
 proof (rule abs_conv_abscissa_leI)
   fix c assume less: "conv_abscissa f + 1 < ereal c"
   define c' where "c' = (if conv_abscissa f = -\<infinity> then c - 2
                      else (c - 1 + real_of_ereal (conv_abscissa f)) / 2)"
   from less have c': "conv_abscissa f < ereal c' \<and> c' < c - 1"
     by (cases "conv_abscissa f") (simp_all add: c'_def field_simps)
   
   from c' have "fds_converges f (of_real c')" 
     by (intro fds_converges) (simp_all add: inner_diff_left dot_square_norm)
   hence "fds_abs_converges f (of_real c)"
     by (rule fds_converges_imp_abs_converges) (insert c', simp_all)
   thus "\<exists>s. s \<bullet> 1 = c \<and> fds_abs_converges f s"
     by (intro exI[of _ "of_real c"]) auto
 qed
 
   
 lemma uniformly_convergent_eval_fds:
   assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > conv_abscissa (f :: 'a fds)"
   shows   "uniformly_convergent_on B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z)"
 proof (cases "B = {}")
   case False
   define \<sigma> where "\<sigma> = Inf ((\<lambda>s. s \<bullet> 1) ` B)"
   have \<sigma>_le: "s \<bullet> 1 \<ge> \<sigma>" if "s \<in> B" for s
     unfolding \<sigma>_def using that 
     by (intro cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded B) auto
   have "\<sigma> \<in> ((\<lambda>s. s \<bullet> 1) ` B)"
     unfolding \<sigma>_def using B \<open>B \<noteq> {}\<close>
     by (intro closed_contains_Inf bounded_inner_imp_bdd_below compact_imp_bounded B 
           compact_imp_closed compact_continuous_image continuous_intros) auto
   with B(2) have \<sigma>_gt: "\<sigma> > conv_abscissa f" by auto
   define s where "s = (if conv_abscissa f = -\<infinity> then \<sigma> - 1 else 
                          (\<sigma> + real_of_ereal (conv_abscissa f)) / 2)"
   from \<sigma>_gt have s: "conv_abscissa f < s \<and> s < \<sigma>"
     by (cases "conv_abscissa f") (auto simp: s_def)
   show ?thesis using s \<open>compact B\<close>
     by (intro uniformly_convergent_eval_fds_aux'[of f "of_real s"] fds_converges)
        (auto dest: \<sigma>_le)
 qed auto
     
 corollary uniformly_convergent_eval_fds':
   assumes B: "compact B" "\<And>z. z \<in> B \<Longrightarrow> z \<bullet> 1 > conv_abscissa (f :: 'a fds)"
   shows   "uniformly_convergent_on B (\<lambda>N z. \<Sum>n<N. fds_nth f n / nat_power n z)"
 proof -
   from uniformly_convergent_eval_fds[OF assms] obtain l where
     "uniform_limit B (\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z) l at_top" 
     by (auto simp: uniformly_convergent_on_def)
   also have "(\<lambda>N z. \<Sum>n\<le>N. fds_nth f n / nat_power n z) = 
                 (\<lambda>N z. \<Sum>n<Suc N. fds_nth f n / nat_power n z)"
     by (simp only: lessThan_Suc_atMost)
   finally have "uniform_limit B (\<lambda>N z. \<Sum>n<N. fds_nth f n / nat_power n z) l at_top" 
     unfolding uniform_limit_iff by (subst (asm) eventually_sequentially_Suc)
   thus ?thesis by (auto simp: uniformly_convergent_on_def)
 qed
 
 
 subsection \<open>Derivative of a Dirichlet series\<close>
 
 lemma fds_converges_deriv_aux:
   assumes conv: "fds_converges f (s0 :: 'a)" and gt: "s \<bullet> 1 > s0 \<bullet> 1"
   shows "fds_converges (fds_deriv f) s"
 proof -
   have "Cauchy (\<lambda>n. \<Sum>k\<le>n. (-ln (real k) *\<^sub>R fds_nth f k) / nat_power k s)"
   proof (rule CauchyI', goal_cases)
     case (1 \<epsilon>)
     define \<delta> where "\<delta> = s \<bullet> 1 - s0 \<bullet> 1"
     define \<delta>' where "\<delta>' = \<delta> / 2"
     from gt have \<delta>_pos: "\<delta> > 0" by (simp add: \<delta>_def)
     define A where "A = sum_upto (\<lambda>k. fds_nth f k / nat_power k s0)"
     from conv have "convergent (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)"
       by (simp add: fds_converges_def summable_iff_convergent')
     hence "Bseq (\<lambda>n. \<Sum>k\<le>n. fds_nth f k / nat_power k s0)" by (rule convergent_imp_Bseq)
     then obtain C_aux where C_aux: "\<And>n. norm (\<Sum>k\<le>n. fds_nth f k / nat_power k s0) \<le> C_aux"
       by (auto simp: Bseq_def)
     define C where "C = max C_aux 1"
     have C_pos: "C > 0" by (simp add: C_def)
     have C: "norm (A x) \<le> C" for x
     proof -
       have "A x = (\<Sum>k\<le>nat \<lfloor>x\<rfloor>. fds_nth f k / nat_power k s0)"
         unfolding A_def sum_upto_altdef by (intro sum.mono_neutral_left) auto
       also have "norm \<dots> \<le> C_aux" by (rule C_aux)
       also have "\<dots> \<le> C" by (simp add: C_def)
       finally show ?thesis .
     qed
     define C' where "C' = 2 * C + C * (norm (s0 - s) * (1 + 1 / \<delta>) + 1) / \<delta>"
   
     have "(\<lambda>m. C' * real m powr (-\<delta>')) \<longlonglongrightarrow> 0" unfolding \<delta>'_def using gt \<delta>_pos
       by (intro tendsto_mult_right_zero tendsto_neg_powr filterlim_real_sequentially) simp_all
     from order_tendstoD(2)[OF this \<open>\<epsilon> > 0\<close>] obtain M1 where
       M1: "\<And>m. m \<ge> M1 \<Longrightarrow> C' * real m powr - \<delta>' < \<epsilon>" 
       by (auto simp: eventually_at_top_linorder)
     have "((\<lambda>x. ln (real x) / real x powr \<delta>') \<longlongrightarrow> 0) at_top" using \<delta>_pos
       by (intro lim_ln_over_power) (simp_all add: \<delta>'_def)
     from order_tendstoD(2)[OF this zero_less_one] eventually_gt_at_top[of "1::nat"]
       have "eventually (\<lambda>n. ln (real n) \<le> n powr \<delta>') at_top" by eventually_elim simp_all
     then obtain M2 where M2: "\<And>n. n \<ge> M2 \<Longrightarrow> ln (real n) \<le> n powr \<delta>'"
       by (auto simp: eventually_at_top_linorder)
     let ?f' = "\<lambda>k. -ln (real k) *\<^sub>R fds_nth f k"
 
     show ?case
     proof (intro exI[of _ "max (max M1 M2) 1"] allI impI, goal_cases)
       case (1 m n)
       hence mn: "m \<ge> M1" "m \<ge> M2" "m > 0" "m < n" by simp_all
       define g :: "real \<Rightarrow> 'a" where "g = (\<lambda>t. real_power t (s0 - s) * of_real (ln t))"
       define g' :: "real \<Rightarrow> 'a"            
         where "g' = (\<lambda>t. real_power t (s0 - s - 1) * ((s0 - s) * of_real (ln t) + 1))"
       define norm_g' :: "real \<Rightarrow> real"
         where "norm_g' = (\<lambda>t. t powr (-\<delta> - 1) * (norm (s0 - s) * ln t + 1))"
       define norm_g :: "real \<Rightarrow> real"
         where "norm_g = (\<lambda>t. -(t powr -\<delta>) * (norm (s0 - s) * (\<delta> * ln t + 1) + \<delta>) / \<delta>^2)"
       have g_g': "(g has_vector_derivative g' t) (at t)" if "t \<in> {real m..real n}" for t 
         using mn that by (auto simp: g_def g'_def real_power_diff field_simps real_power_add 
                                intro!: derivative_eq_intros)
       have [continuous_intros]: "continuous_on {real m..real n} g" using mn
         by (auto simp: g_def intro!: continuous_intros)
         
       let ?S = "\<Sum>k\<in>real -` {real m<..real n}. fds_nth f k / nat_power k s0 * g k"
       have "dist (\<Sum>k\<le>m. ?f' k / nat_power k s) (\<Sum>k\<le>n. ?f' k / nat_power k s) =
               norm (\<Sum>k\<in>{..n} - {..m}. fds_nth f k / nat_power k s * of_real (ln (real k)))" 
         using mn by (subst sum_diff) 
            (simp_all add: dist_norm norm_minus_commute sum_negf scaleR_conv_of_real mult_ac)
       also have "{..n} - {..m} = real -` {real m<..real n}" by auto
       also have "(\<Sum>k\<in>\<dots>. fds_nth f k / nat_power k s * of_real (ln (real k))) = 
         (\<Sum>k\<in>\<dots>. fds_nth f k / nat_power k s0 * g k)" using mn unfolding g_def
         by (intro sum.cong refl) (auto simp: real_power_nat_power field_simps nat_power_diff)
       also have *: "((\<lambda>t. A t * g' t) has_integral
                     (A (real n) * g n - A (real m) * g m - ?S))
                     {real m..real n}" (is "(?h has_integral _) _") unfolding A_def using mn
         by (intro partial_summation_strong[of "{}"])
            (auto intro!: g_g' simp: field_simps continuous_intros)
       hence "?S = A (real n) * g n - A (real m) * g m - integral {real m..real n} ?h"
         using mn by (simp add: has_integral_iff field_simps)
       also have "norm \<dots> \<le> norm (A (real n) * g n) + norm (A (real m) * g m) + 
                              norm (integral {real m..real n} ?h)"
         by (intro order.trans[OF norm_triangle_ineq4] add_right_mono order.refl)
       also have "norm (A (real n) * g n) \<le> C * norm (g n)"
         unfolding norm_mult using mn C_pos by (intro mult_mono C) auto
       also have "norm (g n) \<le> n powr -\<delta> * n powr \<delta>'" using mn M2[of n]
         by (simp add: g_def norm_real_power norm_mult \<delta>_def inner_diff_left)
       also have "\<dots> = n powr -\<delta>'" using mn 
         by (simp add: \<delta>'_def powr_minus field_simps powr_add [symmetric])
       also have "norm (A (real m) * g m) \<le> C * norm (g m)"
         unfolding norm_mult using mn C_pos by (intro mult_mono C) auto
       also have "norm (g m) \<le> m powr -\<delta> * m powr \<delta>'" using mn M2[of m]
         by (simp add: g_def norm_real_power norm_mult \<delta>_def inner_diff_left)
       also have "\<dots> = m powr -\<delta>'" using mn 
         by (simp add: \<delta>'_def powr_minus field_simps powr_add [symmetric])
       also have "C * real n powr - \<delta>' \<le> C * real m powr - \<delta>'" using \<delta>_pos mn C_pos
         by (intro mult_left_mono powr_mono2') (simp_all add: \<delta>'_def)
       also have "\<dots> + \<dots> = 2 * \<dots>" by simp
       also have "norm (integral {m..n} ?h) \<le> integral {m..n} (\<lambda>t. C * norm_g' t)"
       proof (intro integral_norm_bound_integral ballI, goal_cases)
         case 1
         with * show ?case by (simp add: has_integral_iff)
       next
         case 2
         from mn show ?case 
           by (auto intro!: integrable_continuous_real continuous_intros simp: norm_g'_def)
       next
         case (3 t)
         have "norm (g' t) \<le> norm_g' t" unfolding g'_def norm_g'_def using 3 mn
           unfolding norm_mult
           by (intro mult_mono order.trans[OF norm_triangle_ineq])
              (auto simp: norm_real_power inner_diff_left dot_square_norm norm_mult \<delta>_def
                    intro!: mult_left_mono)
         thus ?case unfolding norm_mult using C_pos mn 
           by (intro mult_mono C) simp_all
       qed
       also have "\<dots> = C * integral {m..n} norm_g'"
         unfolding norm_g'_def by (simp add: norm_g'_def \<delta>_def inner_diff_left)
       also {
         have "(norm_g' has_integral (norm_g n - norm_g m)) {m..n}"
           unfolding norm_g'_def norm_g_def power2_eq_square using mn \<delta>_pos
           by (intro fundamental_theorem_of_calculus)
              (auto simp: has_field_derivative_iff_has_vector_derivative [symmetric] 
                  field_simps powr_diff intro!: derivative_eq_intros)
         hence "integral {m..n} norm_g' = norm_g n - norm_g m" by (simp add: has_integral_iff)
         also have "norm_g n \<le> 0" unfolding norm_g_def using \<delta>_pos mn
           by (intro divide_nonpos_pos mult_nonpos_nonneg add_nonneg_nonneg mult_nonneg_nonneg)
              simp_all
         hence "norm_g n - norm_g m \<le> -norm_g m" by simp
         also have "\<dots> = real m powr -\<delta> * ln (real m) * (norm (s0 - s)) / \<delta> +
                         real m powr -\<delta> * ((norm (s0 - s) / \<delta> + 1) / \<delta>)" using \<delta>_pos
           by (simp add: field_simps norm_g_def power2_eq_square)
         also {
           have "ln (real m) \<le> real m powr \<delta>'" using M2[of m] mn by simp
           also have "real m powr -\<delta> * \<dots> = real m powr -\<delta>'"
             by (simp add: powr_add [symmetric] \<delta>'_def)
           finally have "real m powr -\<delta> * ln (real m) * (norm (s0 - s)) / \<delta> \<le>
                           \<dots> * (norm (s0 - s)) / \<delta>" using \<delta>_pos
             by (intro divide_right_mono mult_right_mono) (simp_all add: mult_left_mono)
         }
         also have "real m powr -\<delta> * ((norm (s0 - s) / \<delta> + 1) / \<delta>) \<le>
                      real m powr -\<delta>' * ((norm (s0 - s) / \<delta> + 1) / \<delta>)" using mn \<delta>_pos
           by (intro mult_right_mono powr_mono) (simp_all add: \<delta>'_def)
         also have "real m powr - \<delta>' * norm (s0 - s) / \<delta> + \<dots> =
                      real m powr -\<delta>' * (norm (s0 - s) * (1 + 1 / \<delta>) + 1) / \<delta>" using \<delta>_pos
           by (simp add: field_simps power2_eq_square)
         finally have "integral {real m..real n} norm_g' \<le>
                         real m powr - \<delta>' * (norm (s0 - s) * (1 + 1 / \<delta>) + 1) / \<delta>" by - simp_all
       }
       also have "2 * (C * m powr - \<delta>') + C * (m powr - \<delta>' * (norm (s0 - s) * (1 + 1 / \<delta>) + 1) / \<delta>) =
                    C' * m powr -\<delta>'" by (simp add: algebra_simps C'_def)
       also have "\<dots> < \<epsilon>" using M1[of m] mn by simp
       finally show ?case using C_pos by - simp_all
     qed
   qed
   from Cauchy_convergent[OF this] 
     show ?thesis by (simp add: summable_iff_convergent' fds_converges_def fds_nth_deriv)
 qed
 
 theorem
   assumes "s \<bullet> 1 > conv_abscissa (f :: 'a fds)"
   shows   fds_converges_deriv: "fds_converges (fds_deriv f) s"
     and   has_field_derivative_eval_fds [derivative_intros]:
             "(eval_fds f has_field_derivative eval_fds (fds_deriv f) s) (at s within A)"
 proof -
   define s1 :: real where
     "s1 = (if conv_abscissa f = -\<infinity> then s \<bullet> 1 - 2 else
              (s \<bullet> 1 * 1 / 3 + real_of_ereal (conv_abscissa f) * 2 / 3))"
   define s2 :: real where
     "s2 = (if conv_abscissa f = -\<infinity> then s \<bullet> 1 - 1 else
              (s \<bullet> 1 * 2 / 3 + real_of_ereal (conv_abscissa f) * 1 / 3))"
   from assms have s: "conv_abscissa f < s1 \<and> s1 < s2 \<and> s2 < s \<bullet> 1"
     by (cases "conv_abscissa f") (auto simp: s1_def s2_def field_simps)
   from s have *: "fds_converges f (of_real s1)" by (intro fds_converges) simp_all
   thus conv': "fds_converges (fds_deriv f) s"
     by (rule fds_converges_deriv_aux) (insert s, simp_all)
   from * have conv: "fds_converges (fds_deriv f) (of_real s2)"
     by (rule fds_converges_deriv_aux) (insert s, simp_all)
       
   define \<delta> :: real where "\<delta> = (s \<bullet> 1 - s2) / 2"
   from s have \<delta>_pos: "\<delta> > 0" by (simp add: \<delta>_def)
   
   have "uniformly_convergent_on (cball s \<delta>) 
           (\<lambda>n s. \<Sum>k\<le>n. fds_nth (fds_deriv f) k / nat_power k s)"
   proof (intro uniformly_convergent_eval_fds_aux'[OF conv])
     fix s'' :: 'a assume s'': "s'' \<in> cball s \<delta>"
     have "dist (s \<bullet> 1) (s'' \<bullet> 1) \<le> dist s s''" 
       by (intro Euclidean_dist_upper) (simp_all add: one_in_Basis)
     also from s'' have "\<dots> \<le> \<delta>" by simp
     finally show "s'' \<bullet> 1 > (of_real s2 :: 'a) \<bullet> 1" using s
       by (auto simp: \<delta>_def dist_real_def abs_if split: if_splits)
   qed (insert \<delta>_pos, auto)
   then obtain l where 
      "uniform_limit (cball s \<delta>) (\<lambda>n s. \<Sum>k\<le>n. fds_nth (fds_deriv f) k / nat_power k s) l at_top"
     by (auto simp: uniformly_convergent_on_def)
   also have "(\<lambda>n s. \<Sum>k\<le>n. fds_nth (fds_deriv f) k / nat_power k s) =
                (\<lambda>n s. \<Sum>k<Suc n. fds_nth (fds_deriv f) k / nat_power k s)"
     by (simp only: lessThan_Suc_atMost)
   finally have "uniform_limit (cball s \<delta>) (\<lambda>n s. \<Sum>k<n. fds_nth (fds_deriv f) k / nat_power k s) 
                   l at_top"
     unfolding uniform_limit_iff by (subst (asm) eventually_sequentially_Suc)
   hence *: "uniformly_convergent_on (cball s \<delta>) 
               (\<lambda>n s. \<Sum>k<n. fds_nth (fds_deriv f) k / nat_power k s)"
     unfolding uniformly_convergent_on_def by blast
       
   have "(eval_fds f has_field_derivative eval_fds (fds_deriv f) s) (at s)"
     unfolding eval_fds_def
   proof (rule has_field_derivative_series'(2)[OF _ _ *])
     show "s \<in> cball s \<delta>" "s \<in> interior (cball s \<delta>)" using s by (simp_all add: \<delta>_def)
     show "summable (\<lambda>n. fds_nth f n / nat_power n s)"
       using assms fds_converges[of f s] by (simp add: fds_converges_def)
   next
     fix s' :: 'a and n :: nat
     show "((\<lambda>s. fds_nth f n / nat_power n s) has_field_derivative
             fds_nth (fds_deriv f) n / nat_power n s') (at s' within cball s \<delta>)"
       by (cases "n = 0") 
          (simp, auto intro!: derivative_eq_intros simp: fds_nth_deriv field_simps)
   qed (auto simp: fds_nth_deriv intro!: derivative_eq_intros)
   thus "(eval_fds f has_field_derivative eval_fds (fds_deriv f) s) (at s within A)"
     by (rule has_field_derivative_at_within)
 qed
 
 lemmas has_field_derivative_eval_fds' [derivative_intros] = 
   DERIV_chain2[OF has_field_derivative_eval_fds]
 
 lemma continuous_eval_fds [continuous_intros]:
   assumes "s \<bullet> 1 > conv_abscissa f"
   shows   "continuous (at s within A) (eval_fds (f :: 'a :: dirichlet_series fds))"
 proof -
   have "isCont (eval_fds f) s"
     by (rule has_field_derivative_eval_fds DERIV_isCont assms)+
   thus ?thesis by (rule continuous_within_subset) auto
 qed
 
 lemma continuous_eval_fds' [continuous_intros]:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "continuous (at s within A) g" "g s \<bullet> 1 > conv_abscissa f"
   shows   "continuous (at s within A) (\<lambda>x. eval_fds f (g x))"
   by (rule continuous_within_compose3[OF _ assms(1)] continuous_intros assms)+
 
 lemma continuous_on_eval_fds [continuous_intros]:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "A \<subseteq> {s. s \<bullet> 1 > conv_abscissa f}"
   shows   "continuous_on A (eval_fds f)"
   by (rule DERIV_continuous_on derivative_intros)+ (insert assms, auto)
 
 lemma continuous_on_eval_fds' [continuous_intros]:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "continuous_on A g" "g ` A \<subseteq> {s. s \<bullet> 1 > conv_abscissa f}"
   shows   "continuous_on A (\<lambda>x. eval_fds f (g x))"
   by (rule continuous_on_compose2[OF continuous_on_eval_fds assms(1)])
      (insert assms, auto simp: image_iff)
 
 lemma conv_abscissa_deriv_le:
   fixes f :: "'a fds"
   shows "conv_abscissa (fds_deriv f) \<le> conv_abscissa f"
 proof (rule conv_abscissa_leI)
   fix c' :: real
   assume "ereal c' > conv_abscissa f"
   thus "\<exists>s. s \<bullet> 1 = c' \<and> fds_converges (fds_deriv f) s"
     by (intro exI[of _ "of_real c'"]) (auto simp: fds_converges_deriv)
 qed
 
 lemma abs_conv_abscissa_integral:
   fixes f :: "'a fds"
   shows "abs_conv_abscissa (fds_integral a f) = abs_conv_abscissa f"
 proof (rule antisym)
   show "abs_conv_abscissa (fds_integral a f) \<le> abs_conv_abscissa f"
   proof (rule abs_conv_abscissa_leI, goal_cases)
     case (1 c)
     have "fds_abs_converges (fds_integral a f) (of_real c)"
       unfolding fds_abs_converges_def
     proof (rule summable_comparison_test_ev)
       from 1 have "fds_abs_converges f (of_real c)"
         by (intro fds_abs_converges) auto
       thus "summable (\<lambda>n. norm (fds_nth f n / nat_power n (of_real c)))"
         by (simp add: fds_abs_converges_def)
     next
       show "\<forall>\<^sub>F n in sequentially. norm (norm (fds_nth (fds_integral a f) n / nat_power n (of_real c))) \<le> 
               norm (fds_nth f n / nat_power n (of_real c))"
         using eventually_gt_at_top[of 3]
       proof eventually_elim
         case (elim n)
         from elim and exp_le have "ln (exp 1) \<le> ln (real n)"
           by (subst ln_le_cancel_iff) auto
         hence "1 * norm (fds_nth f n) \<le> ln (real n) * norm (fds_nth f n)"
           by (intro mult_right_mono) auto
         with elim show ?case
           by (simp add: norm_divide norm_nat_power fds_integral_def field_simps)
       qed
     qed
     thus ?case by (intro exI[of _ "of_real c"]) auto
   qed
 next
   show "abs_conv_abscissa f \<le> abs_conv_abscissa (fds_integral a f)" (is "_ \<le> ?s0")
   proof (cases "abs_conv_abscissa (fds_integral a f) = \<infinity>")
     case False
     show ?thesis
     proof (rule abs_conv_abscissa_leI)
       fix c :: real
       define \<epsilon> where "\<epsilon> = (if ?s0 = -\<infinity> then 1 else (c - real_of_ereal ?s0) / 2)"
       assume "ereal c > ?s0"
       with False have \<epsilon>: "\<epsilon> > 0" "c - \<epsilon> > ?s0"
         by (cases ?s0; force simp: \<epsilon>_def field_simps)+
   
       have "fds_abs_converges f (of_real c)"
         unfolding fds_abs_converges_def
       proof (rule summable_comparison_test_ev)
         from \<epsilon> have "fds_abs_converges (fds_integral a f) (of_real (c - \<epsilon>))"
           by (intro fds_abs_converges) (auto simp: algebra_simps)
         thus "summable (\<lambda>n. norm (fds_nth (fds_integral a f) n / nat_power n (of_real (c - \<epsilon>))))"
           by (simp add: fds_abs_converges_def)
       next
         have "\<forall>\<^sub>F n in at_top. ln (real n) / real n powr \<epsilon> < 1"
           by (rule order_tendstoD lim_ln_over_power \<open>\<epsilon> > 0\<close> zero_less_one)+
         thus "\<forall>\<^sub>F n in sequentially. norm (norm (fds_nth f n / nat_power n (of_real c)))
                 \<le> norm (fds_nth (fds_integral a f) n / nat_power n (of_real (c - \<epsilon>)))"
           using eventually_gt_at_top[of 1]
         proof eventually_elim
           case (elim n)
           hence "ln (real n) * norm (fds_nth f n) \<le> real n powr \<epsilon> * norm (fds_nth f n)"
             by (intro mult_right_mono) auto
           with elim show ?case
             by (simp add: norm_divide norm_nat_power field_simps 
                           powr_diff inner_diff_left fds_integral_def)
         qed
       qed
       thus "\<exists>s. s \<bullet> 1 = c \<and> fds_abs_converges f s"
         by (intro exI[of _ "of_real c"]) auto
     qed
   qed auto
 qed
 
 lemma abs_conv_abscissa_ln: 
   "abs_conv_abscissa (fds_ln l (f :: 'a :: dirichlet_series fds)) = 
      abs_conv_abscissa (fds_deriv f / f)"
   by (simp add: fds_ln_def abs_conv_abscissa_integral)
 
 lemma abs_conv_abscissa_deriv:
   fixes f :: "'a fds"
   shows "abs_conv_abscissa (fds_deriv f) = abs_conv_abscissa f"
 proof -
   have "abs_conv_abscissa (fds_deriv f) = 
           abs_conv_abscissa (fds_integral (fds_nth f 1) (fds_deriv f))"
     by (rule abs_conv_abscissa_integral [symmetric])
   also have "fds_integral (fds_nth f 1) (fds_deriv f) = f"
     by (rule fds_integral_fds_deriv)
   finally show ?thesis .
 qed
 
 lemma abs_conv_abscissa_higher_deriv:
   "abs_conv_abscissa ((fds_deriv ^^ n) f) = abs_conv_abscissa (f :: 'a :: dirichlet_series fds)"
   by (induction n) (simp_all add: abs_conv_abscissa_deriv)
 
 lemma conv_abscissa_higher_deriv_le:
   "conv_abscissa ((fds_deriv ^^ n) f) \<le> conv_abscissa (f :: 'a :: dirichlet_series fds)"
   by (induction n) (auto intro: order.trans[OF conv_abscissa_deriv_le])
 
 lemma abs_conv_abscissa_restrict:
   "abs_conv_abscissa (fds_subseries P f) \<le> abs_conv_abscissa f"
   by (rule abs_conv_abscissa_mono) auto
 
 lemma eval_fds_deriv:
   fixes f :: "'a fds"
   assumes "s \<bullet> 1 > conv_abscissa f"
   shows   "eval_fds (fds_deriv f) s = deriv (eval_fds f) s"
   by (intro DERIV_imp_deriv [symmetric] derivative_intros assms)
 
 lemma eval_fds_higher_deriv:
   assumes "(s :: 'a :: dirichlet_series) \<bullet> 1 > conv_abscissa f"
   shows   "eval_fds ((fds_deriv ^^ n) f) s = (deriv ^^ n) (eval_fds f) s"
   using assms
 proof (induction n arbitrary: f s)
   case (Suc n f s)
   have ev: "eventually (\<lambda>s. s \<in> {s. s \<bullet> 1 > conv_abscissa f}) (nhds s)"
     using Suc.prems open_halfspace_gt[of _ "1::'a"]
     by (intro eventually_nhds_in_open, cases "conv_abscissa f")
        (auto simp: open_halfspace_gt inner_commute)
   have "eval_fds ((fds_deriv ^^ Suc n) f) s = eval_fds ((fds_deriv ^^ n) (fds_deriv f)) s"
     by (subst funpow_Suc_right) simp
   also have "\<dots> = (deriv ^^ n) (eval_fds (fds_deriv f)) s"
     by (intro Suc.IH le_less_trans[OF conv_abscissa_deriv_le] Suc.prems)
   also have "\<dots> = (deriv ^^ n) (deriv (eval_fds f)) s"
     by (intro higher_deriv_cong_ev refl eventually_mono[OF ev] eval_fds_deriv) auto
   also have "\<dots> = (deriv ^^ Suc n) (eval_fds f) s"
     by (subst funpow_Suc_right) simp
   finally show ?case .
 qed auto
 
 end
 
 
 subsection \<open>Multiplication of two series\<close>
 
 lemma 
   fixes f g :: "nat \<Rightarrow> 'a :: {banach, real_normed_field, second_countable_topology, nat_power}"
   fixes s :: 'a
   assumes [simp]: "f 0 = 0" "g 0 = 0"
   assumes summable: "summable (\<lambda>n. norm (f n / nat_power n s))"
                     "summable (\<lambda>n. norm (g n / nat_power n s))"
   shows   summable_dirichlet_prod: "summable (\<lambda>n. norm (dirichlet_prod f g n / nat_power n s))"
     and   suminf_dirichlet_prod:
             "(\<Sum>n. dirichlet_prod f g n / nat_power n s) = 
                (\<Sum>n. f n / nat_power n s) * (\<Sum>n. g n / nat_power n s)"
 proof -
   have summable': "(\<lambda>n. f n / nat_power n s) abs_summable_on A"
                   "(\<lambda>n. g n / nat_power n s) abs_summable_on A" for A
     by ((rule abs_summable_on_subset[OF _ subset_UNIV], insert summable, 
         simp add: abs_summable_on_nat_iff'); fail)+
   have f_g: "f a / nat_power a s * (g b / nat_power b s) =
               f a * g b / nat_power (a * b) s" for a b
     by (cases "a * b = 0") (auto simp: nat_power_mult_distrib)
 
   have eq: "(\<Sum>\<^sub>a(m, n)\<in>{(m, n). m * n = x}. f m * g n / nat_power x s) =
               dirichlet_prod f g x / nat_power x s" for x :: nat
   proof (cases "x > 0")
     case False
     hence "(\<Sum>\<^sub>a(m,n) | m * n = x. f m * g n / nat_power x s) = (\<Sum>\<^sub>a(m,n) | m * n = x. 0)"
       by (intro infsetsum_cong) auto
     with False show ?thesis by simp
   next
     case True
     from finite_divisors_nat'[OF this] show ?thesis 
       by (simp add: dirichlet_prod_altdef2 case_prod_unfold sum_divide_distrib)
   qed
 
   have "(\<lambda>(m,n). (f m / nat_power m s) * (g n / nat_power n s)) abs_summable_on UNIV \<times> UNIV"
     using summable' by (intro abs_summable_on_product) auto
   also have "?this \<longleftrightarrow> (\<lambda>(m,n). f m * g n / nat_power (m*n) s) abs_summable_on UNIV"
     using f_g by (intro abs_summable_on_cong) auto
   also have "\<dots> \<longleftrightarrow> (\<lambda>(x,(m,n)). f m * g n / nat_power (m*n) s) abs_summable_on 
                            (SIGMA x:UNIV. {(m,n). m * n = x})"
     unfolding case_prod_unfold
     by (rule abs_summable_on_reindex_bij_betw [symmetric]) 
        (auto simp: bij_betw_def inj_on_def image_iff)
   also have "\<dots> \<longleftrightarrow> (\<lambda>(x,(m,n)). f m * g n / nat_power x s) abs_summable_on 
                            (SIGMA x:UNIV. {(m,n). m * n = x})"
     by (intro abs_summable_on_cong) auto
   finally have summable'': \<dots> .
   from abs_summable_on_Sigma_project1'[OF this]
     show summable''': "summable (\<lambda>n. norm (dirichlet_prod f g n / nat_power n s))"
     by (simp add: eq abs_summable_on_nat_iff')
 
   have "(\<Sum>n. f n / nat_power n s) * (\<Sum>n. g n / nat_power n s) =
           (\<Sum>\<^sub>an. f n / nat_power n s) * (\<Sum>\<^sub>an. g n / nat_power n s)"
     using summable' by (simp add: infsetsum_nat')
   also have "\<dots> = (\<Sum>\<^sub>a(m,n). (f m / nat_power m s) * (g n / nat_power n s))"
     using summable' by (subst infsetsum_product [symmetric]) simp_all
   also have "\<dots> = (\<Sum>\<^sub>a(m,n). f m * g n / nat_power (m * n) s)"
     using f_g by (intro infsetsum_cong refl) auto
   also have "\<dots> = (\<Sum>\<^sub>a(x,(m,n))\<in>(SIGMA x:UNIV. {(m,n). m * n = x}). 
                       f m * g n / nat_power (m * n) s)" 
     unfolding case_prod_unfold
     by (rule infsetsum_reindex_bij_betw [symmetric]) (auto simp: bij_betw_def inj_on_def image_iff)
   also have "\<dots> = (\<Sum>\<^sub>a(x,(m,n))\<in>(SIGMA x:UNIV. {(m,n). m * n = x}). 
                       f m * g n / nat_power x s)"
     by (intro infsetsum_cong refl) (auto simp: case_prod_unfold)
   also have "\<dots> = (\<Sum>\<^sub>ax. dirichlet_prod f g x / nat_power x s)"
     (is "_ = infsetsum ?T _") using summable'' by (subst infsetsum_Sigma) (auto simp: eq)
   also have "\<dots> = (\<Sum>x. dirichlet_prod f g x / nat_power x s)"
     using summable''' by (intro infsetsum_nat') (simp_all add: abs_summable_on_nat_iff')
   finally show "\<dots> = (\<Sum>n. f n / nat_power n s) * (\<Sum>n. g n / nat_power n s)" ..
 qed
 
 lemma 
   fixes f g :: "nat \<Rightarrow> real"
   fixes s :: real
   assumes "f 0 = 0" "g 0 = 0"
   assumes summable: "summable (\<lambda>n. norm (f n / real n powr s))"
                     "summable (\<lambda>n. norm (g n / real n powr s))"
   shows   summable_dirichlet_prod_real: "summable (\<lambda>n. norm (dirichlet_prod f g n / real n powr s))"
     and   suminf_dirichlet_prod_real:
             "(\<Sum>n. dirichlet_prod f g n / real n powr s) = 
                (\<Sum>n. f n / nat_power n s) * (\<Sum>n. g n / real n powr s)"
   using summable_dirichlet_prod[of f g s] suminf_dirichlet_prod[of f g s] assms by simp_all
 
 lemma fds_abs_converges_mult: 
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   assumes "fds_abs_converges f s" "fds_abs_converges g s"
   shows   "fds_abs_converges (f * g) s"
   using summable_dirichlet_prod[OF _ _ assms[unfolded fds_abs_converges_def]]
   by (simp add: fds_abs_converges_def fds_nth_mult)
 
 lemma fds_abs_converges_power: 
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   shows "fds_abs_converges f s \<Longrightarrow> fds_abs_converges (f ^ n) s"
   by (induction n) (auto intro!: fds_abs_converges_mult)
 
 lemma fds_abs_converges_prod: 
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   shows "(\<And>x. x \<in> A \<Longrightarrow> fds_abs_converges (f x) s) \<Longrightarrow> fds_abs_converges (prod f A) s"
   by (induction A rule: infinite_finite_induct) (auto intro!: fds_abs_converges_mult)
 
 lemma abs_conv_abscissa_mult_le:
   "abs_conv_abscissa (f * g :: 'a :: dirichlet_series fds) \<le> 
       max (abs_conv_abscissa f) (abs_conv_abscissa g)"
 proof (rule abs_conv_abscissa_leI, goal_cases)
   case (1 c')
   thus ?case
     by (auto intro!: exI[of _ "of_real c'"] fds_abs_converges_mult intro: fds_abs_converges)
 qed
 
 lemma abs_conv_abscissa_mult_leI:
   "abs_conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow>
    abs_conv_abscissa g \<le> d \<Longrightarrow> abs_conv_abscissa (f * g) \<le> d"
   using abs_conv_abscissa_mult_le[of f g] by (auto simp add: le_max_iff_disj)
 
 lemma abs_conv_abscissa_shift [simp]:
   "abs_conv_abscissa (fds_shift c f) = abs_conv_abscissa (f :: 'a :: dirichlet_series fds) + c \<bullet> 1"
 proof -
   have "abs_conv_abscissa (fds_shift c f) \<le> abs_conv_abscissa f + c \<bullet> 1" for c :: 'a and f
   proof (rule abs_conv_abscissa_leI)
     fix d assume "abs_conv_abscissa f + c \<bullet> 1 < ereal d"
     hence "abs_conv_abscissa f < ereal (d - c \<bullet> 1)" by (cases "abs_conv_abscissa f") auto
     hence "fds_abs_converges (fds_shift c f) (of_real d)"
       by (auto intro!: fds_abs_converges_shift fds_abs_converges simp: algebra_simps)
     thus "\<exists>s. s \<bullet> 1 = d \<and> fds_abs_converges (fds_shift c f) s"
       by (auto intro!: exI[of _ "of_real d"])
   qed
   note * = this[of c f] this[of "-c" "fds_shift c f"]
   show ?thesis by (cases "abs_conv_abscissa (fds_shift c f)"; cases "abs_conv_abscissa f")
                   (insert *, auto intro!: antisym)
 qed
 
 lemma eval_fds_mult:
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   assumes "fds_abs_converges f s" "fds_abs_converges g s"
   shows   "eval_fds (f * g) s = eval_fds f s * eval_fds g s"
   using suminf_dirichlet_prod[OF _ _ assms[unfolded fds_abs_converges_def]]
   by (simp_all add: eval_fds_def fds_nth_mult)
 
 lemma eval_fds_power:
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   assumes "fds_abs_converges f s"
   shows "eval_fds (f ^ n) s = eval_fds f s ^ n"
   using assms by (induction n) (simp_all add: eval_fds_mult fds_abs_converges_power)
 
 lemma eval_fds_prod:
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   assumes "(\<And>x. x \<in> A \<Longrightarrow> fds_abs_converges (f x) s)"
   shows "eval_fds (prod f A) s = (\<Prod>x\<in>A. eval_fds (f x) s)" using assms
   by (induction A rule: infinite_finite_induct) (auto simp: eval_fds_mult fds_abs_converges_prod)
 
 lemma eval_fds_inverse:
   fixes s :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology}"
   assumes "fds_abs_converges f s" "fds_abs_converges (inverse f) s" "fds_nth f 1 \<noteq> 0"
   shows   "eval_fds (inverse f) s = inverse (eval_fds f s)"
 proof -
   have "eval_fds (inverse f * f) s = eval_fds (inverse f) s * eval_fds f s"
     by (intro eval_fds_mult assms)
   also have "inverse f * f = 1" by (intro fds_left_inverse assms)
   also have "eval_fds 1 s = 1" by simp
   finally show ?thesis by (auto simp: divide_simps)
 qed
 
 lemma eval_fds_integral_has_field_derivative:
   fixes s :: "'a :: dirichlet_series"
   assumes "ereal (s \<bullet> 1) > abs_conv_abscissa f"
   assumes "fds_nth f 1 = 0"
   shows   "(eval_fds (fds_integral c f) has_field_derivative eval_fds f s) (at s)"
 proof -
   have "conv_abscissa (fds_integral c f) \<le> abs_conv_abscissa (fds_integral c f)"
     by (rule conv_le_abs_conv_abscissa)
   also from assms have "\<dots> < ereal (s \<bullet> 1)" by (simp add: abs_conv_abscissa_integral)
   finally have "(eval_fds (fds_integral c f) has_field_derivative 
                   eval_fds (fds_deriv (fds_integral c f)) s) (at s)"
     by (intro derivative_eq_intros) auto
   also from assms have "fds_deriv (fds_integral c f) = f"
     by simp
   finally show ?thesis .
 qed
 
 lemma holomorphic_fds_eval [holomorphic_intros]:
   "A \<subseteq> {z. Re z > conv_abscissa f} \<Longrightarrow> eval_fds f holomorphic_on A"
   unfolding holomorphic_on_def field_differentiable_def
   by (rule ballI exI derivative_intros)+ auto
     
 lemma analytic_fds_eval [holomorphic_intros]:
   assumes "A \<subseteq> {z. Re z > conv_abscissa f}"
   shows   "eval_fds f analytic_on A"
 proof -
   have "eval_fds f analytic_on {z. Re z > conv_abscissa f}"
   proof (subst analytic_on_open)
     show "open {z. Re z > conv_abscissa f}"
       by (cases "conv_abscissa f") (simp_all add: open_halfspace_Re_gt)
   qed (intro holomorphic_intros, simp_all)
   from analytic_on_subset[OF this assms] show ?thesis .
 qed
 
 lemma conv_abscissa_0 [simp]: 
   "conv_abscissa (0 :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: conv_abscissa_MInf_iff)
 
 lemma abs_conv_abscissa_0 [simp]: 
   "abs_conv_abscissa (0 :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: abs_conv_abscissa_MInf_iff)
 
 lemma conv_abscissa_1 [simp]: 
   "conv_abscissa (1 :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: conv_abscissa_MInf_iff)
 
 lemma abs_conv_abscissa_1 [simp]: 
   "abs_conv_abscissa (1 :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: abs_conv_abscissa_MInf_iff)
 
 lemma conv_abscissa_const [simp]: 
   "conv_abscissa (fds_const (c :: 'a :: dirichlet_series)) = -\<infinity>"
   by (auto simp: conv_abscissa_MInf_iff)
 
 lemma abs_conv_abscissa_const [simp]: 
   "abs_conv_abscissa (fds_const (c :: 'a :: dirichlet_series)) = -\<infinity>"
   by (auto simp: abs_conv_abscissa_MInf_iff)
 
 lemma conv_abscissa_numeral [simp]: 
   "conv_abscissa (numeral n :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: numeral_fds)
 
 lemma abs_conv_abscissa_numeral [simp]: 
   "abs_conv_abscissa (numeral n :: 'a :: dirichlet_series fds) = -\<infinity>"
   by (auto simp: numeral_fds)
 
 lemma abs_conv_abscissa_power_le:
   "abs_conv_abscissa (f ^ n :: 'a :: dirichlet_series fds) \<le> abs_conv_abscissa f"
   by (induction n) (auto intro!: order.trans[OF abs_conv_abscissa_mult_le])
 
 lemma abs_conv_abscissa_power_leI:
   "abs_conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow> abs_conv_abscissa (f ^ n) \<le> d"
   by (rule order.trans[OF abs_conv_abscissa_power_le])
 
 lemma abs_conv_abscissa_prod_le:
   assumes "\<And>x. x \<in> A \<Longrightarrow> abs_conv_abscissa (f x :: 'a :: dirichlet_series fds) \<le> d"
   shows   "abs_conv_abscissa (prod f A) \<le> d" using assms
   by (induction A rule: infinite_finite_induct) (auto intro!: abs_conv_abscissa_mult_leI)
 
 lemma conv_abscissa_add_le:
   "conv_abscissa (f + g :: 'a :: dirichlet_series fds) \<le> max (conv_abscissa f) (conv_abscissa g)"
   by (rule conv_abscissa_leI_weak) (auto intro!: fds_converges_add intro: fds_converges)
 
 lemma conv_abscissa_add_leI:
   "conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow> conv_abscissa g \<le> d \<Longrightarrow> 
      conv_abscissa (f + g) \<le> d"
   using conv_abscissa_add_le[of f g] by (auto simp: le_max_iff_disj)
 
 lemma conv_abscissa_sum_leI:
   assumes "\<And>x. x \<in> A \<Longrightarrow> conv_abscissa (f x :: 'a :: dirichlet_series fds) \<le> d"
   shows   "conv_abscissa (sum f A) \<le> d" using assms
   by (induction A rule: infinite_finite_induct) (auto intro!: conv_abscissa_add_leI)
 
 lemma abs_conv_abscissa_add_le:
   "abs_conv_abscissa (f + g :: 'a :: dirichlet_series fds) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
   by (rule abs_conv_abscissa_leI_weak) (auto intro!: fds_abs_converges_add intro: fds_abs_converges)
 
 lemma abs_conv_abscissa_add_leI:
   "abs_conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow> abs_conv_abscissa g \<le> d \<Longrightarrow> 
      abs_conv_abscissa (f + g) \<le> d"
   using abs_conv_abscissa_add_le[of f g] by (auto simp: le_max_iff_disj)
 
 lemma abs_conv_abscissa_sum_leI:
   assumes "\<And>x. x \<in> A \<Longrightarrow> abs_conv_abscissa (f x :: 'a :: dirichlet_series fds) \<le> d"
   shows   "abs_conv_abscissa (sum f A) \<le> d" using assms
   by (induction A rule: infinite_finite_induct) (auto intro!: abs_conv_abscissa_add_leI)
 
 lemma fds_converges_cmult_left [intro]:
   assumes "fds_converges f s"
   shows   "fds_converges (fds_const c * f) s"
 proof -
   from assms have "summable (\<lambda>n. c * (fds_nth f n / nat_power n s))"
     by (intro summable_mult) (auto simp: fds_converges_def)
   thus ?thesis by (simp add: fds_converges_def mult_ac)
 qed
 
 lemma fds_converges_cmult_right [intro]:
   assumes "fds_converges f s"
   shows   "fds_converges (f * fds_const c) s"
   using fds_converges_cmult_left[OF assms] by (simp add: mult_ac)
 
 lemma conv_abscissa_cmult_left [simp]:
   fixes c :: "'a :: dirichlet_series" assumes "c \<noteq> 0"
   shows "conv_abscissa (fds_const c * f) = conv_abscissa f"
 proof -
   have "fds_converges (fds_const c * f) s \<longleftrightarrow> fds_converges f s" for s
   proof
     assume "fds_converges (fds_const c * f) s"
     hence "fds_converges (fds_const (inverse c) * (fds_const c * f)) s"
       by (rule fds_converges_cmult_left)
     also have "fds_const (inverse c) * (fds_const c * f) = fds_const (inverse c * c) * f"
       by simp
     also have "inverse c * c = 1"
       using assms by simp
     finally show "fds_converges f s" by simp
   qed auto
   thus ?thesis by (simp add: conv_abscissa_def)
 qed
 
 lemma conv_abscissa_cmult_right [simp]:
   fixes c :: "'a :: dirichlet_series" assumes "c \<noteq> 0"
   shows "conv_abscissa (f * fds_const c) = conv_abscissa f"
   using assms by (subst mult.commute) auto
 
 lemma abs_conv_abscissa_cmult: 
   fixes c :: "'a :: dirichlet_series" assumes "c \<noteq> 0"
   shows "abs_conv_abscissa (fds_const c * f) = abs_conv_abscissa f"
 proof (intro antisym)
   have "abs_conv_abscissa (fds_const (inverse c) * (fds_const c * f)) \<le> 
           abs_conv_abscissa (fds_const c * f)"
     using abs_conv_abscissa_mult_le[of "fds_const (inverse c)" "fds_const c * f"]
     by (auto simp: max_def)
   also have "fds_const (inverse c) * (fds_const c * f) = fds_const (inverse c * c) * f"
     by (simp add: mult_ac)
   also have "inverse c * c = 1" using assms by simp
   finally show "abs_conv_abscissa f \<le> abs_conv_abscissa (fds_const c * f)" by simp
 qed (insert abs_conv_abscissa_mult_le[of "fds_const c" f], auto simp: max_def)
 
 lemma conv_abscissa_minus [simp]:
   fixes f :: "'a :: dirichlet_series fds"
   shows "conv_abscissa (-f) = conv_abscissa f"
   using conv_abscissa_cmult_left[of "-1" f] by simp
 
 lemma abs_conv_abscissa_minus [simp]:
   fixes f :: "'a :: dirichlet_series fds"
   shows "abs_conv_abscissa (-f) = abs_conv_abscissa f"
   using abs_conv_abscissa_cmult[of "-1" f] by simp
 
 lemma conv_abscissa_diff_le:
   "conv_abscissa (f - g :: 'a :: dirichlet_series fds) \<le> max (conv_abscissa f) (conv_abscissa g)"
   using conv_abscissa_add_le[of f "-g"] by simp
 
 lemma conv_abscissa_diff_leI:
   "conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow> conv_abscissa g \<le> d \<Longrightarrow> 
      conv_abscissa (f - g) \<le> d"
   using conv_abscissa_add_le[of f "-g"] by (auto simp: le_max_iff_disj)
 
 lemma abs_conv_abscissa_diff_le:
   "abs_conv_abscissa (f - g :: 'a :: dirichlet_series fds) \<le>
      max (abs_conv_abscissa f) (abs_conv_abscissa g)"
   using abs_conv_abscissa_add_le[of f "-g"] by simp
 
 lemma abs_conv_abscissa_diff_leI:
   "abs_conv_abscissa (f :: 'a :: dirichlet_series fds) \<le> d \<Longrightarrow> abs_conv_abscissa g \<le> d \<Longrightarrow> 
      abs_conv_abscissa (f - g) \<le> d"
   using abs_conv_abscissa_add_le[of f "-g"] by (auto simp: le_max_iff_disj)
 
 lemmas eval_fds_integral_has_field_derivative' [derivative_intros] = 
   DERIV_chain'[OF _ eval_fds_integral_has_field_derivative]
 
 lemma abs_conv_abscissa_completely_multiplicative_log_deriv:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "completely_multiplicative_function (fds_nth f)" "fds_nth f 1 \<noteq> 0"
   shows   "abs_conv_abscissa (fds_deriv f / f) \<le> abs_conv_abscissa f"
 proof -
   have "fds_deriv f = - fds (\<lambda>n. fds_nth f n * mangoldt n) * f"
     using assms by (subst completely_multiplicative_fds_deriv') simp_all
   also have "\<dots> / f = - fds (\<lambda>n. fds_nth f n * mangoldt n) * (f * inverse f)"
     by (simp add: divide_fds_def)
   also have "f * inverse f = 1" using assms by (intro fds_right_inverse)
   finally have "fds_deriv f / f = - fds (\<lambda>n. fds_nth f n * mangoldt n)" by simp
   also have "abs_conv_abscissa \<dots> = 
                abs_conv_abscissa (fds (\<lambda>n. fds_nth f n * mangoldt n))" 
     (is "_ = abs_conv_abscissa ?f") by (rule abs_conv_abscissa_minus)
   also have "\<dots> \<le> abs_conv_abscissa f"
   proof (rule abs_conv_abscissa_leI, goal_cases)
     case (1 c)
     have "fds_abs_converges ?f (of_real c)" unfolding fds_abs_converges_def
     proof (rule summable_comparison_test_ev)
       from 1 have "fds_abs_converges (fds_deriv f) (of_real c)" 
         by (intro fds_abs_converges) (auto simp: abs_conv_abscissa_deriv)
       thus "summable (\<lambda>n. \<bar>ln (real n)\<bar> * norm (fds_nth f n) / norm (nat_power n (of_real c :: 'a)))"
         by (simp add: fds_abs_converges_def fds_deriv_def fds_nth_fds' 
                       scaleR_conv_of_real powr_minus norm_mult norm_divide norm_nat_power)
     next
       show "\<forall>\<^sub>F n in sequentially.
               norm (norm (fds_nth (fds (\<lambda>n. fds_nth f n * mangoldt n)) n /
                  nat_power n (of_real c)))
               \<le> \<bar>ln (real n)\<bar> * norm (fds_nth f n) / norm (nat_power n (of_real c) :: 'a)"
         using eventually_gt_at_top[of 0]
       proof eventually_elim
         case (elim n)
         have "norm (norm (fds_nth (fds (\<lambda>n. fds_nth f n * mangoldt n)) n /
                   nat_power n (of_real c))) =
                 norm (fds_nth f n) * mangoldt n / real n powr c"
           using elim by (simp add: fds_nth_fds' norm_mult norm_divide 
                                    norm_nat_power abs_mult mangoldt_nonneg)
         also have "\<dots> \<le> norm (fds_nth f n) * ln n / real n powr c" using elim
           by (intro mult_left_mono divide_right_mono mangoldt_le)
              (simp_all add: mangoldt_def)
         finally show ?case using elim by (simp add: norm_nat_power algebra_simps)
       qed
     qed
     thus ?case by (intro exI[of _ "of_real c"]) auto
   qed
   finally show ?thesis .
 qed
 
   
 subsection \<open>Uniqueness\<close>
   
 context
   assumes "SORT_CONSTRAINT ('a :: dirichlet_series)"
 begin
 
 lemma norm_dirichlet_series_cutoff_le:
   assumes "fds_abs_converges f (s0 :: 'a)" "N > 0" "s \<bullet> 1 \<ge> c" "c \<ge> s0 \<bullet> 1"
   shows   "summable (\<lambda>n. fds_nth f (n + N) / nat_power (n + N) s)"
           "summable (\<lambda>n. norm (fds_nth f (n + N)) / nat_power (n + N) c)"
     and   "norm (\<Sum>n. fds_nth f (n + N) / nat_power (n + N) s) \<le> 
              (\<Sum>n. norm (fds_nth f (n + N)) / nat_power (n + N) c) / nat_power N (s \<bullet> 1 - c)"
 proof -
   from assms have "fds_abs_converges f (of_real c)" 
     using fds_abs_converges_Re_le[of f s0 "of_real c"] by auto
   hence "summable (\<lambda>n. norm (fds_nth f (n + N) / nat_power (n + N) (of_real c)))"
     unfolding fds_abs_converges_def by (rule summable_ignore_initial_segment)
   also have "?this \<longleftrightarrow> summable (\<lambda>n. norm (fds_nth f (n + N)) / nat_power (n + N) c)"
     by (intro summable_cong eventually_mono[OF eventually_gt_at_top[of "0::nat"]]) 
        (auto simp: norm_divide norm_nat_power)
   finally show *: "summable (\<lambda>n. norm (fds_nth f (n + N)) / nat_power (n + N) c)" .
 
   from assms have "fds_abs_converges f s" using fds_abs_converges_Re_le[of f s0 s] by auto
   hence **: "summable (\<lambda>n. norm (fds_nth f (n + N) / nat_power (n + N) s))"
     unfolding fds_abs_converges_def by (rule summable_ignore_initial_segment)
   thus "summable (\<lambda>n. fds_nth f (n + N) / nat_power (n + N) s)"
     by (rule summable_norm_cancel)
 
   have "norm (\<Sum>n. fds_nth f (n + N) / nat_power (n + N) s)
           \<le> (\<Sum>n. norm (fds_nth f (n + N) / nat_power (n + N) s))"
     by (intro summable_norm **) 
   also have "\<dots> \<le> (\<Sum>n. norm (fds_nth f (n + N)) / nat_power (n + N) c / nat_power N (s \<bullet> 1 - c))"
   proof (intro suminf_le * ** summable_divide allI)
     fix n :: nat
     have "real N powr (s \<bullet> 1 - c) \<le> real (n + N) powr (s \<bullet> 1 - c)"
       using assms by (intro powr_mono2) simp_all
     also have "real (n + N) powr c * \<dots> = real (n + N) powr (s \<bullet> 1)"
       by (simp add: powr_diff)
     finally have "norm (fds_nth f (n + N)) / real (n + N) powr (s \<bullet> 1) \<le>
                     norm (fds_nth f (n + N)) / (real (n + N) powr c * real N powr (s \<bullet> 1 - c))"
       using \<open>N > 0\<close> by (intro divide_left_mono) (simp_all add: mult_left_mono)
     thus "norm (fds_nth f (n + N) / nat_power (n + N) s) \<le> 
             norm (fds_nth f (n + N)) / nat_power (n + N) c / nat_power N (s \<bullet> 1 - c)"
       using \<open>N > 0\<close> by (simp add: norm_divide norm_nat_power )
   qed
   also have "\<dots> = (\<Sum>n. norm (fds_nth f (n + N)) / nat_power (n + N) c) / nat_power N (s \<bullet> 1 - c)"
     using * by (rule suminf_divide)
   finally show "norm (\<Sum>n. fds_nth f (n + N) / nat_power (n + N) s) \<le> \<dots>" .
 qed
 
 lemma eval_fds_zeroD_aux:
   fixes h :: "'a fds"
   assumes conv: "fds_abs_converges h (s0 :: 'a)"
   assumes freq: "frequently (\<lambda>s. eval_fds h s = 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
   shows   "h = 0"
 proof (rule ccontr)
   assume "h \<noteq> 0"
   hence ex: "\<exists>n>0. fds_nth h n \<noteq> 0" by (auto simp: fds_eq_iff)
   define N :: nat where "N = (LEAST n. n > 0 \<and> fds_nth h n \<noteq> 0)"
   have N: "N > 0" "fds_nth h N \<noteq> 0" 
     using LeastI_ex[OF ex, folded N_def] by auto
   have less_N: "fds_nth h n = 0" if "n < N" for n
     using Least_le[of "\<lambda>n. n > 0 \<and> fds_nth h n \<noteq> 0" n, folded N_def] that
     by (cases "n = 0") (auto simp: not_less)
       
   define c where "c = s0 \<bullet> 1"
   define remainder where "remainder = (\<lambda>s. (\<Sum>n. fds_nth h (n + Suc N) / nat_power (n + Suc N) s))"
   define A where "A = (\<Sum>n. norm (fds_nth h (n + Suc N)) / nat_power (n + Suc N) c) *
                          nat_power (Suc N) c"
 
   have eq: "fds_nth h N = nat_power N s * eval_fds h s - nat_power N s * remainder s"
     if "s \<bullet> 1 \<ge> c" for s :: 'a
   proof -
     from conv and that have conv': "fds_abs_converges h s" 
       unfolding c_def by (rule fds_abs_converges_Re_le)
     hence conv'': "fds_converges h s" by blast
     from conv'' have "(\<lambda>n. fds_nth h n / nat_power n s) sums eval_fds h s"
       by (simp add: fds_converges_iff)
     hence "(\<lambda>n. fds_nth h (n + Suc N) / nat_power (n + Suc N) s) sums 
              (eval_fds h s - (\<Sum>n<Suc N. fds_nth h n / nat_power n s))"
       by (rule sums_split_initial_segment)
     also have "(\<Sum>n<Suc N. fds_nth h n / nat_power n s) = 
                  (\<Sum>n<Suc N. if n = N then fds_nth h N / nat_power N s else 0)"
       by (intro sum.cong refl) (auto simp: less_N)
     also have "\<dots> = fds_nth h N / nat_power N s" by (subst sum.delta) auto
     finally show ?thesis unfolding remainder_def using \<open>N > 0\<close> by (auto simp: sums_iff field_simps)
   qed
   
   have remainder_bound: "norm (remainder s) \<le> A / real (Suc N) powr (s \<bullet> 1)"
     if "s \<bullet> 1 \<ge> c" for s :: 'a
   proof -
     note * = norm_dirichlet_series_cutoff_le[of h s0 "Suc N" c s, folded remainder_def]
     have "norm (remainder s) \<le> (\<Sum>n. norm (fds_nth h (n + Suc N)) /
            nat_power (n + Suc N) c) / nat_power (Suc N) (s \<bullet> 1 - c)"
       using that assms unfolding remainder_def by (intro *) (simp_all add: c_def)
     also have "\<dots> = A / real (Suc N) powr (s \<bullet> 1)" by (simp add: A_def powr_diff)
     finally show ?thesis .
   qed
 
   from freq have "\<forall>c. \<exists>s. s \<bullet> 1 \<ge> c \<and> eval_fds h s = 0"
     unfolding frequently_def by (auto simp: eventually_going_to_at_top_linorder)
   hence "\<forall>k. \<exists>s. s \<bullet> 1 \<ge> real k \<and> eval_fds h s = 0" by blast
   then obtain S where S: "\<And>k. S k \<bullet> 1 \<ge> real k \<and> eval_fds h (S k) = 0"
     by metis
   have S_limit: "filterlim (\<lambda>k. S k \<bullet> 1) at_top sequentially"
     by (rule filterlim_at_top_mono[OF filterlim_real_sequentially]) (use S in auto)
 
   have "eventually (\<lambda>k. real k \<ge> c) sequentially" by real_asymp
   hence "eventually (\<lambda>k. norm (fds_nth h N) \<le> 
            (real N / real (Suc N)) powr (S k \<bullet> 1) * A) sequentially"
   proof eventually_elim
     case (elim k)
     hence "norm (fds_nth h N) = real N powr (S k \<bullet> 1) *  norm (remainder (S k))"
       (is "_ = _ * ?X") using \<open>N > 0\<close> S[of k] eq[of "S k"]
       by (auto simp: norm_mult norm_nat_power c_def)
     also have "norm (remainder (S k)) \<le> A / real (Suc N) powr (S k \<bullet> 1)"
       using elim S[of k] by (intro remainder_bound) (simp_all add: c_def)
     finally show ?case
       using N by (simp add: mult_left_mono powr_divide field_simps del: of_nat_Suc)
   qed
   moreover have "((\<lambda>k. (real N / real (Suc N)) powr (S k \<bullet> 1) * A) \<longlongrightarrow> 0) sequentially"
     by (rule filterlim_compose[OF _ S_limit]) (use \<open>N > 0\<close> in real_asymp)
   ultimately have "((\<lambda>_. fds_nth h N) \<longlongrightarrow> 0) sequentially"
     by (rule Lim_null_comparison)
   hence "fds_nth h N = 0" by (simp add: tendsto_const_iff)
   with \<open>fds_nth h N \<noteq> 0\<close> show False by contradiction
 qed
 
 lemma eval_fds_zeroD:
   fixes h :: "'a fds"
   assumes conv: "conv_abscissa h < \<infinity>"
   assumes freq: "frequently (\<lambda>s. eval_fds h s = 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
   shows   "h = 0"
 proof -
   have [simp]: "2 \<bullet> (1 :: 'a) = 2"
     using of_real_inner_1[of 2] unfolding of_real_numeral by simp
   from conv obtain s where "fds_converges h s"
     by auto
   hence "fds_abs_converges h (s + 2)"
     by (rule fds_converges_imp_abs_converges) (auto simp: algebra_simps)
   from this assms(2-) show ?thesis by (rule eval_fds_zeroD_aux)
 qed
 
 lemma eval_fds_eqD:
   fixes f g :: "'a fds"
   assumes conv: "conv_abscissa f < \<infinity>" "conv_abscissa g < \<infinity>"
   assumes eq:   "frequently (\<lambda>s. eval_fds f s = eval_fds g s) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
   shows   "f = g"
 proof -
   have conv': "conv_abscissa (f - g) < \<infinity>"
     using assms by (intro le_less_trans[OF conv_abscissa_diff_le]) (auto simp: max_def)
 
   have "max (conv_abscissa f) (conv_abscissa g) < \<infinity>"
     using conv by (auto simp: max_def)
   from ereal_dense2[OF this] obtain c where c: "max (conv_abscissa f) (conv_abscissa g) < ereal c"
     by auto
 
   (* TODO: something like "frequently_elim" would be great here *)
   have "frequently (\<lambda>s. eval_fds f s = eval_fds g s \<and> s \<bullet> 1 \<ge> c) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
     using eq by (rule frequently_eventually_frequently) auto
   hence *: "frequently (\<lambda>s. eval_fds (f - g) s = 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
   proof (rule frequently_mono [rotated], safe, goal_cases)
     case (1 s)
     thus ?case using c
       by (subst eval_fds_diff) (auto intro!: fds_converges intro: less_le_trans)
   qed
   have "f - g = 0" by (rule eval_fds_zeroD fds_abs_converges_diff assms * conv')+
   thus ?thesis by simp
 qed
 
 end
 
 
 subsection \<open>Limit at infinity\<close>
 
 lemma eval_fds_at_top_tail_bound:
   fixes f :: "'a :: dirichlet_series fds"
   assumes c: "ereal c > abs_conv_abscissa f"
   defines "B \<equiv> (\<Sum>n. norm (fds_nth f (n+2)) / real (n+2) powr c) * 2 powr c"
   assumes s: "s \<bullet> 1 \<ge> c"
   shows   "norm (eval_fds f s - fds_nth f 1) \<le> B / 2 powr (s \<bullet> 1)"
 proof -
   from c have "fds_abs_converges f (of_real c)" by (intro fds_abs_converges) simp_all
   also have "?this \<longleftrightarrow> summable (\<lambda>n. norm (fds_nth f n) / real n powr c)"
     unfolding fds_abs_converges_def
     by (intro summable_cong eventually_mono[OF eventually_gt_at_top[of "0::nat"]]) 
        (auto simp: norm_divide norm_nat_power norm_powr_real_powr)
   finally have summable_c: \<dots> .
 
   note c
   also from s have "ereal c \<le> ereal (s \<bullet> 1)" by simp
   finally have "fds_abs_converges f s" by (intro fds_abs_converges) auto
   hence summable: "summable (\<lambda>n. norm (fds_nth f n / nat_power n s))"
     by (simp add: fds_abs_converges_def)
   from summable_norm_cancel[OF this]
     have "(\<lambda>n. fds_nth f n / nat_power n s) sums eval_fds f s"
     by (simp add: eval_fds_def sums_iff)
   from sums_split_initial_segment[OF this, of "Suc (Suc 0)"]
     have "norm (eval_fds f s - fds_nth f 1) = norm (\<Sum>n. fds_nth f (n+2) / nat_power (n+2) s)"
     by (auto simp: sums_iff)
   also have "\<dots> \<le> (\<Sum>n. norm (fds_nth f (n+2) / nat_power (n+2) s))"
     by (intro summable_norm summable_ignore_initial_segment summable)
   also have "\<dots> \<le> (\<Sum>n. norm (fds_nth f (n+2)) / real (n+2) powr c / 2 powr (s \<bullet> 1 - c))"
   proof (intro suminf_le allI)
     fix n :: nat
     have "norm (fds_nth f (n + 2) / nat_power (n + 2) s) =
             norm (fds_nth f (n + 2)) / real (n+2) powr c / real (n+2) powr (s \<bullet> 1 - c)"
       by (simp add: field_simps powr_diff norm_divide norm_nat_power)
     also have "\<dots> \<le> norm (fds_nth f (n + 2)) / real (n+2) powr c / 2 powr (s \<bullet> 1 - c)" using s
       by (intro divide_left_mono divide_nonneg_pos powr_mono2 mult_pos_pos) simp_all
     finally show "norm (fds_nth f (n + 2) / nat_power (n + 2) s) \<le> \<dots>" .
   qed (intro summable_ignore_initial_segment summable summable_divide summable_c)+
   also have "\<dots> = (\<Sum>n. norm (fds_nth f (n+2)) / real (n+2) powr c) / 2 powr (s \<bullet> 1 - c)"
     by (intro suminf_divide summable_ignore_initial_segment summable_c)
   also have "\<dots> = B / 2 powr (s \<bullet> 1)" by (simp add: B_def powr_diff)
   finally show ?thesis .
 qed
 
 lemma tendsto_eval_fds_Re_at_top:
   assumes "conv_abscissa (f :: 'a :: dirichlet_series fds) \<noteq> \<infinity>"
   assumes lim: "filterlim (\<lambda>x. S x \<bullet> 1) at_top F"
   shows   "((\<lambda>x. eval_fds f (S x)) \<longlongrightarrow> fds_nth f 1) F"
 proof -
   from assms(1) have "abs_conv_abscissa f < \<infinity>"
     using abs_conv_le_conv_abscissa_plus_1[of f] by auto
   from ereal_dense2[OF this] obtain c where c: "abs_conv_abscissa f < ereal c" by auto
   define B where "B = (\<Sum>n. norm (fds_nth f (n+2)) / real (n+2) powr c) * 2 powr c"
 
   have *: "norm (eval_fds f s - fds_nth f 1) \<le> B / 2 powr (s \<bullet> 1)" if s: "s \<bullet> 1 \<ge> c" for s
     using eval_fds_at_top_tail_bound[of f c s] that c by (simp add: B_def)
   moreover from lim have "eventually (\<lambda>x. S x \<bullet> 1 \<ge> c) F" by (auto simp: filterlim_at_top)
   ultimately have "eventually (\<lambda>x. norm (eval_fds f (S x) - fds_nth f 1) \<le> 
                       B / 2 powr (S x \<bullet> 1)) F" by (auto elim!: eventually_mono)
   moreover have "((\<lambda>x. B / 2 powr (S x \<bullet> 1)) \<longlongrightarrow> 0) F"
     using filterlim_tendsto_pos_mult_at_top[OF tendsto_const[of "ln 2"] _ lim]
     by (intro real_tendsto_divide_at_top[OF tendsto_const])
        (auto simp: powr_def mult_ac intro!: filterlim_compose[OF exp_at_top])
   ultimately have "((\<lambda>x. eval_fds f (S x) - fds_nth f 1) \<longlongrightarrow> 0) F"
     by (rule Lim_null_comparison)
   thus ?thesis by (subst (asm) Lim_null [symmetric])
 qed
 
 lemma tendsto_eval_fds_Re_at_top':
   assumes "conv_abscissa (f :: complex fds) \<noteq> \<infinity>"
   shows   "uniform_limit UNIV (\<lambda>\<sigma> t. eval_fds f (of_real \<sigma> + of_real t * \<i>)
                        ) (\<lambda>_ .fds_nth f 1) at_top"
 proof -
   from assms(1) have "abs_conv_abscissa f < \<infinity>"
     using abs_conv_le_conv_abscissa_plus_1[of f] by auto
   from ereal_dense2[OF this] obtain c where c: "abs_conv_abscissa f < ereal c" by auto
   define B where "B \<equiv> (\<Sum>n. norm (fds_nth f (n+2)) / real (n+2) powr c) * 2 powr c"
 
   show ?thesis
     unfolding uniform_limit_iff
   proof safe
     fix \<epsilon> :: real assume "\<epsilon> > 0"
     hence "eventually (\<lambda>\<sigma>. B / 2 powr \<sigma> < \<epsilon>) at_top"
       by real_asymp
     thus "eventually (\<lambda>\<sigma>. \<forall>t\<in>UNIV.
             dist (eval_fds f (of_real \<sigma> + of_real t * \<i>)) (fds_nth f 1) < \<epsilon>) at_top"
       using eventually_ge_at_top[of c]
     proof eventually_elim
       case (elim \<sigma>)
       show ?case
       proof
         fix t :: real
         have "dist (eval_fds f (of_real \<sigma> + of_real t * \<i>)) (fds_nth f 1) \<le> B / 2 powr \<sigma>"
           using eval_fds_at_top_tail_bound[of f c "of_real \<sigma> + of_real t * \<i>"] elim c
           by (simp add: dist_norm B_def)
         also have "\<dots> < \<epsilon>" by fact
         finally show "dist (eval_fds f (of_real \<sigma> + of_real t * \<i>)) (fds_nth f 1) < \<epsilon>" .
       qed
     qed
   qed
 qed
 
 lemma tendsto_eval_fds_Re_going_to_at_top:
   assumes "conv_abscissa (f :: 'a :: dirichlet_series fds) \<noteq> \<infinity>"
   shows   "((\<lambda>s. eval_fds f s) \<longlongrightarrow> fds_nth f 1) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
   using assms by (rule tendsto_eval_fds_Re_at_top) auto
 
 lemma tendsto_eval_fds_Re_going_to_at_top':
   assumes "conv_abscissa (f :: complex fds) \<noteq> \<infinity>"
   shows   "((\<lambda>s. eval_fds f s) \<longlongrightarrow> fds_nth f 1) (Re going_to at_top)"
   using assms by (rule tendsto_eval_fds_Re_at_top) auto
 
 text \<open>
   Any Dirichlet series that is not identically zero and does not diverge everywhere
   has a half-plane in which it converges and is non-zero.
 \<close>
 theorem fds_nonzero_halfplane_exists:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "conv_abscissa f < \<infinity>" "f \<noteq> 0"
   shows   "eventually (\<lambda>s. fds_converges f s \<and> eval_fds f s \<noteq> 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
 proof -
   from ereal_dense2[OF assms(1)] obtain c where c: "conv_abscissa f < ereal c" by auto
   have "eventually (\<lambda>s::'a. s \<bullet> 1 > c) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
     using eventually_gt_at_top[of c] by auto
   hence "eventually (\<lambda>s. fds_converges f s) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
     by eventually_elim (use c in \<open>auto intro!: fds_converges simp: less_le_trans\<close>)
   moreover have "eventually (\<lambda>s. eval_fds f s \<noteq> 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
     using eval_fds_zeroD[OF assms(1)] assms(2) by (auto simp: frequently_def)
   ultimately show ?thesis by (rule eventually_conj)
 qed 
 
 
 subsection \<open>Normed series\<close>
 
 lemma fds_converges_norm_iff [simp]: 
   fixes s :: "'a :: {nat_power_normed_field,banach}"
   shows "fds_converges (fds_norm f) (s \<bullet> 1) \<longleftrightarrow> fds_abs_converges f s"
   unfolding fds_converges_def fds_abs_converges_def
   by (rule summable_cong [OF eventually_mono[OF eventually_gt_at_top[of 0]]])
      (simp add: fds_abs_converges_def fds_norm_def fds_nth_fds' norm_divide norm_nat_power)
 
 lemma fds_abs_converges_norm_iff [simp]: 
   fixes s :: "'a :: {nat_power_normed_field,banach}"
   shows "fds_abs_converges (fds_norm f) (s \<bullet> 1) \<longleftrightarrow> fds_abs_converges f s"
   unfolding fds_abs_converges_def
   by (rule summable_cong [OF eventually_mono[OF eventually_gt_at_top[of 0]]])
      (simp add: fds_abs_converges_def fds_norm_def fds_nth_fds' norm_divide norm_nat_power)
 
 lemma fds_converges_norm_iff': 
   fixes f :: "'a :: {nat_power_normed_field,banach} fds"
   shows "fds_converges (fds_norm f) s \<longleftrightarrow> fds_abs_converges f (of_real s)"
   unfolding fds_converges_def fds_abs_converges_def
   by (rule summable_cong [OF eventually_mono[OF eventually_gt_at_top[of 0]]])
      (simp add: fds_abs_converges_def fds_norm_def fds_nth_fds' norm_divide norm_nat_power)
 
 lemma fds_abs_converges_norm_iff': 
   fixes f :: "'a :: {nat_power_normed_field,banach} fds"
   shows "fds_abs_converges (fds_norm f) s \<longleftrightarrow> fds_abs_converges f (of_real s)"
   unfolding fds_abs_converges_def
   by (rule summable_cong [OF eventually_mono[OF eventually_gt_at_top[of 0]]])
      (simp add: fds_abs_converges_def fds_norm_def fds_nth_fds' norm_divide norm_nat_power)
 
 lemma abs_conv_abscissa_norm [simp]: 
   fixes f :: "'a :: dirichlet_series fds"
   shows "abs_conv_abscissa (fds_norm f) = abs_conv_abscissa f"
 proof (rule antisym)
   show "abs_conv_abscissa f \<le> abs_conv_abscissa (fds_norm f)"
   proof (rule abs_conv_abscissa_leI_weak)
     fix x assume "abs_conv_abscissa (fds_norm f) < ereal x"
     hence "fds_abs_converges (fds_norm f) (of_real x)" by (intro fds_abs_converges) auto
     thus "fds_abs_converges f (of_real x)" by (simp add: fds_abs_converges_norm_iff')
   qed
 qed (auto intro!: abs_conv_abscissa_leI_weak simp: fds_abs_converges_norm_iff' fds_abs_converges)
 
 lemma conv_abscissa_norm [simp]: 
   fixes f :: "'a :: dirichlet_series fds"
   shows "conv_abscissa (fds_norm f) = abs_conv_abscissa f"
 proof (rule antisym)
   show "abs_conv_abscissa f \<le> conv_abscissa (fds_norm f)"
   proof (rule abs_conv_abscissa_leI_weak)
     fix x assume "conv_abscissa (fds_norm f) < ereal x"
     hence "fds_converges (fds_norm f) (of_real x)" by (intro fds_converges) auto
     thus "fds_abs_converges f (of_real x)" by (simp add: fds_converges_norm_iff')
   qed
 qed (auto intro!: conv_abscissa_leI_weak simp: fds_abs_converges)
 
 lemma 
   fixes f g :: "'a :: dirichlet_series fds"
   assumes "fds_abs_converges (fds_norm f) s" "fds_abs_converges (fds_norm g) s"
   shows   fds_abs_converges_norm_mult: "fds_abs_converges (fds_norm (f * g)) s"
   and     eval_fds_norm_mult_le: 
             "eval_fds (fds_norm (f * g)) s \<le> eval_fds (fds_norm f) s * eval_fds (fds_norm g) s"
 proof -
   show conv: "fds_abs_converges (fds_norm (f * g)) s" unfolding fds_abs_converges_def
   proof (rule summable_comparison_test_ev)
     have "fds_abs_converges (fds_norm f * fds_norm g) s" by (rule fds_abs_converges_mult assms)+
     thus "summable (\<lambda>n. norm (fds_nth (fds_norm f * fds_norm g) n) / nat_power n s)"
       by (simp add: fds_abs_converges_def)
   qed (auto intro!: always_eventually divide_right_mono order.trans[OF fds_nth_norm_mult_le] 
             simp: norm_divide)
   have conv': "fds_abs_converges (fds_norm f * fds_norm g) s"
     by (intro fds_abs_converges_mult assms)
   hence "eval_fds (fds_norm (f * g)) s \<le> eval_fds (fds_norm f * fds_norm g) s"
     using conv unfolding eval_fds_def fds_abs_converges_def norm_divide
     by (intro suminf_le allI divide_right_mono) (simp_all add: norm_mult fds_nth_norm_mult_le)
   also have "\<dots> = eval_fds (fds_norm f) s * eval_fds (fds_norm g) s"
     by (intro eval_fds_mult assms)
   finally show "eval_fds (fds_norm (f * g)) s \<le> eval_fds (fds_norm f) s * eval_fds (fds_norm g) s" .
 qed
 
 lemma eval_fds_norm_nonneg:
   assumes "fds_abs_converges (fds_norm f) s"
   shows   "eval_fds (fds_norm f) s \<ge> 0"
   using assms unfolding eval_fds_def fds_abs_converges_def
   by (intro suminf_nonneg) auto
 
 lemma
   fixes f :: "'a :: dirichlet_series fds"
   assumes "fds_abs_converges (fds_norm f) s"
   shows   fds_abs_converges_norm_power: "fds_abs_converges (fds_norm (f ^ n)) s"
   and     eval_fds_norm_power_le: 
             "eval_fds (fds_norm (f ^ n)) s \<le> eval_fds (fds_norm f) s ^ n"
 proof -
   show *: "fds_abs_converges (fds_norm (f ^ n)) s" for n
     by (induction n) (auto intro!: fds_abs_converges_norm_mult assms)
   show "eval_fds (fds_norm (f ^ n)) s \<le> eval_fds (fds_norm f) s ^ n"
     by (induction n) (auto intro!: order.trans[OF eval_fds_norm_mult_le] assms * 
                                    mult_left_mono eval_fds_norm_nonneg)
 qed
 
 
 subsection \<open>Logarithms of Dirichlet series\<close>
 
 (* TODO: Move ? *)
 lemma eventually_gt_ereal_at_top: "c \<noteq> \<infinity> \<Longrightarrow> eventually (\<lambda>x. ereal x > c) at_top"
   by (cases c) auto
 
 lemma eval_fds_log_deriv:
   fixes s :: "'a :: dirichlet_series"
   assumes "fds_nth f 1 \<noteq> 0" "s \<bullet> 1 > abs_conv_abscissa f" 
            "s \<bullet> 1 > abs_conv_abscissa (fds_deriv f / f)"
   assumes "eval_fds f s \<noteq> 0"
   shows   "eval_fds (fds_deriv f / f) s = eval_fds (fds_deriv f) s / eval_fds f s"
 proof -
   have "eval_fds (fds_deriv f / f * f) s = eval_fds (fds_deriv f / f) s * eval_fds f s"
     using assms by (intro eval_fds_mult fds_abs_converges) auto
   also have "fds_deriv f / f * f = fds_deriv f * (f * inverse f)"
     by (simp add: divide_fds_def algebra_simps)
   also have "f * inverse f = 1" using assms by (intro fds_right_inverse)
   finally show ?thesis using assms by simp
 qed
 
 text \<open>
   Given a sufficiently nice absolutely convergent Dirichlet series that converges
   to some function $f(s)$ and a holomorphic branch of $\ln f(s)$, we can construct 
   a Dirichlet series that absolutely converges to that logarithm.
 \<close>
 lemma eval_fds_ln:
   fixes s0 :: ereal
   assumes nz: "\<And>s. Re s > s0 \<Longrightarrow> eval_fds f s \<noteq> 0" "fds_nth f 1 \<noteq> 0"
   assumes l: "exp l = fds_nth f 1" "((g \<circ> of_real) \<longlongrightarrow> l) at_top"
   assumes g: "\<And>s. Re s > s0 \<Longrightarrow> exp (g s) = eval_fds f s"
   assumes holo_g: "g holomorphic_on {s. Re s > s0}"
   assumes "ereal (Re s) > s0"
   assumes "s0 \<ge> abs_conv_abscissa f" and "s0 \<ge> abs_conv_abscissa (fds_deriv f / f)"
   shows   "eval_fds (fds_ln l f) s = g s"
 proof -
   let ?s0 = "abs_conv_abscissa f" and ?s1 = "abs_conv_abscissa (inverse f)"
   let ?h = "\<lambda>s. eval_fds (fds_ln l f) s - g s"
   let ?A = "{s. Re s > s0}"
   have open_A: "open ?A" by (cases s0) (auto simp: open_halfspace_Re_gt)
 
   have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
   moreover from assms have "\<dots> \<noteq> \<infinity>" by auto
   ultimately have "conv_abscissa f \<noteq> \<infinity>" by auto
 
   have "conv_abscissa (fds_ln l f) \<le> abs_conv_abscissa (fds_ln l f)"
     by (rule conv_le_abs_conv_abscissa)
   also have "\<dots> \<le> abs_conv_abscissa (fds_deriv f / f)"
     unfolding fds_ln_def by (simp add: abs_conv_abscissa_integral)
   finally have "conv_abscissa (fds_ln l f) \<noteq> \<infinity>"
     using assms by (auto simp: max_def abs_conv_abscissa_deriv split: if_splits)
 
   have deriv_g [derivative_intros]:
     "(g has_field_derivative eval_fds (fds_deriv f) s / eval_fds f s) (at s within B)"
     if s: "Re s > s0" for s B
   proof -
     have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> \<le> s0" using assms by simp
     also have "\<dots> < Re s" by fact
     finally have s': "Re s > conv_abscissa f" .
 
     have deriv_g: "(g has_field_derivative deriv g s) (at s)"
       using holomorphic_derivI[OF holo_g open_A, of s] s
       by (auto simp: at_within_open[OF _ open_A])
     have "((\<lambda>s. exp (g s)) has_field_derivative eval_fds f s * deriv g s) (at s)" (is ?P)
       by (rule derivative_eq_intros deriv_g s)+ (insert s, simp_all add: g)
     also from s have ev: "eventually (\<lambda>t. t \<in> ?A) (nhds s)"
       by (intro eventually_nhds_in_open open_A) auto
     have "?P \<longleftrightarrow> (eval_fds f has_field_derivative eval_fds f s * deriv g s) (at s)"
       by (intro DERIV_cong_ev refl eventually_mono[OF ev]) (auto simp: g)
     finally have "(eval_fds f has_field_derivative eval_fds f s * deriv g s) (at s)" .
     moreover have "(eval_fds f has_field_derivative eval_fds (fds_deriv f) s) (at s)"
       using s' assms by (intro derivative_intros) auto
     ultimately have "eval_fds f s * deriv g s = eval_fds (fds_deriv f) s"
       by (rule DERIV_unique)
     hence "deriv g s = eval_fds (fds_deriv f) s / eval_fds f s"
       using s nz by (simp add: field_simps)
     with deriv_g show ?thesis by (auto intro: has_field_derivative_at_within)
   qed  
 
   have "\<exists>c. \<forall>z\<in>{z. Re z > s0}. ?h z = c"
   proof (rule has_field_derivative_zero_constant, goal_cases)
     case 1
     show ?case using convex_halfspace_gt[of _ "1::complex"]
       by (cases s0) auto
   next
     case (2 z)
     have "conv_abscissa (fds_ln l f) \<le> abs_conv_abscissa (fds_ln l f)"
       by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> \<le> abs_conv_abscissa (fds_deriv f / f)"
       by (simp add: abs_conv_abscissa_ln)
     also have "\<dots> < Re z" using 2 assms by (auto simp: abs_conv_abscissa_deriv)
     finally have s1: "conv_abscissa (fds_ln l f) < ereal (Re z)" .
 
     have "conv_abscissa f \<le> abs_conv_abscissa f"
       by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> < Re z" using 2 assms by auto
     finally have s2: "conv_abscissa f < ereal (Re z)" .
 
     from l have "fds_nth f 1 \<noteq> 0" by auto
     with 2 assms have *: "eval_fds (fds_deriv f / f) z = eval_fds (fds_deriv f) z / (eval_fds f z)"
       by (auto simp: eval_fds_log_deriv)
     have "eval_fds f z \<noteq> 0" using 2 assms by auto
     show ?case using s1 s2 2 nz
       by (auto intro!: derivative_eq_intros simp: * field_simps)
   qed
   then obtain c where c: "\<And>z. Re z > s0 \<Longrightarrow> ?h z = c" by blast
 
   have "(at_top :: real filter) \<noteq> bot" by simp
   moreover from assms have "s0 \<noteq> \<infinity>" by auto
   have "eventually (\<lambda>x. c = (?h \<circ> of_real) x) at_top"
     using eventually_gt_ereal_at_top[OF \<open>s0 \<noteq> \<infinity>\<close>] by eventually_elim (simp add: c)
   hence "((?h \<circ> of_real) \<longlongrightarrow> c) at_top"
     by (force intro: Lim_transform_eventually) 
   moreover have "((?h \<circ> of_real) \<longlongrightarrow> fds_nth (fds_ln l f) 1 - l) at_top"
     using \<open>conv_abscissa (fds_ln l f) \<noteq> \<infinity>\<close> and l unfolding o_def
     by (intro tendsto_intros tendsto_eval_fds_Re_at_top) (auto simp: filterlim_ident)
   ultimately have "c = fds_nth (fds_ln l f) 1 - l"
     by (rule tendsto_unique)
   with c[OF \<open>Re s > s0\<close>] and l and nz show ?thesis
     by (simp add: exp_minus field_simps)
 qed
 
 text \<open>
   Less explicitly: For a sufficiently nice absolutely convergent Dirichlet series converging
   to a function $f(s)$, the formal logarithm absolutely converges to some logarithm of $f(s)$.
 \<close>
 lemma eval_fds_ln':
   fixes s0 :: ereal
   assumes "ereal (Re s) > s0"
   assumes "s0 \<ge> abs_conv_abscissa f" and "s0 \<ge> abs_conv_abscissa (fds_deriv f / f)"
       and nz: "\<And>s. Re s > s0 \<Longrightarrow> eval_fds f s \<noteq> 0" "fds_nth f 1 \<noteq> 0"
   assumes l: "exp l = fds_nth f 1"
   shows   "exp (eval_fds (fds_ln l f) s) = eval_fds f s"
 proof -
   let ?s0 = "abs_conv_abscissa f" and ?s1 = "abs_conv_abscissa (inverse f)"
   let ?h = "\<lambda>s. eval_fds f s * exp (-eval_fds (fds_ln l f) s)"
 
   have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
   moreover from assms have "\<dots> \<noteq> \<infinity>" by auto
   ultimately have "conv_abscissa f \<noteq> \<infinity>" by auto
 
   have "conv_abscissa (fds_ln l f) \<le> abs_conv_abscissa (fds_ln l f)"
     by (rule conv_le_abs_conv_abscissa)
   also have "\<dots> \<le> abs_conv_abscissa (fds_deriv f / f)"
     unfolding fds_ln_def by (simp add: abs_conv_abscissa_integral)
   finally have "conv_abscissa (fds_ln l f) \<noteq> \<infinity>"
     using assms by (auto simp: max_def abs_conv_abscissa_deriv split: if_splits)
 
   have "\<exists>c. \<forall>z\<in>{z. Re z > s0}. ?h z = c"
   proof (rule has_field_derivative_zero_constant, goal_cases)
     case 1
     show ?case using convex_halfspace_gt[of _ "1::complex"]
       by (cases s0) auto
   next
     case (2 z)
     have "conv_abscissa (fds_ln l f) \<le> abs_conv_abscissa (fds_ln l f)"
       by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> \<le> abs_conv_abscissa (fds_deriv f / f)"
       unfolding fds_ln_def by (simp add: abs_conv_abscissa_integral)
     also have "\<dots> < Re z" using 2 assms by (auto simp: abs_conv_abscissa_deriv)
     finally have s1: "conv_abscissa (fds_ln l f) < ereal (Re z)" .
 
     have "conv_abscissa f \<le> abs_conv_abscissa f"
       by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> < Re z" using 2 assms by auto
     finally have s2: "conv_abscissa f < ereal (Re z)" .
 
     from l have "fds_nth f 1 \<noteq> 0" by auto
     with 2 assms have *: "eval_fds (fds_deriv f / f) z = eval_fds (fds_deriv f) z / (eval_fds f z)"
       by (subst eval_fds_log_deriv) auto
     have "eval_fds f z \<noteq> 0" using 2 assms by auto
     thus ?case using s1 s2
       by (auto intro!: derivative_eq_intros simp: *)
   qed
   then obtain c where c: "\<And>z. Re z > s0 \<Longrightarrow> ?h z = c" by blast
 
   have "(at_top :: real filter) \<noteq> bot" by simp
   moreover from assms have "s0 \<noteq> \<infinity>" by auto
   have "eventually (\<lambda>x. c = (?h \<circ> of_real) x) at_top"
     using eventually_gt_ereal_at_top[OF \<open>s0 \<noteq> \<infinity>\<close>] by eventually_elim (simp add: c)
   hence "((?h \<circ> of_real) \<longlongrightarrow> c) at_top"
     by (force intro: Lim_transform_eventually) 
   moreover have "((?h \<circ> of_real) \<longlongrightarrow> fds_nth f 1 * exp (-fds_nth (fds_ln l f) 1)) at_top"
     unfolding o_def using \<open>conv_abscissa (fds_ln l f) \<noteq> \<infinity>\<close> and \<open>conv_abscissa f \<noteq> \<infinity>\<close>
     by (intro tendsto_intros tendsto_eval_fds_Re_at_top) (auto simp: filterlim_ident)
   ultimately have "c = fds_nth f 1 * exp (-fds_nth (fds_ln l f) 1)"
     by (rule tendsto_unique)
   with c[OF \<open>Re s > s0\<close>] and l and nz show ?thesis
     by (simp add: exp_minus field_simps)
 qed
 
 lemma fds_ln_completely_multiplicative:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "completely_multiplicative_function (fds_nth f)"
   assumes "fds_nth f 1 \<noteq> 0"
   shows   "fds_ln l f = fds (\<lambda>n. if n = 1 then l else fds_nth f n * mangoldt n /\<^sub>R ln n)"
 proof -
   have "fds_ln l f = fds_integral l (fds_deriv f / f)"
     by (simp add: fds_ln_def)
   also have "fds_deriv f = -fds (\<lambda>n. fds_nth f n * mangoldt n) * f"
     by (intro completely_multiplicative_fds_deriv' assms)
   also have "\<dots> / f = -fds (\<lambda>n. fds_nth f n * mangoldt n) * (f * inverse f)"
     by (simp add: divide_fds_def)
   also from assms have "f * inverse f = 1"
     by (simp add: fds_right_inverse)
   also have "fds_integral l (- fds (\<lambda>n. fds_nth f n * mangoldt n) * 1) = 
                fds (\<lambda>n. if n = 1 then l else fds_nth f n * mangoldt n /\<^sub>R ln n)"
     by (simp add: fds_integral_def cong: if_cong)
   finally show ?thesis .
 qed
 
 lemma eval_fds_ln_completely_multiplicative_strong:
   fixes s :: "'a :: dirichlet_series" and l :: 'a and f :: "'a fds" and g :: "nat \<Rightarrow> 'a"
   defines "h \<equiv> fds (\<lambda>n. fds_nth (fds_ln l f) n * g n)"
   assumes "fds_abs_converges h s"
   assumes "completely_multiplicative_function (fds_nth f)" and "fds_nth f 1 \<noteq> 0"
   shows  "(\<lambda>(p,k). (fds_nth f p / nat_power p s) ^ Suc k * g (p ^ Suc k) / of_nat (Suc k))
             abs_summable_on ({p. prime p} \<times> UNIV)" (is ?th1)
     and  "eval_fds h s = l * g 1 + (\<Sum>\<^sub>a(p, k)\<in>{p. prime p}\<times>UNIV.
             (fds_nth f p / nat_power p s) ^ Suc k * g (p ^ Suc k) / of_nat (Suc k))" (is ?th2)
 proof -
   let ?P = "{p::nat. prime p}"
   interpret f: completely_multiplicative_function "fds_nth f" by fact
   from assms have *: "(\<lambda>n. fds_nth h n / nat_power n s) abs_summable_on UNIV"
     by (auto simp: abs_summable_on_nat_iff' fds_abs_converges_def)
   have eq: "h = fds (\<lambda>n. if n = 1 then l * g 1 else fds_nth f n * g n * mangoldt n /\<^sub>R ln (real n))"
     using fds_ln_completely_multiplicative [OF assms(3), of l] 
     by (simp add: h_def fds_eq_iff)
 
   note *
   also have "(\<lambda>n. fds_nth h n / nat_power n s) abs_summable_on UNIV \<longleftrightarrow>
           (\<lambda>x. if x = Suc 0 then l * g 1 else fds_nth f x * g x * mangoldt x /\<^sub>R ln (real x) /
                 nat_power x s) abs_summable_on {1} \<union> Collect primepow"
     using eq by (intro abs_summable_on_cong_neutral) (auto simp: fds_nth_fds mangoldt_def)
   finally have sum1: "(\<lambda>x. if x = Suc 0 then l * g 1 else 
                         fds_nth f x * g x * mangoldt x /\<^sub>R ln (real x) / nat_power x s)
                       abs_summable_on Collect primepow"
     by (rule abs_summable_on_subset) auto
   also have "?this \<longleftrightarrow> (\<lambda>x. fds_nth f x * g x * mangoldt x /\<^sub>R ln (real x) / nat_power x s) 
                          abs_summable_on Collect primepow" 
     by (intro abs_summable_on_cong) (insert primepow_gt_Suc_0, auto)
   also have "\<dots> \<longleftrightarrow> (\<lambda>(p,k). fds_nth f (p ^ Suc k) * g (p ^ Suc k) * mangoldt (p ^ Suc k)
                  /\<^sub>R ln (real (p ^ Suc k)) / nat_power (p ^ Suc k) s) abs_summable_on (?P \<times> UNIV)"
     using bij_betw_primepows unfolding case_prod_unfold
     by (intro abs_summable_on_reindex_bij_betw [symmetric])
   also have "\<dots> \<longleftrightarrow> ?th1"
     by (intro abs_summable_on_cong)
        (auto simp: f.mult f.power mangoldt_def aprimedivisor_prime_power ln_realpow prime_gt_0_nat 
           nat_power_power_left divide_simps scaleR_conv_of_real simp del: power_Suc)
   finally show ?th1 .
 
   have "eval_fds h s = (\<Sum>\<^sub>an. fds_nth h n / nat_power n s)"
     using * unfolding eval_fds_def by (subst infsetsum_nat') auto
   also have "\<dots> = (\<Sum>\<^sub>an \<in> {1} \<union> {n. primepow n}.
         if n = 1 then l * g 1 else fds_nth f n * g n * mangoldt n /\<^sub>R ln (real n) /  nat_power n s)"
     by (intro infsetsum_cong_neutral) (auto simp: eq fds_nth_fds mangoldt_def)
   also have "\<dots> = l * g 1 + (\<Sum>\<^sub>an | primepow n.
          if n = 1 then l * g 1 else fds_nth f n * g n * mangoldt n /\<^sub>R ln (real n) / nat_power n s)"
     (is "_ = _ + ?x") using sum1 primepow_gt_Suc_0 by (subst infsetsum_Un_disjoint) auto
   also have "?x =
     (\<Sum>\<^sub>an\<in>Collect primepow. fds_nth f n * g n * mangoldt n /\<^sub>R ln (real n) / nat_power n s)"
     (is "_ = infsetsum ?f _") by (intro infsetsum_cong refl) (insert primepow_gt_Suc_0, auto)
   also have "\<dots> = (\<Sum>\<^sub>a(p,k)\<in>(?P \<times> UNIV). fds_nth f (p ^ Suc k) * g (p ^ Suc k) *
                     mangoldt (p ^ Suc k) /\<^sub>R ln (p ^ Suc k) / nat_power (p ^ Suc k) s)"
      using bij_betw_primepows unfolding case_prod_unfold 
      by (intro infsetsum_reindex_bij_betw [symmetric])
    also have "\<dots> = (\<Sum>\<^sub>a(p,k)\<in>(?P \<times> UNIV). 
                      (fds_nth f p / nat_power p s) ^ Suc k * g (p ^ Suc k) / of_nat (Suc k))"
     by (intro infsetsum_cong)
        (auto simp: f.mult f.power mangoldt_def aprimedivisor_prime_power ln_realpow prime_gt_0_nat 
           nat_power_power_left divide_simps scaleR_conv_of_real simp del: power_Suc)
   finally show ?th2 .
 qed    
 
 lemma eval_fds_ln_completely_multiplicative:
   fixes s :: "'a :: dirichlet_series" and l :: 'a and f :: "'a fds"
   assumes "completely_multiplicative_function (fds_nth f)" and "fds_nth f 1 \<noteq> 0"
   assumes "s \<bullet> 1 > abs_conv_abscissa (fds_deriv f / f)"
   shows  "(\<lambda>(p,k). (fds_nth f p / nat_power p s) ^ Suc k / of_nat (Suc k))
             abs_summable_on ({p. prime p} \<times> UNIV)" (is ?th1)
     and  "eval_fds (fds_ln l f) s =
                   l + (\<Sum>\<^sub>a(p, k)\<in>{p. prime p}\<times>UNIV.
                         (fds_nth f p / nat_power p s) ^ Suc k / of_nat (Suc k))" (is ?th2)
 proof -
   from assms have "fds_abs_converges (fds_ln l f) s"
     by (intro fds_abs_converges_ln) (auto intro!: fds_abs_converges_mult intro: fds_abs_converges)
   hence "fds_abs_converges (fds (\<lambda>n. fds_nth (fds_ln l f) n * 1)) s"
     by simp
   from eval_fds_ln_completely_multiplicative_strong [OF this assms(1,2)] show ?th1 ?th2 
     by simp_all
 qed
 
 subsection \<open>Exponential and logarithm\<close>
 
 lemma summable_fds_exp_aux:
   assumes "fds_nth f' 1 = (0 :: 'a :: real_normed_algebra_1)"
   shows   "summable (\<lambda>k. fds_nth (f' ^ k) n /\<^sub>R fact k)"
 proof (rule summable_finite)
   fix k assume "k \<notin> {..n}"
   hence "n < k" by simp
   also have "\<dots> < 2 ^ k" by (induction k) auto
   finally have "fds_nth (f' ^ k) n = 0"
     using assms by (intro fds_nth_power_eq_0) auto
   thus "fds_nth (f' ^ k) n /\<^sub>R fact k = 0" by simp
 qed auto
 
 lemma
   fixes f :: "'a :: dirichlet_series fds"
   assumes "fds_abs_converges f s"
   shows   fds_abs_converges_exp: "fds_abs_converges (fds_exp f) s"
   and     eval_fds_exp: "eval_fds (fds_exp f) s = exp (eval_fds f s)"
 proof -
   have conv: "fds_abs_converges (fds_exp f) s" and ev: "eval_fds (fds_exp f) s = exp (eval_fds f s)"
     if "fds_abs_converges f s" and [simp]: "fds_nth f (Suc 0) = 0" for f
   proof -
     have [simp]: "fds (\<lambda>n. if n = Suc 0 then 0 else fds_nth f n) = f"
       by (intro fds_eqI) simp_all
     have  "(\<lambda>(k,n). fds_nth (f ^ k) n / fact k / nat_power n s) abs_summable_on (UNIV \<times> {1..})"
     proof (subst abs_summable_on_Sigma_iff, safe, goal_cases)
       case (3 k)
       from that have "fds_abs_converges (f ^ k) s" by (intro fds_abs_converges_power)
       hence "(\<lambda>n. fds_nth (f ^ k) n / nat_power n s * inverse (fact k)) abs_summable_on {1..}"
         unfolding fds_abs_converges_altdef by (intro abs_summable_on_cmult_left)
       thus ?case by (simp add: field_simps)
     next
       case 4
       show ?case unfolding abs_summable_on_nat_iff'
       proof (rule summable_comparison_test_ev[OF always_eventually[OF allI]])
         fix k :: nat
         from that have *: "fds_abs_converges (fds_norm (f ^ k)) (s \<bullet> 1)"
           by (auto simp: fds_abs_converges_power)
         have "(\<Sum>\<^sub>an\<in>{1..}. norm (fds_nth (f ^ k) n / fact k / nat_power n s)) =
                 (\<Sum>\<^sub>an\<in>{1..}. fds_nth (fds_norm (f ^ k)) n / nat_power n (s \<bullet> 1) / fact k)" 
           (is "?S = _") by (intro infsetsum_cong) (simp_all add: norm_divide norm_mult norm_nat_power)
         also have "\<dots> = (\<Sum>\<^sub>an\<in>{1..}. fds_nth (fds_norm (f ^ k)) n / nat_power n (s \<bullet> 1)) /\<^sub>R fact k"
           (is "_ = ?S' /\<^sub>R _") using * unfolding fds_abs_converges_altdef
           by (subst infsetsum_cdiv) (auto simp: abs_summable_on_nat_iff scaleR_conv_of_real divide_simps)
         also have "?S' = eval_fds (fds_norm (f ^ k)) (s \<bullet> 1)"
           using * unfolding fds_abs_converges_altdef eval_fds_def
           by (subst infsetsum_nat) (auto intro!: suminf_cong)
         finally have eq: "?S = \<dots> /\<^sub>R fact k" .
         note eq
         also have "?S \<ge> 0" by (intro infsetsum_nonneg) auto
         hence "?S = norm (norm ?S)" by simp
         also have "eval_fds (fds_norm (f ^ k)) (s \<bullet> 1) \<le> eval_fds (fds_norm f) (s \<bullet> 1) ^ k"
           using that by (intro eval_fds_norm_power_le) auto
         finally show "norm (norm (\<Sum>\<^sub>an\<in>{1..}. norm (fds_nth (f ^ k) n / fact k / nat_power n s))) \<le>
                         eval_fds (fds_norm f) (s \<bullet> 1) ^ k /\<^sub>R fact k" 
           by (simp add: divide_right_mono)
       next
         from exp_converges[of "eval_fds (fds_norm f) (s \<bullet> 1)"]
         show "summable (\<lambda>x. eval_fds (fds_norm f) (s \<bullet> 1) ^ x /\<^sub>R fact x)"
           by (simp add: sums_iff)
       qed
     qed auto
     hence summable:
       "(\<lambda>(n,k). fds_nth (f ^ k) n / fact k / nat_power n s) abs_summable_on {1..} \<times> UNIV"
       by (subst abs_summable_on_Times_swap) (simp add: case_prod_unfold)
   
     have summable': "(\<lambda>k. fds_nth (f ^ k) n / fact k) abs_summable_on UNIV" for n
       using abs_summable_on_cmult_left[of "nat_power n s",
               OF abs_summable_on_Sigma_project2 [OF summable, of n]] by (cases "n = 0") simp_all
   
     have "(\<lambda>n. \<Sum>\<^sub>ak. fds_nth (f ^ k) n / fact k / nat_power n s) abs_summable_on {1..}"
       using summable by (rule abs_summable_on_Sigma_project1') auto
     also have "?this \<longleftrightarrow> (\<lambda>n. (\<Sum>k. fds_nth (f ^ k) n / fact k) * inverse (nat_power n s)) 
                            abs_summable_on {1..}"
     proof (intro abs_summable_on_cong refl, goal_cases)
       case (1 n)
       hence "(\<Sum>\<^sub>ak. fds_nth (f ^ k) n / fact k / nat_power n s) =
               (\<Sum>\<^sub>ak. fds_nth (f ^ k) n / fact k) * inverse (nat_power n s)"
         using summable'[of n]
         by (subst infsetsum_cmult_left [symmetric]) (auto simp: field_simps)
       also have "(\<Sum>\<^sub>ak. fds_nth (f ^ k) n / fact k) = (\<Sum>k. fds_nth (f ^ k) n / fact k)"
         using summable'[of n] 1 by (intro abs_summable_on_cong refl infsetsum_nat') auto
       finally show ?case .
     qed
     finally show "fds_abs_converges (fds_exp f) s"
       by (simp add: fds_exp_def fds_nth_fds' abs_summable_on_Sigma_iff scaleR_conv_of_real 
                     fds_abs_converges_altdef field_simps)
   
     have "eval_fds (fds_exp f) s = (\<Sum>n. (\<Sum>k. fds_nth (f ^ k) n /\<^sub>R fact k) / nat_power n s)"
       by (simp add: fds_exp_def eval_fds_def fds_nth_fds')
     also have "\<dots> = (\<Sum>n. (\<Sum>\<^sub>ak. fds_nth (f ^ k) n /\<^sub>R fact k) / nat_power n s)"
     proof (intro suminf_cong, goal_cases)
       case (1 n)
       show ?case
       proof (cases "n = 0")
         case False
         have "(\<Sum>k. fds_nth (f ^ k) n /\<^sub>R fact k) = (\<Sum>\<^sub>ak. fds_nth (f ^ k) n /\<^sub>R fact k)"
           using summable'[of n] False
           by (intro infsetsum_nat' [symmetric]) (auto simp: scaleR_conv_of_real field_simps)
         thus ?thesis by simp
       qed simp_all
     qed
     also have "\<dots> = (\<Sum>\<^sub>an. (\<Sum>\<^sub>ak. fds_nth (f ^ k) n /\<^sub>R fact k) / nat_power n s)"
     proof (intro infsetsum_nat' [symmetric], goal_cases)
       case 1
       have *: "UNIV - {Suc 0..} = {0}" by auto
       have "(\<lambda>x. \<Sum>\<^sub>ay. fds_nth (f ^ y) x / fact y / nat_power x s) abs_summable_on {1..}"
         by (intro abs_summable_on_Sigma_project1'[OF summable]) auto
       also have "?this \<longleftrightarrow> (\<lambda>x. (\<Sum>\<^sub>ay. fds_nth (f ^ y) x / fact y) * inverse (nat_power x s)) 
                     abs_summable_on {1..}"
         using summable' by (intro abs_summable_on_cong refl, subst infsetsum_cmult_left [symmetric])
                            (auto simp: field_simps)
       also have "\<dots> \<longleftrightarrow> (\<lambda>x. (\<Sum>\<^sub>ay. fds_nth (f ^ y) x /\<^sub>R fact y) / (nat_power x s)) 
                     abs_summable_on {1..}" by (simp add: field_simps scaleR_conv_of_real)
       finally show ?case by (rule abs_summable_on_finite_diff) (use * in auto)
     qed
     also have "\<dots> = (\<Sum>\<^sub>an. (\<Sum>\<^sub>ak. fds_nth (f ^ k) n /\<^sub>R fact k * inverse (nat_power n s)))"
       using summable' by (subst infsetsum_cmult_left) (auto simp: field_simps scaleR_conv_of_real)
     also have "\<dots> = (\<Sum>\<^sub>an\<in>{1..}. (\<Sum>\<^sub>ak. fds_nth (f ^ k) n /\<^sub>R fact k * inverse (nat_power n s)))"
       by (intro infsetsum_cong_neutral) (auto simp: Suc_le_eq)
     also have "\<dots> = (\<Sum>\<^sub>ak. \<Sum>\<^sub>an\<in>{1..}. fds_nth (f ^ k) n / nat_power n s /\<^sub>R fact k)" using summable 
       by (subst infsetsum_swap) (auto simp: field_simps scaleR_conv_of_real case_prod_unfold)
     also have "\<dots> = (\<Sum>\<^sub>ak. (\<Sum>\<^sub>an\<in>{1..}. fds_nth (f ^ k) n / nat_power n s) /\<^sub>R fact k)"
       by (subst infsetsum_scaleR_right) simp
     also have "\<dots> = (\<Sum>\<^sub>ak. eval_fds f s ^ k /\<^sub>R fact k)"
     proof (intro infsetsum_cong refl, goal_cases)
       case (1 k)
       have *: "fds_abs_converges (f ^ k) s" by (intro fds_abs_converges_power that)
       have "(\<Sum>\<^sub>an\<in>{1..}. fds_nth (f ^ k) n / nat_power n s) =
               (\<Sum>\<^sub>an. fds_nth (f ^ k) n / nat_power n s)"
         by (intro infsetsum_cong_neutral) (auto simp: Suc_le_eq)
       also have "\<dots> = eval_fds (f ^ k) s" using * unfolding eval_fds_def 
         by (intro infsetsum_nat') (auto simp: fds_abs_converges_def abs_summable_on_nat_iff')
       also from that have "\<dots> = eval_fds f s ^ k" by (simp add: eval_fds_power)
       finally show ?case by simp
     qed
     also have "\<dots> = (\<Sum>k. eval_fds f s ^ k /\<^sub>R fact k)"
       using exp_converges[of "norm (eval_fds f s)"]
       by (intro infsetsum_nat') (auto simp: abs_summable_on_nat_iff' sums_iff field_simps norm_power)
     also have "\<dots> = exp (eval_fds f s)" by (simp add: exp_def)
     finally show "eval_fds (fds_exp f) s = exp (eval_fds f s)" .
   qed
   
   define f' where "f' = f - fds_const (fds_nth f 1)"
   have *: "fds_abs_converges (fds_exp f') s" 
     by (auto simp: f'_def intro!: fds_abs_converges_diff conv assms)
   have "fds_abs_converges (fds_const (exp (fds_nth f 1)) * fds_exp f') s"
     unfolding f'_def
     by (intro fds_abs_converges_mult conv fds_abs_converges_diff assms) auto
   thus "fds_abs_converges (fds_exp f) s" unfolding f'_def
     by (simp add: fds_exp_times_fds_nth_0)
   have "eval_fds (fds_exp f) s = eval_fds (fds_const (exp (fds_nth f 1)) * fds_exp f') s"
     by (simp add: f'_def fds_exp_times_fds_nth_0)
   also have "\<dots> = exp (fds_nth f (Suc 0)) * eval_fds (fds_exp f') s" using *
     using assms by (subst eval_fds_mult) (simp_all)
   also have "\<dots> = exp (eval_fds f s)" using ev[of f'] assms unfolding f'_def
     by (auto simp: fds_abs_converges_diff eval_fds_diff fds_abs_converges_imp_converges exp_diff)
   finally show "eval_fds (fds_exp f) s = exp (eval_fds f s)" .
 qed
 
 lemma fds_exp_add:
   fixes f :: "'a :: dirichlet_series fds"
   shows   "fds_exp (f + g) = fds_exp f * fds_exp g"
 proof (rule fds_eqI_truncate)
   fix m :: nat assume m: "m > 0"
   let ?T = "fds_truncate m"
   have "?T (fds_exp (f + g)) = ?T (fds_exp (?T f + ?T g))"
     by (simp add: fds_truncate_exp fds_truncate_add_strong [symmetric])
   also have "fds_exp (?T f + ?T g) = fds_exp (?T f) * fds_exp (?T g)"
   proof (rule eval_fds_eqD)
     have "fds_abs_converges (fds_exp (?T f + ?T g)) 0"
       by (intro fds_abs_converges_exp fds_abs_converges_add) auto
     thus "conv_abscissa (fds_exp (?T f + ?T g)) < \<infinity>"
       using conv_abscissa_PInf_iff by blast
     hence "fds_abs_converges (fds_exp (fds_truncate m f) * fds_exp (fds_truncate m g)) 0"
       by (intro fds_abs_converges_mult fds_abs_converges_exp) auto
     thus "conv_abscissa (fds_exp (fds_truncate m f) * fds_exp (fds_truncate m g)) < \<infinity>"
       using conv_abscissa_PInf_iff by blast
     show "frequently (\<lambda>s. eval_fds (fds_exp (fds_truncate m f + fds_truncate m g)) s =
                           eval_fds (fds_exp (fds_truncate m f) * fds_exp (fds_truncate m g)) s)
             ((\<lambda>s. s \<bullet> 1) going_to at_top)"
       by (auto simp: eval_fds_add eval_fds_mult eval_fds_exp fds_abs_converges_add 
                      fds_abs_converges_exp exp_add)
   qed
   also have "?T \<dots> = ?T (fds_exp f * fds_exp g)"
     by (subst fds_truncate_mult [symmetric], subst (1 2) fds_truncate_exp)
        (simp add: fds_truncate_mult)
   finally show "?T (fds_exp (f + g)) = \<dots>" .
 qed
 
 lemma fds_exp_minus:
   fixes f :: "'a :: dirichlet_series fds"
   shows   "fds_exp (-f) = inverse (fds_exp f)"
 proof (rule fds_right_inverse_unique)
   have "fds_exp f * fds_exp (- f) = fds_exp (f + (-f))"
     by (subst fds_exp_add) simp_all
   also have "f + (-f) = 0" by simp
   also have "fds_exp \<dots> = 1" by simp
   finally show "fds_exp f * fds_exp (-f) = 1" .
 qed
 
 lemma abs_conv_abscissa_exp: 
   fixes f :: "'a :: dirichlet_series fds"
   shows "abs_conv_abscissa (fds_exp f) \<le> abs_conv_abscissa f"
   by (intro abs_conv_abscissa_mono fds_abs_converges_exp)
 
 lemma fds_deriv_exp [simp]:
   fixes f :: "'a :: dirichlet_series fds"
   shows   "fds_deriv (fds_exp f) = fds_exp f * fds_deriv f"
 proof (rule fds_eqI_truncate)
   fix m :: nat assume m: "m > 0"
   let ?T = "fds_truncate m"
   have "abs_conv_abscissa (fds_deriv (?T f)) = -\<infinity>"
     by (simp add: abs_conv_abscissa_deriv)
 
   have "?T (fds_deriv (fds_exp f)) = ?T (fds_deriv (fds_exp (?T f)))"
     by (simp add: fds_truncate_deriv fds_truncate_exp)
   also have "fds_deriv (fds_exp (?T f)) = fds_exp (?T f) * fds_deriv (?T f)"
   proof (rule eval_fds_eqD)
     note abscissa = conv_le_abs_conv_abscissa abs_conv_abscissa_exp
     note abscissa' = abscissa[THEN le_less_trans]
     have "fds_abs_converges (fds_deriv (fds_exp (fds_truncate m f))) 0"
       by (intro fds_abs_converges )
          (auto simp: abs_conv_abscissa_deriv intro: le_less_trans[OF abs_conv_abscissa_exp])
     thus "conv_abscissa (fds_deriv (fds_exp (fds_truncate m f))) < \<infinity>"
       using conv_abscissa_PInf_iff by blast
     have "fds_abs_converges (fds_exp (fds_truncate m f) * fds_deriv (fds_truncate m f)) 0"
       by (intro fds_abs_converges_mult fds_abs_converges_exp)
          (auto intro: fds_abs_converges simp add: fds_truncate_deriv [symmetric])
     thus "conv_abscissa (fds_exp (fds_truncate m f) * fds_deriv (fds_truncate m f)) < \<infinity>"
       using conv_abscissa_PInf_iff by blast
     show "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> 1) going_to at_top.
             eval_fds (fds_deriv (fds_exp (?T f))) s =
             eval_fds (fds_exp (?T f) * fds_deriv (?T f)) s"
     proof (intro always_eventually eventually_frequently allI, goal_cases)
       case (2 s)
       have "eval_fds (fds_deriv (fds_exp (?T f))) s =
               deriv (eval_fds (fds_exp (?T f))) s"
         by (auto simp: eval_fds_exp eval_fds_mult fds_abs_converges_mult fds_abs_converges_exp
                   fds_abs_converges eval_fds_deriv abscissa')
       also have "eval_fds (fds_exp (?T f)) = (\<lambda>s. exp (eval_fds (?T f) s))"
         by (intro ext eval_fds_exp) auto
       also have "deriv \<dots>  = (\<lambda>s. exp (eval_fds (?T f) s) * deriv (eval_fds (?T f)) s)"
         by (auto intro!: DERIV_imp_deriv derivative_eq_intros simp: eval_fds_deriv)
       also have "\<dots> = eval_fds (fds_exp (?T f) * fds_deriv (?T f))"
         by (auto simp: eval_fds_exp eval_fds_mult fds_abs_converges_mult fds_abs_converges_exp
                        fds_abs_converges eval_fds_deriv abs_conv_abscissa_deriv)
       finally show ?case .
     qed auto
   qed
   also have "?T \<dots> = ?T (fds_exp f * fds_deriv f)"
     by (subst fds_truncate_mult [symmetric])
        (simp add: fds_truncate_exp fds_truncate_deriv [symmetric], simp add: fds_truncate_mult)
   finally show "?T (fds_deriv (fds_exp f)) = \<dots>" .
 qed
 
 lemma fds_exp_ln_strong:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "fds_nth f (Suc 0) \<noteq> 0"
   shows   "fds_exp (fds_ln l f) = fds_const (exp l / fds_nth f (Suc 0)) * f"
 proof -
   let ?c = "exp l / fds_nth f (Suc 0)"
   have "f * fds_const ?c = f * (fds_exp (-fds_ln l f) * fds_exp (fds_ln l f)) * fds_const ?c" 
     (is "_ = _ * (?g * ?h) * _") by (subst fds_exp_add [symmetric]) simp
   also have "\<dots> = fds_const ?c * (f * ?g) * ?h" by (simp add: mult_ac)
   also have "f * ?g = fds_const (inverse ?c)"
   proof (rule fds_deriv_eq_imp_eq)
     have "fds_deriv (f * fds_exp (-fds_ln l f)) = 
             fds_exp (- fds_ln l f) * fds_deriv f * (1 - f / f)"
       by (simp add: divide_fds_def algebra_simps)
     also from assms have "f / f = 1" by (simp add: divide_fds_def fds_right_inverse)
     finally show "fds_deriv (f * fds_exp (-fds_ln l f)) = fds_deriv (fds_const (inverse ?c))" 
       by simp
   qed (insert assms, auto simp: exp_minus field_simps)
   also have "fds_const ?c * fds_const (inverse ?c) = 1"
     using assms by (subst fds_const_mult [symmetric]) (simp add: divide_simps)
   finally show ?thesis by (simp add: mult_ac)
 qed
 
 lemma fds_exp_ln [simp]:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "exp l = fds_nth f (Suc 0)"
   shows   "fds_exp (fds_ln l f) = f"
   using assms by (subst fds_exp_ln_strong) auto
 
 lemma fds_ln_exp [simp]:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "l = fds_nth f (Suc 0)"
   shows   "fds_ln l (fds_exp f) = f"
 proof (rule fds_deriv_eq_imp_eq)
   have "fds_deriv (fds_ln l (fds_exp f)) = fds_deriv f * (fds_exp f / fds_exp f)"
     by (simp add: algebra_simps divide_fds_def)
   also have "fds_exp f / fds_exp f = 1" by (simp add: divide_fds_def fds_right_inverse)
   finally show "fds_deriv (fds_ln l (fds_exp f)) = fds_deriv f" by simp
 qed (insert assms, auto simp: field_simps)
 
 
 subsection \<open>Euler products\<close>
 
 lemma fds_euler_product_LIMSEQ:
   fixes f :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology} fds"
   assumes "multiplicative_function (fds_nth f)" and "fds_abs_converges f s"
   shows   "(\<lambda>n. \<Prod>p\<le>n. if prime p then \<Sum>i. fds_nth f (p ^ i) / nat_power (p ^ i) s else 1) \<longlonglongrightarrow> 
              eval_fds f s"
   unfolding eval_fds_def
 proof (rule euler_product_LIMSEQ)
   show "multiplicative_function (\<lambda>n. fds_nth f n / nat_power n s)"
     by (rule multiplicative_function_divide_nat_power) fact+
 qed (insert assms, auto simp: fds_abs_converges_def)
 
 lemma fds_euler_product_LIMSEQ':
   fixes f :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology} fds"
   assumes "completely_multiplicative_function (fds_nth f)" and "fds_abs_converges f s"
   shows   "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - fds_nth f p / nat_power p s) else 1) \<longlonglongrightarrow> 
              eval_fds f s"
   unfolding eval_fds_def
 proof (rule euler_product_LIMSEQ')
   show "completely_multiplicative_function (\<lambda>n. fds_nth f n / nat_power n s)"
     by (rule completely_multiplicative_function_divide_nat_power) fact+
 qed (insert assms, auto simp: fds_abs_converges_def)
 
 lemma fds_abs_convergent_euler_product:
   fixes f :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology} fds"
   assumes "multiplicative_function (fds_nth f)" and "fds_abs_converges f s"
   shows   "abs_convergent_prod 
              (\<lambda>p. if prime p then \<Sum>i. fds_nth f (p ^ i) / nat_power (p ^ i) s else 1)"
   unfolding eval_fds_def
 proof (rule abs_convergent_euler_product)
   show "multiplicative_function (\<lambda>n. fds_nth f n / nat_power n s)"
     by (rule multiplicative_function_divide_nat_power) fact+
 qed (insert assms, auto simp: fds_abs_converges_def)
 
 lemma fds_abs_convergent_euler_product':
   fixes f :: "'a :: {nat_power, real_normed_field, banach, second_countable_topology} fds"
   assumes "completely_multiplicative_function (fds_nth f)" and "fds_abs_converges f s"
   shows   "abs_convergent_prod 
              (\<lambda>p. if prime p then inverse (1 - fds_nth f p / nat_power p s) else 1)"
   unfolding eval_fds_def
 proof (rule abs_convergent_euler_product')
   show "completely_multiplicative_function (\<lambda>n. fds_nth f n / nat_power n s)"
     by (rule completely_multiplicative_function_divide_nat_power) fact+
 qed (insert assms, auto simp: fds_abs_converges_def)
 
 lemma fds_abs_convergent_zero_iff:
   fixes f :: "'a :: {nat_power_field, real_normed_field, banach, second_countable_topology} fds"
   assumes "completely_multiplicative_function (fds_nth f)"
   assumes "fds_abs_converges f s"
   shows   "eval_fds f s = 0 \<longleftrightarrow> (\<exists>p. prime p \<and> fds_nth f p = nat_power p s)"
 proof -
   let ?g = "\<lambda>p. if prime p then inverse (1 - fds_nth f p / nat_power p s) else 1" 
   have lim: "(\<lambda>n. \<Prod>p\<le>n. ?g p) \<longlonglongrightarrow> eval_fds f s"
     by (intro fds_euler_product_LIMSEQ' assms)
   have conv: "convergent_prod ?g"
     by (intro abs_convergent_prod_imp_convergent_prod fds_abs_convergent_euler_product' assms)
 
   {
     assume "eval_fds f s = 0"
     from convergent_prod_to_zero_iff[OF conv] and this and lim
       have "\<exists>p. prime p \<and> fds_nth f p = nat_power p s" 
       by (auto split: if_splits)
   } moreover {
     assume "\<exists>p. prime p \<and> fds_nth f p = nat_power p s"
     then obtain p where "prime p" "fds_nth f p = nat_power p s" by blast
     moreover from this have "nat_power p s \<noteq> 0" 
       by (intro nat_power_nonzero) (auto simp: prime_gt_0_nat)
     ultimately have "(\<lambda>n. \<Prod>p\<le>n. ?g p) \<longlonglongrightarrow> 0"
       using convergent_prod_to_zero_iff[OF conv]
       by (auto intro!: exI[of _ p] split: if_splits)
     from tendsto_unique[OF _ lim this] have "eval_fds f s = 0"
       by simp
   }
   ultimately show ?thesis by blast
 qed
 
 lemma 
   fixes s :: "'a :: {nat_power_normed_field,banach,euclidean_space}"
   assumes "s \<bullet> 1 > 1"
   shows   euler_product_fds_zeta: 
             "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / nat_power p s) else 1)
                 \<longlonglongrightarrow> eval_fds fds_zeta s" (is ?th1)
   and     eval_fds_zeta_nonzero: "eval_fds fds_zeta s \<noteq> 0"
 proof -
   have *: "completely_multiplicative_function (fds_nth fds_zeta)"
     by standard auto
   have lim: "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - fds_nth fds_zeta p / nat_power p s) else 1) 
           \<longlonglongrightarrow> eval_fds fds_zeta s" (is "filterlim ?g _ _")
     using assms by (intro fds_euler_product_LIMSEQ' * fds_abs_summable_zeta)
   also have "?g = (\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / nat_power p s) else 1)"
     by (intro ext prod.cong refl) (auto simp: fds_zeta_def fds_nth_fds)
   finally show ?th1 .
 
   {
     fix p :: nat assume "prime p"
     from this have "p > 1" by (simp add: prime_gt_Suc_0_nat)
     hence "norm (nat_power p s) = real p powr (s \<bullet> 1)" 
       by (simp add: norm_nat_power)
     also have "\<dots> > real p powr 0" using assms and \<open>p > 1\<close> 
       by (intro powr_less_mono) auto
     finally have "nat_power p s \<noteq> 1"
       using \<open>p > 1\<close> by auto
   }
   hence **: "\<nexists>p. prime p \<and> fds_nth fds_zeta p = nat_power p s"
     by (auto simp: fds_zeta_def fds_nth_fds)
   show "eval_fds fds_zeta s \<noteq> 0"
     using assms * ** by (subst fds_abs_convergent_zero_iff) simp_all
 qed
 
 lemma fds_primepow_subseries_euler_product_cm:
   fixes f :: "'a :: dirichlet_series fds"
   assumes "completely_multiplicative_function (fds_nth f)" "prime p"
   assumes "s \<bullet> 1 > abs_conv_abscissa f"
   shows   "eval_fds (fds_primepow_subseries p f) s = 1 / (1 - fds_nth f p / nat_power p s)"
 proof -
   let ?f = "(\<lambda>n. \<Prod>pa\<le>n. if prime pa then inverse (1 - fds_nth (fds_primepow_subseries p f) pa /
                  nat_power pa s) else 1)"
   have "sequentially \<noteq> bot" by simp
   moreover have "?f \<longlonglongrightarrow> eval_fds (fds_primepow_subseries p f) s" 
     by (intro fds_euler_product_LIMSEQ' completely_multiplicative_function_only_pows assms
           fds_abs_converges_subseries) (insert assms, auto intro!: fds_abs_converges)
   moreover have "eventually (\<lambda>n. ?f n = 1 / (1 - fds_nth f p / nat_power p s)) at_top"
     using eventually_ge_at_top[of p]
   proof eventually_elim
     case (elim n)
     have "(\<Prod>pa\<le>n. if prime pa then inverse (1 - fds_nth (fds_primepow_subseries p f) pa /
               nat_power pa s) else 1) = 
           (\<Prod>q\<le>n. if q = p then inverse (1 - fds_nth f p / nat_power p s) else 1)" using \<open>prime p\<close>
       by (intro prod.cong) (auto simp: fds_nth_subseries prime_prime_factors)
     also have "\<dots> = 1 / (1 - fds_nth f p / nat_power p s)"
       using elim by (subst prod.delta) (auto simp: divide_simps)
     finally show ?case .
   qed
   hence "?f \<longlonglongrightarrow> 1 / (1 - fds_nth f p / nat_power p s)" by (rule tendsto_eventually)
   ultimately show ?thesis by (rule tendsto_unique)
 qed
 
 
 subsection \<open>Non-negative Dirichlet series\<close>
 
 lemma nonneg_Reals_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>\<^sub>\<ge>\<^sub>0) \<Longrightarrow> sum f A \<in> \<real>\<^sub>\<ge>\<^sub>0"
   by (induction A rule: infinite_finite_induct) auto
 
 locale nonneg_dirichlet_series =
   fixes f :: "'a :: dirichlet_series fds"
   assumes nonneg_coeffs_aux: "n > 0 \<Longrightarrow> fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0"
 begin
 
 lemma nonneg_coeffs: "fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0"
   using nonneg_coeffs_aux[of n] by (cases "n = 0") auto
 
 end
 
 lemma nonneg_dirichlet_series_0 [simp,intro]: "nonneg_dirichlet_series 0"
   by standard (auto simp: zero_fds_def)
 
 lemma nonneg_dirichlet_series_1 [simp,intro]: "nonneg_dirichlet_series 1"
   by standard (auto simp: one_fds_def)
 
 lemma nonneg_dirichlet_series_const [simp,intro]: 
   "c \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> nonneg_dirichlet_series (fds_const c)"
   by standard (auto simp: fds_const_def)
 
 lemma nonneg_dirichlet_series_add [intro]:
   assumes "nonneg_dirichlet_series f" "nonneg_dirichlet_series g"
   shows   "nonneg_dirichlet_series (f + g)"
 proof -
   interpret f: nonneg_dirichlet_series f by fact
   interpret g: nonneg_dirichlet_series g by fact
   show ?thesis
     by standard (auto intro!: nonneg_Reals_add_I f.nonneg_coeffs g.nonneg_coeffs)
 qed
 
 lemma nonneg_dirichlet_series_mult [intro]:
   assumes "nonneg_dirichlet_series f" "nonneg_dirichlet_series g"
   shows   "nonneg_dirichlet_series (f * g)"
 proof -
   interpret f: nonneg_dirichlet_series f by fact
   interpret g: nonneg_dirichlet_series g by fact
   show ?thesis
     by standard (auto intro!: nonneg_Reals_sum nonneg_Reals_mult_I f.nonneg_coeffs g.nonneg_coeffs 
                       simp: fds_nth_mult dirichlet_prod_def)
 qed
 
 lemma nonneg_dirichlet_series_power [intro]:
   assumes "nonneg_dirichlet_series f"
   shows   "nonneg_dirichlet_series (f ^ n)"
   using assms by (induction n) auto
 
 context nonneg_dirichlet_series
 begin
 
 lemma nonneg_exp [intro]: "nonneg_dirichlet_series (fds_exp f)"
 proof
   fix n :: nat assume "n > 0"
   define c where "c = exp (fds_nth f (Suc 0))"
   define f' where "f' = fds (\<lambda>n. if n = Suc 0 then 0 else fds_nth f n)"
   from nonneg_coeffs[of 1] obtain c' where "fds_nth f (Suc 0) = of_real c'"
     by (auto elim!: nonneg_Reals_cases)
   hence "c = of_real (exp c')" by (simp add: c_def exp_of_real)
   hence c: "c \<in> \<real>\<^sub>\<ge>\<^sub>0" by simp
   have less: "n < 2 ^ k" if "n < k" for k
   proof -
     have "n < k" by fact
     also have "\<dots> < 2 ^ k" by (induction k) auto
     finally show ?thesis .
   qed
   have nonneg_power: "fds_nth (f' ^ k) n \<in> \<real>\<^sub>\<ge>\<^sub>0" for k
   proof -
     have "nonneg_dirichlet_series f'"
       by standard (insert nonneg_coeffs, auto simp: f'_def)
     interpret nonneg_dirichlet_series "f' ^ k"
       by (intro nonneg_dirichlet_series_power) fact+
     from nonneg_coeffs[of n] show ?thesis .
   qed
   hence "fds_nth (fds_exp f) n = c * (\<Sum>k. fds_nth (f' ^ k) n /\<^sub>R fact k)"
     by (simp add: fds_exp_def fds_nth_fds' f'_def c_def)
   also have "(\<Sum>k. fds_nth (f' ^ k) n /\<^sub>R fact k) = (\<Sum>k\<le>n. fds_nth (f' ^ k) n /\<^sub>R fact k)"
     by (intro suminf_finite) (auto intro!: fds_nth_power_eq_0 less simp: f'_def not_le)
   also have "c * \<dots> \<in> \<real>\<^sub>\<ge>\<^sub>0" unfolding scaleR_conv_of_real
     by (intro nonneg_Reals_mult_I nonneg_Reals_sum nonneg_power, unfold nonneg_Reals_of_real_iff )
        (auto simp: c)
   finally show "fds_nth (fds_exp f) n \<in> \<real>\<^sub>\<ge>\<^sub>0" .
 qed
 
 end
 
 lemma nonneg_dirichlet_series_lnD: 
   assumes "nonneg_dirichlet_series (fds_ln l f)" "exp l = fds_nth f (Suc 0)"
   shows   "nonneg_dirichlet_series f"
 proof -
   from assms have "nonneg_dirichlet_series (fds_exp (fds_ln l f))"
     by (intro nonneg_dirichlet_series.nonneg_exp)
   thus ?thesis using assms by simp
 qed
 
 context nonneg_dirichlet_series
 begin
 
 lemma fds_of_real_norm: "fds_of_real (fds_norm f) = f"
 proof (rule fds_eqI)
   fix n :: nat assume n: "n > 0"
   show "fds_nth (fds_of_real (fds_norm f)) n = fds_nth f n"
     using nonneg_coeffs[of n] by (auto elim!: nonneg_Reals_cases)
 qed
 
 end
 
 (* TODO: Simplify proof *)
 lemma pringsheim_landau_aux:
   fixes c :: real and f :: "complex fds"
   assumes "nonneg_dirichlet_series f"
   assumes abscissa: "c \<ge> abs_conv_abscissa f"
   assumes g: "\<And>s. s \<in> A \<Longrightarrow> Re s > c \<Longrightarrow> g s = eval_fds f s"
   assumes "g holomorphic_on A" "open A" "c \<in> A"
   shows   "\<exists>x. x < c \<and> fds_abs_converges f (of_real x)"
 proof -
   interpret nonneg_dirichlet_series f by fact
   define a where "a = 1 + c"
 
   define g' where "g' = (\<lambda>s. if s \<in> {s. Re s > c} then eval_fds f s else g s)"
 
   \<comment> \<open>We can find some $\varepsilon > 0$ such that the Dirichlet series can be continued
       analytically in a ball of radius $1 + \varepsilon$ around $a$.\<close>
   from \<open>open A\<close> \<open>c \<in> A\<close> obtain \<delta> where \<delta>: "\<delta> > 0" "ball c \<delta> \<subseteq> A"
     by (auto simp: open_contains_ball)
   define \<epsilon> where "\<epsilon> = sqrt (1 + \<delta>^2) - 1"
   from \<delta> have \<epsilon>: "\<epsilon> > 0" by (simp add: \<epsilon>_def)
 
   have ball_a_subset: "ball a (1 + \<epsilon>) \<subseteq> {s. Re s > c} \<union> A"
   proof (intro subsetI)
     fix s :: complex assume s: "s \<in> ball a (1 + \<epsilon>)"
     define x y where "x = Re s" and "y = Im s"
     have [simp]: "s = x + \<i> * y" by (simp add: complex_eq_iff x_def y_def)
     show "s \<in> {s. Re s > c} \<union> A"
     proof (cases "Re s \<le> c")
       case True
       hence "(c - x)\<^sup>2 + y\<^sup>2 \<le> (1 + c - x)\<^sup>2 + y\<^sup>2 - 1"
         by (simp add: power2_eq_square algebra_simps)
       also from s have "(1 + c - x)\<^sup>2 + y\<^sup>2 - 1 < \<delta>\<^sup>2"
         by (auto simp: dist_norm cmod_def a_def \<epsilon>_def)
       finally have "sqrt ((c - x)\<^sup>2 + y\<^sup>2) < \<delta>" using \<delta>
         by (intro real_less_lsqrt) auto
       hence "s \<in> ball c \<delta>" by (auto simp: dist_norm cmod_def)
       also have "\<dots> \<subseteq> A" by fact
       finally show ?thesis ..
     qed auto
   qed
 
   have holo: "g' holomorphic_on ball a (1 + \<epsilon>)" unfolding g'_def
   proof (intro holomorphic_on_subset[OF _ ball_a_subset] holomorphic_on_If_Un)
     have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
     also have "\<dots> \<le> ereal c" by fact
     finally have*:  "conv_abscissa f \<le> ereal c" .
     show "eval_fds f holomorphic_on {s. c < Re s}"
       by (intro holomorphic_intros) (auto intro: le_less_trans[OF *])
   qed (insert assms, auto intro!: holomorphic_intros open_halfspace_Re_gt)
 
   define f' where "f' = fds_norm f"
   have f_f': "f = fds_of_real f'" by (simp add: f'_def fds_of_real_norm)
   have f'_nonneg: "fds_nth f' n \<ge> 0" for n
     using nonneg_coeffs[of n] by (auto elim!: nonneg_Reals_cases simp: f'_def)
 
   have deriv: "(\<lambda>n. (deriv ^^ n) g' a) = (\<lambda>n. eval_fds ((fds_deriv ^^ n) f) a)"
   proof
     fix n :: nat
     have ev: "eventually (\<lambda>s. s \<in> {s. Re s > c}) (nhds (complex_of_real a))"
       by (intro eventually_nhds_in_open open_halfspace_Re_gt) (auto simp: a_def)
     have "(deriv ^^ n) g' a = (deriv ^^ n) (eval_fds f) a"
       by (intro higher_deriv_cong_ev refl eventually_mono[OF ev]) (auto simp: g'_def)
     also have "\<dots> = eval_fds ((fds_deriv ^^ n) f) a"
     proof (intro eval_fds_higher_deriv [symmetric])
       have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
       also have "\<dots> \<le> ereal c" by (rule assms)
       also have "\<dots> < a" by (simp add: a_def)
       finally show "conv_abscissa f < ereal (complex_of_real a \<bullet> 1)" by simp
     qed
     finally show "(deriv ^^ n) g' a = eval_fds ((fds_deriv ^^ n) f) a" .
   qed
 
   have nth_deriv_conv: "fds_abs_converges ((fds_deriv ^^ n) f) (of_real a)" for n
     by (intro fds_abs_converges)
        (auto simp: abs_conv_abscissa_higher_deriv a_def intro!: le_less_trans[OF abscissa])
 
   have nth_deriv_eq: "(fds_deriv ^^ n) f = fds (\<lambda>k. (-1) ^ n * fds_nth f k * ln (real k) ^ n)" for n
   proof -
     have "fds_nth ((fds_deriv ^^ n) f) k = (-1) ^ n * fds_nth f k * ln (real k) ^ n" for k
       by (induction n) (simp_all add: fds_deriv_def fds_eq_iff fds_nth_fds' scaleR_conv_of_real)
     thus ?thesis by (intro fds_eqI) simp_all
   qed
 
   have deriv': "(\<lambda>n. eval_fds ((fds_deriv ^^ n) f) (complex_of_real a)) =
       (\<lambda>n. (- 1) ^ n * complex_of_real (\<Sum>\<^sub>ak. fds_nth f' k * ln (real k) ^ n / real k powr a))"
   proof
     fix n
     have "eval_fds ((fds_deriv ^^ n) f) (of_real a) = 
                  (\<Sum>\<^sub>ak. fds_nth ((fds_deriv ^^ n) f) k / of_nat k powr complex_of_real a)"
       using nth_deriv_conv by (subst eval_fds_altdef) auto
     hence "eval_fds ((fds_deriv ^^ n) f) (of_real a) = 
                  (\<Sum>\<^sub>ak. (- 1) ^ n *\<^sub>R (fds_nth f k * ln (real k) ^ n / k powr a))"
       by (simp add: nth_deriv_eq fds_nth_fds' powr_Reals_eq scaleR_conv_of_real algebra_simps)
     also have "\<dots> = (- 1) ^ n * (\<Sum>\<^sub>ak. of_real (fds_nth f' k * ln (real k) ^ n / k powr a))"
       by (subst infsetsum_scaleR_right) (simp_all add: scaleR_conv_of_real f_f')
     also have "\<dots> = (- 1) ^ n * of_real (\<Sum>\<^sub>ak. fds_nth f' k * ln (real k) ^ n / k powr a)"
       by (subst infsetsum_of_real) (rule refl)
     finally show "eval_fds ((fds_deriv ^^ n) f) (complex_of_real a) =
       (- 1) ^ n * complex_of_real (\<Sum>\<^sub>ak. fds_nth f' k * ln (real k) ^ n / real k powr a)" .
   qed
 
   define s :: complex where "s = c - \<epsilon> / 2"
   have s: "Re s < c" using assms \<delta> by (simp_all add: s_def \<epsilon>_def field_simps)
   have "s \<in> ball a (1 + \<epsilon>)" using s by (simp add: a_def dist_norm cmod_def s_def)
   from holomorphic_power_series[OF holo this]
     have sums: "(\<lambda>n. (deriv ^^ n) g' a / fact n * (s - a) ^ n) sums g' s" by simp
   also note deriv
   also have "s - a = -of_real (1 + \<epsilon> / 2)" by (simp add: s_def a_def)
   also have "(\<lambda>n. \<dots> ^ n) = (\<lambda>n. of_real ((-1) ^ n * (1 + \<epsilon> / 2) ^ n))"
     by (intro ext) (subst power_minus, auto)
   also have "(\<lambda>n. eval_fds ((fds_deriv ^^ n) f) a / fact n * \<dots> n) =
                (\<lambda>n. of_real ((-1) ^ n * eval_fds ((fds_deriv ^^ n) f') a / fact n * (1+\<epsilon>/2) ^ n))"
     using nth_deriv_conv by (simp add: f_f' fds_abs_converges_imp_converges mult_ac)
   finally have "summable \<dots>" by (simp add: sums_iff)
   hence summable: "summable (\<lambda>n. (-1)^n * eval_fds ((fds_deriv ^^ n) f') a / fact n * (1+\<epsilon>/2)^n)"
     by (subst (asm) summable_of_real_iff)
 
   have "(\<lambda>(n,k). (-1)^n * fds_nth f k * ln (real k) ^ n / (real k powr a) * ((s-a) ^ n / fact n))
           abs_summable_on (UNIV \<times> UNIV)"
   proof (subst abs_summable_on_Sigma_iff, safe, goal_cases)
     case (3 n)
     from nth_deriv_conv[of n] show ?case
       unfolding fds_abs_converges_altdef'
       by (intro abs_summable_on_cmult_left) (simp add: nth_deriv_eq fds_nth_fds' powr_Reals_eq)
   next
     case 4
     have nth_deriv_f_f': "(fds_deriv ^^ n) f = fds_of_real ((fds_deriv ^^ n) f')" for n
       by (induction n) (auto simp: f'_def fds_of_real_norm)
     have norm_nth_deriv_f: "norm (fds_nth ((fds_deriv ^^ n) f) k) = 
                                  (-1) ^ n * of_real (fds_nth ((fds_deriv ^^ n) f') k)" for n k
     proof (induction n)
       case (Suc n)
       thus ?case by (cases k) (auto simp: f_f' fds_nth_deriv scaleR_conv_of_real norm_mult)
     qed (auto simp: f'_nonneg f_f')
 
     note summable
     also have "(\<lambda>n. (-1)^n * eval_fds ((fds_deriv ^^ n) f') a / fact n * (1+\<epsilon>/2)^n) =
                (\<lambda>n. \<Sum>\<^sub>ak. norm ((- 1) ^ n * fds_nth f k * ln (real k) ^ n /
                     (real k powr a) * ((s - a) ^ n / fact n)))" (is "_ = ?h")
     proof (rule ext, goal_cases)
       case (1 n)
       have "(\<Sum>\<^sub>ak. norm ((- 1) ^ n * fds_nth f k * ln (real k) ^ n /
                     (real k powr a) * ((s - a) ^ n / fact n))) =
             (norm ((s - a) ^ n / fact n) * (-1) ^ n) *\<^sub>R 
               (\<Sum>\<^sub>ak. (-1) ^ n * norm (fds_nth ((fds_deriv ^^ n) f) k / real k powr a))" (is "_ = _ *\<^sub>R ?S")
         by (subst infsetsum_scaleR_right [symmetric])
            (auto simp: norm_mult norm_divide norm_power mult_ac nth_deriv_eq fds_nth_fds')
       also have "?S = (\<Sum>\<^sub>ak. fds_nth ((fds_deriv ^^ n) f') k / real k powr a)"
         by (intro infsetsum_cong) (auto simp: norm_mult norm_divide norm_power norm_nth_deriv_f)
       also have "\<dots> = eval_fds ((fds_deriv ^^ n) f') a"
         using nth_deriv_conv[of n] by (subst eval_fds_altdef) (auto simp: f'_def nth_deriv_f_f')
       also have "(norm ((s - a) ^ n / fact n) * (- 1) ^ n) *\<^sub>R eval_fds ((fds_deriv ^^ n) f') a =
                   (-1) ^ n * eval_fds ((fds_deriv ^^ n) f') a / fact n * norm (s - a) ^ n"
         by (simp add: norm_divide norm_power)
       also have s_a: "s - a = -of_real (1 + \<epsilon> / 2)" by (simp add: s_def a_def)
       have "norm (s - a) = 1 + \<epsilon> / 2" unfolding s_a norm_minus_cancel norm_of_real using \<epsilon> by simp
       finally show ?case ..
     qed
     also have "?h n \<ge> 0" for n by (intro infsetsum_nonneg) auto
     hence "?h = (\<lambda>n. norm (?h n))" by simp
     finally show ?case unfolding abs_summable_on_nat_iff' .
   qed auto
   hence "(\<lambda>(k,n). (-1)^n * fds_nth f k * ln (real k) ^ n / (real k powr a) * ((s-a) ^ n / fact n))
           abs_summable_on (UNIV \<times> UNIV)"
     by (subst (asm) abs_summable_on_Times_swap) (simp add: case_prod_unfold)
   hence "(\<lambda>k. \<Sum>\<^sub>an. (- 1) ^ n * fds_nth f k * ln (real k) ^ n / (k powr a) *
            ((s - a) ^ n / fact n)) abs_summable_on UNIV" (is "?h abs_summable_on _")
     by (rule abs_summable_on_Sigma_project1') auto
   also have "?this \<longleftrightarrow> (\<lambda>k. fds_nth f k / nat_power k s) abs_summable_on UNIV"
   proof (intro abs_summable_on_cong refl, goal_cases)
     case (1 k)
     have "?h k = (fds_nth f' k / k powr a) *\<^sub>R (\<Sum>\<^sub>an. (-ln (real k) * (s - a)) ^ n / fact n)"
       by (subst infsetsum_scaleR_right [symmetric], rule infsetsum_cong) 
          (simp_all add: scaleR_conv_of_real f_f' power_minus' power_mult_distrib divide_simps)
     also have "(\<Sum>\<^sub>an. (-ln (real k) * (s - a)) ^ n / fact n) = exp (-ln (real k) * (s - a))"
       using exp_converges[of "-ln k * (s - a)"] exp_converges[of "norm (-ln k * (s - a))"]
       by (subst infsetsum_nat') (auto simp: abs_summable_on_nat_iff' sums_iff scaleR_conv_of_real
                                             divide_simps norm_divide norm_mult norm_power)
     also have "(fds_nth f' k / k powr a) *\<^sub>R \<dots> = fds_nth f k / nat_power k s"
       by (auto simp: scaleR_conv_of_real f_f' powr_def exp_minus
                      field_simps exp_of_real [symmetric] exp_diff)
     finally show ?case .
   qed
   finally have "fds_abs_converges f s"
     by (simp add: fds_abs_converges_def abs_summable_on_nat_iff')
   thus ?thesis by (intro exI[of _ "(c - \<epsilon> / 2)"]) (auto simp: s_def a_def \<epsilon>)
 qed
 
 theorem pringsheim_landau:
   fixes c :: real and f :: "complex fds"
   assumes "nonneg_dirichlet_series f"
   assumes abscissa: "abs_conv_abscissa f = c"
   assumes g: "\<And>s. s \<in> A \<Longrightarrow> Re s > c \<Longrightarrow> g s = eval_fds f s"
   assumes "g holomorphic_on A" "open A" "c \<in> A"
   shows   False
 proof -
   have "\<exists>x<c. fds_abs_converges f (complex_of_real x)"
     by (rule pringsheim_landau_aux[where g = g and A = A]) (insert assms, auto)
   then obtain x where x: "x < c" "fds_abs_converges f (complex_of_real x)" by blast
   hence "abs_conv_abscissa f \<le> complex_of_real x \<bullet> 1" 
     unfolding abs_conv_abscissa_def 
     by (intro Inf_lower) (auto simp: image_iff intro!: exI[of _ "of_real x"])
   also have "\<dots> < abs_conv_abscissa f" using assms x by simp
   finally show False by simp
 qed    
 
 corollary entire_continuation_imp_abs_conv_abscissa_MInfty:
   assumes "nonneg_dirichlet_series f"
   assumes c: "c \<ge> abs_conv_abscissa f"
   assumes g: "\<And>s. Re s > c \<Longrightarrow> g s = eval_fds f s"
   assumes holo: "g holomorphic_on UNIV"
   shows   "abs_conv_abscissa f = -\<infinity>"
 proof (rule ccontr)
   assume "abs_conv_abscissa f \<noteq> -\<infinity>"
   with c obtain a where abscissa [simp]: "abs_conv_abscissa f = ereal a"
     by (cases "abs_conv_abscissa f") auto
   show False
   proof (rule pringsheim_landau[OF assms(1) abscissa _ holo])
     fix s assume s: "Re s > a"
     show "g s = eval_fds f s"
     proof (rule sym, rule analytic_continuation_open[of _ _ _ g])
       show "g holomorphic_on {s. Re s > a}" by (rule holomorphic_on_subset[OF holo]) auto
       from assms show "{s. Re s > c} \<subseteq> {s. Re s > a}" by auto
     next
       have "conv_abscissa f \<le> abs_conv_abscissa f" by (rule conv_le_abs_conv_abscissa)
       also have "\<dots> = ereal a" by simp
       finally show "eval_fds f holomorphic_on {s. Re s > a}"
         by (intro holomorphic_intros) (auto intro: le_less_trans)
     qed (insert assms s, auto intro!: exI[of _ "of_real (c + 1)"] 
            open_halfspace_Re_gt convex_connected convex_halfspace_Re_gt)
   qed auto
 qed
 
 
 
 subsection \<open>Convergence of the $\zeta$ and M\"{o}bius $\mu$ series\<close>
 
 lemma fds_abs_summable_zeta_real_iff [simp]:
   "fds_abs_converges fds_zeta s \<longleftrightarrow> s > (1 :: real)"
 proof -
   have "fds_abs_converges fds_zeta s \<longleftrightarrow> summable (\<lambda>n. real n powr -s)"
     unfolding fds_abs_converges_def
     by (intro summable_cong always_eventually)
        (auto simp: fds_nth_zeta powr_minus divide_simps)
   also have "\<dots> \<longleftrightarrow> s > 1" by (simp add: summable_real_powr_iff)
   finally show ?thesis .
 qed
 
 lemma fds_abs_summable_zeta_real: "s > (1 :: real) \<Longrightarrow> fds_abs_converges fds_zeta s"
   by simp
 
 lemma fds_abs_converges_moebius_mu_real: 
   assumes "s > (1 :: real)"
   shows   "fds_abs_converges (fds moebius_mu) s"
   unfolding fds_abs_converges_def
 proof (rule summable_comparison_test, intro exI allI impI)
   fix n :: nat
   show "norm (norm (fds_nth (fds moebius_mu) n / nat_power n s)) \<le> n powr (-s)"
     by (simp add: powr_minus divide_simps abs_moebius_mu_le)
 next
   from assms show "summable (\<lambda>n. real n powr -s)" by (simp add: summable_real_powr_iff)
 qed
 
 
 subsection \<open>Application to the M\"{o}bius $\mu$ function\<close>
 
 lemma inverse_squares_sums': "(\<lambda>n. 1 / real n ^ 2) sums (pi ^ 2 / 6)"
   using inverse_squares_sums sums_Suc_iff[of "\<lambda>n. 1 / real n ^ 2" "pi^2 / 6"] by simp
 
 lemma norm_summable_moebius_over_square: 
   "summable (\<lambda>n. norm (moebius_mu n / real n ^ 2))"
 proof (subst summable_Suc_iff [symmetric], rule summable_comparison_test)
   show "summable (\<lambda>n. 1 / real (Suc n) ^ 2)"
     using inverse_squares_sums by (simp add: sums_iff)
 qed (auto simp del: of_nat_Suc simp: field_simps abs_moebius_mu_le)
 
 lemma summable_moebius_over_square:
   "summable (\<lambda>n. moebius_mu n / real n ^ 2)"
   using norm_summable_moebius_over_square by (rule summable_norm_cancel)
 
 lemma moebius_over_square_sums: "(\<lambda>n. moebius_mu n / n\<^sup>2) sums (6 / pi\<^sup>2)"
 proof -
   have "1 = eval_fds (1 :: real fds) 2" by simp
   also have "(1 :: real fds) = fds_zeta * fds moebius_mu" 
     by (rule fds_zeta_times_moebius_mu [symmetric])
   also have "eval_fds \<dots> 2 = eval_fds fds_zeta 2 * eval_fds (fds moebius_mu) 2"
     by (intro eval_fds_mult fds_abs_converges_moebius_mu_real) simp_all
   also have "\<dots> = pi ^ 2 / 6 * (\<Sum>n. moebius_mu n / (real n)\<^sup>2)"
     using inverse_squares_sums' by (simp add: eval_fds_at_numeral suminf_fds_zeta_aux sums_iff)
   finally have "(\<Sum>n. moebius_mu n / (real n)\<^sup>2) = 6 / pi ^ 2" by (simp add: field_simps)
   with summable_moebius_over_square show ?thesis by (simp add: sums_iff)
 qed
 
 end
diff --git a/thys/Dirichlet_Series/ROOT b/thys/Dirichlet_Series/ROOT
--- a/thys/Dirichlet_Series/ROOT
+++ b/thys/Dirichlet_Series/ROOT
@@ -1,18 +1,18 @@
 chapter AFP
 
-session Dirichlet_Series (AFP) = "HOL-Analysis" +
+session Dirichlet_Series (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 1200]
   sessions
     "HOL-Analysis"
     "HOL-Number_Theory"
     "HOL-Computational_Algebra"
     "HOL-Real_Asymp"
     "Polynomial_Factorization"
     "Landau_Symbols"
     "Euler_MacLaurin"
   theories
     Arithmetic_Summatory_Asymptotics
     Dirichlet_Efficient_Code
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/E_Transcendental/E_Transcendental.thy b/thys/E_Transcendental/E_Transcendental.thy
--- a/thys/E_Transcendental/E_Transcendental.thy
+++ b/thys/E_Transcendental/E_Transcendental.thy
@@ -1,687 +1,687 @@
 (*
   File:       E_Transcendental.thy
   Author:     Manuel Eberl <eberlm@in.tum.de>
 
   A proof that e (Euler's number) is transcendental.
   Could possibly be extended to a transcendence proof for pi or
   the very general Lindemann-Weierstrass theorem.
 *)
 section \<open>Proof of the Transcendence of $e$\<close>
 theory E_Transcendental
   imports
-    "HOL-Analysis.Analysis"
+    "HOL-Complex_Analysis.Complex_Analysis"
     "HOL-Number_Theory.Number_Theory"
     "HOL-Computational_Algebra.Polynomial"
 begin
 
 (* TODO: Lots of stuff to move to the distribution *)
   
 subsection \<open>Various auxiliary facts\<close>
 
 lemma fact_dvd_pochhammer:
   assumes "m \<le> n + 1"
   shows   "fact m dvd pochhammer (int n - int m + 1) m"
 proof -
   have "(real n gchoose m) * fact m = of_int (pochhammer (int n - int m + 1) m)"
     by (simp add: gbinomial_pochhammer' pochhammer_of_int [symmetric])
   also have "(real n gchoose m) * fact m = of_int (int (n choose m) * fact m)"
     by (simp add: binomial_gbinomial)
   finally have "int (n choose m) * fact m = pochhammer (int n - int m + 1) m"
     by (subst (asm) of_int_eq_iff)
   from this [symmetric] show ?thesis by simp
 qed
 
 lemma of_nat_eq_1_iff [simp]: "of_nat x = (1 :: 'a :: semiring_char_0) \<longleftrightarrow> x = 1"
   by (fact of_nat_eq_1_iff)
 
 lemma prime_elem_int_not_dvd_neg1_power:
   "prime_elem (p :: int) \<Longrightarrow> \<not>p dvd (-1) ^ n"
   by (rule notI, frule (1) prime_elem_dvd_power, cases "p \<ge> 0") (auto simp: prime_elem_def)
 
 lemma nat_fact [simp]: "nat (fact n) = fact n"
   by (subst of_nat_fact [symmetric]) (rule nat_int)
 
 lemma prime_dvd_fact_iff_int:
   "p dvd fact n \<longleftrightarrow> p \<le> int n" if "prime p"
   using that prime_dvd_fact_iff [of "nat \<bar>p\<bar>" n]
   by auto (simp add: prime_ge_0_int)
 
 lemma filterlim_minus_nat_at_top:
   "filterlim (\<lambda>n. n - k :: nat) at_top at_top"
 proof -
   have "sequentially = filtermap (\<lambda>n. n + k) at_top"
     by (auto simp: filter_eq_iff eventually_filtermap)
   also have "filterlim (\<lambda>n. n - k :: nat) at_top \<dots>"
     by (simp add: filterlim_filtermap filterlim_ident)
   finally show ?thesis .
 qed
 
 lemma power_over_fact_tendsto_0:
   "(\<lambda>n. (x :: real) ^ n / fact n) \<longlonglongrightarrow> 0"
   using summable_exp[of x] by (intro summable_LIMSEQ_zero) (simp add: sums_iff field_simps)
 
 lemma power_over_fact_tendsto_0':
   "(\<lambda>n. c * (x :: real) ^ n / fact n) \<longlonglongrightarrow> 0"
   using tendsto_mult[OF tendsto_const[of c] power_over_fact_tendsto_0[of x]] by simp
 
 
 subsection \<open>Lifting integer polynomials\<close>
 
 lift_definition of_int_poly :: "int poly \<Rightarrow> 'a :: comm_ring_1 poly" is "\<lambda>g x. of_int (g x)"
   by (auto elim: eventually_mono)
 
 lemma coeff_of_int_poly [simp]: "coeff (of_int_poly p) n = of_int (coeff p n)"
   by transfer' simp
 
 lemma of_int_poly_0 [simp]: "of_int_poly 0 = 0"
   by transfer (simp add: fun_eq_iff)
 
 lemma of_int_poly_pCons [simp]: "of_int_poly (pCons c p) = pCons (of_int c) (of_int_poly p)"
   by transfer' (simp add: fun_eq_iff split: nat.splits)
 
 lemma of_int_poly_smult [simp]: "of_int_poly (smult c p) = smult (of_int c) (of_int_poly p)"
   by transfer simp
 
 lemma of_int_poly_1 [simp]: "of_int_poly 1 = 1"
   by (simp add: one_pCons)
 
 lemma of_int_poly_add [simp]: "of_int_poly (p + q) = of_int_poly p + of_int_poly q"
   by transfer' (simp add: fun_eq_iff)
 
 lemma of_int_poly_mult [simp]: "of_int_poly (p * q) = (of_int_poly p * of_int_poly q)"
   by (induction p) simp_all
 
 lemma of_int_poly_sum [simp]: "of_int_poly (sum f A) = sum (\<lambda>x. of_int_poly (f x)) A"
   by (induction A rule: infinite_finite_induct) simp_all
 
 lemma of_int_poly_prod [simp]: "of_int_poly (prod f A) = prod (\<lambda>x. of_int_poly (f x)) A"
   by (induction A rule: infinite_finite_induct) simp_all
 
 lemma of_int_poly_power [simp]: "of_int_poly (p ^ n) = of_int_poly p ^ n"
   by (induction n) simp_all
 
 lemma of_int_poly_monom [simp]: "of_int_poly (monom c n) = monom (of_int c) n"
   by transfer (simp add: fun_eq_iff)
 
 lemma poly_of_int_poly [simp]: "poly (of_int_poly p) (of_int x) = of_int (poly p x)"
   by (induction p) simp_all
 
 lemma poly_of_int_poly_of_nat [simp]: "poly (of_int_poly p) (of_nat x) = of_int (poly p (int x))"
   by (induction p) simp_all
 
 lemma poly_of_int_poly_0 [simp]: "poly (of_int_poly p) 0 = of_int (poly p 0)"
   by (induction p) simp_all
 
 lemma poly_of_int_poly_1 [simp]: "poly (of_int_poly p) 1 = of_int (poly p 1)"
   by (induction p) simp_all
 
 lemma poly_of_int_poly_of_real [simp]:
     "poly (of_int_poly p) (of_real x) = of_real (poly (of_int_poly p) x)"
   by (induction p) simp_all
 
 lemma of_int_poly_eq_iff [simp]:
   "of_int_poly p = (of_int_poly q :: 'a :: {comm_ring_1, ring_char_0} poly) \<longleftrightarrow> p = q"
   by (simp add: poly_eq_iff)
 
 lemma of_int_poly_eq_0_iff [simp]:
   "of_int_poly p = (0 :: 'a :: {comm_ring_1, ring_char_0} poly) \<longleftrightarrow> p = 0"
   using of_int_poly_eq_iff[of p 0] by (simp del: of_int_poly_eq_iff)
 
 lemma degree_of_int_poly [simp]:
   "degree (of_int_poly p :: 'a :: {comm_ring_1, ring_char_0} poly) = degree p"
   by (simp add: degree_def)
 
 lemma pderiv_of_int_poly [simp]: "pderiv (of_int_poly p) = of_int_poly (pderiv p)"
   by (induction p) (simp_all add: pderiv_pCons)
 
 lemma higher_pderiv_of_int_poly [simp]:
   "(pderiv ^^ n) (of_int_poly p) = of_int_poly ((pderiv ^^ n) p)"
   by (induction n) simp_all
 
 lemma int_polyE:
   assumes "\<And>n. coeff (p :: 'a :: {comm_ring_1, ring_char_0} poly) n \<in> \<int>"
   obtains p' where "p = of_int_poly p'"
 proof -
   from assms have "\<forall>n. \<exists>c. coeff p n = of_int c" by (auto simp: Ints_def)
   hence "\<exists>c. \<forall>n. of_int (c n) = coeff p n" by (simp add: choice_iff eq_commute)
   then obtain c where c: "of_int (c n) = coeff p n" for n by blast
   have [simp]: "coeff (Abs_poly c) = c"
   proof (rule poly.Abs_poly_inverse, clarify)
     have "eventually (\<lambda>n. n > degree p) at_top" by (rule eventually_gt_at_top)
     hence "eventually (\<lambda>n. coeff p n = 0) at_top"
       by eventually_elim (simp add: coeff_eq_0)
     thus "eventually (\<lambda>n. c n = 0) cofinite"
       by (simp add: c [symmetric] cofinite_eq_sequentially)
   qed
   have "p = of_int_poly (Abs_poly c)"
     by (rule poly_eqI) (simp add: c)
   thus ?thesis by (rule that)
 qed
 
 
 subsection \<open>General facts about polynomials\<close>
 
 lemma pderiv_power:
   "pderiv (p ^ n) = smult (of_nat n) (p ^ (n - 1) * pderiv p)"
   by (cases n) (simp_all add: pderiv_power_Suc del: power_Suc)
 
 lemma degree_prod_sum_eq:
   "(\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0) \<Longrightarrow>
      degree (prod f A :: 'a :: idom poly) = (\<Sum>x\<in>A. degree (f x))"
   by (induction A rule: infinite_finite_induct) (auto simp: degree_mult_eq)
 
 lemma pderiv_monom:
   "pderiv (monom c n) = monom (of_nat n * c) (n - 1)"
   by (cases n)
      (simp_all add: monom_altdef pderiv_power_Suc pderiv_smult pderiv_pCons mult_ac del: power_Suc)
 
 lemma power_poly_const [simp]: "[:c:] ^ n = [:c ^ n:]"
   by (induction n) (simp_all add: power_commutes)
 
 lemma monom_power: "monom c n ^ k = monom (c ^ k) (n * k)"
   by (induction k) (simp_all add: mult_monom)
 
 lemma coeff_higher_pderiv:
   "coeff ((pderiv ^^ m) f) n = pochhammer (of_nat (Suc n)) m * coeff f (n + m)"
   by (induction m arbitrary: n) (simp_all add: coeff_pderiv pochhammer_rec algebra_simps)
 
 lemma higher_pderiv_add: "(pderiv ^^ n) (p + q) = (pderiv ^^ n) p + (pderiv ^^ n) q"
   by (induction n arbitrary: p q) (simp_all del: funpow.simps add: funpow_Suc_right pderiv_add)
 
 lemma higher_pderiv_smult: "(pderiv ^^ n) (smult c p) = smult c ((pderiv ^^ n) p)"
   by (induction n arbitrary: p) (simp_all del: funpow.simps add: funpow_Suc_right pderiv_smult)
 
 lemma higher_pderiv_0 [simp]: "(pderiv ^^ n) 0 = 0"
   by (induction n) simp_all
 
 lemma higher_pderiv_monom:
   "m \<le> n + 1 \<Longrightarrow> (pderiv ^^ m) (monom c n) = monom (pochhammer (int n - int m + 1) m * c) (n - m)"
 proof (induction m arbitrary: c n)
   case (Suc m)
   thus ?case
     by (cases n)
        (simp_all del: funpow.simps add: funpow_Suc_right pderiv_monom pochhammer_rec' Suc.IH)
 qed simp_all
 
 lemma higher_pderiv_monom_eq_zero:
   "m > n + 1 \<Longrightarrow> (pderiv ^^ m) (monom c n) = 0"
 proof (induction m arbitrary: c n)
   case (Suc m)
   thus ?case
     by (cases n)
        (simp_all del: funpow.simps add: funpow_Suc_right pderiv_monom pochhammer_rec' Suc.IH)
 qed simp_all
 
 lemma higher_pderiv_sum: "(pderiv ^^ n) (sum f A) = (\<Sum>x\<in>A. (pderiv ^^ n) (f x))"
   by (induction A rule: infinite_finite_induct) (simp_all add: higher_pderiv_add)
 
 lemma fact_dvd_higher_pderiv:
   "[:fact n :: int:] dvd (pderiv ^^ n) p"
 proof -
   have "[:fact n:] dvd (pderiv ^^ n) (monom c k)" for c :: int and k :: nat
     by (cases "n \<le> k + 1")
        (simp_all add: higher_pderiv_monom higher_pderiv_monom_eq_zero
           fact_dvd_pochhammer const_poly_dvd_iff)
   hence "[:fact n:] dvd (pderiv ^^ n) (\<Sum>k\<le>degree p. monom (coeff p k) k)"
     by (simp_all add: higher_pderiv_sum dvd_sum)
   thus ?thesis by (simp add: poly_as_sum_of_monoms)
 qed
 
 lemma fact_dvd_poly_higher_pderiv_aux:
   "(fact n :: int) dvd poly ((pderiv ^^ n) p) x"
 proof -
   have "[:fact n:] dvd (pderiv ^^ n) p" by (rule fact_dvd_higher_pderiv)
   then obtain q where "(pderiv ^^ n) p = [:fact n:] * q" by (erule dvdE)
   thus ?thesis by simp
 qed
 
 lemma fact_dvd_poly_higher_pderiv_aux':
   "m \<le> n \<Longrightarrow> (fact m :: int) dvd poly ((pderiv ^^ n) p) x"
   by (rule dvd_trans[OF fact_dvd fact_dvd_poly_higher_pderiv_aux]) simp_all
 
 lemma algebraicE':
   assumes "algebraic (x :: 'a :: field_char_0)"
   obtains p where "p \<noteq> 0" "poly (of_int_poly p) x = 0"
 proof -
   from assms obtain q where "\<And>i. coeff q i \<in> \<int>" "q \<noteq> 0" "poly q x = 0"
     by (erule algebraicE)
   moreover from this(1) obtain q' where "q = of_int_poly q'" by (erule int_polyE)
   ultimately show ?thesis by (intro that[of q']) simp_all
 qed
 
 lemma algebraicE'_nonzero:
   assumes "algebraic (x :: 'a :: field_char_0)" "x \<noteq> 0"
   obtains p where "p \<noteq> 0" "coeff p 0 \<noteq> 0" "poly (of_int_poly p) x = 0"
 proof -
   from assms(1) obtain p where p: "p \<noteq> 0" "poly (of_int_poly p) x = 0"
     by (erule algebraicE')
   define n :: nat where "n = order 0 p"
   have "monom 1 n dvd p" by (simp add: monom_1_dvd_iff p n_def)
   then obtain q where q: "p = monom 1 n * q" by (erule dvdE)
   from p have "q \<noteq> 0" "poly (of_int_poly q) x = 0" by (auto simp: q poly_monom assms(2))
   moreover from this have "order 0 p = n + order 0 q" by (simp add: q order_mult)
   hence "order 0 q = 0" by (simp add: n_def)
   with \<open>q \<noteq> 0\<close> have "poly q 0 \<noteq> 0" by (simp add: order_root)
   ultimately show ?thesis using that[of q] by (auto simp: poly_0_coeff_0)
 qed
 
 lemma algebraic_of_real_iff [simp]:
    "algebraic (of_real x :: 'a :: {real_algebra_1,field_char_0}) \<longleftrightarrow> algebraic x"
 proof
   assume "algebraic (of_real x :: 'a)"
   then obtain p where "p \<noteq> 0" "poly (of_int_poly p) (of_real x :: 'a) = 0"
     by (erule algebraicE')
   hence "(of_int_poly p :: real poly) \<noteq> 0"
         "poly (of_int_poly p :: real poly) x = 0" by simp_all
   thus "algebraic x" by (intro algebraicI[of "of_int_poly p"]) simp_all
 next
   assume "algebraic x"
   then obtain p where "p \<noteq> 0" "poly (of_int_poly p) x = 0" by (erule algebraicE')
   hence "of_int_poly p \<noteq> (0 :: 'a poly)" "poly (of_int_poly p) (of_real x :: 'a) = 0"
     by simp_all
   thus "algebraic (of_real x)" by (intro algebraicI[of "of_int_poly p"]) simp_all
 qed
 
 
 subsection \<open>Main proof\<close>
 
 lemma lindemann_weierstrass_integral:
   fixes u :: complex and f :: "complex poly"
   defines "df \<equiv> \<lambda>n. (pderiv ^^ n) f"
   defines "m \<equiv> degree f"
   defines "I \<equiv> \<lambda>f u. exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) -
                        (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)"
   shows "((\<lambda>t. exp (u - t) * poly f t) has_contour_integral I f u) (linepath 0 u)"
 proof -
   note [derivative_intros] =
     exp_scaleR_has_vector_derivative_right vector_diff_chain_within
   let ?g = "\<lambda>t. 1 - t" and ?f = "\<lambda>t. -exp (t *\<^sub>R u)"
   have "((\<lambda>t. exp ((1 - t) *\<^sub>R u) * u) has_integral
           (?f \<circ> ?g) 1 - (?f \<circ> ?g) 0) {0..1}"
     by (rule fundamental_theorem_of_calculus)
        (auto intro!: derivative_eq_intros simp del: o_apply)
   hence aux_integral: "((\<lambda>t. exp (u - t *\<^sub>R u) * u) has_integral exp u - 1) {0..1}"
     by (simp add: algebra_simps)
 
   have "((\<lambda>t. exp (u - t *\<^sub>R u) * u * poly f (t *\<^sub>R u)) has_integral I f u) {0..1}"
     unfolding df_def m_def
   proof (induction "degree f" arbitrary: f)
     case 0
     then obtain c where c: "f = [:c:]" by (auto elim: degree_eq_zeroE)
     have "((\<lambda>t. c * (exp (u - t *\<^sub>R u) * u)) has_integral c * (exp u - 1)) {0..1}"
       using aux_integral by (rule has_integral_mult_right)
     with c show ?case by (simp add: algebra_simps I_def)
   next
     case (Suc m)
     define df where "df = (\<lambda>j. (pderiv ^^ j) f)"
     show ?case
     proof (rule integration_by_parts[OF bounded_bilinear_mult])
       fix t :: real assume "t \<in> {0..1}"
       have "((?f \<circ> ?g) has_vector_derivative exp (u - t *\<^sub>R u) * u) (at t)"
         by (auto intro!: derivative_eq_intros simp: algebra_simps simp del: o_apply)
       thus "((\<lambda>t. -exp (u - t *\<^sub>R u)) has_vector_derivative exp (u - t *\<^sub>R u) * u) (at t)"
         by (simp add: algebra_simps o_def)
     next
       fix t :: real assume "t \<in> {0..1}"
       have "(poly f \<circ> (\<lambda>t. t *\<^sub>R u) has_vector_derivative u * poly (pderiv f) (t *\<^sub>R u)) (at t)"
         by (rule field_vector_diff_chain_at) (auto intro!: derivative_eq_intros)
       thus "((\<lambda>t. poly f (t *\<^sub>R u)) has_vector_derivative u * poly (pderiv f) (t *\<^sub>R u)) (at t)"
         by (simp add: o_def)
     next
       from Suc(2) have m: "m = degree (pderiv f)" by (simp add: degree_pderiv)
       from Suc(1)[OF this] this
         have "((\<lambda>t. exp (u - t *\<^sub>R u) * u * poly (pderiv f) (t *\<^sub>R u)) has_integral
                 exp u * (\<Sum>j=0..m. poly (df (Suc j)) 0) - (\<Sum>j=0..m. poly (df (Suc j)) u)) {0..1}"
         by (simp add: df_def funpow_swap1 atMost_atLeast0 I_def)
       also have "(\<Sum>j=0..m. poly (df (Suc j)) 0) = (\<Sum>j=Suc 0..Suc m. poly (df j) 0)"
         by (rule sum.shift_bounds_cl_Suc_ivl [symmetric])
       also have "\<dots> = (\<Sum>j=0..Suc m. poly (df j) 0) - poly f 0"
         by (subst (2) sum.atLeast_Suc_atMost) (simp_all add: df_def)
       also have "(\<Sum>j=0..m. poly (df (Suc j)) u) = (\<Sum>j=Suc 0..Suc m. poly (df j) u)"
         by (rule sum.shift_bounds_cl_Suc_ivl [symmetric])
       also have "\<dots> = (\<Sum>j=0..Suc m. poly (df j) u) - poly f u"
         by (subst (2) sum.atLeast_Suc_atMost) (simp_all add: df_def)
       finally have "((\<lambda>t. - (exp (u - t *\<^sub>R u) * u * poly (pderiv f) (t *\<^sub>R u))) has_integral
                         -(exp u * ((\<Sum>j = 0..Suc m. poly (df j) 0) - poly f 0) -
                                   ((\<Sum>j = 0..Suc m. poly (df j) u) - poly f u))) {0..1}"
           (is "(_ has_integral ?I) _") by (rule has_integral_neg)
       also have "?I = - exp (u - 1 *\<^sub>R u) * poly f (1 *\<^sub>R u) -
                        - exp (u - 0 *\<^sub>R u) * poly f (0 *\<^sub>R u) - I f u"
         by (simp add: df_def algebra_simps Suc(2) atMost_atLeast0 I_def)
       finally show "((\<lambda>t. - exp (u - t *\<^sub>R u) * (u * poly (pderiv f) (t *\<^sub>R u)))
                         has_integral \<dots>) {0..1}" by (simp add: algebra_simps)
     qed (auto intro!: continuous_intros)
   qed
   thus ?thesis by (simp add: has_contour_integral_linepath algebra_simps)
 qed
 
 locale lindemann_weierstrass_aux =
   fixes f :: "complex poly"
 begin
 
 definition I :: "complex \<Rightarrow> complex" where
   "I u = exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) -
                        (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)"
 
 lemma lindemann_weierstrass_integral_bound:
   fixes u :: complex
   assumes "C \<ge> 0" "\<And>t. t \<in> closed_segment 0 u \<Longrightarrow> norm (poly f t) \<le> C"
   shows "norm (I u) \<le> norm u * exp (norm u) * C"
 proof -
   have "I u = contour_integral (linepath 0 u) (\<lambda>t. exp (u - t) * poly f t)"
     using contour_integral_unique[OF lindemann_weierstrass_integral[of u f]] unfolding I_def ..
   also have "norm \<dots> \<le> exp (norm u) * C * norm (u - 0)"
   proof (intro contour_integral_bound_linepath)
     fix t assume t: "t \<in> closed_segment 0 u"
     then obtain s where s: "s \<in> {0..1}" "t = s *\<^sub>R u" by (auto simp: closed_segment_def)
     hence "s * norm u \<le> 1 * norm u" by (intro mult_right_mono) simp_all
     with s have norm_t: "norm t \<le> norm u" by auto
 
     from s have "Re u - Re t = (1 - s) * Re u" by (simp add: algebra_simps)
     also have "\<dots> \<le> norm u"
     proof (cases "Re u \<ge> 0")
       case True
       with \<open>s \<in> {0..1}\<close> have "(1 - s) * Re u \<le> 1 * Re u" by (intro mult_right_mono) simp_all
       also have "Re u \<le> norm u" by (rule complex_Re_le_cmod)
       finally show ?thesis by simp
     next
       case False
       with \<open>s \<in> {0..1}\<close> have "(1 - s) * Re u \<le> 0" by (intro mult_nonneg_nonpos) simp_all
       also have "\<dots> \<le> norm u" by simp
       finally show ?thesis .
     qed
     finally have "exp (Re u - Re t) \<le> exp (norm u)" by simp
 
     hence "exp (Re u - Re t) * norm (poly f t) \<le> exp (norm u) * C"
       using assms t norm_t by (intro mult_mono) simp_all
     thus "norm (exp (u - t) * poly f t) \<le> exp (norm u) * C"
       by (simp add: norm_mult exp_diff norm_divide field_simps)
   qed (auto simp: intro!: mult_nonneg_nonneg contour_integrable_continuous_linepath
                           continuous_intros assms)
   finally show ?thesis by (simp add: mult_ac)
 qed
 
 end
 
 lemma poly_higher_pderiv_aux1:
   fixes c :: "'a :: idom"
   assumes "k < n"
   shows   "poly ((pderiv ^^ k) ([:-c, 1:] ^ n * p)) c = 0"
   using assms
 proof (induction k arbitrary: n p)
   case (Suc k n p)
   from Suc.prems obtain n' where n: "n = Suc n'" by (cases n) auto
   from Suc.prems n have "k < n'" by simp
   have "(pderiv ^^ Suc k) ([:- c, 1:] ^ n * p) =
           (pderiv ^^ k) ([:- c, 1:] ^ n * pderiv p + [:- c, 1:] ^ n' * smult (of_nat n) p)"
     by (simp only: funpow_Suc_right o_def pderiv_mult n pderiv_power_Suc,
         simp only: n [symmetric]) (simp add: pderiv_pCons mult_ac)
   also from Suc.prems \<open>k < n'\<close> have "poly \<dots> c = 0"
     by (simp add: higher_pderiv_add Suc.IH del: mult_smult_right)
   finally show ?case .
 qed simp_all
 
 lemma poly_higher_pderiv_aux1':
   fixes c :: "'a :: idom"
   assumes "k < n" "[:-c, 1:] ^ n dvd p"
   shows   "poly ((pderiv ^^ k) p) c = 0"
 proof -
   from assms(2) obtain q where "p = [:-c, 1:] ^ n * q" by (elim dvdE)
   also from assms(1) have "poly ((pderiv ^^ k) \<dots>) c = 0"
     by (rule poly_higher_pderiv_aux1)
   finally show ?thesis .
 qed
 
 lemma poly_higher_pderiv_aux2:
   fixes c :: "'a :: {idom, semiring_char_0}"
   shows   "poly ((pderiv ^^ n) ([:-c, 1:] ^ n * p)) c = fact n * poly p c"
 proof (induction n arbitrary: p)
   case (Suc n p)
   have "(pderiv ^^ Suc n) ([:- c, 1:] ^ Suc n * p) =
           (pderiv ^^ n) ([:- c, 1:] ^ Suc n * pderiv p) +
             (pderiv ^^ n) ([:- c, 1:] ^ n * smult (1 + of_nat n) p)"
     by (simp del: funpow.simps power_Suc add: funpow_Suc_right pderiv_mult
           pderiv_power_Suc higher_pderiv_add pderiv_pCons mult_ac)
   also have "[:- c, 1:] ^ Suc n * pderiv p = [:- c, 1:] ^ n * ([:-c, 1:] * pderiv p)"
     by (simp add: algebra_simps)
   finally show ?case by (simp add: Suc.IH del: mult_smult_right power_Suc)
 qed simp_all
 
 lemma poly_higher_pderiv_aux3:
   fixes c :: "'a :: {idom,semiring_char_0}"
   assumes "k \<ge> n"
   shows   "\<exists>q. poly ((pderiv ^^ k) ([:-c, 1:] ^ n * p)) c = fact n * poly q c"
   using assms
 proof (induction k arbitrary: n p)
   case (Suc k n p)
   show ?case
   proof (cases n)
     fix n' assume n: "n = Suc n'"
     have "poly ((pderiv ^^ Suc k) ([:-c, 1:] ^ n * p)) c =
             poly ((pderiv ^^ k) ([:- c, 1:] ^ n * pderiv p)) c +
               of_nat n * poly ((pderiv ^^ k) ([:-c, 1:] ^ n' * p)) c"
       by (simp del: funpow.simps power_Suc add: funpow_Suc_right pderiv_power_Suc
             pderiv_mult n pderiv_pCons higher_pderiv_add mult_ac higher_pderiv_smult)
     also have "\<exists>q1. poly ((pderiv ^^ k) ([:-c, 1:] ^ n * pderiv p)) c = fact n * poly q1 c"
       using Suc.prems Suc.IH[of n "pderiv p"]
       by (cases "n' = k") (auto simp: n poly_higher_pderiv_aux1 simp del: power_Suc of_nat_Suc
                                 intro: exI[of _ "0::'a poly"])
     then obtain q1
       where "poly ((pderiv ^^ k) ([:-c, 1:] ^ n * pderiv p)) c = fact n * poly q1 c" ..
     also from Suc.IH[of n' p] Suc.prems obtain q2
       where "poly ((pderiv ^^ k) ([:-c, 1:] ^ n' * p)) c = fact n' * poly q2 c"
       by (auto simp: n)
     finally show ?case by (auto intro!: exI[of _ "q1 + q2"] simp: n algebra_simps)
   qed auto
 qed auto
 
 lemma poly_higher_pderiv_aux3':
   fixes c :: "'a :: {idom, semiring_char_0}"
   assumes "k \<ge> n" "[:-c, 1:] ^ n dvd p"
   shows   "fact n dvd poly ((pderiv ^^ k) p) c"
 proof -
   from assms(2) obtain q where "p = [:-c, 1:] ^ n * q" by (elim dvdE)
   with poly_higher_pderiv_aux3[OF assms(1), of c q] show ?thesis by auto
 qed
 
 lemma e_transcendental_aux_bound:
   obtains C where "C \<ge> 0"
     "\<And>x. x \<in> closed_segment 0 (of_nat n) \<Longrightarrow>
         norm (\<Prod>k\<in>{1..n}. (x - of_nat k :: complex)) \<le> C"
 proof -
   let ?f = "\<lambda>x. (\<Prod>k\<in>{1..n}. (x - of_nat k))"
   define C where "C = max 0 (Sup (cmod ` ?f ` closed_segment 0 (of_nat n)))"
   have "C \<ge> 0" by (simp add: C_def)
   moreover {
     fix x :: complex assume "x \<in> closed_segment 0 (of_nat n)"
     hence "cmod (?f x) \<le> Sup ((cmod \<circ> ?f) ` closed_segment 0 (of_nat n))"
       by (intro cSup_upper bounded_imp_bdd_above compact_imp_bounded compact_continuous_image)
          (auto intro!: continuous_intros)
     also have "\<dots> \<le> C" by (simp add: C_def image_comp)
     finally have "cmod (?f x) \<le> C" .
   }
   ultimately show ?thesis by (rule that)
 qed
 
 
 theorem e_transcendental_complex: "\<not> algebraic (exp 1 :: complex)"
 proof
   assume "algebraic (exp 1 :: complex)"
   then obtain q :: "int poly"
     where q: "q \<noteq> 0" "coeff q 0 \<noteq> 0" "poly (of_int_poly q) (exp 1 :: complex) = 0"
       by (elim algebraicE'_nonzero) simp_all
 
   define n :: nat where "n = degree q"
   from q have [simp]: "n \<noteq> 0" by (intro notI) (auto simp: n_def elim!: degree_eq_zeroE)
   define qmax where "qmax = Max (insert 0 (abs ` set (coeffs q)))"
   have qmax_nonneg [simp]: "qmax \<ge> 0" by (simp add: qmax_def)
   have qmax: "\<bar>coeff q k\<bar> \<le> qmax" for k
     by (cases "k \<le> degree q")
        (auto simp: qmax_def coeff_eq_0 coeffs_def simp del: upt_Suc intro: Max.coboundedI)
   obtain C where C: "C \<ge> 0"
     "\<And>x. x \<in> closed_segment 0 (of_nat n) \<Longrightarrow> norm (\<Prod>k\<in>{1..n}. (x - of_nat k :: complex)) \<le> C"
     by (erule e_transcendental_aux_bound)
   define E where "E = (1 + real n) * real_of_int qmax * real n * exp (real n) / real n"
   define F where "F = real n * C"
 
   have ineq: "fact (p - 1) \<le> E * F ^ p" if p: "prime p" "p > n" "p > abs (coeff q 0)" for p
   proof -
     from p(1) have p_pos: "p > 0" by (simp add: prime_gt_0_nat)
     define f :: "int poly"
       where "f = monom 1 (p - 1) * (\<Prod>k\<in>{1..n}. [:-of_nat k, 1:] ^ p)"
     have poly_f: "poly (of_int_poly f) x = x ^ (p - 1) * (\<Prod>k\<in>{1..n}. (x - of_nat k)) ^ p"
       for x :: complex by (simp add: f_def poly_prod poly_monom prod_power_distrib)
     define m :: nat where "m = degree f"
     from p_pos have m: "m = (n + 1) * p - 1"
       by (simp add: m_def f_def degree_mult_eq degree_monom_eq degree_prod_sum_eq degree_linear_power)
 
     define M :: int where "M = (- 1) ^ (n * p) * fact n ^ p"
     with p have p_not_dvd_M: "\<not>int p dvd M"
       by (auto simp: M_def prime_elem_int_not_dvd_neg1_power prime_dvd_power_iff
             prime_gt_0_nat prime_dvd_fact_iff_int prime_dvd_mult_iff)
 
     interpret lindemann_weierstrass_aux "of_int_poly f" .
     define J :: complex where "J = (\<Sum>k\<le>n. of_int (coeff q k) * I (of_nat k))"
     define idxs where "idxs = ({..n}\<times>{..m}) - {(0, p - 1)}"
 
     hence "J = (\<Sum>k\<le>n. of_int (coeff q k) * exp 1 ^ k) * (\<Sum>n\<le>m. of_int (poly ((pderiv ^^ n) f) 0)) -
                  of_int (\<Sum>k\<le>n. \<Sum>n\<le>m. coeff q k * poly ((pderiv ^^ n) f) (int k))"
       by (simp add: J_def I_def algebra_simps sum_subtractf sum_distrib_left m_def
                     exp_of_nat_mult [symmetric])
     also have "(\<Sum>k\<le>n. of_int (coeff q k) * exp 1 ^ k) = poly (of_int_poly q) (exp 1 :: complex)"
       by (simp add: poly_altdef n_def)
     also have "\<dots> = 0" by fact
     finally have "J = of_int (-(\<Sum>(k,n)\<in>{..n}\<times>{..m}. coeff q k * poly ((pderiv ^^ n) f) (int k)))"
       by (simp add: sum.cartesian_product)
     also have "{..n}\<times>{..m} = insert (0, p - 1) idxs" by (auto simp: m idxs_def)
     also have "-(\<Sum>(k,n)\<in>\<dots>. coeff q k * poly ((pderiv ^^ n) f) (int k)) =
        - (coeff q 0 * poly ((pderiv ^^ (p - 1)) f) 0) -
          (\<Sum>(k, n)\<in>idxs. coeff q k * poly ((pderiv ^^ n) f) (of_nat k))"
       by (subst sum.insert) (simp_all add: idxs_def)
     also have "coeff q 0 * poly ((pderiv ^^ (p - 1)) f) 0 = coeff q 0 * M * fact (p - 1)"
     proof -
       have "f = [:-0, 1:] ^ (p - 1) * (\<Prod>k = 1..n. [:- of_nat k, 1:] ^ p)"
         by (simp add: f_def monom_altdef)
       also have "poly ((pderiv ^^ (p - 1)) \<dots>) 0 =
                    fact (p - 1) * poly (\<Prod>k = 1..n. [:- of_nat k, 1:] ^ p) 0"
         by (rule poly_higher_pderiv_aux2)
       also have "poly (\<Prod>k = 1..n. [:- of_nat k :: int, 1:] ^ p) 0 = (-1)^(n*p) * fact n ^ p"
         by (induction n) (simp_all add: prod.nat_ivl_Suc' power_mult_distrib mult_ac
                             power_minus' power_add del: of_nat_Suc)
       finally show ?thesis by (simp add: mult_ac M_def)
     qed
     also have "\<exists>N. (\<Sum>(k, n)\<in>idxs. coeff q k * poly ((pderiv ^^ n) f) (int k)) = fact p * N"
     proof -
       have "\<forall>(k, n)\<in>idxs. fact p dvd poly ((pderiv ^^ n) f) (of_nat k)"
       proof clarify
         fix k j assume idxs: "(k, j) \<in> idxs"
         then consider "k = 0" "j < p - 1" | "k = 0" "j > p - 1" | "k \<noteq> 0" "j < p" | "k \<noteq> 0" "j \<ge> p"
           by (fastforce simp: idxs_def)
         thus "fact p dvd poly ((pderiv ^^ j) f) (of_nat k)"
         proof cases
           case 1
           thus ?thesis
             by (simp add: f_def poly_higher_pderiv_aux1' monom_altdef)
         next
           case 2
           thus ?thesis
             by (simp add: f_def poly_higher_pderiv_aux3' monom_altdef fact_dvd_poly_higher_pderiv_aux')
         next
           case 3
           thus ?thesis unfolding f_def
             by (subst poly_higher_pderiv_aux1'[of _ p])
                (insert idxs, auto simp: idxs_def intro!: dvd_mult)
         next
           case 4
           thus ?thesis unfolding f_def
             by (intro poly_higher_pderiv_aux3') (insert idxs, auto intro!: dvd_mult simp: idxs_def)
         qed
       qed
       hence "fact p dvd (\<Sum>(k, n)\<in>idxs. coeff q k * poly ((pderiv ^^ n) f) (int k))"
         by (auto intro!: dvd_sum dvd_mult simp del: of_int_fact)
       thus ?thesis by (blast elim: dvdE)
     qed
     then guess N .. note N = this
     also from p have "- (coeff q 0 * M * fact (p - 1)) - fact p * N =
                         - fact (p - 1) * (coeff q 0 * M + p * N)"
       by (subst fact_reduce[of p]) (simp_all add: algebra_simps)
     finally have J: "J = -of_int (fact (p - 1) * (coeff q 0 * M + p * N))" by simp
 
     from p q(2) have "\<not>p dvd coeff q 0 * M + p * N"
       by (auto simp: dvd_add_left_iff p_not_dvd_M prime_dvd_fact_iff_int prime_dvd_mult_iff
                dest: dvd_imp_le_int)
     hence "coeff q 0 * M + p * N \<noteq> 0" by (intro notI) simp_all
     hence "abs (coeff q 0 * M + p * N) \<ge> 1" by simp
     hence "norm (of_int (coeff q 0 * M + p * N) :: complex) \<ge> 1" by (simp only: norm_of_int)
     hence "fact (p - 1) * \<dots> \<ge> fact (p - 1) * 1" by (intro mult_left_mono) simp_all
     hence J_lower: "norm J \<ge> fact (p - 1)" unfolding J norm_minus_cancel of_int_mult of_int_fact
       by (simp add: norm_mult)
 
     have "norm J \<le> (\<Sum>k\<le>n. norm (of_int (coeff q k) * I (of_nat k)))"
       unfolding J_def by (rule norm_sum)
     also have "\<dots> \<le> (\<Sum>k\<le>n. of_int qmax * (real n * exp (real n) * real n ^ (p - 1) * C ^ p))"
     proof (intro sum_mono)
       fix k assume k: "k \<in> {..n}"
       have "n > 0" by (rule ccontr) simp
       {
         fix x :: complex assume x: "x \<in> closed_segment 0 (of_nat k)"
         then obtain t where t: "t \<ge> 0" "t \<le> 1" "x = of_real t * of_nat k"
           by (auto simp: closed_segment_def scaleR_conv_of_real)
         hence "norm x = t * real k" by (simp add: norm_mult)
         also from \<open>t \<le> 1\<close> k have *: "\<dots> \<le> 1 * real n" by (intro mult_mono) simp_all
         finally have x': "norm x \<le> real n" by simp
         from t \<open>n > 0\<close> * have x'': "x \<in> closed_segment 0 (of_nat n)"
           by (auto simp: closed_segment_def scaleR_conv_of_real field_simps
                    intro!: exI[of _ "t * real k / real n"] )
         have "norm (poly (of_int_poly f) x) =
                 norm x ^ (p - 1) * cmod (\<Prod>i = 1..n. x - i) ^ p"
           by (simp add: poly_f norm_mult norm_power)
         also from x x' x'' have "\<dots> \<le> of_nat n ^ (p - 1) * C ^ p"
           by (intro mult_mono C power_mono) simp_all
         finally have "norm (poly (of_int_poly f) x) \<le> real n ^ (p - 1) * C ^ p" .
       } note A = this
 
       have "norm (I (of_nat k)) \<le>
                       cmod (of_nat k) * exp (cmod (of_nat k)) * (of_nat n ^ (p - 1) * C ^ p)"
         by (intro lindemann_weierstrass_integral_bound[OF _ A]
               C mult_nonneg_nonneg zero_le_power) auto
       also have "\<dots> \<le> cmod (of_nat n) * exp (cmod (of_nat n)) * (of_nat n ^ (p - 1) * C ^ p)"
         using k by (intro mult_mono zero_le_power mult_nonneg_nonneg C) simp_all
       finally show "cmod (of_int (coeff q k) * I (of_nat k)) \<le>
                       of_int qmax * (real n * exp (real n) * real n ^ (p - 1) * C ^ p)"
         unfolding norm_mult
         by (intro mult_mono) (simp_all add: qmax of_int_abs [symmetric] del: of_int_abs)
     qed
     also have "\<dots> = E * F ^ p" using p_pos
       by (simp add: power_diff power_mult_distrib E_def F_def)
     finally show "fact (p - 1) \<le> E * F ^ p" using J_lower by linarith
   qed
 
   have "(\<lambda>n. E * F * F ^ (n - 1) / fact (n - 1)) \<longlonglongrightarrow> 0" (is ?P)
     by (intro filterlim_compose[OF power_over_fact_tendsto_0' filterlim_minus_nat_at_top])
   also have "?P \<longleftrightarrow> (\<lambda>n. E * F ^ n / fact (n - 1)) \<longlonglongrightarrow> 0"
     by (intro filterlim_cong refl eventually_mono[OF eventually_gt_at_top[of "0::nat"]])
        (auto simp: power_Suc [symmetric] simp del: power_Suc)
   finally have "eventually (\<lambda>n. E * F ^ n / fact (n - 1) < 1) at_top"
     by (rule order_tendstoD) simp_all
   hence "eventually (\<lambda>n. E * F ^ n < fact (n - 1)) at_top" by eventually_elim simp
   then obtain P where P: "\<And>n. n \<ge> P \<Longrightarrow> E * F ^ n < fact (n - 1)"
     by (auto simp: eventually_at_top_linorder)
 
   have "\<exists>p. prime p \<and> p > Max {nat (abs (coeff q 0)), n, P}" by (rule bigger_prime)
   then obtain p where "prime p" "p > Max {nat (abs (coeff q 0)), n, P}" by blast
   hence "int p > abs (coeff q 0)" "p > n" "p \<ge> P" by auto
   with ineq[of p] \<open>prime p\<close> have "fact (p - 1) \<le> E * F ^ p" by simp
   moreover from \<open>p \<ge> P\<close> have "fact (p - 1) > E * F ^ p" by (rule P)
   ultimately show False by linarith
 qed
 
 corollary e_transcendental_real: "\<not> algebraic (exp 1 :: real)"
 proof -
   have "\<not>algebraic (exp 1 :: complex)" by (rule e_transcendental_complex)
   also have "(exp 1 :: complex) = of_real (exp 1)" using exp_of_real[of 1] by simp
   also have "algebraic \<dots> \<longleftrightarrow> algebraic (exp 1 :: real)" by simp
   finally show ?thesis .
 qed
 
 end
diff --git a/thys/E_Transcendental/ROOT b/thys/E_Transcendental/ROOT
--- a/thys/E_Transcendental/ROOT
+++ b/thys/E_Transcendental/ROOT
@@ -1,11 +1,11 @@
 chapter AFP
 
-session E_Transcendental (AFP) = "HOL-Analysis" +
+session E_Transcendental (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 600]
   sessions
     "HOL-Number_Theory"
   theories
     E_Transcendental
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/Error_Function/Error_Function.thy b/thys/Error_Function/Error_Function.thy
--- a/thys/Error_Function/Error_Function.thy
+++ b/thys/Error_Function/Error_Function.thy
@@ -1,655 +1,655 @@
 (*
   File:     Error_Function.thy
   Author:   Manuel Eberl, TU München
 
   The complex and real error function (most results are on the real error function though)
 *)
 section \<open>The complex and real error function\<close>
 theory Error_Function
-  imports "HOL-Analysis.Analysis" "HOL-Library.Landau_Symbols"
+  imports "HOL-Complex_Analysis.Complex_Analysis" "HOL-Library.Landau_Symbols"
 begin
 
 subsection \<open>Auxiliary Facts\<close>
 
 lemma tendsto_sandwich_mono:
   assumes "(\<lambda>n. f (real n)) \<longlonglongrightarrow> (c::real)"
   assumes "eventually (\<lambda>x. \<forall>y z. x \<le> y \<and> y \<le> z \<longrightarrow> f y \<le> f z) at_top"
   shows   "(f \<longlongrightarrow> c) at_top"
 proof (rule tendsto_sandwich)
   from assms(2) obtain C where C: "\<And>x y. C \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
     by (auto simp: eventually_at_top_linorder)
   show "eventually (\<lambda>x. f (real (nat \<lfloor>x\<rfloor>)) \<le> f x) at_top"
     using eventually_gt_at_top[of "0::real"] eventually_gt_at_top[of "C+1::real"]
     by eventually_elim (rule C, linarith+)
   show "eventually (\<lambda>x. f (real (Suc (nat \<lfloor>x\<rfloor>))) \<ge> f x) at_top"
     using eventually_gt_at_top[of "0::real"] eventually_gt_at_top[of "C+1::real"]
     by eventually_elim (rule C, linarith+)
 qed (auto intro!: filterlim_compose[OF assms(1)]
                   filterlim_compose[OF filterlim_nat_sequentially]
                   filterlim_compose[OF filterlim_Suc] filterlim_floor_sequentially
           simp del: of_nat_Suc)
 
 lemma tendsto_sandwich_antimono:
   assumes "(\<lambda>n. f (real n)) \<longlonglongrightarrow> (c::real)"
   assumes "eventually (\<lambda>x. \<forall>y z. x \<le> y \<and> y \<le> z \<longrightarrow> f y \<ge> f z) at_top"
   shows   "(f \<longlongrightarrow> c) at_top"
 proof (rule tendsto_sandwich)
   from assms(2) obtain C where C: "\<And>x y. C \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
     by (auto simp: eventually_at_top_linorder)
   show "eventually (\<lambda>x. f (real (nat \<lfloor>x\<rfloor>)) \<ge> f x) at_top"
     using eventually_gt_at_top[of "0::real"] eventually_gt_at_top[of "C+1::real"]
     by eventually_elim (rule C, linarith+)
   show "eventually (\<lambda>x. f (real (Suc (nat \<lfloor>x\<rfloor>))) \<le> f x) at_top"
     using eventually_gt_at_top[of "0::real"] eventually_gt_at_top[of "C+1::real"]
     by eventually_elim (rule C, linarith+)
 qed (auto intro!: filterlim_compose[OF assms(1)]
                   filterlim_compose[OF filterlim_nat_sequentially]
                   filterlim_compose[OF filterlim_Suc] filterlim_floor_sequentially
           simp del: of_nat_Suc)
 
 lemma has_bochner_integral_completion [intro]:
   fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
   shows "has_bochner_integral M f I \<Longrightarrow> has_bochner_integral (completion M) f I"
   by (auto simp: has_bochner_integral_iff integrable_completion integral_completion 
                  borel_measurable_integrable)
 
 lemma has_bochner_integral_imp_has_integral:
   "has_bochner_integral lebesgue (\<lambda>x. indicator S x *\<^sub>R f x) I \<Longrightarrow> 
      (f has_integral (I :: 'b :: euclidean_space)) S"
   using has_integral_set_lebesgue[of S f] 
   by (simp add: has_bochner_integral_iff set_integrable_def set_lebesgue_integral_def) 
     
 lemma has_bochner_integral_imp_has_integral':
   "has_bochner_integral lborel (\<lambda>x. indicator S x *\<^sub>R f x) I \<Longrightarrow> 
      (f has_integral (I :: 'b :: euclidean_space)) S"
   by (intro has_bochner_integral_imp_has_integral has_bochner_integral_completion)
 
 lemma has_bochner_integral_erf_aux:
   "has_bochner_integral lborel (\<lambda>x. indicator {0..} x *\<^sub>R exp (- x\<^sup>2)) (sqrt pi / 2)"
 proof -
   let ?pI = "\<lambda>f. (\<integral>\<^sup>+s. f (s::real) * indicator {0..} s \<partial>lborel)"
   let ?gauss = "\<lambda>x. exp (- x\<^sup>2)"
   let ?I = "indicator {0<..} :: real \<Rightarrow> real"
   let ?ff= "\<lambda>x s. x * exp (- x\<^sup>2 * (1 + s\<^sup>2)) :: real"
   have *: "?pI ?gauss = (\<integral>\<^sup>+x. ?gauss x * ?I x \<partial>lborel)"
     by (intro nn_integral_cong_AE AE_I[where N="{0}"]) (auto split: split_indicator)
 
   have "?pI ?gauss * ?pI ?gauss = (\<integral>\<^sup>+x. \<integral>\<^sup>+s. ?gauss x * ?gauss s * ?I s * ?I x \<partial>lborel \<partial>lborel)"
     by (auto simp: nn_integral_cmult[symmetric] nn_integral_multc[symmetric] * 
                    ennreal_mult[symmetric] intro!: nn_integral_cong split: split_indicator)
   also have "\<dots> = (\<integral>\<^sup>+x. \<integral>\<^sup>+s. ?ff x s * ?I s * ?I x \<partial>lborel \<partial>lborel)"
   proof (rule nn_integral_cong, cases)
     fix x :: real assume "x \<noteq> 0"
     then show "(\<integral>\<^sup>+s. ?gauss x * ?gauss s * ?I s * ?I x \<partial>lborel) =
                  (\<integral>\<^sup>+s. ?ff x s * ?I s * ?I x \<partial>lborel)"
       by (subst nn_integral_real_affine[where t="0" and c="x"])
          (auto simp: mult_exp_exp nn_integral_cmult[symmetric] field_simps zero_less_mult_iff ennreal_mult[symmetric]
                intro!: nn_integral_cong split: split_indicator)
   qed simp
   also have "... = \<integral>\<^sup>+s. \<integral>\<^sup>+x. ?ff x s * ?I s * ?I x \<partial>lborel \<partial>lborel"
     by (rule lborel_pair.Fubini'[symmetric]) auto
   also have "... = ?pI (\<lambda>s. ?pI (\<lambda>x. ?ff x s))"
     by (rule nn_integral_cong_AE)
        (auto intro!: nn_integral_cong_AE AE_I[where N="{0}"] split: split_indicator_asm)
   also have "\<dots> = ?pI (\<lambda>s. ennreal (1 / (2 * (1 + s\<^sup>2))))"
   proof (intro nn_integral_cong ennreal_mult_right_cong)
     fix s :: real show "?pI (\<lambda>x. ?ff x s) = ennreal (1 / (2 * (1 + s\<^sup>2)))"
     proof (subst nn_integral_FTC_atLeast)
       have "((\<lambda>a. - (exp (- ((1 + s\<^sup>2) * a\<^sup>2)) / (2 + 2 * s\<^sup>2))) \<longlongrightarrow> (- (0 / (2 + 2 * s\<^sup>2)))) at_top"
         by (intro tendsto_intros filterlim_compose[OF exp_at_bot]
                   filterlim_compose[OF filterlim_uminus_at_bot_at_top]
                   filterlim_tendsto_pos_mult_at_top)
            (auto intro!: filterlim_at_top_mult_at_top[OF filterlim_ident filterlim_ident]
                  simp: add_pos_nonneg  power2_eq_square add_nonneg_eq_0_iff)
       then show "((\<lambda>a. - (exp (- a\<^sup>2 - s\<^sup>2 * a\<^sup>2) / (2 + 2 * s\<^sup>2))) \<longlongrightarrow> 0) at_top"
         by (simp add: field_simps)
     qed (auto intro!: derivative_eq_intros simp: field_simps add_nonneg_eq_0_iff)
   qed
   also have "... = ennreal (pi / 4)"
   proof (subst nn_integral_FTC_atLeast)
     show "((\<lambda>a. arctan a / 2) \<longlongrightarrow> (pi / 2) / 2 ) at_top"
       by (intro tendsto_intros) (simp_all add: tendsto_arctan_at_top)
   qed (auto intro!: derivative_eq_intros simp: add_nonneg_eq_0_iff field_simps power2_eq_square)
   finally have "?pI ?gauss^2 = pi / 4"
     by (simp add: power2_eq_square)
   then have "?pI ?gauss = sqrt (pi / 4)"
     using power_eq_iff_eq_base[of 2 "enn2real (?pI ?gauss)" "sqrt (pi / 4)"]
     by (cases "?pI ?gauss") (auto simp: power2_eq_square ennreal_mult[symmetric] ennreal_top_mult)
   also have "?pI ?gauss = (\<integral>\<^sup>+x. indicator {0..} x *\<^sub>R exp (- x\<^sup>2) \<partial>lborel)"
     by (intro nn_integral_cong) (simp split: split_indicator)
   also have "sqrt (pi / 4) = sqrt pi / 2"
     by (simp add: real_sqrt_divide)
   finally show ?thesis
     by (rule has_bochner_integral_nn_integral[rotated 3])
        auto
 qed
 
 lemma has_integral_erf_aux: "((\<lambda>t::real. exp (-t\<^sup>2)) has_integral (sqrt pi / 2)) {0..}"
   by (intro has_bochner_integral_imp_has_integral' has_bochner_integral_erf_aux)
 
 lemma contour_integrable_on_linepath_neg_exp_squared [simp, intro]:
   "(\<lambda>t. exp (-(t^2))) contour_integrable_on linepath 0 z"
   by (auto intro!: contour_integrable_continuous_linepath continuous_intros)
 
 lemma holomorphic_on_chain:
   "g holomorphic_on t \<Longrightarrow> f holomorphic_on s \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> 
     (\<lambda>x. g (f x)) holomorphic_on s"
   using holomorphic_on_compose_gen[of f s g t] by (simp add: o_def)
     
 lemma holomorphic_on_chain_UNIV:
   "g holomorphic_on UNIV \<Longrightarrow> f holomorphic_on s\<Longrightarrow> 
     (\<lambda>x. g (f x)) holomorphic_on s"
   using holomorphic_on_compose_gen[of f s g UNIV] by (simp add: o_def)
 
 lemmas holomorphic_on_exp' [holomorphic_intros] = 
   holomorphic_on_exp [THEN holomorphic_on_chain_UNIV]
 
 lemma leibniz_rule_field_derivative_real:
   fixes f::"'a::{real_normed_field, banach} \<Rightarrow> real \<Rightarrow> 'a"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> {a..b} \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on {a..b}"
   assumes cont_fx: "continuous_on (U \<times> {a..b}) (\<lambda>(x, t). fx x t)"
   assumes U: "x0 \<in> U" "convex U"
   shows "((\<lambda>x. integral {a..b} (f x)) has_field_derivative integral {a..b} (fx x0)) (at x0 within U)"    
   using leibniz_rule_field_derivative[of U a b f fx x0] assms by simp
 
 lemma has_vector_derivative_linepath_within [derivative_intros]:
   assumes [derivative_intros]: 
     "(f has_vector_derivative f') (at x within S)" "(g has_vector_derivative g') (at x within S)"
     "(h has_real_derivative h') (at x within S)"
   shows "((\<lambda>x. linepath (f x) (g x) (h x)) has_vector_derivative 
            (1 - h x) *\<^sub>R f' + h x *\<^sub>R g' - h' *\<^sub>R (f x - g x)) (at x within S)"
   unfolding linepath_def [abs_def]
   by (auto intro!: derivative_eq_intros simp: field_simps scaleR_diff_right)
     
 lemma has_field_derivative_linepath_within [derivative_intros]:
   assumes [derivative_intros]: 
     "(f has_field_derivative f') (at x within S)" "(g has_field_derivative g') (at x within S)"
     "(h has_real_derivative h') (at x within S)"
   shows "((\<lambda>x. linepath (f x) (g x) (h x)) has_field_derivative 
            (1 - h x) *\<^sub>R f' + h x *\<^sub>R g' - h' *\<^sub>R (f x - g x)) (at x within S)"
   unfolding linepath_def [abs_def]
   by (auto intro!: derivative_eq_intros simp: field_simps scaleR_diff_right)
 
 lemma continuous_on_linepath' [continuous_intros]:
   assumes [continuous_intros]: "continuous_on A f" "continuous_on A g" "continuous_on A h"
   shows   "continuous_on A (\<lambda>x. linepath (f x) (g x) (h x))"
   using assms unfolding linepath_def by (auto intro!: continuous_intros)
     
 lemma contour_integral_has_field_derivative:
   assumes A: "open A" "convex A" "a \<in> A" "z \<in> A"
   assumes integrable: "\<And>z. z \<in> A \<Longrightarrow> f contour_integrable_on linepath a z"
   assumes holo: "f holomorphic_on A"
   shows   "((\<lambda>z. contour_integral (linepath a z) f) has_field_derivative f z) (at z within B)"
 proof -
   have "(f has_field_derivative deriv f z) (at z)" if "z \<in> A" for z
     using that assms by (auto intro!: holomorphic_derivI)
   note [derivative_intros] = DERIV_chain2[OF this]
   note [continuous_intros] = 
     continuous_on_compose2[OF holomorphic_on_imp_continuous_on [OF holo]]
     continuous_on_compose2[OF holomorphic_on_imp_continuous_on [OF holomorphic_deriv[OF holo]]]
   have [derivative_intros]:
       "((\<lambda>x. linepath a x t) has_field_derivative of_real t) (at x within A)" for t x
     by (auto simp: linepath_def scaleR_conv_of_real intro!: derivative_eq_intros)
  
   have *: "linepath a b t \<in> A" if "a \<in> A" "b \<in> A" "t \<in> {0..1}" for a b t
     using that linepath_in_convex_hull[of a A b t] \<open>convex A\<close> by (simp add: hull_same)
     
   have "((\<lambda>z. integral {0..1} (\<lambda>x. f (linepath a z x)) * (z - a)) has_field_derivative 
       integral {0..1} (\<lambda>t. deriv f (linepath a z t) * of_real t) * (z - a) +
       integral {0..1} (\<lambda>x. f (linepath a z x))) (at z within A)" 
       (is "(_ has_field_derivative ?I) _")
     by (rule derivative_eq_intros leibniz_rule_field_derivative_real)+
        (insert assms,
         auto intro!: derivative_eq_intros leibniz_rule_field_derivative_real
                      integrable_continuous_real continuous_intros 
              simp:   split_beta scaleR_conv_of_real *)
   also have "(\<lambda>z. integral {0..1} (\<lambda>x. f (linepath a z x)) * (z - a)) = 
                (\<lambda>z. contour_integral (linepath a z) f)"
     by (simp add: contour_integral_integral)
   also have "?I = integral {0..1} (\<lambda>x. deriv f (linepath a z x) * of_real x * (z - a) + 
                      f (linepath a z x))" (is "_ = integral _ ?g")
     by (subst integral_mult_left [symmetric], subst integral_add [symmetric])
        (insert assms, auto intro!: integrable_continuous_real continuous_intros simp: *)
   also have "(?g has_integral of_real 1 * f (linepath a z 1) - of_real 0 * f (linepath a z 0)) {0..1}"
     using * A
     by (intro fundamental_theorem_of_calculus)
        (auto intro!: derivative_eq_intros has_vector_derivative_real_field 
              simp: linepath_def scaleR_conv_of_real)
   hence "integral {0..1} ?g = f (linepath a z 1)" by (simp add: has_integral_iff)
   also have "linepath a z 1 = z" by (simp add: linepath_def)
   also from \<open>z \<in> A\<close> and \<open>open A\<close> have "at z within A = at z" by (rule at_within_open)
   finally show ?thesis by (rule DERIV_subset) simp_all
 qed
 
 
 subsection \<open>Definition of the error function\<close>
 
 definition erf_coeffs :: "nat \<Rightarrow> real" where
   "erf_coeffs n = 
      (if odd n then 2 / sqrt pi * (-1) ^ (n div 2) / (of_nat n * fact (n div 2)) 
       else 0)"
 
 lemma summable_erf:
   fixes z :: "'a :: {real_normed_div_algebra, banach}"
   shows "summable (\<lambda>n. of_real (erf_coeffs n) * z ^ n)"
 proof -
   define b where "b = (\<lambda>n. 2 / sqrt pi * (if odd n then inverse (fact (n div 2)) else 0))"
   show ?thesis
   proof (rule summable_comparison_test[OF exI[of _ 1]], clarify)
     fix n :: nat assume n: "n \<ge> 1"
     hence "norm (of_real (erf_coeffs n) * z ^ n) \<le> b n * norm z ^ n"
       unfolding norm_mult norm_power erf_coeffs_def b_def
       by (intro mult_right_mono) (auto simp: field_simps norm_divide abs_mult)
     thus "norm (of_real (erf_coeffs n) * z ^ n) \<le> b n * norm z ^ n"
       by (simp add: mult_ac)
   next
     have "summable (\<lambda>n. (norm z * 2 / sqrt pi) * (inverse (fact n) * norm z ^ (2*n)))" 
       (is "summable ?c") unfolding power_mult by (intro summable_mult summable_exp)
     also have "?c = (\<lambda>n. b (2*n+1) * norm z ^ (2*n+1))"
       unfolding b_def by (auto simp: fun_eq_iff b_def)
     also have "summable \<dots> \<longleftrightarrow> summable (\<lambda>n. b n * norm z ^ n)"
       using summable_mono_reindex [of "\<lambda>n. 2*n+1"]
       by (intro summable_mono_reindex [of "\<lambda>n. 2*n+1"]) 
          (auto elim!: oddE simp: strict_mono_def b_def)
     finally show \<dots> .
   qed
 qed
   
 definition erf :: "('a :: {real_normed_field, banach}) \<Rightarrow> 'a" where
   "erf z = (\<Sum>n. of_real (erf_coeffs n) * z ^ n)"
 
 lemma erf_converges: "(\<lambda>n. of_real (erf_coeffs n) * z ^ n) sums erf z"
   using summable_erf by (simp add: sums_iff erf_def)
 
 lemma erf_0 [simp]: "erf 0 = 0"
   unfolding erf_def powser_zero by (simp add: erf_coeffs_def)
 
 lemma erf_minus [simp]: "erf (-z) = - erf z"
   unfolding erf_def
   by (subst suminf_minus [OF summable_erf, symmetric], rule suminf_cong)
      (simp_all add: erf_coeffs_def)
 
 lemma erf_of_real [simp]: "erf (of_real x) = of_real (erf x)"
   unfolding erf_def using summable_erf[of x]
   by (subst suminf_of_real) (simp_all add: summable_erf)
     
 lemma of_real_erf_numeral [simp]: "of_real (erf (numeral n)) = erf (numeral n)"
   by (simp only: erf_of_real [symmetric] of_real_numeral)
 
 lemma of_real_erf_1 [simp]: "of_real (erf 1) = erf 1"
   by (simp only: erf_of_real [symmetric] of_real_1)
 
 
 lemma erf_has_field_derivative:
   "(erf has_field_derivative of_real (2 / sqrt pi) * exp (-(z^2))) (at z within A)"
 proof -
     define a' where "a' = (\<lambda>n. 2 / sqrt pi *
      (if even n then (- 1) ^ (n div 2) / fact (n div 2) else 0))"
 
   have "(erf has_field_derivative
            (\<Sum>n. diffs (\<lambda>n. of_real (erf_coeffs n)) n * z ^ n)) (at z)"
     using summable_erf unfolding erf_def by (rule termdiffs_strong_converges_everywhere)
   also have "diffs (\<lambda>n. of_real (erf_coeffs n)) = (\<lambda>n. of_real (a' n) :: 'a)"
     by (simp add: erf_coeffs_def a'_def diffs_def fun_eq_iff del: of_nat_Suc)
   hence "(\<Sum>n. diffs (\<lambda>n. of_real (erf_coeffs n)) n * z ^ n) = 
            (\<Sum>n. of_real (a' n) * z ^ n)" by simp
   also have "\<dots> = (\<Sum>n. of_real (a' (2*n)) * z ^ (2*n))"
     by (intro suminf_mono_reindex [symmetric]) (auto simp: strict_mono_def a'_def elim!: evenE)
   also have "(\<lambda>n. of_real (a' (2*n)) * z ^ (2*n)) = 
                (\<lambda>n. of_real (2 / sqrt pi) * (inverse (fact n) * (-(z^2))^n))"
     by (simp add: fun_eq_iff power_mult [symmetric] a'_def field_simps power_minus')
   also have "suminf \<dots> = of_real (2 / sqrt pi) * exp (-(z^2))"
     by (subst suminf_mult, intro summable_exp) 
        (auto simp: field_simps scaleR_conv_of_real exp_def)
   finally show ?thesis by (rule DERIV_subset) simp_all
 qed
 
 lemmas erf_has_field_derivative' [derivative_intros] =
   erf_has_field_derivative [THEN DERIV_chain2]
   
 lemma erf_continuous_on: "continuous_on A erf"
   by (rule DERIV_continuous_on erf_has_field_derivative)+
 
 lemma continuous_on_compose2_UNIV:
   "continuous_on UNIV g \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
   by (rule continuous_on_compose2[of UNIV g s f]) simp_all
     
 lemmas erf_continuous_on' [continuous_intros] = 
   erf_continuous_on [THEN continuous_on_compose2_UNIV]
 
 lemma erf_continuous [continuous_intros]: "continuous (at x within A) erf"
   by (rule continuous_within_subset[OF _ subset_UNIV])
      (insert erf_continuous_on[of UNIV], auto simp: continuous_on_eq_continuous_at)
 
 lemmas erf_continuous' [continuous_intros] = 
   continuous_within_compose2[OF _ erf_continuous]
   
 lemmas tendsto_erf [tendsto_intros] = isCont_tendsto_compose[OF erf_continuous]
 
 lemma erf_cnj [simp]: "erf (cnj z) = cnj (erf z)"
 proof -
   interpret bounded_linear cnj by (rule bounded_linear_cnj)
   from suminf[OF summable_erf] show ?thesis by (simp add: erf_def erf_coeffs_def)
 qed
 
 
 lemma integral_exp_minus_squared_real:
   assumes "a \<le> b"
   shows   "((\<lambda>t. exp (-(t^2))) has_integral (sqrt pi / 2 * (erf b - erf a))) {a..b}"
 proof -
   have "((\<lambda>t. exp (-(t^2))) has_integral (sqrt pi / 2 * erf b - sqrt pi / 2 * erf a)) {a..b}"
     using assms
     by (intro fundamental_theorem_of_calculus)
        (auto intro!: derivative_eq_intros 
              simp: has_field_derivative_iff_has_vector_derivative [symmetric])
   thus ?thesis by (simp add: field_simps)
 qed
 
 lemma erf_real_altdef_nonneg:
   "x \<ge> 0 \<Longrightarrow> erf (x::real) = 2 / sqrt pi * integral {0..x} (\<lambda>t. exp (-(t^2)))"
   using integral_exp_minus_squared_real[of 0 x]
   by (simp add: has_integral_iff field_simps)
     
 lemma erf_real_altdef_nonpos:
   "x \<le> 0 \<Longrightarrow> erf (x::real) = -2 / sqrt pi * integral {0..-x} (\<lambda>t. exp (-(t^2)))"
   using erf_real_altdef_nonneg[of "-x"] by simp
 
 lemma less_imp_erf_real_less:
   assumes "a < (b::real)"
   shows   "erf a < erf b"
 proof -
   from assms have "\<exists>z. z > a \<and> z < b \<and> erf b - erf a = (b - a) * (2 / sqrt pi * exp (- z\<^sup>2))"
     by (intro MVT2) (auto intro!: derivative_eq_intros simp: field_simps)
   then guess z by (elim exE conjE) note z = this
   note z(3)
   also from assms have "(b - a) * (2 / sqrt pi * exp (- z\<^sup>2)) > 0"
     by (intro mult_pos_pos divide_pos_pos) simp_all
   finally show ?thesis by simp
 qed
   
 lemma le_imp_erf_real_le: "a \<le> (b::real) \<Longrightarrow> erf a \<le> erf b"
   by (cases "a < b") (auto dest: less_imp_erf_real_less)
   
 lemma erf_real_less_cancel [simp]: "(erf (a :: real) < erf b) \<longleftrightarrow> a < b"
   using less_imp_erf_real_less[of a b] less_imp_erf_real_less[of b a]
   by (cases a b rule: linorder_cases) simp_all
 
 lemma erf_real_eq_iff [simp]: "erf (a::real) = erf b \<longleftrightarrow> a = b"
   by (cases a b rule: linorder_cases) (auto dest: less_imp_erf_real_less)
     
 lemma erf_real_le_cancel [simp]: "(erf (a :: real) \<le> erf b) \<longleftrightarrow> a \<le> b"
   by (cases a b rule: linorder_cases) (auto simp: less_eq_real_def)
     
 lemma inj_on_erf_real [intro]: "inj_on (erf :: real \<Rightarrow> real) A"
   by (auto simp: inj_on_def)
     
 lemma strict_mono_erf_real [intro]: "strict_mono (erf :: real \<Rightarrow> real)"
   by (auto simp: strict_mono_def)
     
 lemma mono_erf_real [intro]: "mono (erf :: real \<Rightarrow> real)"
   by (auto simp: mono_def)
 
 lemma erf_real_ge_0_iff [simp]: "erf (x::real) \<ge> 0 \<longleftrightarrow> x \<ge> 0"
   using erf_real_le_cancel[of 0 x] unfolding erf_0 .
 
 lemma erf_real_le_0_iff [simp]: "erf (x::real) \<le> 0 \<longleftrightarrow> x \<le> 0"
   using erf_real_le_cancel[of x 0] unfolding erf_0 .
     
 lemma erf_real_gt_0_iff [simp]: "erf (x::real) > 0 \<longleftrightarrow> x > 0"
   using erf_real_less_cancel[of 0 x] unfolding erf_0 .
 
 lemma erf_real_less_0_iff [simp]: "erf (x::real) < 0 \<longleftrightarrow> x < 0"
   using erf_real_less_cancel[of x 0] unfolding erf_0 .
     
 
 
 lemma erf_at_top [tendsto_intros]: "((erf :: real \<Rightarrow> real) \<longlongrightarrow> 1) at_top"
 proof -
   have *: "(\<Union>n. {0..real n}) = {0..}" by (auto intro!: real_nat_ceiling_ge)
   let ?f = "\<lambda>t::real. exp (-t\<^sup>2)"
   have "(\<lambda>n. set_lebesgue_integral lborel {0..real n} ?f)
           \<longlonglongrightarrow> set_lebesgue_integral lborel (\<Union>n. {0..real n}) ?f"
     using has_bochner_integral_erf_aux     
     by (intro set_integral_cont_up )
        (insert *, auto simp: incseq_def has_bochner_integral_iff set_integrable_def)
   also note *
   also have "(\<lambda>n. set_lebesgue_integral lborel {0..real n} ?f) = (\<lambda>n. integral {0..real n} ?f)"
   proof -
     have "\<And>n. set_integrable lborel {0..real n} (\<lambda>x. exp (- x\<^sup>2))"
       unfolding set_integrable_def
       by (intro borel_integrable_compact) (auto intro!: continuous_intros)
     then show ?thesis
       by (intro set_borel_integral_eq_integral ext)
   qed
   also have "\<dots> = (\<lambda>n. sqrt pi / 2 * erf (real n))" by (simp add: erf_real_altdef_nonneg)
   also have "set_lebesgue_integral lborel {0..} ?f = sqrt pi / 2"
     using has_bochner_integral_erf_aux by (simp add: has_bochner_integral_iff set_lebesgue_integral_def)
   finally have "(\<lambda>n. 2 / sqrt pi * (sqrt pi / 2 * erf (real n))) \<longlonglongrightarrow> 
                   (2 / sqrt pi) * (sqrt pi / 2)" by (intro tendsto_intros)
   hence "(\<lambda>n. erf (real n)) \<longlonglongrightarrow> 1" by simp
   thus ?thesis by (rule tendsto_sandwich_mono) auto
 qed
   
 lemma erf_at_bot [tendsto_intros]: "((erf :: real \<Rightarrow> real) \<longlongrightarrow> -1) at_bot"
   by (simp add: filterlim_at_bot_mirror tendsto_minus_cancel_left erf_at_top)
 
 lemmas tendsto_erf_at_top [tendsto_intros] = filterlim_compose[OF erf_at_top]
 lemmas tendsto_erf_at_bot [tendsto_intros] = filterlim_compose[OF erf_at_bot]
 
 
 subsection \<open>The complimentary error function\<close>  
   
 definition erfc where "erfc z = 1 - erf z"
   
 lemma erf_conv_erfc: "erf z = 1 - erfc z" by (simp add: erfc_def)
 
 lemma erfc_0 [simp]: "erfc 0 = 1"
   by (simp add: erfc_def)
 
 lemma erfc_minus: "erfc (-z) = 2 - erfc z"
   by (simp add: erfc_def)
 
 lemma erfc_of_real [simp]: "erfc (of_real x) = of_real (erfc x)"
   by (simp add: erfc_def)
     
 lemma of_real_erfc_numeral [simp]: "of_real (erfc (numeral n)) = erfc (numeral n)"
   by (simp add: erfc_def)
 
 lemma of_real_erfc_1 [simp]: "of_real (erfc 1) = erfc 1"
   by (simp add: erfc_def)
     
 
 lemma less_imp_erfc_real_less: "a < (b::real) \<Longrightarrow> erfc a > erfc b"
   by (simp add: erfc_def)
   
 lemma le_imp_erfc_real_le: "a \<le> (b::real) \<Longrightarrow> erfc a \<ge> erfc b"
   by (simp add: erfc_def)
   
 lemma erfc_real_less_cancel [simp]: "(erfc (a :: real) < erfc b) \<longleftrightarrow> a > b"
   by (simp add: erfc_def)
 
 lemma erfc_real_eq_iff [simp]: "erfc (a::real) = erfc b \<longleftrightarrow> a = b"
   by (simp add: erfc_def)
     
 lemma erfc_real_le_cancel [simp]: "(erfc (a :: real) \<le> erfc b) \<longleftrightarrow> a \<ge> b"
   by (simp add: erfc_def)
     
 lemma inj_on_erfc_real [intro]: "inj_on (erfc :: real \<Rightarrow> real) A"
   by (auto simp: inj_on_def)
 
 lemma antimono_erfc_real [intro]: "antimono (erfc :: real \<Rightarrow> real)"
   by (auto simp: antimono_def)
 
 lemma erfc_real_ge_0_iff [simp]: "erfc (x::real) \<ge> 1 \<longleftrightarrow> x \<le> 0"
   by (simp add: erfc_def)
 
 lemma erfc_real_le_0_iff [simp]: "erfc (x::real) \<le> 1 \<longleftrightarrow> x \<ge> 0"
   by (simp add: erfc_def)
     
 lemma erfc_real_gt_0_iff [simp]: "erfc (x::real) > 1 \<longleftrightarrow> x < 0"
   by (simp add: erfc_def)
 
 lemma erfc_real_less_0_iff [simp]: "erfc (x::real) < 1 \<longleftrightarrow> x > 0"
   by (simp add: erfc_def)
 
 
 lemma erfc_has_field_derivative:
   "(erfc has_field_derivative -of_real (2 / sqrt pi) * exp (-(z^2))) (at z within A)"
   unfolding erfc_def [abs_def] by (auto intro!: derivative_eq_intros)
 
 lemmas erfc_has_field_derivative' [derivative_intros] =
   erfc_has_field_derivative [THEN DERIV_chain2]
   
 lemma erfc_continuous_on: "continuous_on A erfc"
   by (rule DERIV_continuous_on erfc_has_field_derivative)+
 
 lemmas erfc_continuous_on' [continuous_intros] = 
   erfc_continuous_on [THEN continuous_on_compose2_UNIV]
 
 lemma erfc_continuous [continuous_intros]: "continuous (at x within A) erfc"
   by (rule continuous_within_subset[OF _ subset_UNIV])
      (insert erfc_continuous_on[of UNIV], auto simp: continuous_on_eq_continuous_at)
 
 lemmas erfc_continuous' [continuous_intros] = 
   continuous_within_compose2[OF _ erfc_continuous]
   
 lemmas tendsto_erfc [tendsto_intros] = isCont_tendsto_compose[OF erfc_continuous]
 
   
 lemma erfc_at_top [tendsto_intros]: "((erfc :: real \<Rightarrow> real) \<longlongrightarrow> 0) at_top"
   unfolding erfc_def [abs_def] by (auto intro!: tendsto_eq_intros)
   
 lemma erfc_at_bot [tendsto_intros]: "((erfc :: real \<Rightarrow> real) \<longlongrightarrow> 2) at_bot"
   unfolding erfc_def [abs_def] by (auto intro!: tendsto_eq_intros)
 
 lemmas tendsto_erfc_at_top [tendsto_intros] = filterlim_compose[OF erfc_at_top]
 lemmas tendsto_erfc_at_bot [tendsto_intros] = filterlim_compose[OF erfc_at_bot]  
 
 lemma integrable_exp_minus_squared:
   assumes "A \<subseteq> {0..}" "A \<in> sets lborel"
   shows   "set_integrable lborel A (\<lambda>t::real. exp (-t\<^sup>2))" (is ?thesis1)
     and   "(\<lambda>t::real. exp (-t\<^sup>2)) integrable_on A" (is ?thesis2)
 proof -
   show ?thesis1 
     by (rule set_integrable_subset[of _ "{0..}"]) 
        (insert assms has_bochner_integral_erf_aux, auto simp: has_bochner_integral_iff set_integrable_def)
   thus ?thesis2 by (rule set_borel_integral_eq_integral)
 qed
 
 lemma
   assumes "x \<ge> 0"
   shows   erfc_real_altdef_nonneg: "erfc x = 2 / sqrt pi * integral {x..} (\<lambda>t. exp (-t\<^sup>2))"
     and   has_integral_erfc:       "((\<lambda>t. exp (-t\<^sup>2)) has_integral (sqrt pi / 2 * erfc x)) {x..}"
 proof -
   let ?f = "\<lambda>t::real. exp (-t\<^sup>2)"
   have int: "set_integrable lborel {0..} ?f"
     using has_bochner_integral_erf_aux by (simp add: has_bochner_integral_iff set_integrable_def)
   from assms have *: "{(0::real)..} = {0..x} \<union> {x..}" by auto
   have "set_lebesgue_integral lborel ({0..x} \<union> {x..}) ?f = 
                set_lebesgue_integral lborel {0..x} ?f + set_lebesgue_integral lborel {x..} ?f"
     by (subst set_integral_Un_AE; (rule set_integrable_subset[OF int])?)
        (insert assms AE_lborel_singleton[of x], auto elim!: eventually_mono)
   also note * [symmetric]
   also have "set_lebesgue_integral lborel {0..} ?f = sqrt pi / 2"
     using has_bochner_integral_erf_aux by (simp add: has_bochner_integral_iff set_lebesgue_integral_def)
   also have "set_lebesgue_integral lborel {0..x} ?f = sqrt pi / 2 * erf x"
     by (subst set_borel_integral_eq_integral(2)[OF set_integrable_subset[OF int]])
        (insert assms, auto simp: erf_real_altdef_nonneg)
   also have "set_lebesgue_integral lborel {x..} ?f = integral {x..} ?f"
     by (subst set_borel_integral_eq_integral(2)[OF set_integrable_subset[OF int]])
        (insert assms, auto) 
   finally show "erfc x = 2 / sqrt pi * integral {x..} ?f" by (simp add: field_simps erfc_def)
   with integrable_exp_minus_squared(2)[of "{x..}"] assms
   show "(?f has_integral (sqrt pi / 2 * erfc x)) {x..}"
     by (simp add: has_integral_iff)
 qed
 
 
 lemma erfc_real_gt_0 [simp, intro]: "erfc (x::real) > 0"
 proof (cases "x \<ge> 0")
   case True
   have "0 < integral {x..x+1} (\<lambda>t. exp (-(x+1)\<^sup>2))" by simp
   also from True have "\<dots> \<le> integral {x..x+1} (\<lambda>t. exp (-t\<^sup>2))"
     by (intro integral_le) 
        (auto intro!: integrable_continuous_real continuous_intros power_mono)
   also have "\<dots> \<le> sqrt pi / 2 * erfc x"
     by (rule has_integral_subset_le[OF _ integrable_integral has_integral_erfc])
        (auto intro!: integrable_continuous_real continuous_intros True)
   finally have "sqrt pi / 2 * erfc x > 0" .
   hence "\<dots> * (2 / sqrt pi) > 0" by (rule mult_pos_pos) simp_all
   thus "erfc x > 0" by simp
 next
   case False
   have "0 \<le> (1::real)" by simp
   also from False have "\<dots> < erfc x" by simp
   finally show ?thesis .
 qed
 
 lemma erfc_real_less_2 [intro]: "erfc (x::real) < 2"
   using erfc_real_gt_0[of "-x"] unfolding erfc_minus by simp
     
 lemma erf_real_gt_neg1 [intro]: "erf (x::real) > -1"
   using erfc_real_less_2[of x] unfolding erfc_def by simp
     
 lemma erf_real_less_1 [intro]: "erf (x::real) < 1"
   using erfc_real_gt_0[of x] unfolding erfc_def by simp
 
 lemma erfc_cnj [simp]: "erfc (cnj z) = cnj (erfc z)"
   by (simp add: erfc_def)
 
 
 subsection \<open>Specific facts about the complex case\<close>  
   
 lemma erf_complex_altdef:
   "erf z = of_real (2 / sqrt pi) * contour_integral (linepath 0 z) (\<lambda>t. exp (-(t^2)))"
 proof -
   define A where "A = (\<lambda>z. contour_integral (linepath 0 z) (\<lambda>t. exp (-(t^2))))"
   have [derivative_intros]: "(A has_field_derivative exp (-(z^2))) (at z)" for z :: complex
     unfolding A_def
     by (rule contour_integral_has_field_derivative[where A = UNIV])
        (auto intro!: holomorphic_intros)
   have "erf z - 2 / sqrt pi * A z = erf 0 - 2 / sqrt pi * A 0"
     by (rule has_derivative_zero_unique [where f = "\<lambda>z. erf z - 2 / sqrt pi * A z" and s = UNIV])
        (auto intro!: has_field_derivative_imp_has_derivative derivative_eq_intros)
   also have "A 0 = 0" by (simp only: A_def contour_integral_trivial)
   finally show ?thesis unfolding A_def by (simp add: algebra_simps)
 qed
 
 lemma erf_holomorphic_on: "erf holomorphic_on A"
   by (auto simp: holomorphic_on_def field_differentiable_def intro!: erf_has_field_derivative)
 
 lemmas erf_holomorphic_on' [holomorphic_intros] = 
   erf_holomorphic_on [THEN holomorphic_on_chain_UNIV]
   
 lemma erf_analytic_on: "erf analytic_on A"
   by (auto simp: analytic_on_def) (auto intro!: exI[of _ 1] holomorphic_intros)
 
 lemma erf_analytic_on'  [analytic_intros]:
   assumes "f analytic_on A"
   shows   "(\<lambda>x. erf (f x)) analytic_on A"
 proof -
   from assms and erf_analytic_on have "erf \<circ> f analytic_on A"
     by (rule analytic_on_compose_gen) auto
   thus ?thesis by (simp add: o_def)
 qed
 
 lemma erfc_holomorphic_on: "erfc holomorphic_on A"
   by (auto simp: holomorphic_on_def field_differentiable_def intro!: erfc_has_field_derivative)
 
 lemmas erfc_holomorphic_on' [holomorphic_intros] = 
   erfc_holomorphic_on [THEN holomorphic_on_chain_UNIV]
 
 lemma erfc_analytic_on: "erfc analytic_on A"
   by (auto simp: analytic_on_def) (auto intro!: exI[of _ 1] holomorphic_intros)
 
 lemma erfc_analytic_on' [analytic_intros]:
   assumes "f analytic_on A"
   shows   "(\<lambda>x. erfc (f x)) analytic_on A"
 proof -
   from assms and erfc_analytic_on have "erfc \<circ> f analytic_on A"
     by (rule analytic_on_compose_gen) auto
   thus ?thesis by (simp add: o_def)
 qed
    
 end
diff --git a/thys/Error_Function/ROOT b/thys/Error_Function/ROOT
--- a/thys/Error_Function/ROOT
+++ b/thys/Error_Function/ROOT
@@ -1,11 +1,11 @@
 chapter AFP
 
-session Error_Function (AFP) = "HOL-Analysis" +
+session Error_Function (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 300]
   sessions
     Landau_Symbols
   theories
     Error_Function
     Error_Function_Asymptotics
   document_files
     "root.tex"
diff --git a/thys/Euler_MacLaurin/Euler_MacLaurin.thy b/thys/Euler_MacLaurin/Euler_MacLaurin.thy
--- a/thys/Euler_MacLaurin/Euler_MacLaurin.thy
+++ b/thys/Euler_MacLaurin/Euler_MacLaurin.thy
@@ -1,1618 +1,1617 @@
 section \<open>The Euler--MacLaurin summation formula\<close>
 theory Euler_MacLaurin
 imports 
-  "HOL-Analysis.Analysis"
-  "HOL-Library.Multiset"
+  "HOL-Complex_Analysis.Complex_Analysis"
   Bernoulli.Periodic_Bernpoly
   Bernoulli.Bernoulli_FPS
 begin
   
 subsection \<open>Auxiliary facts\<close>
 
 (* TODO Move? *)
 lemma pbernpoly_of_int [simp]: "pbernpoly n (of_int a) = bernoulli n"
   by (simp add: pbernpoly_def)
 
 lemma continuous_on_bernpoly' [continuous_intros]:
   assumes "continuous_on A f"
   shows   "continuous_on A (\<lambda>x. bernpoly n (f x) :: 'a :: real_normed_algebra_1)"
   using continuous_on_compose2[OF continuous_on_bernpoly assms, of UNIV n] by auto
 
 lemma sum_atLeastAtMost_int_last:
   assumes "a < (b :: int)"
   shows   "sum f {a..b} = sum f {a..<b} + f b"
 proof -
   from assms have "{a..b} = insert b {a..<b}" by auto
   also have "sum f \<dots> = sum f {a..<b} + f b"
     by (subst sum.insert) (auto simp: add_ac)
   finally show ?thesis .
 qed
 
 lemma sum_atLeastAtMost_int_head:
   assumes "a < (b :: int)"
   shows   "sum f {a..b} = f a + sum f {a<..b}"
 proof -
   from assms have "{a..b} = insert a {a<..b}" by auto
   also have "sum f \<dots> = f a + sum f {a<..b}"
     by (subst sum.insert) auto
   finally show ?thesis .
 qed
 
 lemma not_in_nonpos_Reals_imp_add_nonzero:
   assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" "x \<ge> 0"
   shows   "z + of_real x \<noteq> 0"
   using assms by (auto simp: add_eq_0_iff2)
 (* END TODO *)
 
 lemma negligible_atLeastAtMostI: "b \<le> a \<Longrightarrow> negligible {a..(b::real)}"
   by (cases "b < a") auto
     
 lemma integrable_on_negligible: 
  "negligible A \<Longrightarrow> (f :: 'n :: euclidean_space \<Rightarrow> 'a :: banach) integrable_on A"
   by (subst integrable_spike_set_eq[of _ "{}"]) (simp_all add: integrable_on_empty)  
     
 lemma Union_atLeastAtMost_real_of_int:
   assumes "a < b" 
   shows   "(\<Union>n\<in>{a..<b}. {real_of_int n..real_of_int (n + 1)}) = {real_of_int a..real_of_int b}"
 proof (intro equalityI subsetI)
   fix x assume x: "x \<in> {real_of_int a..real_of_int b}"
   thus "x \<in> (\<Union>n\<in>{a..<b}. {real_of_int n..real_of_int (n + 1)})"
   proof (cases "x = real_of_int b")
     case True
     with assms show ?thesis by (auto intro!: bexI[of _ "b - 1"])
   next
     case False
     with x have x: "x \<ge> real_of_int a" "x < real_of_int b" by simp_all
     hence "x \<ge> of_int \<lfloor>x\<rfloor>" "x \<le> of_int \<lfloor>x\<rfloor> + 1" by linarith+
     moreover from x have "\<lfloor>x\<rfloor> \<ge> a" "\<lfloor>x\<rfloor> < b" by linarith+ 
     ultimately have "\<exists>n\<in>{a..<b}. x \<in> {of_int n..of_int (n + 1)}"
       by (intro bexI[of _ "\<lfloor>x\<rfloor>"]) simp_all
     thus ?thesis by blast
   qed
 qed auto
 
 
 subsection \<open>The remainder terms\<close>
 
 text \<open>
   The following describes the remainder term in the classical version of the Euler--MacLaurin
   formula.
 \<close>
 definition EM_remainder' :: "nat \<Rightarrow> (real \<Rightarrow> 'a :: banach) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> 'a" where
   "EM_remainder' n f a b = ((-1) ^ Suc n / fact n) *\<^sub>R integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t)"
 
 
 text \<open>
   Next, we define the remainder term that occurs when one lets the right bound of summation 
   in the Euler--MacLaurin formula tend to infinity.
 \<close>
 definition EM_remainder_converges :: "nat \<Rightarrow> (real \<Rightarrow> 'a :: banach) \<Rightarrow> int \<Rightarrow> bool" where
   "EM_remainder_converges n f a \<longleftrightarrow> (\<exists>L. ((\<lambda>x. EM_remainder' n f a (of_int x)) \<longlongrightarrow> L) at_top)"
 
 definition EM_remainder :: "nat \<Rightarrow> (real \<Rightarrow> 'a :: banach) \<Rightarrow> int \<Rightarrow> 'a" where
   "EM_remainder n f a = 
      (if EM_remainder_converges n f a then
         Lim at_top (\<lambda>x. EM_remainder' n f a (of_int x)) else 0)"
 
 text \<open>
   The following lemmas are fairly obvious -- but tedious to prove -- 
   properties of the remainder terms.
 \<close>
 
 lemma EM_remainder_eqI: 
   fixes L
   assumes "((\<lambda>x. EM_remainder' n f b (of_int x)) \<longlongrightarrow> L) at_top"
   shows   "EM_remainder n f b = L"
   using assms by (auto simp: EM_remainder_def EM_remainder_converges_def intro!: tendsto_Lim)
 
 lemma integrable_EM_remainder'_int:
   fixes a b :: int and f :: "real \<Rightarrow> 'a :: banach"
   assumes "continuous_on {of_int a..of_int b} f"
   shows   "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {a..b}"
 proof -
   have [continuous_intros]: "continuous_on A f" if "A \<subseteq> {of_int a..of_int b}" for A
     using continuous_on_subset[OF assms that] .
   consider "a > b" | "a = b" | "a < b" "n = 1" | "a < b" "n \<noteq> 1"
     by (cases a b rule: linorder_cases) auto
   thus ?thesis
   proof cases
     assume "a < b" and "n \<noteq> 1"
     thus ?thesis by (intro integrable_continuous_real continuous_intros) auto
   next
     assume ab: "a < b" and [simp]: "n = 1"
     let ?A = "(\<lambda>n. {real_of_int n..real_of_int (n+1)}) ` {a..<b}"
     show ?thesis
     proof (rule integrable_combine_division; (intro ballI)?)
       show "?A division_of {of_int a..of_int b}"
         using Union_atLeastAtMost_real_of_int[OF ab] by (simp add: division_of_def)
     next
       fix I assume "I \<in> ?A"
       then obtain i where i: "i \<in> {a..<b}" "I = {of_int i..of_int (i + 1)}" by auto
       show "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on I"
       proof (rule integrable_spike)
         show "(\<lambda>t. (t - of_int i - 1/2) *\<^sub>R f t) integrable_on I"
           using i by (auto intro!: integrable_continuous_real continuous_intros)
       next
         fix x assume "x \<in> I - {of_int (i + 1)}"
         with i have "of_int i \<le> x" "x < of_int i + 1" by simp_all
         hence "floor x = i" by linarith
         thus "pbernpoly n x *\<^sub>R f x = (x - of_int i - 1 / 2) *\<^sub>R f x"
           by (simp add: pbernpoly_def bernpoly_def frac_def)
       qed simp_all
     qed
   qed (simp_all add: integrable_on_negligible)
 qed
 
 lemma integrable_EM_remainder':
   fixes a b :: real and f :: "real \<Rightarrow> 'a :: banach"
   assumes "continuous_on {a..b} f"
   shows   "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {a..b}"
 proof (cases "\<lceil>a\<rceil> \<le> \<lfloor>b\<rfloor>")
   case True
   define a' b' where "a' = \<lceil>a\<rceil>" and "b' = \<lfloor>b\<rfloor>"
   from True have *: "a' \<le> b'" "a' \<ge> a" "b' \<le> b" by (auto simp: a'_def b'_def)
   from * have A: "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on ({a'..b'})"
     by (intro integrable_EM_remainder'_int continuous_on_subset[OF assms]) auto
   have B: "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on ({a..a'})"
   proof (rule integrable_spike)
     show "pbernpoly n x *\<^sub>R f x = bernpoly n (x - of_int (floor a)) *\<^sub>R f x" 
          if "x \<in> {a..real_of_int a'} - {real_of_int a'}"for x
     proof -
       have "x \<ge> a" "x <real_of_int a'" using that by auto
       with True have "floor x = floor a" unfolding a'_def
         using ceiling_diff_floor_le_1[of a] by (intro floor_unique; linarith) 
       thus ?thesis by (simp add: pbernpoly_def frac_def)
     qed
   qed (insert *, auto intro!: continuous_intros integrable_continuous_real 
          continuous_on_subset[OF assms])
   have C: "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on ({b'..b})"
   proof (rule integrable_spike)
     show "pbernpoly n x *\<^sub>R f x = bernpoly n (x - of_int b') *\<^sub>R f x"
          if "x \<in> {real_of_int b'..b} - {real_of_int b'}" for x
     proof -
       have "x \<le> b" "x > real_of_int b'" using that by auto
       with True have "floor x = b'" unfolding b'_def by (intro floor_unique; linarith) 
       thus ?thesis by (simp add: pbernpoly_def frac_def)
     qed
   qed (insert *, auto intro!: continuous_intros integrable_continuous_real 
          continuous_on_subset[OF assms])
   have "(\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on ({a..a'} \<union> {a'..b'} \<union> {b'..b})" using * A B C
     by (intro integrable_Un; (subst ivl_disj_un)?)
        (auto simp: ivl_disj_un max_def min_def)
   also have "{a..a'} \<union> {a'..b'} \<union> {b'..b} = {a..b}" using * by auto
   finally show ?thesis .
 next
   assume *: "\<not>ceiling a \<le> floor b"
   show ?thesis
   proof (rule integrable_spike)
     show "(\<lambda>t. bernpoly n (t - floor a) *\<^sub>R f t) integrable_on {a..b}" using *
       by (auto intro!: integrable_continuous_real continuous_intros assms)
   next
     show "pbernpoly n x *\<^sub>R f x = bernpoly n (x - floor a) *\<^sub>R f x"
          if "x \<in> {a..b} - {}" for x
     proof -
       from * have **: "b < floor a + 1" 
         unfolding ceiling_altdef by (auto split: if_splits simp: le_floor_iff)
       from that have x: "x \<ge> a" "x \<le> b" by simp_all
       with * ** have "floor x = floor a" by linarith
       thus ?thesis by (simp add: pbernpoly_def frac_def)
     qed
   qed simp_all
 qed
 
 lemma EM_remainder'_bounded_linear:
   assumes "bounded_linear h"
   assumes "continuous_on {a..b} f"
   shows   "EM_remainder' n (\<lambda>x. h (f x)) a b = h (EM_remainder' n f a b)"
 proof -
   have "integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R h (f t)) = 
            integral {a..b} (\<lambda>t. h (pbernpoly n t *\<^sub>R f t))" using assms
     by (simp add: linear_simps)
   also have "\<dots> = h (integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t))"
     by (subst integral_linear [OF _ assms(1), symmetric])
        (auto intro!: integrable_EM_remainder' assms(2) simp: o_def)
   finally show ?thesis using assms(1)
     by (simp add: EM_remainder'_def linear_simps)
 qed
 
 lemma EM_remainder_converges_of_real:
   assumes "EM_remainder_converges n f a" "continuous_on {of_int a..} f"
   shows   "EM_remainder_converges n (\<lambda>x. of_real (f x)) a"
 proof -
   from assms obtain L 
     where L: "((\<lambda>b. EM_remainder' n f (real_of_int a) (real_of_int b)) \<longlongrightarrow> L) at_top"
     by (auto simp: EM_remainder_converges_def)
   have "((\<lambda>b. EM_remainder' n (\<lambda>x. of_real (f x)) (of_int a) (of_int b)) \<longlongrightarrow> of_real L) at_top"
   proof (rule Lim_transform_eventually)
     show "eventually (\<lambda>b. of_real (EM_remainder' n f (of_int a) (of_int b)) = 
             EM_remainder' n (\<lambda>x. of_real (f x)) (of_int a) (of_int b)) at_top"
       using eventually_ge_at_top[of a]
       by eventually_elim
          (intro EM_remainder'_bounded_linear [OF bounded_linear_of_real, symmetric]
                 continuous_on_subset[OF assms(2)], auto)
   qed (intro tendsto_intros L)
   thus ?thesis unfolding EM_remainder_converges_def ..
 qed
 
 lemma EM_remainder_converges_of_real_iff:
   fixes f :: "real \<Rightarrow> real"
   assumes "continuous_on {of_int a..} f"
   shows   "EM_remainder_converges n (\<lambda>x. of_real (f x) :: 
             'a :: {banach, real_normed_algebra_1, real_inner}) a \<longleftrightarrow> EM_remainder_converges n f a"
 proof
   assume "EM_remainder_converges n (\<lambda>x. of_real (f x) :: 'a) a"
   then obtain L :: 'a
     where L: "((\<lambda>b. EM_remainder' n (\<lambda>x. of_real (f x)) (of_int a) (of_int b)) \<longlongrightarrow> L) at_top"
     by (auto simp: EM_remainder_converges_def)
   have "((\<lambda>b. EM_remainder' n f (of_int a) (of_int b)) \<longlongrightarrow> L \<bullet> 1) at_top"
   proof (rule Lim_transform_eventually)
     show "eventually (\<lambda>b. EM_remainder' n (\<lambda>x. of_real (f x) :: 'a) (of_int a) (of_int b) \<bullet> 1 =
             EM_remainder' n f (of_int a) (of_int b)) at_top" using eventually_ge_at_top[of a]
       by eventually_elim
          (subst EM_remainder'_bounded_linear [OF bounded_linear_of_real],
           auto intro!: continuous_on_subset[OF assms])
   qed (intro tendsto_intros L)
   thus "EM_remainder_converges n f a" unfolding EM_remainder_converges_def ..
 qed (intro EM_remainder_converges_of_real assms)
 
 lemma EM_remainder_of_real:
   assumes "continuous_on {a..} f"
   shows   "EM_remainder n (\<lambda>x. of_real (f x) :: 
              'a :: {banach, real_normed_algebra_1, real_inner}) a = 
              of_real (EM_remainder n f a)"
 proof -
   have eq: "EM_remainder' n (\<lambda>x. of_real (f x) :: 'a) (real_of_int a) = 
               (\<lambda>x::int. of_real (EM_remainder' n f a x))"
       by (intro ext EM_remainder'_bounded_linear[OF bounded_linear_of_real]
             continuous_on_subset[OF assms]) auto
   show ?thesis
   proof (cases "EM_remainder_converges n f a")
     case False
     with EM_remainder_converges_of_real_iff[OF assms, of n] show ?thesis
       by (auto simp: EM_remainder_def)
   next
     case True
     then obtain L where L: "((\<lambda>x. EM_remainder' n f a (real_of_int x)) \<longlongrightarrow> L) at_top" 
       by (auto simp: EM_remainder_converges_def)
     have L': "((\<lambda>x. EM_remainder' n (\<lambda>x. of_real (f x) :: 'a) a 
                 (real_of_int x)) \<longlongrightarrow> of_real L) at_top" unfolding eq by (intro tendsto_of_real L)
     from L L' tendsto_Lim[OF _ L] tendsto_Lim[OF _ L'] show ?thesis
       by (auto simp: EM_remainder_def EM_remainder_converges_def)
   qed
 qed
 
 lemma EM_remainder'_cong:
   assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> f x = g x" "n = n'" "a = a'" "b = b'"
   shows   "EM_remainder' n f a b = EM_remainder' n' g a' b'"
 proof -
   have "integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t) = integral {a'..b'} (\<lambda>t. pbernpoly n' t *\<^sub>R g t)"
     unfolding assms using assms by (intro integral_cong) auto
   with assms show ?thesis by (simp add: EM_remainder'_def)
 qed
 
 lemma EM_remainder_converges_cong:
   assumes "\<And>x. x \<ge> of_int a \<Longrightarrow> f x = g x" "n = n'" "a = a'"
   shows   "EM_remainder_converges n f a = EM_remainder_converges n' g a'"
   unfolding EM_remainder_converges_def
   by (subst EM_remainder'_cong[OF _ refl refl refl, of _ _ f g]) (use assms in auto)
 
 lemma EM_remainder_cong:
   assumes "\<And>x. x \<ge> of_int a \<Longrightarrow> f x = g x" "n = n'" "a = a'"
   shows   "EM_remainder n f a = EM_remainder n' g a'"
 proof -
   have *: "EM_remainder_converges n f a = EM_remainder_converges n' g a'"
     using assms by (intro EM_remainder_converges_cong) auto
   show ?thesis unfolding EM_remainder_def
     by (subst EM_remainder'_cong[OF _ refl refl refl, of _ _ f g]) (use assms * in auto)
 qed
 
 lemma EM_remainder_converges_cnj: 
   assumes "continuous_on {a..} f" and "EM_remainder_converges n f a"
   shows   "EM_remainder_converges n (\<lambda>x. cnj (f x)) a"
 proof -
   interpret bounded_linear cnj by (rule bounded_linear_cnj)
   obtain L where L: "((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) \<longlongrightarrow> L) at_top"
     using assms unfolding EM_remainder_converges_def by blast
   note tendsto_cnj [OF this]
   also have "(\<lambda>x. cnj (EM_remainder' n f (real_of_int a) (real_of_int x))) =
                (\<lambda>x. EM_remainder' n (\<lambda>x. cnj (f x)) (real_of_int a) (real_of_int x))"
     by (subst EM_remainder'_bounded_linear [OF bounded_linear_cnj])
        (rule continuous_on_subset [OF assms(1)], auto)
   finally have L': "(\<dots> \<longlongrightarrow> cnj L) at_top" .
   thus "EM_remainder_converges n (\<lambda>x. cnj (f x)) a"
     by (auto simp: EM_remainder_converges_def)
 qed
 
 lemma EM_remainder_converges_cnj_iff:
   assumes "continuous_on {of_int a..} f"
   shows   "EM_remainder_converges n (\<lambda>x. cnj (f x)) a \<longleftrightarrow> EM_remainder_converges n f a"
 proof
   assume "EM_remainder_converges n (\<lambda>x. cnj (f x)) a"
   hence "EM_remainder_converges n (\<lambda>x. cnj (cnj (f x))) a"
     by (rule EM_remainder_converges_cnj [rotated]) (auto intro: continuous_intros assms)
   thus "EM_remainder_converges n f a" by simp
 qed (intro EM_remainder_converges_cnj assms)
 
 lemma EM_remainder_cnj: 
   assumes "continuous_on {a..} f"
   shows   "EM_remainder n (\<lambda>x. cnj (f x)) a = cnj (EM_remainder n f a)"
 proof (cases "EM_remainder_converges n f a")
   case False
   hence "\<not>EM_remainder_converges n (\<lambda>x. cnj (f x)) a"
     by (subst EM_remainder_converges_cnj_iff [OF assms])
   with False show ?thesis by (simp add: EM_remainder_def)
 next
   case True
   then obtain L where L: "((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) \<longlongrightarrow> L) at_top"
     unfolding EM_remainder_converges_def by blast
   note tendsto_cnj [OF this]
   also have "(\<lambda>x. cnj (EM_remainder' n f (real_of_int a) (real_of_int x))) =
                (\<lambda>x. EM_remainder' n (\<lambda>x. cnj (f x)) (real_of_int a) (real_of_int x))"
     by (subst EM_remainder'_bounded_linear [OF bounded_linear_cnj])
        (rule continuous_on_subset [OF assms(1)], auto)
   finally have L': "(\<dots> \<longlongrightarrow> cnj L) at_top" .
   moreover from assms and L have "EM_remainder n f a = L"
     by (intro EM_remainder_eqI)
   ultimately show "EM_remainder n (\<lambda>x. cnj (f x)) a = cnj (EM_remainder n f a)"
     using L' by (intro EM_remainder_eqI) simp_all
 qed
 
 lemma EM_remainder'_combine:
   fixes f :: "real \<Rightarrow> 'a :: banach"
   assumes [continuous_intros]: "continuous_on {a..c} f"
   assumes "a \<le> b" "b \<le> c"
   shows   "EM_remainder' n f a b + EM_remainder' n f b c = EM_remainder' n f a c"
 proof -
   have "integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t) + integral {b..c} (\<lambda>t. pbernpoly n t *\<^sub>R f t) =
           integral {a..c} (\<lambda>t. pbernpoly n t *\<^sub>R f t)"
     by (intro Henstock_Kurzweil_Integration.integral_combine assms integrable_EM_remainder')
   from this [symmetric] show ?thesis by (simp add: EM_remainder'_def algebra_simps) 
 qed
 
 lemma uniformly_convergent_EM_remainder':
   fixes f :: "'a \<Rightarrow> real \<Rightarrow> 'b :: {banach,real_normed_algebra}"
   assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"
   assumes integrable: "\<And>a' b y. y \<in> A \<Longrightarrow>  a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> 
                          (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
   assumes conv: "convergent (\<lambda>y. G (real y))"
   assumes bound: "eventually (\<lambda>x. \<forall>y\<in>A. norm (f y x) \<le> g x) at_top"
   shows   "uniformly_convergent_on A (\<lambda>b s. EM_remainder' n (f s) a b)"
 proof -
   interpret bounded_linear "\<lambda>x::'b. ((- 1) ^ Suc n / fact n) *\<^sub>R x"
     by (rule bounded_linear_scaleR_right)
 
   from bounded_pbernpoly[of n] guess C . note C = this
   from C[of 0] have [simp]: "C \<ge> 0" by simp
 
   show ?thesis unfolding EM_remainder'_def
   proof (intro uniformly_convergent_on uniformly_convergent_improper_integral')
     fix x assume "x \<ge> a"
     thus "((\<lambda>x. C * G x) has_real_derivative C * g x) (at x within {a..})"
       by (intro DERIV_cmult deriv)
   next
     fix y a' b assume "y \<in> A" "a \<le> a'" "a' \<le> b"
     thus "(\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
       by (rule integrable)
   next
     from conv obtain L where "(\<lambda>y. G (real y)) \<longlonglongrightarrow> L"
       by (auto simp: convergent_def)
     from tendsto_mult[OF tendsto_const[of C] this]
       show "convergent (\<lambda>y. C * G (real y))"
         by (auto simp: convergent_def)
   next
     show "\<forall>\<^sub>F x in at_top. \<forall>y\<in>A. norm (pbernpoly n x *\<^sub>R f y x) \<le> C * g x"
       using C unfolding norm_scaleR 
       by (intro eventually_mono[OF bound] ballI mult_mono) auto
   qed
 qed
 
 lemma uniform_limit_EM_remainder:
   fixes f :: "'a \<Rightarrow> real \<Rightarrow> 'b :: {banach,real_normed_algebra}"
   assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"
   assumes integrable: "\<And>a' b y. y \<in> A \<Longrightarrow>  a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> 
                          (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
   assumes conv: "convergent (\<lambda>y. G (real y))"
   assumes bound: "eventually (\<lambda>x. \<forall>y\<in>A. norm (f y x) \<le> g x) at_top"
   shows   "uniform_limit A (\<lambda>b s. EM_remainder' n (f s) a b) 
              (\<lambda>s. EM_remainder n (f s) a) sequentially"
 proof -
   have *: "uniformly_convergent_on A (\<lambda>b s. EM_remainder' n (f s) a b)"
     by (rule uniformly_convergent_EM_remainder'[OF assms])
   also have "?this \<longleftrightarrow> ?thesis"
     unfolding uniformly_convergent_uniform_limit_iff
   proof (intro uniform_limit_cong refl always_eventually allI ballI)
     fix s assume "s \<in> A"
     with * have **: "convergent (\<lambda>b. EM_remainder' n (f s) a b)"
       by (rule uniformly_convergent_imp_convergent)
     show "lim (\<lambda>b. EM_remainder' n (f s) a b) = EM_remainder n (f s) a"
     proof (rule sym, rule EM_remainder_eqI)
       have "((\<lambda>x. EM_remainder' n (f s) (real_of_int a) (real x)) \<longlongrightarrow>
                lim (\<lambda>x. EM_remainder' n (f s) (real_of_int a) (real x))) at_top"
         (is "(_ \<longlongrightarrow> ?L) _") using ** unfolding convergent_LIMSEQ_iff by blast
       hence "((\<lambda>x. EM_remainder' n (f s) (real_of_int a) (real (nat x))) \<longlongrightarrow> ?L) at_top"
         by (rule filterlim_compose) (fact filterlim_nat_sequentially)
       thus "((\<lambda>x. EM_remainder' n (f s) (real_of_int a) (real_of_int x)) \<longlongrightarrow> ?L) at_top"
         by (rule Lim_transform_eventually)
            (auto intro: eventually_mono[OF eventually_ge_at_top[of 0]])
     qed
   qed
   finally show \<dots> .
 qed
 
 lemma tendsto_EM_remainder:
   fixes f :: "real \<Rightarrow> 'b :: {banach,real_normed_algebra}"
   assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"
   assumes integrable: "\<And>a' b . a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> 
                          (\<lambda>t. pbernpoly n t *\<^sub>R f t) integrable_on {a'..b}"
   assumes conv: "convergent (\<lambda>y. G (real y))"
   assumes bound: "eventually (\<lambda>x. norm (f x) \<le> g x) at_top"
   shows   "filterlim (\<lambda>b. EM_remainder' n f a b) 
              (nhds (EM_remainder n f a)) sequentially"
 proof -
   have "uniform_limit {()} (\<lambda>b s. EM_remainder' n f a b) 
              (\<lambda>s. EM_remainder n f a) sequentially"
     using assms by (intro uniform_limit_EM_remainder[where G = G and g = g]) auto
   moreover have "() \<in> {()}" by simp
   ultimately show ?thesis by (rule tendsto_uniform_limitI)
 qed
 
 lemma EM_remainder_0 [simp]: "EM_remainder n (\<lambda>x. 0) a = 0"
   by (rule EM_remainder_eqI) (simp add: EM_remainder'_def)
 
 lemma holomorphic_EM_remainder':
   assumes deriv: 
     "\<And>z t. z \<in> U \<Longrightarrow> t \<in> {a..x} \<Longrightarrow> 
        ((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"
   assumes int: "\<And>b c z e. a \<le> b \<Longrightarrow> c \<le> x \<Longrightarrow> z \<in> U \<Longrightarrow>
                   (\<lambda>t. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"
   assumes cont: "continuous_on (U \<times> {a..x}) (\<lambda>(z, t). f' z t)"
   assumes "convex U"
   shows "(\<lambda>s. EM_remainder' n (f s) a x) holomorphic_on U"
   unfolding EM_remainder'_def scaleR_conv_of_real
 proof (intro holomorphic_intros)
   have holo: "(\<lambda>z. integral (cbox b c) (\<lambda>t. of_real (bernpoly n (t - e)) * f z t)) holomorphic_on U"
     if "b \<ge> a" "c \<le> x" for b c e :: real
   proof (rule leibniz_rule_holomorphic)
     fix z t assume "z \<in> U" "t \<in> cbox b c"
     thus "((\<lambda>z. complex_of_real (bernpoly n (t - e)) * f z t) has_field_derivative
              complex_of_real (bernpoly n (t - e)) * f' z t) (at z within U)"
       using that by (intro DERIV_cmult deriv) auto
   next
     fix z assume "z \<in> U"
     thus "(\<lambda>t. complex_of_real (bernpoly n (t - e)) * f z t) integrable_on cbox b c"
       using that int[of b c z] by auto
   next
     have "continuous_on (U \<times> {b..c}) (\<lambda>(z, t). f' z t)"
       using cont by (rule continuous_on_subset) (insert that, auto)
     thus "continuous_on (U \<times> cbox b c) (\<lambda>(z, t).
             complex_of_real (bernpoly n (t - e)) * f' z t)"
       by (auto simp: case_prod_unfold intro!: continuous_intros)
   qed fact+
 
   consider "a > x" | "a \<le> x" "floor x \<le> a" | "a \<le> x" "floor x > a" by force
   hence "(\<lambda>z. integral (cbox a x) (\<lambda>t. of_real (pbernpoly n t) * f z t)) holomorphic_on U"
     (is "?f a x holomorphic_on _")
   proof cases
     case 2
     have "(\<lambda>z. integral (cbox a x) (\<lambda>t. of_real (bernpoly n (t - of_int \<lfloor>x\<rfloor>)) * f z t))
                    holomorphic_on U" 
       by (intro holo) auto
     also have "(\<lambda>z. integral (cbox a x) (\<lambda>t. of_real (bernpoly n (t - of_int \<lfloor>x\<rfloor>)) * f z t)) = ?f a x"
     proof (intro ext integral_cong, goal_cases)
       case (1 z t)
       hence "t \<ge> a" "t \<le> x" by auto
       hence "floor t = floor x" using 2 by linarith
       thus ?case by (simp add: pbernpoly_def frac_def)
     qed
     finally show ?thesis .
   next
     case 3
           define N :: "int set" where "N = {\<lceil>a\<rceil>..<\<lfloor>x\<rfloor>}"
     define A where "A = insert {a..of_int \<lceil>a\<rceil>} (insert {of_int \<lfloor>x\<rfloor>..x} 
                           ((\<lambda>n. {of_int n..of_int n + 1}) ` N))"
     {
       fix X assume "X \<in> A"
       then consider "X = {a..of_int \<lceil>a\<rceil>}" | "X = {of_int \<lfloor>x\<rfloor>..x}" |
              n where "X = {of_int n..of_int n + 1}" "n \<in> N" by (auto simp: A_def)
     } note A_cases = this
 
     have division: "A division_of {a..x}"
     proof (rule division_ofI)
       show "finite A" by (auto simp: A_def N_def)
       fix K assume K: "K \<in> A"
       from 3 have "of_int \<lceil>a\<rceil> \<le> x" 
         using ceiling_le[of a "floor x"] by linarith
       moreover from 3 have "of_int \<lfloor>x\<rfloor> \<ge> a" by linarith
       ultimately show "K \<subseteq> {a..x}" using K 3 by (auto simp: A_def N_def) linarith+
       from K show "K \<noteq> {}" and "\<exists>a b. K = cbox a b" by (auto simp: A_def)
     next
       fix K1 K2 assume K: "K1 \<in> A" "K2 \<in> A" "K1 \<noteq> K2"
       have F1: "interior {a..\<lceil>a\<rceil>} \<inter> interior {\<lfloor>x\<rfloor>..x} = {}" using 3 ceiling_le[of a "floor x"]
         by (auto simp: min_def max_def) 
       hence F2: "interior {\<lfloor>x\<rfloor>..x} \<inter> interior {a..\<lceil>a\<rceil>} = {}" by simp
       have F3: "interior {a..\<lceil>a\<rceil>} \<inter> interior {of_int n..of_int n+1} = {}"
                "interior {\<lfloor>x\<rfloor>..x} \<inter> interior {of_int n..of_int n+1} = {}"
                "interior {of_int n..of_int n+1} \<inter> interior {a..\<lceil>a\<rceil>} = {}"
                "interior {of_int n..of_int n+1} \<inter> interior {\<lfloor>x\<rfloor>..x} = {}"if "n \<in> N" for n
         using 3 ceiling_le[of a "floor x"] that by (auto simp: min_def max_def N_def)
       have F4: "interior {real_of_int n..of_int n+1} \<inter> interior {of_int m..of_int m+1} = {}"
         if "{real_of_int n..of_int n+1} \<noteq> {of_int m..of_int m+1}" for m n
       proof -
         from that have "n \<noteq> m" by auto
         thus ?thesis by simp
       qed
       from F1 F2 F3 F4 K show "interior K1 \<inter> interior K2 = {}"
         by (elim A_cases) (simp_all only: not_False_eq_True)
     next
       show "\<Union>A = {a..x}"
       proof (cases "\<lceil>a\<rceil> = \<lfloor>x\<rfloor>")
         case True
         thus ?thesis using 3 by (auto simp: A_def N_def intro: order.trans) linarith+
       next
         case False
         with 3 have *: "\<lceil>a\<rceil> < \<lfloor>x\<rfloor>" by linarith
         have "\<Union>A = {a..of_int \<lceil>a\<rceil>} \<union> (\<Union>n\<in>N. {of_int n..of_int (n + 1)}) \<union> {of_int \<lfloor>x\<rfloor>..x}"
           by (simp add: A_def Un_ac)
         also have "(\<Union>n\<in>N. {of_int n..of_int (n + 1)}) = {of_int \<lceil>a\<rceil>..real_of_int \<lfloor>x\<rfloor>}"
           using * unfolding N_def by (intro Union_atLeastAtMost_real_of_int)
         also have "{a..of_int \<lceil>a\<rceil>} \<union> \<dots> = {a..real_of_int \<lfloor>x\<rfloor>}"
           using 3 * by (intro ivl_disj_un) auto
         also have "\<dots> \<union> {of_int \<lfloor>x\<rfloor>..x} = {a..x}"
           using 3 * by (intro ivl_disj_un) auto
         finally show ?thesis .
       qed
     qed
 
 
     have "(\<lambda>z. \<Sum>X\<in>A. integral X (\<lambda>t. of_real (bernpoly n (t - \<lfloor>Inf X\<rfloor>)) * f z t)) 
             holomorphic_on U"
     proof (intro holomorphic_on_sum holo, goal_cases)
       case (1 X)
       from 1 and division have subset: "X \<subseteq> {a..x}" by (auto simp: division_of_def)
       from 1 obtain b c where [simp]: "X = cbox b c" "b \<le> c" by (auto simp: A_def)
       from subset have "b \<ge> a" "c \<le> x" by auto
       hence "(\<lambda>x. integral (cbox b c) (\<lambda>t. of_real (bernpoly n (t - \<lfloor>Inf {b..c}\<rfloor>)) * f x t)) 
                holomorphic_on U" by (intro holo) auto
       thus ?case by simp
     qed
     also have "?this \<longleftrightarrow> (\<lambda>z. integral {a..x} (\<lambda>t. of_real (pbernpoly n t) * f z t)) 
                            holomorphic_on U"
     proof (intro holomorphic_cong refl, goal_cases)
       case (1 z)
       have "((\<lambda>t. of_real (pbernpoly n t) * f z t) has_integral
               (\<Sum>X\<in>A. integral X (\<lambda>t. of_real (bernpoly n (t - \<lfloor>Inf X\<rfloor>)) * f z t))) {a..x}"
         using division
       proof (rule has_integral_combine_division)
         fix X assume X: "X \<in> A"
         then obtain b c where X': "X = {b..c}" "b \<le> c" by (elim A_cases) auto
         from X and division have "X \<subseteq> {a..x}" by (auto simp: division_of_def)
         with X' have bc: "b \<ge> a" "c \<le> x" by auto
         have "((\<lambda>t. of_real (bernpoly n (t - of_int \<lfloor>Inf X\<rfloor>)) * f z t) has_integral
                  integral X (\<lambda>t. of_real (bernpoly n (t - of_int \<lfloor>Inf X\<rfloor>)) * f z t)) X"
           unfolding X' using \<open>z \<in> U\<close> bc by (intro integrable_integral int)
         also have "?this \<longleftrightarrow> ((\<lambda>t. of_real (pbernpoly n t) * f z t) has_integral
                  integral X (\<lambda>t. of_real (bernpoly n (t - of_int \<lfloor>Inf X\<rfloor>)) * f z t)) X"
         proof (rule has_integral_spike_eq[of "{Sup X}"], goal_cases)
           case (2 t)
           note t = this
           from \<open>X \<in> A\<close> have "\<lfloor>t\<rfloor> = \<lfloor>Inf X\<rfloor>"
           proof (cases rule: A_cases [consumes 1])
             case 1
             with t show ?thesis
               by (intro floor_unique) (auto simp: ceiling_altdef split: if_splits, (linarith+)?)
           next
             case 2
             with t show ?thesis
               by (intro floor_unique) (auto simp: ceiling_altdef split: if_splits, (linarith+)?)
           next
             case 3
             with t show ?thesis
               by (intro floor_unique) (auto simp: ceiling_altdef N_def split: if_splits)
           qed
           thus ?case by (simp add: pbernpoly_def frac_def)
         qed auto
         finally show \<dots> .
       qed
       thus ?case by (simp add: has_integral_iff)
     qed
     finally show ?thesis by simp
   qed auto
   thus "(\<lambda>z. integral {a..x} (\<lambda>t. of_real (pbernpoly n t) * f z t)) holomorphic_on U"
     by simp
 qed
 
 lemma
   assumes deriv: "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"
   assumes deriv': 
     "\<And>z t x. z \<in> U \<Longrightarrow> x \<ge> a \<Longrightarrow> t \<in> {a..x} \<Longrightarrow> 
        ((\<lambda>z. f z t) has_field_derivative f' z t) (at z within U)"
   assumes cont: "continuous_on (U \<times> {of_int a..}) (\<lambda>(z, t). f' z t)"
   assumes int: "\<And>b c z e. a \<le> b \<Longrightarrow> z \<in> U \<Longrightarrow>
                   (\<lambda>t. of_real (bernpoly n (t - e)) * f z t) integrable_on {b..c}"
   assumes int': "\<And>a' b y. y \<in> U \<Longrightarrow>  a \<le> a' \<Longrightarrow> a' \<le> b \<Longrightarrow> 
                          (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"
   assumes conv: "convergent (\<lambda>y. G (real y))"
   assumes bound: "eventually (\<lambda>x. \<forall>y\<in>U. norm (f y x) \<le> g x) at_top"
   assumes "open U" 
   shows analytic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) analytic_on U"
     and holomorphic_EM_remainder: "(\<lambda>s::complex. EM_remainder n (f s) a) holomorphic_on U"
 proof -
   show "(\<lambda>s::complex. EM_remainder n (f s) a) analytic_on U"
   unfolding analytic_on_def
   proof
     fix z assume "z \<in> U"
     from \<open>z \<in> U\<close> and \<open>open U\<close> obtain \<epsilon> where \<epsilon>: "\<epsilon> > 0" "ball z \<epsilon> \<subseteq> U"
       by (auto simp: open_contains_ball)
     have "(\<lambda>s. EM_remainder n (f s) a) holomorphic_on ball z \<epsilon>"
     proof (rule holomorphic_uniform_sequence)
       fix x :: nat
       show "(\<lambda>s. EM_remainder' n (f s) a x) holomorphic_on ball z \<epsilon>"
       proof (rule holomorphic_EM_remainder', goal_cases)
         fix s t assume "s \<in> ball z \<epsilon>" "t \<in> {real_of_int a..real x}"
         thus "((\<lambda>z. f z t) has_field_derivative f' s t) (at s within ball z \<epsilon>)"
           using \<epsilon> by (intro DERIV_subset[OF deriv'[of _ x]]) auto
       next
         case (2 b c s e)
         with \<epsilon> have "s \<in> U" by blast
         with 2 show ?case using \<epsilon> int[of b s e c] by (cases "a \<le> x") auto
       next
         from cont show "continuous_on (ball z \<epsilon> \<times> {real_of_int a..real x}) (\<lambda>(z, t). f' z t)"
           by (rule continuous_on_subset) (insert \<epsilon>, auto)
       qed (auto)
     next
       fix s assume s: "s \<in> ball z \<epsilon>"
       have "open (ball z \<epsilon>)" by simp
       with s obtain \<delta> where \<delta>: "\<delta> > 0" "cball s \<delta> \<subseteq> ball z \<epsilon>" 
         unfolding open_contains_cball by blast
       moreover have bound': "eventually (\<lambda>x. \<forall>y\<in>cball s \<delta>. norm (f y x) \<le> g x) at_top"
         by (intro eventually_mono [OF bound]) (insert \<delta> \<epsilon>, auto)
       have "uniform_limit (cball s \<delta>) (\<lambda>x s. EM_remainder' n (f s) (real_of_int a) (real x))
                         (\<lambda>s. EM_remainder n (f s) a) sequentially"
         by (rule uniform_limit_EM_remainder[OF deriv int' conv bound']) (insert \<delta> \<epsilon> s, auto)
       ultimately show "\<exists>\<delta>>0. cball s \<delta> \<subseteq> ball z \<epsilon> \<and> uniform_limit (cball s \<delta>)
                           (\<lambda>x s. EM_remainder' n (f s) (real_of_int a) (real x))
                           (\<lambda>s. EM_remainder n (f s) a) sequentially" by blast
     qed auto
     with \<epsilon> show "\<exists>\<epsilon>>0. (\<lambda>s. EM_remainder n (f s) a) holomorphic_on ball z \<epsilon>"
       by blast
   qed
   thus "(\<lambda>s::complex. EM_remainder n (f s) a) holomorphic_on U"
     by (rule analytic_imp_holomorphic)
 qed
 
 
 
 text \<open>
   The following lemma is the first step in the proof of the Euler--MacLaurin formula: 
   We show the relationship between the first-order remainder term and the difference of 
   the integral and the sum.
 \<close>
   
 context
   fixes f f' :: "real \<Rightarrow> 'a :: banach"
   fixes a b :: int and I S :: 'a
   fixes Y :: "real set"
   assumes "a \<le> b"
   assumes fin: "finite Y"
   assumes cont: "continuous_on {real_of_int a..real_of_int b} f"
   assumes deriv [derivative_intros]: 
             "\<And>x::real. x \<in> {a..b} - Y \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
   defines S_def: "S \<equiv> (\<Sum>i\<in>{a<..b}. f i)" and I_def: "I \<equiv> integral {a..b} f"
 begin
   
 lemma
   diff_sum_integral_has_integral_int:
     "((\<lambda>t. (frac t - 1/2) *\<^sub>R f' t) has_integral (S - I - (f b - f a) /\<^sub>R 2)) {a..b}"
 proof (cases "a = b")
   case True
   thus ?thesis by (simp_all add: S_def I_def has_integral_refl)
 next
   case False
   with \<open>a \<le> b\<close> have ab: "a < b" by simp
   let ?A = "(\<lambda>n. {real_of_int n..real_of_int (n+1)}) ` {a..<b}"
   have division: "?A division_of {of_int a..of_int b}"
     using Union_atLeastAtMost_real_of_int[OF ab] by (simp add: division_of_def)
   have cont' [continuous_intros]: "continuous_on A f" if "A \<subseteq> {of_int a..of_int b}" for A
     using continuous_on_subset[OF cont that] .
   
   define d where "d = (\<lambda>x. (f x + f (x + 1)) /\<^sub>R 2 - integral {x..x+1} f)"    
   have "((\<lambda>t. (frac t - 1/2) *\<^sub>R f' t) has_integral d i) {of_int i..of_int (i+1)}"
     if i: "i \<in> {a..<b}" for i
   proof (rule has_integral_spike)
     show "(frac x - 1 / 2) *\<^sub>R f' x = (x - of_int i - 1 / 2) *\<^sub>R f' x"
          if "x \<in> {of_int i..of_int (i + 1)} - {of_int (i + 1)}" for x
     proof -
       have "x \<ge> of_int i" "x < of_int (i + 1)" using that by auto
       hence "floor x = of_int i" by (subst floor_unique) auto
       thus ?thesis by (simp add: frac_def)
     qed
   next
     define h where "h = (\<lambda>x::real. (x - of_int i - 1 / 2) *\<^sub>R f' x)"
     define g where "g = (\<lambda>x::real. (x - of_int i - 1/2) *\<^sub>R f x - integral {of_int i..x} f)"
     have *: "((\<lambda>x. integral {real_of_int i..x} f) has_vector_derivative f x) (at x within {i..i+1})"
       if "x \<in> {of_int i<..<of_int i + 1}" for x using that i
       by (intro integral_has_vector_derivative cont') auto
     have "((\<lambda>x. integral {real_of_int i..x} f) has_vector_derivative f x) (at x)" 
       if "x \<in> {of_int i<..<of_int i + 1}" for x 
       using that i at_within_interior[of x "{of_int i..of_int (i + 1)}"] *[of x] by simp
     hence "(h has_integral g (of_int (i + 1)) - g (of_int i)) {of_int i..of_int (i+1)}"
       unfolding g_def h_def using that 
       by (intro fundamental_theorem_of_calculus_interior_strong[OF fin])
          (auto intro!: derivative_eq_intros continuous_intros indefinite_integral_continuous_1
              integrable_continuous_real)
     also have "g (of_int (i + 1)) - g (of_int i) = d i"
       by (simp add: g_def scaleR_add_right [symmetric] d_def)
     finally show "(h has_integral d i) {of_int i..of_int (i + 1)}" .
   qed simp_all
   hence *: "\<And>I. I\<in>?A \<Longrightarrow> ((\<lambda>x. (frac x - 1 / 2) *\<^sub>R f' x) has_integral d (\<lfloor>Inf I\<rfloor>)) I"
     by (auto simp: add_ac)
   have "((\<lambda>x::real. (frac x - 1 / 2) *\<^sub>R f' x) has_integral (\<Sum>I\<in>?A. d (\<lfloor>Inf I\<rfloor>))) (\<Union>?A)"
     by (intro has_integral_Union * finite_imageI) (force intro!: negligible_atLeastAtMostI pairwiseI)+
   also have "\<Union>?A = {of_int a..of_int b}"
     by (intro Union_atLeastAtMost_real_of_int ab)
   also have "(\<Sum>I\<in>?A. d (\<lfloor>Inf I\<rfloor>)) = (\<Sum>i=a..<b. d i)"
     by (subst sum.reindex) (auto simp: inj_on_def)
   also have "\<dots> = (1 / 2) *\<^sub>R ((\<Sum>i = a..<b. f (real_of_int i)) +
                     (\<Sum>i = a..<b. f (real_of_int (i + 1)))) -
                   (\<Sum>i = a..<b. integral {real_of_int i..1 + real_of_int i} f)"
     (is "_ = _ *\<^sub>R (?S1 + ?S2) - ?S3") 
     by (simp add: d_def algebra_simps sum.distrib sum_subtractf scaleR_sum_right)
   also have "?S1 = (\<Sum>i = a..b. f (real_of_int i)) - f b"
     unfolding S_def using ab by (subst sum_atLeastAtMost_int_last) auto
   also have "(\<Sum>i = a..b. f (real_of_int i)) = S + f a"
     unfolding S_def using ab by (subst sum_atLeastAtMost_int_head) auto
   also have "?S2 = S" unfolding S_def
     by (intro sum.reindex_bij_witness[of _ "\<lambda>i. i-1" "\<lambda>i. i+1"]) auto
   also have "(1 / 2) *\<^sub>R (S + f a - f b + S) = 
                 (1/2) *\<^sub>R S + (1/2) *\<^sub>R S - (f b - f a) /\<^sub>R 2"
     by (simp add: algebra_simps)
   also have "(1/2) *\<^sub>R S + (1/2) *\<^sub>R S = S" by (simp add: scaleR_add_right [symmetric])
       
   also have "?S3 = (\<Sum>I\<in>?A. integral I f)"
     by (subst sum.reindex) (auto simp: inj_on_def add_ac)
   also have "\<dots> = I" unfolding I_def
     by (intro integral_combine_division_topdown [symmetric] division integrable_continuous_real 
               continuous_intros) simp_all
   finally show ?thesis by (simp add: algebra_simps)
 qed
 
 lemma diff_sum_integral_has_integral_int':
   "((\<lambda>t. pbernpoly 1 t *\<^sub>R f' t) has_integral (S - I - (f b - f a) /\<^sub>R 2 )) {a..b}"
   using diff_sum_integral_has_integral_int by (simp add: pbernpoly_def bernpoly_def)
   
 lemma EM_remainder'_Suc_0: "EM_remainder' (Suc 0) f' a b = S - I - (f b - f a) /\<^sub>R 2"
   using diff_sum_integral_has_integral_int' by (simp add: has_integral_iff EM_remainder'_def)
 
 end
 
 
 text \<open>
   Next, we show that the $n$-th-order remainder can be expressed in terms of the $n+1$-th-order 
   remainder term. Iterating this essentially yields the Euler--MacLaurin formula.
 \<close>
 
 context
   fixes f f' :: "real \<Rightarrow> 'a :: banach" and a b :: int and n :: nat and A :: "real set"
   assumes ab: "a \<le> b" and n: "n > 0"
   assumes fin:   "finite A"
   assumes cont:  "continuous_on {of_int a..of_int b} f"
   assumes cont': "continuous_on {of_int a..of_int b} f'"
   assumes deriv: "\<And>x. x \<in> {of_int a<..<of_int b} - A \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
 begin
   
 lemma EM_remainder'_integral_conv_Suc:
   shows   "integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t) =
               (bernoulli (Suc n) / real (Suc n)) *\<^sub>R (f b - f a) -
               integral {a..b} (\<lambda>t. pbernpoly (Suc n) t *\<^sub>R f' t) /\<^sub>R real (Suc n)"
   unfolding EM_remainder'_def
 proof -
   let ?h = "\<lambda>i. (pbernpoly (Suc n) (real_of_int i) / real (Suc n)) *\<^sub>R f (real_of_int i)"
   define T where "T = integral {a..b} (\<lambda>t. (pbernpoly (Suc n) t / real (Suc n)) *\<^sub>R f' t)"
   note [derivative_intros] = has_field_derivative_pbernpoly_Suc'
   let ?A = "real_of_int ` {a..b} \<union> A"
     
   have "((\<lambda>t. pbernpoly n t *\<^sub>R f t) has_integral (-T + (?h b - ?h a))) {a..b}"
   proof (rule integration_by_parts_interior_strong[OF bounded_bilinear_scaleR])
     from fin show "finite ?A" by simp
     from \<open>n > 0\<close> show "continuous_on {of_int a..of_int b} (\<lambda>t. pbernpoly (Suc n) t / real (Suc n))"
       by (intro continuous_intros) auto
     show "continuous_on {of_int a..of_int b} f" by fact
     show "(f has_vector_derivative f' t) (at t)" if "t \<in> {of_int a<..<of_int b} - ?A" for t
       using deriv[of t] that by auto
     have "(\<lambda>t. pbernpoly (Suc n) t *\<^sub>R f' t) integrable_on {a..b}"
       by (intro integrable_EM_remainder' cont')
     hence "(\<lambda>t. (1 / real (Suc n)) *\<^sub>R pbernpoly (Suc n) t *\<^sub>R f' t) integrable_on {a..b}" 
       by (rule integrable_cmul)
     also have "(\<lambda>t. (1 / real (Suc n)) *\<^sub>R pbernpoly (Suc n) t *\<^sub>R f' t) =
                  (\<lambda>t. (pbernpoly (Suc n) t / real (Suc n)) *\<^sub>R f' t)"
       by (rule ext) (simp add: algebra_simps)
     finally show "((\<lambda>t. (pbernpoly (Suc n) t / real (Suc n)) *\<^sub>R f' t) 
                     has_integral ?h b - ?h a - (- T + (?h b - ?h a))) {a..b}"
       using integrable_EM_remainder'[of a b f' "Suc n"]
         by (simp add: has_integral_iff T_def)
     qed (insert ab n, auto intro!: derivative_eq_intros
            simp: has_field_derivative_iff_has_vector_derivative [symmetric] not_le elim!: Ints_cases)
     also have "?h b - ?h a = (bernoulli (Suc n) / real (Suc n)) *\<^sub>R (f b - f a)"
       using n by (simp add: algebra_simps bernoulli'_def)
     finally have "integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t) = \<dots> - T"
       by (simp add: has_integral_iff)
     also have "T = integral {a..b} (\<lambda>t. (1 / real (Suc n)) *\<^sub>R (pbernpoly (Suc n) t) *\<^sub>R f' t)"
       by (simp add: T_def)
     also have "\<dots> = integral {a..b} (\<lambda>t. pbernpoly (Suc n) t *\<^sub>R f' t) /\<^sub>R real (Suc n)"
       by (subst integral_cmul) (simp_all add: divide_simps)
     finally show ?thesis .
 qed 
 
 lemma EM_remainder'_conv_Suc: 
   "EM_remainder' n f a b = 
      ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (f b - f a) + 
      EM_remainder' (Suc n) f' a b"
   by (simp add: EM_remainder'_def EM_remainder'_integral_conv_Suc scaleR_diff_right 
                 scaleR_add_right field_simps del: of_nat_Suc)
 
 end
 
 context
   fixes f f' :: "real \<Rightarrow> 'a :: banach" and a :: int and n :: nat and A :: "real set" and C
   assumes n: "n > 0"
   assumes fin:   "finite A"
   assumes cont:  "continuous_on {of_int a..} f"
   assumes cont': "continuous_on {of_int a..} f'"
   assumes lim:   "(f \<longlongrightarrow> C) at_top"
   assumes deriv: "\<And>x. x \<in> {of_int a<..} - A \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
 begin  
   
 lemma
   shows EM_remainder_converges_iff_Suc_converges:
           "EM_remainder_converges n f a \<longleftrightarrow> EM_remainder_converges (Suc n) f' a"
   and   EM_remainder_conv_Suc: 
            "EM_remainder_converges n f a \<Longrightarrow> 
               EM_remainder n f a = 
                   ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (C - f a) + 
                   EM_remainder (Suc n) f' a"
 proof (rule iffI)
   define g where "g = (\<lambda>x. ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (f x - f a))"
   define G where "G = ((-1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (C - f a)"
   have limit_g: "(g \<longlongrightarrow> G) at_top" unfolding g_def G_def by (intro tendsto_intros lim)
   have *: "eventually (\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x) = 
               g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)) at_top"
     using eventually_ge_at_top[of a]
   proof eventually_elim
     case (elim b)
     thus ?case
     using EM_remainder'_conv_Suc[OF elim n fin continuous_on_subset[OF cont] 
             continuous_on_subset[OF cont'] deriv] by (auto simp: g_def)
   qed
   
   {
     assume "EM_remainder_converges n f a"
     then obtain L
       where L: "((\<lambda>b. EM_remainder' n f (real_of_int a) (real_of_int b)) \<longlongrightarrow> L) at_top"
       by (auto simp: EM_remainder_converges_def)
     have *: "((\<lambda>b. EM_remainder' (Suc n) f' (real_of_int a) (real_of_int b)) \<longlongrightarrow> L - G) at_top"
     proof (rule Lim_transform_eventually)
       show "\<forall>\<^sub>F x in at_top. EM_remainder' n f (real_of_int a) (real_of_int x) - g x =
                EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)"
         using * by (simp add: algebra_simps)
       show "((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x) - g x) \<longlongrightarrow> L - G) at_top"
         by (intro tendsto_intros filterlim_compose[OF limit_g] L)
     qed
     from * show "EM_remainder_converges (Suc n) f' a" unfolding EM_remainder_converges_def ..
     from * have "EM_remainder (Suc n) f' a = L - G" by (rule EM_remainder_eqI)
     moreover from L have "EM_remainder n f a = L" by (rule EM_remainder_eqI)
     ultimately show "EM_remainder n f a = G + EM_remainder (Suc n) f' a" by (simp add: G_def)
   }
   {
     assume "EM_remainder_converges (Suc n) f' a" 
     then obtain L
       where L: "((\<lambda>b. EM_remainder' (Suc n) f' (real_of_int a) (real_of_int b)) \<longlongrightarrow> L) at_top"
       by (auto simp: EM_remainder_converges_def)
     have *: "((\<lambda>b. EM_remainder' n f (real_of_int a) (real_of_int b)) \<longlongrightarrow> G + L) at_top"
     proof (rule Lim_transform_eventually)
       show "\<forall>\<^sub>F x in at_top. g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x) =
                 EM_remainder' n f (real_of_int a) (real_of_int x)"
         using * by (subst eq_commute)
       show "((\<lambda>x. g x + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int x)) \<longlongrightarrow> G + L) at_top"
         by (intro tendsto_intros filterlim_compose[OF limit_g] L)
     qed
     thus "EM_remainder_converges n f a" unfolding EM_remainder_converges_def ..
   }
 qed
 
 end
   
 
 
 subsection \<open>The conventional version of the Euler--MacLaurin formula\<close>
 
 text \<open>
   The following theorems are the classic Euler--MacLaurin formula that can be found,
   with slight variations, in many sources (e.\,g.\ \cite{AS_HMF, apostol99, GKP_CM}).
 \<close>
 
 context
   fixes f :: "real \<Rightarrow> 'a :: banach"
   fixes fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"
   fixes a b :: int assumes ab: "a \<le> b"
   fixes N :: nat assumes N: "N > 0"
   fixes Y :: "real set" assumes fin: "finite Y"
   assumes fs_0 [simp]: "fs 0 = f"
   assumes fs_cont [continuous_intros]:  
     "\<And>k. k \<le> N \<Longrightarrow> continuous_on {real_of_int a..real_of_int b} (fs k)"
   assumes fs_deriv [derivative_intros]: 
     "\<And>k x. k < N \<Longrightarrow> x \<in> {a..b} - Y \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)"
 begin
 
 theorem euler_maclaurin_raw_strong_int:
   defines "S \<equiv> (\<Sum>i\<in>{a<..b}. f (of_int i))"
   defines "I \<equiv> integral {of_int a..of_int b} f"
   defines "c' \<equiv> \<lambda>k. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k b - fs k a)"
   shows   "S - I = (\<Sum>k<N. c' k) + EM_remainder' N (fs N) a b"
 proof -
   define c :: "nat \<Rightarrow> 'a" 
     where "c = (\<lambda>k. ((-1) ^ (Suc k) * bernoulli (Suc k) / fact (Suc k)) *\<^sub>R (fs k b - fs k a))"
   have "S - I = (\<Sum>k<m. c k) + EM_remainder' m (fs m) a b" if "m \<ge> 1" "m \<le> N" for m
   using that
   proof (induction m rule: dec_induct)
     case base
     with ab fin fs_cont[of 0] show ?case using fs_deriv[of 0] N unfolding One_nat_def
       by (subst EM_remainder'_Suc_0[of _ _ Y f]) (simp_all add: algebra_simps S_def I_def c_def)
   next
     case (step n)
     from step.prems have "S - I = (\<Sum>k<n. c k) + EM_remainder' n (fs n) a b"
       by (intro step.IH) simp_all
     also have "(\<Sum>k<n. c k) = (\<Sum>k<Suc n. c k) +
                   (((-1) ^ n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (fs n b - fs n a))"
       (is "_ = _ + ?c") by (simp add: EM_remainder'_Suc_0 c_def)
     also have "\<dots> + EM_remainder' n (fs n) a b = (\<Sum>k<Suc n. c k) + (?c + EM_remainder' n (fs n) a b)"
       by (simp add: add.assoc)
     also from step.prems step.hyps ab fin
       have "?c + EM_remainder' n (fs n) a b = EM_remainder' (Suc n) (fs (Suc n)) a b"
       by (subst EM_remainder'_conv_Suc [where A = Y]) 
          (auto intro!: fs_deriv fs_cont)
     finally show ?case .
   qed
   from this[of N] and N 
     have "S - I = sum c {..<N} + EM_remainder' N (fs N) (real_of_int a) (real_of_int b)" by simp
   also have "sum c {..<N} = sum c' {..<N}"
   proof (intro sum.cong refl)
     fix k :: nat
     show "c k = c' k"
       by (cases "even k")
          (auto simp: c_def c'_def bernoulli'_def algebra_simps bernoulli_odd_eq_0)
   qed
   finally show ?thesis .
 qed
 
 end
 
 theorem euler_maclaurin_strong_raw_nat:
   assumes "a \<le> b" "0 < N" "finite Y" "fs 0 = f"
     "(\<And>k. k \<le> N \<Longrightarrow> continuous_on {real a..real b} (fs k))"
     "(\<And>k x. k < N \<Longrightarrow> x \<in> {real a..real b} - Y \<Longrightarrow>
        (fs k has_vector_derivative fs (Suc k) x) (at x))"
   shows "(\<Sum>i\<in>{a<..b}. f (real i)) - integral {real a..real b} f =
            (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (real b) - fs k (real a))) +
            EM_remainder' N (fs N) (real a) (real b)"
 proof -
   have "(\<Sum>i\<in>{int a<..int b}. f (real_of_int i)) - 
            integral {real_of_int (int a)..real_of_int (int b)} f =
            (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R 
              (fs k (real_of_int (int b)) - fs k (real_of_int (int a)))) +
            EM_remainder' N (fs N) (real_of_int (int a)) (real_of_int (int b))"
     using assms by (intro euler_maclaurin_raw_strong_int[where Y = Y] assms) simp_all
   also have "(\<Sum>i\<in>{int a<..int b}. f (real_of_int i)) = (\<Sum>i\<in>{a<..b}. f (real i))"
     by (intro sum.reindex_bij_witness[of _ int nat]) auto
   finally show ?thesis by simp
 qed
 
   
 subsection \<open>The ``Concrete Mathematics'' version of the Euler--MacLaurin formula\<close>
 
 text \<open>
   As explained in \textit{Concrete Mathematics}~\cite{GKP_CM}, the above form of the 
   formula has some drawbacks: When applying it to determine the asymptotics of some concrete 
   function, one is usually left with several different unwieldy constant terms that are difficult 
   to get rid of.
 
   There is no general way to determine what these constant terms are, but in concrete applications, 
   they can often be determined or estimated by other means. We can therefore simply group all the 
   constant terms into a single constant and have the user provide a proof of what it is.
 \<close>
 
 locale euler_maclaurin_int =
   fixes F f :: "real \<Rightarrow> 'a :: banach"
   fixes fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"
   fixes a :: int
   fixes N :: nat assumes N: "N > 0"
   fixes C :: 'a
   fixes Y :: "real set" assumes fin: "finite Y"
   assumes fs_0 [simp]: "fs 0 = f"
   assumes fs_cont [continuous_intros]:  
     "\<And>k. k \<le> N \<Longrightarrow> continuous_on {real_of_int a..} (fs k)"
   assumes fs_deriv [derivative_intros]: 
     "\<And>k x. k < N \<Longrightarrow> x \<in> {of_int a..} - Y \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)"
   assumes F_cont [continuous_intros]: "continuous_on {of_int a..} F"
   assumes F_deriv [derivative_intros]: 
     "\<And>x. x \<in> {of_int a..} - Y \<Longrightarrow> (F has_vector_derivative f x) (at x)" 
   assumes limit: 
     "((\<lambda>b. (\<Sum>k=a..b. f k) - F (of_int b) - 
          (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b))) \<longlongrightarrow> C) at_top"
 begin
 
 context
   fixes C' T
   defines "C' \<equiv> -f a + F a + C + (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (of_int a)))"
       and "T \<equiv> (\<lambda>x. \<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i x)"
 begin
   
 lemma euler_maclaurin_strong_int_aux:
   assumes ab: "a \<le> b"
   defines "S \<equiv> (\<Sum>k=a..b. f (of_int k))"
   shows "S - F (of_int b) - T (of_int b) = EM_remainder' N (fs N) (of_int a) (of_int b) + (C - C')"
 proof (cases "a = b")
   case True
   thus ?thesis unfolding C'_def by (simp add: S_def EM_remainder'_def T_def)
 next
   case False
   with assms have ab: "a < b" by simp
   define T' where "T' = (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (of_int a)))"
   have "(\<Sum>i\<in>{a<..b}. f (of_int i)) - integral {of_int a..of_int b} f =
           (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (of_int b) - fs k (of_int a))) +
           EM_remainder' N (fs N) (of_int a) (of_int b)" using ab
     by (intro euler_maclaurin_raw_strong_int [where Y = Y] N fin fs_0
               continuous_on_subset[OF fs_cont] fs_deriv) auto
   also have "(f has_integral (F b - F a)) {of_int a..of_int b}" using ab
     by (intro fundamental_theorem_of_calculus_strong[OF fin])
        (auto intro!: continuous_on_subset[OF F_cont] derivative_intros)
   hence "integral {of_int a..of_int b} f = F (of_int b) - F (of_int a)"
     by (simp add: has_integral_iff)
   also have "(\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R (fs k (of_int b) - fs k (of_int a))) =
                T (of_int b) - T'"
     by (simp add: T_def T'_def algebra_simps sum_subtractf)
   also have "(\<Sum>i\<in>{a<..b}. f (of_int i)) = S - f (of_int a)"
     unfolding S_def using ab by (subst sum_atLeastAtMost_int_head) auto
   finally show ?thesis by (simp add: algebra_simps C'_def T'_def)  
 qed
 
 lemma EM_remainder_limit:
   assumes ab: "a \<le> b"
   defines "D \<equiv> EM_remainder' N (fs N) (of_int a) (of_int b)"
   shows "EM_remainder N (fs N) b = C' - D"
     and EM_remainder_converges: "EM_remainder_converges N (fs N) b"
 proof -
   note limit
   also have "((\<lambda>b. (\<Sum>k = a..b. f (of_int k)) - F (of_int b) -
                (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b))) \<longlongrightarrow> C) at_top =
              ((\<lambda>b. (\<Sum>k = a..b. f (of_int k)) - F (of_int b) - T (of_int b)) \<longlongrightarrow> C) at_top"
     unfolding T_def ..
   also have "eventually (\<lambda>x. (\<Sum>k=a..x. f k) - F (of_int x) - T (of_int x) = 
                EM_remainder' N (fs N) (of_int a) (of_int x) + (C - C')) at_top"
     (is "eventually (\<lambda>x. ?f x = ?g x) _")
     using eventually_gt_at_top[of b]
     by eventually_elim (rule euler_maclaurin_strong_int_aux, insert ab, simp_all)
   hence "(?f \<longlongrightarrow> C) at_top \<longleftrightarrow> (?g \<longlongrightarrow> C) at_top" by (intro filterlim_cong refl)
   finally have "((\<lambda>x. ?g x - (C - C')) \<longlongrightarrow> (C - (C - C'))) at_top"
     by (rule tendsto_diff[OF _ tendsto_const])
   hence *: "((\<lambda>x. EM_remainder' N (fs N) (of_int a) (of_int x)) \<longlongrightarrow> C') at_top"
     by simp
   
   have "((\<lambda>x. EM_remainder' N (fs N) (of_int a) (of_int x) - D) \<longlongrightarrow> C' - D) at_top"
     by (intro tendsto_intros *)
   also have "eventually (\<lambda>x. EM_remainder' N (fs N) (of_int a) (of_int x) - D = 
                           EM_remainder' N (fs N) (of_int b) (of_int x)) at_top" 
     (is "eventually (\<lambda>x. ?f x = ?g x) _") using eventually_ge_at_top[of b]
   proof eventually_elim
     case (elim x)
     have "EM_remainder' N (fs N) (of_int a) (of_int x) = 
             D + EM_remainder' N (fs N) (of_int b) (of_int x)" 
       using elim ab unfolding D_def
       by (intro EM_remainder'_combine [symmetric] continuous_on_subset[OF fs_cont]) auto
     thus ?case by simp
   qed
   hence "(?f \<longlongrightarrow> C' - D) at_top \<longleftrightarrow> (?g \<longlongrightarrow> C' - D) at_top" by (intro filterlim_cong refl)
   finally have *: \<dots> .
   from * show "EM_remainder_converges N (fs N) b" unfolding EM_remainder_converges_def ..
   from * show "EM_remainder N (fs N) b = C' - D"
     by (rule EM_remainder_eqI)
 qed
 
 theorem euler_maclaurin_strong_int:
   assumes ab: "a \<le> b"
   defines "S \<equiv> (\<Sum>k=a..b. f (of_int k))"
   shows "S = F (of_int b) + C + T (of_int b) - EM_remainder N (fs N) b"
 proof -
   have "S = F (of_int b) + T (of_int b) + - (C' - EM_remainder' N (fs N) (of_int a) (of_int b)) + C"
     using euler_maclaurin_strong_int_aux[OF ab] by (simp add: algebra_simps S_def)
   also have "C' - EM_remainder' N (fs N) (of_int a) (of_int b) = EM_remainder N (fs N) b"
     using ab by (rule EM_remainder_limit(1) [symmetric])
   finally show ?thesis by simp
 qed
   
 end
 end
 
 
 text \<open>
   The following version of the formula removes all the terms where the associated
   Bernoulli numbers vanish. 
 \<close>
 
 locale euler_maclaurin_int' =
   fixes F f :: "real \<Rightarrow> 'a :: banach"
   fixes fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"
   fixes a :: int
   fixes N :: nat
   fixes C :: 'a
   fixes Y :: "real set" assumes fin: "finite Y"
   assumes fs_0 [simp]: "fs 0 = f"
   assumes fs_cont [continuous_intros]:  
     "\<And>k. k \<le> 2*N+1 \<Longrightarrow> continuous_on {real_of_int a..} (fs k)"
   assumes fs_deriv [derivative_intros]: 
     "\<And>k x. k \<le> 2*N \<Longrightarrow> x \<in> {of_int a..} - Y \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)"
   assumes F_cont [continuous_intros]: "continuous_on {of_int a..} F"
   assumes F_deriv [derivative_intros]: 
     "\<And>x. x \<in> {of_int a..} - Y \<Longrightarrow> (F has_vector_derivative f x) (at x)" 
   assumes limit: 
     "((\<lambda>b. (\<Sum>k=a..b. f k) - F (of_int b) -
          (\<Sum>i<2*N+1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b))) \<longlongrightarrow> C) at_top"
 begin
 
 sublocale euler_maclaurin_int F f fs a "2*N+1" C Y
   by standard (insert fin fs_0 fs_cont fs_deriv F_cont F_deriv limit, simp_all)
 
 theorem euler_maclaurin_strong_int':
   assumes "a \<le> b"
   shows   "(\<Sum>k=a..b. f (of_int k)) = 
              F (of_int b) + C + (1 / 2) *\<^sub>R f (of_int b) +
              (\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (of_int b)) - 
              EM_remainder (2*N+1) (fs (2*N+1)) b"
 proof -
   have "(\<Sum>k=a..b. f (real_of_int k)) = 
                F (of_int b) + C + (\<Sum>i<2*N+1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b)) - 
                EM_remainder (2*N+1) (fs (2*N+1)) b"
     by (rule euler_maclaurin_strong_int)
        (simp_all only: lessThan_Suc_atMost Suc_eq_plus1 [symmetric] assms)
   also have "{..<2*N+1} = insert 0 {1..2*N}" by auto
   also have "(\<Sum>i\<in>\<dots>. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b)) =
                (1 / 2) *\<^sub>R f (of_int b) + 
                (\<Sum>i\<in>{1..2*N}. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b))"
     by (subst sum.insert) (auto simp: assms bernoulli'_def)
   also have "(\<Sum>i\<in>{1..2*N}. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (of_int b)) =
                (\<Sum>i\<in>{1..N}. (bernoulli' (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (of_int b))"
   proof (rule sym, rule sum.reindex_bij_witness_not_neutral)
     fix i assume "i \<in> {1..2*N} - {i\<in>{1..2*N}. even i}"
     thus "2 * ((i + 1) div 2) - 1 = i" "(i + 1) div 2 \<in> {1..N} - {}" 
       by (auto elim!: oddE)
   qed (auto simp: bernoulli_odd_eq_0 bernoulli'_def algebra_simps)
   also have "\<dots> = (\<Sum>i\<in>{1..N}. (bernoulli (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (of_int b))"
     by (intro sum.cong refl) (auto simp: bernoulli'_def)
   finally show ?thesis by (simp only: add_ac)
 qed
 
 end
 
 
 text \<open>
   For convenience, we also offer a version where the sum ranges over natural numbers
   instead of integers.
 \<close>
 
 lemma sum_atLeastAtMost_of_int_nat_transfer:
   "(\<Sum>k=int a..int b. f (of_int k)) = (\<Sum>k=a..b. f (of_nat k))"
   by (intro sum.reindex_bij_witness[of _ int nat]) auto
   
 lemma euler_maclaurin_nat_int_transfer:
   fixes F and f :: "real \<Rightarrow> 'a :: real_normed_vector"
   assumes "((\<lambda>b. (\<Sum>k=a..b. f (real k)) - F (real b) - T (real b)) \<longlongrightarrow> C) at_top"
   shows   "((\<lambda>b. (\<Sum>k=int a..b. f (of_int k)) - F (of_int b) - T (of_int b)) \<longlongrightarrow> C) at_top"
 proof -
   have *: "((\<lambda>b. (\<Sum>k=int a..int b. f (of_int k)) - F (of_int (int b)) - T (of_int (int b)))
             \<longlongrightarrow> C) at_top" using assms by (subst sum_atLeastAtMost_of_int_nat_transfer) simp
   thus ?thesis by (rule filterlim_int_of_nat_at_topD)
 qed
   
 locale euler_maclaurin_nat =
   fixes F f :: "real \<Rightarrow> 'a :: banach"
   fixes fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"
   fixes a :: nat
   fixes N :: nat assumes N: "N > 0"
   fixes C :: 'a
   fixes Y :: "real set" assumes fin: "finite Y"
   assumes fs_0 [simp]: "fs 0 = f"
   assumes fs_cont [continuous_intros]:  
     "\<And>k. k \<le> N \<Longrightarrow> continuous_on {real a..} (fs k)"
   assumes fs_deriv [derivative_intros]: 
     "\<And>k x. k < N \<Longrightarrow> x \<in> {real a..} - Y \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)"
   assumes F_cont [continuous_intros]: "continuous_on {real a..} F"
   assumes F_deriv [derivative_intros]: 
     "\<And>x. x \<in> {real a..} - Y \<Longrightarrow> (F has_vector_derivative f x) (at x)" 
   assumes limit: 
     "((\<lambda>b. (\<Sum>k=a..b. f k) - F (real b) - 
          (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlongrightarrow> C) at_top"
 begin
 
 sublocale euler_maclaurin_int F f fs "int a" N C Y
   by standard (insert N fin fs_cont fs_deriv F_cont F_deriv 
                  euler_maclaurin_nat_int_transfer[OF limit], simp_all)
 
 theorem euler_maclaurin_strong_nat:
   assumes ab: "a \<le> b"
   defines "S \<equiv> (\<Sum>k=a..b. f (real k))"
   shows "S = F (real b) + C + (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b)) - 
                EM_remainder N (fs N) (int b)"
   using euler_maclaurin_strong_int[of "int b"]
   by (simp add: assms sum_atLeastAtMost_of_int_nat_transfer)
 
 end
 
 locale euler_maclaurin_nat' =
   fixes F f :: "real \<Rightarrow> 'a :: banach"
   fixes fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"
   fixes a :: nat
   fixes N :: nat
   fixes C :: 'a
   fixes Y :: "real set" assumes fin: "finite Y"
   assumes fs_0 [simp]: "fs 0 = f"
   assumes fs_cont [continuous_intros]:  
     "\<And>k. k \<le> 2*N+1 \<Longrightarrow> continuous_on {real a..} (fs k)"
   assumes fs_deriv [derivative_intros]: 
     "\<And>k x. k \<le> 2*N \<Longrightarrow> x \<in> {real a..} - Y \<Longrightarrow> (fs k has_vector_derivative fs (Suc k) x) (at x)"
   assumes F_cont [continuous_intros]: "continuous_on {real a..} F"
   assumes F_deriv [derivative_intros]: 
     "\<And>x. x \<in> {real a..} - Y \<Longrightarrow> (F has_vector_derivative f x) (at x)" 
   assumes limit: 
     "((\<lambda>b. (\<Sum>k=a..b. f k) - F (real b) - 
         (\<Sum>i<2*N+1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlongrightarrow> C) at_top"
 begin
 
 sublocale euler_maclaurin_int' F f fs "int a" N C Y
   by standard (insert fin fs_cont fs_deriv F_cont F_deriv 
                  euler_maclaurin_nat_int_transfer[OF limit], simp_all)
 
 theorem euler_maclaurin_strong_nat':
   assumes "a \<le> b"
   shows   "(\<Sum>k=a..b. f (real k)) = 
              F (real b) + C + (1 / 2) *\<^sub>R f (real b) +
              (\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (real b)) - 
              EM_remainder (2*N+1) (fs (2*N+1)) b"
   using euler_maclaurin_strong_int'[of b]
   by (simp add: assms sum_atLeastAtMost_of_int_nat_transfer)
     
 end
 
 
 subsection \<open>Bounds on the remainder term\<close>
 
 text \<open>
   The following theorems provide some simple means to bound the remainder terms.
   In practice, better bounds can often be obtained e.\,g. for the $n$-th remainder term
   by expanding it to the sum of the first discarded term in the expansion and the $n+1$-th 
   remainder term.
 \<close>
 
 lemma
   fixes f :: "real \<Rightarrow> 'a :: {real_normed_field, banach}"
     and g g' :: "real \<Rightarrow> real"
   assumes fin:     "finite Y"
   assumes pbernpoly_bound: "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"
   assumes cont_f:  "continuous_on {a..} f"
   assumes cont_g:  "continuous_on {a..} g"
   assumes cont_g': "continuous_on {a..} g'"
   assumes limit_g: "(g \<longlongrightarrow> C) at_top"
   assumes f_bound: "\<And>x. x \<ge> a \<Longrightarrow> norm (f x) \<le> g' x"
   assumes deriv:   "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_field_derivative g' x) (at x)"
   shows   norm_EM_remainder_le_strong_int:
             "\<forall>x. of_int x \<ge> a \<longrightarrow> norm (EM_remainder n f x) \<le> D / fact n * (C - g x)"
   and     norm_EM_remainder_le_strong_nat:
             "\<forall>x. real x \<ge> a \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * (C - g x)"
 proof -
   from pbernpoly_bound have D: "norm (pbernpoly n x) \<le> D" for x by auto 
   from this[of 0] have D_nonneg: "D \<ge> 0" by simp
   define D' where "D' = D / fact n"
   from D_nonneg have D'_nonneg: "D' \<ge> 0" by (simp add: D'_def)
 
   have bound: "norm (EM_remainder' n f x y) \<le> D' * (g y - g x)"
     if xy: "x \<ge> a" "x \<le> y" for x y :: real
   proof -
     have "norm (EM_remainder' n f x y) = norm (integral {x..y} (\<lambda>t. pbernpoly n t *\<^sub>R f t)) / fact n"
       by (simp add: EM_remainder'_def)
     also have "(\<lambda>t. D * g' t) integrable_on {x..y}" using xy
       by (intro integrable_continuous_real continuous_intros continuous_on_subset[OF cont_g'])
          auto
     hence "norm (integral {x..y} (\<lambda>t. pbernpoly n t *\<^sub>R f t)) \<le>
             integral {x..y} (\<lambda>t. D * g' t)" using D D_nonneg xy
       by (intro integral_norm_bound_integral integrable_EM_remainder' 
                continuous_on_subset[OF cont_f]) (auto intro!: mult_mono f_bound)
     also have "\<dots> = D * integral {x..y} g'" by simp
     also have "(g' has_integral (g y - g x)) {x..y}" using xy
       by (intro fundamental_theorem_of_calculus_strong[OF fin] continuous_on_subset[OF cont_g])
          (auto simp: has_field_derivative_iff_has_vector_derivative [symmetric] intro!: deriv)
     hence "integral {x..y} g' = g y - g x" by (simp add: has_integral_iff)
     finally show ?thesis by (simp add: D'_def divide_simps)
   qed
   
   have lim: "((\<lambda>y. EM_remainder' n f x (of_int y)) \<longlongrightarrow> EM_remainder n f x) at_top" 
     if x: "x \<ge> a" for x :: int
   proof -
     have "(\<lambda>n. g (real n)) \<longlonglongrightarrow> C" 
       by (rule filterlim_compose[OF limit_g filterlim_real_sequentially])
     hence Cauchy: "Cauchy (\<lambda>n. g (real n))" using convergent_eq_Cauchy by blast
     have "Cauchy (\<lambda>y. EM_remainder' n f x (int y))"
     proof (rule CauchyI', goal_cases)
       case (1 \<epsilon>)
       define \<epsilon>' where "\<epsilon>' = (if D' = 0 then 1 else \<epsilon> / (2*D'))"
       from \<open>\<epsilon> > 0\<close> D'_nonneg have \<epsilon>': "\<epsilon>' > 0" by (simp add: \<epsilon>'_def divide_simps)
       from CauchyD[OF Cauchy this] obtain M 
         where M: "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> norm (g (real m) - g (real n)) < \<epsilon>'" by blast
       show ?case
       proof (intro CauchyI' exI[of _ "max (max 0 M) (nat x)"] allI impI, goal_cases)
         case (1 k l)
         have "EM_remainder' n f x k + EM_remainder' n f k l = EM_remainder' n f x l"
           using 1 x by (intro EM_remainder'_combine continuous_on_subset[OF cont_f]) auto
         hence "EM_remainder' n f x l - EM_remainder' n f x k = EM_remainder' n f k l" 
           by (simp add: algebra_simps)
         also from 1 x have "norm \<dots> \<le> D' * (g l - g k)" by (intro bound) auto
         also have "g l - g k \<le> norm (g l - g k)" by simp
         also from 1 have "\<dots> \<le> \<epsilon>'" using M[of l k] by auto
         also from \<open>\<epsilon> > 0\<close> have "D' * \<epsilon>' \<le> \<epsilon> / 2" by (simp add: \<epsilon>'_def)
         also from \<open>\<epsilon> > 0\<close> have "\<dots> < \<epsilon>" by simp
         finally show ?case by (simp add: D'_nonneg mult_left_mono dist_norm norm_minus_commute)
       qed
     qed
     then obtain L where "(\<lambda>y. EM_remainder' n f x (int y)) \<longlonglongrightarrow> L"
       by (auto simp: Cauchy_convergent_iff convergent_def)
     from filterlim_int_of_nat_at_topD[OF this]
       have "((\<lambda>y. EM_remainder' n f x (of_int y)) \<longlongrightarrow> L) at_top" by simp
     moreover from this have "EM_remainder n f x = L" by (rule EM_remainder_eqI)
     ultimately show "((\<lambda>y. EM_remainder' n f x (of_int y)) \<longlongrightarrow> EM_remainder n f x) at_top"
       by simp
   qed
   
   have *: "norm (EM_remainder n f x) \<le> D' * (C - g x)" if x: "x \<ge> a" for x :: int
   proof (rule tendsto_le)
     show "((\<lambda>y. D' * (g (of_int y) - g (of_int x))) \<longlongrightarrow> D' * (C - g (of_int x))) at_top"
       by (intro tendsto_intros filterlim_compose[OF limit_g])
     show "((\<lambda>y. norm (EM_remainder' n f x (of_int y))) \<longlongrightarrow> norm (EM_remainder n f x)) at_top"
       using x by (intro tendsto_norm lim)
     show "eventually (\<lambda>y. norm (EM_remainder' n f (of_int x) (of_int y))
              \<le> D' * (g (of_int y) - g (of_int x))) at_top" 
       using eventually_ge_at_top[of x] by eventually_elim (rule bound, insert x, simp_all)
   qed simp_all
   thus "\<forall>x. of_int x \<ge> a \<longrightarrow> norm (EM_remainder n f x) \<le> D' * (C - g x)" by blast
   
   have "norm (EM_remainder n f x) \<le> D' * (C - g x)" if x: "x \<ge> a" for x :: nat
     using x *[of "int x"] by simp
   thus "\<forall>x. real x \<ge> a \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D' * (C - g x)" by blast
 qed
 
 lemma
   fixes f :: "real \<Rightarrow> 'a :: {real_normed_field, banach}"
     and g g' :: "real \<Rightarrow> real"
   assumes fin:     "finite Y"
   assumes pbernpoly_bound: "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"
   assumes cont_f:  "continuous_on {a..} f"
   assumes cont_g:  "continuous_on {a..} g"
   assumes cont_g': "continuous_on {a..} g'"
   assumes limit_g: "(g \<longlongrightarrow> 0) at_top"
   assumes f_bound: "\<And>x. x \<ge> a \<Longrightarrow> norm (f x) \<le> g' x"
   assumes deriv:   "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_field_derivative -g' x) (at x)"
   shows   norm_EM_remainder_le_strong_int': 
             "\<forall>x. of_int x \<ge> a \<longrightarrow> norm (EM_remainder n f x) \<le> D / fact n * g x"
     and   norm_EM_remainder_le_strong_nat': 
             "\<forall>x. real x \<ge> a \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * g x"
 proof -
   have "\<forall>x. of_int x \<ge> a \<longrightarrow> norm (EM_remainder n f x) \<le> D / fact n * (0 - (-g x))" using assms
     by (intro norm_EM_remainder_le_strong_int[OF fin pbernpoly_bound _ _ cont_g'])
        (auto intro!: continuous_intros tendsto_eq_intros derivative_eq_intros)
   thus "\<forall>x. of_int x \<ge> a \<longrightarrow> norm (EM_remainder n f x) \<le> D / fact n * g x" by auto
 next
   have "\<forall>x. real x \<ge> a \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * (0 - (-g x))" using assms
     by (intro norm_EM_remainder_le_strong_nat[OF fin pbernpoly_bound _ _ cont_g'])
        (auto intro!: continuous_intros tendsto_eq_intros derivative_eq_intros)
   thus "\<forall>x. real x \<ge> a \<longrightarrow> norm (EM_remainder n f (int x)) \<le> D / fact n * g x" by auto
 qed
 
 lemma norm_EM_remainder'_le:
   fixes f :: "real \<Rightarrow> 'a :: {real_normed_field, banach}"
     and g g' :: "real \<Rightarrow> real"
   assumes cont_f: "continuous_on {a..} f"
   assumes cont_g': "continuous_on {a..} g'"
   assumes f_bigo: "eventually (\<lambda>x. norm (f x) \<le> g' x) at_top"
   assumes deriv:  "eventually (\<lambda>x. (g has_field_derivative g' x) (at x)) at_top"
   obtains C D where
     "eventually (\<lambda>x. norm (EM_remainder' n f a x) \<le> C + D * g x) at_top"
 proof -
   note cont = continuous_on_subset[OF cont_f] continuous_on_subset[OF cont_g']
   from bounded_pbernpoly[of n] obtain D where D: "\<And>x. norm (pbernpoly n x) \<le> D" by blast
   from this[of 0] have D_nonneg: "D \<ge> 0" by simp
   from eventually_conj[OF f_bigo eventually_conj[OF deriv eventually_ge_at_top[of a]]] 
     obtain x0 where x0:
       "x0 \<ge> a" "\<And>x. x \<ge> x0 \<Longrightarrow> norm (f x) \<le> g' x" 
       "\<And>x. x \<ge> x0 \<Longrightarrow> (g has_field_derivative g' x) (at x)"
       by (auto simp: eventually_at_top_linorder)
   define C where "C = (norm (integral {a..x0} (\<lambda>t. pbernpoly n t *\<^sub>R f t)) - D * g x0) / fact n"
       
   have "eventually (\<lambda>x. norm (EM_remainder' n f a x) \<le> C + D / fact n * g x) at_top"
     using eventually_ge_at_top[of x0]
   proof eventually_elim
     case (elim x)
     have "integral {a..x} (\<lambda>t. pbernpoly n t *\<^sub>R f t) =
             integral {a..x0} (\<lambda>t. pbernpoly n t *\<^sub>R f t) +
             integral {x0..x} (\<lambda>t. pbernpoly n t *\<^sub>R f t)" (is "_ = ?I1 + ?I2") using elim x0(1)
       by (intro Henstock_Kurzweil_Integration.integral_combine [symmetric] integrable_EM_remainder' cont) auto
     also have "norm \<dots> \<le> norm ?I1 + norm ?I2" by (rule norm_triangle_ineq)
     also have "norm ?I2 \<le> integral {x0..x} (\<lambda>t. D * g' t)"
       using x0 D D_nonneg
       by (intro integral_norm_bound_integral integrable_EM_remainder')
          (auto intro!: integrable_continuous_real continuous_intros cont mult_mono)
     also have "\<dots> = D * integral {x0..x} g'" by simp
     also from elim have "(g' has_integral (g x - g x0)) {x0..x}"
       by (intro fundamental_theorem_of_calculus) 
          (auto intro!: has_field_derivative_at_within[OF x0(3)] 
                simp: has_field_derivative_iff_has_vector_derivative [symmetric])
     hence "integral {x0..x} g' = g x - g x0" by (simp add: has_integral_iff)
     finally have "norm (integral {a..x} (\<lambda>t. pbernpoly n t *\<^sub>R f t)) \<le> norm ?I1 + D * (g x - g x0)"
       by simp_all
     thus "norm (EM_remainder' n f a x) \<le> C + D / fact n * g x"
       by (simp add: EM_remainder'_def field_simps C_def)
   qed
   thus ?thesis by (rule that)
 qed
 
 
 subsection \<open>Application to harmonic numbers\<close>
 
 text \<open>
   As a first application, we can apply the machinery we have developed to the 
   harmonic numbers.
 \<close>
 
 definition harm_remainder :: "nat \<Rightarrow> nat \<Rightarrow> real" where
   "harm_remainder N n = EM_remainder (2*N+1) (\<lambda>x. -fact (2*N+1) / x ^ (2*N+2)) (int n)"
 
 lemma harm_expansion:
   assumes n: "n > 0" and N: "N > 0"
   shows "harm n = ln n + euler_mascheroni + 1 / (2*n) -
                      (\<Sum>i=1..N. bernoulli (2*i) / ((2*i) * n ^ (2*i))) - harm_remainder N n"
 proof -
   define fs where "fs = (\<lambda>k x. (-1) ^ k * fact k / x ^ (Suc k) :: real)"    
   interpret euler_maclaurin_nat' ln "\<lambda>x. 1/x" fs 1 N euler_mascheroni "{}"
   proof
     fix k x assume "k \<le> 2*N" "x \<in> {real 1..} - {}"
     thus "(fs k has_vector_derivative fs (Suc k) x) (at x)"
       by (cases "k = 0")
          (auto intro!: derivative_eq_intros 
                simp: fs_def has_field_derivative_iff_has_vector_derivative [symmetric] 
                      field_simps power_diff)
   next
     have "(\<lambda>b. harm b - ln (real b) -
                  (\<Sum>i<2*N+1. bernoulli' (Suc i) * (- 1) ^ i / (real (Suc i) * (real b ^ Suc i))))
             \<longlonglongrightarrow> (euler_mascheroni - (\<Sum>i<2*N+1. 0))"
       by (intro tendsto_diff euler_mascheroni_LIMSEQ tendsto_sum 
             real_tendsto_divide_at_top[OF tendsto_const] 
             filterlim_tendsto_pos_mult_at_top[OF tendsto_const] filterlim_pow_at_top
             filterlim_real_sequentially) auto
     thus "(\<lambda>b. (\<Sum>k = 1..b. 1 / real k) - ln (real b) - 
             (\<Sum>i<2*N+1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlonglongrightarrow> euler_mascheroni"
       by (simp add: harm_def divide_simps fs_def)
   qed (insert n N, auto intro!: continuous_intros derivative_eq_intros
          simp: fs_def has_field_derivative_iff_has_vector_derivative [symmetric])
        
   have "harm n = (\<Sum>k=1..n. 1 / real k)" by (simp add: harm_def divide_simps)
   also have "\<dots> = ln (real n) + euler_mascheroni + (1/2) *\<^sub>R (1 / real n) +
                      (\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (real n)) -
                      EM_remainder (2*N+1) (fs (2*N+1)) (int n)" using n N
     using n by (intro euler_maclaurin_strong_nat') simp_all
   also have "(\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R (fs (2*i-1) (real n))) =
                (\<Sum>i=1..N. -(bernoulli (2*i) / (real (2*i) * real n ^ (2*i))))"
     by (intro sum.cong refl) 
        (simp_all add: fs_def divide_simps fact_reduce del: of_nat_Suc power_Suc)
   also have "\<dots> = -(\<Sum>i=1..N. bernoulli (2*i) / (real (2*i) * real n ^ (2*i)))" 
     by (simp add: sum_negf)
   finally show ?thesis unfolding fs_def by (simp add: harm_remainder_def)
 qed
 
 lemma of_nat_ge_1_iff: "of_nat x \<ge> (1 :: 'a :: linordered_semidom) \<longleftrightarrow> x \<ge> 1"
   using of_nat_le_iff[of 1 x] by (simp del: of_nat_le_iff)
 
 lemma harm_remainder_bound:
   fixes N :: nat
   assumes N: "N > 0"
   shows "\<exists>C. \<forall>n\<ge>1. norm (harm_remainder N n) \<le> C / real n ^ (2*N+1)"
 proof -
   from bounded_pbernpoly[of "2*N+1"] obtain D where D: "\<forall>x. \<bar>pbernpoly (2*N+1) x\<bar> \<le> D" by auto
   have "\<forall>x. 1 \<le> real x \<longrightarrow> norm (harm_remainder N x) \<le>
           D / fact (2*N+1) * (fact (2*N) / x ^ (2*N+1))"
     unfolding harm_remainder_def of_int_of_nat_eq
   proof (rule norm_EM_remainder_le_strong_nat'[of "{}"])
     fix x :: real assume x: "x \<ge> 1"
     show "norm (-fact (2*N+1) / x ^ (2 * N + 2)) \<le> fact (2*N+1) / x ^ (2*N+2)"
       using x by simp
   next
     show "((\<lambda>x::real. fact (2 * N) / x ^ (2 * N + 1)) \<longlongrightarrow> 0) at_top"
       by (intro real_tendsto_divide_at_top[OF tendsto_const] filterlim_pow_at_top filterlim_ident)
          simp_all
   qed (insert N D, auto intro!: derivative_eq_intros continuous_intros simp: field_simps power_diff)
   hence "\<forall>x. 1 \<le> x \<longrightarrow> norm (harm_remainder N x) \<le> D / (2*N+1) / real x ^ (2*N+1)" by simp
   thus ?thesis by blast
 qed
   
 
 subsection \<open>Application to sums of inverse squares\<close>
 
 text \<open>
   In the same vein, we can derive the asymptotics of the partial sum of inverse squares.
 \<close>
 
 lemma sum_inverse_squares_expansion:
   assumes n: "n > 0" and N: "N > 0"
   shows "(\<Sum>k=1..n. 1 / real k ^ 2) = 
             pi ^ 2 / 6 - 1 / real n + 1 / (2 * real n ^ 2) -
                      (\<Sum>i=1..N. bernoulli (2*i) / n ^ (2*i+1)) - 
                     EM_remainder (2*N+1) (\<lambda>x. -fact (2*N+2) / x ^ (2*N+3)) (int n)"
 proof -
   have 3: "3 = Suc (Suc (Suc 0))" by (simp add: eval_nat_numeral)
   define fs where "fs = (\<lambda>k x. (-1) ^ k * fact (Suc k) / x ^ (k+2) :: real)"    
   interpret euler_maclaurin_nat' "\<lambda>x. -1/x" "\<lambda>x. 1/x^2" fs 1 N "pi^2/6" "{}"
   proof
     fix k x assume "k \<le> 2*N" "x \<in> {real 1..} - {}"
     thus "(fs k has_vector_derivative fs (Suc k) x) (at x)"
       by (cases "k = 0")
          (auto intro!: derivative_eq_intros 
                simp: fs_def has_field_derivative_iff_has_vector_derivative [symmetric] 
                      field_simps power_diff)
   next
     from inverse_squares_sums
       have "(\<lambda>n. \<Sum>k<n. 1 / real (Suc k) ^ 2) \<longlonglongrightarrow> pi\<^sup>2 / 6" by (simp add: sums_def)
     also have "(\<lambda>n. \<Sum>k<n. 1 / real (Suc k) ^ 2) = (\<lambda>n. \<Sum>k=1..n. 1 / real k ^ 2)"
       by (intro ext sum.reindex_bij_witness[of _ "\<lambda>n. n - 1" Suc]) auto
     finally have "(\<lambda>b. (\<Sum>k = 1..b. 1 / real k^2) + 1 / real b -
                  (\<Sum>i<2*N+1. bernoulli' (Suc i) * (- 1) ^ i / (real b ^ (i+2))))
             \<longlonglongrightarrow> (pi^2/6 + 0 - (\<Sum>i<2*N+1. 0))"
       by (intro tendsto_diff tendsto_add real_tendsto_divide_at_top[OF tendsto_const] 
             filterlim_tendsto_pos_mult_at_top[OF tendsto_const] filterlim_pow_at_top
             filterlim_real_sequentially tendsto_sum) auto
     thus "(\<lambda>b. (\<Sum>k = 1..b. 1 / real k^2) - (- 1 / real b) - 
             (\<Sum>i<2*N+1. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlonglongrightarrow> pi^2/6"
       by (simp add: harm_def field_simps fs_def del: power_Suc of_nat_Suc)
   qed (insert n N, auto intro!: continuous_intros derivative_eq_intros
          simp: fs_def has_field_derivative_iff_has_vector_derivative [symmetric] power2_eq_square)
 
   have "(\<Sum>k=1..n. 1 / real k ^ 2) = - 1 / real n + pi^2/6 + (1/2) *\<^sub>R (1 / real n^2) +
                (\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R fs (2*i-1) (real n)) -
                EM_remainder (2*N+1) (fs (2*N+1)) (int n)" using n N
     using n by (intro euler_maclaurin_strong_nat') simp_all
   also have "(\<Sum>i=1..N. (bernoulli (2*i) / fact (2*i)) *\<^sub>R (fs (2*i-1) (real n))) =
                (\<Sum>i=1..N. -(bernoulli (2*i) / (real n ^ (2*i+1))))"
     by (intro sum.cong refl) 
        (simp_all add: fs_def divide_simps fact_reduce del: of_nat_Suc power_Suc)
   also have "\<dots> = -(\<Sum>i=1..N. bernoulli (2*i) / real n ^ (2*i+1))"
     by (simp add: sum_negf)
   finally show ?thesis unfolding fs_def by (simp add: fs_def 3)
 qed
 
 lemma sum_inverse_squares_remainder_bound:
   fixes N :: nat
   assumes N: "N > 0"
   defines "R \<equiv> (\<lambda>n. EM_remainder (2*N+1) (\<lambda>x. -fact (2*N+2) / x ^ (2*N+3)) (int n))"
   shows   "\<exists>C. \<forall>n\<ge>1. norm (R n) \<le> C / real n ^ (2*N+2)"
 proof -
   have 3: "3 = Suc (Suc (Suc 0))" by simp
   from bounded_pbernpoly[of "2*N+1"] obtain D where D: "\<forall>x. \<bar>pbernpoly (2*N+1) x\<bar> \<le> D" by auto
   have "\<forall>x. 1 \<le> real x \<longrightarrow> norm (R x) \<le> D / fact (2*N+1) * (fact (2*N+1) / x ^ (2*N+2))"
     unfolding R_def of_int_of_nat_eq
   proof (rule norm_EM_remainder_le_strong_nat'[of "{}"])
     fix x :: real assume x: "x \<ge> 1"
     show "norm (-fact (2*N+2) / x ^ (2*N+3)) \<le> fact (2*N+2) / x ^ (2*N+3)"
       using x by simp
   next
     show "((\<lambda>x::real. fact (2*N+1) / x ^ (2*N+2)) \<longlongrightarrow> 0) at_top"
       by (intro real_tendsto_divide_at_top[OF tendsto_const] filterlim_pow_at_top filterlim_ident)
          simp_all
   qed (insert N D, auto intro!: derivative_eq_intros continuous_intros simp: field_simps power_diff 3)
   hence "\<forall>x\<ge>1. norm (R x) \<le> D / real x ^ (2 * N + 2)" by simp
   thus ?thesis by blast
 qed
 
 end
diff --git a/thys/Euler_MacLaurin/ROOT b/thys/Euler_MacLaurin/ROOT
--- a/thys/Euler_MacLaurin/ROOT
+++ b/thys/Euler_MacLaurin/ROOT
@@ -1,13 +1,13 @@
 chapter AFP
 
-session Euler_MacLaurin (AFP) = "HOL-Analysis" +
+session Euler_MacLaurin (AFP) = Bernoulli +
   options [timeout = 600]
   sessions
-    Bernoulli
+    "HOL-Complex_Analysis"
     Landau_Symbols
   theories
     Euler_MacLaurin
     Euler_MacLaurin_Landau
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/Green/Derivs.thy b/thys/Green/Derivs.thy
--- a/thys/Green/Derivs.thy
+++ b/thys/Green/Derivs.thy
@@ -1,654 +1,651 @@
 theory Derivs
   imports General_Utils
 begin
 
 lemma field_simp_has_vector_derivative [derivative_intros]:
   "(f has_field_derivative y) F \<Longrightarrow> (f has_vector_derivative y) F"
   by (simp add: has_field_derivative_iff_has_vector_derivative)
 
 lemma continuous_on_cases_empty [continuous_intros]:
   "\<lbrakk>closed S; continuous_on S f; \<And>x. \<lbrakk>x \<in> S; \<not> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow>
     continuous_on S (\<lambda>x. if P x then f x else g x)"
   using continuous_on_cases [of _ "{}"] by force
 
 lemma inj_on_cases:
   assumes "inj_on f (Collect P \<inter> S)" "inj_on g (Collect (Not \<circ> P) \<inter> S)"
     "f ` (Collect P \<inter> S) \<inter> g ` (Collect (Not \<circ> P) \<inter> S) = {}"
   shows "inj_on (\<lambda>x. if P x then f x else g x) S"
   using assms by (force simp: inj_on_def)
 
 lemma inj_on_arccos: "S \<subseteq> {-1..1} \<Longrightarrow> inj_on arccos S"
   by (metis atLeastAtMost_iff cos_arccos inj_onI subsetCE)
 
 lemma has_vector_derivative_componentwise_within:
   "(f has_vector_derivative f') (at a within S) \<longleftrightarrow>
     (\<forall>i \<in> Basis. ((\<lambda>x. f x \<bullet> i) has_vector_derivative (f' \<bullet> i)) (at a within S))"
   apply (simp add: has_vector_derivative_def)
   apply (subst has_derivative_componentwise_within)
   apply simp
   done
 
 lemma has_vector_derivative_pair_within:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> 'b::euclidean_space"
   assumes "\<And>u. u \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> u) has_vector_derivative f' \<bullet> u) (at x within S)"
     "\<And>u. u \<in> Basis \<Longrightarrow> ((\<lambda>x. g x \<bullet> u) has_vector_derivative g' \<bullet> u) (at x within S)"
   shows "((\<lambda>x. (f x, g x)) has_vector_derivative (f',g')) (at x within S)"
   apply (subst has_vector_derivative_componentwise_within)
   apply (auto simp: assms Basis_prod_def)
   done
 
 lemma piecewise_C1_differentiable_const:
   shows "(\<lambda>x. c) piecewise_C1_differentiable_on s"
   using continuous_on_const
   by (auto simp add: piecewise_C1_differentiable_on_def)
 
 declare piecewise_C1_differentiable_const [simp, derivative_intros]
 declare piecewise_C1_differentiable_neg [simp, derivative_intros]
 declare piecewise_C1_differentiable_add [simp, derivative_intros]
 declare piecewise_C1_differentiable_diff [simp, derivative_intros]
 
-(*move to Cauchy_Integral_Theorem*)
+(*move to Derivative*)
 
 lemma piecewise_C1_differentiable_on_ident [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   shows "(\<lambda>x. x) piecewise_C1_differentiable_on S"
   unfolding piecewise_C1_differentiable_on_def using C1_differentiable_on_ident
   by (blast intro: continuous_on_id C1_differentiable_on_ident)
 
 lemma piecewise_C1_differentiable_on_mult [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a::real_normed_algebra"
   assumes "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on S"
   shows "(\<lambda>x. f x * g x) piecewise_C1_differentiable_on S"
   using assms
   unfolding piecewise_C1_differentiable_on_def
   apply safe
    apply (blast intro: continuous_intros)
   apply (rename_tac A B)
   apply (rule_tac x="A \<union> B" in exI)
   apply (auto intro: C1_differentiable_on_mult C1_differentiable_on_subset)
   done
 
 lemma C1_differentiable_on_cdiv [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a :: real_normed_field"
   shows "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x / c) C1_differentiable_on S"
   by (simp add: divide_inverse)
 
 lemma piecewise_C1_differentiable_on_cdiv [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a::real_normed_field"
   assumes "f piecewise_C1_differentiable_on S"
   shows "(\<lambda>x. f x / c) piecewise_C1_differentiable_on S"
   by (simp add: divide_inverse piecewise_C1_differentiable_const piecewise_C1_differentiable_on_mult assms)
 
 lemma sqrt_C1_differentiable [simp, derivative_intros]:
   assumes f: "f C1_differentiable_on S" and fim: "f ` S \<subseteq> {0<..}"
   shows "(\<lambda>x. sqrt (f x)) C1_differentiable_on S"
 proof -
   have contf: "continuous_on S f"
     by (simp add: C1_differentiable_imp_continuous_on f)
   show ?thesis
     using assms
     unfolding C1_differentiable_on_def has_field_derivative_iff_has_vector_derivative [symmetric]
     by (fastforce intro!: contf continuous_intros derivative_intros)
 qed
 
 lemma sqrt_piecewise_C1_differentiable [simp, derivative_intros]:
   assumes f: "f piecewise_C1_differentiable_on S" and fim: "f ` S \<subseteq> {0<..}"
   shows "(\<lambda>x. sqrt (f x)) piecewise_C1_differentiable_on S"
   using assms
   unfolding piecewise_C1_differentiable_on_def
   by (fastforce intro!: continuous_intros derivative_intros)
 
 lemma
   fixes f :: "real \<Rightarrow> 'a::{banach,real_normed_field}"
   assumes f: "f C1_differentiable_on S"
   shows sin_C1_differentiable [simp, derivative_intros]: "(\<lambda>x. sin (f x)) C1_differentiable_on S"
   and   cos_C1_differentiable [simp, derivative_intros]: "(\<lambda>x. cos (f x)) C1_differentiable_on S"
 proof -
   have contf: "continuous_on S f"
     by (simp add: C1_differentiable_imp_continuous_on f)
   note df_sin = field_vector_diff_chain_at [where g=sin, unfolded o_def]
   note df_cos = field_vector_diff_chain_at [where g=cos, unfolded o_def]
   show  "(\<lambda>x. sin (f x)) C1_differentiable_on S" "(\<lambda>x. cos (f x)) C1_differentiable_on S"
     using assms
     unfolding C1_differentiable_on_def has_field_derivative_iff_has_vector_derivative [symmetric]
     apply auto
     by (rule contf continuous_intros df_sin df_cos derivative_intros exI conjI ballI | force)+
 qed
 
 lemma has_derivative_abs:
   fixes a::real
   assumes "a \<noteq> 0"
   shows "(abs has_derivative ((*) (sgn a))) (at a)"
 proof -
   have [simp]: "norm = abs"
     using real_norm_def by force
   show ?thesis
     using has_derivative_norm [where 'a=real, simplified] assms
     by (simp add: mult_commute_abs)
 qed
 
 lemma abs_C1_differentiable [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f C1_differentiable_on S" and "0 \<notin> f ` S"
   shows "(\<lambda>x. abs (f x)) C1_differentiable_on S"
 proof -
   have contf: "continuous_on S f"
     by (simp add: C1_differentiable_imp_continuous_on f)
   note df = DERIV_chain [where f=abs and g=f, unfolded o_def]
   show ?thesis
     using assms
     unfolding C1_differentiable_on_def has_field_derivative_iff_has_vector_derivative [symmetric]
     apply clarify
     apply (rule df exI conjI ballI)+
       apply (force simp: has_field_derivative_def intro: has_derivative_abs continuous_intros contf)+
     done
 qed
 
 lemma C1_differentiable_on_pair [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> 'b::euclidean_space"
   assumes "f C1_differentiable_on S" "g C1_differentiable_on S"
   shows "(\<lambda>x. (f x, g x)) C1_differentiable_on S"
   using assms unfolding C1_differentiable_on_def
   apply safe
   apply (rename_tac A B)
   apply (intro exI ballI conjI)
    apply (rule_tac f'="A x" and g'="B x" in has_vector_derivative_pair_within)
   using  has_vector_derivative_componentwise_within
   by (blast intro: continuous_on_Pair)+
 
 lemma piecewise_C1_differentiable_on_pair [simp, derivative_intros]:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> 'b::euclidean_space"
   assumes "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on S"
   shows "(\<lambda>x. (f x, g x)) piecewise_C1_differentiable_on S"
   using assms unfolding piecewise_C1_differentiable_on_def
   by (blast intro!: continuous_intros C1_differentiable_on_pair del: continuous_on_discrete
       intro: C1_differentiable_on_subset)
 
 lemma test2:
   assumes s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x"
     and fs: "finite s" and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
     and "x \<in> {0..1}" "x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s"
   shows "vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x within {0..1}) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u) within{0..1})"
 proof -
   have i:"(g has_vector_derivative vector_derivative g (at ((v - u) * x + u))) (at ((v-u) * x + u))"
     using assms s [of "(v - u) * x + u"] uv mult_left_le [of x "v-u"]
     by (auto simp:  vector_derivative_works)
   have ii:"((\<lambda>x. g ((v - u) * x + u)) has_vector_derivative (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u))) (at x)"
     by (intro vector_diff_chain_at [simplified o_def] derivative_eq_intros | simp add: i)+
   have 0: "0 \<le> (v - u) * x + u"
     using assms uv by auto
   have 1: "(v - u) * x + u \<le> 1"
     using assms uv
     by simp (metis add.commute atLeastAtMost_iff atLeastatMost_empty_iff diff_ge_0_iff_ge empty_iff le_diff_eq mult_left_le)
   have iii: "vector_derivative g (at ((v - u) * x + u) within {0..1}) = vector_derivative g (at ((v - u) * x + u))"
     using Derivative.vector_derivative_at_within_ivl[OF i, of "0" "1", OF 0 1]
     by auto
   have iv: "vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x within {0..1}) = (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u))"
     using Derivative.vector_derivative_at_within_ivl[OF ii, of "0" "1"] assms
     by auto
   show ?thesis
     using iii iv by auto
 qed
 
 lemma C1_differentiable_on_components:
   assumes "\<And>i. i \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s"
   shows "f C1_differentiable_on s"
 proof (clarsimp simp add: C1_differentiable_on_def has_vector_derivative_def)
   have *:"\<forall>f i x. x *\<^sub>R (f \<bullet> i) = (x *\<^sub>R f) \<bullet> i" by auto
   have "\<exists>f'. \<forall>i\<in>Basis. \<forall>x\<in>s. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>z. z *\<^sub>R f' x \<bullet> i)) (at x) \<and> continuous_on s f'"
     using assms lambda_skolem_euclidean[of "\<lambda>i D. (\<forall>x\<in>s. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>z. z *\<^sub>R D x)) (at x)) \<and> continuous_on s D"]
     apply (simp only: C1_differentiable_on_def has_vector_derivative_def *)
     using continuous_on_componentwise[of "s"]
     by metis
   then obtain f' where f': "\<forall>i\<in>Basis. \<forall>x\<in>s. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>z. z *\<^sub>R f' x \<bullet> i)) (at x) \<and> continuous_on s f'"
     by auto
   then have 0: "(\<forall>x\<in>s. (f has_derivative (\<lambda>z. z *\<^sub>R f' x)) (at x)) \<and> continuous_on s f'"
     using f' has_derivative_componentwise_within[of "f", where S= UNIV]
     by auto
   then show "\<exists>D. (\<forall>x\<in>s. (f has_derivative (\<lambda>z. z *\<^sub>R D x)) (at x)) \<and> continuous_on s D" by metis
 qed
 
 lemma piecewise_C1_differentiable_on_components:
   assumes "finite t"
     "\<And>i. i \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s - t"
     "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on s (\<lambda>x. f x \<bullet> i)"
   shows "f piecewise_C1_differentiable_on s"
   using C1_differentiable_on_components assms continuous_on_componentwise piecewise_C1_differentiable_on_def by blast
 
 lemma all_components_smooth_one_pw_smooth_is_pw_smooth:
   assumes "\<And>i. i \<in> Basis - {j} \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s"
   assumes "(\<lambda>x. f x \<bullet> j) piecewise_C1_differentiable_on s"
   shows "f piecewise_C1_differentiable_on s"
 proof -
   have is_cont: "\<forall>i\<in>Basis. continuous_on s (\<lambda>x. f x \<bullet> i)"
     using assms C1_differentiable_imp_continuous_on piecewise_C1_differentiable_on_def
     by fastforce
   obtain t where t:"(finite t \<and> (\<lambda>x. f x \<bullet> j) C1_differentiable_on s - t)" using assms(2) piecewise_C1_differentiable_on_def by auto
   show ?thesis
     using piecewise_C1_differentiable_on_components[where ?f = "f"]
     using assms(2) piecewise_C1_differentiable_on_def
       C1_differentiable_on_subset[OF assms(1) Diff_subset, where ?B1 ="t"] t is_cont
     by fastforce
 qed
 
 lemma derivative_component_fun_component:
   fixes i::"'a::euclidean_space"
   assumes "f differentiable (at x)"
   shows "((vector_derivative f (at x)) \<bullet> i) = ((vector_derivative (\<lambda>x. (f x) \<bullet> i) (at x)) )"
 proof -
   have "((\<lambda>x. f x \<bullet> i) has_vector_derivative vector_derivative f (at x) \<bullet> i) (at x)"
     using assms and bounded_linear.has_vector_derivative[of "(\<lambda>x. x \<bullet> i)" "f" "(vector_derivative f (at x))" "(at x)"] and
       bounded_linear_inner_left[of "i"] and vector_derivative_works[of "f" "(at x)"]
     by blast
   then show "((vector_derivative f (at x)) \<bullet> i) = ((vector_derivative (\<lambda>x. (f x) \<bullet> i) (at x)) )"
     using vector_derivative_works[of "(\<lambda>x. (f x) \<bullet> i)" "(at x)"] and
       differentiableI_vector[of "(\<lambda>x. (f x) \<bullet> i)" "(vector_derivative f (at x) \<bullet> i)" "(at x)"] and
       Derivative.vector_derivative_at
     by force
 qed
 
 lemma gamma_deriv_at_within:
   assumes a_leq_b: "a < b"  and
     x_within_bounds: "x \<in> {a..b}" and
     gamma_differentiable: "\<forall>x \<in> {a .. b}. \<gamma> differentiable at x"
   shows  "vector_derivative \<gamma> (at x within {a..b}) =  vector_derivative \<gamma> (at x)"
   using Derivative.vector_derivative_at_within_ivl[of "\<gamma>" "vector_derivative \<gamma> (at x)" "x" "a" "b"]
     gamma_differentiable x_within_bounds a_leq_b
   by (auto simp add: vector_derivative_works)
 
 lemma islimpt_diff_finite:
   assumes "finite (t::'a::t1_space set)"
   shows "x islimpt s - t = x islimpt s"
 proof-
   have iii: "s - t = s - (t \<inter> s)" by auto
   have "(t \<inter> s) \<subseteq> s" by auto
   have ii:"finite (t \<inter> s)" using assms(1) by auto
   have i: "(t \<inter> s) \<union> (s - (t \<inter> s)) = ( s)"
     using assms by auto
   then have "x islimpt s - (t \<inter> s) = x islimpt s"
     using islimpt_Un_finite[OF ii,where ?t = "s - (t \<inter> s)"] i assms(1)
     by force
   then show ?thesis using iii by auto
 qed
 
 lemma ivl_limpt_diff:
   assumes "finite s" "a < b"  "(x::real) \<in> {a..b} - s"
   shows "x islimpt {a..b} - s"
 proof-
   have "x islimpt {a..b}"
   proof (cases "x \<in>{a,b}")
     have i: "finite {a,b}" and ii:  "{a, b} \<union> {a<..<b} = {a..b}" using assms by auto
     assume "x \<in>{a,b}"
     then show ?thesis
       using islimpt_Un_finite[OF i, where ?t= "{a<..<b}"]
       by (metis assms(2) empty_iff ii insert_iff islimpt_greaterThanLessThan1 islimpt_greaterThanLessThan2)
   next
     assume "x \<notin>{a,b}"
     then show "x islimpt {a..b}" using assms by auto
   qed
   then show "x islimpt {a..b} - s" using islimpt_diff_finite[OF assms(1)] assms
     by fastforce
 qed
 
 lemma ivl_closure_diff_del:
   assumes "finite s" "a < b"  "(x::real) \<in> {a..b} - s"
   shows "x \<in> closure (({a..b} - s) - {x})"
   using ivl_limpt_diff islimpt_in_closure assms by blast
 
 lemma ivl_not_trivial_limit_within:
   assumes "finite s"
     "a < b"
     "(x::real) \<in> {a..b} - s"
   shows "at x within {a..b} - s \<noteq> bot"
   using assms ivl_closure_diff_del not_trivial_limit_within
   by blast
 
 lemma vector_derivative_at_within_non_trivial_limit:
   "at x within s \<noteq> bot \<and> (f has_vector_derivative f') (at x) \<Longrightarrow>
      vector_derivative f (at x within s) = f'"
   using has_vector_derivative_at_within vector_derivative_within by fastforce
 
 lemma vector_derivative_at_within_ivl_diff:
   "finite s \<and> a < b \<and> (x::real) \<in> {a..b} - s \<and> (f has_vector_derivative f') (at x) \<Longrightarrow>
      vector_derivative f (at x within {a..b} - s) = f'"
   using vector_derivative_at_within_non_trivial_limit ivl_not_trivial_limit_within by fastforce
 
 lemma gamma_deriv_at_within_diff:
   assumes a_leq_b: "a < b"  and
     x_within_bounds: "x \<in> {a..b} - s" and
     gamma_differentiable: "\<forall>x \<in> {a .. b} - s. \<gamma> differentiable at x" and
     s_subset: "s \<subseteq> {a..b}" and
     finite_s: "finite s"
   shows "vector_derivative \<gamma> (at x within {a..b} - s)
                            =  vector_derivative \<gamma> (at x)"
   using vector_derivative_at_within_ivl_diff [of "s" "a" "b" "x" "\<gamma>" "vector_derivative \<gamma> (at x)"]
     gamma_differentiable
     x_within_bounds a_leq_b s_subset finite_s
   by (auto simp add: vector_derivative_works)
 
 lemma gamma_deriv_at_within_gen:
   assumes a_leq_b: "a < b"  and
     x_within_bounds: "x \<in> s" and
     s_subset: "s \<subseteq> {a..b}" and
     gamma_differentiable: "\<forall>x \<in> s. \<gamma> differentiable at x"
   shows "vector_derivative \<gamma> (at x within ({a..b})) =  vector_derivative \<gamma> (at x)"
   using Derivative.vector_derivative_at_within_ivl[of "\<gamma>" "vector_derivative \<gamma> (at x)" "x" "a" "b"]
     gamma_differentiable  x_within_bounds a_leq_b s_subset
   by (auto simp add: vector_derivative_works)
 
 lemma derivative_component_fun_component_at_within_gen:
   assumes gamma_differentiable: "\<forall>x \<in> s. \<gamma> differentiable at x" and s_subset: "s \<subseteq> {0..1}"
   shows "\<forall>x \<in> s. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) \<bullet> (i::'a:: euclidean_space)
            =  vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
 proof -
   have gamma_i_component_smooth:
     "\<forall>x \<in> s. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
     using gamma_differentiable
     by auto
   show "\<forall>x \<in> s. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) \<bullet> i
                                   =  vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
   proof
     fix x::real
     assume x_within_bounds: "x \<in> s"
     have gamma_deriv_at_within:
         "vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) = vector_derivative (\<lambda>x. \<gamma> x) (at x)"
       using gamma_deriv_at_within_gen[of "0" "1"] x_within_bounds
         gamma_differentiable s_subset
       by (auto simp add: vector_derivative_works)
     then have gamma_component_deriv_at_within:
       "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x)
        = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
       using gamma_deriv_at_within_gen[of "0" "1", where \<gamma> = "(\<lambda>x. \<gamma> x \<bullet> i)"] x_within_bounds
         gamma_i_component_smooth s_subset
       by (auto simp add: vector_derivative_works)
     have gamma_component_deriv_eq_gamma_deriv_component:
       "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x) = vector_derivative (\<lambda>x. \<gamma> x) (at x) \<bullet> i"
       using derivative_component_fun_component[of "\<gamma>" "x" "i"] gamma_differentiable x_within_bounds
       by auto
     show "vector_derivative \<gamma> (at x within {0..1}) \<bullet> i = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
       using gamma_component_deriv_eq_gamma_deriv_component gamma_component_deriv_at_within gamma_deriv_at_within
       by auto
   qed
 qed
 
 lemma derivative_component_fun_component_at_within:
   assumes gamma_differentiable: "\<forall>x \<in> {0 .. 1}. \<gamma> differentiable at x"
   shows "\<forall>x \<in> {0..1}. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) \<bullet> (i::'a:: euclidean_space)
                                        =  vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
 proof -
   have gamma_i_component_smooth:
     "\<forall>x \<in> {0 .. 1}. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
     using gamma_differentiable by auto
   show "\<forall>x \<in> {0..1}. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) \<bullet> i
                                   =  vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
   proof
     fix x::real
     assume x_within_bounds: "x \<in> {0..1}"
     have gamma_deriv_at_within:
       "vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) =  vector_derivative (\<lambda>x. \<gamma> x) (at x)"
       using gamma_deriv_at_within[of "0" "1"] x_within_bounds
         gamma_differentiable
       by (auto simp add: vector_derivative_works)
     have gamma_component_deriv_at_within:
       "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x) = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
       using Derivative.vector_derivative_at_within_ivl[of "(\<lambda>x. (\<gamma> x) \<bullet> i)" "vector_derivative (\<lambda>x. (\<gamma> x)  \<bullet> i) (at x)" "x" "0" "1"]
         has_vector_derivative_at_within[of "(\<lambda>x. \<gamma> x \<bullet> i)" "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x)" "x" "{0..1}"]
         gamma_i_component_smooth x_within_bounds
       by (simp add: vector_derivative_works)
     have gamma_component_deriv_eq_gamma_deriv_component:
       "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x) = vector_derivative (\<lambda>x. \<gamma> x) (at x) \<bullet> i"
       using derivative_component_fun_component[of "\<gamma>" "x" "i"] gamma_differentiable x_within_bounds
       by auto
     show "vector_derivative \<gamma> (at x within {0..1}) \<bullet> i = vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1})"
       using gamma_component_deriv_eq_gamma_deriv_component gamma_component_deriv_at_within gamma_deriv_at_within
       by auto
   qed
 qed
 
 lemma straight_path_diffrentiable_x:
   fixes b :: "real" and y1 ::"real"
   assumes gamma_def: "\<gamma> = (\<lambda>x. (b, y2 + y1 * x))"
   shows "\<forall>x. \<gamma> differentiable at x"
   unfolding gamma_def differentiable_def
   by (fast intro!: derivative_intros)
 
 lemma straight_path_diffrentiable_y:
   fixes b :: "real" and
     y1 y2 ::"real"
   assumes gamma_def: "\<gamma> = (\<lambda>x. (y2 + y1 * x , b))"
   shows "\<forall>x. \<gamma> differentiable at x"
   unfolding gamma_def differentiable_def
   by (fast intro!: derivative_intros)
 
 lemma piecewise_C1_differentiable_on_imp_continuous_on:
   assumes "f piecewise_C1_differentiable_on s"
   shows "continuous_on s f"
   using assms
   by (auto simp add: piecewise_C1_differentiable_on_def)
 
 lemma boring_lemma1:
   fixes f :: "real\<Rightarrow>real"
   assumes "(f has_vector_derivative D) (at x)"
   shows "((\<lambda>x. (f x, 0)) has_vector_derivative ((D,0::real))) (at x)"
 proof-
   have *: "((\<lambda>x. (f x) *\<^sub>R (1,0)) has_vector_derivative (D *\<^sub>R (1,0))) (at x)"
     using bounded_linear.has_vector_derivative[OF Real_Vector_Spaces.bounded_linear_scaleR_left assms(1),
         of "(1,0)"] by auto
   have "((\<lambda>x. (f x) *\<^sub>R (1,0)) has_vector_derivative (D,0)) (at x)"
   proof -
     have "(D, 0::'a) = D *\<^sub>R (1, 0)"
       by simp
     then show ?thesis
       by (metis (no_types) *)
   qed
   then show ?thesis by auto
 qed
 
 lemma boring_lemma2:
   fixes f :: "real\<Rightarrow>real"
   assumes "(f has_vector_derivative D) (at x)"
   shows "((\<lambda>x. (0, f x)) has_vector_derivative (0, D)) (at x)"
 proof-
   have *: "((\<lambda>x. (f x) *\<^sub>R (0,1)) has_vector_derivative (D *\<^sub>R (0,1))) (at x)"
     using bounded_linear.has_vector_derivative[OF Real_Vector_Spaces.bounded_linear_scaleR_left assms(1),
         of "(0,1)"] by auto
   then have "((\<lambda>x. (f x) *\<^sub>R (0,1)) has_vector_derivative ((0,D))) (at x)"
     using scaleR_Pair Real_Vector_Spaces.real_scaleR_def
   proof -
     have "(0::'b, D) = D *\<^sub>R (0, 1)"
       by auto
     then show ?thesis
       by (metis (no_types) *)
   qed
   then show ?thesis by auto
 qed
 
 lemma pair_prod_smooth_pw_smooth:
   assumes "(f::real\<Rightarrow>real) C1_differentiable_on s" "(g::real\<Rightarrow>real) piecewise_C1_differentiable_on s"
   shows "(\<lambda>x. (f x, g x)) piecewise_C1_differentiable_on s"
 proof -
   have f_cont: "continuous_on s f"
     using assms(1) C1_differentiable_imp_continuous_on
     by fastforce
   have g_cont: "continuous_on s g"
     using assms(2)  by (auto simp add: piecewise_C1_differentiable_on_def)
   obtain t where t:"(finite t \<and> g C1_differentiable_on s - t)" using assms(2) piecewise_C1_differentiable_on_def by auto
   show ?thesis
     using piecewise_C1_differentiable_on_components[where ?f = "(\<lambda>x. (f x, g x))"]
     apply (simp add: real_pair_basis)
     using assms(2) piecewise_C1_differentiable_on_def
       C1_differentiable_on_subset[OF assms(1) Diff_subset, where ?B1 ="t"] t
       f_cont g_cont
     by fastforce
 qed
 
 lemma scale_shift_smooth:
   shows "(\<lambda>x. a + b * x) C1_differentiable_on s"
 proof -
   show "(\<lambda>x. a + b * x) C1_differentiable_on s"
-    using Cauchy_Integral_Theorem.C1_differentiable_on_mult
-      Cauchy_Integral_Theorem.C1_differentiable_on_add
-      Cauchy_Integral_Theorem.C1_differentiable_on_const
-      Cauchy_Integral_Theorem.C1_differentiable_on_ident
-    by auto
+    using C1_differentiable_on_mult C1_differentiable_on_add C1_differentiable_on_const
+          C1_differentiable_on_ident by auto
 qed
 
 lemma open_diff:
   assumes "finite (t::'a::t1_space set)"
     "open (s::'a set)"
   shows "open (s -t)"
   using assms
 proof(induction "t")
   show "open s \<Longrightarrow> open (s - {})" by auto
 next
   fix x::"'a::t1_space"
   fix F::"'a::t1_space set"
   assume step: "finite F " " x \<notin> F" "open s"
   then have i: "(s - insert x F) = (s - F) - {x}" by auto
   assume ind_hyp: " (open s \<Longrightarrow> open (s - F))"
   show "open (s - insert x F)"
     apply (simp only: i)
     using open_delete[of "s -F"] ind_hyp[OF step(3)] by auto
 qed
 
 lemma has_derivative_transform_within:
   assumes "0 < d"
     and "x \<in> s"
     and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
     and "(f has_derivative f') (at x within s)"
   shows "(g has_derivative f') (at x within s)"
   using assms
   unfolding has_derivative_within
   by (force simp add: intro: Lim_transform_within)
 
 lemma has_derivative_transform_within_ivl:
   assumes "(0::real) < b"
     and     "\<forall>x\<in>{a..b} -s. f x = g x"
     and "x \<in> {a..b} -s"
     and "(f has_derivative f') (at x within {a..b} - s)"
   shows "(g has_derivative f') (at x within {a..b} - s)"
   using has_derivative_transform_within[of "b" "x" "{a..b} - s"] assms
   by auto
 
 lemma has_vector_derivative_transform_within_ivl:
   assumes "(0::real) < b"
     and     "\<forall>x\<in>{a..b} -s . f x = g x"
     and "x \<in> {a..b} - s"
     and "(f has_vector_derivative f') (at x within {a..b} - s)"
   shows "(g has_vector_derivative f') (at x within {a..b} - s)"
   using assms has_derivative_transform_within_ivl
   apply (auto simp add: has_vector_derivative_def)
   by blast
 
 lemma has_derivative_transform_at:
   assumes "0 < d"
     and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
     and "(f has_derivative f') (at x)"
   shows "(g has_derivative f') (at x)"
   using has_derivative_transform_within [of d x UNIV f g f'] assms
   by simp
 
 lemma has_vector_derivative_transform_at:
   assumes "0 < d"
     and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
     and "(f has_vector_derivative f') (at x)"
   shows "(g has_vector_derivative f') (at x)"
   using assms
   unfolding has_vector_derivative_def
   by (rule has_derivative_transform_at)
 
 lemma C1_diff_components_2:
   assumes "b \<in> Basis"
   assumes "f C1_differentiable_on s"
   shows "(\<lambda>x. f x \<bullet> b) C1_differentiable_on s"
 proof -
   obtain D where D:"(\<forall>x\<in>s. (f has_derivative (\<lambda>z. z *\<^sub>R D x)) (at x))" "continuous_on s D"
     using assms(2) by (fastforce simp add: C1_differentiable_on_def has_vector_derivative_def)
   show ?thesis
   proof (simp add: C1_differentiable_on_def has_vector_derivative_def, intro exI conjI)
     show "continuous_on s (\<lambda>x. D x \<bullet> b)" using D continuous_on_componentwise assms(1) by fastforce
     show "(\<forall>x\<in>s. ((\<lambda>x. f x \<bullet> b) has_derivative (\<lambda>y. y * (\<lambda>x. D x \<bullet> b) x)) (at x))"
       using has_derivative_inner_left D(1) by fastforce
   qed
 qed
 
 lemma eq_smooth:
   assumes "0 < d"
     "\<forall>x\<in>s. \<forall>y. dist x y < d \<longrightarrow> f y = g y" (*This crappy condition cannot be loosened :( *)
     "f C1_differentiable_on s"
   shows "g C1_differentiable_on s"
 proof (simp add: C1_differentiable_on_def)
   obtain D where D: "(\<forall>x\<in>s. (f has_vector_derivative D x) (at x)) \<and> continuous_on s D"
     using assms by (auto simp add: C1_differentiable_on_def)
   then have f: "(\<forall>x\<in>s. (g has_vector_derivative D x) (at x))"
     using assms(1-2)
     by (metis dist_commute has_vector_derivative_transform_at)
   have "(\<forall>x\<in>s. (g has_vector_derivative D x) (at x)) \<and> continuous_on s D" using D f by auto
   then show "\<exists>D. (\<forall>x\<in>s. (g has_vector_derivative D x) (at x)) \<and> continuous_on s D" by metis
 qed
 
 lemma eq_pw_smooth:
   assumes "0 < d"
     "\<forall>x\<in>s. \<forall>y. dist x y < d \<longrightarrow> f y = g y" (*This crappy condition cannot be loosened :( *)
     "\<forall>x\<in>s. f x = g x"
     "f piecewise_C1_differentiable_on s"
   shows "g piecewise_C1_differentiable_on s"
 proof (simp add: piecewise_C1_differentiable_on_def)
   have g_cont: "continuous_on s g" using assms piecewise_C1_differentiable_const
     by (simp add: piecewise_C1_differentiable_on_def)
   obtain t where t: "finite t \<and> f C1_differentiable_on s - t"
     using assms by (auto simp add: piecewise_C1_differentiable_on_def)
   then have "g C1_differentiable_on s - t" using assms eq_smooth by blast
   then show "continuous_on s g \<and> (\<exists>t. finite t \<and> g C1_differentiable_on s - t)"
     using t g_cont by metis
 qed
 
 lemma scale_piecewise_C1_differentiable_on:
   assumes "f piecewise_C1_differentiable_on s"
   shows "(\<lambda>x. (c::real) * (f x)) piecewise_C1_differentiable_on s"
 proof (simp add: piecewise_C1_differentiable_on_def, intro conjI)
   show "continuous_on s (\<lambda>x. c * f x)"
     using assms continuous_on_mult_left
     by (auto simp add: piecewise_C1_differentiable_on_def)
   show "\<exists>t. finite t \<and> (\<lambda>x. c * f x) C1_differentiable_on s - t"
     using assms continuous_on_mult_left
     by (auto simp add: piecewise_C1_differentiable_on_def)
 qed
 
 lemma eq_smooth_gen:
   assumes "f C1_differentiable_on s"
     "\<forall>x. f x = g x"
   shows "g C1_differentiable_on s"
   using assms unfolding C1_differentiable_on_def
   by (metis (no_types, lifting) has_vector_derivative_weaken UNIV_I top_greatest)
 
 lemma subpath_compose:
   shows "(subpath a b \<gamma>) = \<gamma> o (\<lambda>x.  (b - a) * x + a)"
   by (auto simp add: subpath_def)
 
 lemma subpath_smooth:
   assumes "\<gamma> C1_differentiable_on {0..1}" "0 \<le> a" "a < b" "b \<le> 1"
   shows "(subpath a b \<gamma>) C1_differentiable_on {0..1}"
 proof-
   have "\<gamma> C1_differentiable_on {a..b}"
     apply (rule C1_differentiable_on_subset)
     using assms by auto
   then have "\<gamma> C1_differentiable_on (\<lambda>x. (b - a) * x + a) ` {0..1}"
     using \<open>a < b\<close> closed_segment_eq_real_ivl closed_segment_real_eq by auto
   moreover have "finite ({0..1} \<inter> (\<lambda>x. (b - a) * x + a) -` {x})" for x
   proof -
     have "((\<lambda>x. (b - a) * x + a) -` {x}) = {(x -a) /(b-a)}"
       using assms by (auto simp add: divide_simps)
     then show ?thesis
       by auto
   qed
   ultimately show ?thesis
     by (force simp add: subpath_compose intro: C1_differentiable_compose derivative_intros)
 qed
 
 lemma has_vector_derivative_divide[derivative_intros]:
   fixes a :: "'a::real_normed_field"
   shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x / a) has_vector_derivative (x / a)) F"
   unfolding divide_inverse by (fact has_vector_derivative_mult_left)
 
 end
diff --git a/thys/Green/Green.thy b/thys/Green/Green.thy
--- a/thys/Green/Green.thy
+++ b/thys/Green/Green.thy
@@ -1,3117 +1,3117 @@
 theory Green
   imports Paths Derivs Integrals General_Utils
 begin
 
 lemma frontier_Un_subset_Un_frontier:
      "frontier (s \<union> t) \<subseteq> (frontier s) \<union> (frontier t)"
   by (simp add: frontier_def Un_Diff) (auto simp add: closure_def interior_def islimpt_def)
 
 definition has_partial_derivative:: "(('a::euclidean_space) \<Rightarrow> 'b::euclidean_space) \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a) \<Rightarrow> bool" where
   "has_partial_derivative F base_vec F' a
         \<equiv> ((\<lambda>x::'a::euclidean_space. F( (a - ((a \<bullet> base_vec) *\<^sub>R base_vec)) + (x \<bullet> base_vec) *\<^sub>R base_vec ))
                 has_derivative F') (at a)"
 
 definition has_partial_vector_derivative:: "(('a::euclidean_space) \<Rightarrow> 'b::euclidean_space) \<Rightarrow> 'a \<Rightarrow> ( 'b) \<Rightarrow> ('a) \<Rightarrow> bool" where
   "has_partial_vector_derivative F base_vec F' a
         \<equiv> ((\<lambda>x. F( (a - ((a \<bullet> base_vec) *\<^sub>R base_vec)) + x *\<^sub>R base_vec ))
                 has_vector_derivative F') (at (a \<bullet> base_vec))"
 
 definition partially_vector_differentiable where
   "partially_vector_differentiable F base_vec p \<equiv> (\<exists>F'. has_partial_vector_derivative F base_vec F' p)"
 
 definition partial_vector_derivative:: "(('a::euclidean_space) \<Rightarrow> 'b::euclidean_space) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b" where
   "partial_vector_derivative F base_vec a
         \<equiv> (vector_derivative (\<lambda>x. F( (a - ((a \<bullet> base_vec) *\<^sub>R base_vec)) + x *\<^sub>R base_vec)) (at (a \<bullet> base_vec)))"
 
 lemma partial_vector_derivative_works:
   assumes "partially_vector_differentiable F base_vec a"
   shows "has_partial_vector_derivative F base_vec (partial_vector_derivative F base_vec a) a"
 proof -
   obtain F' where F'_prop: "((\<lambda>x. F( (a - ((a \<bullet> base_vec) *\<^sub>R base_vec)) + x *\<^sub>R base_vec ))
                             has_vector_derivative F') (at (a \<bullet> base_vec))"
     using assms partially_vector_differentiable_def has_partial_vector_derivative_def
     by blast
   show ?thesis
     using Derivative.differentiableI_vector[OF F'_prop]
     by(simp add: vector_derivative_works partial_vector_derivative_def[symmetric]
         has_partial_vector_derivative_def[symmetric])
 qed
 
 lemma fundamental_theorem_of_calculus_partial_vector:
   fixes a b:: "real" and
     F:: "('a::euclidean_space \<Rightarrow> 'b::euclidean_space)" and
     i:: "'a" and
     j:: "'b" and
     F_j_i:: "('a::euclidean_space \<Rightarrow> real)"
   assumes a_leq_b: "a \<le> b" and
     Base_vecs: "i \<in> Basis" "j \<in> Basis" and
     no_i_component: "c \<bullet> i = 0 " and
     has_partial_deriv: "\<forall>p \<in> D. has_partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i (F_j_i p) p" and
     domain_subset_of_D: "{x *\<^sub>R i + c |x. a \<le> x \<and> x \<le> b} \<subseteq> D"
   shows "((\<lambda>x. F_j_i( x *\<^sub>R i + c)) has_integral
           F(b *\<^sub>R i + c) \<bullet> j - F(a *\<^sub>R i + c) \<bullet> j) (cbox a b)"
 proof -
   let ?domain = "{v. \<exists>x. a \<le> x \<and> x \<le> b \<and> v = x *\<^sub>R i + c}"
   have "\<forall>x\<in> ?domain.  has_partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i (F_j_i x) x"
     using has_partial_deriv domain_subset_of_D
     by auto
   then have "\<forall>x\<in> (cbox a b).  ((\<lambda>x. F(x *\<^sub>R i + c) \<bullet> j) has_vector_derivative (F_j_i( x *\<^sub>R i + c))) (at x)"
   proof(clarsimp simp add: has_partial_vector_derivative_def)
     fix x::real
     assume range_of_x: "a \<le> x" "x \<le> b"
     assume ass2: "\<forall>x. (\<exists>z\<ge>a. z \<le> b \<and> x = z *\<^sub>R i + c) \<longrightarrow>
                      ((\<lambda>z. F(x - (x \<bullet> i) *\<^sub>R i + z *\<^sub>R i) \<bullet> j) has_vector_derivative F_j_i x) (at (x \<bullet> i))"
     have "((\<lambda>z. F((x *\<^sub>R i + c) - ((x *\<^sub>R i + c) \<bullet> i) *\<^sub>R i + z *\<^sub>R i) \<bullet> j) has_vector_derivative F_j_i (x *\<^sub>R i + c)) (at ((x *\<^sub>R i + c) \<bullet> i)) "
       using range_of_x ass2 by auto
     then have 0: "((\<lambda>x. F( c + x *\<^sub>R i) \<bullet> j) has_vector_derivative F_j_i (x *\<^sub>R i + c)) (at x)"
       by (simp add: assms(2) inner_left_distrib no_i_component)
     have 1: "(\<lambda>x. F(x *\<^sub>R i + c) \<bullet> j) = (\<lambda>x. F(c + x *\<^sub>R i) \<bullet> j)"
       by (simp add: add.commute)
     then show "((\<lambda>x. F(x *\<^sub>R i + c) \<bullet> j) has_vector_derivative F_j_i (x *\<^sub>R i + c)) (at x)" 
       using 0 and 1 by auto
   qed
   then have "\<forall>x\<in> (cbox a b).  ((\<lambda>x. F(x *\<^sub>R i + c) \<bullet> j) has_vector_derivative (F_j_i( x *\<^sub>R i + c))) (at_within x (cbox a b))"
     using has_vector_derivative_at_within
     by blast
   then show "( (\<lambda>x. F_j_i( x *\<^sub>R i + c)) has_integral
              F(b *\<^sub>R i + c) \<bullet> j - F(a *\<^sub>R i + c) \<bullet> j) (cbox a b)"
     using fundamental_theorem_of_calculus[of "a" "b" "(\<lambda>x::real.  F(x *\<^sub>R i + c) \<bullet> j)" "(\<lambda>x::real. F_j_i( x *\<^sub>R i + c))"] and
       a_leq_b
     by auto
 qed
 
 lemma fundamental_theorem_of_calculus_partial_vector_gen:
   fixes k1 k2:: "real" and
     F:: "('a::euclidean_space \<Rightarrow> 'b::euclidean_space)" and
     i:: "'a" and
     F_i:: "('a::euclidean_space \<Rightarrow> 'b)"
   assumes a_leq_b: "k1 \<le> k2" and
     unit_len: "i \<bullet> i = 1" and
     no_i_component: "c \<bullet> i = 0 " and
     has_partial_deriv: "\<forall>p \<in> D. has_partial_vector_derivative F i (F_i p) p" and
     domain_subset_of_D: "{v. \<exists>x. k1 \<le> x \<and> x \<le> k2 \<and> v = x *\<^sub>R i + c} \<subseteq> D"
   shows "( (\<lambda>x. F_i( x *\<^sub>R i + c)) has_integral
                                         F(k2 *\<^sub>R i + c) - F(k1 *\<^sub>R i + c)) (cbox k1 k2)"
 proof -
   let ?domain = "{v. \<exists>x. k1 \<le> x \<and> x \<le> k2 \<and> v = x *\<^sub>R i + c}"
   have "\<forall>x\<in> ?domain.  has_partial_vector_derivative F i (F_i x) x"
     using has_partial_deriv domain_subset_of_D
     by auto
   then have "\<forall>x\<in> (cbox k1 k2).  ((\<lambda>x. F(x *\<^sub>R i + c)) has_vector_derivative (F_i( x *\<^sub>R i + c))) (at x)"
   proof (clarsimp simp add: has_partial_vector_derivative_def)
     fix x::real
     assume range_of_x: "k1 \<le> x" "x \<le> k2"
     assume ass2: "\<forall>x. (\<exists>z\<ge>k1. z \<le> k2 \<and> x = z *\<^sub>R i + c) \<longrightarrow>
                      ((\<lambda>z. F(x - (x \<bullet> i) *\<^sub>R i + z *\<^sub>R i)) has_vector_derivative F_i x) (at (x \<bullet> i))"
     have "((\<lambda>z. F((x *\<^sub>R i + c) - ((x *\<^sub>R i + c) \<bullet> i) *\<^sub>R i + z *\<^sub>R i)) has_vector_derivative F_i (x *\<^sub>R i + c)) (at ((x *\<^sub>R i + c) \<bullet> i)) "
       using range_of_x ass2 by auto
     then have 0: "((\<lambda>x. F( c + x *\<^sub>R i)) has_vector_derivative F_i (x *\<^sub>R i + c)) (at x)"
       by (simp add: inner_commute inner_right_distrib no_i_component unit_len)
     have 1: "(\<lambda>x. F(x *\<^sub>R i + c)) = (\<lambda>x. F(c + x *\<^sub>R i))"
       by (simp add: add.commute)
     then show "((\<lambda>x. F(x *\<^sub>R i + c)) has_vector_derivative F_i (x *\<^sub>R i + c)) (at x)" using 0 and 1 by auto
   qed
   then have "\<forall>x\<in> (cbox k1 k2).  ((\<lambda>x. F(x *\<^sub>R i + c) ) has_vector_derivative (F_i( x *\<^sub>R i + c))) (at_within x (cbox k1 k2))"
     using has_vector_derivative_at_within
     by blast
   then show "( (\<lambda>x. F_i( x *\<^sub>R i + c)) has_integral
                                         F(k2 *\<^sub>R i + c) - F(k1 *\<^sub>R i + c)) (cbox k1 k2)"
     using fundamental_theorem_of_calculus[of "k1" "k2" "(\<lambda>x::real.  F(x *\<^sub>R i + c))" "(\<lambda>x::real. F_i( x *\<^sub>R i + c))"] and
       a_leq_b
     by auto
 qed
 
 lemma add_scale_img:
   assumes "a < b" shows "(\<lambda>x::real. a + (b - a) * x) ` {0 .. 1} = {a .. b}"
   using assms 
   apply (auto simp: algebra_simps affine_ineq image_iff)
   using less_eq_real_def apply force
   apply (rule_tac x="(x-a)/(b-a)" in bexI)
    apply (auto simp: field_simps)
   done
 
 lemma add_scale_img':
   assumes "a \<le> b"
   shows "(\<lambda>x::real. a + (b - a) * x) ` {0 .. 1} = {a .. b}"
 proof (cases "a = b")
   case True
   then show ?thesis by force
 next
   case False
   then show ?thesis
     using add_scale_img assms by auto
 qed
 
 definition analytically_valid:: "'a::euclidean_space set \<Rightarrow> ('a \<Rightarrow> 'b::{euclidean_space,times,one}) \<Rightarrow> 'a \<Rightarrow> bool" where
   "analytically_valid s F i \<equiv>
        (\<forall>a \<in> s. partially_vector_differentiable F i a) \<and>
        continuous_on s F \<and> \<comment> \<open>TODO: should we replace this with saying that \<open>F\<close> is partially diffrerentiable on \<open>Dy\<close>,\<close>
                            \<comment> \<open>i.e. there is a partial derivative on every dimension\<close>
        integrable lborel (\<lambda>p. (partial_vector_derivative F i) p * indicator s p) \<and>
        (\<lambda>x. integral UNIV (\<lambda>y. (partial_vector_derivative F i (y *\<^sub>R i + x *\<^sub>R (\<Sum> b \<in>(Basis - {i}). b)))
             * (indicator s (y *\<^sub>R i + x *\<^sub>R (\<Sum>b \<in> Basis - {i}. b)) ))) \<in> borel_measurable lborel"
   (*(\<lambda>x. integral UNIV (\<lambda>y. ((partial_vector_derivative F i) (y, x)) * (indicator s (y, x)))) \<in> borel_measurable lborel)"*)
 
 
 lemma analytically_valid_imp_part_deriv_integrable_on:
   assumes "analytically_valid (s::(real*real) set) (f::(real*real)\<Rightarrow> real) i"
   shows "(partial_vector_derivative f i) integrable_on s"
 proof -
   have "integrable lborel (\<lambda>p. partial_vector_derivative f i p * indicator s p)"
     using assms
     by(simp add:  analytically_valid_def indic_ident)
   then have "integrable lborel (\<lambda>p::(real*real). if p \<in> s then partial_vector_derivative f i p else 0)"
     using indic_ident[of "partial_vector_derivative f i"]
     by (simp add: indic_ident)
   then have "(\<lambda>x. if x \<in> s then partial_vector_derivative f i x else 0) integrable_on UNIV"
     using Equivalence_Lebesgue_Henstock_Integration.integrable_on_lborel
     by auto
   then show "(partial_vector_derivative f i) integrable_on s"
     using integrable_restrict_UNIV
     by auto
 qed
 
 (*******************************************************************************)
 
 definition typeII_twoCube :: "((real * real) \<Rightarrow> (real * real)) \<Rightarrow> bool" where
   "typeII_twoCube twoC
          \<equiv> \<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a..b}. g2 x \<le> g1 x) \<and>
                        twoC = (\<lambda>(y, x). ((1 - y) * (g2 ((1-x)*a + x*b)) + y * (g1 ((1-x)*a + x*b)),
                                         (1-x)*a + x*b)) \<and>
                        g1 piecewise_C1_differentiable_on {a .. b} \<and>
                        g2 piecewise_C1_differentiable_on {a .. b}"
 
 abbreviation unit_cube where "unit_cube \<equiv> cbox (0,0) (1::real,1::real)" 
 
 definition cubeImage:: "two_cube \<Rightarrow> ((real*real) set)" where
   "cubeImage twoC \<equiv> (twoC ` unit_cube)"
 
 lemma typeII_twoCubeImg:
   assumes "typeII_twoCube twoC"
   shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a .. b}. g2 x \<le> g1 x) \<and>
                       cubeImage twoC = {(y,x). x \<in> {a..b} \<and> y \<in> {g2 x .. g1 x}}
                       \<and> twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))
                       \<and> g1 piecewise_C1_differentiable_on {a .. b} \<and> g2 piecewise_C1_differentiable_on {a .. b} "
   using assms
 proof (simp add: typeII_twoCube_def cubeImage_def image_def)
   assume " \<exists>a b. a < b \<and> (\<exists>g1 g2. (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and>
                             twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b)) \<and>
                             g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b})"
   then obtain a b g1 g2 where
     twoCisTypeII: "a < b"
     "(\<forall>x\<in>{a..b}. g2 x \<le> g1 x)"
     "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     by auto
   have ab1: "a - x * a + x * b \<le> b" if a1: "0 \<le> x" "x \<le> 1" for x
     using that apply (simp add: algebra_simps)
     by (metis affine_ineq less_eq_real_def mult.commute twoCisTypeII(1))
   have ex: "\<exists>z\<in>Green.unit_cube.
                (d, c) =
                (case z of
                 (y, x) \<Rightarrow>
                   (g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b),
                    a - x * a + x * b))" 
     if c_bounds: "a \<le> c" "c \<le> b" and d_bounds: "g2 c \<le> d" "d \<le> g1 c" for c d
   proof -
     have b_minus_a_nzero: "b - a \<noteq> 0" using twoCisTypeII(1) by auto
     have x_witness: "\<exists>k1. c = (1 - k1)*a + k1 * b \<and> 0 \<le> k1 \<and> k1 \<le> 1"
       apply (rule_tac x="(c - a)/(b - a)" in exI)
       using c_bounds \<open>a < b\<close> apply (simp add: divide_simps algebra_simps)
       using sum_sqs_eq by blast
     have y_witness: "\<exists>k2. d = (1 - k2)*(g2 c) + k2 * (g1 c) \<and> 0 \<le> k2 \<and> k2 \<le> 1"
     proof (cases "g1 c - g2 c = 0")
       case True
       with d_bounds show ?thesis by (fastforce simp add: algebra_simps)
     next
       case False
       let ?k2 = "(d - g2 c)/(g1 c - g2 c)"
       have k2_in_bounds: "0 \<le> ?k2 \<and> ?k2 \<le> 1" 
         using twoCisTypeII(2) c_bounds d_bounds False by simp
       have "d = (1 - ?k2) * g2 c + ?k2 * g1 c" 
         using False sum_sqs_eq by (fastforce simp add: divide_simps algebra_simps)
       with k2_in_bounds show ?thesis 
         by fastforce
     qed
     show "\<exists>x\<in>unit_cube. (d, c) = (case x of (y, x) \<Rightarrow> (g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b), a - x * a + x * b))"
       using x_witness y_witness by (force simp add: left_diff_distrib)
   qed
   have "{y. \<exists>x\<in>unit_cube. y = twoC x} = {(y, x). a \<le> x \<and> x \<le> b \<and> g2 x \<le> y \<and> y \<le> g1 x}"
     apply (auto simp add: twoCisTypeII ab1 left_diff_distrib ex)
     using order.order_iff_strict twoCisTypeII(1) apply fastforce
      apply (smt affine_ineq atLeastAtMost_iff mult_left_mono twoCisTypeII)+
     done
   then show "\<exists>a b. a < b \<and> (\<exists>g1 g2. (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and>
                             {y. \<exists>x\<in>unit_cube. y = twoC x} = {(y, x). a \<le> x \<and> x \<le> b \<and> g2 x \<le> y \<and> y \<le> g1 x} \<and>
                             twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b)) \<and>
                             g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b})"
     using twoCisTypeII by blast
 qed
 
 definition horizontal_boundary :: "two_cube \<Rightarrow> one_chain" where
   "horizontal_boundary twoC \<equiv> {(1, (\<lambda>x. twoC(x,0))), (-1, (\<lambda>x. twoC(x,1)))}"
 
 definition vertical_boundary :: "two_cube \<Rightarrow> one_chain" where
   "vertical_boundary twoC \<equiv> {(-1, (\<lambda>y. twoC(0,y))), (1, (\<lambda>y. twoC(1,y)))}"
 
 definition boundary :: "two_cube \<Rightarrow> one_chain" where
   "boundary twoC \<equiv> horizontal_boundary twoC \<union> vertical_boundary twoC"
 
 definition valid_two_cube where
   "valid_two_cube twoC \<equiv> card (boundary twoC) = 4"
 
 definition two_chain_integral:: "two_chain \<Rightarrow> ((real*real)\<Rightarrow>(real)) \<Rightarrow> real" where
   "two_chain_integral twoChain F \<equiv> \<Sum>C\<in>twoChain. (integral (cubeImage C) F)"
 
 definition valid_two_chain where
   "valid_two_chain twoChain \<equiv> (\<forall>twoCube \<in> twoChain. valid_two_cube twoCube) \<and> pairwise (\<lambda>c1 c2. ((boundary c1) \<inter> (boundary c2)) = {}) twoChain \<and> inj_on cubeImage twoChain"
 
 definition two_chain_boundary:: "two_chain \<Rightarrow> one_chain" where
   "two_chain_boundary twoChain == \<Union>(boundary ` twoChain)"
 
 definition gen_division where
   "gen_division s S \<equiv> (finite S \<and> (\<Union>S = s) \<and> pairwise (\<lambda>X Y. negligible (X \<inter> Y)) S)"
 
 
 definition two_chain_horizontal_boundary:: "two_chain \<Rightarrow> one_chain" where
   "two_chain_horizontal_boundary twoChain  \<equiv> \<Union>(horizontal_boundary ` twoChain)"
 
 definition two_chain_vertical_boundary:: "two_chain \<Rightarrow> one_chain" where
   "two_chain_vertical_boundary twoChain  \<equiv> \<Union>(vertical_boundary ` twoChain)"
 
 definition only_horizontal_division where
   "only_horizontal_division one_chain two_chain 
       \<equiv> \<exists>\<H> \<V>. finite \<H> \<and> finite \<V> \<and>
                (\<forall>(k,\<gamma>) \<in> \<H>.
                  (\<exists>(k', \<gamma>') \<in> two_chain_horizontal_boundary two_chain.
                      (\<exists>a \<in> {0..1}. \<exists>b \<in> {0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>))) \<and>
                (common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V>
                 \<or> common_reparam_exists \<V> (two_chain_vertical_boundary two_chain)) \<and>
                boundary_chain \<V> \<and>
                one_chain = \<H> \<union> \<V> \<and> (\<forall>(k,\<gamma>)\<in>\<V>. valid_path \<gamma>)"
 
 lemma sum_zero_set:
   assumes "\<forall>x \<in> s. f x = 0" "finite s" "finite t"
   shows "sum f (s \<union> t) = sum f t"
   using assms
   by (simp add: IntE sum.union_inter_neutral sup_commute)
 
 abbreviation "valid_typeII_division s twoChain \<equiv> ((\<forall>twoCube \<in> twoChain. typeII_twoCube twoCube) \<and>
                                                 (gen_division s (cubeImage ` twoChain)) \<and>
                                                 (valid_two_chain twoChain))"
 
 lemma two_chain_vertical_boundary_is_boundary_chain:
   shows "boundary_chain (two_chain_vertical_boundary twoChain)"
   by(simp add: boundary_chain_def two_chain_vertical_boundary_def vertical_boundary_def)
 
 lemma two_chain_horizontal_boundary_is_boundary_chain:
   shows "boundary_chain (two_chain_horizontal_boundary twoChain)"
   by(simp add: boundary_chain_def two_chain_horizontal_boundary_def horizontal_boundary_def)
 
 definition typeI_twoCube :: "two_cube \<Rightarrow> bool" where
   "typeI_twoCube (twoC::two_cube)
         \<equiv> \<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a..b}. g2 x \<le> g1 x) \<and>
                        twoC = (\<lambda>(x,y). ((1-x)*a + x*b,
                                         (1 - y) * (g2 ((1-x)*a + x*b)) + y * (g1 ((1-x)*a + x*b)))) \<and>
                        g1 piecewise_C1_differentiable_on {a..b} \<and>
                        g2 piecewise_C1_differentiable_on {a..b}"
 
 lemma typeI_twoCubeImg:
   assumes "typeI_twoCube twoC"
   shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a .. b}. g2 x \<le> g1 x) \<and>
                       cubeImage twoC = {(x,y). x \<in> {a..b} \<and> y \<in> {g2 x .. g1 x}} \<and>
                       twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b))) \<and>
                       g1 piecewise_C1_differentiable_on {a .. b} \<and> g2 piecewise_C1_differentiable_on {a .. b} "
 proof -
   have "\<exists>a b. a < b \<and>
           (\<exists>g1 g2. (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and>
                    twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))\<and>
                    g1 piecewise_C1_differentiable_on {a .. b} \<and> g2 piecewise_C1_differentiable_on {a .. b})"
     using assms by (simp add: typeI_twoCube_def)
   then obtain a b g1 g2 where
     twoCisTypeI: "a < b"
         "(\<forall>x\<in>{a..b}. g2 x \<le> g1 x)"
         "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
         "g1 piecewise_C1_differentiable_on {a .. b}"
         "g2 piecewise_C1_differentiable_on {a .. b}"
     by auto
   have ex: "\<exists>z\<in>Green.unit_cube.
                (c, d) =
                (case z of
                 (x, y) \<Rightarrow>
                   (a - x * a + x * b,
                    g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b)))"
     if c_bounds: "a \<le> c" "c \<le> b" and  d_bounds: "g2 c \<le> d" "d \<le> g1 c" for c d
   proof -
     have x_witness: "\<exists>k1. c = (1 - k1)*a + k1 * b \<and> 0 \<le> k1 \<and> k1 \<le> 1"
     proof -
       let ?k1 = "(c - a)/(b - a)"
       have k1_in_bounds: "0 \<le> (c - a)/(b - a) \<and> (c - a)/(b - a) \<le> 1" 
         using twoCisTypeI(1) c_bounds by simp
       have "c = (1 - ?k1)*a + ?k1 * b" 
         using twoCisTypeI(1) sum_sqs_eq
         by (auto simp add: divide_simps algebra_simps)
       then show ?thesis
         using twoCisTypeI k1_in_bounds by fastforce
     qed
     have y_witness: "\<exists>k2. d = (1 - k2)*(g2 c) + k2 * (g1 c) \<and> 0 \<le> k2 \<and> k2 \<le> 1"
     proof (cases "g1 c - g2 c = 0")
       case True
       with d_bounds show ?thesis
         by force
     next
       case False
       let ?k2 = "(d - g2 c)/(g1 c - g2 c)"
       have k2_in_bounds: "0 \<le> ?k2 \<and> ?k2 \<le> 1" using twoCisTypeI(2) c_bounds d_bounds False by simp
       have "d = (1 - ?k2) * g2 c + ?k2 * g1 c"
         using False apply (simp add: divide_simps algebra_simps)
         using sum_sqs_eq by fastforce        
       then show ?thesis using k2_in_bounds by fastforce
     qed
     show "\<exists>x\<in>unit_cube. (c, d) =
             (case x of (x, y) \<Rightarrow> (a - x * a + x * b, g2 (a - x * a + x * b) - y * g2 (a - x * a + x * b) + y * g1 (a - x * a + x * b)))"
       using x_witness y_witness by (force simp add: left_diff_distrib)
   qed
   have "{y. \<exists>x\<in>unit_cube. y = twoC x} = {(x, y). a \<le> x \<and> x \<le> b \<and> g2 x \<le> y \<and> y \<le> g1 x}"
     apply (auto simp add: twoCisTypeI left_diff_distrib ex)
     using less_eq_real_def twoCisTypeI(1) apply auto[1]
        apply (smt affine_ineq twoCisTypeI)
       apply (smt affine_ineq atLeastAtMost_iff mult_left_mono twoCisTypeI)+
     done
   then show ?thesis
     unfolding cubeImage_def image_def using twoCisTypeI by auto
 qed
 
 lemma typeI_cube_explicit_spec:
   assumes "typeI_twoCube twoC"
   shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a .. b}. g2 x \<le> g1 x) \<and>
                       cubeImage twoC = {(x,y). x \<in> {a..b} \<and> y \<in> {g2 x .. g1 x}}
                       \<and> twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))
                       \<and> g1 piecewise_C1_differentiable_on {a .. b} \<and> g2 piecewise_C1_differentiable_on {a .. b}
                       \<and> (\<lambda>x. twoC(x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))
                       \<and> (\<lambda>y. twoC(1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))
                       \<and> (\<lambda>x. twoC(x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))
                       \<and> (\<lambda>y. twoC(0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(x, 0))"
   let ?right_edge = "(\<lambda>y. twoC(1, y))"
   let ?top_edge = "(\<lambda>x. twoC(x, 1))"
   let ?left_edge = "(\<lambda>y. twoC(0, y))"
   obtain a b g1 g2 where
     twoCisTypeI: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(x,y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     using assms and typeI_twoCubeImg[of"twoC"] by auto
   have bottom_edge_explicit: "?bottom_edge = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
     by (simp add: twoCisTypeI(4) algebra_simps)
   have right_edge_explicit: "?right_edge = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
     by (simp add: twoCisTypeI(4) algebra_simps)
   have top_edge_explicit: "?top_edge = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
     by (simp add: twoCisTypeI(4) algebra_simps)
   have left_edge_explicit: "?left_edge = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
     by (simp add: twoCisTypeI(4) algebra_simps)
   show ?thesis
     using bottom_edge_explicit right_edge_explicit top_edge_explicit left_edge_explicit twoCisTypeI
     by auto
 qed
 
 lemma typeI_twoCube_smooth_edges:
   assumes "typeI_twoCube twoC"
     "(k,\<gamma>) \<in> boundary twoC"
   shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(x, 0))"
   let ?right_edge = "(\<lambda>y. twoC(1, y))"
   let ?top_edge = "(\<lambda>x. twoC(x, 1))"
   let ?left_edge = "(\<lambda>y. twoC(0, y))"
   obtain a b g1 g2 where
     twoCisTypeI: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(x,y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     "(\<lambda>x. twoC(x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
     "(\<lambda>x. twoC(x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
     using assms and typeI_cube_explicit_spec[of"twoC"]
     by auto
   have bottom_edge_smooth: "(\<lambda>x. twoC (x, 0)) piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x))-` {x} = {(x - a)/(b -a)}"
       using twoCisTypeI(1)
       by(auto simp add: Set.vimage_def)
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     have scale_shif_smth: "(\<lambda>x. (a + (b - a) * x)) C1_differentiable_on {0..1}" using scale_shift_smooth by auto
     then have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" using C1_differentiable_imp_piecewise by blast
     have g2_smooth: "g2 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeI(1)] twoCisTypeI(6) by auto
     have "(\<lambda>x. g2 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
-      using Cauchy_Integral_Theorem.piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
+      using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x::real. (a + (b - a) * x, g2 (a + (b - a) * x))) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x::real. (a + (b - a) * x, g2 (a + (b - a) * x)))"]
       apply (simp only: real_pair_basis)
       by fastforce
     then show ?thesis using twoCisTypeI(7) by auto
   qed
   have top_edge_smooth: "?top_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x))-` {x} = {(x - a)/(b -a)}"
       using twoCisTypeI(1)
       by(auto simp add: Set.vimage_def)
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     have scale_shif_smth: "(\<lambda>x. (a + (b - a) * x)) C1_differentiable_on {0..1}" using scale_shift_smooth by auto
     then have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" using C1_differentiable_imp_piecewise by blast
     have g1_smooth: "g1 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeI(1)] twoCisTypeI(5) by auto
     have "(\<lambda>x. g1 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
-      using Cauchy_Integral_Theorem.piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g1_smooth finite_vimg]
+      using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g1_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x))) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"]
       apply (simp only: real_pair_basis)
       by fastforce
     then show ?thesis using twoCisTypeI(9) by auto
   qed
   have right_edge_smooth: "?right_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "(\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b))) C1_differentiable_on {0..1}"
       using scale_shift_smooth C1_differentiable_imp_piecewise by auto
     then have "(\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b))) piecewise_C1_differentiable_on {0..1}"
       using C1_differentiable_imp_piecewise by fastforce
     then have "(\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b))) piecewise_C1_differentiable_on {0..1}"
       using pair_prod_smooth_pw_smooth by auto
     then show ?thesis
       using twoCisTypeI(8) by auto
   qed
   have left_edge_smooth: "?left_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "(\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a))) C1_differentiable_on {0..1}"
       using scale_shift_smooth by auto
     then have "(\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a))) piecewise_C1_differentiable_on {0..1}"
       using C1_differentiable_imp_piecewise by fastforce
     then have "(\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a))) piecewise_C1_differentiable_on {0..1}"
       using pair_prod_smooth_pw_smooth by auto
     then show ?thesis
       using twoCisTypeI(10) by auto
   qed
   have "\<gamma> = ?bottom_edge \<or> \<gamma> = ?right_edge \<or> \<gamma> = ?top_edge \<or> \<gamma> = ?left_edge"
     using assms by (auto simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   then show ?thesis
     using left_edge_smooth right_edge_smooth top_edge_smooth bottom_edge_smooth by auto
 qed
 
 lemma two_chain_integral_eq_integral_divisable:
   assumes f_integrable: "\<forall>twoCube \<in> twoChain. F integrable_on cubeImage twoCube" and
     gen_division: "gen_division s (cubeImage ` twoChain)" and
     valid_two_chain: "valid_two_chain twoChain"
   shows "integral s F = two_chain_integral twoChain F"
 proof -
   show "integral s F = two_chain_integral twoChain F"
   proof (simp add: two_chain_integral_def)
     have partial_deriv_integrable:
       "\<forall>twoCube \<in> twoChain. ((F) has_integral (integral (cubeImage twoCube) (F))) (cubeImage twoCube)"
       using f_integrable by auto
     then have partial_deriv_integrable:
       "\<And>twoCubeImg. twoCubeImg \<in> cubeImage ` twoChain \<Longrightarrow> (F has_integral (integral (twoCubeImg) F)) (twoCubeImg)"
       using Henstock_Kurzweil_Integration.integrable_neg by force
     have finite_images: "finite (cubeImage ` twoChain)"
       using gen_division gen_division_def by auto
     have negligible_images: "pairwise (\<lambda>S S'. negligible (S \<inter> S')) (cubeImage ` twoChain)"
       using gen_division  by (auto simp add: gen_division_def pairwise_def)
     have inj: "inj_on cubeImage twoChain"
       using valid_two_chain by (simp add: inj_on_def valid_two_chain_def)
     have "integral s F = (\<Sum>twoCubeImg\<in>cubeImage ` twoChain. integral twoCubeImg F)"
       using has_integral_Union[OF finite_images partial_deriv_integrable negligible_images] gen_division
       by (auto simp add: gen_division_def)
     also have "\<dots> = (\<Sum>C\<in>twoChain. integral (cubeImage C) F)"
       using sum.reindex inj by auto
     finally show "integral s F = (\<Sum>C\<in>twoChain. integral (cubeImage C) F)" .
   qed
 qed
 
 definition only_vertical_division where
   "only_vertical_division one_chain two_chain \<equiv>
        \<exists>\<V> \<H>. finite \<H> \<and> finite \<V> \<and>
                (\<forall>(k,\<gamma>) \<in> \<V>.
                  (\<exists>(k',\<gamma>') \<in> two_chain_vertical_boundary two_chain.
                      (\<exists>a \<in> {0..1}. \<exists>b \<in> {0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>))) \<and>
                (common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H>
                 \<or> common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)) \<and>
                boundary_chain \<H> \<and> one_chain = \<V> \<union> \<H> \<and>
                (\<forall>(k,\<gamma>)\<in>\<H>. valid_path \<gamma>)"
 
 abbreviation "valid_typeI_division s twoChain 
    \<equiv> (\<forall>twoCube \<in> twoChain. typeI_twoCube twoCube) \<and>
       gen_division s (cubeImage ` twoChain) \<and> valid_two_chain twoChain"
 
 
 lemma field_cont_on_typeI_region_cont_on_edges:
   assumes typeI_twoC: "typeI_twoCube twoC" 
     and field_cont: "continuous_on (cubeImage twoC) F" 
     and member_of_boundary: "(k,\<gamma>) \<in> boundary twoC"
   shows "continuous_on (\<gamma> ` {0 .. 1}) F"
 proof -
   obtain a b g1 g2 where
     twoCisTypeI: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(x,y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     "(\<lambda>x. twoC(x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
     "(\<lambda>x. twoC(x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
     using typeI_twoC and typeI_cube_explicit_spec[of"twoC"]
     by auto
   let ?bottom_edge = "(\<lambda>x. twoC(x, 0))"
   let ?right_edge = "(\<lambda>y. twoC(1, y))"
   let ?top_edge = "(\<lambda>x. twoC(x, 1))"
   let ?left_edge = "(\<lambda>y. twoC(0, y))"
   let ?Dg1 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1(x))}"
   have line_is_pair_img: "?Dg1 = (\<lambda>x. (x, g1(x))) ` (cbox a b)"
     using image_def by auto
   have field_cont_on_top_edge_image: "continuous_on ?Dg1  F"
     by (rule continuous_on_subset [OF field_cont]) (auto simp: twoCisTypeI(2) twoCisTypeI(3))
   have top_edge_is_compos_of_scal_and_g1:
     "(\<lambda>x. twoC(x, 1)) = (\<lambda>x. (x, g1(x))) \<circ> (\<lambda>x. a + (b - a) * x)"
     using twoCisTypeI  by auto
   have Dg1_is_bot_edge_pathimg: "path_image (\<lambda>x. twoC(x, 1)) = ?Dg1"
     using line_is_pair_img and top_edge_is_compos_of_scal_and_g1 image_comp path_image_def add_scale_img and twoCisTypeI(1)
     by (metis (no_types, lifting) cbox_interval)
   then have cont_on_top: "continuous_on (path_image ?top_edge) F"
     using field_cont_on_top_edge_image by auto
   let ?Dg2 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2(x))}"
   have line_is_pair_img: "?Dg2 = (\<lambda>x. (x, g2(x))) ` (cbox a b)"
     using image_def by auto
   have field_cont_on_bot_edge_image: "continuous_on ?Dg2 F"
     apply (rule continuous_on_subset [OF field_cont])
     using twoCisTypeI(2) twoCisTypeI(3) by auto
   have bot_edge_is_compos_of_scal_and_g2: "(\<lambda>x. twoC(x, 0)) = (\<lambda>x. (x, g2(x))) \<circ> (\<lambda>x. a + (b - a) * x)"
     using twoCisTypeI by auto
   have Dg2_is_bot_edge_pathimg:
     "path_image (\<lambda>x. twoC(x, 0)) = ?Dg2"
     using line_is_pair_img and bot_edge_is_compos_of_scal_and_g2 image_comp path_image_def add_scale_img and twoCisTypeI(1)
     by (metis (no_types, lifting) cbox_interval)
   then have cont_on_bot: "continuous_on (path_image ?bottom_edge) F"
     using field_cont_on_bot_edge_image by auto
   let ?D_left_edge = "{p. \<exists>y. y \<in> cbox (g2 a) (g1 a) \<and> p = (a, y)}"
   have field_cont_on_left_edge_image: "continuous_on ?D_left_edge  F"
     apply (rule continuous_on_subset [OF field_cont])
     using twoCisTypeI(1) twoCisTypeI(3) by auto
   have "g2 a \<le> g1 a" using twoCisTypeI(1) twoCisTypeI(2) by auto
   then have "(\<lambda>x. g2 a + (g1 a - g2 a) * x) ` {(0::real)..1} = {g2 a .. g1 a}"
     using add_scale_img'[of "g2 a" "g1 a"] by blast
   then have left_eq: "?D_left_edge = ?left_edge ` {0..1}"
     unfolding twoCisTypeI(10)
     by(auto simp add: subset_iff image_def set_eq_iff Semiring_Normalization.comm_semiring_1_class.semiring_normalization_rules(7))
   then have cont_on_left: "continuous_on (path_image ?left_edge) F"
     using field_cont_on_left_edge_image path_image_def
     by (metis left_eq field_cont_on_left_edge_image path_image_def)
   let ?D_right_edge =  "{p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (b, y)}"
   have field_cont_on_left_edge_image: "continuous_on ?D_right_edge F"
     apply (rule continuous_on_subset [OF field_cont])
     using twoCisTypeI(1) twoCisTypeI(3) by auto
   have "g2 b \<le> g1 b" using twoCisTypeI(1)  twoCisTypeI(2) by auto
   then have "(\<lambda>x. g2 b + (g1 b - g2 b) * x) ` {(0::real)..1} = {g2 b .. g1 b}"
     using add_scale_img'[of "g2 b" "g1 b"] by blast
   then have right_eq: "?D_right_edge = ?right_edge ` {0..1}"
     unfolding twoCisTypeI(8) 
     by(auto simp add: subset_iff image_def set_eq_iff Semiring_Normalization.comm_semiring_1_class.semiring_normalization_rules(7))
   then have cont_on_right:
     "continuous_on (path_image ?right_edge) F"
     using field_cont_on_left_edge_image path_image_def
     by (metis right_eq field_cont_on_left_edge_image path_image_def)
   have all_edge_cases:
     "(\<gamma> = ?bottom_edge \<or> \<gamma> = ?right_edge \<or> \<gamma> = ?top_edge \<or> \<gamma> = ?left_edge)"
     using assms by (auto simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   show ?thesis
     apply (simp add: path_image_def[symmetric])
     using cont_on_top cont_on_bot cont_on_right cont_on_left all_edge_cases
     by blast
 qed
 
 lemma typeII_cube_explicit_spec:
   assumes "typeII_twoCube twoC"
   shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x \<in> {a .. b}. g2 x \<le> g1 x) \<and>
                      cubeImage twoC = {(y, x). x \<in> {a..b} \<and> y \<in> {g2 x .. g1 x}}
                   \<and> twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))
                   \<and> g1 piecewise_C1_differentiable_on {a .. b} \<and> g2 piecewise_C1_differentiable_on {a .. b}
                   \<and> (\<lambda>x. twoC(0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))
                   \<and> (\<lambda>y. twoC(y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))
                   \<and> (\<lambda>x. twoC(1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))
                   \<and> (\<lambda>y. twoC(y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(0, x))"
   let ?right_edge = "(\<lambda>y. twoC(y, 1))"
   let ?top_edge = "(\<lambda>x. twoC(1, x))"
   let ?left_edge = "(\<lambda>y. twoC(y, 0))"
   obtain a b g1 g2 where
     twoCisTypeII: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(y, x). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     using assms and typeII_twoCubeImg[of"twoC"] by auto
   have bottom_edge_explicit: "?bottom_edge = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   have right_edge_explicit: "?right_edge = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   have top_edge_explicit: "?top_edge = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   have left_edge_explicit: "?left_edge = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   show ?thesis
     using bottom_edge_explicit right_edge_explicit top_edge_explicit left_edge_explicit twoCisTypeII
     by auto
 qed
 
 lemma typeII_twoCube_smooth_edges:
   assumes "typeII_twoCube twoC" "(k,\<gamma>) \<in> boundary twoC"
   shows "\<gamma> piecewise_C1_differentiable_on {0..1}"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(0, x))"
   let ?right_edge = "(\<lambda>y. twoC(y, 1))"
   let ?top_edge = "(\<lambda>x. twoC(1, x))"
   let ?left_edge = "(\<lambda>y. twoC(y, 0))"
   obtain a b g1 g2 where
     twoCisTypeII: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(y, x). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     "(\<lambda>x. twoC(0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
     "(\<lambda>y. twoC(y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
     "(\<lambda>x. twoC(1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
     "(\<lambda>y. twoC(y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
     using assms and typeII_cube_explicit_spec[of"twoC"]
     by auto
   have bottom_edge_smooth: "?bottom_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x)) -` {x} = {(x - a)/(b -a)}"
       using twoCisTypeII(1)  by auto
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" 
       using scale_shift_smooth C1_differentiable_imp_piecewise by blast
     have g2_smooth: "g2 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeII(1)] twoCisTypeII(6) by auto
     have "(\<lambda>x. g2 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
-      using Cauchy_Integral_Theorem.piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
+      using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x::real. (g2 (a + (b - a) * x), a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x::real. (g2 (a + (b - a) * x), a + (b - a) * x))"]
       by (fastforce simp add: real_pair_basis)
     then show ?thesis using twoCisTypeII(7) by auto
   qed
   have top_edge_smooth: "?top_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x))-` {x} = {(x - a)/(b -a)}"
       using twoCisTypeII(1) by auto
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" 
       using scale_shift_smooth C1_differentiable_imp_piecewise by blast
     have g1_smooth: "g1 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeII(1)] twoCisTypeII(5) by auto
     have "(\<lambda>x. g1 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g1_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x::real. (g1 (a + (b - a) * x), a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x::real. (g1 (a + (b - a) * x), a + (b - a) * x))"]
       by (fastforce simp add: real_pair_basis)
     then show ?thesis using twoCisTypeII(9) by auto
   qed
   have right_edge_smooth: "?right_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "(\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b))) piecewise_C1_differentiable_on {0..1}"
       by (simp add: C1_differentiable_imp_piecewise)
     then have "(\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b)) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[of "(1,0)" "(\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"]
       by (auto simp add: real_pair_basis)
     then show ?thesis
       using twoCisTypeII(8) by auto
   qed
   have left_edge_smooth: "?left_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have 0: "(\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a))) C1_differentiable_on {0..1}"
       using C1_differentiable_imp_piecewise by fastforce
     have "(\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a)) piecewise_C1_differentiable_on {0..1}"
       using C1_differentiable_imp_piecewise[OF C1_differentiable_on_pair[OF 0 C1_differentiable_on_const[of "a" "{0..1}"]]]
       by force
     then show ?thesis
       using twoCisTypeII(10) by auto
   qed
   have "\<gamma> = ?bottom_edge \<or> \<gamma> = ?right_edge \<or> \<gamma> = ?top_edge \<or> \<gamma> = ?left_edge"
     using assms by (auto simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   then show ?thesis
     using left_edge_smooth right_edge_smooth top_edge_smooth bottom_edge_smooth by auto
 qed
 
 lemma field_cont_on_typeII_region_cont_on_edges:
   assumes typeII_twoC:
     "typeII_twoCube twoC" and
     field_cont:
     "continuous_on (cubeImage twoC) F" and
     member_of_boundary:
     "(k,\<gamma>) \<in> boundary twoC"
   shows "continuous_on (\<gamma> ` {0 .. 1}) F"
 proof -
   obtain a b g1 g2 where
     twoCisTypeII: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(y, x). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     "(\<lambda>x. twoC(0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
     "(\<lambda>y. twoC(y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
     "(\<lambda>x. twoC(1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
     "(\<lambda>y. twoC(y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
     using typeII_twoC and typeII_cube_explicit_spec[of"twoC"]
     by auto
   let ?bottom_edge = "(\<lambda>x. twoC(0, x))"
   let ?right_edge = "(\<lambda>y. twoC(y, 1))"
   let ?top_edge = "(\<lambda>x. twoC(1, x))"
   let ?left_edge = "(\<lambda>y. twoC(y, 0))"
   let ?Dg1 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (g1(x), x)}"
   have line_is_pair_img: "?Dg1 = (\<lambda>x. (g1(x), x)) ` (cbox a b)"
     using image_def by auto
   have field_cont_on_top_edge_image: "continuous_on ?Dg1 F"
     by (rule continuous_on_subset [OF field_cont]) (auto simp: twoCisTypeII(2) twoCisTypeII(3))
   have top_edge_is_compos_of_scal_and_g1:
     "(\<lambda>x. twoC(1, x)) = (\<lambda>x. (g1(x), x)) \<circ> (\<lambda>x. a + (b - a) * x)"
     using twoCisTypeII by auto
   have Dg1_is_bot_edge_pathimg:
     "path_image (\<lambda>x. twoC(1, x)) = ?Dg1"
     using line_is_pair_img and top_edge_is_compos_of_scal_and_g1 image_comp path_image_def add_scale_img and twoCisTypeII(1)
     by (metis (no_types, lifting) cbox_interval)
   then have cont_on_top: "continuous_on (path_image ?top_edge) F"
     using field_cont_on_top_edge_image by auto
   let ?Dg2 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (g2(x), x)}"
   have line_is_pair_img: "?Dg2 = (\<lambda>x. (g2(x), x)) ` (cbox a b)"
     using image_def by auto
   have field_cont_on_bot_edge_image: "continuous_on ?Dg2 F"
     by (rule continuous_on_subset [OF field_cont]) (auto simp add: twoCisTypeII(2) twoCisTypeII(3))
   have bot_edge_is_compos_of_scal_and_g2:
     "(\<lambda>x. twoC(0, x)) = (\<lambda>x. (g2(x), x)) \<circ> (\<lambda>x. a + (b - a) * x)"
     using twoCisTypeII by auto
   have Dg2_is_bot_edge_pathimg: "path_image (\<lambda>x. twoC(0, x)) = ?Dg2"
     unfolding path_image_def
     using line_is_pair_img and bot_edge_is_compos_of_scal_and_g2 image_comp  add_scale_img [OF \<open>a < b\<close>]
     by (metis (no_types, lifting) box_real(2))
   then have cont_on_bot: "continuous_on (path_image ?bottom_edge) F"
     using field_cont_on_bot_edge_image
     by auto
   let ?D_left_edge =  "{p. \<exists>y. y \<in> cbox (g2 a) (g1 a) \<and> p = (y, a)}"
   have field_cont_on_left_edge_image: "continuous_on ?D_left_edge F"
     apply (rule continuous_on_subset [OF field_cont])
     using twoCisTypeII(1) twoCisTypeII(3) by auto
   have "g2 a \<le> g1 a" using twoCisTypeII(1) twoCisTypeII(2) by auto
   then have "(\<lambda>x. g2 a + x * (g1 a - g2 a)) ` {(0::real)..1} = {g2 a .. g1 a}"
     using add_scale_img'[of "g2 a" "g1 a"] by (auto simp add: ac_simps)
   with \<open>g2 a \<le> g1 a\<close> have left_eq: "?D_left_edge = ?left_edge ` {0..1}"
     by (simp only: twoCisTypeII(10)) auto
   then have cont_on_left: "continuous_on (path_image ?left_edge) F"
     using field_cont_on_left_edge_image path_image_def
     by (metis left_eq field_cont_on_left_edge_image path_image_def)
   let ?D_right_edge = "{p. \<exists>y. y \<in> cbox (g2 b) (g1 b) \<and> p = (y, b)}"
   have field_cont_on_left_edge_image: "continuous_on ?D_right_edge F"
     apply (rule continuous_on_subset [OF field_cont])
     using twoCisTypeII(1) twoCisTypeII(3) by auto
   have "g2 b \<le> g1 b" using twoCisTypeII(1) twoCisTypeII(2) by auto
   then have "(\<lambda>x. g2 b + x * (g1 b - g2 b)) ` {(0::real)..1} = {g2 b .. g1 b}"
     using add_scale_img'[of "g2 b" "g1 b"] by (auto simp add: ac_simps)
   with \<open>g2 b \<le> g1 b\<close> have right_eq: "?D_right_edge = ?right_edge ` {0..1}"
     by (simp only: twoCisTypeII(8)) auto
   then have cont_on_right:
     "continuous_on (path_image ?right_edge) F"
     using field_cont_on_left_edge_image path_image_def
     by (metis right_eq field_cont_on_left_edge_image path_image_def)
   have all_edge_cases:
     "(\<gamma> = ?bottom_edge \<or> \<gamma> = ?right_edge \<or> \<gamma> = ?top_edge \<or> \<gamma> = ?left_edge)"
     using assms unfolding boundary_def horizontal_boundary_def vertical_boundary_def by blast
   show ?thesis
     apply (simp add: path_image_def[symmetric])
     using cont_on_top cont_on_bot cont_on_right cont_on_left all_edge_cases
     by blast
 qed
 
 lemma two_cube_boundary_is_boundary: "boundary_chain (boundary C)"
   by (auto simp add: boundary_chain_def boundary_def horizontal_boundary_def vertical_boundary_def)
 
 lemma common_boundary_subdiv_exists_refl:
   assumes "\<forall>(k,\<gamma>)\<in>boundary twoC. valid_path \<gamma>"
   shows "common_boundary_sudivision_exists (boundary twoC) (boundary twoC)"
   using assms chain_subdiv_chain_refl common_boundary_sudivision_exists_def two_cube_boundary_is_boundary by blast
 
 lemma common_boundary_subdiv_exists_refl':
   assumes "\<forall>(k,\<gamma>)\<in>C. valid_path \<gamma>"
     "boundary_chain (C::(int \<times> (real \<Rightarrow> real \<times> real)) set)"
   shows "common_boundary_sudivision_exists (C) (C)"
   using assms chain_subdiv_chain_refl common_boundary_sudivision_exists_def by blast
 
 lemma gen_common_boundary_subdiv_exists_refl_twochain_boundary:
   assumes "\<forall>(k,\<gamma>)\<in>C. valid_path \<gamma>"
     "boundary_chain (C::(int \<times> (real \<Rightarrow> real \<times> real)) set)"
   shows "common_sudiv_exists (C) (C)"
   using assms chain_subdiv_chain_refl common_boundary_sudivision_exists_def common_subdiv_imp_gen_common_subdiv by blast
 
 lemma two_chain_boundary_is_boundary_chain:
   shows "boundary_chain (two_chain_boundary twoChain)"
   by (simp add: boundary_chain_def two_chain_boundary_def boundary_def horizontal_boundary_def vertical_boundary_def)
 
 lemma typeI_edges_are_valid_paths:
   assumes "typeI_twoCube twoC" "(k,\<gamma>) \<in> boundary twoC"
   shows "valid_path \<gamma>"
   using typeI_twoCube_smooth_edges[OF assms] C1_differentiable_imp_piecewise
   by (auto simp: valid_path_def)
 
 lemma typeII_edges_are_valid_paths:
   assumes "typeII_twoCube twoC" "(k,\<gamma>) \<in> boundary twoC"
   shows "valid_path \<gamma>"
   using typeII_twoCube_smooth_edges[OF assms] C1_differentiable_imp_piecewise
   by (auto simp: valid_path_def)
 
 lemma finite_two_chain_vertical_boundary:
   assumes "finite two_chain"
   shows "finite (two_chain_vertical_boundary two_chain)"
   using assms  by (simp add: two_chain_vertical_boundary_def vertical_boundary_def)
 
 lemma finite_two_chain_horizontal_boundary:
   assumes "finite two_chain"
   shows "finite (two_chain_horizontal_boundary two_chain)"
   using assms  by (simp add: two_chain_horizontal_boundary_def horizontal_boundary_def)
 
 locale R2 =
   fixes i j
   assumes i_is_x_axis: "i = (1::real,0::real)" and
     j_is_y_axis: "j = (0::real, 1::real)"
 begin
 
 lemma analytically_valid_y:
   assumes "analytically_valid s F i"
   shows "(\<lambda>x. integral UNIV (\<lambda>y. (partial_vector_derivative F i) (y, x) * (indicator s (y, x)))) \<in> borel_measurable lborel"
 proof -
   have "{(1::real, 0::real), (0, 1)} - {(1, 0)} = {(0::real,1::real)}"
     by force
   with assms show ?thesis
     using assms  by(simp add: real_pair_basis analytically_valid_def i_is_x_axis j_is_y_axis)
 qed
 
 lemma analytically_valid_x:
   assumes "analytically_valid s F j"
   shows "(\<lambda>x. integral UNIV (\<lambda>y. ((partial_vector_derivative F j) (x, y)) * (indicator s (x, y)))) \<in> borel_measurable lborel"
 proof -
   have "{(1::real, 0::real), (0, 1)} - {(0, 1)} = {(1::real, 0::real)}"
     by force
   with assms show ?thesis
     by(simp add: real_pair_basis analytically_valid_def i_is_x_axis j_is_y_axis)
 qed
 
 lemma Greens_thm_type_I:
   fixes F:: "((real*real) \<Rightarrow> (real * real))" and
     gamma1 gamma2 gamma3 gamma4 :: "(real \<Rightarrow> (real * real))" and
     a:: "real" and b:: "real" and
     g1:: "(real \<Rightarrow> real)" and g2:: "(real \<Rightarrow> real)"
   assumes Dy_def: "Dy_pair = {(x::real,y) . x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}" and
     gamma1_def: "gamma1 = (\<lambda>x. (a + (b - a) * x, g2(a + (b - a) * x)))" and
     gamma1_smooth: "gamma1 piecewise_C1_differentiable_on {0..1}" and (*TODO: This should be piecewise smooth*)
     gamma2_def: "gamma2 = (\<lambda>x. (b, g2(b) + x  *\<^sub>R (g1(b) - g2(b))))" and
     gamma3_def: "gamma3 = (\<lambda>x. (a + (b - a) * x, g1(a + (b - a) * x)))" and
     gamma3_smooth: "gamma3 piecewise_C1_differentiable_on {0..1}" and
     gamma4_def: "gamma4 = (\<lambda>x. (a,  g2(a) + x *\<^sub>R (g1(a) - g2(a))))" and
     F_i_analytically_valid: "analytically_valid Dy_pair (\<lambda>p. F(p) \<bullet> i) j" and
     g2_leq_g1: "\<forall>x \<in> cbox a b. (g2 x) \<le> (g1 x)" and (*This is needed otherwise what would Dy be?*)
     a_lt_b: "a < b"
   shows "(line_integral F {i} gamma1) +
          (line_integral F {i} gamma2) -
          (line_integral F {i} gamma3) -
          (line_integral F {i} gamma4)
                  = (integral Dy_pair (\<lambda>a. - (partial_vector_derivative (\<lambda>p. F(p) \<bullet> i) j a)))"
     "line_integral_exists F {i} gamma4"
     "line_integral_exists F {i} gamma3"
     "line_integral_exists F {i} gamma2"
     "line_integral_exists F {i} gamma1"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>a. (F a) \<bullet> i)  j"
   have F_first_is_continuous: "continuous_on Dy_pair (\<lambda>a. F(a) \<bullet> i)"
     using F_i_analytically_valid
     by (auto simp add: analytically_valid_def)
   let ?f = "(\<lambda>x. if x \<in> (Dy_pair) then (partial_vector_derivative (\<lambda>a. (F a) \<bullet> i)  j) x else 0)"
   have f_lesbegue_integrable: "integrable lborel ?f"
     using F_i_analytically_valid
     by (auto simp add: analytically_valid_def indic_ident)
   have partially_vec_diff: "\<forall>a \<in> Dy_pair. partially_vector_differentiable (\<lambda>a. (F a) \<bullet> i) j a"
     using F_i_analytically_valid
     by (auto simp add: analytically_valid_def indicator_def)
   have x_axis_integral_measurable: "(\<lambda>x. integral UNIV (\<lambda>y. ?f(x, y))) \<in> borel_measurable lborel"
   proof -
     have "(\<lambda>p. (?F_b' p) * indicator (Dy_pair) p) = (\<lambda>x. if x \<in> (Dy_pair) then (?F_b') x else 0)"
       using indic_ident[of "?F_b'"] by auto
     then have "\<And>x y. ?F_b' (x,y) * indicator (Dy_pair) (x,y) = (\<lambda>x. if x \<in> (Dy_pair) then (?F_b') x else 0) (x,y)"
       by auto
     then show ?thesis
       using analytically_valid_x[OF F_i_analytically_valid]
       by (auto simp add: indicator_def)
   qed
   have F_partially_differentiable: "\<forall>a\<in>Dy_pair. has_partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j (?F_b' a) a"
     using partial_vector_derivative_works partially_vec_diff
     by fastforce
   have g1_g2_continuous: "continuous_on (cbox a b) g1"
     "continuous_on (cbox a b) g2"
   proof -
     have shift_scale_cont: "continuous_on {a..b} (\<lambda>x. (x - a)*(1/(b-a)))"
       by (intro continuous_intros)
     have shift_scale_inv: "(\<lambda>x. a + (b - a) * x) \<circ> (\<lambda>x. (x - a)*(1/(b-a))) = id"
       using a_lt_b by (auto simp add: o_def)
     have img_shift_scale: "(\<lambda>x. (x - a)*(1/(b-a)))`{a..b} = {0..1}"
       using a_lt_b apply (auto simp: divide_simps image_iff)
       apply (rule_tac x="x * (b - a) + a" in bexI)
       using le_diff_eq by fastforce+
     have gamma1_y_component: "(\<lambda>x. g2(a + (b - a) * x)) = g2 \<circ> (\<lambda>x.(a + (b - a) * x))"
       by auto
     have "continuous_on {0..1} (\<lambda>x. g2(a + (b - a) * x))"
       using continuous_on_inner[OF piecewise_C1_differentiable_on_imp_continuous_on[OF gamma1_smooth], of "(\<lambda>x. j)", OF continuous_on_const]
       by (simp add: gamma1_def j_is_y_axis)
     then have "continuous_on {a..b} ((\<lambda>x. g2(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using img_shift_scale continuous_on_compose shift_scale_cont
       by force
     then have "continuous_on {a..b} (g2 \<circ> (\<lambda>x.(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using gamma1_y_component by auto
     then show "continuous_on (cbox a b) g2"
       using a_lt_b by (simp add: shift_scale_inv)
     have gamma3_y_component: "(\<lambda>x. g1(a + (b - a) * x)) = g1 \<circ> (\<lambda>x.(a + (b - a) * x))"
       by auto
     have "continuous_on {0..1} (\<lambda>x. g1(a + (b - a) * x))"
       using continuous_on_inner[OF piecewise_C1_differentiable_on_imp_continuous_on[OF gamma3_smooth], of "(\<lambda>x. j)", OF continuous_on_const]
       by (simp add: gamma3_def j_is_y_axis)
     then have "continuous_on {a..b} ((\<lambda>x. g1(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using img_shift_scale continuous_on_compose shift_scale_cont
       by force
     then have "continuous_on {a..b} (g1 \<circ> (\<lambda>x.(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using gamma3_y_component by auto
     then show "continuous_on (cbox a b) g1"
       using a_lt_b by (simp add: shift_scale_inv)
   qed
   have g2_scale_j_contin: "continuous_on (cbox a b) (\<lambda>x. (0, g2 x))"
     by (intro continuous_intros g1_g2_continuous)
   let ?Dg2 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (x, g2(x))}"
   have line_is_pair_img: "?Dg2 = (\<lambda>x. (x, g2(x))) ` (cbox a b)"
     using image_def by auto
   have g2_path_continuous: "continuous_on (cbox a b) (\<lambda>x. (x, g2(x)))"
     by (intro continuous_intros g1_g2_continuous)
   have field_cont_on_gamma1_image: "continuous_on ?Dg2  (\<lambda>a. F(a) \<bullet> i)"
     apply (rule continuous_on_subset [OF F_first_is_continuous])
     by (auto simp add: Dy_def g2_leq_g1)
   have gamma1_is_compos_of_scal_and_g2:
     "gamma1 = (\<lambda>x. (x, g2(x))) \<circ> (\<lambda>x. a + (b - a) * x)"
     using gamma1_def by auto
   have add_scale_img:
     "(\<lambda>x. a + (b - a) * x) ` {0 .. 1} = {a .. b}" using add_scale_img and a_lt_b by auto
   then have Dg2_is_gamma1_pathimg: "path_image gamma1 = ?Dg2"
     by (metis (no_types, lifting) box_real(2) gamma1_is_compos_of_scal_and_g2 image_comp line_is_pair_img path_image_def)
   have Base_vecs: "i \<in> Basis" "j \<in> Basis" "i \<noteq> j"
     using real_pair_basis and i_is_x_axis and j_is_y_axis by auto
   have gamma1_as_euclid_space_fun: "gamma1 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R i + (0, g2 (a + (b - a) * x)))"
     using i_is_x_axis gamma1_def by auto
   have 0: "line_integral F {i} gamma1 = integral (cbox a b) (\<lambda>x. F(x, g2(x)) \<bullet> i )"
           "line_integral_exists F {i} gamma1"
     using line_integral_on_pair_path_strong [OF norm_Basis[OF Base_vecs(1)] _ gamma1_as_euclid_space_fun, of "F"] 
       gamma1_def gamma1_smooth g2_scale_j_contin a_lt_b add_scale_img
       Dg2_is_gamma1_pathimg and field_cont_on_gamma1_image
     by (auto simp: pathstart_def pathfinish_def i_is_x_axis)
   then show "(line_integral_exists F {i} gamma1)" by metis
   have gamma2_x_const: "\<forall>x. gamma2 x \<bullet> i = b"
     by (simp add: i_is_x_axis gamma2_def)
   have 1: "(line_integral F {i} gamma2) = 0" "(line_integral_exists F {i} gamma2)"
       using line_integral_on_pair_straight_path[OF gamma2_x_const] straight_path_diffrentiable_x gamma2_def
       by (auto simp add: mult.commute)
   then show "(line_integral_exists F {i} gamma2)" by metis
   have "continuous_on (cbox a b) (\<lambda>x. F(x, g2(x))  \<bullet> i)"
     using line_is_pair_img and g2_path_continuous and field_cont_on_gamma1_image
       Topological_Spaces.continuous_on_compose i_is_x_axis j_is_y_axis
     by auto
   then have 6: "(\<lambda>x. F(x, g2(x))  \<bullet> i) integrable_on (cbox a b)"
     using integrable_continuous [of "a" "b" "(\<lambda>x. F(x, g2(x))  \<bullet> i)"] by auto
   have g1_scale_j_contin: "continuous_on (cbox a b) (\<lambda>x. (0, g1 x))"
     by (intro continuous_intros g1_g2_continuous)
   let ?Dg1 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1(x))}"
   have line_is_pair_img: "?Dg1 = (\<lambda>x. (x, g1(x)) ) ` (cbox a b)"
     using image_def  by auto
   have g1_path_continuous: "continuous_on (cbox a b) (\<lambda>x. (x, g1(x)))"
     by (intro continuous_intros g1_g2_continuous)
   have field_cont_on_gamma3_image: "continuous_on ?Dg1  (\<lambda>a. F(a)  \<bullet> i)"
     apply (rule continuous_on_subset [OF F_first_is_continuous])
     by (auto simp add: Dy_def g2_leq_g1)
   have gamma3_is_compos_of_scal_and_g1:
     "gamma3 = (\<lambda>x. (x, g1(x))) \<circ> (\<lambda>x. a + (b - a) * x)"
     using gamma3_def by auto
   then have Dg1_is_gamma3_pathimg: "path_image gamma3 = ?Dg1"
     by (metis (no_types, lifting) box_real(2) image_comp line_is_pair_img local.add_scale_img path_image_def)
   have Base_vecs: "i \<in> Basis" "j \<in> Basis" "i \<noteq> j"
     using real_pair_basis and i_is_x_axis and j_is_y_axis by auto
   have gamma3_as_euclid_space_fun: "gamma3 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R i + (0, g1 (a + (b - a) * x)))"
     using i_is_x_axis gamma3_def by auto
   have 2: "line_integral F {i} gamma3 = integral (cbox a b) (\<lambda>x. F(x, g1(x)) \<bullet> i )"
          "line_integral_exists F {i} gamma3"
     using line_integral_on_pair_path_strong [OF norm_Basis[OF Base_vecs(1)] _ gamma3_as_euclid_space_fun, of "F"]
       gamma3_def and gamma3_smooth and g1_scale_j_contin and a_lt_b and add_scale_img
       Dg1_is_gamma3_pathimg and field_cont_on_gamma3_image
     by (auto simp: pathstart_def pathfinish_def i_is_x_axis)
   then show "(line_integral_exists F {i} gamma3)" by metis
   have gamma4_x_const: "\<forall>x. gamma4 x \<bullet> i = a"
     using gamma4_def
     by (auto simp add: real_inner_class.inner_add_left inner_not_same_Basis  i_is_x_axis)
   have 3: "(line_integral F {i} gamma4) = 0" "(line_integral_exists F {i} gamma4)"
     using line_integral_on_pair_straight_path[OF gamma4_x_const] straight_path_diffrentiable_x gamma4_def
     by (auto simp add: mult.commute)
   then show "(line_integral_exists F {i} gamma4)" 
     by metis
   have "continuous_on (cbox a b) (\<lambda>x. F(x, g1(x)) \<bullet> i)"
     using line_is_pair_img and g1_path_continuous and field_cont_on_gamma3_image
       continuous_on_compose i_is_x_axis j_is_y_axis
     by auto
   then have 7: "(\<lambda>x. F(x, g1(x))  \<bullet> i) integrable_on (cbox a b)"
     using integrable_continuous [of "a" "b" "(\<lambda>x. F(x, g1(x))  \<bullet> i)"]
     by auto
   have partial_deriv_one_d_integrable:
     "((\<lambda>y. ?F_b'(xc, y)) has_integral 
         F(xc,g1(xc)) \<bullet> i - F(xc, g2(xc)) \<bullet> i) (cbox (g2 xc) (g1 xc))" 
      if "xc \<in> cbox a b" for xc
   proof -
     have "{(xc', y).  y \<in> cbox (g2 xc) (g1 xc) \<and> xc' = xc} \<subseteq> Dy_pair"
       using that by (auto simp add: Dy_def)
     then show "((\<lambda>y. ?F_b' (xc, y)) has_integral F(xc, g1 xc) \<bullet> i - F(xc, g2 xc) \<bullet> i) (cbox (g2 xc) (g1 xc))"
       using that and Base_vecs and F_partially_differentiable
         and Dy_def [symmetric] and g2_leq_g1
         and fundamental_theorem_of_calculus_partial_vector
         [of "g2 xc" "g1 xc" "j" "i" "xc *\<^sub>R i" "Dy_pair" "F" "?F_b'" ]
       by (auto simp add:  Groups.ab_semigroup_add_class.add.commute i_is_x_axis j_is_y_axis)
   qed
   have partial_deriv_integrable: "(?F_b') integrable_on Dy_pair"
     by (simp add: F_i_analytically_valid analytically_valid_imp_part_deriv_integrable_on)
   have 4: "integral Dy_pair ?F_b'
            = integral (cbox a b) (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. ?F_b' (x, y)))"
   proof -
     have x_axis_gauge_integrable:
       "\<And>x. (\<lambda>y. ?f(x,y)) integrable_on UNIV"
     proof -
       fix x::real
       have "\<forall>x. x \<notin> cbox a b \<longrightarrow>  (\<lambda>y. ?f (x, y)) = (\<lambda>y. 0)"
         by (auto simp add: Dy_def)
       then have f_integrable_x_not_in_range:
         "\<forall>x. x \<notin> cbox a b \<longrightarrow>  (\<lambda>y. ?f (x, y)) integrable_on UNIV"
         by (simp add: integrable_0)
       let ?F_b'_oneD = "(\<lambda>x. (\<lambda>y. if y \<in> (cbox (g2 x) ( g1 x)) then ?F_b' (x,y) else 0))"
       have f_value_x_in_range: "\<forall>x \<in> cbox a b. ?F_b'_oneD x = (\<lambda>y. ?f(x,y))"
         by (auto simp add: Dy_def)
       have "\<forall>x \<in> cbox a b. ?F_b'_oneD x integrable_on UNIV"
         using has_integral_integrable integrable_restrict_UNIV partial_deriv_one_d_integrable by blast
       then have f_integrable_x_in_range:
         "\<forall>x. x \<in> cbox a b \<longrightarrow>  (\<lambda>y. ?f (x, y)) integrable_on UNIV"
         using f_value_x_in_range by auto
       show "(\<lambda>y. ?f (x, y)) integrable_on UNIV"
         using f_integrable_x_not_in_range and f_integrable_x_in_range by auto
     qed
     have arg: "(\<lambda>a. if a \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j a else 0) =
                (\<lambda>x. if x \<in> Dy_pair then if x \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x else 0 else 0)"
       by auto
     have arg2: "Dy_pair = {(x, y). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g2 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g1 x \<bullet> i)}"
       using Dy_def  by auto
     have arg3: "\<And> x. x \<in> Dy_pair \<Longrightarrow> (\<lambda>x. if x \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x else 0) x
                            = (\<lambda>x. partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) x"
       by auto
     have arg4: "\<And>x. x \<in> (cbox a b) \<Longrightarrow>
                             (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. if (x, y) \<in> Dy_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> i) j (x, y) else 0)) x =
                                       (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> i) j (x, y))) x"
       apply (simp add: Dy_def)
       by (smt Henstock_Kurzweil_Integration.integral_cong atLeastAtMost_iff)
     show ?thesis
       using gauge_integral_Fubini_curve_bounded_region_x
         [OF f_lesbegue_integrable x_axis_gauge_integrable x_axis_integral_measurable arg arg2]
         Henstock_Kurzweil_Integration.integral_cong[OF arg3, of "Dy_pair" "(\<lambda>x. x)"]
         Henstock_Kurzweil_Integration.integral_cong[OF arg4, of "cbox a b" "(\<lambda>x. x)"]
       by auto
   qed
   have 5: "(integral Dy_pair (\<lambda>a. (?F_b' a))
         = integral (cbox a b) (\<lambda>x. F(x, g1(x))  \<bullet> i - F(x, g2(x))  \<bullet> i))"
     using 4 Henstock_Kurzweil_Integration.integral_cong partial_deriv_one_d_integrable integrable_def
     by (smt integral_unique)
   show "(line_integral F {i} gamma1) + (line_integral F {i} gamma2) -
           (line_integral F {i} gamma3) - (line_integral F {i} gamma4)
         = (integral Dy_pair (\<lambda>a. - (?F_b' a)))"
     using "0"(1) "1"(1) "2"(1) "3"(1) 5 "6" "7" 
     by (simp add: Henstock_Kurzweil_Integration.integral_diff)    
 qed
 
 theorem Greens_thm_type_II:
   fixes F:: "((real*real) \<Rightarrow> (real * real))" and
     gamma4 gamma3 gamma2 gamma1 :: "(real \<Rightarrow> (real * real))" and
     a:: "real" and b:: "real" and
     g1:: "(real \<Rightarrow> real)" and g2:: "(real \<Rightarrow> real)"
   assumes Dx_def: "Dx_pair = {(x::real,y) . y \<in> cbox a b \<and> x \<in> cbox (g2 y) (g1 y)}" and
     gamma4_def: "gamma4 = (\<lambda>x. (g2(a + (b - a) * x), a + (b - a) * x))" and
     gamma4_smooth: "gamma4 piecewise_C1_differentiable_on {0..1}" and (*TODO: This should be piecewise smooth*)
     gamma3_def: "gamma3 = (\<lambda>x. (g2(b) + x  *\<^sub>R (g1(b) - g2(b)), b))" and
     gamma2_def: "gamma2 = (\<lambda>x. (g1(a + (b - a) * x), a + (b - a) * x))" and
     gamma2_smooth: "gamma2 piecewise_C1_differentiable_on {0..1}" and
     gamma1_def: "gamma1 = (\<lambda>x. (g2(a) + x *\<^sub>R (g1(a) - g2(a)), a))" and
     F_j_analytically_valid: "analytically_valid Dx_pair (\<lambda>p. F(p) \<bullet> j) i" and
     g2_leq_g1: "\<forall>x \<in> cbox a b. (g2 x) \<le> (g1 x)" and (*This is needed otherwise what would Dy be?*)
     a_lt_b: "a < b"
   shows "-(line_integral F {j} gamma4) -
          (line_integral F {j} gamma3) +
          (line_integral F {j} gamma2) +
          (line_integral F {j} gamma1)
                  = (integral Dx_pair (\<lambda>a. (partial_vector_derivative (\<lambda>a. (F a) \<bullet> j)  i a)))"
     "line_integral_exists F {j} gamma4"
     "line_integral_exists F {j} gamma3"
     "line_integral_exists F {j} gamma2"
     "line_integral_exists F {j} gamma1"
 proof -
   let ?F_a' = "partial_vector_derivative (\<lambda>a. (F a) \<bullet> j)  i"
   have F_first_is_continuous: "continuous_on Dx_pair (\<lambda>a. F(a) \<bullet> j)"
     using F_j_analytically_valid
     by (auto simp add: analytically_valid_def)
   let ?f = "(\<lambda>x. if x \<in> (Dx_pair) then (partial_vector_derivative (\<lambda>a. (F a) \<bullet> j)  i) x else 0)"
   have f_lesbegue_integrable: "integrable lborel ?f"
     using F_j_analytically_valid
     by (auto simp add: analytically_valid_def indic_ident)
   have partially_vec_diff: "\<forall>a \<in> Dx_pair. partially_vector_differentiable (\<lambda>a. (F a) \<bullet> j) i a"
     using F_j_analytically_valid
     by (auto simp add: analytically_valid_def indicator_def)
   have "\<And>x y. ?F_a' (x,y) * indicator (Dx_pair) (x,y) = (\<lambda>x. if x \<in> (Dx_pair) then ?F_a' x else 0) (x,y)"
     using indic_ident[of "?F_a'"] by auto
   then have y_axis_integral_measurable: "(\<lambda>x. integral UNIV (\<lambda>y. ?f(y, x))) \<in> borel_measurable lborel"
     using analytically_valid_y[OF F_j_analytically_valid]
     by (auto simp add: indicator_def)
   have F_partially_differentiable: "\<forall>a\<in>Dx_pair. has_partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i (?F_a' a) a"
     using partial_vector_derivative_works partially_vec_diff by fastforce
   have g1_g2_continuous: "continuous_on (cbox a b) g1" "continuous_on (cbox a b) g2"
   proof -
     have shift_scale_cont: "continuous_on {a..b} (\<lambda>x. (x - a)*(1/(b-a)))"
       by (intro continuous_intros)
     have shift_scale_inv: "(\<lambda>x. a + (b - a) * x) \<circ> (\<lambda>x. (x - a)*(1/(b-a))) = id"
       using a_lt_b by (auto simp add: o_def)
     have img_shift_scale:
       "(\<lambda>x. (x - a)*(1/(b-a)))`{a..b} = {0..1}"
       apply (auto simp: divide_simps image_iff)
        apply (rule_tac x="x * (b - a) + a" in bexI)
       using a_lt_b by (auto simp: algebra_simps mult_le_cancel_left affine_ineq)
     have "continuous_on {0..1} (\<lambda>x. g2(a + (b - a) * x))"
       using continuous_on_inner[OF piecewise_C1_differentiable_on_imp_continuous_on[OF gamma4_smooth], of "(\<lambda>x. i)", OF continuous_on_const]
       by (simp add: gamma4_def i_is_x_axis)
     then have "continuous_on {a..b} ((\<lambda>x. g2(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using img_shift_scale continuous_on_compose shift_scale_cont by force
     then show "continuous_on (cbox a b) g2"
       using a_lt_b by (simp add: shift_scale_inv)
     have "continuous_on {0..1} (\<lambda>x. g1(a + (b - a) * x))"
       using continuous_on_inner[OF piecewise_C1_differentiable_on_imp_continuous_on[OF gamma2_smooth], of "(\<lambda>x. i)", OF continuous_on_const]
       by (simp add: gamma2_def i_is_x_axis)
     then have "continuous_on {a..b} ((\<lambda>x. g1(a + (b - a) * x)) \<circ> (\<lambda>x. (x - a)*(1/(b-a))))"
       using img_shift_scale continuous_on_compose shift_scale_cont by force
     then show "continuous_on (cbox a b) g1"
       using a_lt_b by (simp add: shift_scale_inv)
   qed
   have g2_scale_i_contin: "continuous_on (cbox a b) (\<lambda>x. (g2 x, 0))"
     by (intro continuous_intros g1_g2_continuous)
   let ?Dg2 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (g2(x), x)}"
   have line_is_pair_img: "?Dg2 = (\<lambda>x. (g2(x), x) ) ` (cbox a b)"
     using image_def  by auto
   have g2_path_continuous: "continuous_on (cbox a b) (\<lambda>x. (g2(x), x))"
     by (intro continuous_intros g1_g2_continuous)
   have field_cont_on_gamma4_image: "continuous_on ?Dg2 (\<lambda>a. F(a) \<bullet> j)"
     by (rule continuous_on_subset [OF F_first_is_continuous]) (auto simp: Dx_def g2_leq_g1)
   have gamma4_is_compos_of_scal_and_g2: "gamma4 = (\<lambda>x. (g2(x), x)) \<circ> (\<lambda>x. a + (b - a) * x)"
     using gamma4_def  by auto
   have add_scale_img:
     "(\<lambda>x. a + (b - a) * x) ` {0 .. 1} = {a .. b}" using add_scale_img and a_lt_b by auto
   then have Dg2_is_gamma4_pathimg: "path_image gamma4 = ?Dg2"
     using line_is_pair_img and gamma4_is_compos_of_scal_and_g2 image_comp path_image_def
     by (metis (no_types, lifting) cbox_interval)
   have Base_vecs: "i \<in> Basis" "j \<in> Basis" "i \<noteq> j"
     using real_pair_basis and i_is_x_axis and j_is_y_axis by auto
   have gamma4_as_euclid_space_fun: "gamma4 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R j + (g2 (a + (b - a) * x), 0))"
     using j_is_y_axis gamma4_def
     by auto
   have 0: "(line_integral F {j} gamma4) = integral (cbox a b) (\<lambda>x. F(g2(x), x) \<bullet> j)"
           "line_integral_exists F {j} gamma4"
     using line_integral_on_pair_path_strong [OF norm_Basis[OF Base_vecs(2)] _ gamma4_as_euclid_space_fun]
       gamma4_def gamma4_smooth g2_scale_i_contin a_lt_b add_scale_img
       Dg2_is_gamma4_pathimg and field_cont_on_gamma4_image
     by (auto simp: pathstart_def pathfinish_def j_is_y_axis)
   then show "line_integral_exists F {j} gamma4" by metis
   have gamma3_y_const: "\<forall>x. gamma3 x \<bullet> j = b"
     by (simp add: gamma3_def j_is_y_axis)
   have 1: "(line_integral F {j} gamma3) = 0" "(line_integral_exists F {j} gamma3)"
     using line_integral_on_pair_straight_path[OF gamma3_y_const] straight_path_diffrentiable_y gamma3_def
     by (auto simp add: mult.commute)
   then show "line_integral_exists F {j} gamma3" by auto
   have "continuous_on (cbox a b) (\<lambda>x. F(g2(x), x) \<bullet> j)"
     by (smt Collect_mono_iff continuous_on_compose2 continuous_on_eq field_cont_on_gamma4_image g2_path_continuous line_is_pair_img)
   then have 6: "(\<lambda>x. F(g2(x), x)  \<bullet> j) integrable_on (cbox a b)"
     using integrable_continuous by blast
   have g1_scale_i_contin: "continuous_on (cbox a b) (\<lambda>x. (g1 x, 0))"
     by (intro continuous_intros g1_g2_continuous)
   let ?Dg1 =  "{p. \<exists>x. x \<in> cbox a b \<and> p = (g1(x), x)}"
   have line_is_pair_img: "?Dg1 = (\<lambda>x. (g1(x), x) ) ` (cbox a b)"
     using image_def by auto
   have g1_path_continuous: "continuous_on (cbox a b) (\<lambda>x. (g1(x), x))"
     by (intro continuous_intros g1_g2_continuous)
   have field_cont_on_gamma2_image: "continuous_on ?Dg1  (\<lambda>a. F(a) \<bullet> j)"
     by (rule continuous_on_subset [OF F_first_is_continuous]) (auto simp: Dx_def g2_leq_g1)
   have "gamma2 = (\<lambda>x. (g1(x), x)) \<circ> (\<lambda>x. a + (b - a) * x)"
     using gamma2_def by auto
   then have Dg1_is_gamma2_pathimg: "path_image gamma2 = ?Dg1"
     using line_is_pair_img image_comp path_image_def add_scale_img
     by (metis (no_types, lifting) cbox_interval)
   have Base_vecs: "i \<in> Basis" "j \<in> Basis" "i \<noteq> j"
     using real_pair_basis and i_is_x_axis and j_is_y_axis by auto
   have gamma2_as_euclid_space_fun: "gamma2 = (\<lambda>x. (a + (b - a) * x) *\<^sub>R j + (g1 (a + (b - a) * x), 0))"
     using j_is_y_axis gamma2_def by auto
   have 2: "(line_integral F {j} gamma2) = integral (cbox a b) (\<lambda>x. F(g1(x), x) \<bullet> j)"
     "(line_integral_exists F {j} gamma2)"
     using line_integral_on_pair_path_strong [OF norm_Basis[OF Base_vecs(2)] _ gamma2_as_euclid_space_fun]
       gamma2_def and gamma2_smooth and g1_scale_i_contin and a_lt_b and add_scale_img
       Dg1_is_gamma2_pathimg and field_cont_on_gamma2_image
     by (auto simp: pathstart_def pathfinish_def j_is_y_axis)
   then show "line_integral_exists F {j} gamma2" by metis
   have gamma1_y_const: "\<forall>x. gamma1 x \<bullet> j = a"
     using gamma1_def
     by (auto simp add: real_inner_class.inner_add_left
         euclidean_space_class.inner_not_same_Basis j_is_y_axis)
   have 3: "(line_integral F {j} gamma1) = 0" "(line_integral_exists F {j} gamma1)"
     using line_integral_on_pair_straight_path[OF gamma1_y_const] straight_path_diffrentiable_y gamma1_def
     by (auto simp add: mult.commute)
   then show "line_integral_exists F {j} gamma1" by auto
   have "continuous_on (cbox a b) (\<lambda>x. F(g1(x), x) \<bullet> j)"
     by (smt Collect_mono_iff continuous_on_compose2 continuous_on_eq field_cont_on_gamma2_image g1_path_continuous line_is_pair_img)
   then have 7: "(\<lambda>x. F(g1(x), x) \<bullet> j) integrable_on (cbox a b)"
     using integrable_continuous [of "a" "b" "(\<lambda>x. F(g1(x), x) \<bullet> j)"]
     by auto
   have partial_deriv_one_d_integrable:
     "((\<lambda>y. ?F_a'(y, xc)) has_integral F(g1(xc), xc) \<bullet> j - F(g2(xc), xc) \<bullet> j) (cbox (g2 xc) (g1 xc))"
     if "xc \<in> cbox a b" for xc::real
   proof -
     have "{(y, xc').  y \<in> cbox (g2 xc) (g1 xc) \<and> xc' = xc} \<subseteq> Dx_pair"
       using that by (auto simp add: Dx_def)
     then show ?thesis
       using that and Base_vecs and F_partially_differentiable
         and Dx_def [symmetric] and g2_leq_g1
         and fundamental_theorem_of_calculus_partial_vector
         [of "g2 xc" "g1 xc" "i" "j" "xc *\<^sub>R j" "Dx_pair" "F" "?F_a'" ]
       by (auto simp add:  Groups.ab_semigroup_add_class.add.commute i_is_x_axis j_is_y_axis)
   qed
   have "?f integrable_on UNIV"
     by (simp add: f_lesbegue_integrable integrable_on_lborel)
   then have partial_deriv_integrable: "?F_a' integrable_on Dx_pair"
     using integrable_restrict_UNIV by auto
   have 4: "integral Dx_pair ?F_a' = integral (cbox a b) (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. ?F_a' (y, x)))"
   proof -
     have y_axis_gauge_integrable: "(\<lambda>y. ?f(y, x)) integrable_on UNIV" for x
     proof -
       let ?F_a'_oneD = "(\<lambda>x. (\<lambda>y. if y \<in> (cbox (g2 x) ( g1 x)) then ?F_a' (y, x) else 0))"
       have "\<forall>x. x \<notin> cbox a b \<longrightarrow>  (\<lambda>y. ?f (y, x)) = (\<lambda>y. 0)"
         by (auto simp add: Dx_def)
       then have f_integrable_x_not_in_range:
         "\<forall>x. x \<notin> cbox a b \<longrightarrow>  (\<lambda>y. ?f (y, x)) integrable_on UNIV"
         by (simp add: integrable_0)
       have "\<forall>x \<in> cbox a b. ?F_a'_oneD x = (\<lambda>y. ?f(y, x))"
         using g2_leq_g1 by (auto simp add: Dx_def)
       moreover have "\<forall>x \<in> cbox a b. ?F_a'_oneD x integrable_on UNIV"
         using has_integral_integrable integrable_restrict_UNIV partial_deriv_one_d_integrable by blast
       ultimately have "\<forall>x. x \<in> cbox a b \<longrightarrow>  (\<lambda>y. ?f (y, x)) integrable_on UNIV"
         by auto
       then show "(\<lambda>y. ?f (y, x)) integrable_on UNIV"
         using f_integrable_x_not_in_range by auto
     qed
     have arg: "(\<lambda>a. if a \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i a else 0) =
                (\<lambda>x. if x \<in> Dx_pair then if x \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x else 0 else 0)"
       by auto
     have arg2: "Dx_pair = {(y, x). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g2 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g1 x \<bullet> i)}"
       using Dx_def by auto
     have arg3: "\<And> x. x \<in> Dx_pair \<Longrightarrow> (\<lambda>x. if x \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x else 0) x
                            = (\<lambda>x. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x) x"
       by auto
     have arg4: "\<And>x. x \<in> (cbox a b) \<Longrightarrow>
                       (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. if (y, x) \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i (y, x) else 0)) x =
                       (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i (y, x))) x"
       apply (clarsimp simp: Dx_def)
       by (smt Henstock_Kurzweil_Integration.integral_cong atLeastAtMost_iff)
     show ?thesis
       using gauge_integral_Fubini_curve_bounded_region_y
         [OF f_lesbegue_integrable y_axis_gauge_integrable y_axis_integral_measurable arg arg2]
         Henstock_Kurzweil_Integration.integral_cong[OF arg3, of "Dx_pair" "(\<lambda>x. x)"]
         Henstock_Kurzweil_Integration.integral_cong[OF arg4, of "cbox a b" "(\<lambda>x. x)"]
       by auto
   qed
   have "((integral Dx_pair (\<lambda>a. (?F_a' a)))
         = integral (cbox a b) (\<lambda>x. F(g1(x), x) \<bullet> j - F(g2(x), x) \<bullet> j))"
     using 4 Henstock_Kurzweil_Integration.integral_cong partial_deriv_one_d_integrable integrable_def
     by (smt integral_unique)
   then have "integral Dx_pair (\<lambda>a. - (?F_a' a))
               = - integral (cbox a b) (\<lambda>x. F(g1(x), x) \<bullet> j - F(g2(x), x) \<bullet> j)"
     using partial_deriv_integrable and integral_neg by auto
   then have 5: "integral Dx_pair (\<lambda>a. - (?F_a' a))
                 = integral (cbox a b) (\<lambda>x. ( F(g2(x), x) \<bullet> j - F(g1(x), x) \<bullet> j))"
     using 6 7  
       and integral_neg [of _ "(\<lambda>x. F(g1 x, x) \<bullet> j - F(g2 x, x) \<bullet> j)"] by auto
   have "(line_integral F {j} gamma4) + (line_integral F {j} gamma3) -
         (line_integral F {j} gamma2) - (line_integral F {j} gamma1)
         = (integral Dx_pair (\<lambda>a. -(?F_a' a)))"
     using 0 1 2 3 5 6 7
       Henstock_Kurzweil_Integration.integral_diff[of "(\<lambda>x. F(g2(x), x)  \<bullet> j)"
         "cbox a b" "(\<lambda>x. F(g1(x), x) \<bullet> j)"] by (auto)
   then show "-(line_integral F {j} gamma4) -
          (line_integral F {j} gamma3) +
          (line_integral F {j} gamma2) +
          (line_integral F {j} gamma1)
                  = (integral Dx_pair (\<lambda>a. (?F_a' a)))"
     by (simp add: \<open>integral Dx_pair (\<lambda>a. - ?F_a' a) = - integral (cbox a b) (\<lambda>x. F(g1 x, x) \<bullet> j - F(g2 x, x) \<bullet> j)\<close> \<open>integral Dx_pair ?F_a' = integral (cbox a b) (\<lambda>x. F(g1 x, x) \<bullet> j - F(g2 x, x) \<bullet> j)\<close>)
 qed
 
 end
 
 locale green_typeII_cube =  R2 + 
   fixes twoC F
   assumes 
     two_cube: "typeII_twoCube twoC" and
     valid_two_cube: "valid_two_cube twoC" and
     f_analytically_valid: "analytically_valid (cubeImage twoC) (\<lambda>x. (F x) \<bullet> j) i"
 begin
 
 lemma GreenThm_typeII_twoCube:
   shows "integral (cubeImage twoC) (\<lambda>a. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i  a) = one_chain_line_integral F {j} (boundary twoC)"
     "\<forall>(k,\<gamma>) \<in> boundary twoC. line_integral_exists F {j} \<gamma>"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(x, 0))"
   let ?right_edge = "(\<lambda>y. twoC(1, y))"
   let ?top_edge = "(\<lambda>x. twoC(x, 1))"
   let ?left_edge = "(\<lambda>y. twoC(0, y))"
   have line_integral_around_boundary: 
       "one_chain_line_integral F {j} (boundary twoC) =  
          line_integral F {j} ?bottom_edge + line_integral F {j} ?right_edge
          - line_integral F {j} ?top_edge - line_integral F {j} ?left_edge"
   proof (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def)
     have finite1: "finite {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))}" by auto
     then have sum_step1: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (1, \<lambda>x. twoC (x, 0)), (- 1, \<lambda>x. twoC (x, 1))}. k * line_integral F {j} g) =
                        line_integral F {j} (\<lambda>x. twoC (x, 0)) + (\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))}. k * line_integral F {j} g)"
       using sum.insert_remove [OF finite1]
       using valid_two_cube
       apply (simp only: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def valid_two_cube_def)
       by (auto simp: card_insert_if split: if_split_asm)
     have three_distinct_edges: "card {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))} = 3"
       using valid_two_cube
       apply (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def valid_two_cube_def)
       by (auto simp: card_insert_if split: if_split_asm)
     have finite2: "finite {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}" by auto
     then have sum_step2: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (-1, \<lambda>x. twoC (x, 1))}. k * line_integral F {j} g) =
                        (- line_integral F {j} (\<lambda>x. twoC (x, 1))) + (\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}. k * line_integral F {j} g)"
       using sum.insert_remove [OF finite2] three_distinct_edges
       by (auto simp: card_insert_if split: if_split_asm)
     have two_distinct_edges: "card {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))} = 2"
       using three_distinct_edges
       by (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def)
     have finite3: "finite {(- 1::int, \<lambda>y. twoC (0, y))}" by auto
     then have sum_step3: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}. k * line_integral F {j} g) =
                        ( line_integral F {j} (\<lambda>y. twoC (1, y))) + (\<Sum>(k, g)\<in>{(- (1::real), \<lambda>y. twoC (0, y))}. k * line_integral F {j} g)"
       using sum.insert_remove [OF finite2] three_distinct_edges
       by (auto simp: card_insert_if split: if_split_asm)
     show "(\<Sum>x\<in>{(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (1, \<lambda>x. twoC (x, 0)), (- 1, \<lambda>x. twoC (x, 1))}. case x of (k, g) \<Rightarrow> k * line_integral F {j} g) =
                   line_integral F {j} (\<lambda>x. twoC (x, 0)) + line_integral F {j} (\<lambda>y. twoC (1, y)) - line_integral F {j} (\<lambda>x. twoC (x, 1)) - line_integral F {j} (\<lambda>y. twoC (0, y))"
       using sum_step1 sum_step2 sum_step3  by auto
   qed
   obtain a b g1 g2 where
     twoCisTypeII: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(y, x). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     using two_cube and typeII_twoCubeImg[of"twoC"]
     by auto
   have left_edge_explicit: "?left_edge = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   have left_edge_smooth: "?left_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x))-` {x} = {(x - a)/(b -a)}"
       using twoCisTypeII(1) by(auto simp add: Set.vimage_def)
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     have scale_shif_smth: "(\<lambda>x. (a + (b - a) * x)) C1_differentiable_on {0..1}" using scale_shift_smooth by auto
     then have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" using C1_differentiable_imp_piecewise by blast
     have g2_smooth: "g2 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeII(1)] twoCisTypeII(6) by auto
     have "(\<lambda>x. g2 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
-      using Cauchy_Integral_Theorem.piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
+      using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g2_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x::real. (g2 (a + (b - a) * x), a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x::real. (g2 (a + (b - a) * x), a + (b - a) * x))"]
       by (fastforce simp add: real_pair_basis)
     then show ?thesis using left_edge_explicit by auto
   qed
   have top_edge_explicit: "?top_edge = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
    and right_edge_explicit: "?right_edge = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x))"
     by (simp_all add: twoCisTypeII(4) algebra_simps)
   have right_edge_smooth: "?right_edge piecewise_C1_differentiable_on {0..1}"
   proof -
     have "\<forall>x. (\<lambda>x. (a + (b - a) * x))-` {x} = {(x - a)/(b -a)}"
       using twoCisTypeII(1) by(auto simp add: Set.vimage_def)
     then have finite_vimg: "\<And>x. finite({0..1} \<inter> (\<lambda>x. (a + (b - a) * x))-` {x})" by auto
     then have scale_shif_pw_smth: "(\<lambda>x. (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}" 
       using C1_differentiable_imp_piecewise [OF scale_shift_smooth] by auto
     have g1_smooth: "g1 piecewise_C1_differentiable_on (\<lambda>x. a + (b - a) * x) ` {0..1}" using add_scale_img[OF twoCisTypeII(1)] twoCisTypeII(5) by auto
     have "(\<lambda>x. g1 (a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using piecewise_C1_differentiable_compose[OF scale_shif_pw_smth g1_smooth finite_vimg]
       by (auto simp add: o_def)
     then have "(\<lambda>x::real. (g1 (a + (b - a) * x), a + (b - a) * x)) piecewise_C1_differentiable_on {0..1}"
       using all_components_smooth_one_pw_smooth_is_pw_smooth[where f = "(\<lambda>x::real. (g1 (a + (b - a) * x), a + (b - a) * x))"]
       by (fastforce simp add: real_pair_basis)
     then show ?thesis using right_edge_explicit by auto
   qed
   have bottom_edge_explicit: "?bottom_edge = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
     by (simp add: twoCisTypeII(4) algebra_simps)
   show "integral (cubeImage twoC) (\<lambda>a. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i  a) = one_chain_line_integral F {j} (boundary twoC)"
     using Greens_thm_type_II[OF twoCisTypeII(3) left_edge_explicit left_edge_smooth
         top_edge_explicit right_edge_explicit right_edge_smooth
         bottom_edge_explicit f_analytically_valid
         twoCisTypeII(2) twoCisTypeII(1)]
       line_integral_around_boundary
     by auto
   have "line_integral_exists F {j} \<gamma>" if "(k,\<gamma>) \<in> boundary twoC" for k \<gamma>
   proof -
     have "line_integral_exists F {j} (\<lambda>y. twoC (0, y))"
          "line_integral_exists F {j} (\<lambda>x. twoC (x, 1))"
          "line_integral_exists F {j} (\<lambda>y. twoC (1, y))"
          "line_integral_exists F {j} (\<lambda>x. twoC (x, 0))"
       using Greens_thm_type_II[OF twoCisTypeII(3) left_edge_explicit left_edge_smooth
           top_edge_explicit right_edge_explicit right_edge_smooth
           bottom_edge_explicit f_analytically_valid
           twoCisTypeII(2) twoCisTypeII(1)] line_integral_around_boundary
       by auto
     then show "line_integral_exists F {j} \<gamma>"
       using that by (auto simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   qed
   then show "\<forall>(k,\<gamma>) \<in> boundary twoC. line_integral_exists F {j} \<gamma>" by auto
 qed
 
 lemma line_integral_exists_on_typeII_Cube_boundaries':
   assumes "(k,\<gamma>) \<in> boundary twoC"
   shows "line_integral_exists F {j} \<gamma>"
   using assms GreenThm_typeII_twoCube(2) by blast
 
 end
 
 locale green_typeII_chain =  R2 + 
   fixes F two_chain s
   assumes valid_typeII_div: "valid_typeII_division s two_chain" and
           F_anal_valid: "\<forall>twoC \<in> two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. (F x) \<bullet> j) i"
 begin
 
 lemma two_chain_valid_valid_cubes: "\<forall>two_cube \<in> two_chain. valid_two_cube two_cube" using valid_typeII_div
   by (auto simp add: valid_two_chain_def)
 
 lemma typeII_chain_line_integral_exists_boundary':
   shows "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
     using green_typeII_cube.line_integral_exists_on_typeII_Cube_boundaries'[of i j] F_anal_valid valid_typeII_div 
     apply(auto simp add:  two_chain_boundary_def)
     using R2_axioms green_typeII_cube_axioms_def green_typeII_cube_def two_chain_valid_valid_cubes by blast
   then show integ_exis_vert:
     "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
     by (simp add: two_chain_boundary_def two_chain_vertical_boundary_def boundary_def)
 qed
 
 lemma typeII_chain_line_integral_exists_boundary'':
      "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {j} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
     using green_typeII_cube.line_integral_exists_on_typeII_Cube_boundaries'[of i j] valid_typeII_div
     apply (simp add: two_chain_boundary_def boundary_def)
     using F_anal_valid R2_axioms green_typeII_cube_axioms_def green_typeII_cube_def two_chain_valid_valid_cubes by fastforce
   then show integ_exis_vert:
     "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {j} \<gamma>"
     by (simp add: two_chain_boundary_def two_chain_horizontal_boundary_def boundary_def)
 qed
 
 lemma typeII_cube_line_integral_exists_boundary:
      "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
   using valid_typeII_div typeII_chain_line_integral_exists_boundary' typeII_chain_line_integral_exists_boundary''
   apply (auto simp add: two_chain_boundary_def two_chain_horizontal_boundary_def two_chain_vertical_boundary_def)
   using boundary_def by auto
 
 lemma type_II_chain_horiz_bound_valid:
      "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. valid_path \<gamma>"
   using valid_typeII_div typeII_edges_are_valid_paths
   by (force simp add: two_chain_boundary_def two_chain_horizontal_boundary_def boundary_def)
 
 lemma type_II_chain_vert_bound_valid: (*This and the previous one need to be used in all proofs*)
      "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. valid_path \<gamma>"
   using typeII_edges_are_valid_paths valid_typeII_div
   by (force simp add: two_chain_boundary_def two_chain_vertical_boundary_def boundary_def)
 
 lemma members_of_only_horiz_div_line_integrable':
   assumes "only_horizontal_division one_chain two_chain"
     "(k::int, \<gamma>)\<in>one_chain"
     "(k::int, \<gamma>)\<in>one_chain"
     "finite two_chain"
     "\<forall>two_cube \<in> two_chain. valid_two_cube two_cube"
   shows "line_integral_exists F {j} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
     using typeII_cube_line_integral_exists_boundary by blast
   have integ_exis_vert:
     "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
     using typeII_chain_line_integral_exists_boundary' assms by auto
   have integ_exis_horiz:
     "(\<And>k \<gamma>. (\<exists>(k', \<gamma>') \<in> two_chain_horizontal_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<Longrightarrow>
                             line_integral_exists F {j} \<gamma>)"
     using integ_exis type_II_chain_horiz_bound_valid
     using line_integral_exists_subpath[of "F" "{j}"]
     by (fastforce simp add: two_chain_boundary_def two_chain_horizontal_boundary_def
         two_chain_vertical_boundary_def boundary_def)
   obtain \<H> \<V> where hv_props: "finite \<H>"
         "(\<forall>(k,\<gamma>) \<in> \<H>. (\<exists>(k', \<gamma>') \<in> two_chain_horizontal_boundary two_chain.
             (\<exists>a \<in> {0 .. 1}. \<exists>b \<in> {0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>)))"
         "one_chain = \<H> \<union> \<V>"
         "((common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V>)
           \<or> common_reparam_exists \<V> (two_chain_vertical_boundary two_chain))"
     "finite \<V>"
     "boundary_chain \<V>"
     "\<forall>(k,\<gamma>)\<in>\<V>. valid_path \<gamma>"
     using assms(1) unfolding only_horizontal_division_def  by blast
   have finite_j: "finite {j}" by auto
   show "line_integral_exists F {j} \<gamma>"
   proof(cases "common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V>")
     case True
     show ?thesis
       using gen_common_subdivision_imp_eq_line_integral(2)[OF True two_chain_vertical_boundary_is_boundary_chain hv_props(6) integ_exis_vert finite_two_chain_vertical_boundary[OF assms(4)] hv_props(5) finite_j]
         integ_exis_horiz[of "\<gamma>"] assms(3) case_prod_conv hv_props(2) hv_props(3)
       by fastforce
   next
     case False
     have i: "{j} \<subseteq> Basis" using j_is_y_axis real_pair_basis by auto
     have ii: " \<forall>(k2, \<gamma>2)\<in>two_chain_vertical_boundary two_chain. \<forall>b\<in>{j}. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
       using F_anal_valid field_cont_on_typeII_region_cont_on_edges valid_typeII_div
       by (fastforce simp add: analytically_valid_def two_chain_vertical_boundary_def boundary_def path_image_def)
     show "line_integral_exists F {j} \<gamma>"
       using common_reparam_exists_imp_eq_line_integral(2)[OF finite_j hv_props(5) 
            finite_two_chain_vertical_boundary[OF assms(4)] hv_props(6) two_chain_vertical_boundary_is_boundary_chain ii _ hv_props(7) type_II_chain_vert_bound_valid]
         integ_exis_horiz[of "\<gamma>"] assms(3) hv_props False 
       by fastforce
   qed
 qed
 
 lemma GreenThm_typeII_twoChain:
   shows "two_chain_integral two_chain (partial_vector_derivative (\<lambda>a. (F a) \<bullet> j)  i) = one_chain_line_integral F {j} (two_chain_boundary two_chain)"
 proof (simp add: two_chain_boundary_def one_chain_line_integral_def two_chain_integral_def)
   let ?F_a' = "partial_vector_derivative (\<lambda>a. (F a) \<bullet> j)  i"
   have "(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {j} g) = integral (cubeImage twoCube) (\<lambda>a. ?F_a' a)"
     if "twoCube \<in> two_chain" for twoCube
     using green_typeII_cube.GreenThm_typeII_twoCube(1) valid_typeII_div F_anal_valid one_chain_line_integral_def valid_two_chain_def
     by (simp add: R2_axioms green_typeII_cube_axioms_def green_typeII_cube_def that)
   then have double_sum_eq_sum:
     "(\<Sum>twoCube\<in>(two_chain).(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {j} g))
                      =  (\<Sum>twoCube\<in>(two_chain). integral (cubeImage twoCube) (\<lambda>a. ?F_a' a))"
     using Finite_Cartesian_Product.sum_cong_aux by auto
   have pairwise_disjoint_boundaries: "\<forall>x\<in> (boundary ` two_chain). (\<forall>y\<in> (boundary ` two_chain). (x \<noteq> y \<longrightarrow> (x \<inter> y = {})))"
     using valid_typeII_div by (fastforce simp add:  image_def valid_two_chain_def pairwise_def)
   have finite_boundaries: "\<forall>B \<in> (boundary` two_chain). finite B"
     using valid_typeII_div image_iff by (fastforce simp add: valid_two_cube_def valid_two_chain_def)
   have boundary_inj: "inj_on boundary two_chain"
     using valid_typeII_div by (force simp add: valid_two_cube_def valid_two_chain_def pairwise_def inj_on_def)
   have "(\<Sum>x\<in>(\<Union>x\<in>two_chain. boundary x). case x of (k, g) \<Rightarrow> k * line_integral F {j} g) =
                       (\<Sum>twoCube\<in>(two_chain).(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {j} g))"
     using sum.reindex[OF boundary_inj,  of "\<lambda>x. (\<Sum>(k,g)\<in> x. k * line_integral F {j} g)"]
     using sum.Union_disjoint[OF finite_boundaries
         pairwise_disjoint_boundaries,
         of "\<lambda>x. case x of (k, g) \<Rightarrow> (k::int) * line_integral F {j} g"]
     by auto
   then show "(\<Sum>C\<in>two_chain. integral (cubeImage C) (\<lambda>a. ?F_a' a)) = (\<Sum>x\<in>(\<Union>x\<in>two_chain. boundary x). case x of (k, g) \<Rightarrow> k * line_integral F {j} g)"
     using double_sum_eq_sum  by auto
 qed
 
 lemma GreenThm_typeII_divisible:
   assumes 
     gen_division: "gen_division s (cubeImage ` two_chain)"    (*This should follow from the assumption that images are not negligible*)
   shows "integral s (partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i) = one_chain_line_integral F {j} (two_chain_boundary two_chain)"
 proof -
   let ?F_a' = "(partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i)"
   have "integral s (\<lambda>x. ?F_a' x) = two_chain_integral two_chain (\<lambda>a. ?F_a' a)"
   proof (simp add: two_chain_integral_def)
     have partial_deriv_integrable:
       "(?F_a' has_integral (integral (cubeImage twoCube) ?F_a')) (cubeImage twoCube)"
       if "twoCube \<in> two_chain" for twoCube
       by (simp add: analytically_valid_imp_part_deriv_integrable_on F_anal_valid integrable_integral that)
     then have partial_deriv_integrable:
       "\<And>twoCubeImg. twoCubeImg \<in> cubeImage ` two_chain \<Longrightarrow> ((\<lambda>x. ?F_a' x) has_integral (integral (twoCubeImg) (\<lambda>x. ?F_a' x))) (twoCubeImg)"
       using integrable_neg by force
     have finite_images: "finite (cubeImage ` two_chain)"
       using gen_division gen_division_def by auto
     have negligible_images: "pairwise (\<lambda>S S'. negligible (S \<inter> S')) (cubeImage ` two_chain)"
       using gen_division by (auto simp add: gen_division_def pairwise_def)
     have "inj_on cubeImage two_chain" using valid_typeII_div valid_two_chain_def by auto
     then have "(\<Sum>twoCubeImg\<in>cubeImage ` two_chain. integral twoCubeImg (\<lambda>x. ?F_a' x))
              = (\<Sum>C\<in>two_chain. integral (cubeImage C) (\<lambda>a. ?F_a' a))"
       using sum.reindex by auto
     then show "integral s (\<lambda>x. ?F_a' x) = (\<Sum>C\<in>two_chain. integral (cubeImage C) (\<lambda>a. ?F_a' a))"
       using has_integral_Union[OF finite_images partial_deriv_integrable negligible_images] gen_division
       by (auto simp add: gen_division_def)
   qed
   then show ?thesis
     using GreenThm_typeII_twoChain F_anal_valid
     by auto
 qed
 
 lemma GreenThm_typeII_divisible_region_boundary_gen:
   assumes only_horizontal_division: "only_horizontal_division \<gamma> two_chain"
   shows "integral s (partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
 proof -
   let ?F_a' = "(partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i)"
     (*Proving that line_integral on the x axis is 0 for the vertical 1-cubes*)
   have horiz_line_integral_zero:
     "one_chain_line_integral F {j} (two_chain_horizontal_boundary two_chain) = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {j} (snd oneCube) = 0"
       if "oneCube \<in> two_chain_horizontal_boundary(two_chain)" for oneCube
     proof -
       from that obtain x y1 y2 k 
         where horiz_edge_def: "oneCube = (k, (\<lambda>t::real. ((1 - t) * (y2) + t * y1, x::real)))"
         using valid_typeII_div
         by (auto simp add: typeII_twoCube_def two_chain_horizontal_boundary_def horizontal_boundary_def)
       let ?horiz_edge = "(snd oneCube)"
       have horiz_edge_y_const: "\<forall>t. (?horiz_edge t) \<bullet> j = x"
         by (auto simp add: horiz_edge_def real_inner_class.inner_add_left
             euclidean_space_class.inner_not_same_Basis j_is_y_axis)
       have horiz_edge_is_straight_path: "?horiz_edge = (\<lambda>t. (y2 + t * (y1 - y2), x))"
         by (auto simp: horiz_edge_def algebra_simps)
       have "\<forall>x. ?horiz_edge differentiable at x"
         using horiz_edge_is_straight_path straight_path_diffrentiable_y
         by (metis mult_commute_abs)
       then show "line_integral F {j} (snd oneCube) = 0"
         using line_integral_on_pair_straight_path(1) j_is_y_axis real_pair_basis horiz_edge_y_const
         by blast
     qed
     then show "(\<Sum>x\<in>two_chain_horizontal_boundary two_chain. case x of (k, g) \<Rightarrow> k * line_integral F {j} g) = 0"
       by (force intro: sum.neutral)
   qed
     (*then, the x axis line_integral on the boundaries of the twoCube is equal to the line_integral on the horizontal boundaries of the twoCube \<longrightarrow> 1*)
   have boundary_is_finite: "finite (two_chain_boundary two_chain)"
     unfolding two_chain_boundary_def
   proof (rule finite_UN_I)
     show "finite two_chain"
       using valid_typeII_div finite_image_iff gen_division_def valid_two_chain_def by auto
     show "\<And>a. a \<in> two_chain \<Longrightarrow> finite (boundary a)"
       by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   qed
   have boundary_is_vert_hor:
      "two_chain_boundary two_chain =
       (two_chain_vertical_boundary two_chain) \<union>
       (two_chain_horizontal_boundary two_chain)"
     by(auto simp add: two_chain_boundary_def two_chain_vertical_boundary_def two_chain_horizontal_boundary_def
         boundary_def)
   then have hor_vert_finite:
     "finite (two_chain_vertical_boundary two_chain)"
     "finite (two_chain_horizontal_boundary two_chain)"
     using boundary_is_finite by auto
   have horiz_verti_disjoint:
     "(two_chain_vertical_boundary two_chain) \<inter>
      (two_chain_horizontal_boundary two_chain) = {}"
   proof (simp add: two_chain_vertical_boundary_def two_chain_horizontal_boundary_def horizontal_boundary_def
         vertical_boundary_def)
     show "(\<Union>x\<in>two_chain. {(- 1, \<lambda>y. x (0, y)), (1::int, \<lambda>y. x (1::real, y))}) \<inter> (\<Union>x\<in>two_chain. {(1, \<lambda>z. x (z, 0)), (- 1, \<lambda>z. x (z, 1))}) = {}"
     proof -
       have "{(- 1, \<lambda>y. twoCube (0, y)), (1::int, \<lambda>y. twoCube (1, y))} \<inter>
             {(1, \<lambda>z. twoCube2 (z, 0)), (- 1, \<lambda>z. twoCube2 (z, 1))} = {}"
         if "twoCube\<in>two_chain" "twoCube2\<in>two_chain" for twoCube twoCube2
       proof (cases "twoCube = twoCube2")
         case True
         have card_4: "card {(- 1, \<lambda>y. twoCube2 (0::real, y)), (1::int, \<lambda>y. twoCube2 (1, y)), (1, \<lambda>x. twoCube2 (x, 0)), (- 1, \<lambda>x. twoCube2 (x, 1))} = 4"
           using valid_typeII_div valid_two_chain_def that(2)
           by (auto simp add: boundary_def vertical_boundary_def horizontal_boundary_def valid_two_cube_def)
         show ?thesis
           using card_4 by (auto simp: True card_insert_if split: if_split_asm)
       next
         case False
         show ?thesis
           using valid_typeII_div valid_two_chain_def
           by (simp add: boundary_def vertical_boundary_def horizontal_boundary_def pairwise_def \<open>twoCube \<noteq> twoCube2\<close> that)
       qed
       then have "\<Union> ((\<lambda>twoCube. {(- 1, \<lambda>y. twoCube (0::real, y)), (1::real, \<lambda>y. twoCube (1::real, y))}) ` two_chain)
                                           \<inter> \<Union> ((\<lambda>twoCube. {(1::int, \<lambda>z. twoCube (z, 0::real)), (- 1, \<lambda>z. twoCube (z, 1::real))}) ` two_chain)
                                               = {}"
         by (fastforce simp add: Union_disjoint)
       then show ?thesis by force
     qed
   qed
   have "one_chain_line_integral F {j} (two_chain_boundary two_chain)
        = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain) +
          one_chain_line_integral F {j} (two_chain_horizontal_boundary two_chain)"
     using boundary_is_vert_hor horiz_verti_disjoint
     by (simp add: hor_vert_finite sum.union_disjoint one_chain_line_integral_def)
   then have y_axis_line_integral_is_only_vertical:
     "one_chain_line_integral F {j} (two_chain_boundary two_chain)
                          = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
     using horiz_line_integral_zero
     by auto
       (*Since \<gamma> \<subseteq> the boundary of the 2-chain and the horizontal boundaries are \<subseteq> \<gamma>, then there is some \<H> \<subseteq> \<partial>\<^sub>H(2-\<C>) such that \<gamma> = \<H> \<union> \<partial>\<^sub>v(2-\<C>)*)
   obtain \<H> \<V> where hv_props: "finite \<H>"
     "(\<forall>(k,\<gamma>) \<in> \<H>. (\<exists>(k', \<gamma>') \<in> two_chain_horizontal_boundary two_chain.
                       (\<exists>a \<in> {0 .. 1}.
                             \<exists>b \<in> {0..1}.
                                 a \<le> b \<and> subpath a b \<gamma>' = \<gamma>)))"
     "\<gamma> = \<H> \<union> \<V>"
     "(common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V>
       \<or> common_reparam_exists \<V> (two_chain_vertical_boundary two_chain))"
     "finite \<V>"
     "boundary_chain \<V>"
     "\<forall>(k,\<gamma>)\<in>\<V>. valid_path \<gamma>"
     using only_horizontal_division
     by(fastforce simp add:  only_horizontal_division_def)
   have "finite {j}" by auto
   then have eq_integrals: " one_chain_line_integral F {j} \<V> = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
   proof(cases "common_sudiv_exists (two_chain_vertical_boundary two_chain) \<V>")
     case True then show ?thesis
       using gen_common_subdivision_imp_eq_line_integral(1)[OF True two_chain_vertical_boundary_is_boundary_chain hv_props(6) _ hor_vert_finite(1) hv_props(5)]
         typeII_chain_line_integral_exists_boundary'
       by force
   next
     case False
     have integ_exis_vert:
       "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
       using typeII_chain_line_integral_exists_boundary' assms
       by (fastforce simp add: valid_two_chain_def)
     have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {j} \<gamma>"
       using typeII_cube_line_integral_exists_boundary by blast
     have valid_paths: "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. valid_path \<gamma>"
       using typeII_edges_are_valid_paths valid_typeII_div
       by (fastforce simp add: two_chain_boundary_def two_chain_horizontal_boundary_def boundary_def)
     have integ_exis_horiz:
       "(\<And>k \<gamma>. (\<exists>(k', \<gamma>')\<in>two_chain_horizontal_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<Longrightarrow>
                 line_integral_exists F {j} \<gamma>)"
       using integ_exis valid_paths line_integral_exists_subpath[of "F" "{j}"]
       by (fastforce simp add: two_chain_boundary_def two_chain_horizontal_boundary_def
           two_chain_vertical_boundary_def boundary_def)
     have finite_j: "finite {j}" by auto
     have i: "{j} \<subseteq> Basis" using j_is_y_axis real_pair_basis by auto
     have ii: " \<forall>(k2, \<gamma>2)\<in>two_chain_vertical_boundary two_chain. \<forall>b\<in>{j}. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
       using valid_typeII_div field_cont_on_typeII_region_cont_on_edges F_anal_valid
       by (fastforce simp add: analytically_valid_def two_chain_vertical_boundary_def boundary_def path_image_def)
     show "one_chain_line_integral F {j} \<V> = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
       using hv_props(4) False common_reparam_exists_imp_eq_line_integral(1)[OF finite_j hv_props(5) hor_vert_finite(1) hv_props(6) two_chain_vertical_boundary_is_boundary_chain ii 
           _ hv_props(7) type_II_chain_vert_bound_valid] 
       by fastforce
   qed
     (*Since \<H> \<subseteq> \<partial>\<^sub>H(2-\<C>), then the line_integral on the x axis along \<H> is 0, and from 1 Q.E.D. *)
   have line_integral_on_path:
     "one_chain_line_integral F {j} \<gamma> =
      one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
   proof (simp only: one_chain_line_integral_def)
     have "line_integral F {j} (snd oneCube) = 0" if oneCube: "oneCube \<in> \<H>" for oneCube
     proof -
       obtain k \<gamma> where k_gamma: "(k,\<gamma>) = oneCube"
         using prod.collapse by blast
       then obtain k' \<gamma>' a b where kp_gammap:
         "(k'::int, \<gamma>') \<in> two_chain_horizontal_boundary two_chain"
         "a \<in> {0 .. 1}"
         "b \<in> {0..1}"
         "subpath a b \<gamma>' = \<gamma>"
         using hv_props oneCube
         by (smt case_prodE split_conv)
       obtain x y1 y2 where horiz_edge_def: "(k',\<gamma>') = (k', (\<lambda>t::real. ((1 - t) * (y2) + t * y1, x::real)))"
         using valid_typeII_div kp_gammap
         by (auto simp add: typeII_twoCube_def two_chain_horizontal_boundary_def horizontal_boundary_def)
       have horiz_edge_y_const: "\<forall>t. \<gamma> (t) \<bullet> j = x"
         using horiz_edge_def kp_gammap(4)
         by (auto simp add: real_inner_class.inner_add_left
             euclidean_space_class.inner_not_same_Basis j_is_y_axis subpath_def)
       have horiz_edge_is_straight_path:
         "\<gamma> = (\<lambda>t::real. ((1*y2 - a*y2) +  a*y1 + ((b-a)*y1 - (b - a)*y2)*t, x::real))"
       proof -
         fix t::real
         have "(1 - (b - a)*t + a) * (y2) + ((b-a)*t + a) * y1 = (1 - (b - a)*t + a) * (y2) + ((b-a)*t + a) * y1"
           by auto
         then have "\<gamma> = (\<lambda>t::real. ((1 - (b - a)*t - a) * (y2) + ((b-a)*t + a) * y1, x::real))"
           using horiz_edge_def Product_Type.snd_conv Product_Type.fst_conv kp_gammap(4)
           by (simp add: subpath_def diff_diff_eq[symmetric])
         also have "\<dots> = (\<lambda>t::real. ((1*y2 - (b - a)*y2*t - a*y2) + ((b-a)*y1*t + a*y1), x::real))"
           by(auto simp add: ring_class.ring_distribs(2) Groups.diff_diff_eq left_diff_distrib)
         also have "... = (\<lambda>t::real. ((1*y2 - a*y2) +  a*y1 + ((b-a)*y1 - (b - a)*y2)*t, x::real))"
           by (force simp add: left_diff_distrib)
         finally show "\<gamma> = (\<lambda>t::real. ((1*y2 - a*y2) +  a*y1 + ((b-a)*y1 - (b - a)*y2)*t, x::real))" .
       qed
       show "line_integral F {j} (snd oneCube) = 0"
       proof -
         have "\<forall>x. \<gamma> differentiable at x"
           by (simp add: horiz_edge_is_straight_path straight_path_diffrentiable_y)
         then have "line_integral F {j} \<gamma> = 0"
           by (simp add: horiz_edge_y_const line_integral_on_pair_straight_path(1))
         then show ?thesis
           using Product_Type.snd_conv k_gamma by auto
       qed
     qed
     then have "\<forall>x\<in>\<H>. (case x of (k, g) \<Rightarrow> (k::int) * line_integral F {j} g) = 0"
       by auto
     then show "(\<Sum>x\<in>\<gamma>. case x of (k, g) \<Rightarrow> real_of_int k * line_integral F {j} g) =
                (\<Sum>x\<in>two_chain_vertical_boundary two_chain. case x of (k, g) \<Rightarrow> real_of_int k * line_integral F {j} g)"
       using hv_props(1) hv_props(3) hv_props(5) sum_zero_set hor_vert_finite(1) eq_integrals
       by (clarsimp simp add: one_chain_line_integral_def sum_zero_set)
   qed
   then have "one_chain_line_integral F {j} \<gamma> =
                            one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
     using line_integral_on_path by auto
   then have "one_chain_line_integral F {j} \<gamma> =
                            one_chain_line_integral F {j} (two_chain_boundary two_chain)"
     using y_axis_line_integral_is_only_vertical by auto
   then show ?thesis
     using valid_typeII_div GreenThm_typeII_divisible by auto
 qed
 
 lemma GreenThm_typeII_divisible_region_boundary:
   assumes
     two_cubes_trace_vertical_boundaries: 
     "two_chain_vertical_boundary two_chain \<subseteq> \<gamma>" and
     boundary_of_region_is_subset_of_partition_boundary:
     "\<gamma> \<subseteq> two_chain_boundary two_chain"
   shows "integral s (partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
 proof -
   let ?F_a' = "(partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i)"
     (*Proving that line_integral on the x axis is 0 for the vertical 1-cubes*)
   have horiz_line_integral_zero:
     "one_chain_line_integral F {j} (two_chain_horizontal_boundary two_chain) = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {j} (snd oneCube) = 0"
       if one: "oneCube \<in> two_chain_horizontal_boundary(two_chain)" for oneCube
     proof -
       obtain x y1 y2 k where horiz_edge_def: "oneCube = (k, (\<lambda>t::real. ((1 - t) * (y2) + t * y1, x::real)))"
         using valid_typeII_div one
         by (auto simp add: typeII_twoCube_def two_chain_horizontal_boundary_def horizontal_boundary_def)
       let ?horiz_edge = "(snd oneCube)"
       have horiz_edge_y_const: "\<forall>t. (?horiz_edge t) \<bullet> j = x"
         using horiz_edge_def
         by (auto simp add: real_inner_class.inner_add_left
             euclidean_space_class.inner_not_same_Basis j_is_y_axis)
       have horiz_edge_is_straight_path:
         "?horiz_edge = (\<lambda>t. (y2 + t * (y1 - y2), x))"
         by (simp add: add_diff_eq diff_add_eq mult.commute right_diff_distrib horiz_edge_def)
       show "line_integral F {j} (snd oneCube) = 0"
         by (metis horiz_edge_is_straight_path horiz_edge_y_const line_integral_on_pair_straight_path(1) mult.commute straight_path_diffrentiable_y)
     qed
     then show "(\<Sum>x\<in>two_chain_horizontal_boundary two_chain. case x of (k, g) \<Rightarrow> k * line_integral F {j} g) = 0"
       by (simp add: prod.case_eq_if)
   qed
     (*then, the x axis line_integral on the boundaries of the twoCube is equal to the line_integral on the horizontal boundaries of the twoCube \<longrightarrow> 1*)
   have boundary_is_finite: "finite (two_chain_boundary two_chain)"
     unfolding two_chain_boundary_def
   proof (rule finite_UN_I)
     show "finite two_chain"
       using valid_typeII_div finite_image_iff gen_division_def valid_two_chain_def by auto
     show "\<And>a. a \<in> two_chain \<Longrightarrow> finite (boundary a)"
       by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   qed
   have boundary_is_vert_hor:
     "two_chain_boundary two_chain = (two_chain_vertical_boundary two_chain) \<union> (two_chain_horizontal_boundary two_chain)"
     by(auto simp add: two_chain_boundary_def two_chain_vertical_boundary_def two_chain_horizontal_boundary_def boundary_def)
   then have hor_vert_finite:
     "finite (two_chain_vertical_boundary two_chain)"
     "finite (two_chain_horizontal_boundary two_chain)"
     using boundary_is_finite by auto
   have horiz_verti_disjoint:
     "(two_chain_vertical_boundary two_chain) \<inter> (two_chain_horizontal_boundary two_chain) = {}"
   proof (simp add: two_chain_vertical_boundary_def two_chain_horizontal_boundary_def horizontal_boundary_def
         vertical_boundary_def)
     show "(\<Union>x\<in>two_chain. {(- 1, \<lambda>y. x (0, y)), (1::int, \<lambda>y. x (1::real, y))}) \<inter> (\<Union>x\<in>two_chain. {(1, \<lambda>z. x (z, 0)), (- 1, \<lambda>z. x (z, 1))}) = {}"
     proof -
       have "{(- 1, \<lambda>y. twoCube (0, y)), (1::int, \<lambda>y. twoCube (1, y))} \<inter>
             {(1, \<lambda>y. twoCube2 (y, 0)), (- 1, \<lambda>y. twoCube2 (y, 1))} = {}"
         if  "twoCube\<in>two_chain" "twoCube2\<in>two_chain" for twoCube twoCube2
       proof (cases "twoCube = twoCube2")
         case True
         have "card {(- 1, \<lambda>y. twoCube2 (0::real, y)), (1::int, \<lambda>y. twoCube2 (1, y)), (1, \<lambda>x. twoCube2 (x, 0)), (- 1, \<lambda>x. twoCube2 (x, 1))} = 4"
           using valid_typeII_div valid_two_chain_def that(2)
           by (auto simp add: boundary_def vertical_boundary_def horizontal_boundary_def valid_two_cube_def)
         then show ?thesis
           by (auto simp: True card_insert_if split: if_split_asm)
       next
         case False show ?thesis
           using valid_typeII_div valid_two_chain_def
           by (simp add: boundary_def vertical_boundary_def horizontal_boundary_def pairwise_def \<open>twoCube \<noteq> twoCube2\<close> that(1) that(2))
       qed
       then have "\<Union> ((\<lambda>twoCube. {(- 1, \<lambda>y. twoCube (0::real, y)), (1::real, \<lambda>y. twoCube (1::real, y))}) ` two_chain)
                                           \<inter> \<Union> ((\<lambda>twoCube. {(1::int, \<lambda>z. twoCube (z, 0::real)), (- 1, \<lambda>z. twoCube (z, 1::real))}) ` two_chain)
                                               = {}"
         by (force simp add: Complete_Lattices.Union_disjoint)
       then show ?thesis by force
     qed
   qed
   have "one_chain_line_integral F {j} (two_chain_boundary two_chain)
         = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain) +
           one_chain_line_integral F {j} (two_chain_horizontal_boundary two_chain)"
     using boundary_is_vert_hor horiz_verti_disjoint
     by (auto simp add: one_chain_line_integral_def hor_vert_finite sum.union_disjoint)
   then have y_axis_line_integral_is_only_vertical:
     "one_chain_line_integral F {j} (two_chain_boundary two_chain)
      = one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
     using horiz_line_integral_zero by auto
       (*Since \<gamma> \<subseteq> the boundary of the 2-chain and the horizontal boundaries are \<subseteq> \<gamma>, then there is some \<H> \<subseteq> \<partial>\<^sub>H(2-\<C>) such that \<gamma> = \<H> \<union> \<partial>\<^sub>v(2-\<C>)*)
   have "\<exists>\<H>. \<H> \<subseteq> (two_chain_horizontal_boundary two_chain) \<and>
              \<gamma> = \<H> \<union> (two_chain_vertical_boundary two_chain)"
   proof
     let ?\<H> = "\<gamma> - (two_chain_vertical_boundary two_chain)"
     show "?\<H> \<subseteq> two_chain_horizontal_boundary two_chain \<and> \<gamma> = ?\<H> \<union> two_chain_vertical_boundary two_chain"
       using two_cubes_trace_vertical_boundaries
         boundary_of_region_is_subset_of_partition_boundary boundary_is_vert_hor
       by blast
   qed
   then obtain \<H> where
     h_props: "\<H> \<subseteq> (two_chain_horizontal_boundary two_chain)"
     "\<gamma> = \<H> \<union> (two_chain_vertical_boundary two_chain)"
     by auto
   have h_vert_disj: "\<H> \<inter> (two_chain_vertical_boundary two_chain) = {}"
     using horiz_verti_disjoint h_props(1) by auto
   have h_finite: "finite \<H>"
     using hor_vert_finite h_props(1) Finite_Set.rev_finite_subset by force
   have line_integral_on_path:
     "one_chain_line_integral F {j} \<gamma> =
      one_chain_line_integral F {j} \<H> + one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
     by (auto simp add: one_chain_line_integral_def  h_props sum.union_disjoint[OF h_finite hor_vert_finite(1) h_vert_disj])
       (*Since \<H> \<subseteq> \<partial>\<^sub>H(2-\<C>), then the line_integral on the x axis along \<H> is 0, and from 1 Q.E.D. *)
   have "one_chain_line_integral F {j} \<H> = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {j} (snd oneCube) = 0"
       if oneCube: "oneCube \<in> two_chain_horizontal_boundary(two_chain)" for oneCube
     proof -
       obtain x y1 y2 k where horiz_edge_def: "oneCube = (k, (\<lambda>t::real. ((1 - t) * (y2) + t * y1, x::real)))"
         using valid_typeII_div oneCube
         by (auto simp add: typeII_twoCube_def two_chain_horizontal_boundary_def horizontal_boundary_def)
       let ?horiz_edge = "(snd oneCube)"
       have horiz_edge_y_const: "\<forall>t. (?horiz_edge t) \<bullet> j = x"
         by (simp add: j_is_y_axis horiz_edge_def)
       have horiz_edge_is_straight_path:
         "?horiz_edge = (\<lambda>t. (y2 + t * (y1 - y2), x))"
         by (simp add: add_diff_eq diff_add_eq mult.commute right_diff_distrib horiz_edge_def)
       show "line_integral F {j} (snd oneCube) = 0"
         by (simp add: horiz_edge_is_straight_path j_is_y_axis line_integral_on_pair_straight_path(1) mult.commute straight_path_diffrentiable_y)
     qed
     then have "\<forall>oneCube \<in> \<H>. line_integral F {j} (snd oneCube) = 0"
       using h_props by auto
     then have "\<forall>x\<in>\<H>. (case x of (k, g) \<Rightarrow> (k::int) * line_integral F {j} g) = 0"
       by auto
     then show "(\<Sum>x\<in>\<H>. case x of (k, g) \<Rightarrow> k * line_integral F {j} g) = 0"
       by simp
   qed
   then have "one_chain_line_integral F {j} \<gamma> =
                            one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
     using line_integral_on_path by auto
   then have "one_chain_line_integral F {j} \<gamma> =
              one_chain_line_integral F {j} (two_chain_boundary two_chain)"
     using y_axis_line_integral_is_only_vertical by auto
   then show ?thesis
     using valid_typeII_div GreenThm_typeII_divisible by auto
 qed
 
 end
 
 locale green_typeI_cube =  R2 +
   fixes twoC F
   assumes 
     two_cube: "typeI_twoCube twoC" and
     valid_two_cube: "valid_two_cube twoC" and
     f_analytically_valid: "analytically_valid (cubeImage twoC) (\<lambda>x. (F x) \<bullet> i) j"
 begin
 
 lemma GreenThm_typeI_twoCube:
   shows "integral (cubeImage twoC) (\<lambda>a. - partial_vector_derivative (\<lambda>p. F p \<bullet> i) j  a) = one_chain_line_integral F {i} (boundary twoC)"
     "\<forall>(k,\<gamma>) \<in> boundary twoC. line_integral_exists F {i} \<gamma>"
 proof -
   let ?bottom_edge = "(\<lambda>x. twoC(x, 0))"
   let ?right_edge = "(\<lambda>y. twoC(1, y))"
   let ?top_edge = "(\<lambda>x. twoC(x, 1))"
   let ?left_edge = "(\<lambda>y. twoC(0, y))"
   have line_integral_around_boundary:
     "one_chain_line_integral F {i} (boundary twoC) = 
      line_integral F {i} ?bottom_edge + line_integral F {i} ?right_edge
      -  line_integral F {i} ?top_edge - line_integral F {i} ?left_edge"
   proof (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def)
     have finite1: "finite {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))}" by auto
     have sum_step1: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (1, \<lambda>x. twoC (x, 0)), (- 1, \<lambda>x. twoC (x, 1))}. k * line_integral F {i} g) =
                        line_integral F {i} (\<lambda>x. twoC (x, 0)) + (\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))}. k * line_integral F {i} g)"
       using sum.insert_remove [OF finite1] valid_two_cube
       by (auto simp: horizontal_boundary_def vertical_boundary_def boundary_def valid_two_cube_def card_insert_if split: if_split_asm)
     have three_distinct_edges: "card {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (- 1, \<lambda>x. twoC (x, 1))} = 3"
       using valid_two_cube
       apply (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def valid_two_cube_def)
       by (auto simp: card_insert_if split: if_split_asm)
     have finite2: "finite {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}" by auto
     have sum_step2: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (-1, \<lambda>x. twoC (x, 1))}. k * line_integral F {i} g) =
                           (- line_integral F {i} (\<lambda>x. twoC (x, 1))) + (\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}. k * line_integral F {i} g)"
       using sum.insert_remove [OF finite2] three_distinct_edges
       by (auto simp: card_insert_if split: if_split_asm)
     have two_distinct_edges: "card {(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))} = 2"
       using three_distinct_edges
       by (simp add: one_chain_line_integral_def horizontal_boundary_def vertical_boundary_def boundary_def)
     have finite3: "finite {(- 1::int, \<lambda>y. twoC (0, y))}" by auto
     have sum_step3: "(\<Sum>(k, g)\<in>{(- (1::int), \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y))}. k * line_integral F {i} g) =
                           (line_integral F {i} (\<lambda>y. twoC (1, y))) + (\<Sum>(k, g)\<in>{(- (1::real), \<lambda>y. twoC (0, y))}. k * line_integral F {i} g)"
       using sum.insert_remove [OF finite2] three_distinct_edges 
       by (auto simp: card_insert_if split: if_split_asm)
     show "(\<Sum>x\<in>{(- 1::int, \<lambda>y. twoC (0, y)), (1, \<lambda>y. twoC (1, y)), (1, \<lambda>x. twoC (x, 0)), (- 1, \<lambda>x. twoC (x, 1))}. case x of (k, g) \<Rightarrow> k * line_integral F {i} g) =
                                line_integral F {i} (\<lambda>x. twoC (x, 0)) + line_integral F {i} (\<lambda>y. twoC (1, y)) - line_integral F {i} (\<lambda>x. twoC (x, 1)) - line_integral F {i} (\<lambda>y. twoC (0, y))"
       using sum_step1 sum_step2 sum_step3 by auto
   qed
   obtain a b g1 g2 where
     twoCisTypeI: "a < b"
     "(\<forall>x \<in> cbox a b. g2 x \<le> g1 x)"
     "cubeImage twoC = {(x,y). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
     "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
     "g1 piecewise_C1_differentiable_on {a .. b}"
     "g2 piecewise_C1_differentiable_on {a .. b}"
     "(\<lambda>x. twoC(x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(1, y)) = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
     "(\<lambda>x. twoC(x, 1)) = (\<lambda>x. (a + (b - a) * x, g1 (a + (b - a) * x)))"
     "(\<lambda>y. twoC(0, y)) = (\<lambda>x. (a, g2 a + x *\<^sub>R (g1 a - g2 a)))"
     using two_cube and typeI_cube_explicit_spec[of"twoC"]  by auto
   have bottom_edge_smooth: "(\<lambda>x. twoC (x, 0)) piecewise_C1_differentiable_on {0..1}"
     using typeI_twoCube_smooth_edges two_cube boundary_def vertical_boundary_def horizontal_boundary_def
     by auto
   have top_edge_smooth: "?top_edge piecewise_C1_differentiable_on {0..1}"
     using typeI_twoCube_smooth_edges two_cube boundary_def vertical_boundary_def horizontal_boundary_def
     by auto
   show "integral (cubeImage twoC) (\<lambda>a. - partial_vector_derivative (\<lambda>p. F p \<bullet> i) j  a) = one_chain_line_integral F {i} (boundary twoC)"
     using Greens_thm_type_I[OF twoCisTypeI(3) twoCisTypeI(7) bottom_edge_smooth
         twoCisTypeI(8) twoCisTypeI(9) top_edge_smooth
         twoCisTypeI(10) f_analytically_valid twoCisTypeI(2) twoCisTypeI(1)]
         line_integral_around_boundary
     by auto
   have "line_integral_exists F {i} (\<lambda>y. twoC (0, y))"
        "line_integral_exists F {i} (\<lambda>x. twoC (x, 1))"
        "line_integral_exists F {i} (\<lambda>y. twoC (1, y))"
        "line_integral_exists F {i} (\<lambda>x. twoC (x, 0))"
     using Greens_thm_type_I[OF twoCisTypeI(3) twoCisTypeI(7) bottom_edge_smooth
         twoCisTypeI(8) twoCisTypeI(9) top_edge_smooth
         twoCisTypeI(10) f_analytically_valid twoCisTypeI(2) twoCisTypeI(1)]
       line_integral_around_boundary
     by auto
   then have "line_integral_exists F {i} \<gamma>" if "(k,\<gamma>) \<in> boundary twoC" for k \<gamma>
     using that  by (auto simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   then show "\<forall>(k,\<gamma>) \<in> boundary twoC. line_integral_exists F {i} \<gamma>" by auto
 qed
 
 lemma line_integral_exists_on_typeI_Cube_boundaries':
   assumes "(k,\<gamma>) \<in> boundary twoC"
   shows "line_integral_exists F {i} \<gamma>"
   using assms two_cube valid_two_cube f_analytically_valid GreenThm_typeI_twoCube(2) by blast
 
 end
 
 locale green_typeI_chain = R2 + 
   fixes F two_chain s
   assumes valid_typeI_div: "valid_typeI_division s two_chain" and
           F_anal_valid: "\<forall>twoC \<in> two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. (F x) \<bullet> i) j"
 begin
 
 lemma two_chain_valid_valid_cubes: "\<forall>two_cube \<in> two_chain. valid_two_cube two_cube" using valid_typeI_div
       by (auto simp add: valid_two_chain_def)
 
 lemma typeI_cube_line_integral_exists_boundary':
   assumes "\<forall>two_cube \<in> two_chain. typeI_twoCube two_cube"
   assumes "\<forall>twoC \<in> two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. (F x) \<bullet> i) j"
   assumes "\<forall>two_cube \<in> two_chain. valid_two_cube two_cube"
   shows "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {i} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
     using green_typeI_cube.line_integral_exists_on_typeI_Cube_boundaries'[of i j] assms
     using R2_axioms green_typeI_cube_axioms_def green_typeI_cube_def two_chain_boundary_def by fastforce
   then show integ_exis_vert:
     "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. line_integral_exists F {i} \<gamma>"
     by (simp add: two_chain_boundary_def two_chain_vertical_boundary_def boundary_def)
 qed
 
 lemma typeI_cube_line_integral_exists_boundary'':
   "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
     using green_typeI_cube.line_integral_exists_on_typeI_Cube_boundaries'[of i j] valid_typeI_div
     apply (simp add: two_chain_boundary_def boundary_def)
     using F_anal_valid R2_axioms green_typeI_cube_axioms_def green_typeI_cube_def two_chain_valid_valid_cubes by fastforce
   then show integ_exis_horiz:
     "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
     by (simp add: two_chain_boundary_def two_chain_horizontal_boundary_def boundary_def)
 qed
 
 lemma typeI_cube_line_integral_exists_boundary:
   "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
   using  typeI_cube_line_integral_exists_boundary' typeI_cube_line_integral_exists_boundary''
   apply (auto simp add: two_chain_boundary_def two_chain_horizontal_boundary_def two_chain_vertical_boundary_def)
   by (meson R2_axioms green_typeI_chain.F_anal_valid green_typeI_chain_axioms green_typeI_cube.line_integral_exists_on_typeI_Cube_boundaries' green_typeI_cube_axioms_def green_typeI_cube_def two_chain_valid_valid_cubes valid_typeI_div)
 
 lemma type_I_chain_horiz_bound_valid:
   "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. valid_path \<gamma>"
   using typeI_edges_are_valid_paths valid_typeI_div
   by (force simp add: two_chain_boundary_def two_chain_horizontal_boundary_def boundary_def)
 
 lemma type_I_chain_vert_bound_valid: (*This and the previous one need to be used in all proofs*)
   assumes "\<forall>two_cube \<in> two_chain. typeI_twoCube two_cube"
   shows "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. valid_path \<gamma>"
   using typeI_edges_are_valid_paths assms(1)
   by (force simp add: two_chain_boundary_def two_chain_vertical_boundary_def boundary_def)
 
 lemma members_of_only_vertical_div_line_integrable':
   assumes "only_vertical_division one_chain two_chain"
     "(k::int, \<gamma>)\<in>one_chain"
     "(k::int, \<gamma>)\<in>one_chain"
     "finite two_chain"
   shows "line_integral_exists F {i} \<gamma>"
 proof -
   have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
     using typeI_cube_line_integral_exists_boundary by blast
   have integ_exis_horiz:
     "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
     using typeI_cube_line_integral_exists_boundary'' assms by auto
   have valid_paths: "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. valid_path \<gamma>"
     using type_I_chain_vert_bound_valid valid_typeI_div by linarith
   have integ_exis_vert:
     "(\<And>k \<gamma>. (\<exists>(k', \<gamma>')\<in>two_chain_vertical_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<Longrightarrow>
                             line_integral_exists F {i} \<gamma>)"
     using integ_exis valid_paths line_integral_exists_subpath[of "F" "{i}"]
     by (fastforce simp add: two_chain_boundary_def two_chain_horizontal_boundary_def
         two_chain_vertical_boundary_def boundary_def)
   obtain \<H> \<V> where hv_props: "finite \<V>"
         "(\<forall>(k,\<gamma>) \<in> \<V>. (\<exists>(k', \<gamma>') \<in> two_chain_vertical_boundary two_chain.
                         (\<exists>a \<in> {0 .. 1}. \<exists>b \<in> {0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>)))"
         "one_chain = \<H> \<union> \<V>"
         "(common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H>)
          \<or> common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)"
         "finite \<H>"
         "boundary_chain \<H>"
         "\<forall>(k,\<gamma>)\<in>\<H>. valid_path \<gamma>"
     using assms(1) unfolding only_vertical_division_def by blast
   have finite_i: "finite {i}" by auto
   show "line_integral_exists F {i} \<gamma>"
   proof(cases "common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H>")
     case True
     show ?thesis
       using gen_common_subdivision_imp_eq_line_integral(2)[OF True two_chain_horizontal_boundary_is_boundary_chain hv_props(6) integ_exis_horiz finite_two_chain_horizontal_boundary[OF assms(4)] hv_props(5) finite_i]
         integ_exis_vert[of "\<gamma>"] assms(3) case_prod_conv hv_props(2) hv_props(3)
       by fastforce
   next
     case False
     have i: "{i} \<subseteq> Basis" using i_is_x_axis real_pair_basis by auto
     have ii: " \<forall>(k2, \<gamma>2)\<in>two_chain_horizontal_boundary two_chain. \<forall>b\<in>{i}. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
       using assms field_cont_on_typeI_region_cont_on_edges F_anal_valid valid_typeI_div
       by (fastforce simp add: analytically_valid_def two_chain_horizontal_boundary_def boundary_def path_image_def)
     show "line_integral_exists F {i} \<gamma>"
       using common_reparam_exists_imp_eq_line_integral(2)[OF finite_i hv_props(5) finite_two_chain_horizontal_boundary[OF assms(4)] hv_props(6) two_chain_horizontal_boundary_is_boundary_chain ii 
             _ hv_props(7) type_I_chain_horiz_bound_valid]
         integ_exis_vert[of "\<gamma>"] False
         assms(3) hv_props(2-4) by fastforce
   qed
 qed
 
 lemma GreenThm_typeI_two_chain:
    "two_chain_integral two_chain (\<lambda>a. - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j a) = one_chain_line_integral F {i} (two_chain_boundary two_chain)"
 proof (simp add: two_chain_boundary_def one_chain_line_integral_def two_chain_integral_def)
   let ?F_b' = "partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j"
   have all_two_cubes_have_four_distict_edges: "\<forall>twoCube \<in> two_chain. card (boundary twoCube) = 4"
     using valid_typeI_div valid_two_chain_def valid_two_cube_def by auto
   have no_shared_edges_have_similar_orientations:
     "\<forall>twoCube1 \<in> two_chain. \<forall>twoCube2 \<in> two_chain.  twoCube1 \<noteq> twoCube2 \<longrightarrow>
                           boundary twoCube1 \<inter> boundary twoCube2 = {}"
     using valid_typeI_div valid_two_chain_def
     by (auto simp add: pairwise_def)
   have "(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {i} g) = integral (cubeImage twoCube) (\<lambda>a. - ?F_b' a)"
         if "twoCube \<in> two_chain" for twoCube
   proof -
     have analytically_val: " analytically_valid (cubeImage twoCube) (\<lambda>x. F x \<bullet> i) j"
       using that F_anal_valid by auto
     show "(\<Sum>(k, g)\<in>boundary twoCube. k * line_integral F {i} g) = integral (cubeImage twoCube) (\<lambda>a. - ?F_b' a)"
       using green_typeI_cube.GreenThm_typeI_twoCube
       apply (simp add: one_chain_line_integral_def)
       by (simp add: R2_axioms analytically_val green_typeI_cube_axioms_def green_typeI_cube_def that two_chain_valid_valid_cubes valid_typeI_div)
   qed
   then have double_sum_eq_sum:
     "(\<Sum>twoCube\<in>(two_chain).(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {i} g))
                      =  (\<Sum>twoCube\<in>(two_chain). integral (cubeImage twoCube) (\<lambda>a. - ?F_b' a))"
     using Finite_Cartesian_Product.sum_cong_aux by auto
   have pairwise_disjoint_boundaries: "\<forall>x\<in> (boundary ` two_chain). (\<forall>y\<in> (boundary ` two_chain). (x \<noteq> y \<longrightarrow> (x \<inter> y = {})))"
     using no_shared_edges_have_similar_orientations
     by (force simp add: image_def disjoint_iff_not_equal)
   have finite_boundaries: "\<forall>B \<in> (boundary` two_chain). finite B"
     using all_two_cubes_have_four_distict_edges
     using image_iff by fastforce
   have boundary_inj: "inj_on boundary two_chain"
     using  all_two_cubes_have_four_distict_edges and no_shared_edges_have_similar_orientations
     by (force simp add: inj_on_def)
   have "(\<Sum>x\<in>(\<Union>(boundary` two_chain)). case x of (k, g) \<Rightarrow> k * line_integral F {i} g) 
         = (sum \<circ> sum) (\<lambda>x. case x of (k, g) \<Rightarrow> (k::int) * line_integral F {i} g) (boundary ` two_chain)"
     using sum.Union_disjoint[OF finite_boundaries pairwise_disjoint_boundaries]
     by simp
   also have "... = (\<Sum>twoCube\<in>(two_chain).(\<Sum>(k,g)\<in> boundary twoCube. k * line_integral F {i} g))"
     using sum.reindex[OF boundary_inj, of "\<lambda>x. (\<Sum>(k,g)\<in> x. k * line_integral F {i} g)"]
     by auto
   finally show "(\<Sum>C\<in>two_chain. - integral (cubeImage C) (partial_vector_derivative (\<lambda>x. F x \<bullet> i) j)) = (\<Sum>x\<in>(\<Union>x\<in>two_chain. boundary x). case x of (k, g) \<Rightarrow> real_of_int k * line_integral F {i} g)"
     using double_sum_eq_sum by auto
 qed
 
 lemma GreenThm_typeI_divisible:
   assumes gen_division: "gen_division s (cubeImage ` two_chain)"
   shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F(a) \<bullet> i) j x) = one_chain_line_integral F {i} (two_chain_boundary two_chain)"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>a. F(a) \<bullet> i) j"
   have "integral s (\<lambda>x. - ?F_b' x) = two_chain_integral two_chain (\<lambda>a. - ?F_b' a)"
   proof (simp add: two_chain_integral_def)
     have "(\<Sum>f\<in>two_chain. integral (cubeImage f) (partial_vector_derivative (\<lambda>p. F p \<bullet> i) j)) = integral s (partial_vector_derivative (\<lambda>p. F p \<bullet> i) j)"
       by (metis analytically_valid_imp_part_deriv_integrable_on F_anal_valid gen_division two_chain_integral_def two_chain_integral_eq_integral_divisable valid_typeI_div)
     then show "- integral s (partial_vector_derivative (\<lambda>a. F a \<bullet> i) j) = (\<Sum>C\<in>two_chain. - integral (cubeImage C) (partial_vector_derivative (\<lambda>a. F a \<bullet> i) j))"
       by (simp add: sum_negf)
   qed
   then show ?thesis
     using GreenThm_typeI_two_chain assms by auto
 qed
 
 lemma GreenThm_typeI_divisible_region_boundary:
   assumes 
     gen_division: "gen_division s (cubeImage ` two_chain)" and
     two_cubes_trace_horizontal_boundaries:
     "two_chain_horizontal_boundary two_chain \<subseteq> \<gamma>" and
     boundary_of_region_is_subset_of_partition_boundary:
     "\<gamma> \<subseteq> two_chain_boundary two_chain"
   shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F(a) \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>a. F(a) \<bullet> i)"
   have all_two_cubes_have_four_distict_edges: "\<forall>twoCube \<in> two_chain. card (boundary twoCube) = 4"
     using valid_typeI_div valid_two_chain_def valid_two_cube_def by auto
   have no_shared_edges_have_similar_orientations:
     "\<forall>twoCube1 \<in> two_chain. \<forall>twoCube2 \<in> two_chain.
         twoCube1 \<noteq> twoCube2 \<longrightarrow> boundary twoCube1 \<inter> boundary twoCube2 = {}"
     using valid_typeI_div valid_two_chain_def by (auto simp add: pairwise_def)
       (*Proving that line_integral on the x axis is 0 for the vertical 1-cubes*)
   have vert_line_integral_zero:
     "one_chain_line_integral F {i} (two_chain_vertical_boundary two_chain) = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {i} (snd oneCube) = 0"
       if oneCube: "oneCube \<in> two_chain_vertical_boundary(two_chain)" for oneCube
     proof -
       obtain x y1 y2 k where vert_edge_def: "oneCube = (k, (\<lambda>t::real. (x::real, (1 - t) * (y2) + t * y1)))"
         using valid_typeI_div oneCube
         by (auto simp add: typeI_twoCube_def two_chain_vertical_boundary_def vertical_boundary_def)
       let ?vert_edge = "(snd oneCube)"
       have vert_edge_x_const: "\<forall>t. (?vert_edge t) \<bullet> i = x"
         by (simp add: i_is_x_axis vert_edge_def)
       have vert_edge_is_straight_path: "?vert_edge = (\<lambda>t. (x, y2 + t * (y1 - y2)))"
         using vert_edge_def Product_Type.snd_conv
         by (auto simp add: mult.commute right_diff_distrib')
       show ?thesis
         by (simp add: i_is_x_axis line_integral_on_pair_straight_path(1) mult.commute straight_path_diffrentiable_x vert_edge_is_straight_path)
     qed
     then show "(\<Sum>x\<in>two_chain_vertical_boundary two_chain. case x of (k, g) \<Rightarrow> k * line_integral F {i} g) = 0"
       using comm_monoid_add_class.sum.neutral by (simp add: prod.case_eq_if)
   qed
     (*then, the x axis line_integral on the boundaries of the twoCube is equal to the line_integral on the horizontal boundaries of the twoCube \<longrightarrow> 1*)
   have boundary_is_finite: "finite (two_chain_boundary two_chain)"
     unfolding two_chain_boundary_def
     by (metis all_two_cubes_have_four_distict_edges card.infinite finite_UN_I finite_imageD 
               gen_division gen_division_def zero_neq_numeral valid_typeI_div valid_two_chain_def)
   have boundary_is_vert_hor: "(two_chain_boundary two_chain) =
                               (two_chain_vertical_boundary two_chain) \<union>
                               (two_chain_horizontal_boundary two_chain)"
   by(auto simp add: two_chain_boundary_def two_chain_vertical_boundary_def two_chain_horizontal_boundary_def
         boundary_def)
   then have hor_vert_finite:
     "finite (two_chain_vertical_boundary two_chain)"
     "finite (two_chain_horizontal_boundary two_chain)"
     using boundary_is_finite by auto
   have horiz_verti_disjoint:
     "(two_chain_vertical_boundary two_chain) \<inter> (two_chain_horizontal_boundary two_chain) = {}"
   proof (simp add: two_chain_vertical_boundary_def two_chain_horizontal_boundary_def horizontal_boundary_def
         vertical_boundary_def)
     show "(\<Union>x\<in>two_chain. {(- 1, \<lambda>y. x (0, y)), (1::int, \<lambda>y. x (1::real, y))}) \<inter> (\<Union>x\<in>two_chain. {(1, \<lambda>z. x (z, 0)), (- 1, \<lambda>z. x (z, 1))}) = {}"
     proof -
       have "{(- 1, \<lambda>y. twoCube (0, y)), (1::int, \<lambda>y. twoCube (1, y))} \<inter>
             {(1, \<lambda>z. twoCube2 (z, 0)), (- 1, \<lambda>z. twoCube2 (z, 1))} = {}"
         if "twoCube\<in>two_chain" "twoCube2\<in>two_chain" for twoCube twoCube2
       proof (cases "twoCube = twoCube2")
         case True
         have "card {(- 1::int, \<lambda>y. twoCube2 (0::real, y)), (1::int, \<lambda>y. twoCube2 (1, y)), (1, \<lambda>x. twoCube2 (x, 0)), (- 1, \<lambda>x. twoCube2 (x, 1))} = 4"
           using all_two_cubes_have_four_distict_edges that(2)
           by (auto simp add: boundary_def vertical_boundary_def horizontal_boundary_def)
         then show ?thesis
           by (auto simp: True card_insert_if split: if_split_asm)
       next
         case False
         then show ?thesis
           using no_shared_edges_have_similar_orientations
           by (simp add: that boundary_def vertical_boundary_def horizontal_boundary_def)
       qed
       then have "\<Union> ((\<lambda>twoCube. {(-1::int, \<lambda>y. twoCube (0,y)), (1, \<lambda>y. twoCube (1, y))}) ` two_chain)
                \<inter> \<Union> ((\<lambda>twoCube. {(1, \<lambda>y. twoCube (y, 0)), (-1, \<lambda>z. twoCube (z, 1))}) ` two_chain) = {}"
         using Complete_Lattices.Union_disjoint by force
       then show ?thesis by force
     qed
   qed
   have "one_chain_line_integral F {i} (two_chain_boundary two_chain)
         = one_chain_line_integral F {i} (two_chain_vertical_boundary two_chain) +
           one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
     using boundary_is_vert_hor horiz_verti_disjoint
     by (auto simp add: one_chain_line_integral_def hor_vert_finite sum.union_disjoint)
   then have x_axis_line_integral_is_only_horizontal:
     "one_chain_line_integral F {i} (two_chain_boundary two_chain)
      = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
     using vert_line_integral_zero by auto
       (*Since \<gamma> \<subseteq> the boundary of the 2-chain and the horizontal boundaries are \<subseteq> \<gamma>, then there is some \<V> \<subseteq> \<partial>\<^sub>V(2-\<C>) such that \<gamma> = \<V> \<union> \<partial>\<^sub>H(2-\<C>)*)
   have "\<exists>\<V>. \<V> \<subseteq> (two_chain_vertical_boundary two_chain) \<and> \<gamma> = \<V> \<union> (two_chain_horizontal_boundary two_chain)"
   proof
     let ?\<V> = "\<gamma> - (two_chain_horizontal_boundary two_chain)"
     show "?\<V> \<subseteq> two_chain_vertical_boundary two_chain \<and> \<gamma> = ?\<V> \<union> two_chain_horizontal_boundary two_chain"
       using two_cubes_trace_horizontal_boundaries
         boundary_of_region_is_subset_of_partition_boundary boundary_is_vert_hor
       by blast
   qed
   then obtain \<V> where
     v_props: "\<V> \<subseteq> (two_chain_vertical_boundary two_chain)" "\<gamma> = \<V> \<union> (two_chain_horizontal_boundary two_chain)"
     by auto
   have v_horiz_disj: "\<V> \<inter> (two_chain_horizontal_boundary two_chain) = {}"
     using horiz_verti_disjoint v_props(1) by auto
   have v_finite: "finite \<V>"
     using hor_vert_finite v_props(1) Finite_Set.rev_finite_subset by force
   have line_integral_on_path: "one_chain_line_integral F {i} \<gamma> =
                                one_chain_line_integral F {i} \<V> + one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
     by(auto simp add: one_chain_line_integral_def v_props sum.union_disjoint[OF v_finite hor_vert_finite(2) v_horiz_disj])
       (*Since \<V> \<subseteq> \<partial>\<^sub>V(2-\<C>), then the line_integral on the x axis along \<H> is 0, and from 1 Q.E.D. *)
   have "one_chain_line_integral F {i} \<V> = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {i} (snd oneCube) = 0"
       if oneCube: "oneCube \<in> two_chain_vertical_boundary(two_chain)" for oneCube
     proof -
       obtain x y1 y2 k where vert_edge_def: "oneCube = (k, (\<lambda>t::real. (x::real, (1 - t) * (y2) + t * y1)))"
         using valid_typeI_div oneCube
         by (auto simp add: typeI_twoCube_def two_chain_vertical_boundary_def vertical_boundary_def)
       let ?vert_edge = "(snd oneCube)"
       have vert_edge_x_const: "\<forall>t. (?vert_edge t) \<bullet> i = x"
         by (simp add: i_is_x_axis vert_edge_def)
       have vert_edge_is_straight_path:
         "?vert_edge = (\<lambda>t. (x, y2 + t * (y1 - y2)))"
         by (auto simp: vert_edge_def algebra_simps)
       have "\<forall>x. ?vert_edge differentiable at x"
         by (metis mult.commute vert_edge_is_straight_path straight_path_diffrentiable_x)
       then show "line_integral F {i} (snd oneCube) = 0"
         using line_integral_on_pair_straight_path(1) vert_edge_x_const by blast
     qed
     then have "\<forall>oneCube \<in> \<V>. line_integral F {i} (snd oneCube) = 0"
       using v_props by auto
     then show "(\<Sum>x\<in>\<V>. case x of (k, g) \<Rightarrow> k * line_integral F {i} g) = 0"
       using comm_monoid_add_class.sum.neutral by (simp add: prod.case_eq_if) 
   qed
   then have "one_chain_line_integral F {i} \<gamma> =
              one_chain_line_integral F {i} (two_chain_boundary two_chain)"
     using x_axis_line_integral_is_only_horizontal by (simp add: line_integral_on_path)
   then show ?thesis
     using assms and GreenThm_typeI_divisible by auto
 qed
 
 lemma GreenThm_typeI_divisible_region_boundary_gen:
   assumes valid_typeI_div: "valid_typeI_division s two_chain" and
     f_analytically_valid: "\<forall>twoC \<in> two_chain. analytically_valid (cubeImage twoC) (\<lambda>a. F(a) \<bullet> i) j" and
     only_vertical_division:
     "only_vertical_division \<gamma> two_chain"
   shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F(a) \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>a. F(a) \<bullet> i)"
   have all_two_cubes_have_four_distict_edges: "\<forall>twoCube \<in> two_chain. card (boundary twoCube) = 4"
     using valid_typeI_div valid_two_chain_def valid_two_cube_def
     by auto
   have no_shared_edges_have_similar_orientations:
     "\<forall>twoCube1 \<in> two_chain. \<forall>twoCube2 \<in> two_chain.
          twoCube1 \<noteq> twoCube2 \<longrightarrow> boundary twoCube1 \<inter> boundary twoCube2 = {}"
     using valid_typeI_div valid_two_chain_def by (auto simp add: pairwise_def)
       (*Proving that line_integral on the x axis is 0 for the vertical 1-cubes*)
   have vert_line_integral_zero:
     "one_chain_line_integral F {i} (two_chain_vertical_boundary two_chain) = 0"
   proof (simp add: one_chain_line_integral_def)
     have "line_integral F {i} (snd oneCube) = 0"
       if oneCube: "oneCube \<in> two_chain_vertical_boundary(two_chain)" for oneCube
     proof -
       obtain x y1 y2 k where vert_edge_def: "oneCube = (k, (\<lambda>t::real. (x::real, (1 - t) * (y2) + t * y1)))"
         using valid_typeI_div oneCube
         by (auto simp add: typeI_twoCube_def two_chain_vertical_boundary_def vertical_boundary_def)
       let ?vert_edge = "(snd oneCube)"
       have vert_edge_x_const: "\<forall>t. (?vert_edge t) \<bullet> i = x"
         by (simp add: i_is_x_axis vert_edge_def)
       have vert_edge_is_straight_path: "?vert_edge = (\<lambda>t. (x, y2 + t * (y1 - y2)))"
         by (auto simp: vert_edge_def algebra_simps)
       show ?thesis
         by (simp add: i_is_x_axis line_integral_on_pair_straight_path(1) mult.commute straight_path_diffrentiable_x vert_edge_is_straight_path)
     qed
     then show "(\<Sum>x\<in>two_chain_vertical_boundary two_chain. case x of (k, g) \<Rightarrow> k * line_integral F {i} g) = 0"
       using comm_monoid_add_class.sum.neutral by (simp add: prod.case_eq_if)
   qed
     (*then, the x axis line_integral on the boundaries of the twoCube is equal to the line_integral on the horizontal boundaries of the twoCube \<longrightarrow> 1*)
   have boundary_is_finite: "finite (two_chain_boundary two_chain)"
     unfolding two_chain_boundary_def
   proof (rule finite_UN_I)
     show "finite two_chain"
       using assms(1) finite_imageD gen_division_def valid_two_chain_def by auto
     show "\<And>a. a \<in> two_chain \<Longrightarrow> finite (boundary a)"
       by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
   qed
   have boundary_is_vert_hor: "two_chain_boundary two_chain =
                               (two_chain_vertical_boundary two_chain) \<union> (two_chain_horizontal_boundary two_chain)"
     by (auto simp add: two_chain_boundary_def two_chain_vertical_boundary_def two_chain_horizontal_boundary_def boundary_def)
   then have hor_vert_finite:
     "finite (two_chain_vertical_boundary two_chain)"
     "finite (two_chain_horizontal_boundary two_chain)"
     using boundary_is_finite by auto
   have horiz_verti_disjoint:
     "(two_chain_vertical_boundary two_chain) \<inter> (two_chain_horizontal_boundary two_chain) = {}"
   proof (simp add: two_chain_vertical_boundary_def two_chain_horizontal_boundary_def horizontal_boundary_def
         vertical_boundary_def)
     show "(\<Union>x\<in>two_chain. {(- 1, \<lambda>y. x (0, y)), (1::int, \<lambda>y. x (1::real, y))}) 
         \<inter> (\<Union>x\<in>two_chain. {(1, \<lambda>y. x (y, 0)), (- 1, \<lambda>y. x (y, 1))}) = {}"
     proof -
       have "{(- 1, \<lambda>y. twoCube (0, y)), (1::int, \<lambda>y. twoCube (1, y))} \<inter>
             {(1, \<lambda>y. twoCube2 (y, 0)), (- 1, \<lambda>y. twoCube2 (y, 1))} = {}"
         if "twoCube \<in> two_chain" "twoCube2 \<in> two_chain" for twoCube twoCube2
       proof (cases "twoCube = twoCube2")
         case True
         have "card {(- 1::int, \<lambda>y. twoCube2 (0, y)), (1::int, \<lambda>y. twoCube2 (1, y)), (1, \<lambda>x. twoCube2 (x, 0)), (- 1, \<lambda>x. twoCube2 (x, 1))} = 4"
           using all_two_cubes_have_four_distict_edges that(2)
           by (auto simp add: boundary_def vertical_boundary_def horizontal_boundary_def)
         then show ?thesis
           by (auto simp: card_insert_if True split: if_split_asm)
       next
         case False
         then show ?thesis
           using no_shared_edges_have_similar_orientations
           by (simp add: that boundary_def vertical_boundary_def horizontal_boundary_def)
       qed
       then have "\<Union> ((\<lambda>twoCube. {(- 1, \<lambda>y. twoCube (0, y)), (1, \<lambda>y. twoCube (1, y))}) ` two_chain)
                \<inter> \<Union> ((\<lambda>twoCube. {(1::int, \<lambda>y. twoCube (y, 0)), (- 1, \<lambda>y. twoCube (y, 1))}) ` two_chain)
                  = {}"
         using Complete_Lattices.Union_disjoint by force
       then show ?thesis by force
     qed
   qed
   have "one_chain_line_integral F {i} (two_chain_boundary two_chain)
          = one_chain_line_integral F {i} (two_chain_vertical_boundary two_chain) +
            one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
     using boundary_is_vert_hor horiz_verti_disjoint
     by (auto simp add: one_chain_line_integral_def hor_vert_finite sum.union_disjoint)
   then have x_axis_line_integral_is_only_horizontal:
     "one_chain_line_integral F {i} (two_chain_boundary two_chain)
      = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
     using vert_line_integral_zero by auto
       (*Since \<gamma> \<subseteq> the boundary of the 2-chain and the horizontal boundaries are \<subseteq> \<gamma>, then there is some \<V> \<subseteq> \<partial>\<^sub>V(2-\<C>) such that \<gamma> = \<V> \<union> \<partial>\<^sub>H(2-\<C>)*)
   obtain \<H> \<V> where hv_props: "finite \<V>"
     "(\<forall>(k,\<gamma>) \<in> \<V>. (\<exists>(k', \<gamma>') \<in> two_chain_vertical_boundary two_chain.
         (\<exists>a \<in> {0 .. 1}. \<exists>b \<in> {0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>)))" "\<gamma> = \<H> \<union> \<V>"
     "(common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H>)
      \<or> common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)"
     "finite \<H>"
     "boundary_chain \<H>"
     "\<forall>(k,\<gamma>)\<in>\<H>. valid_path \<gamma>"
     using only_vertical_division by (auto simp add:  only_vertical_division_def)
   have "finite {i}" by auto
   then have eq_integrals: " one_chain_line_integral F {i} \<H> = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
   proof(cases "common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H>")
     case True
     then show ?thesis
       using gen_common_subdivision_imp_eq_line_integral(1)[OF True two_chain_horizontal_boundary_is_boundary_chain hv_props(6) _ hor_vert_finite(2) hv_props(5)]
         typeI_cube_line_integral_exists_boundary''
       by force
   next
     case False
     have integ_exis_horiz:
       "\<forall>(k,\<gamma>) \<in> two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
       using typeI_cube_line_integral_exists_boundary'' assms
       by (fastforce simp add: valid_two_chain_def)
     have integ_exis: "\<forall>(k,\<gamma>) \<in> two_chain_boundary two_chain. line_integral_exists F {i} \<gamma>"
       using typeI_cube_line_integral_exists_boundary by blast
     have valid_paths: "\<forall>(k,\<gamma>) \<in> two_chain_vertical_boundary two_chain. valid_path \<gamma>"
       using typeI_edges_are_valid_paths assms
       by (fastforce simp add: two_chain_boundary_def two_chain_vertical_boundary_def boundary_def)
     have integ_exis_vert:
       "(\<And>k \<gamma>. (\<exists>(k', \<gamma>') \<in> two_chain_vertical_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>) \<Longrightarrow>
                                     line_integral_exists F {i} \<gamma>)"
       using integ_exis valid_paths line_integral_exists_subpath[of "F" "{i}"]
       by (fastforce simp add: two_chain_boundary_def two_chain_vertical_boundary_def
           two_chain_horizontal_boundary_def boundary_def)
     have finite_i: "finite {i}" by auto
     have i: "{i} \<subseteq> Basis" using i_is_x_axis real_pair_basis by auto
     have ii: " \<forall>(k2, \<gamma>2)\<in>two_chain_horizontal_boundary two_chain. \<forall>b\<in>{i}. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
       using assms(1) field_cont_on_typeI_region_cont_on_edges assms(2)
       by (fastforce simp add: analytically_valid_def two_chain_horizontal_boundary_def boundary_def path_image_def)
     have *: "common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)"
       using hv_props(4) False by auto
     show "one_chain_line_integral F {i} \<H> = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
       using common_reparam_exists_imp_eq_line_integral(1)[OF finite_i hv_props(5) hor_vert_finite(2) hv_props(6) two_chain_horizontal_boundary_is_boundary_chain ii * hv_props(7) type_I_chain_horiz_bound_valid]
       by fastforce
   qed
     (*Since \<V> \<subseteq> \<partial>\<^sub>H(2-\<C>), then the line_integral on the x axis along \<V> is 0, and from 1 Q.E.D. *)
   have line_integral_on_path:
     "one_chain_line_integral F {i} \<gamma> =
      one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
   proof (auto simp add: one_chain_line_integral_def)
     have "line_integral F {i} (snd oneCube) = 0" if oneCube: "oneCube \<in> \<V>" for oneCube
     proof -
       obtain k \<gamma> where k_gamma: "(k,\<gamma>) = oneCube"
         by (metis coeff_cube_to_path.cases)
       then obtain k' \<gamma>' a b where kp_gammap:
         "(k'::int, \<gamma>') \<in> two_chain_vertical_boundary two_chain"
         "a \<in> {0 .. 1}"
         "b \<in> {0..1}"
         "subpath a b \<gamma>' = \<gamma>"
         using hv_props oneCube
         by (smt case_prodE split_conv)
       obtain x y1 y2 where vert_edge_def: "(k',\<gamma>') = (k', (\<lambda>t::real. (x::real, (1 - t) * (y2) + t * y1)))"
         using valid_typeI_div kp_gammap
         by (auto simp add: typeI_twoCube_def two_chain_vertical_boundary_def vertical_boundary_def)
       have vert_edge_x_const: "\<forall>t. \<gamma> (t) \<bullet> i = x"
         by (metis (no_types, lifting) Pair_inject fstI i_is_x_axis inner_Pair_0(2) kp_gammap(4) real_inner_1_right subpath_def vert_edge_def)
       have "\<gamma> = (\<lambda>t::real. (x::real, (1 - (b - a)*t - a) * (y2) + ((b-a)*t + a) * y1))"
         using vert_edge_def Product_Type.snd_conv Product_Type.fst_conv kp_gammap(4)
         by (simp add: subpath_def diff_diff_eq[symmetric])
       also have "... = (\<lambda>t::real. (x::real, (1*y2 - a*y2) +  a*y1 + ((b-a)*y1 - (b - a)*y2)*t))"
         by (simp add: algebra_simps)
       finally have vert_edge_is_straight_path:
         "\<gamma> = (\<lambda>t::real. (x::real, (1*y2 - a*y2) +  a*y1 + ((b-a)*y1 - (b - a)*y2)*t))" .
       show "line_integral F {i} (snd oneCube) = 0"
       proof -
         have "\<forall>x. \<gamma> differentiable at x"
           by (simp add: straight_path_diffrentiable_x vert_edge_is_straight_path)
         then have "line_integral F {i} \<gamma> = 0"
           using line_integral_on_pair_straight_path(1) vert_edge_x_const by blast
         then show ?thesis
           using Product_Type.snd_conv k_gamma by auto
       qed
     qed
     then have "\<forall>x\<in>\<V>. (case x of (k, g) \<Rightarrow> (k::int) * line_integral F {i} g) = 0"
       by auto
     then show "(\<Sum>x\<in>\<gamma>. case x of (k, g) \<Rightarrow> real_of_int k * line_integral F {i} g) =
                (\<Sum>x\<in>two_chain_horizontal_boundary two_chain. case x of (k, g) \<Rightarrow> of_int k * line_integral F {i} g)"
       using hv_props(1) hv_props(3) hv_props(5) sum_zero_set hor_vert_finite(2) eq_integrals
       apply(auto simp add: one_chain_line_integral_def)
       by (smt Un_commute sum_zero_set)
   qed
   then have "one_chain_line_integral F {i} \<gamma> =
              one_chain_line_integral F {i} (two_chain_boundary two_chain)"
     using x_axis_line_integral_is_only_horizontal line_integral_on_path by auto
   then show ?thesis
     using assms GreenThm_typeI_divisible by auto
 qed
 
 end
 
 locale green_typeI_typeII_chain = R2: R2 i j + T1: green_typeI_chain i j F two_chain_typeI + T2: green_typeII_chain i j F two_chain_typeII for i j F two_chain_typeI two_chain_typeII
 begin
 
 lemma GreenThm_typeI_typeII_divisible_region_boundary:
   assumes 
     gen_divisions: "gen_division s (cubeImage ` two_chain_typeI)"
     "gen_division s (cubeImage ` two_chain_typeII)" and
     typeI_two_cubes_trace_horizontal_boundaries:
     "two_chain_horizontal_boundary two_chain_typeI \<subseteq> \<gamma>" and
     typeII_two_cubes_trace_vertical_boundaries:
     "two_chain_vertical_boundary two_chain_typeII \<subseteq> \<gamma>" and
     boundary_of_region_is_subset_of_partition_boundaries:
     "\<gamma> \<subseteq> two_chain_boundary two_chain_typeI"
     "\<gamma> \<subseteq> two_chain_boundary two_chain_typeII"
   shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x)
          = one_chain_line_integral F {i, j} \<gamma>"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>a. F a \<bullet> i) j"
   let ?F_a' = "partial_vector_derivative (\<lambda>a. F a \<bullet> j) i"
   have typeI_regions_integral: "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} \<gamma>"
     using T1.GreenThm_typeI_divisible_region_boundary
         gen_divisions(1) typeI_two_cubes_trace_horizontal_boundaries
         boundary_of_region_is_subset_of_partition_boundaries(1)
     by blast
   have typeII_regions_integral: "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} \<gamma>"
     using T2.GreenThm_typeII_divisible_region_boundary gen_divisions(2)
          typeII_two_cubes_trace_vertical_boundaries
         boundary_of_region_is_subset_of_partition_boundaries(2)
     by auto
   have integral_dis: "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = integral s (\<lambda>x. ?F_a' x) + integral s (\<lambda>x. - ?F_b' x)"
   proof -
     have "\<forall>twoCube \<in> two_chain_typeII. (?F_a' has_integral integral (cubeImage twoCube) ?F_a') (cubeImage twoCube)"
       by (simp add: analytically_valid_imp_part_deriv_integrable_on T2.F_anal_valid has_integral_iff)
     then have "\<And>u. u \<in> (cubeImage ` two_chain_typeII) \<Longrightarrow> (?F_a' has_integral integral u ?F_a') u"
       by auto
     then have "(?F_a' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeII. integral img ?F_a')) s"
       using gen_divisions(2) unfolding gen_division_def
       by (metis has_integral_Union)
     then have F_a'_integrable: "(?F_a' integrable_on s)" by auto
     have "\<forall>twoCube \<in> two_chain_typeI. (?F_b' has_integral integral (cubeImage twoCube) ?F_b') (cubeImage twoCube)"
       using analytically_valid_imp_part_deriv_integrable_on T1.F_anal_valid by blast
     then have "\<And>u. u \<in> (cubeImage ` two_chain_typeI) \<Longrightarrow> (?F_b' has_integral integral u ?F_b') u"
       by auto
     then have "(?F_b' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeI. integral img ?F_b')) s"
       using gen_divisions(1) unfolding gen_division_def
       by (metis has_integral_Union)
     then show ?thesis
       by (simp add: F_a'_integrable Henstock_Kurzweil_Integration.integral_diff has_integral_iff)
   qed
   have line_integral_dist: "one_chain_line_integral F {i, j} \<gamma> = one_chain_line_integral F {i} \<gamma> + one_chain_line_integral F {j} \<gamma>"
   proof (simp add: one_chain_line_integral_def)
     have "k * line_integral F {i, j} g = k * line_integral F {i} g +  k * line_integral F {j} g"
       if kg: "(k,g) \<in> \<gamma>" for k g
     proof -
       obtain twoCube_typeI where twoCube_typeI_props:
         "twoCube_typeI \<in> two_chain_typeI"
         "(k, g) \<in> boundary twoCube_typeI"
         "typeI_twoCube twoCube_typeI"
         "continuous_on (cubeImage twoCube_typeI) (\<lambda>x. F(x) \<bullet> i)"
         using boundary_of_region_is_subset_of_partition_boundaries(1) two_chain_boundary_def T1.valid_typeI_div
           T1.F_anal_valid kg
         by (auto simp add: analytically_valid_def)
       obtain twoCube_typeII where twoCube_typeII_props:
         "twoCube_typeII \<in> two_chain_typeII"
         "(k, g) \<in> boundary twoCube_typeII"
         "typeII_twoCube twoCube_typeII"
         "continuous_on (cubeImage twoCube_typeII) (\<lambda>x. F(x) \<bullet> j)"
         using boundary_of_region_is_subset_of_partition_boundaries(2) two_chain_boundary_def T2.valid_typeII_div
           kg T2.F_anal_valid
         by (auto simp add: analytically_valid_def)
       have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
       proof -
         have int_exists_i: "line_integral_exists F {i} g"
           using T1.typeI_cube_line_integral_exists_boundary assms kg
           by (auto simp add: valid_two_chain_def)
         have int_exists_j: "line_integral_exists F {j} g"
           using T2.typeII_cube_line_integral_exists_boundary assms kg
           by (auto simp add: valid_two_chain_def)
         have finite: "finite {i, j}" by auto
         show ?thesis
           using line_integral_sum_gen[OF finite int_exists_i int_exists_j] R2.i_is_x_axis R2.j_is_y_axis
           by auto
       qed
       then show "k * line_integral F {i, j} g = k * line_integral F {i} g + k * line_integral F {j} g"
         by (simp add: distrib_left)
     qed
     then have line_integral_distrib: 
           "(\<Sum>(k,g)\<in>\<gamma>. k * line_integral F {i, j} g) =
            (\<Sum>(k,g)\<in>\<gamma>. k * line_integral F {i} g +  k * line_integral F {j} g)"
       by (force intro: sum.cong split_cong)
     have "(\<lambda>x. (case x of (k, g) \<Rightarrow> (k::int) * line_integral F {i} g) + (case x of (k, g) \<Rightarrow> (k::int) * line_integral F {j} g)) =
                                      (\<lambda>x. (case x of (k, g) \<Rightarrow> (k * line_integral F {i} g) +  (k::int) * line_integral F {j} g))"
       using comm_monoid_add_class.sum.distrib by auto
     then show "(\<Sum>(k, g)\<in>\<gamma>. k * line_integral F {i, j} g) =
           (\<Sum>(k, g)\<in>\<gamma>. (k::int) * line_integral F {i} g) + (\<Sum>(k, g)\<in>\<gamma>. (k::int) * line_integral F {j} g)"
       using comm_monoid_add_class.sum.distrib[of "(\<lambda>(k, g).  k * line_integral F {i} g)" "(\<lambda>(k, g).  k * line_integral F {j} g)" \<gamma>]
         line_integral_distrib
       by presburger
   qed
   show ?thesis
     using integral_dis line_integral_dist typeI_regions_integral typeII_regions_integral
     by auto
 qed
 
 lemma GreenThm_typeI_typeII_divisible_region':
   assumes 
     only_vertical_division:
     "only_vertical_division one_chain_typeI two_chain_typeI"
     "boundary_chain one_chain_typeI" and
     only_horizontal_division:
     "only_horizontal_division one_chain_typeII two_chain_typeII"
     "boundary_chain one_chain_typeII" and
     typeI_and_typII_one_chains_have_gen_common_subdiv:
     "common_sudiv_exists one_chain_typeI one_chain_typeII"
   shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
     "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j"
   let ?F_a' = "partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i"
   have one_chain_i_integrals:
     "one_chain_line_integral F {i} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeII \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>) \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>)"
   proof (intro conjI)
     have "finite two_chain_typeI"
       using T1.valid_typeI_div finite_image_iff
       by (auto simp add: gen_division_def valid_two_chain_def)
     then show ii: "\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>"
       using T1.members_of_only_vertical_div_line_integrable' assms 
       by fastforce
     have "finite (two_chain_horizontal_boundary two_chain_typeI)"
       by (meson T1.valid_typeI_div finite_imageD finite_two_chain_horizontal_boundary gen_division_def valid_two_chain_def)
     then have "finite one_chain_typeI"
       using only_vertical_division(1) only_vertical_division_def by auto
     moreover have "finite one_chain_typeII"
       using only_horizontal_division(1) only_horizontal_division_def by auto
     ultimately show "one_chain_line_integral F {i} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeII"
                and "\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>"
       using gen_common_subdivision_imp_eq_line_integral[OF typeI_and_typII_one_chains_have_gen_common_subdiv
           only_vertical_division(2) only_horizontal_division(2)] ii 
       by auto
   qed
   have one_chain_j_integrals:
     "one_chain_line_integral F {j} one_chain_typeII = one_chain_line_integral F {j} one_chain_typeI \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>) \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>)"
   proof (intro conjI)
     have "finite two_chain_typeII"
       using T2.valid_typeII_div finite_image_iff
       by (auto simp add: gen_division_def valid_two_chain_def)
     then show ii: "\<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>"
       using T2.members_of_only_horiz_div_line_integrable' assms T2.two_chain_valid_valid_cubes by blast
     have typeII_and_typI_one_chains_have_common_subdiv: "common_sudiv_exists one_chain_typeII one_chain_typeI"
       by (simp add: common_sudiv_exists_comm typeI_and_typII_one_chains_have_gen_common_subdiv)
     have iv: "finite one_chain_typeI"
       using only_vertical_division(1) only_vertical_division_def by auto
     moreover have iv': "finite one_chain_typeII"
       using only_horizontal_division(1) only_horizontal_division_def by auto
     ultimately show "one_chain_line_integral F {j} one_chain_typeII =
                      one_chain_line_integral F {j} one_chain_typeI"
                     "\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>"
       using gen_common_subdivision_imp_eq_line_integral[OF typeII_and_typI_one_chains_have_common_subdiv
           only_horizontal_division(2) only_vertical_division(2) ii] ii 
       by auto
   qed
   have typeI_regions_integral:
     "integral s (\<lambda>x. - ?F_b' x) = one_chain_line_integral F {i} one_chain_typeI"
     using T1.GreenThm_typeI_divisible_region_boundary_gen T1.valid_typeI_div
           T1.F_anal_valid  only_vertical_division(1)
     by auto
   have typeII_regions_integral:
     "integral s ?F_a' = one_chain_line_integral F {j} one_chain_typeII"
     using T2.GreenThm_typeII_divisible_region_boundary_gen T2.valid_typeII_div
         T2.F_anal_valid  only_horizontal_division(1)
     by auto
   have line_integral_dist:
     "one_chain_line_integral F {i, j} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeI + one_chain_line_integral F {j} one_chain_typeI \<and>
      one_chain_line_integral F {i, j} one_chain_typeII = one_chain_line_integral F {i} one_chain_typeII + one_chain_line_integral F {j} one_chain_typeII"
   proof (simp add: one_chain_line_integral_def)
     have line_integral_distrib:
       "(\<Sum>(k,g)\<in>one_chain_typeI. k * line_integral F {i, j} g) =
        (\<Sum>(k,g)\<in>one_chain_typeI. k * line_integral F {i} g +  k * line_integral F {j} g) \<and>
        (\<Sum>(k,g)\<in>one_chain_typeII. k * line_integral F {i, j} g) =
        (\<Sum>(k,g)\<in>one_chain_typeII. k * line_integral F {i} g +  k * line_integral F {j} g)"
     proof -
       have 0: "k * line_integral F {i, j} g = k * line_integral F {i} g +  k * line_integral F {j} g"
         if  "(k,g) \<in> one_chain_typeII" for k g
       proof -
         have "line_integral_exists F {i} g" "line_integral_exists F {j} g" "finite {i, j}"
           using one_chain_i_integrals one_chain_j_integrals that by fastforce+
         moreover have "{i} \<inter> {j} = {}"
           by (simp add: R2.i_is_x_axis R2.j_is_y_axis)
         ultimately have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
           by (metis insert_is_Un line_integral_sum_gen(1))
         then show "k * line_integral F {i, j} g = k * line_integral F {i} g + k * line_integral F {j} g"
           by (simp add: distrib_left)
       qed
       have "k * line_integral F {i, j} g = k * line_integral F {i} g +  k * line_integral F {j} g"
         if "(k,g) \<in> one_chain_typeI" for k g
       proof -
         have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
           by (smt that disjoint_insert(2) finite.emptyI finite.insertI R2.i_is_x_axis inf_bot_right insert_absorb insert_commute insert_is_Un R2.j_is_y_axis line_integral_sum_gen(1) one_chain_i_integrals one_chain_j_integrals prod.case_eq_if singleton_inject snd_conv)
         then show "k * line_integral F {i, j} g = k * line_integral F {i} g + k * line_integral F {j} g"
           by (simp add: distrib_left)
       qed
       then show ?thesis
         using 0 by (smt sum.cong split_cong)
     qed
     show "(\<Sum>(k::int, g)\<in>one_chain_typeI. k * line_integral F {i, j} g) =
           (\<Sum>(k, g)\<in>one_chain_typeI. k * line_integral F {i} g) + (\<Sum>(k::int, g)\<in>one_chain_typeI. k * line_integral F {j} g) \<and>
           (\<Sum>(k::int, g)\<in>one_chain_typeII. k * line_integral F {i, j} g) =
           (\<Sum>(k, g)\<in>one_chain_typeII. k * line_integral F {i} g) + (\<Sum>(k::int, g)\<in>one_chain_typeII. k * line_integral F {j} g)"
     proof -
       have 0: "(\<lambda>x. (case x of (k::int, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k::int, g) \<Rightarrow> k * line_integral F {j} g)) =
                                   (\<lambda>x. (case x of (k::int, g) \<Rightarrow> (k * line_integral F {i} g) +  k * line_integral F {j} g))"
         using comm_monoid_add_class.sum.distrib by auto
       then have 1: "(\<Sum>x\<in>one_chain_typeI. (case x of (k::int, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k::int, g) \<Rightarrow> k * line_integral F {j} g)) =
                     (\<Sum>x\<in>one_chain_typeI. (case x of (k::int, g) \<Rightarrow>( k * line_integral F {i} g + k * line_integral F {j} g)))"
         by presburger
       have "(\<Sum>x\<in>one_chain_typeII. (case x of (k, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k, g) \<Rightarrow> k * line_integral F {j} g)) =
             (\<Sum>x\<in>one_chain_typeII. (case x of (k, g) \<Rightarrow>( k * line_integral F {i} g + k * line_integral F {j} g)))"
         using 0 by presburger
       then show ?thesis
         using sum.distrib[of "(\<lambda>(k, g).  k * line_integral F {i} g)" "(\<lambda>(k, g).  k * line_integral F {j} g)" "one_chain_typeI"]
           sum.distrib[of "(\<lambda>(k, g).  k * line_integral F {i} g)" "(\<lambda>(k, g).  k * line_integral F {j} g)" "one_chain_typeII"]
           line_integral_distrib 1
         by auto
     qed
   qed
   have integral_dis: "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = integral s (\<lambda>x. ?F_a' x) + integral s (\<lambda>x. - ?F_b' x)"
   proof -
     have "(?F_a' has_integral integral (cubeImage twoCube) ?F_a') (cubeImage twoCube)"
       if "twoCube \<in> two_chain_typeII" for twoCube
       by (simp add: analytically_valid_imp_part_deriv_integrable_on T2.F_anal_valid has_integral_integrable_integral that)
     then have "\<And>u. u \<in> (cubeImage ` two_chain_typeII) \<Longrightarrow> (?F_a' has_integral integral u ?F_a') u"
       by auto
     then have "(?F_a' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeII. integral img ?F_a')) s"
       using T2.valid_typeII_div unfolding gen_division_def
       by (metis has_integral_Union)
     then have F_a'_integrable:
       "(?F_a' integrable_on s)" by auto
     have "\<forall>twoCube \<in> two_chain_typeI. (?F_b' has_integral integral (cubeImage twoCube) ?F_b') (cubeImage twoCube)"
       using analytically_valid_imp_part_deriv_integrable_on T1.F_anal_valid by blast
     then have "\<And>u. u \<in> (cubeImage ` two_chain_typeI) \<Longrightarrow> (?F_b' has_integral integral u ?F_b') u"
       by auto
     then have "(?F_b' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeI. integral img ?F_b')) s"
       using T1.valid_typeI_div unfolding gen_division_def
       by (metis has_integral_Union)
     then show ?thesis
       by (simp add: F_a'_integrable Henstock_Kurzweil_Integration.integral_diff has_integral_iff)
   qed
   show "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = one_chain_line_integral F {i, j} one_chain_typeI"
     using one_chain_j_integrals integral_dis line_integral_dist typeI_regions_integral typeII_regions_integral
     by auto
   show "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = one_chain_line_integral F {i, j} one_chain_typeII"
     using one_chain_i_integrals integral_dis line_integral_dist typeI_regions_integral typeII_regions_integral
     by auto
 qed
 
 lemma GreenThm_typeI_typeII_divisible_region:
   assumes only_vertical_division:
     "only_vertical_division one_chain_typeI two_chain_typeI"
     "boundary_chain one_chain_typeI" and
     only_horizontal_division:
     "only_horizontal_division one_chain_typeII two_chain_typeII"
     "boundary_chain one_chain_typeII" and
     typeI_and_typII_one_chains_have_common_subdiv:
     "common_boundary_sudivision_exists one_chain_typeI one_chain_typeII"
   shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
     "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"
   using GreenThm_typeI_typeII_divisible_region' only_vertical_division only_horizontal_division common_subdiv_imp_gen_common_subdiv[OF typeI_and_typII_one_chains_have_common_subdiv]
   by auto
 
 lemma GreenThm_typeI_typeII_divisible_region_finite_holes:
   assumes valid_cube_boundary: "\<forall>(k,\<gamma>)\<in>boundary C. valid_path \<gamma>" and
     only_vertical_division:
     "only_vertical_division (boundary C) two_chain_typeI" and
     only_horizontal_division:
     "only_horizontal_division (boundary C) two_chain_typeII" and
     s_is_oneCube: "s = cubeImage C"
   shows "integral (cubeImage C) (\<lambda>x. partial_vector_derivative (\<lambda>x. F x \<bullet> j) i x - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) =
                      one_chain_line_integral F {i, j} (boundary C)"
   using GreenThm_typeI_typeII_divisible_region[OF only_vertical_division
       two_cube_boundary_is_boundary only_horizontal_division two_cube_boundary_is_boundary
       common_boundary_subdiv_exists_refl[OF assms(1)]] s_is_oneCube
   by auto
 
 lemma GreenThm_typeI_typeII_divisible_region_equivallent_boundary:
   assumes 
     gen_divisions: "gen_division s (cubeImage ` two_chain_typeI)"
     "gen_division s (cubeImage ` two_chain_typeII)" and
     typeI_two_cubes_trace_horizontal_boundaries:
     "two_chain_horizontal_boundary two_chain_typeI \<subseteq> one_chain_typeI" and
     typeII_two_cubes_trace_vertical_boundaries:
     "two_chain_vertical_boundary two_chain_typeII \<subseteq> one_chain_typeII" and
     boundary_of_region_is_subset_of_partition_boundaries:
     "one_chain_typeI \<subseteq> two_chain_boundary two_chain_typeI"
     "one_chain_typeII \<subseteq> two_chain_boundary two_chain_typeII" and
     typeI_and_typII_one_chains_have_common_subdiv:
     "common_boundary_sudivision_exists one_chain_typeI one_chain_typeII"
   shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
     "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"
 proof -
   let ?F_b' = "partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j"
   let ?F_a' = "partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i"
   have one_chain_i_integrals:
     "one_chain_line_integral F {i} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeII \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>) \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>)"
   proof (intro conjI)
     have i: "boundary_chain one_chain_typeI"
       using two_chain_boundary_is_boundary_chain boundary_chain_def
         boundary_of_region_is_subset_of_partition_boundaries(1)
       by blast
     have i': "boundary_chain one_chain_typeII"
       using two_chain_boundary_is_boundary_chain boundary_chain_def
         boundary_of_region_is_subset_of_partition_boundaries(2)
       by blast
     have "\<And>k \<gamma>. (k,\<gamma>)\<in>one_chain_typeI \<Longrightarrow> line_integral_exists F {i} \<gamma>"
       using T1.typeI_cube_line_integral_exists_boundary assms
       by (fastforce simp add: valid_two_chain_def)
     then show ii: "\<forall>(k,\<gamma>)\<in>one_chain_typeI. line_integral_exists F {i} \<gamma>" by auto
     have "finite (two_chain_boundary two_chain_typeI)"
       unfolding two_chain_boundary_def
     proof (rule finite_UN_I)
       show "finite two_chain_typeI"
         using T1.valid_typeI_div finite_imageD gen_division_def valid_two_chain_def by auto
       show "\<And>a. a \<in> two_chain_typeI \<Longrightarrow> finite (boundary a)"
         by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
     qed
     then have "finite one_chain_typeI"
       using boundary_of_region_is_subset_of_partition_boundaries(1) finite_subset by fastforce
     moreover have "finite (two_chain_boundary two_chain_typeII)"
       unfolding two_chain_boundary_def
     proof (rule finite_UN_I)
       show "finite two_chain_typeII"
         using T2.valid_typeII_div finite_imageD gen_division_def valid_two_chain_def by auto
       show "\<And>a. a \<in> two_chain_typeII \<Longrightarrow> finite (boundary a)"
         by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
     qed
     then have "finite one_chain_typeII"
       using boundary_of_region_is_subset_of_partition_boundaries(2) finite_subset by fastforce
     ultimately show "one_chain_line_integral F {i} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeII"
       "\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {i} \<gamma>"
       using ii common_subdivision_imp_eq_line_integral[OF typeI_and_typII_one_chains_have_common_subdiv
           i i' ii]
       by auto
   qed
   have one_chain_j_integrals:
     "one_chain_line_integral F {j} one_chain_typeI = one_chain_line_integral F {j} one_chain_typeII \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>) \<and>
               (\<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>)"
   proof (intro conjI)
     have i: "boundary_chain one_chain_typeI" and i': "boundary_chain one_chain_typeII"
       using two_chain_boundary_is_boundary_chain boundary_of_region_is_subset_of_partition_boundaries
       unfolding boundary_chain_def by blast+
     have "line_integral_exists F {j} \<gamma>" if "(k,\<gamma>)\<in>one_chain_typeII" for k \<gamma>
     proof -
       have F_is_continuous: "\<forall>twoC \<in> two_chain_typeII. continuous_on (cubeImage twoC) (\<lambda>a. F(a) \<bullet> j)"
         using T2.F_anal_valid by(simp add: analytically_valid_def)
       show "line_integral_exists F {j} \<gamma>"
         using that T2.valid_typeII_div
           boundary_of_region_is_subset_of_partition_boundaries(2)
         using green_typeII_cube.line_integral_exists_on_typeII_Cube_boundaries' assms valid_two_chain_def
         apply (simp add: two_chain_boundary_def)
         by (metis T2.typeII_cube_line_integral_exists_boundary case_prodD subset_iff that two_chain_boundary_def)
     qed
     then show ii: " \<forall>(k,\<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>" by auto
     have "finite (two_chain_boundary two_chain_typeI)"
       unfolding two_chain_boundary_def
     proof (rule finite_UN_I)
       show "finite two_chain_typeI"
         using T1.valid_typeI_div finite_imageD gen_division_def valid_two_chain_def by auto
       show "\<And>a. a \<in> two_chain_typeI \<Longrightarrow> finite (boundary a)"
         by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
     qed
     then have iv: "finite one_chain_typeI"
       using boundary_of_region_is_subset_of_partition_boundaries(1) finite_subset
       by fastforce
     have "finite (two_chain_boundary two_chain_typeII)"
       unfolding two_chain_boundary_def
     proof (rule finite_UN_I)
       show "finite two_chain_typeII"
         using T2.valid_typeII_div finite_imageD gen_division_def valid_two_chain_def by auto
       show "\<And>a. a \<in> two_chain_typeII \<Longrightarrow> finite (boundary a)"
         by (simp add: boundary_def horizontal_boundary_def vertical_boundary_def)
     qed
     then have iv': "finite one_chain_typeII"
       using boundary_of_region_is_subset_of_partition_boundaries(2) finite_subset
       by fastforce
     have typeII_and_typI_one_chains_have_common_subdiv:
       "common_boundary_sudivision_exists one_chain_typeII one_chain_typeI"
       using typeI_and_typII_one_chains_have_common_subdiv
         common_boundary_sudivision_commutative
       by auto
     show "one_chain_line_integral F {j} one_chain_typeI = one_chain_line_integral F {j} one_chain_typeII" 
           "\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>"
       using common_subdivision_imp_eq_line_integral[OF typeII_and_typI_one_chains_have_common_subdiv
           i' i ii iv' iv] ii
       by auto
   qed
   have typeI_regions_integral:
     "integral s (\<lambda>x. - ?F_b' x) = one_chain_line_integral F {i} one_chain_typeI"
     using T1.GreenThm_typeI_divisible_region_boundary gen_divisions(1)
          typeI_two_cubes_trace_horizontal_boundaries
         boundary_of_region_is_subset_of_partition_boundaries(1)
     by auto
   have typeII_regions_integral:
     "integral s ?F_a' = one_chain_line_integral F {j} one_chain_typeII"
     using T2.GreenThm_typeII_divisible_region_boundary gen_divisions(2) 
         typeII_two_cubes_trace_vertical_boundaries
         boundary_of_region_is_subset_of_partition_boundaries(2)
     by auto
   have line_integral_dist:
     "one_chain_line_integral F {i, j} one_chain_typeI = one_chain_line_integral F {i} one_chain_typeI + one_chain_line_integral F {j} one_chain_typeI \<and>
      one_chain_line_integral F {i, j} one_chain_typeII = one_chain_line_integral F {i} one_chain_typeII + one_chain_line_integral F {j} one_chain_typeII"
   proof (simp add: one_chain_line_integral_def)
     have line_integral_distrib:
       "(\<Sum>(k,g)\<in>one_chain_typeI. k * line_integral F {i, j} g) =
        (\<Sum>(k,g)\<in>one_chain_typeI. k * line_integral F {i} g +  k * line_integral F {j} g) \<and>
        (\<Sum>(k,g)\<in>one_chain_typeII. k * line_integral F {i, j} g) =
        (\<Sum>(k,g)\<in>one_chain_typeII. k * line_integral F {i} g +  k * line_integral F {j} g)"
     proof -
       have 0: "k * line_integral F {i, j} g = k * line_integral F {i} g +  k * line_integral F {j} g"
         if "(k,g) \<in> one_chain_typeII" for k g
       proof -
         have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
         proof -
           have finite: "finite {i, j}" by auto
           have line_integral_all: "\<forall>i\<in>{i, j}. line_integral_exists F {i} g"
             using one_chain_i_integrals one_chain_j_integrals that by auto
           show ?thesis
             using line_integral_sum_gen[OF finite] R2.i_is_x_axis R2.j_is_y_axis line_integral_all by auto
         qed
         then show "k * line_integral F {i, j} g = k * line_integral F {i} g + k * line_integral F {j} g"
           by (simp add: distrib_left)
       qed
       have "k * line_integral F {i, j} g = k * line_integral F {i} g +  k * line_integral F {j} g"
         if "(k,g) \<in> one_chain_typeI" for k g
       proof -
         have finite: "finite {i, j}" by auto
         have line_integral_all: "\<forall>i\<in>{i, j}. line_integral_exists F {i} g"
           using one_chain_i_integrals one_chain_j_integrals that by auto
         have "line_integral F {i, j} g = line_integral F {i} g + line_integral F {j} g"
           using line_integral_sum_gen[OF finite] R2.i_is_x_axis R2.j_is_y_axis line_integral_all by auto
         then show "k * line_integral F {i, j} g = k * line_integral F {i} g + k * line_integral F {j} g"
           by (simp add: distrib_left)
       qed
       then show ?thesis
         using 0  by (smt sum.cong split_cong)
     qed
     show "(\<Sum>(k::int, g)\<in>one_chain_typeI. k * line_integral F {i, j} g) =
           (\<Sum>(k, g)\<in>one_chain_typeI. k * line_integral F {i} g) + (\<Sum>(k::int, g)\<in>one_chain_typeI. k * line_integral F {j} g) \<and>
           (\<Sum>(k::int, g)\<in>one_chain_typeII. k * line_integral F {i, j} g) =
           (\<Sum>(k, g)\<in>one_chain_typeII. k * line_integral F {i} g) + (\<Sum>(k::int, g)\<in>one_chain_typeII. k * line_integral F {j} g)"
     proof -
       have 0: "(\<lambda>x. (case x of (k::int, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k::int, g) \<Rightarrow> k * line_integral F {j} g)) =
                                   (\<lambda>x. (case x of (k::int, g) \<Rightarrow> (k * line_integral F {i} g) +  k * line_integral F {j} g))"
         using comm_monoid_add_class.sum.distrib by auto
       then have 1: "(\<Sum>x\<in>one_chain_typeI. (case x of (k::int, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k::int, g) \<Rightarrow> k * line_integral F {j} g)) =
                     (\<Sum>x\<in>one_chain_typeI. (case x of (k::int, g) \<Rightarrow>( k * line_integral F {i} g + k * line_integral F {j} g)))"
         by presburger
       have "(\<Sum>x\<in>one_chain_typeII. (case x of (k, g) \<Rightarrow> k * line_integral F {i} g) + (case x of (k, g) \<Rightarrow> k * line_integral F {j} g)) =
             (\<Sum>x\<in>one_chain_typeII. (case x of (k, g) \<Rightarrow>( k * line_integral F {i} g + k * line_integral F {j} g)))"
         using 0 by presburger
       then show ?thesis
         using sum.distrib[of "(\<lambda>(k, g).  k * line_integral F {i} g)" "(\<lambda>(k, g).  k * line_integral F {j} g)" "one_chain_typeI"]
           sum.distrib[of "(\<lambda>(k, g).  k * line_integral F {i} g)" "(\<lambda>(k, g).  k * line_integral F {j} g)" "one_chain_typeII"]
           line_integral_distrib
           1
         by auto
     qed
   qed
   have integral_dis: "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = integral s (\<lambda>x. ?F_a' x) + integral s (\<lambda>x. - ?F_b' x)"
   proof -
     have "(?F_a' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeII. integral img ?F_a')) s"
     proof -
       have "(?F_a' has_integral integral (cubeImage twoCube) ?F_a') (cubeImage twoCube)"
         if "twoCube \<in> two_chain_typeII" for twoCube
         by (simp add: analytically_valid_imp_part_deriv_integrable_on T2.F_anal_valid has_integral_integrable_integral that) 
       then have  "\<And>u. u \<in> (cubeImage ` two_chain_typeII) \<Longrightarrow> (?F_a' has_integral integral u ?F_a') u"
         by auto
       then show ?thesis
       using gen_divisions(2) unfolding gen_division_def
       by (metis has_integral_Union)
     qed
     then have F_a'_integrable:
       "(?F_a' integrable_on s)" by auto
     have "(?F_b' has_integral (\<Sum>img\<in>cubeImage ` two_chain_typeI. integral img ?F_b')) s"
     proof -
       have "\<forall>twoCube \<in> two_chain_typeI. (?F_b' has_integral integral (cubeImage twoCube) ?F_b') (cubeImage twoCube)"
         by (simp add: analytically_valid_imp_part_deriv_integrable_on T1.F_anal_valid has_integral_integrable_integral) 
       then have "\<And>u. u \<in> (cubeImage ` two_chain_typeI) \<Longrightarrow> (?F_b' has_integral integral u ?F_b') u"
         by auto
       then show ?thesis
       using gen_divisions(1) unfolding gen_division_def
       by (metis has_integral_Union)
     qed
     then show ?thesis
       using F_a'_integrable Henstock_Kurzweil_Integration.integral_diff by auto
   qed
   show "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = one_chain_line_integral F {i, j} one_chain_typeI"
     using one_chain_j_integrals integral_dis line_integral_dist typeI_regions_integral typeII_regions_integral
     by auto
   show "integral s (\<lambda>x. ?F_a' x - ?F_b' x) = one_chain_line_integral F {i, j} one_chain_typeII"
     using one_chain_i_integrals integral_dis line_integral_dist typeI_regions_integral typeII_regions_integral
     by auto
 qed
 
 end
 end
diff --git a/thys/Green/Paths.thy b/thys/Green/Paths.thy
--- a/thys/Green/Paths.thy
+++ b/thys/Green/Paths.thy
@@ -1,3584 +1,3584 @@
 theory Paths
   imports Derivs General_Utils Integrals
 begin
 (*This has everything related to paths purely*)
 
 lemma reverse_subpaths_join:
   shows " subpath 1 (1 / 2) p +++ subpath (1 / 2) 0 p = reversepath p"
   using reversepath_subpath join_subpaths_middle pathfinish_subpath pathstart_subpath reversepath_joinpaths
   by (metis (no_types, lifting))
 
 
 (*Below F cannot be from 'a \<Rightarrow> 'b because the dot product won't work.
   We have that g returns 'a and then F takes the output of g, so F should start from 'a
   Then we have to compute the dot product of the vector b with both the derivative of g, and F.
   Since the derivative of g returns the same type as g, accordingly F should return the same type as g, i.e. 'a.
  *)
 definition line_integral:: "('a::euclidean_space \<Rightarrow> 'a::euclidean_space) \<Rightarrow> (('a) set) \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> real" where
 "line_integral F basis g \<equiv> integral {0 .. 1} (\<lambda>x. \<Sum>b\<in>basis. (F(g x) \<bullet> b) * (vector_derivative g (at x within {0..1}) \<bullet> b))"
 
 definition line_integral_exists where
   "line_integral_exists F basis \<gamma> \<equiv> (\<lambda>x. \<Sum>b\<in>basis. F(\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
 
 lemma line_integral_on_pair_straight_path:
   fixes F::"('a::euclidean_space) \<Rightarrow> 'a" and g :: "real \<Rightarrow> real" and \<gamma>
   assumes gamma_const: "\<forall>x. \<gamma>(x)\<bullet> i = a"
       and gamma_smooth: "\<forall>x \<in> {0 .. 1}. \<gamma> differentiable at x"          
   shows "(line_integral F {i} \<gamma>) = 0" "(line_integral_exists F {i} \<gamma>)"
 proof (simp add: line_integral_def)
   have *: "F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i) = 0"
     if "0 \<le> x \<and> x \<le> 1" for x
   proof -
     have "((\<lambda>x. \<gamma>(x)\<bullet> i) has_vector_derivative 0) (at x)"
       using vector_derivative_const_at[of "a" "x"] and gamma_const
       by auto
     then have "(vector_derivative \<gamma> (at x) \<bullet> i) = 0"
       using derivative_component_fun_component[ of "\<gamma>" "x" "i"]
         and gamma_smooth and that
       by (simp add: vector_derivative_at)
     then have "(vector_derivative \<gamma> (at x within {0 .. 1}) \<bullet> i) = 0"
       using has_vector_derivative_at_within vector_derivative_at_within_ivl that
       by (metis atLeastAtMost_iff gamma_smooth vector_derivative_works zero_less_one)
     then show ?thesis
       by auto
   qed
   then have "((\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) has_integral 0) {0..1}"
     using has_integral_is_0[of "{0 .. 1}" "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i))"]
     by auto
   then have "((\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) integrable_on {0..1})"
     by auto
   then show "line_integral_exists F {i} \<gamma>" by (auto simp add:line_integral_exists_def)
   show "integral {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) = 0"
     using * has_integral_is_0[of "{0 .. 1}" "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i))"]
     by auto
 qed
 
 lemma line_integral_on_pair_path_strong:
   fixes F::"('a::euclidean_space) \<Rightarrow> ('a)" and
     g::"real \<Rightarrow> 'a" and
     \<gamma>::"(real \<Rightarrow> 'a)" and
     i::'a 
   assumes i_norm_1: "norm i = 1" and
     g_orthogonal_to_i: "\<forall>x. g(x) \<bullet> i = 0" and  
     gamma_is_in_terms_of_i: "\<gamma> = (\<lambda>x. f(x) *\<^sub>R i + g(f(x)))" and
     gamma_smooth: "\<gamma> piecewise_C1_differentiable_on {0 .. 1}" and
     g_continuous_on_f: "continuous_on (f ` {0..1}) g" and
     path_start_le_path_end: "(pathstart \<gamma>) \<bullet> i \<le> (pathfinish \<gamma>) \<bullet> i" and
     field_i_comp_cont: "continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> i)"
   shows "line_integral F {i} \<gamma> 
          = integral (cbox ((pathstart \<gamma>) \<bullet> i) ((pathfinish \<gamma>) \<bullet> i)) (\<lambda>f_var. (F (f_var *\<^sub>R i + g(f_var)) \<bullet> i))"
     "line_integral_exists F {i} \<gamma>"
 proof (simp add: line_integral_def)
   obtain s where gamma_differentiable: "finite s" "(\<forall>x \<in> {0 .. 1} - s. \<gamma> differentiable at x)"
     using gamma_smooth
     by (auto simp add: C1_differentiable_on_eq piecewise_C1_differentiable_on_def)
   then have gamma_i_component_smooth: "\<forall>x \<in> {0 .. 1} - s. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
     by auto
   have field_cont_on_path: "continuous_on ((\<lambda>x. \<gamma> x \<bullet> i) ` {0..1}) (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
   proof - 
     have 0: "(\<lambda>x. \<gamma> x \<bullet> i) = f"
     proof 
       fix x
       show "\<gamma> x \<bullet> i = f x"
         using g_orthogonal_to_i i_norm_1
         by (simp only: gamma_is_in_terms_of_i  real_inner_class.inner_add_left g_orthogonal_to_i inner_scaleR_left inner_same_Basis norm_eq_1)
     qed
     show ?thesis 
       unfolding 0 
       apply (rule continuous_on_compose2 [of _ "(\<lambda>x. F(x)  \<bullet> i)" "f ` { 0..1}" "(\<lambda>x. x *\<^sub>R i + g x)"]
           field_i_comp_cont g_continuous_on_f field_i_comp_cont continuous_intros)+
       by (auto simp add: gamma_is_in_terms_of_i path_image_def)
   qed
   have path_start_le_path_end': "\<gamma> 0 \<bullet> i \<le> \<gamma> 1 \<bullet> i" using path_start_le_path_end by (auto simp add: pathstart_def pathfinish_def)
   have gamm_cont: "continuous_on {0..1} (\<lambda>a. \<gamma> a \<bullet> i)"
     apply(rule continuous_on_inner)
     using gamma_smooth 
      apply (simp add: piecewise_C1_differentiable_on_def)
     using continuous_on_const by auto
   then obtain c d where cd: "c \<le> d" "(\<lambda>a. \<gamma> a \<bullet> i) ` {0..1} = {c..d}"
     by (meson continuous_image_closed_interval zero_le_one)
   then have subset_cd: "(\<lambda>a. \<gamma> a \<bullet> i) ` {0..1} \<subseteq> {c..d}" by auto
   have field_cont_on_path_cd:
     "continuous_on {c..d} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
     using field_cont_on_path cd by auto
   have path_vector_deriv_line_integrals:
     "\<forall>x\<in>{0..1} - s. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative 
                                           vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x))
                                                   (at x)"
     using gamma_i_component_smooth and derivative_component_fun_component and
       vector_derivative_works
     by blast       
   then have "\<forall>x\<in>{0..1} - s. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative 
                                           vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within ({0..1})))
                                                   (at x within ({0..1}))"
     using has_vector_derivative_at_within vector_derivative_at_within_ivl
     by fastforce
   then have has_int:"((\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) *\<^sub>R (F ((\<gamma> x \<bullet> i) *\<^sub>R i + g (\<gamma> x \<bullet> i)) \<bullet> i)) has_integral
            integral {\<gamma> 0 \<bullet> i..\<gamma> 1 \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)) {0..1}"
     using has_integral_substitution_strong[OF gamma_differentiable(1) rel_simps(44)
         path_start_le_path_end' subset_cd field_cont_on_path_cd gamm_cont,
         of "(\<lambda>x. vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within ({0..1})))"]
       gamma_is_in_terms_of_i
     by (auto simp only: has_field_derivative_iff_has_vector_derivative)
   then have has_int':"((\<lambda>x. (F(\<gamma>(x)) \<bullet> i)*(vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within ({0..1})))) has_integral
            integral {((pathstart \<gamma>) \<bullet> i)..((pathfinish \<gamma>) \<bullet> i)} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)) {0..1}"
     using  gamma_is_in_terms_of_i i_norm_1
     apply (auto simp add: pathstart_def pathfinish_def)
     apply (simp only:  real_inner_class.inner_add_left inner_not_same_Basis g_orthogonal_to_i inner_scaleR_left norm_eq_1)
     by (auto simp add: algebra_simps)
   have substitute:
     "integral ({((pathstart \<gamma>) \<bullet> i)..((pathfinish \<gamma>) \<bullet> i)}) (\<lambda>f_var. (F (f_var *\<^sub>R i + g(f_var)) \<bullet> i)) 
                  = integral ({0..1}) (\<lambda>x. (F(\<gamma>(x)) \<bullet> i)*(vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within ({0..1}))))"
     using gamma_is_in_terms_of_i integral_unique[OF has_int] i_norm_1
     apply (auto simp add: pathstart_def pathfinish_def)
     apply (simp only:  real_inner_class.inner_add_left inner_not_same_Basis g_orthogonal_to_i inner_scaleR_left norm_eq_1)
     by (auto simp add: algebra_simps)
   have comp_in_eq_comp_out: "\<forall>x \<in> {0..1} - s.
             (vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1}))
                 = (vector_derivative \<gamma> (at x within {0..1})) \<bullet> i"
   proof
     fix x:: real
     assume ass:"x \<in> {0..1} -s"
     then have x_bounds:"x \<in> {0..1}" by auto
     have "\<gamma> differentiable at x" using ass gamma_differentiable by auto
     then have dotprod_in_is_out:
       "vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x)
                          = (vector_derivative \<gamma> (at x)) \<bullet> i"
       using derivative_component_fun_component 
       by force
     then have 0: "(vector_derivative \<gamma> (at x)) \<bullet> i = (vector_derivative \<gamma> (at x within {0..1})) \<bullet> i"
     proof -
       have "(\<gamma> has_vector_derivative vector_derivative \<gamma> (at x)) (at x)"
         using \<open>\<gamma> differentiable at x\<close> vector_derivative_works by blast
       moreover have "0 \<le> x \<and> x \<le> 1"
         using x_bounds by presburger
       ultimately show ?thesis
         by (simp add: vector_derivative_at_within_ivl)
     qed
     have 1: "vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x) = vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1})"
       using path_vector_deriv_line_integrals and vector_derivative_at_within_ivl and
         x_bounds
       by (metis ass atLeastAtMost_iff zero_less_one)
     show "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) = vector_derivative \<gamma> (at x within {0..1}) \<bullet> i"
       using 0 and 1 and dotprod_in_is_out
       by auto
   qed
   show "integral {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) =
                    integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
     using substitute and comp_in_eq_comp_out and negligible_finite
       Henstock_Kurzweil_Integration.integral_spike
       [of "s" "{0..1}" "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i))"
         "(\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"]
     by (metis (no_types, lifting) gamma_differentiable(1))
   have "((\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) has_integral
                    integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)) {0..1}"
     using has_int' and comp_in_eq_comp_out and negligible_finite
       Henstock_Kurzweil_Integration.has_integral_spike
       [of "s" "{0..1}" "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i))"
         "(\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"
         "integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"]
     by (metis (no_types, lifting) gamma_differentiable(1))
   then have "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) integrable_on {0..1}"
     using integrable_on_def by auto
   then show "line_integral_exists F {i} \<gamma>"
     by (auto simp add: line_integral_exists_def)
 qed
 
 lemma line_integral_on_pair_path:
   fixes F::"('a::euclidean_space) \<Rightarrow> ('a)" and
     g::"real \<Rightarrow> 'a" and
     \<gamma>::"(real \<Rightarrow> 'a)" and
     i::'a 
   assumes i_norm_1: "norm i = 1" and
     g_orthogonal_to_i: "\<forall>x. g(x) \<bullet> i = 0" and  
     gamma_is_in_terms_of_i: "\<gamma> = (\<lambda>x. f(x) *\<^sub>R i + g(f(x)))" and
     gamma_smooth: "\<gamma> C1_differentiable_on {0 .. 1}" and
     g_continuous_on_f: "continuous_on (f ` {0..1}) g" and
     path_start_le_path_end: "(pathstart \<gamma>) \<bullet> i \<le> (pathfinish \<gamma>) \<bullet> i" and
     field_i_comp_cont: "continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> i)"
   shows "(line_integral F {i} \<gamma>) 
                  = integral (cbox ((pathstart \<gamma>) \<bullet> i) ((pathfinish \<gamma>) \<bullet> i)) (\<lambda>f_var. (F (f_var *\<^sub>R i + g(f_var)) \<bullet> i))"
 proof (simp add: line_integral_def)
   have gamma_differentiable: "\<forall>x \<in> {0 .. 1}. \<gamma> differentiable at x"
     using gamma_smooth  C1_differentiable_on_eq by blast
   then have gamma_i_component_smooth:
     "\<forall>x \<in> {0 .. 1}. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
     by auto
   have vec_deriv_i_comp_cont:
     "continuous_on {0..1} (\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"
   proof -
     have "continuous_on {0..1} (\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}))"
       using gamma_smooth C1_differentiable_on_eq
       by (smt C1_differentiable_on_def atLeastAtMost_iff continuous_on_eq vector_derivative_at_within_ivl)
     then have deriv_comp_cont:
       "continuous_on {0..1} (\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x) (at x within {0..1}) \<bullet> i)"
       by (simp add: continuous_intros)
     show ?thesis
       using derivative_component_fun_component_at_within[OF gamma_differentiable, of "i"]                    
         continuous_on_eq[OF deriv_comp_cont, of "(\<lambda>x. vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"]
       by fastforce
   qed
   have field_cont_on_path:
     "continuous_on ((\<lambda>x. \<gamma> x \<bullet> i) ` {0..1}) (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
   proof - 
     have 0: "(\<lambda>x. \<gamma> x \<bullet> i) = f"
     proof 
       fix x
       show "\<gamma> x \<bullet> i = f x"
         using g_orthogonal_to_i i_norm_1
         by (simp only: gamma_is_in_terms_of_i  real_inner_class.inner_add_left g_orthogonal_to_i inner_scaleR_left inner_same_Basis norm_eq_1)
     qed
     show ?thesis 
       unfolding 0 
       apply (rule continuous_on_compose2 [of _ "(\<lambda>x. F(x)  \<bullet> i)" "f ` { 0..1}" "(\<lambda>x. x *\<^sub>R i + g x)"]
           field_i_comp_cont g_continuous_on_f field_i_comp_cont continuous_intros)+
       by (auto simp add: gamma_is_in_terms_of_i path_image_def)
   qed
   have path_vector_deriv_line_integrals:
     "\<forall>x\<in>{0..1}. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative 
                                           vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x))
                                                   (at x)"
     using gamma_i_component_smooth and derivative_component_fun_component and
       vector_derivative_works
     by blast       
   then have "\<forall>x\<in>{0..1}. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative 
                                           vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))
                                                   (at x within {0..1})"
     using has_vector_derivative_at_within vector_derivative_at_within_ivl
     by fastforce
   then have substitute:
     "integral (cbox ((pathstart \<gamma>) \<bullet> i) ((pathfinish \<gamma>) \<bullet> i)) (\<lambda>f_var. (F (f_var *\<^sub>R i + g(f_var)) \<bullet> i)) 
                  = integral (cbox 0 1) (\<lambda>x. (F(\<gamma>(x)) \<bullet> i)*(vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1})))"
     using gauge_integral_by_substitution
       [of "0" "1" "(\<lambda>x. (\<gamma> x) \<bullet> i)"
         "(\<lambda>x. vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1}))"
         "(\<lambda>f_var. (F (f_var *\<^sub>R i + g(f_var)) \<bullet> i))"] and
       path_start_le_path_end and vec_deriv_i_comp_cont and field_cont_on_path and 
       gamma_is_in_terms_of_i i_norm_1
     apply (auto simp add: pathstart_def pathfinish_def)
     apply (simp only:  real_inner_class.inner_add_left inner_not_same_Basis g_orthogonal_to_i inner_scaleR_left norm_eq_1)
     by (auto)
       (*integration by substitution*)
   have comp_in_eq_comp_out: "\<forall>x \<in> {0..1}.
             (vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1}))
                 = (vector_derivative \<gamma> (at x within {0..1})) \<bullet> i"
   proof
     fix x:: real
     assume x_bounds: "x \<in> {0..1}"
     then have "\<gamma> differentiable at x" using gamma_differentiable by auto
     then have dotprod_in_is_out:
       "vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x)
                          = (vector_derivative \<gamma> (at x)) \<bullet> i"
       using derivative_component_fun_component 
       by force
     then have 0: "(vector_derivative \<gamma> (at x)) \<bullet> i
                          = (vector_derivative \<gamma> (at x within {0..1})) \<bullet> i"
       using has_vector_derivative_at_within and x_bounds and vector_derivative_at_within_ivl
       by (smt atLeastAtMost_iff gamma_differentiable inner_commute vector_derivative_works)
     have 1: "vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x) = vector_derivative (\<lambda>x. \<gamma>(x) \<bullet> i) (at x within {0..1})"
       using path_vector_deriv_line_integrals and vector_derivative_at_within_ivl and
         x_bounds
       by fastforce
     show "vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}) = vector_derivative \<gamma> (at x within {0..1}) \<bullet> i"
       using 0 and 1 and dotprod_in_is_out
       by auto
   qed
   show "integral {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i)) =
                    integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i)"
     using substitute and comp_in_eq_comp_out and 
       Henstock_Kurzweil_Integration.integral_cong
       [of "{0..1}" "(\<lambda>x. F (\<gamma> x) \<bullet> i * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> i))"
         "(\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"]
     by auto
 qed
 
 lemma content_box_cases:
   "content (box a b) = (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then prod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
   by (simp add: measure_lborel_box_eq inner_diff)
 
 lemma content_box_cbox:
   shows "content (box a b) = content (cbox a b)"
   by (simp add: content_box_cases content_cbox_cases)
 
 lemma content_eq_0: "content (box a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
   by (auto simp: content_box_cases not_le intro: less_imp_le antisym eq_refl)
 
 lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
   by (auto simp add: content_cbox_cases less_le prod_nonneg)
 
 lemma content_lt_nz: "0 < content (box a b) \<longleftrightarrow> content (box a b) \<noteq> 0"
   using Paths.content_eq_0 zero_less_measure_iff by blast
 
 lemma content_subset: "cbox a b \<subseteq> box c d \<Longrightarrow> content (cbox a b) \<le> content (box c d)"
   unfolding measure_def
   by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq emeasure_lborel_box_eq)
 
 lemma sum_content_null:
   assumes "content (box a b) = 0"
     and "p tagged_division_of (box a b)"
   shows "sum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
 proof (rule sum.neutral, rule)
   fix y
   assume y: "y \<in> p"
   obtain x k where xk: "y = (x, k)"
     using surj_pair[of y] by blast
   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   from this(2) obtain c d where k: "k = cbox c d" by blast
   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
     unfolding xk by auto
   also have "\<dots> = 0"
     using content_subset[OF assm(1)[unfolded k]] content_pos_le[of "cbox c d"]
     unfolding assms(1) k
     by auto
   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
 qed
 
 
 lemma has_integral_null [intro]: "content(box a b) = 0 \<Longrightarrow> (f has_integral 0) (box a b)"
   by (simp add: content_box_cbox content_eq_0_interior)
 
 lemma line_integral_distrib:
   assumes "line_integral_exists f basis g1"
     "line_integral_exists f basis g2"
     "valid_path g1" "valid_path g2"
   shows "line_integral f basis (g1 +++ g2) = line_integral f basis g1 + line_integral f basis g2"
     "line_integral_exists f basis (g1 +++ g2)"
 proof -
   obtain s1 s2
     where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
       and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
     using assms
     by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   obtain i1 i2
     where 1: "((\<lambda>x. \<Sum>b\<in>basis. f (g1 x) \<bullet> b * (vector_derivative g1 (at x within {0..1}) \<bullet> b)) has_integral i1) {0..1}"
       and 2: "((\<lambda>x. \<Sum>b\<in>basis. f (g2 x) \<bullet> b * (vector_derivative g2 (at x within {0..1}) \<bullet> b)) has_integral i2) {0..1}"
     using assms
     by (auto simp: line_integral_exists_def)
   have i1: "((\<lambda>x. 2 * (\<Sum>b\<in>basis. f (g1 (2 * x)) \<bullet> b * (vector_derivative g1 (at (2 * x) within {0..1}) \<bullet> b))) has_integral i1) {0..1/2}"
    and i2: "((\<lambda>x. 2 * (\<Sum>b\<in>basis. f (g2 (2 * x - 1)) \<bullet> b * (vector_derivative g2 (at ((2 * x) - 1) within {0..1}) \<bullet> b))) has_integral i2) {1/2..1}"
     using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
       has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
     by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
   have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z within {0..1}) =
             2 *\<^sub>R vector_derivative g1 (at (z*2) within {0..1})" for z
   proof -
     have i:"\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
               vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2))" for z
     proof-
       have g1_at_z:"\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
                          ((\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) has_vector_derivative
                          2 *\<^sub>R vector_derivative g1 (at (z*2))) (at z)" for z
         apply (rule has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g1(2*x))"])
           apply (simp_all add: dist_real_def abs_if split: if_split_asm)
         apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
          apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
         using s1
         apply (auto simp: algebra_simps vector_derivative_works)
         done
       assume ass: "0 \<le> z" "z*2 < 1" "z*2 \<notin> s1"
       then have z_ge: "z\<le> 1" by auto
       show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2))"
         using Derivative.vector_derivative_at_within_ivl[OF g1_at_z[OF ass] ass(1) z_ge]
         by auto
     qed
     assume ass: "0 \<le> z" "z*2 < 1" "z*2 \<notin> s1"
     then have "(g1 has_vector_derivative ((vector_derivative g1 (at (z*2))))) (at (z*2))"
       using s1 by (auto simp: algebra_simps vector_derivative_works)
     then have ii: "(vector_derivative g1 (at (z*2) within {0..1})) = (vector_derivative g1 (at (z*2)))"
       using Derivative.vector_derivative_at_within_ivl ass
       by force
     show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2) within {0..1})"
       using i[OF ass] ii
       by auto
   qed
   have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z within {0..1}) =
             2 *\<^sub>R vector_derivative g2 (at (z*2 - 1) within {0..1})" for z       proof -
     have i:"\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
               vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1})
                    = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1))" for z
     proof-
       have g2_at_z:"\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
                       ((\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) has_vector_derivative 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))) (at z)" for z
         apply (rule has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g2 (2*x - 1))"])
           apply (simp_all add: dist_real_def abs_if split: if_split_asm)
         apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
          apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
         using s2
         apply (auto simp: algebra_simps vector_derivative_works)
         done
       assume ass: "1 < z*2" "z \<le> 1" "z*2 - 1 \<notin> s2"
       then have z_le: "z\<le> 1" by auto
       have z_ge: "0 \<le> z" using ass by auto
       show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1})
                                 = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1))"
         using Derivative.vector_derivative_at_within_ivl[OF g2_at_z[OF ass] z_ge z_le]
         by auto
     qed
     assume ass: "1 < z*2" "z \<le> 1" "z*2 - 1 \<notin> s2"
     then have "(g2 has_vector_derivative ((vector_derivative g2 (at (z*2 - 1))))) (at (z*2 - 1))"
       using s2 by (auto simp: algebra_simps vector_derivative_works)
     then have ii: "(vector_derivative g2 (at (z*2 - 1) within {0..1})) = (vector_derivative g2 (at (z*2 - 1)))"
       using Derivative.vector_derivative_at_within_ivl ass
       by force
     show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1) within {0..1})"
       using i[OF ass] ii
       by auto
   qed
   have lem1: "((\<lambda>x. \<Sum>b\<in>basis. f ((g1+++g2) x) \<bullet> b * (vector_derivative (g1+++g2) (at x within {0..1}) \<bullet> b)) has_integral i1) {0..1/2}"
     apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
     using s1
      apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
     apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
     by (simp add: sum_distrib_left)
   moreover have lem2: "((\<lambda>x. \<Sum>b\<in>basis. f ((g1+++g2) x) \<bullet> b * (vector_derivative (g1+++g2) (at x within {0..1}) \<bullet> b)) has_integral i2) {1/2..1}"
     apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
     using s2
      apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
     apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
     by (simp add: sum_distrib_left)
   ultimately
   show "line_integral f basis (g1 +++ g2) = line_integral f basis g1 + line_integral f basis g2"
     apply (simp add: line_integral_def)
     apply (rule integral_unique [OF has_integral_combine [where c = "1/2"]])
     using 1 2 integral_unique apply auto
     done
   show "line_integral_exists f basis (g1 +++ g2)"
   proof (simp add: line_integral_exists_def integrable_on_def)
     have "(1::real) \<le> 1 * 2 \<and> (0::real) \<le> 1 / 2"
       by simp
     then show "\<exists>r. ((\<lambda>r. \<Sum>a\<in>basis. f ((g1 +++ g2) r) \<bullet> a * (vector_derivative (g1 +++ g2) (at r within {0..1}) \<bullet> a)) has_integral r) {0..1}"
     using has_integral_combine [where c = "1/2"] 1 2 divide_le_eq_numeral1(1) lem1 lem2 by blast
   qed
 qed
 
 lemma line_integral_exists_joinD1:
   assumes "line_integral_exists f basis (g1 +++ g2)" "valid_path g1"
   shows "line_integral_exists f basis g1"
 proof -
   obtain s1
     where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   have "(\<lambda>x. \<Sum>b\<in>basis. f ((g1 +++ g2) (x/2)) \<bullet> b * (vector_derivative (g1 +++ g2) (at (x/2) within {0..1}) \<bullet> b)) integrable_on {0..1}"
     using assms
     apply (auto simp: line_integral_exists_def)
     apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
      apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
     done
   then have *:"(\<lambda>x. \<Sum>b\<in>basis. ((f ((g1 +++ g2) (x/2)) \<bullet> b) / 2)* (vector_derivative (g1 +++ g2) (at (x/2) within {0..1}) \<bullet> b)) integrable_on {0..1}"
     by (auto simp: Groups_Big.sum_distrib_left dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
   have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z within {0..1}) =
             2 *\<^sub>R vector_derivative g1 (at (z*2) within {0..1})" for z
   proof -
     have i:"\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
               vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2))" for z
     proof-
       have g1_at_z:"\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
                          ((\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) has_vector_derivative
                          2 *\<^sub>R vector_derivative g1 (at (z*2))) (at z)" for z
         apply (rule has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g1(2*x))"])
           apply (simp_all add: dist_real_def abs_if split: if_split_asm)
         apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
          apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
         using s1
         apply (auto simp: algebra_simps vector_derivative_works)
         done
       assume ass: "0 \<le> z" "z*2 < 1" "z*2 \<notin> s1"
       then have z_ge: "z\<le> 1" by auto
       show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2))"
         using Derivative.vector_derivative_at_within_ivl[OF g1_at_z[OF ass] ass(1) z_ge]
         by auto
     qed
     assume ass: "0 \<le> z" "z*2 < 1" "z*2 \<notin> s1"
     then have "(g1 has_vector_derivative ((vector_derivative g1 (at (z*2))))) (at (z*2))"
       using s1 by (auto simp: algebra_simps vector_derivative_works)
     then have ii: "(vector_derivative g1 (at (z*2) within {0..1})) = (vector_derivative g1 (at (z*2)))"
       using Derivative.vector_derivative_at_within_ivl ass by force
     show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g1 (at (z * 2) within {0..1})"
       using i[OF ass] ii by auto
   qed
   show ?thesis
     using s1
     apply (auto simp: line_integral_exists_def)
     apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
      apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
     done
 qed
 
 lemma line_integral_exists_joinD2:
   assumes "line_integral_exists f basis (g1 +++ g2)" "valid_path g2"
   shows "line_integral_exists f basis g2"
 proof -
   obtain s2
     where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
   have "(\<lambda>x. \<Sum>b\<in>basis. f ((g1 +++ g2) (x/2 + 1/2)) \<bullet> b * (vector_derivative (g1 +++ g2) (at (x/2 + 1/2) within {0..1}) \<bullet> b)) integrable_on {0..1}"
     using assms
     apply (auto simp: line_integral_exists_def)
     apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
     apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
     apply (simp add: image_affinity_atLeastAtMost_diff)
     done
   then have *:"(\<lambda>x. \<Sum>b\<in>basis. ((f ((g1 +++ g2) (x/2 + 1/2)) \<bullet> b) / 2)* (vector_derivative (g1 +++ g2) (at (x/2 + 1/2) within {0..1}) \<bullet> b)) integrable_on {0..1}"
     by (auto simp: Groups_Big.sum_distrib_left dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
   have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z within {0..1}) =
             2 *\<^sub>R vector_derivative g2 (at (z*2 - 1) within {0..1})" for z       proof -
     have i:"\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
               vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1})
                    = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1))" for z
     proof-
       have g2_at_z:"\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
                       ((\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) has_vector_derivative 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))) (at z)" for z
         apply (rule has_vector_derivative_transform_at [of "\<bar>z - 1/2\<bar>" _ "(\<lambda>x. g2 (2*x - 1))"])
           apply (simp_all add: dist_real_def abs_if split: if_split_asm)
         apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
          apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
         using s2
         apply (auto simp: algebra_simps vector_derivative_works)
         done
       assume ass: "1 < z*2" "z \<le> 1" "z*2 - 1 \<notin> s2"
       then have z_le: "z\<le> 1" by auto
       have z_ge: "0 \<le> z" using ass by auto
       show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1})
                                 = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1))"
         using Derivative.vector_derivative_at_within_ivl[OF g2_at_z[OF ass] z_ge z_le]
         by auto
     qed
     assume ass: "1 < z*2" "z \<le> 1" "z*2 - 1 \<notin> s2"
     then have "(g2 has_vector_derivative ((vector_derivative g2 (at (z*2 - 1))))) (at (z*2 - 1))"
       using s2 by (auto simp: algebra_simps vector_derivative_works)
     then have ii: "(vector_derivative g2 (at (z*2 - 1) within {0..1})) = (vector_derivative g2 (at (z*2 - 1)))"
       using Derivative.vector_derivative_at_within_ivl ass
       by force
     show "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at z within {0..1}) = 2 *\<^sub>R vector_derivative g2 (at (z * 2 - 1) within {0..1})"
       using i[OF ass] ii
       by auto
   qed
   show ?thesis
     using s2
     apply (auto simp: line_integral_exists_def)
     apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
      apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
     done
 qed
 
 lemma has_line_integral_on_reverse_path:
   assumes g: "valid_path g" and int:
     "((\<lambda>x. \<Sum>b\<in>basis. F (g x) \<bullet> b * (vector_derivative g (at x within {0..1}) \<bullet> b)) has_integral c){0..1}"
   shows  "((\<lambda>x. \<Sum>b\<in>basis. F ((reversepath g) x) \<bullet> b * (vector_derivative (reversepath g) (at x within {0..1}) \<bullet> b)) has_integral -c){0..1}"
 proof -
   { fix s x
     assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> (-) 1 ` s" "0 \<le> x" "x \<le> 1"
     have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
                 - vector_derivative g (at (1 - x) within {0..1})"
     proof -
       obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
         using xs
         by (force simp: has_vector_derivative_def C1_differentiable_on_def)
       have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
         apply (rule vector_diff_chain_within)
          apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
         apply (rule has_vector_derivative_at_within [OF f'])
         done
       then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
         by (simp add: o_def)
       show ?thesis
         using xs
         by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
     qed
   } note * = this
   obtain S where "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
     using g
     by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
   then show ?thesis 
     using has_integral_affinity01 [OF int, where m= "-1" and c=1]
     unfolding reversepath_def
     by (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite) (auto simp: * has_integral_neg Groups_Big.sum_negf)
 qed
 
 lemma line_integral_on_reverse_path:
   assumes "valid_path \<gamma>" "line_integral_exists F basis \<gamma>"
   shows "line_integral F basis \<gamma> = - (line_integral F basis (reversepath \<gamma>))"
         "line_integral_exists F basis (reversepath \<gamma>)"
 proof -
   obtain i where
     0: "((\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) has_integral i){0..1}"
     using assms unfolding integrable_on_def line_integral_exists_def by auto
   then have 1: "((\<lambda>x. \<Sum>b\<in>basis. F ((reversepath \<gamma>) x) \<bullet> b * (vector_derivative (reversepath \<gamma>) (at x within {0..1}) \<bullet> b)) has_integral -i){0..1}"
     using has_line_integral_on_reverse_path assms
     by auto
   then have rev_line_integral:"line_integral F basis (reversepath \<gamma>) = -i"
     using line_integral_def Henstock_Kurzweil_Integration.integral_unique
     by (metis (no_types))
   have line_integral: "line_integral F basis \<gamma> = i"
     using line_integral_def 0 Henstock_Kurzweil_Integration.integral_unique
     by blast
   show "line_integral F basis \<gamma> = - (line_integral F basis (reversepath \<gamma>))"
     using line_integral rev_line_integral
     by auto
   show "line_integral_exists F basis (reversepath \<gamma>)"
     using 1 line_integral_exists_def
     by auto
 qed
 
 lemma line_integral_exists_on_degenerate_path:
   assumes "finite basis"
   shows "line_integral_exists F basis (\<lambda>x. c)"
 proof-
   have every_component_integrable:
     "\<forall>b\<in>basis. (\<lambda>x. F ((\<lambda>x. c) x) \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
   proof
     fix b
     assume b_in_basis: "b \<in> basis"
     have cont_field_zero_one: "continuous_on {0..1} (\<lambda>x. F ((\<lambda>x. c) x) \<bullet> b)"
       using continuous_on_const by fastforce
     have cont_path_zero_one:
       "continuous_on {0..1} (\<lambda>x. (vector_derivative (\<lambda>x. c) (at x within {0..1})) \<bullet> b)"
     proof -
       have "((vector_derivative (\<lambda>x. c) (at x within {0..1})) \<bullet> b) = 0" if "x \<in> {0..1}" for x
       proof -
         have "vector_derivative (\<lambda>x. c) (at x within {0..1}) = 0"
           using that gamma_deriv_at_within[of "0" "1"] differentiable_const vector_derivative_const_at
           by fastforce
         then show "vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b = 0"
           by auto
       qed
       then show "continuous_on {0..1} (\<lambda>x. (vector_derivative (\<lambda>x. c) (at x within {0..1})) \<bullet> b)"
         using continuous_on_const[of "{0..1}" "0"] continuous_on_eq[of "{0..1}" "\<lambda>x. 0" "(\<lambda>x. (vector_derivative (\<lambda>x. c) (at x within {0..1})) \<bullet> b)"]
         by auto
     qed
     show "(\<lambda>x. F (c) \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
       using cont_field_zero_one cont_path_zero_one continuous_on_mult integrable_continuous_real
       by blast
   qed
   have integrable_sum': "\<And>t f s. finite t \<Longrightarrow>
                 \<forall>a\<in>t. f a integrable_on s \<Longrightarrow> (\<lambda>x. \<Sum>a\<in>t. f a x) integrable_on s"
     using integrable_sum by metis
   have field_integrable_on_basis:
     "(\<lambda>x. \<Sum>b\<in>basis. F (c) \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
     using integrable_sum'[OF assms(1) every_component_integrable]
     by auto
   then show ?thesis
     using line_integral_exists_def by auto
 qed
 
 lemma degenerate_path_is_valid_path: "valid_path (\<lambda>x. c)"
   by (auto simp add: valid_path_def piecewise_C1_differentiable_on_def continuous_on_const)
 
 lemma line_integral_degenerate_path:
   assumes "finite basis"
   shows "line_integral F basis (\<lambda>x. c) = 0"
 proof (simp add: line_integral_def)
   have "((vector_derivative (\<lambda>x. c) (at x within {0..1})) \<bullet> b) = 0" if "x \<in> {0..1}" for x b
   proof -
     have "vector_derivative (\<lambda>x. c) (at x within {0..1}) = 0"
       using that gamma_deriv_at_within[of "0" "1"] differentiable_const vector_derivative_const_at
       by fastforce
     then show "vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b = 0"
       by auto
   qed
   then have 0: "\<And>x. x \<in> {0..1} \<Longrightarrow> (\<lambda>x. \<Sum>b\<in>basis. F c \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) x = (\<lambda>x. 0) x"
     by auto
   then show "integral {0..1} (\<lambda>x. \<Sum>b\<in>basis. F c \<bullet> b * (vector_derivative (\<lambda>x. c) (at x within {0..1}) \<bullet> b)) = 0"
     using integral_cong[of "{0..1}", OF 0] integral_0 by auto
 qed
 
 definition point_path where
     "point_path \<gamma> \<equiv> \<exists>c. \<gamma> = (\<lambda>x. c)"
 
 lemma line_integral_point_path:
   assumes "point_path \<gamma>"
   assumes "finite basis"
   shows "line_integral F basis \<gamma> = 0"
   using assms(1) point_path_def line_integral_degenerate_path[OF assms(2)]
   by force
 
 lemma line_integral_exists_point_path:
   assumes "finite basis" "point_path \<gamma>"
   shows "line_integral_exists F basis \<gamma>"
   using assms
   apply(simp add: point_path_def)
   using line_integral_exists_on_degenerate_path by auto
 
 lemma line_integral_exists_subpath:
   assumes f: "line_integral_exists f basis g" and g: "valid_path g"
     and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
   shows "(line_integral_exists f basis (subpath u v g))"
 proof (cases "v=u")
   case tr: True
   have zero: "(\<Sum>b\<in>basis. f (g u) \<bullet> b * (vector_derivative (\<lambda>x. g u) (at x within {0..1}) \<bullet> b)) = 0" if "x \<in> {0..1}" for x::real
   proof -
     have "(vector_derivative (\<lambda>x. g u) (at x within {0..1})) = 0"
       using Deriv.has_vector_derivative_const that Derivative.vector_derivative_at_within_ivl
       by fastforce
     then show "(\<Sum>b\<in>basis. f (g u) \<bullet> b * (vector_derivative (\<lambda>x. g u) (at x within {0..1}) \<bullet> b)) = 0"
       by auto
   qed
   then have "((\<lambda>x. \<Sum>b\<in>basis. f (g u)  \<bullet> b * (vector_derivative (\<lambda>x. g u) (at x within {0..1})  \<bullet> b)) has_integral 0) {0..1}"
     by (meson has_integral_is_0)
   then show ?thesis
     using f tr by (auto simp add: line_integral_def line_integral_exists_def subpath_def)
 next
   case False
   obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
     using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
   have *: "((\<lambda>x. \<Sum>b\<in>basis. f (g ((v - u) * x + u))  \<bullet> b * (vector_derivative g (at ((v - u) * x + u) within {0..1})  \<bullet> b))
             has_integral (1 / (v - u)) * integral {u..v} (\<lambda>x. \<Sum>b\<in>basis. f (g (x))  \<bullet> b * (vector_derivative g (at x within {0..1})  \<bullet> b)))
            {0..1}"
     using f uv
     apply (simp add: line_integral_exists_def subpath_def)
     apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
      apply (simp_all add: has_integral_integral)
     apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
      apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
     apply (simp add: divide_simps False)
     done
   have vd:"\<And>x. x \<in> {0..1} \<Longrightarrow>
            x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
            vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x within {0..1}) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u) within {0..1})"
     using test2[OF s fs uv]
     by auto
   have arg:"\<And>x. (\<Sum>n\<in>basis. (v - u) * (f (g ((v - u) * x + u)) \<bullet> n) * (vector_derivative g (at ((v - u) * x + u) within {0..1}) \<bullet> n))
               =    (\<Sum>b\<in>basis. f (g ((v - u) * x + u)) \<bullet> b * (v - u) * (vector_derivative g (at ((v - u) * x + u) within {0..1}) \<bullet> b))"
     by (simp add: mult.commute)
   have"((\<lambda>x. \<Sum>b\<in>basis. f (g ((v - u) * x + u))  \<bullet> b * (vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x within {0..1})  \<bullet> b)) has_integral
           (integral {u..v} (\<lambda>x. \<Sum>b\<in>basis. f (g (x))  \<bullet> b * (vector_derivative g (at x within {0..1}) \<bullet> b)))) {0..1}"
     apply (cut_tac Henstock_Kurzweil_Integration.has_integral_mult_right [OF *, where c = "v-u"])
     using fs assms
     apply (simp add: False subpath_def line_integral_exists_def)
     apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
       apply (auto simp: inj_on_def False vd finite_vimageI scaleR_conv_of_real Groups_Big.sum_distrib_left
         mult.assoc[symmetric] arg)
     done
   then show "(line_integral_exists f basis (subpath u v g))"
     by(auto simp add: line_integral_exists_def subpath_def integrable_on_def)
 qed
 
 (*This should have everything that has to do with onecubes*)
 
    type_synonym path = "real \<Rightarrow> (real * real)"
    type_synonym one_cube = "(real \<Rightarrow> (real * real))"
    type_synonym one_chain = "(int * path) set"
    type_synonym two_cube = "(real * real) \<Rightarrow> (real * real)"
    type_synonym two_chain = "two_cube set"
 
 
 definition one_chain_line_integral :: "((real * real) \<Rightarrow> (real * real)) => ((real*real) set) \<Rightarrow> one_chain \<Rightarrow> real" where
     "one_chain_line_integral F b C \<equiv> (\<Sum>(k,g)\<in>C. k * (line_integral F b g))"
 
 definition boundary_chain where
     "boundary_chain s \<equiv> (\<forall>(k, \<gamma>) \<in> s. k = 1 \<or> k = -1)"
 
 fun coeff_cube_to_path::"(int * one_cube) \<Rightarrow> path" 
   where "coeff_cube_to_path (k, \<gamma>) = (if k = 1 then \<gamma> else (reversepath \<gamma>))"
 
 fun rec_join :: "(int*path) list \<Rightarrow> path" where   
   "rec_join [] = (\<lambda>x.0)" |
   "rec_join [oneC] = coeff_cube_to_path oneC" |
   "rec_join (oneC # xs) = coeff_cube_to_path oneC +++ (rec_join xs)"
 
 fun valid_chain_list where
   "valid_chain_list [] = True" |
   "valid_chain_list [oneC] = True" |
   "valid_chain_list (oneC#l) = (pathfinish (coeff_cube_to_path (oneC)) = pathstart (rec_join l) \<and> valid_chain_list l)"
 
 lemma joined_is_valid:
   assumes boundary_chain: "boundary_chain (set l)" and
     valid_path: "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>" and
     valid_chain_list_ass: "valid_chain_list l"
   shows "valid_path (rec_join l)"
   using assms
 proof(induction l)
   case Nil
   then show ?case
     using C1_differentiable_imp_piecewise[OF C1_differentiable_on_const[of "0" "{0..1}"]]
     by (auto simp add: valid_path_def)
 next
   case (Cons a l)
   have *: "valid_path (rec_join ((k::int, \<gamma>) # l))"
     if "boundary_chain (set (l))"
       "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set l \<Longrightarrow> valid_path \<gamma>')"
       "valid_chain_list l"
       "valid_path (rec_join l) "
       "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((k, \<gamma>) # l) \<Longrightarrow> valid_path \<gamma>')"
       "valid_chain_list ((k, \<gamma>) # l)"
       "boundary_chain (set ((k,\<gamma>) # l))" for k \<gamma> l
   proof (cases "l = []")
     case True
     with that show "valid_path (rec_join ((k, \<gamma>) # l))"
       by auto
   next
     case False
     then obtain h l' where l_nempty: "l = h#l'"
       by (meson rec_join.elims)
     then show "valid_path (rec_join ((k, \<gamma>) # l))"
     proof (simp, intro conjI impI)
       assume k_eq_1: "k = 1"
       have 0:"valid_path \<gamma>"
         using that by auto
       have 1:"pathfinish \<gamma> = pathstart (rec_join (h#l'))"
         using that(6) k_eq_1 l_nempty by auto
       show "valid_path (\<gamma> +++ rec_join (h#l'))"
         using 0 1 valid_path_join that(4) l_nempty by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg_1: "k = -1"
         using that(7)
         by (auto simp add: boundary_chain_def)
       have "valid_path \<gamma>"
         using that by auto
       then have 0: "valid_path (reversepath \<gamma>)"
-        using Cauchy_Integral_Theorem.valid_path_imp_reverse
+        using valid_path_imp_reverse
         by auto
       have 1: "pathfinish (reversepath \<gamma>) = pathstart (rec_join (h#l'))"
         using that(6) k_eq_neg_1 l_nempty by auto
       show "valid_path ((reversepath \<gamma>) +++ rec_join (h#l'))"
-        using 0 1 Cauchy_Integral_Theorem.valid_path_join that(4) l_nempty by blast
+        using 0 1 valid_path_join that(4) l_nempty by blast
     qed
   qed
   have "\<forall>ps. valid_chain_list ps \<or> (\<exists>i f p psa. ps = (i, f) # p # psa \<and> ((i = 1 \<and> pathfinish f \<noteq> pathstart (rec_join (p # psa)) \<or> i \<noteq> 1 \<and> pathfinish (reversepath f) \<noteq> pathstart (rec_join (p # psa))) \<or> \<not> valid_chain_list (p # psa)))"
     by (smt coeff_cube_to_path.elims valid_chain_list.elims(3)) 
   moreover have "boundary_chain (set l)"
     by (meson Cons.prems(1) boundary_chain_def set_subset_Cons subset_eq)
   ultimately show ?case
     using * Cons by (metis (no_types) list.set_intros(2) prod.collapse valid_chain_list.simps(3))
 qed
 
 lemma pathstart_rec_join_1:
   "pathstart (rec_join ((1, \<gamma>) # l)) = pathstart \<gamma>"
 proof (cases "l = []")
   case True
   then show "pathstart (rec_join ((1, \<gamma>) # l)) = pathstart \<gamma>"
     by simp
 next
   case False
   then obtain h l' where "l = h#l'"
     by (meson rec_join.elims)
   then show "pathstart (rec_join ((1, \<gamma>) # l)) = pathstart \<gamma>"
     by simp
 qed
 
 lemma pathstart_rec_join_2:
   "pathstart (rec_join ((-1, \<gamma>) # l)) = pathstart (reversepath \<gamma>)"
 proof cases
   assume "l = []"
   then show "pathstart (rec_join ((- 1, \<gamma>) # l)) = pathstart (reversepath \<gamma>)"
     by simp
 next
   assume "l \<noteq> [] "
   then obtain h l' where "l = h#l'"
     by (meson rec_join.elims)
   then show "pathstart (rec_join ((- 1, \<gamma>) # l)) = pathstart (reversepath \<gamma>)"
     by simp
 qed
 
 lemma pathstart_rec_join:
   "pathstart (rec_join ((1, \<gamma>) # l)) = pathstart \<gamma>"
   "pathstart (rec_join ((-1, \<gamma>) # l)) = pathstart (reversepath \<gamma>)"
   using pathstart_rec_join_1 pathstart_rec_join_2
   by auto
 
 lemma line_integral_exists_on_rec_join:
   assumes boundary_chain: "boundary_chain (set l)" and
     valid_chain_list: "valid_chain_list l" and
     valid_path: "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>" and
     line_integral_exists: "\<forall>(k, \<gamma>) \<in> set l. line_integral_exists F basis \<gamma>"
   shows "line_integral_exists F basis (rec_join l)"
   using assms
 proof (induction l)
   case Nil
   then show ?case
   proof (simp add: line_integral_exists_def)
     have "\<forall>x. (vector_derivative (\<lambda>x. 0) (at x)) = 0"
       using Derivative.vector_derivative_const_at
       by auto
     then have "\<forall>x. ((\<lambda>x. 0) has_vector_derivative 0) (at x)"
       using Derivative.vector_derivative_const_at
       by auto
     then have "\<forall>x. ((\<lambda>x. 0) has_vector_derivative 0) (at x within {0..1})"
       using Derivative.vector_derivative_const_at
       by auto
     then have 0: "\<forall>x\<in>{0..1}. (vector_derivative (\<lambda>x. 0) (at x within{0..1})) = 0"
       by (simp add: gamma_deriv_at_within)
     have "(\<forall>x\<in>{0..1}. (\<Sum>b\<in>basis. F 0 \<bullet> b * (vector_derivative (\<lambda>x. 0) (at x within {0..1}) \<bullet> b)) = 0)"
       by (simp add: 0)
     then have " ((\<lambda>x. \<Sum>b\<in>basis. F 0 \<bullet> b * (vector_derivative (\<lambda>x. 0) (at x within {0..1}) \<bullet> b)) has_integral 0) {0..1}"
       by (meson has_integral_is_0)
     then show "(\<lambda>x. \<Sum>b\<in>basis. F 0 \<bullet> b * (vector_derivative (\<lambda>x. 0) (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
       by auto
   qed
 next
   case (Cons a l)
   obtain k \<gamma> where aeq: "a = (k,\<gamma>)"
     by force
   show ?case
     unfolding aeq
   proof cases
     assume l_empty: "l = []"
     then show "line_integral_exists F basis (rec_join ((k, \<gamma>) # l))"
       using Cons.prems aeq line_integral_on_reverse_path(2) by fastforce
   next
     assume "l \<noteq> []"
     then obtain h l' where l_nempty: "l = h#l'"
       by (meson rec_join.elims)
     show "line_integral_exists F basis (rec_join ((k, \<gamma>) # l))"
     proof (auto simp add: l_nempty)
       assume k_eq_1: "k = 1"
       have 0: "line_integral_exists F basis \<gamma>"
         using Cons.prems(4) aeq by auto
       have 1: "line_integral_exists F basis (rec_join l)"
         by (metis (mono_tags) Cons boundary_chain_def list.set_intros(2) valid_chain_list.elims(3) valid_chain_list.simps(3))
       have 2: "valid_path \<gamma>"
         using Cons aeq by auto
       have 3:"valid_path (rec_join l)"
         by (metis (no_types) Cons.prems boundary_chain_def joined_is_valid l_nempty set_subset_Cons subsetCE valid_chain_list.simps(3))
       show "line_integral_exists F basis (\<gamma> +++ rec_join (h#l'))"
         using line_integral_distrib(2)[OF 0 1 2 3] assms l_nempty by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg_1: "k = -1"
         using Cons aeq by (simp add: boundary_chain_def)
       have gamma_valid: "valid_path \<gamma>"
         using Cons aeq by auto
       then have 2: "valid_path (reversepath \<gamma>)"
-        using Cauchy_Integral_Theorem.valid_path_imp_reverse by auto
+        using valid_path_imp_reverse by auto
       have "line_integral_exists F basis \<gamma>"
         using Cons aeq by auto
       then have 0: "line_integral_exists F basis (reversepath \<gamma>)"
         using line_integral_on_reverse_path(2) gamma_valid
         by auto
       have 1: "line_integral_exists F basis (rec_join l)"
         using Cons aeq
         by (metis (mono_tags) boundary_chain_def insert_iff list.set(2) list.set_intros(2) valid_chain_list.elims(3) valid_chain_list.simps(3))
       have 3:"valid_path (rec_join l)"
         by (metis (no_types) Cons.prems(1) Cons.prems(2) Cons.prems(3) boundary_chain_def joined_is_valid l_nempty set_subset_Cons subsetCE valid_chain_list.simps(3))
       show "line_integral_exists F basis ((reversepath \<gamma>) +++ rec_join (h#l'))"
         using line_integral_distrib(2)[OF 0 1 2 3] assms l_nempty
         by auto
     qed
   qed
 qed
 
 lemma line_integral_exists_rec_join_cons:
   assumes "line_integral_exists F basis (rec_join ((1,\<gamma>) # l))"
     "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((1,\<gamma>) # l) \<Longrightarrow> valid_path \<gamma>')"
     "finite basis"
   shows "line_integral_exists F basis (\<gamma> +++ (rec_join l))"
 proof cases
   assume l_empty: "l = []"
   show "line_integral_exists F basis (\<gamma> +++ rec_join l)"
     using assms(2) line_integral_distrib(2)[OF assms(1) line_integral_exists_on_degenerate_path[OF assms(3)], of "0"]
     using degenerate_path_is_valid_path
     by (fastforce simp add: l_empty)
 next
   assume "l \<noteq> []"
   then obtain h l' where "l = h#l'"
     by (meson rec_join.elims)
   then show "line_integral_exists F basis (\<gamma> +++ rec_join l)"
     using assms by auto
 qed
 
 lemma line_integral_exists_rec_join_cons_2:
   assumes "line_integral_exists F basis (rec_join ((-1,\<gamma>) # l))"
     "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((1,\<gamma>) # l) \<Longrightarrow> valid_path \<gamma>')"
     "finite basis"
   shows "line_integral_exists F basis ((reversepath \<gamma>) +++ (rec_join l))"
 proof cases
   assume l_empty: "l = []"
   show "line_integral_exists F basis ((reversepath \<gamma>) +++ rec_join l)"
     using assms(2) line_integral_distrib(2)[OF assms(1) line_integral_exists_on_degenerate_path[OF assms(3)], of "0"]
     using degenerate_path_is_valid_path
     by (auto simp add: l_empty)
 next
   assume "l \<noteq> []"
   then obtain h l' where  "l = h#l'"
     by (meson rec_join.elims)
   with assms show "line_integral_exists F basis ((reversepath \<gamma>) +++ rec_join l)"
     using assms by auto
 qed
 
 lemma line_integral_exists_on_rec_join':
   assumes boundary_chain: "boundary_chain (set l)" and
     valid_chain_list: "valid_chain_list l" and
     valid_path: "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>" and
     line_integral_exists: "line_integral_exists F basis (rec_join l)" and
     finite_basis: "finite basis"
   shows "\<forall>(k, \<gamma>) \<in> set l. line_integral_exists F basis \<gamma>"
   using assms
 proof (induction l)
   case Nil
   show ?case
     by (simp add: line_integral_exists_def)
 next
   case ass: (Cons a l)          
   obtain k \<gamma> where k_gamma:"a = (k,\<gamma>)"
     by fastforce
   show ?case
     apply (auto simp add: k_gamma)
   proof -
     show "line_integral_exists F basis \<gamma>"
     proof(cases "k = 1")
       assume k_eq_1: "k = 1"
       have 0: "line_integral_exists F basis (\<gamma> +++ (rec_join l))"
         using line_integral_exists_rec_join_cons k_eq_1 k_gamma ass(4) ass(5) ass(6)
         by auto
       have 2: "valid_path \<gamma>"
         using ass k_gamma by auto
       show "line_integral_exists F basis \<gamma>"
         using line_integral_exists_joinD1[OF 0 2]
         by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg_1: "k = -1"
         using ass k_gamma
         by (simp add: boundary_chain_def)
       have 0: "line_integral_exists F basis ((reversepath \<gamma>) +++ (rec_join l))"
         using line_integral_exists_rec_join_cons_2[OF ] k_eq_neg_1 k_gamma ass(4) ass(5) ass(6)
         by fastforce
       have gamma_valid:
         "valid_path \<gamma>"
         using ass k_gamma by auto
       then have 2: "valid_path (reversepath \<gamma>)"
-        using Cauchy_Integral_Theorem.valid_path_imp_reverse by auto
+        using valid_path_imp_reverse by auto
       have "line_integral_exists F basis (reversepath \<gamma>)"
         using line_integral_exists_joinD1[OF 0 2] by auto
       then show "line_integral_exists F basis (\<gamma>)"
         using line_integral_on_reverse_path(2)[OF 2] reversepath_reversepath
         by auto
     qed
   next
     have 0:"boundary_chain (set l)"
       using ass(2)
       by (auto simp add: boundary_chain_def)
     have 1:"valid_chain_list l"
       using ass(3)
       apply (auto simp add: k_gamma)
       by (metis valid_chain_list.elims(3) valid_chain_list.simps(3))
     have 2:"(\<And>k \<gamma>. (k, \<gamma>) \<in> set (l) \<Longrightarrow> valid_path \<gamma>)"
       using ass(4) by auto
     have 3: "valid_path (rec_join l)"
       using joined_is_valid[OF 0] 1 2 by auto
     have 4: "line_integral_exists F basis (rec_join l)"
     proof(cases "k = 1")
       assume k_eq_1: "k = 1"
       have 0: "line_integral_exists F basis (\<gamma> +++ (rec_join l))"
         using line_integral_exists_rec_join_cons k_eq_1 k_gamma ass(4) ass(5) ass(6) by auto
       show "line_integral_exists F basis (rec_join l)"
         using line_integral_exists_joinD2[OF 0 3] by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg_1: "k = -1"
         using ass k_gamma
         by (simp add: boundary_chain_def)
       have 0: "line_integral_exists F basis ((reversepath \<gamma>) +++ (rec_join l))"
         using line_integral_exists_rec_join_cons_2[OF ] k_eq_neg_1 k_gamma ass(4) ass(5) ass(6)
         by fastforce
       show "line_integral_exists F basis (rec_join l)"
         using line_integral_exists_joinD2[OF 0 3]
         by auto
     qed
     show "\<And>a b. (a, b) \<in> set l \<Longrightarrow> line_integral_exists F basis b"
       using 0 1 2 3 4 ass(1)[OF 0 1 2] ass(6)
       by fastforce
   qed
 qed
 
 inductive chain_subdiv_path
   where I: "chain_subdiv_path \<gamma> (set l)" if "distinct l" "rec_join l = \<gamma>" "valid_chain_list l"
 
 lemma valid_path_equiv_valid_chain_list:
   assumes path_eq_chain: "chain_subdiv_path \<gamma> one_chain" 
     and "boundary_chain one_chain" "\<forall>(k, \<gamma>) \<in> one_chain. valid_path \<gamma>"
   shows "valid_path \<gamma>"
 proof -
   obtain l where l_props: "set l = one_chain" "distinct l" "rec_join l = \<gamma>" "valid_chain_list l"
     using chain_subdiv_path.cases path_eq_chain by force
   show "valid_path \<gamma>"
     using joined_is_valid assms l_props by blast
 qed
 
 lemma line_integral_rec_join_cons:
   assumes "line_integral_exists F basis \<gamma>"
     "line_integral_exists F basis (rec_join ((l)))"
     "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((1,\<gamma>) # l) \<Longrightarrow> valid_path \<gamma>')"
     "finite basis"
   shows "line_integral F basis (rec_join ((1,\<gamma>) # l)) = line_integral F basis (\<gamma> +++ (rec_join l))"
 proof cases
   assume l_empty: "l = []"
   show "line_integral F basis (rec_join ((1,\<gamma>) # l)) = line_integral F basis (\<gamma> +++ (rec_join l))"
     using assms line_integral_distrib(1)[OF assms(1) line_integral_exists_on_degenerate_path[OF assms(4)], of "0"]
     apply (auto simp add: l_empty)
     using degenerate_path_is_valid_path line_integral_degenerate_path
     by fastforce
 next
   assume "l \<noteq> []"
   then obtain h l' where l_nempty: "l = h#l'"
     by (meson rec_join.elims)
   show "line_integral F basis (rec_join ((1,\<gamma>) # l)) = line_integral F basis (\<gamma> +++ (rec_join l))"
     using assms  by (auto simp add: l_nempty)
 qed
 
 lemma line_integral_rec_join_cons_2:
   assumes "line_integral_exists F basis \<gamma>"
     "line_integral_exists F basis (rec_join ((l)))"
     "(\<And>k' \<gamma>'. (k', \<gamma>') \<in> set ((-1,\<gamma>) # l) \<Longrightarrow> valid_path \<gamma>')"
     "finite basis"
   shows "line_integral F basis (rec_join ((-1,\<gamma>) # l)) = line_integral F basis ((reversepath \<gamma>) +++ (rec_join l))"
 proof cases
   assume l_empty: "l = []"
   have 0: "line_integral_exists F basis (reversepath \<gamma>)"
     using assms line_integral_on_reverse_path(2) by fastforce
   have 1: "valid_path (reversepath \<gamma>)"
     using assms  by fastforce
   show "line_integral F basis (rec_join ((-1,\<gamma>) # l)) = line_integral F basis ((reversepath \<gamma>) +++ (rec_join l))"
     using assms line_integral_distrib(1)[OF 0 line_integral_exists_on_degenerate_path[OF assms(4)], of "0"]
     apply (auto simp add: l_empty)
     using degenerate_path_is_valid_path line_integral_degenerate_path
     by fastforce
 next
   assume "l \<noteq> []"
   then obtain h l' where l_nempty: "l = h#l'"
     by (meson rec_join.elims)
   show "line_integral F basis (rec_join ((-1,\<gamma>) # l)) = line_integral F basis ((reversepath \<gamma>) +++ (rec_join l))"
     using assms  by (auto simp add: l_nempty)
 qed
 
 lemma one_chain_line_integral_rec_join:
   assumes l_props: "set l = one_chain" "distinct l" "valid_chain_list l" and
     boundary_chain: "boundary_chain one_chain" and
     line_integral_exists: "\<forall>(k::int, \<gamma>) \<in> one_chain. line_integral_exists F basis \<gamma>" and
     valid_path: "\<forall>(k::int, \<gamma>) \<in> one_chain. valid_path \<gamma>" and
     finite_basis: "finite basis"
   shows "line_integral F basis (rec_join l) = one_chain_line_integral F basis one_chain"
 proof -
   have 0: "sum_list (map (\<lambda>((k::int), \<gamma>). (k::int) * (line_integral F basis \<gamma>)) l) = one_chain_line_integral F basis one_chain"
     unfolding one_chain_line_integral_def
     using l_props Groups_List.comm_monoid_add_class.sum.distinct_set_conv_list[OF l_props(2), of "(\<lambda>(k, \<gamma>). (k::int) * (line_integral F basis \<gamma>))"]
     by auto
   have "valid_chain_list l \<Longrightarrow>
                     boundary_chain (set l) \<Longrightarrow>
                     (\<forall>(k::int, \<gamma>) \<in> set l. line_integral_exists F basis \<gamma>) \<Longrightarrow>
                     (\<forall>(k::int, \<gamma>) \<in> set l. valid_path \<gamma>) \<Longrightarrow>
                     line_integral F basis (rec_join l) = sum_list (map (\<lambda>(k::int, \<gamma>). k * (line_integral F basis \<gamma>)) l)"
   proof (induction l)
     case Nil
     show ?case
       unfolding line_integral_def boundary_chain_def
       apply (auto)
     proof
       have "\<forall>x. (vector_derivative (\<lambda>x. 0) (at x)) = 0"
         using Derivative.vector_derivative_const_at
         by auto
       then have "\<forall>x. ((\<lambda>x. 0) has_vector_derivative 0) (at x)"
         using Derivative.vector_derivative_const_at
         by auto
       then have "\<forall>x. ((\<lambda>x. 0) has_vector_derivative 0) (at x within {0..1})"
         using Derivative.vector_derivative_const_at
         by auto
       then have 0: "\<forall>x\<in>{0..1}. (vector_derivative (\<lambda>x. 0) (at x within{0..1})) = 0"
         by (metis (no_types) box_real(2) vector_derivative_within_cbox zero_less_one)
       have "(\<forall>x\<in>{0..1}. (\<Sum>b\<in>basis. F 0 \<bullet> b * (vector_derivative (\<lambda>x. 0) (at x within {0..1}) \<bullet> b)) = 0)"
         by (simp add: 0)
       then show "((\<lambda>x. \<Sum>b\<in>basis. F 0 \<bullet> b * (vector_derivative (\<lambda>x. 0) (at x within {0..1}) \<bullet> b)) has_integral 0) {0..1}"
         by (meson has_integral_is_0)
     qed
   next
     case ass: (Cons a l)
     obtain k::"int" and
       \<gamma>::"one_cube" where props: "a = (k,\<gamma>)"
     proof
       let ?k2 = "fst a"
       let ?\<gamma>2 = "snd a"
       show "a = (?k2, ?\<gamma>2)"
         by auto
     qed
     have "line_integral_exists F basis (rec_join (a # l))"
       using line_integral_exists_on_rec_join[OF ass(3) ass(2)] ass(5) ass(4)
       by blast
     have "boundary_chain (set l)"
       by (meson ass.prems(2) boundary_chain_def list.set_intros(2))
     have val_l: "\<And>f i. (i,f) \<in> set l \<Longrightarrow> valid_path f"
       using ass.prems(4) by fastforce
     have vcl_l: "valid_chain_list l"
       by (metis (no_types) ass.prems(1) valid_chain_list.elims(3) valid_chain_list.simps(3))
     have line_integral_exists_on_joined:
       "line_integral_exists F basis (rec_join l)"
       by (metis \<open>boundary_chain (set l)\<close> \<open>line_integral_exists F basis (rec_join (a # l))\<close> emptyE val_l vcl_l joined_is_valid line_integral_exists_joinD2 line_integral_exists_on_rec_join list.set(1) neq_Nil_conv rec_join.simps(3))
     have "valid_path (rec_join (a # l))"
       using joined_is_valid ass(5) ass(3) ass(2) by blast
     then have joined_is_valid: "valid_path (rec_join l)"
       using \<open>boundary_chain (set l)\<close> val_l vcl_l joined_is_valid by blast
     show ?case
     proof (clarsimp, cases)
       assume k_eq_1: "(k::int) = 1"
       have line_integral_exists_on_gamma: "line_integral_exists F basis \<gamma>"
         using ass props by auto
       have gamma_is_valid: "valid_path \<gamma>"
         using ass props by auto
       have line_int_rw: "line_integral F basis (rec_join ((k, \<gamma>) # l)) = line_integral F basis (\<gamma> +++ rec_join l)"
       proof -
         have gam_int: "line_integral_exists F basis \<gamma>" using ass props by auto
         have rec_join_int: "line_integral_exists F basis (rec_join l)"
           using line_integral_exists_on_rec_join
           using line_integral_exists_on_joined by blast
         show ?thesis
           using line_integral_rec_join_cons[OF gam_int rec_join_int] ass k_eq_1 finite_basis props by force
       qed
       show "line_integral F basis (rec_join (a # l)) =
             (case a of (x, \<gamma>) \<Rightarrow> real_of_int x * line_integral F basis \<gamma>) + (\<Sum>(x, \<gamma>)\<leftarrow>l. real_of_int x * line_integral F basis \<gamma>)"
         apply (simp add: props line_int_rw)
         using line_integral_distrib[OF line_integral_exists_on_gamma line_integral_exists_on_joined gamma_is_valid joined_is_valid]
           ass k_eq_1 vcl_l
         by (auto simp: boundary_chain_def props)
     next
       assume "k \<noteq> 1"
       then have k_eq_neg_1: "k = -1"
         using ass props
         by (auto simp add: boundary_chain_def)
       have line_integral_exists_on_gamma:
         "line_integral_exists F basis (reversepath \<gamma>)"
         using line_integral_on_reverse_path ass props
         by auto
       have gamma_is_valid: "valid_path (reversepath \<gamma>)"
-        using Cauchy_Integral_Theorem.valid_path_imp_reverse ass props by auto
+        using valid_path_imp_reverse ass props by auto
       have line_int_rw: "line_integral F basis (rec_join ((k, \<gamma>) # l)) = line_integral F basis ((reversepath \<gamma>) +++ rec_join l)"
       proof -
         have gam_int: "line_integral_exists F basis \<gamma>" using ass props by auto
         have rec_join_int: "line_integral_exists F basis (rec_join l)"
           using line_integral_exists_on_rec_join
           using line_integral_exists_on_joined by blast
         show ?thesis
           using line_integral_rec_join_cons_2[OF gam_int rec_join_int]
           using ass k_eq_neg_1
           using finite_basis props by blast
       qed
       show "line_integral F basis (rec_join (a # l)) =
             (case a of (x, \<gamma>) \<Rightarrow> real_of_int x * line_integral F basis \<gamma>) + (\<Sum>(x, \<gamma>)\<leftarrow>l. real_of_int x * line_integral F basis \<gamma>)"
         apply (simp add: props line_int_rw)
         using line_integral_distrib[OF line_integral_exists_on_gamma line_integral_exists_on_joined gamma_is_valid joined_is_valid]
           props ass line_integral_on_reverse_path(1)[of "\<gamma>" "F" "basis"] k_eq_neg_1
         using \<open>boundary_chain (set l)\<close> vcl_l by auto
     qed
   qed
   then have 1:"line_integral F basis (rec_join l) = sum_list (map (\<lambda>(k::int, \<gamma>). k * (line_integral F basis \<gamma>)) l)"
     using l_props assms by auto
   then show ?thesis
     using 0 1 by auto
 qed
 
 lemma line_integral_on_path_eq_line_integral_on_equiv_chain:
   assumes path_eq_chain: "chain_subdiv_path \<gamma> one_chain" and
     boundary_chain: "boundary_chain one_chain" and
     line_integral_exists: "\<forall>(k::int, \<gamma>) \<in> one_chain. line_integral_exists F basis \<gamma>" and
     valid_path: "\<forall>(k::int, \<gamma>) \<in> one_chain. valid_path \<gamma>" and
     finite_basis: "finite basis"
   shows "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
     "line_integral_exists F basis \<gamma>"
     "valid_path \<gamma>"
 proof -
   obtain l where l_props: "set l = one_chain" "distinct l" "rec_join l = \<gamma>" "valid_chain_list l"
     using chain_subdiv_path.cases path_eq_chain by force
   show "line_integral_exists F basis \<gamma>"
     using line_integral_exists_on_rec_join assms l_props
     by blast
   show "valid_path \<gamma>"
     using joined_is_valid assms l_props
     by blast
   have "line_integral F basis (rec_join l) = one_chain_line_integral F basis one_chain"
     using one_chain_line_integral_rec_join l_props assms by auto
   then show "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
     using l_props
     by auto
 qed
 
 lemma line_integral_on_path_eq_line_integral_on_equiv_chain':
   assumes path_eq_chain: "chain_subdiv_path \<gamma> one_chain" and
     boundary_chain: "boundary_chain one_chain" and
     line_integral_exists: "line_integral_exists F basis \<gamma>" and
     valid_path: "\<forall>(k, \<gamma>) \<in> one_chain. valid_path \<gamma>" and
     finite_basis: "finite basis"
   shows "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
     "\<forall>(k, \<gamma>) \<in> one_chain. line_integral_exists F basis \<gamma>"
 proof -
   obtain l where l_props: "set l = one_chain" "distinct l" "rec_join l = \<gamma>" "valid_chain_list l"
     using chain_subdiv_path.cases path_eq_chain by force
   show 0: "\<forall>(k, \<gamma>) \<in> one_chain. line_integral_exists F basis \<gamma>"
     using line_integral_exists_on_rec_join' assms l_props
     by blast
   show "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
     using line_integral_on_path_eq_line_integral_on_equiv_chain(1)[OF assms(1) assms(2) 0 assms(4) assms(5)] by auto
 qed
 
 definition chain_subdiv_chain where
   "chain_subdiv_chain one_chain1 subdiv
         \<equiv> \<exists>f. (\<Union>(f ` one_chain1)) = subdiv \<and>
               (\<forall>c\<in>one_chain1. chain_subdiv_path (coeff_cube_to_path c) (f c)) \<and>
               pairwise (\<lambda> p p'. f p \<inter> f p' = {}) one_chain1 \<and>
               (\<forall>x \<in> one_chain1. finite (f x))"
 
 lemma chain_subdiv_chain_character:
   shows "chain_subdiv_chain one_chain1 subdiv \<longleftrightarrow>
         (\<exists>f. \<Union>(f ` one_chain1) = subdiv \<and>
              (\<forall>(k, \<gamma>)\<in>one_chain1.
                  if k = 1
                  then chain_subdiv_path \<gamma> (f (k, \<gamma>))
                  else chain_subdiv_path (reversepath \<gamma>) (f (k, \<gamma>))) \<and>
              (\<forall>p\<in>one_chain1.
                  \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {}) \<and>
              (\<forall>x\<in>one_chain1. finite (f x)))"  
   unfolding chain_subdiv_chain_def
   by (safe; intro exI conjI iffI; fastforce simp add: pairwise_def)
 
 lemma chain_subdiv_chain_imp_finite_subdiv:
   assumes "finite one_chain1"
     "chain_subdiv_chain one_chain1 subdiv"
   shows "finite subdiv"
   using assms by(auto simp add: chain_subdiv_chain_def)
 
 lemma valid_subdiv_imp_valid_one_chain:
   assumes chain1_eq_chain2: "chain_subdiv_chain one_chain1 subdiv" and
     boundary_chain1: "boundary_chain one_chain1" and
     boundary_chain2: "boundary_chain subdiv" and
     valid_path: "\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>"
   shows "\<forall>(k, \<gamma>) \<in> one_chain1. valid_path \<gamma>"
 proof -
   obtain f where f_props:
     "((\<Union>(f ` one_chain1)) = subdiv)"
     "(\<forall>(k,\<gamma>)\<in>one_chain1. if k = 1 then chain_subdiv_path \<gamma> (f(k,\<gamma>)) else chain_subdiv_path (reversepath \<gamma>) (f(k,\<gamma>)))"
     "(\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {})"
     using chain1_eq_chain2 chain_subdiv_chain_character by auto
   have "\<And> k \<gamma>. (k,\<gamma>) \<in> one_chain1\<Longrightarrow> valid_path \<gamma>"
   proof-
     fix k \<gamma>
     assume ass: "(k,\<gamma>) \<in> one_chain1"
     show "valid_path \<gamma>"
     proof (cases "k = 1")
       assume k_eq_1: "k = 1"
       then have i:"chain_subdiv_path \<gamma> (f(k,\<gamma>))"
         using f_props(2) ass by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms
         apply (simp add: boundary_chain_def)
         by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms
         by blast
       show ?thesis
         using valid_path_equiv_valid_chain_list[OF i ii iv]
         by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg1: "k = -1"
         using ass boundary_chain1
         by (auto simp add: boundary_chain_def)
       then have i:"chain_subdiv_path (reversepath \<gamma>) (f(k,\<gamma>))"
         using f_props(2) ass using \<open>k \<noteq> 1\<close> by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms by (auto simp add: boundary_chain_def)
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms
         by blast
       have "valid_path (reversepath \<gamma>)"
         using valid_path_equiv_valid_chain_list[OF i ii iv]
         by auto
       then show ?thesis
-        using  reversepath_reversepath Cauchy_Integral_Theorem.valid_path_imp_reverse
+        using  reversepath_reversepath valid_path_imp_reverse
         by force
     qed
   qed
   then show valid_path1: "\<forall>(k, \<gamma>) \<in> one_chain1. valid_path \<gamma>"
     by auto
 qed
 
 lemma one_chain_line_integral_eq_line_integral_on_sudivision:
   assumes chain1_eq_chain2: "chain_subdiv_chain one_chain1 subdiv" and
     boundary_chain1: "boundary_chain one_chain1" and
     boundary_chain2: "boundary_chain subdiv" and
     line_integral_exists_on_chain2: "\<forall>(k, \<gamma>) \<in> subdiv. line_integral_exists F basis \<gamma>" and
     valid_path: "\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>" and
     finite_chain1: "finite one_chain1" and
     finite_basis: "finite basis"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
     "\<forall>(k, \<gamma>) \<in> one_chain1. line_integral_exists F basis \<gamma>"
 proof -
   obtain f where f_props:
     "((\<Union>(f ` one_chain1)) = subdiv)"
     "(\<forall>(k,\<gamma>)\<in>one_chain1. if k = 1 then chain_subdiv_path \<gamma> (f(k,\<gamma>)) else chain_subdiv_path (reversepath \<gamma>) (f(k,\<gamma>)))"
     "(\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {})"
     "(\<forall>x \<in> one_chain1. finite (f x))"
     using chain1_eq_chain2 chain_subdiv_chain_character by (auto simp add: pairwise_def chain_subdiv_chain_def)
   then have 0: "one_chain_line_integral F basis subdiv = one_chain_line_integral F basis (\<Union>(f ` one_chain1))"
     by auto
   have finite_chain2: "finite subdiv"
     using finite_chain1 f_props(1) f_props(4)
     apply (simp add: image_def)
     using f_props(1) by auto
   have "\<And> k \<gamma>. (k,\<gamma>) \<in> one_chain1\<Longrightarrow> line_integral_exists F basis \<gamma>"
   proof-
     fix k \<gamma>
     assume ass: "(k,\<gamma>) \<in> one_chain1"
     show "line_integral_exists F basis \<gamma>"
     proof (cases "k = 1")
       assume k_eq_1: "k = 1"
       then have i:"chain_subdiv_path \<gamma> (f(k,\<gamma>))"
         using f_props(2) ass by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms by (auto simp add: boundary_chain_def)
       have iii:"\<forall>(k, \<gamma>)\<in>f (k, \<gamma>). line_integral_exists F basis \<gamma>"
         using f_props(1) ass assms
         by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms
         by blast
       show ?thesis
         using line_integral_on_path_eq_line_integral_on_equiv_chain(2)[OF i ii iii iv finite_basis]
         by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg1: "k = -1"
         using ass boundary_chain1
         by (auto simp add: boundary_chain_def)
       then have i:"chain_subdiv_path (reversepath \<gamma>) (f(k,\<gamma>))"
         using f_props(2) ass by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms by (auto simp add: boundary_chain_def)
       have iii:"\<forall>(k, \<gamma>)\<in>f (k, \<gamma>). line_integral_exists F basis \<gamma>"
         using f_props(1) ass assms
         by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms
         by blast
       have x:"line_integral_exists F basis (reversepath \<gamma>)"
         using line_integral_on_path_eq_line_integral_on_equiv_chain(2)[OF i ii iii iv finite_basis]
         by auto
       have "valid_path (reversepath \<gamma>)"
         using line_integral_on_path_eq_line_integral_on_equiv_chain(3)[OF i ii iii iv finite_basis]
         by auto
       then show ?thesis
         using line_integral_on_reverse_path(2) reversepath_reversepath x
         by fastforce
     qed
   qed
   then show line_integral_exists_on_chain1: "\<forall>(k, \<gamma>) \<in> one_chain1. line_integral_exists F basis \<gamma>"
     by auto
   have 1: "one_chain_line_integral F basis (\<Union>(f ` one_chain1)) = one_chain_line_integral F basis one_chain1"
   proof -
     have 0:"one_chain_line_integral F basis (\<Union>(f ` one_chain1)) =
                            (\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain)"
     proof -
       have finite: "\<forall>chain \<in> (f ` one_chain1). finite chain"
         using f_props(1) finite_chain2
         by (meson Sup_upper finite_subset)
       have disj: " \<forall>A\<in>f ` one_chain1. \<forall>B\<in>f ` one_chain1. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
         by (metis (no_types, hide_lams) f_props(3) image_iff)
       show "one_chain_line_integral F basis (\<Union>(f ` one_chain1)) =
                                 (\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain)"
         using Groups_Big.comm_monoid_add_class.sum.Union_disjoint[OF finite disj]
           one_chain_line_integral_def
         by auto
     qed
     have 1:"(\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain) =
                             one_chain_line_integral F basis one_chain1"
     proof -
       have "(\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain) =
                                      (\<Sum>(k,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>))"
       proof -
         have i:"(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                        (\<Sum>(k,\<gamma>)\<in>one_chain1 - {p. f p = {}}. k*(line_integral F basis \<gamma>))"
         proof -
           have "inj_on f (one_chain1 - {p. f p = {}})"
             unfolding inj_on_def using f_props(3) by blast
           then have 0: "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain)
                                                        = (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). one_chain_line_integral F basis (f (k, \<gamma>)))"
             using Groups_Big.comm_monoid_add_class.sum.reindex
             by auto
           have "\<And> k \<gamma>. (k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}) \<Longrightarrow>
                         one_chain_line_integral F basis (f(k, \<gamma>)) = k* (line_integral F basis \<gamma>)"
           proof-
             fix k \<gamma>
             assume ass: "(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}})"
             have bchain: "boundary_chain (f(k,\<gamma>))"
               using f_props(1) boundary_chain2 ass
               by (auto simp add: boundary_chain_def)
             have wexist: "\<forall>(k, \<gamma>)\<in>(f(k,\<gamma>)). line_integral_exists F basis \<gamma>"
               using f_props(1) line_integral_exists_on_chain2 ass
               by blast
             have vpath: " \<forall>(k, \<gamma>)\<in>(f(k, \<gamma>)). valid_path \<gamma>"
               using f_props(1) assms ass
               by blast
             show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
             proof(cases "k = 1")
               assume k_eq_1: "k = 1"
               have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis \<gamma>"
                 using f_props(2) k_eq_1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
                 by auto
               then show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
                 using k_eq_1  by auto
             next
               assume "k \<noteq> 1"
               then have k_eq_neg1: "k = -1"
                 using ass boundary_chain1
                 by (auto simp add: boundary_chain_def)
               have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis (reversepath \<gamma>)"
                 using f_props(2) k_eq_neg1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
                 by auto
               then have "one_chain_line_integral F basis (f (k, \<gamma>)) = - (line_integral F basis \<gamma>)"
                 using line_integral_on_reverse_path(1) ass line_integral_exists_on_chain1
                   valid_subdiv_imp_valid_one_chain[OF chain1_eq_chain2 boundary_chain1 boundary_chain2 valid_path]
                 by force
               then show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
                 using k_eq_neg1 by auto
             qed
           qed
           then have "(\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). one_chain_line_integral F basis (f (k, \<gamma>)))
                    = (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). k* (line_integral F basis \<gamma>))"
             by (auto intro!: Finite_Cartesian_Product.sum_cong_aux)
           then show "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                      (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). k* (line_integral F basis \<gamma>))"
             using 0  by auto
         qed
         have "\<And> (k::int) \<gamma>. (k, \<gamma>) \<in> one_chain1 \<Longrightarrow> (f (k, \<gamma>) = {}) \<Longrightarrow> (k, \<gamma>) \<in> {(k',\<gamma>'). k'* (line_integral F basis \<gamma>') = 0}"
         proof-
           fix k::"int"
           fix \<gamma>::"one_cube"
           assume ass:"(k, \<gamma>) \<in> one_chain1"
             "(f (k, \<gamma>) = {})"
           then have zero_line_integral:"one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
             using one_chain_line_integral_def
             by auto
           have bchain: "boundary_chain (f(k,\<gamma>))"
             using f_props(1) boundary_chain2 ass
             by (auto simp add: boundary_chain_def)
           have wexist: "\<forall>(k, \<gamma>)\<in>(f(k,\<gamma>)). line_integral_exists F basis \<gamma>"
             using f_props(1) line_integral_exists_on_chain2 ass
             by blast
           have vpath: " \<forall>(k, \<gamma>)\<in>(f(k, \<gamma>)). valid_path \<gamma>"
             using f_props(1) assms ass by blast
           have "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
           proof(cases "k = 1")
             assume k_eq_1: "k = 1"
             have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis \<gamma>"
               using f_props(2) k_eq_1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
               by auto
             then show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
               using k_eq_1
               by auto
           next
             assume "k \<noteq> 1"
             then have k_eq_neg1: "k = -1"
               using ass boundary_chain1
               by (auto simp add: boundary_chain_def)
             have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis (reversepath \<gamma>)"
               using f_props(2) k_eq_neg1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
               by auto
             then have "one_chain_line_integral F basis (f (k, \<gamma>)) = - (line_integral F basis \<gamma>)"
               using line_integral_on_reverse_path(1)  ass line_integral_exists_on_chain1
                 valid_subdiv_imp_valid_one_chain[OF chain1_eq_chain2 boundary_chain1 boundary_chain2 valid_path]
               by force
             then show "one_chain_line_integral F basis (f (k::int, \<gamma>)) = k * line_integral F basis \<gamma>"
               using k_eq_neg1  by auto
           qed
           then show "(k, \<gamma>) \<in> {(k'::int, \<gamma>'). k' * line_integral F basis \<gamma>' = 0}"
             using zero_line_integral by auto
         qed
         then have ii:"(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                           (\<Sum>one_chain \<in> (f ` (one_chain1)). one_chain_line_integral F basis one_chain)"
         proof -
           have "\<And>one_chain. one_chain \<in> (f ` (one_chain1)) - (f ` (one_chain1 - {p. f p = {}})) \<Longrightarrow>
                                                          one_chain_line_integral F basis one_chain = 0"
           proof -
             fix one_chain
             assume "one_chain \<in> (f ` (one_chain1)) - (f ` (one_chain1 - {p. f p = {}}))"
             then show "one_chain_line_integral F basis one_chain = 0"
               by (auto simp add: one_chain_line_integral_def)
           qed
           then have 0:"(\<Sum>one_chain \<in> f ` (one_chain1) - (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain)
                                                        = 0"
             using comm_monoid_add_class.sum.neutral by auto
           then have "(\<Sum>one_chain \<in> f ` (one_chain1). one_chain_line_integral F basis one_chain)
                                                       - (\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain)
                                                        = 0"
           proof -
             have finte: "finite (f ` one_chain1)" using finite_chain1 by auto
             show ?thesis
               using Groups_Big.sum_diff[OF finte, of " (f ` (one_chain1 - {p. f p = {}}))"
                   "one_chain_line_integral F basis"]
                 0
               by auto
           qed
           then show "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                           (\<Sum>one_chain \<in> (f ` (one_chain1)). one_chain_line_integral F basis one_chain)"
             by auto
         qed
         have "\<And> (k::int) \<gamma>. (k, \<gamma>) \<in> one_chain1 \<Longrightarrow> (f (k, \<gamma>) = {}) \<Longrightarrow> f (k, \<gamma>) \<in> {chain. one_chain_line_integral F basis chain = 0}"
         proof-
           fix k::"int"
           fix \<gamma>::"one_cube"
           assume ass:"(k, \<gamma>) \<in> one_chain1" "(f (k, \<gamma>) = {})"
           then have "one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
             using one_chain_line_integral_def  by auto
           then show "f (k, \<gamma>) \<in> {p'. (one_chain_line_integral F basis p' = 0)}"
             by auto
         qed
         then have iii:"(\<Sum>(k::int,\<gamma>)\<in>one_chain1 - {p. f p = {}}. k*(line_integral F basis \<gamma>))
                                                  = (\<Sum>(k::int,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>))"
         proof-
           have "\<And> k \<gamma>. (k,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}})
                                                     \<Longrightarrow> k* (line_integral F basis \<gamma>) = 0"
           proof-
             fix k \<gamma>
             assume ass: "(k,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}})"
             then have "f(k, \<gamma>) = {}"
               by auto
             then have "one_chain_line_integral F basis (f(k, \<gamma>)) = 0"
               by (auto simp add: one_chain_line_integral_def)
             then have zero_line_integral:"one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
               using one_chain_line_integral_def by auto
             have bchain: "boundary_chain (f(k,\<gamma>))"
               using f_props(1) boundary_chain2 ass
               by (auto simp add: boundary_chain_def)
             have wexist: "\<forall>(k, \<gamma>)\<in>(f(k,\<gamma>)). line_integral_exists F basis \<gamma>"
               using f_props(1) line_integral_exists_on_chain2 ass
               by blast
             have vpath: " \<forall>(k, \<gamma>)\<in>(f(k, \<gamma>)). valid_path \<gamma>"
               using f_props(1) assms ass
               by blast
             have "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
             proof(cases "k = 1")
               assume k_eq_1: "k = 1"
               have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis \<gamma>"
                 using f_props(2) k_eq_1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
                 by auto
               then show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
                 using k_eq_1
                 by auto
             next
               assume "k \<noteq> 1"
               then have k_eq_neg1: "k = -1"
                 using ass boundary_chain1
                 by (auto simp add: boundary_chain_def)
               have "one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis (reversepath \<gamma>)"
                 using f_props(2) k_eq_neg1 line_integral_on_path_eq_line_integral_on_equiv_chain bchain wexist vpath ass finite_basis
                 by auto
               then have "one_chain_line_integral F basis (f (k, \<gamma>)) = - (line_integral F basis \<gamma>)"
                 using line_integral_on_reverse_path(1)  ass line_integral_exists_on_chain1
                   valid_subdiv_imp_valid_one_chain[OF chain1_eq_chain2 boundary_chain1 boundary_chain2 valid_path]
                 by force
               then show "one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
                 using k_eq_neg1
                 by auto
             qed
             then show "k* (line_integral F basis \<gamma>) = 0"
               using zero_line_integral
               by auto
           qed
           then have "\<forall>(k::int,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}).
                                                       k* (line_integral F basis \<gamma>) = 0" by auto
           then have "(\<Sum>(k::int,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}). k*(line_integral F basis \<gamma>)) = 0"
             using Groups_Big.comm_monoid_add_class.sum.neutral
               [of "one_chain1 - (one_chain1 - {p. f p = {}})" "(\<lambda>(k::int,\<gamma>). k* (line_integral F basis \<gamma>))"]
             by (simp add: split_beta)
           then have "(\<Sum>(k::int,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>)) -
                      (\<Sum>(k::int,\<gamma>)\<in> (one_chain1 - {p. f p = {}}). k*(line_integral F basis \<gamma>)) = 0"
             using Groups_Big.sum_diff[OF finite_chain1]
             by (metis (no_types) Diff_subset \<open>(\<Sum>(k, \<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}). k * line_integral F basis \<gamma>) = 0\<close> \<open>\<And>f B. B \<subseteq> one_chain1 \<Longrightarrow> sum f (one_chain1 - B) = sum f one_chain1 - sum f B\<close>)
           then show ?thesis by auto
         qed
         show ?thesis using i ii iii by auto
       qed
       then show ?thesis using one_chain_line_integral_def by auto
     qed
     show ?thesis using 0 1 by auto
   qed
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv" using 0 1 by auto
 qed
 
 lemma one_chain_line_integral_eq_line_integral_on_sudivision':
   assumes chain1_eq_chain2: "chain_subdiv_chain one_chain1 subdiv" and
     boundary_chain1: "boundary_chain one_chain1" and
     boundary_chain2: "boundary_chain subdiv" and
     line_integral_exists_on_chain1: "\<forall>(k, \<gamma>) \<in> one_chain1. line_integral_exists F basis \<gamma>" and
     valid_path: "\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>" and
     finite_chain1: "finite one_chain1" and
     finite_basis: "finite basis"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
     "\<forall>(k, \<gamma>) \<in> subdiv. line_integral_exists F basis \<gamma>"
 proof -
   obtain f where f_props:
     "((\<Union>(f ` one_chain1)) = subdiv)"
     "(\<forall>(k,\<gamma>)\<in>one_chain1. if k = 1 then chain_subdiv_path \<gamma> (f(k,\<gamma>)) else chain_subdiv_path (reversepath \<gamma>) (f(k,\<gamma>)))"
     "(\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {})"
     "(\<forall>x \<in> one_chain1. finite (f x))"
     using chain1_eq_chain2 chain_subdiv_chain_character by (auto simp add: pairwise_def chain_subdiv_chain_def)
   have finite_chain2: "finite subdiv"
     using finite_chain1 f_props(1) f_props(4) by blast
   have "\<And>k \<gamma>. (k, \<gamma>) \<in> subdiv \<Longrightarrow> line_integral_exists F basis \<gamma>"
   proof -
     fix k \<gamma>
     assume ass: "(k, \<gamma>) \<in> subdiv"
     then obtain k' \<gamma>' where kp_gammap: "(k',\<gamma>') \<in> one_chain1" "(k,\<gamma>) \<in> f(k',\<gamma>')"
       using f_props(1) by fastforce
     show "line_integral_exists F basis \<gamma>"
     proof (cases "k' = 1")
       assume k_eq_1: "k' = 1"
       then have i:"chain_subdiv_path \<gamma>' (f(k',\<gamma>'))"
         using f_props(2) kp_gammap ass  by auto
       have ii:"boundary_chain (f(k',\<gamma>'))"
         using f_props(1) ass assms kp_gammap  by (meson UN_I boundary_chain_def)
       have iii:"line_integral_exists F basis \<gamma>'"
         using assms kp_gammap by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k', \<gamma>'). valid_path \<gamma>"
         using f_props(1) ass assms kp_gammap by blast
       show ?thesis
         using line_integral_on_path_eq_line_integral_on_equiv_chain'(2)[OF i ii iii iv finite_basis] kp_gammap
         by auto
     next
       assume "k' \<noteq> 1"
       then have k_eq_neg1: "k' = -1"
         using boundary_chain1 kp_gammap
         by (auto simp add: boundary_chain_def)
       then have i:"chain_subdiv_path (reversepath \<gamma>') (f(k',\<gamma>'))"
         using f_props(2) kp_gammap by auto
       have ii:"boundary_chain (f(k',\<gamma>'))"
         using f_props(1) assms kp_gammap  by (meson UN_I boundary_chain_def)
       have iii: " \<forall>(k, \<gamma>)\<in>f (k', \<gamma>'). valid_path \<gamma>"
         using f_props(1) ass assms kp_gammap by blast
       have iv: "valid_path (reversepath \<gamma>')"
         using valid_path_equiv_valid_chain_list[OF i ii iii]
         by force
       have "line_integral_exists F basis \<gamma>'"
         using assms kp_gammap  by blast
       then have x: "line_integral_exists F basis (reversepath \<gamma>')"
         using iv line_integral_on_reverse_path(2) valid_path_reversepath
         by fastforce
       show ?thesis
         using line_integral_on_path_eq_line_integral_on_equiv_chain'(2)[OF i ii x iii finite_basis] kp_gammap
         by auto
     qed
   qed
   then show "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>" by auto
   then show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
     using one_chain_line_integral_eq_line_integral_on_sudivision(1) assms
     by auto
 qed
 
 lemma line_integral_sum_gen:
   assumes finite_basis:
     "finite basis" and
     line_integral_exists:
     "line_integral_exists F basis1 \<gamma>"
     "line_integral_exists F basis2 \<gamma>" and
     basis_partition:
     "basis1 \<union> basis2 = basis" "basis1 \<inter> basis2 = {}"
   shows "line_integral F basis \<gamma> = (line_integral F  basis1 \<gamma>) + (line_integral F basis2 \<gamma>)"
     "line_integral_exists F basis \<gamma>"
    apply (simp add: line_integral_def)
 proof -
   have 0: "integral {0..1} (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                                        (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) =
                     integral {0..1} (\<lambda>x. \<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                            integral {0..1} (\<lambda>x. \<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))"
     using Henstock_Kurzweil_Integration.integral_add line_integral_exists
     by (auto simp add: line_integral_exists_def)
   have 1: "integral {0..1} (\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) =
                        integral {0..1} (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                                             (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)))"
     by (metis (mono_tags, lifting) basis_partition finite_Un finite_basis sum.union_disjoint)
   show "integral {0..1} (\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) =
                   integral {0..1} (\<lambda>x. \<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                   integral {0..1} (\<lambda>x. \<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))"
     using 0 1
     by auto
   have 2: "((\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                                        (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) has_integral
                     integral {0..1} (\<lambda>x. \<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                            integral {0..1} (\<lambda>x. \<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))) {0..1}"
     using Henstock_Kurzweil_Integration.has_integral_add line_integral_exists has_integral_integral
     apply (auto simp add: line_integral_exists_def)
     by blast
   have 3: "(\<lambda>x. \<Sum>b\<in>basis. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) =
                        (\<lambda>x. (\<Sum>b\<in>basis1. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) +
                                             (\<Sum>b\<in>basis2. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)))"
     by (metis (mono_tags, lifting) basis_partition finite_Un finite_basis sum.union_disjoint)
   show "line_integral_exists F basis \<gamma>"
     apply(auto simp add: line_integral_exists_def has_integral_integral)
     using 2 3
     using has_integral_integrable_integral by fastforce
 qed
 
 definition common_boundary_sudivision_exists where
     "common_boundary_sudivision_exists one_chain1 one_chain2 \<equiv>
              \<exists>subdiv. chain_subdiv_chain one_chain1 subdiv \<and>
                       chain_subdiv_chain one_chain2 subdiv \<and>
                       (\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>)\<and>
                       boundary_chain subdiv"
 
 lemma common_boundary_sudivision_commutative:
   "(common_boundary_sudivision_exists one_chain1 one_chain2) = (common_boundary_sudivision_exists one_chain2 one_chain1)"
   apply (simp add: common_boundary_sudivision_exists_def)
   by blast
 
 lemma common_subdivision_imp_eq_line_integral:
   assumes "(common_boundary_sudivision_exists one_chain1 one_chain2)"
     "boundary_chain one_chain1"
     "boundary_chain one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     "finite one_chain1"
     "finite one_chain2"
     "finite basis"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F basis \<gamma>"
 proof -
   obtain subdiv where subdiv_props:
     "chain_subdiv_chain one_chain1 subdiv"
     "chain_subdiv_chain one_chain2 subdiv"
     "(\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>)"
     "(boundary_chain subdiv)"
     using assms
     by (auto simp add: common_boundary_sudivision_exists_def)
   have i: "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"
     using one_chain_line_integral_eq_line_integral_on_sudivision'(2)[OF subdiv_props(1) assms(2) subdiv_props(4) assms(4) subdiv_props(3) assms(5) assms(7)]
     by auto
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     using one_chain_line_integral_eq_line_integral_on_sudivision'(1)[OF subdiv_props(1) assms(2) subdiv_props(4) assms(4) subdiv_props(3) assms(5) assms(7)]
       one_chain_line_integral_eq_line_integral_on_sudivision(1)[OF subdiv_props(2) assms(3) subdiv_props(4) i subdiv_props(3) assms(6) assms(7)]
     by auto
   show "\<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F basis \<gamma>"
     using one_chain_line_integral_eq_line_integral_on_sudivision(2)[OF subdiv_props(2) assms(3) subdiv_props(4) i subdiv_props(3) assms(6) assms(7)]
     by auto
 qed
 
 definition common_sudiv_exists where
     "common_sudiv_exists one_chain1 one_chain2 \<equiv>
         \<exists>subdiv ps1 ps2. chain_subdiv_chain (one_chain1 - ps1) subdiv \<and>
                   chain_subdiv_chain (one_chain2 - ps2) subdiv \<and>
                   (\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>) \<and>
                   (boundary_chain subdiv) \<and>
                   (\<forall>(k, \<gamma>) \<in> ps1. point_path \<gamma>) \<and>
                   (\<forall>(k, \<gamma>) \<in> ps2. point_path \<gamma>)"
 
 lemma common_sudiv_exists_comm:
   shows "common_sudiv_exists C1 C2 = common_sudiv_exists C2 C1"
   by (auto simp add: common_sudiv_exists_def)
 
 lemma line_integral_degenerate_chain:
   assumes "(\<forall>(k, \<gamma>) \<in> chain. point_path \<gamma>)"
   assumes "finite basis"
   shows "one_chain_line_integral F basis chain = 0"
 proof (simp add: one_chain_line_integral_def)
   have "\<forall>(k,g)\<in>chain. line_integral F basis g = 0"
     using assms line_integral_point_path
     by blast
   then have "\<forall>(k,g)\<in>chain. real_of_int k * line_integral F basis g = 0" by auto
   then have "\<And>p. p \<in> chain \<Longrightarrow> (case p of (i, f) \<Rightarrow> real_of_int i * line_integral F basis f) = 0"
     by fastforce
   then show "(\<Sum>x\<in>chain. case x of (k, g) \<Rightarrow> real_of_int k * line_integral F basis g) = 0"
     by simp
 qed
 
 lemma gen_common_subdiv_imp_common_subdiv:
   shows "(common_sudiv_exists one_chain1 one_chain2) = (\<exists>ps1 ps2. (common_boundary_sudivision_exists (one_chain1 - ps1) (one_chain2 - ps2)) \<and> (\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>))"
   by (auto simp add: common_sudiv_exists_def common_boundary_sudivision_exists_def)
 
 lemma common_subdiv_imp_gen_common_subdiv:
   assumes "(common_boundary_sudivision_exists one_chain1 one_chain2)"
   shows "(common_sudiv_exists one_chain1 one_chain2)"
   using assms
   apply (auto simp add: common_sudiv_exists_def common_boundary_sudivision_exists_def)
   by (metis Diff_empty all_not_in_conv)
 
 lemma one_chain_line_integral_point_paths:
   assumes "finite one_chain"
   assumes "finite basis"
   assumes "(\<forall>(k, \<gamma>)\<in>ps. point_path \<gamma>)"
   shows "one_chain_line_integral F basis (one_chain - ps) = one_chain_line_integral F basis (one_chain)"
 proof-
   have 0:"(\<forall>x\<in>ps. case x of (k, g) \<Rightarrow>  (real_of_int k * line_integral F basis g) = 0)"
     using line_integral_point_path assms
     by force
   show "one_chain_line_integral F basis (one_chain - ps) = one_chain_line_integral F basis one_chain"
     unfolding one_chain_line_integral_def using 0 \<open>finite one_chain\<close>
     by (force simp add:  intro: comm_monoid_add_class.sum.mono_neutral_left)
 qed
 
 lemma boundary_chain_diff:
   assumes "boundary_chain one_chain"
   shows "boundary_chain (one_chain - s)"
   using assms
   by (auto simp add: boundary_chain_def)
 
 lemma gen_common_subdivision_imp_eq_line_integral:
   assumes "(common_sudiv_exists one_chain1 one_chain2)"
     "boundary_chain one_chain1"
     "boundary_chain one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     "finite one_chain1"
     "finite one_chain2"
     "finite basis"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F basis \<gamma>"
 proof -
   obtain ps1 ps2 where gen_subdiv: "(common_boundary_sudivision_exists (one_chain1 - ps1) (one_chain2 - ps2))" "(\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>)" " (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>)"
     using assms(1) gen_common_subdiv_imp_common_subdiv
     by blast
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     using one_chain_line_integral_point_paths gen_common_subdiv_imp_common_subdiv
       assms(2-7) gen_subdiv
       common_subdivision_imp_eq_line_integral(1)[OF gen_subdiv(1) boundary_chain_diff[OF assms(2)] boundary_chain_diff[OF assms(3)]]
     by auto
   show "\<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F basis \<gamma>"
   proof-
     obtain subdiv where subdiv_props:
       "chain_subdiv_chain (one_chain1-ps1) subdiv"
       "chain_subdiv_chain (one_chain2-ps2) subdiv"
       "(\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>)"
       "(boundary_chain subdiv)"
       using gen_subdiv(1)
       by (auto simp add: common_boundary_sudivision_exists_def)
     have "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"
       using one_chain_line_integral_eq_line_integral_on_sudivision'(2)[OF subdiv_props(1) boundary_chain_diff[OF assms(2)] subdiv_props(4)] assms(4) subdiv_props(3) assms(5) assms(7)
       by blast
     then have i: "\<forall>(k, \<gamma>)\<in>one_chain2-ps2. line_integral_exists F basis \<gamma>"
       using one_chain_line_integral_eq_line_integral_on_sudivision(2)[OF subdiv_props(2) boundary_chain_diff[OF assms(3)] subdiv_props(4)] subdiv_props(3) assms(6) assms(7)
       by blast
     then show ?thesis
       using  gen_subdiv(3) line_integral_exists_point_path[OF assms(7)]
       by blast
   qed      
 qed
 
 lemma common_sudiv_exists_refl:
       assumes "common_sudiv_exists C1 C2"
       shows "common_sudiv_exists C2 C1"
       using assms
       apply(simp add: common_sudiv_exists_def)
       by auto
 
 lemma chain_subdiv_path_singleton:
   shows "chain_subdiv_path \<gamma> {(1,\<gamma>)}" 
 proof -
   have "rec_join [(1,\<gamma>)] = \<gamma>"
     by (simp add: joinpaths_def)
   then have "set [(1,\<gamma>)] = {(1, \<gamma>)}" "distinct [(1,\<gamma>)]" "rec_join [(1,\<gamma>)] = \<gamma>" "valid_chain_list [(1,\<gamma>)]"
     by auto
   then show ?thesis
     by (metis (no_types) chain_subdiv_path.intros)
 qed
 
 lemma chain_subdiv_path_singleton_reverse:
   shows "chain_subdiv_path (reversepath \<gamma>) {(-1,\<gamma>)}"
 proof -
   have "rec_join [(-1,\<gamma>)] = reversepath \<gamma>"
     by (simp add: joinpaths_def)
   then have "set [(-1,\<gamma>)] = {(- 1, \<gamma>)}" "distinct [(-1,\<gamma>)]" 
              "rec_join [(-1,\<gamma>)] = reversepath \<gamma>" "valid_chain_list [(-1,\<gamma>)]"
     by auto
   then  have "chain_subdiv_path (reversepath \<gamma>) (set [(- 1, \<gamma>)])"
     using chain_subdiv_path.intros by blast
   then show ?thesis
     by simp
 qed
 
 lemma chain_subdiv_chain_refl:
   assumes "boundary_chain C"
   shows "chain_subdiv_chain C C"
   using chain_subdiv_path_singleton chain_subdiv_path_singleton_reverse assms
   unfolding chain_subdiv_chain_def boundary_chain_def pairwise_def using case_prodI2 coeff_cube_to_path.simps 
   by (rule_tac x="\<lambda>x. {x}" in exI) auto
 
 (*path reparam_weaketrization*)
 
 definition reparam_weak where
   "reparam_weak \<gamma>1 \<gamma>2 \<equiv> \<exists>\<phi>. (\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 o \<phi>) x) \<and> \<phi> piecewise_C1_differentiable_on {0..1} \<and> \<phi>(0) = 0 \<and> \<phi>(1) = 1 \<and> \<phi> ` {0..1} = {0..1}"
 
 definition reparam where
   "reparam \<gamma>1 \<gamma>2 \<equiv> \<exists>\<phi>. (\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 o \<phi>) x) \<and> \<phi> piecewise_C1_differentiable_on {0..1} \<and> \<phi>(0) = 0 \<and> \<phi>(1) = 1 \<and> bij_betw \<phi> {0..1} {0..1} \<and> \<phi> -` {0..1} \<subseteq> {0..1} \<and> (\<forall>x\<in>{0..1}. finite (\<phi> -` {x}))"
 
 lemma reparam_weak_eq_refl:
   shows "reparam_weak \<gamma>1 \<gamma>1"
   unfolding reparam_weak_def
   apply (rule_tac x="id" in exI)
   by (auto simp add: id_def piecewise_C1_differentiable_on_def C1_differentiable_on_def continuous_on_id)
 
 lemma line_integral_exists_smooth_one_base:
   assumes "\<gamma> C1_differentiable_on {0..1}" (*To generalise this to valid_path we need a version of has_integral_substitution_strong that allows finite discontinuities of f*)
     "continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
   shows "line_integral_exists F {b} \<gamma>"
 proof-
   have gamma2_differentiable: "(\<forall>x \<in> {0 .. 1}. \<gamma> differentiable at x)"
     using assms(1) 
     by (auto simp add: valid_path_def C1_differentiable_on_eq)
   then have gamma2_b_component_differentiable: "(\<forall>x \<in> {0 .. 1}. (\<lambda>x. (\<gamma> x) \<bullet> b) differentiable at x)"
     by auto
   then have "(\<lambda>x. (\<gamma> x) \<bullet> b) differentiable_on {0..1}"
     using differentiable_at_withinI
     by (auto simp add: differentiable_on_def)
   then have gama2_cont_comp: "continuous_on {0..1} (\<lambda>x. (\<gamma> x) \<bullet> b)"
     using differentiable_imp_continuous_on
     by auto
   have gamma2_cont:"continuous_on {0..1} \<gamma>"
     using assms(1) C1_differentiable_imp_continuous_on
     by (auto simp add: valid_path_def)
   have iii: "continuous_on {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b))"
   proof-
     have 0: "continuous_on {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> b)"
       using assms(2) continuous_on_compose[OF gamma2_cont]
       by (auto simp add: path_image_def) 
     obtain D where D: "(\<forall>x\<in>{0..1}. (\<gamma> has_vector_derivative D x) (at x)) \<and> continuous_on {0..1} D"
       using assms(1)
       by (auto simp add: C1_differentiable_on_def)
     then have *:"\<forall>x\<in>{0..1}. vector_derivative \<gamma> (at x within{0..1}) = D x"
       using vector_derivative_at vector_derivative_at_within_ivl
       by fastforce
     then have "continuous_on {0..1} (\<lambda>x. vector_derivative \<gamma> (at x within{0..1}))"
       using continuous_on_eq D by force
     then have 1: "continuous_on {0..1} (\<lambda>x. (vector_derivative \<gamma> (at x within{0..1})) \<bullet> b)"
       by(auto intro!: continuous_intros)
     show ?thesis
       using continuous_on_mult[OF 0 1]  by auto
   qed
   then have "(\<lambda>x. F (\<gamma> x) \<bullet> b * (vector_derivative \<gamma> (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
     using integrable_continuous_real
     by auto
   then show "line_integral_exists F {b} \<gamma>"
     by(auto simp add: line_integral_exists_def)
 qed
 
 lemma contour_integral_primitive_lemma:
   fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
   assumes "a \<le> b"
       and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
       and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
     shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
              has_integral (f(g b) - f(g a))) {a..b}"
 proof -
   obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
     using assms by (auto simp: piecewise_differentiable_on_def)
   have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
     apply (rule continuous_on_compose [OF cg, unfolded o_def])
     using assms
     apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
     done
   { fix x::real
     assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
     then have "g differentiable at x within {a..b}"
       using k by (simp add: differentiable_at_withinI)
     then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
       by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
     then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
       by (simp add: has_vector_derivative_def scaleR_conv_of_real)
     have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
       using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
     then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
       by (simp add: has_field_derivative_def)
     have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
       using diff_chain_within [OF gdiff fdiff]
       by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
   } note * = this
   show ?thesis
     apply (rule fundamental_theorem_of_calculus_interior_strong)
     using k assms cfg *
     apply (auto simp: at_within_Icc_at)
     done
 qed
 
 lemma line_integral_primitive_lemma: (*Only for real normed fields, i.e. vectors with products*)
   fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a::{euclidean_space,real_normed_field}" and
         g :: "real \<Rightarrow> 'a"
   assumes "\<And>(a::'a). a \<in> s \<Longrightarrow> (f has_field_derivative (f' a)) (at a within s)"
       and "g piecewise_differentiable_on {0::real..1}"  "\<And>x. x \<in> {0..1} \<Longrightarrow> g x \<in> s"
       and "base_vec \<in> Basis"
     shows "((\<lambda>x. ((f'(g x)) * (vector_derivative g (at x within {0..1}))) \<bullet> base_vec)
              has_integral (((f(g 1)) \<bullet> base_vec - (f(g 0)) \<bullet> base_vec))) {0..1}"
 proof -
   obtain k where k: "finite k" "\<forall>x\<in>{0..1} - k. g differentiable (at x within {0..1})" and cg: "continuous_on {0..1} g"
     using assms by (auto simp: piecewise_differentiable_on_def)
   have cfg: "continuous_on {0..1} (\<lambda>x. f (g x))"
     apply (rule continuous_on_compose [OF cg, unfolded o_def])
     using assms
     apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
     done
   { fix x::real
     assume a: "0 < x" and b: "x < 1" and xk: "x \<notin> k"
     then have "g differentiable at x within {0..1}"
       using k by (simp add: differentiable_at_withinI)
     then have "(g has_vector_derivative vector_derivative g (at x within {0..1})) (at x within {0..1})"
       by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
     then have gdiff: "(g has_derivative (\<lambda>u. of_real u * vector_derivative g (at x within {0..1}))) (at x within {0..1})"
       by (simp add: has_vector_derivative_def scaleR_conv_of_real)
     have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {0..1})"
       using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
     then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {0..1})"
       by (simp add: has_field_derivative_def)
     have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {0..1})) (at x within {0..1})"
       using diff_chain_within [OF gdiff fdiff]
       by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
   }
   then have *: "\<And>x. x\<in>{0<..<1} - k \<Longrightarrow> ((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {0..1})) (at x within {0..1})"
        by auto
   have "((\<lambda>x. ((f'(g x))) * ((vector_derivative g (at x within {0..1}))))
              has_integral (((f(g 1)) - (f(g 0))))) {0..1}"
     using fundamental_theorem_of_calculus_interior_strong[OF k(1) zero_le_one _ cfg]
     using k assms cfg * by (auto simp: at_within_Icc_at)
   then have "((\<lambda>x. (((f'(g x))) * ((vector_derivative g (at x within {0..1})))) \<bullet> base_vec)
              has_integral (((f(g 1)) - (f(g 0)))) \<bullet> base_vec) {0..1}"
        using has_integral_componentwise_iff assms(4)
        by fastforce
   then show ?thesis using inner_mult_left'
        by (simp add: inner_mult_left' inner_diff_left)
 qed
 
 lemma reparam_eq_line_integrals:
   assumes reparam: "reparam \<gamma>1 \<gamma>2" and
     pw_smooth: "\<gamma>2 piecewise_C1_differentiable_on {0..1}" and (*To generalise this to valid_path we need a version of has_integral_substitution_strong that allows finite discontinuities of f*)
     cont: "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)" and
     line_integral_ex: "line_integral_exists F {b} \<gamma>2" (*We need to remove this and work on special cases like conservative fields and field/line combinations that satisfy the improper integrals theorem assumptions*)
   shows "line_integral F {b} \<gamma>1 = line_integral F {b} \<gamma>2"
     "line_integral_exists F {b} \<gamma>1"
 proof-
   obtain \<phi> where phi: "(\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 o \<phi>) x)"" \<phi> piecewise_C1_differentiable_on {0..1}"" \<phi>(0) = 0"" \<phi>(1) = 1""bij_betw \<phi> {0..1} {0..1}" "\<phi> -` {0..1} \<subseteq> {0..1}" "\<forall>x\<in>{0..1}. finite (\<phi> -` {x})"
     using reparam
     by (auto simp add: reparam_def)
   obtain s where s: "finite s" "\<phi> C1_differentiable_on {0..1} - s"
     using phi
     by(auto simp add: reparam_def piecewise_C1_differentiable_on_def)
   let ?s = "s \<inter> {0..1}"
   have s_inter: "finite ?s" "\<phi> C1_differentiable_on {0..1} - ?s"
     using s
      apply blast
     by (metis Diff_Compl Diff_Diff_Int Diff_eq inf.commute s(2))
   have cont_phi: "continuous_on {0..1} \<phi>"
     using phi
     by(auto simp add: reparam_def piecewise_C1_differentiable_on_imp_continuous_on)
   obtain s' D where s'_D: "finite s'" "(\<forall>x \<in> {0 .. 1} - s'. \<gamma>2 differentiable at x)" "(\<forall>x\<in>{0..1} - s'. (\<gamma>2 has_vector_derivative D x) (at x)) \<and> continuous_on ({0..1} - s') D"
     using pw_smooth 
     apply (auto simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
     by (simp add: vector_derivative_works)
   let ?s' = "s' \<inter> {0..1}"
   have gamma2_differentiable: "finite ?s'" "(\<forall>x \<in> {0 .. 1} - ?s'. \<gamma>2 differentiable at x)" "(\<forall>x\<in>{0..1} - ?s'. (\<gamma>2 has_vector_derivative D x) (at x)) \<and> continuous_on ({0..1} - ?s') D"
     using s'_D
       apply blast
     using s'_D(2) apply auto[1]
     by (metis Diff_Int2 inf_top.left_neutral s'_D(3))
   then have gamma2_b_component_differentiable: "(\<forall>x \<in> {0 .. 1} - ?s'. (\<lambda>x. (\<gamma>2 x) \<bullet> b) differentiable at x)"
     using differentiable_inner by force
   then have "(\<lambda>x. (\<gamma>2 x) \<bullet> b) differentiable_on {0..1} - ?s'"
     using differentiable_at_withinI
     by (auto simp add: differentiable_on_def)
   then have gama2_cont_comp: "continuous_on ({0..1} - ?s') (\<lambda>x. (\<gamma>2 x) \<bullet> b)"
     using differentiable_imp_continuous_on
     by auto
       (**********Additional needed assumptions ************)
   have s_in01: "?s \<subseteq> {0..1}" by auto
   have s'_in01: "?s' \<subseteq> {0..1}" by auto
   have phi_backimg_s': "\<phi> -` {0..1} \<subseteq> {0..1}" using phi by auto
   have "inj_on \<phi> {0..1}" using phi(5) by (auto simp add: bij_betw_def)
   have bij_phi: "bij_betw \<phi> {0..1} {0..1}" using phi(5) by auto
   have finite_bck_img_single: "\<forall>x\<in>{0..1}. finite (\<phi> -` {x})" using phi by auto
   then have finite_bck_img_single_s': " \<forall>x\<in>?s'. finite (\<phi> -` {x})" by auto
   have gamma2_line_integrable: "(\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)) integrable_on {0..1}"
     using line_integral_ex
     by (simp add: line_integral_exists_def)
       (****************************************************************)
   have finite_neg_img: "finite (\<phi> -` ?s')"
     using finite_bck_img_single
     by (metis Int_iff finite_Int gamma2_differentiable(1) image_vimage_eq inf_img_fin_dom')
   have gamma2_cont:"continuous_on ({0..1} - ?s') \<gamma>2"
     using  gamma2_differentiable
     by (meson continuous_at_imp_continuous_on differentiable_imp_continuous_within)
   have iii: "continuous_on ({0..1} - ?s') (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
   proof-
     have 0: "continuous_on ({0..1} - ?s') (\<lambda>x. F (\<gamma>2 x) \<bullet> b)"
       using cont continuous_on_compose[OF gamma2_cont] continuous_on_compose2 gamma2_cont
       unfolding path_image_def  by fastforce
     have D: "(\<forall>x\<in>{0..1} - ?s'. (\<gamma>2 has_vector_derivative D x) (at x)) \<and> continuous_on ({0..1} - ?s') D"
       using gamma2_differentiable
       by auto                       
     then have *:"\<forall>x\<in>{0..1} - ?s'. vector_derivative \<gamma>2 (at x within{0..1}) = D x"
       using vector_derivative_at vector_derivative_at_within_ivl
       by fastforce
     then have "continuous_on ({0..1} - ?s') (\<lambda>x. vector_derivative \<gamma>2 (at x within{0..1}))"
       using continuous_on_eq D
       by metis
     then have 1: "continuous_on ({0..1} - ?s') (\<lambda>x. (vector_derivative \<gamma>2 (at x within{0..1})) \<bullet> b)"
       by (auto intro!: continuous_intros)
     show ?thesis
       using continuous_on_mult[OF 0 1]
       by auto
   qed
   have iv: "\<phi>(0) \<le> \<phi>(1)"
     using phi(3) phi(4)
     by (simp add: reparam_def)
   have v: "\<phi>`{0..1} \<subseteq> {0..1}"
     using phi
     by (auto simp add: reparam_def bij_betw_def)
   obtain D where D_props: "(\<forall>x\<in>{0..1} - ?s. (\<phi> has_vector_derivative D x) (at x))"
     using s
     by (auto simp add: C1_differentiable_on_def)
   then have "(\<And>x. x \<in> ({0..1} - ?s) \<Longrightarrow> (\<phi> has_vector_derivative D x) (at x within {0..1}))"
     using has_vector_derivative_at_within
     by blast
   then have vi: "(\<And>x. x \<in> ({0..1} - ?s) \<Longrightarrow> (\<phi> has_real_derivative D x) (at x within {0..1}))"
     using has_field_derivative_iff_has_vector_derivative
     by blast
   have a:"((\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) has_integral
               integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
               ({0..1})"
   proof-
     have a: "integral {\<phi> 1..\<phi> 0} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)) = 0" using integral_singleton integral_empty iv
       by (simp add: phi(3) phi(4))
     have b: "((\<lambda>x. D x *\<^sub>R (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) has_integral
                        integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)) - integral {\<phi> 1..\<phi> 0} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
                            {0..1}"
       apply(rule  has_integral_substitution_general_'[OF s_inter(1) zero_le_one gamma2_differentiable(1) v gamma2_line_integrable iii cont_phi finite_bck_img_single_s'])
     proof-
       have "surj_on {0..1} \<phi>"
         using bij_phi
         by (metis (full_types) bij_betw_def image_subsetI rangeI)
       then show "surj_on ?s' \<phi>" using bij_phi s'_in01
         by blast
       show "inj_on \<phi> ({0..1} \<union> (?s \<union> \<phi> -` ?s'))"
       proof-
         have i: "inj_on \<phi> {0..1}" using  bij_phi 
           using bij_betw_def by blast
         have ii: "({0..1} \<union> (?s \<union> \<phi> -` ?s')) = {0..1}" using phi_backimg_s' s_in01 s'_in01
           by blast
         show ?thesis using i ii by auto
       qed
       show "\<And>x. x \<in> {0..1} - ?s \<Longrightarrow> (\<phi> has_real_derivative D x) (at x within {0..1})"
         using vi by blast
     qed
     show ?thesis using a b by auto
   qed
   then have b: "integral {0..1} (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = 
                        integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
     by auto
   have gamma2_vec_diffable: "\<And>x::real. x \<in> {0..1} - ((\<phi> -` ?s') \<union> ?s) \<Longrightarrow> ((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
   proof-
     fix x::real 
     assume ass: "x \<in> {0..1} -   ((\<phi> -` ?s') \<union> ?s)"
     have zer_le_x_le_1:"0\<le> x \<and> x \<le> 1" using ass
       by simp
     show "((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
     proof-
       have **: "\<gamma>2 differentiable at (\<phi> x)"
         using  gamma2_differentiable(2) ass v 
         by blast
       have ***: " \<phi> differentiable at x"
         using s ass
         by (auto simp add: C1_differentiable_on_eq)
       then show "((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
         using differentiable_chain_at[OF *** **] 
         by (auto simp add: vector_derivative_works)
     qed
   qed
   then have gamma2_vec_deriv_within: "\<And>x::real. x \<in> {0..1} - ((\<phi> -` ?s') \<union> ?s) \<Longrightarrow> vector_derivative (\<gamma>2 \<circ> \<phi>) (at x) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
     using vector_derivative_at_within_ivl[OF gamma2_vec_diffable]
     by auto
   have "\<forall>x\<in>{0..1} - ((\<phi> -` ?s') \<union> ?s). D x * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b) = (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
   proof
     fix x::real
     assume ass: "x \<in> {0..1} -((\<phi> -` ?s') \<union> ?s)"
     then have 0: "\<phi> differentiable (at x)"
       using s by (auto simp add: C1_differentiable_on_def differentiable_def has_vector_derivative_def)
     obtain D2 where "(\<phi> has_vector_derivative D2) (at x)"
       using D_props ass  by blast
     have "\<phi> x \<in> {0..1} - ?s'"
       using phi(5) ass by (metis Diff_Un Diff_iff Int_iff bij_betw_def image_eqI vimageI)
     then have 1: "\<gamma>2 differentiable (at (\<phi> x))"
       using gamma2_differentiable
       by auto
     have 3:" vector_derivative \<gamma>2 (at (\<phi> x)) =  vector_derivative \<gamma>2 (at (\<phi> x) within {0..1})"
     proof-
       have *:"0\<le> \<phi> x \<and> \<phi> x \<le> 1" using phi(5) ass
         using \<open>\<phi> x \<in> {0..1} - s' \<inter> {0..1}\<close> by auto
       then have **:"(\<gamma>2 has_vector_derivative (vector_derivative \<gamma>2 (at (\<phi> x)))) (at (\<phi> x))"
         using 1 vector_derivative_works  by auto
       show ?thesis
         using * vector_derivative_at_within_ivl[OF **]  by auto
     qed
     show "D x * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b"
       using vector_derivative_chain_at[OF 0 1]
       apply (auto simp add: gamma2_vec_deriv_within[OF ass, symmetric] 3[symmetric])
       using D_props ass vector_derivative_at
       by fastforce
   qed
   then have c:"\<And>x.  x\<in>({0..1} -((\<phi> -` ?s') \<union> ?s)) \<Longrightarrow> D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = 
                                     F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
     by auto
   then have d: "integral ({0..1}) (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = 
                                             integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b))"
   proof-
     have "negligible ((\<phi> -` ?s') \<union> ?s)" using finite_neg_img s(1) by auto
     then show ?thesis
       using c integral_spike  by (metis(no_types,lifting))
   qed
   have phi_in_int: "(\<And>x. x \<in> {0..1} \<Longrightarrow> \<phi> x \<in> {0..1})" using phi
     using v by blast
   then have e: "((\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) has_integral
                              integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))){0..1}"
   proof-
     have "negligible ?s" using s_inter(1) by auto
     have 0: "negligible ((\<phi> -` ?s') \<union> ?s)" using finite_neg_img s(1) by auto
     have c':"\<forall> x\<in> {0..1} - ((\<phi> -` ?s') \<union> ?s). D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = 
                                     F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
       using c  by blast
     have has_integral_spike_eq': "\<And>s t f g y. negligible s \<Longrightarrow>
                                                        \<forall>x\<in>t - s. g x = f x \<Longrightarrow> (f has_integral y) t = (g has_integral y) t"
       using has_integral_spike_eq by metis
     show ?thesis
       using a has_integral_spike_eq'[OF 0 c']  by blast
   qed
   then have f: "((\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative  \<gamma>1 (at x within {0..1}) \<bullet> b)) has_integral
                      integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
                        {0..1}"
   proof-
     assume ass: "((\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) has_integral
                             integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
                              {0..1}"
     have *:"\<forall>x\<in>{0..1} - ( ((\<phi> -` ?s') \<union> ?s) \<union> {0,1}). (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) x =
                                                   (\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative \<gamma>1 (at x within {0..1}) \<bullet> b)) x"
     proof-
       have "\<forall>x\<in>{0<..<1}  - (\<phi> -` ?s' \<union> ?s). (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b) = (vector_derivative  (\<gamma>1) (at x within {0..1}) \<bullet> b)"
       proof
         have i: "open ({0<..<1}  - ((\<phi> -` ?s') \<union> ?s))" using open_diff s(1) open_greaterThanLessThan finite_neg_img
           by (simp add: open_diff)
         have ii: "\<forall>x\<in>{0<..<1::real}  - (\<phi> -` ?s' \<union> ?s). (\<gamma>2 \<circ> \<phi>) x = \<gamma>1 x" using phi(1)
           by auto                              
         fix x::real
         assume ass: " x \<in> {0<..<1::real} -  ((\<phi> -` ?s') \<union> ?s)"
         then have iii: "(\<gamma>2 \<circ> \<phi> has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})) (at x)"
             by (metis (no_types) Diff_iff add.commute add_strict_mono ass atLeastAtMost_iff gamma2_vec_deriv_within gamma2_vec_diffable greaterThanLessThan_iff less_irrefl not_le)
             (*Most crucial but annoying step*)
         then have iv:"(\<gamma>1 has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})) (at x)"
           using has_derivative_transform_within_open i ii ass
           apply(auto simp add: has_vector_derivative_def)
             apply (meson ass has_derivative_transform_within_open i ii)
            apply (meson ass has_derivative_transform_within_open i ii)
           by (meson ass has_derivative_transform_within_open i ii)
         have v: "0 \<le> x" "x \<le> 1" using ass by auto
         have 0: "vector_derivative \<gamma>1 (at x within {0..1}) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
           using vector_derivative_at_within_ivl[OF iv v(1) v(2) zero_less_one]
           by force
         have 1: "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
           using vector_derivative_at_within_ivl[OF iii v(1) v(2) zero_less_one]
           by force
         then have "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) = vector_derivative \<gamma>1 (at x within {0..1})"
           using 0 1 by auto
         then show "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b = vector_derivative \<gamma>1 (at x within {0..1}) \<bullet> b" by auto
       qed
       then have i: "\<forall>x\<in>{0..1} - ( ((\<phi> -` ?s') \<union> ?s)\<union>{0,1}). (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b) = (vector_derivative  (\<gamma>1) (at x within {0..1}) \<bullet> b)"
         by auto
       have ii: "\<forall>x\<in>{0..1} - (((\<phi> -` ?s') \<union> ?s)\<union>{0,1}).  F (\<gamma>1 x) \<bullet> b =  F (\<gamma>2 (\<phi> x)) \<bullet> b"
         using phi(1)
         by auto
       show ?thesis 
         using i ii  by metis
     qed
     have **: "negligible ((\<phi> -` ?s') \<union> ?s \<union> {0, 1})" using s(1) finite_neg_img by auto
     have has_integral_spike_eq': "\<And>s t g f y. negligible s \<Longrightarrow>
                                 \<forall>x\<in>t - s. g x = f x \<Longrightarrow> (f has_integral y) t = (g has_integral y) t"
       using has_integral_spike_eq by metis
     show ?thesis
       using has_integral_spike_eq'[OF ** *] ass
       by blast
   qed
   then show "line_integral_exists F {b} \<gamma>1"
     using phi  by(auto simp add: line_integral_exists_def)
   have "integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) =
                       integral ({0..1}) (\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative \<gamma>1  (at x within {0..1}) \<bullet> b))"
     using integral_unique[OF e] integral_unique[OF f]
     by metis
   moreover have "integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) =
                          integral ({0..1}) (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
     using b d phi by (auto simp add:)
   ultimately show "line_integral F {b} \<gamma>1 = line_integral F {b} \<gamma>2"
     using phi  by(auto simp add: line_integral_def)
 qed
 
 lemma reparam_weak_eq_line_integrals:
   assumes "reparam_weak \<gamma>1 \<gamma>2"
     "\<gamma>2 C1_differentiable_on {0..1}" (*To generalise this to valid_path we need a version of has_integral_substitution_strong that allows finite discontinuities of f*)
     "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
   shows "line_integral F {b} \<gamma>1 = line_integral F {b} \<gamma>2"
     "line_integral_exists F {b} \<gamma>1"
 proof-
   obtain \<phi> where phi: "(\<forall>x\<in>{0..1}. \<gamma>1 x = (\<gamma>2 o \<phi>) x)"" \<phi> piecewise_C1_differentiable_on {0..1}"" \<phi>(0) = 0"" \<phi>(1) = 1"" \<phi> ` {0..1} = {0..1}"
     using assms(1)
     by (auto simp add: reparam_weak_def)
   obtain s where s: "finite s" "\<phi> C1_differentiable_on {0..1} - s"
     using phi
     by(auto simp add: reparam_weak_def piecewise_C1_differentiable_on_def)
   have cont_phi: "continuous_on {0..1} \<phi>"
     using phi
     by(auto simp add: reparam_weak_def piecewise_C1_differentiable_on_imp_continuous_on)
   have gamma2_differentiable: "(\<forall>x \<in> {0 .. 1}. \<gamma>2 differentiable at x)"
     using assms(2) 
     by (auto simp add: valid_path_def C1_differentiable_on_eq)
   then have gamma2_b_component_differentiable: "(\<forall>x \<in> {0 .. 1}. (\<lambda>x. (\<gamma>2 x) \<bullet> b) differentiable at x)"
     by auto
   then have "(\<lambda>x. (\<gamma>2 x) \<bullet> b) differentiable_on {0..1}"
     using differentiable_at_withinI
     by (auto simp add: differentiable_on_def)
   then have gama2_cont_comp: "continuous_on {0..1} (\<lambda>x. (\<gamma>2 x) \<bullet> b)"
     using differentiable_imp_continuous_on
     by auto
   have gamma2_cont:"continuous_on {0..1} \<gamma>2"
     using assms(2) C1_differentiable_imp_continuous_on
     by (auto simp add: valid_path_def)
   have iii: "continuous_on {0..1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
   proof-
     have 0: "continuous_on {0..1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b)"
       using assms(3) continuous_on_compose[OF gamma2_cont]
       by (auto simp add: path_image_def) 
     obtain D where D: "(\<forall>x\<in>{0..1}. (\<gamma>2 has_vector_derivative D x) (at x)) \<and> continuous_on {0..1} D"
       using assms(2)  by (auto simp add: C1_differentiable_on_def)
     then have *:"\<forall>x\<in>{0..1}. vector_derivative \<gamma>2 (at x within{0..1}) = D x"
       using vector_derivative_at vector_derivative_at_within_ivl
       by fastforce
     then have "continuous_on {0..1} (\<lambda>x. vector_derivative \<gamma>2 (at x within{0..1}))"
       using continuous_on_eq D  by force
     then have 1: "continuous_on {0..1} (\<lambda>x. (vector_derivative \<gamma>2 (at x within{0..1})) \<bullet> b)"
       by (auto intro!: continuous_intros)
     show ?thesis
       using continuous_on_mult[OF 0 1]  by auto
   qed
   have iv: "\<phi>(0) \<le> \<phi>(1)"
     using phi(3) phi(4)  by (simp add: reparam_weak_def)
   have v: "\<phi>`{0..1} \<subseteq> {0..1}"
     using phi(5)  by (simp add: reparam_weak_def)
   obtain D where D_props: "(\<forall>x\<in>{0..1} - s. (\<phi> has_vector_derivative D x) (at x))"
     using s
     by (auto simp add: C1_differentiable_on_def)
   then have "(\<And>x. x \<in> ({0..1} -s) \<Longrightarrow> (\<phi> has_vector_derivative D x) (at x within {0..1}))"
     using has_vector_derivative_at_within  by blast
   then have vi: "(\<And>x. x \<in> ({0..1} - s) \<Longrightarrow> (\<phi> has_real_derivative D x) (at x within {0..1}))"
     using has_field_derivative_iff_has_vector_derivative
     by blast
   have a:"((\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) has_integral
               integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
               {0..1}"
     using has_integral_substitution_strong[OF s(1) zero_le_one iv v iii cont_phi vi]
     by simp
   then have b: "integral {0..1} (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = 
                        integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
     by auto
   have gamma2_vec_diffable: "\<And>x::real. x \<in> {0..1} -s \<Longrightarrow> ((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
   proof-
     fix x::real 
     assume ass: "x \<in> {0..1} -s"
     have zer_le_x_le_1:"0\<le> x \<and> x \<le> 1" using ass by auto
     show "((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
     proof-
       have **: "\<gamma>2 differentiable at (\<phi> x)"
         using phi gamma2_differentiable
         by (auto simp add: zer_le_x_le_1)
       have ***: " \<phi> differentiable at x"
         using s ass
         by (auto simp add: C1_differentiable_on_eq)
       then show "((\<gamma>2 o \<phi>) has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x)) (at x)"
         using differentiable_chain_at[OF *** **] 
         by (auto simp add: vector_derivative_works)
     qed
   qed
   then have gamma2_vec_deriv_within: "\<And>x::real. x \<in> {0..1} -s \<Longrightarrow> vector_derivative (\<gamma>2 \<circ> \<phi>) (at x) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
     using vector_derivative_at_within_ivl[OF gamma2_vec_diffable]
     by auto
   have "\<forall>x\<in>{0..1} - s. D x * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b) = (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
   proof
     fix x::real
     assume ass: "x \<in> {0..1} -s"
     then have 0: "\<phi> differentiable (at x)"
       using s
       by (auto simp add: C1_differentiable_on_def differentiable_def has_vector_derivative_def)
     obtain D2 where "(\<phi> has_vector_derivative D2) (at x)"
       using D_props ass
       by blast
     have "\<phi> x \<in> {0..1}"
       using phi(5) ass 
       by (auto simp add: reparam_weak_def)
     then have 1: "\<gamma>2 differentiable (at (\<phi> x))"
       using gamma2_differentiable
       by auto
     have 3:" vector_derivative \<gamma>2 (at (\<phi> x)) =  vector_derivative \<gamma>2 (at (\<phi> x) within {0..1})"
     proof-
       have *:"0\<le> \<phi> x \<and> \<phi> x \<le> 1" using phi(5) ass by auto
       then have **:"(\<gamma>2 has_vector_derivative (vector_derivative \<gamma>2 (at (\<phi> x)))) (at (\<phi> x))"
         using 1 vector_derivative_works
         by auto
       show ?thesis
         using * vector_derivative_at_within_ivl[OF **]
         by auto
     qed
     show "D x * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b"
       using vector_derivative_chain_at[OF 0 1]
       apply (auto simp add: gamma2_vec_deriv_within[OF ass, symmetric] 3[symmetric])
       using D_props ass vector_derivative_at
       by fastforce
   qed
   then have c:"\<And>x.  x\<in>({0..1} -s) \<Longrightarrow> D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = 
                                     F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
     by auto
   then have d: "integral ({0..1}) (\<lambda>x. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b))) = 
                                             integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b))"
   proof-
     have "negligible s" using s(1) by auto
     then show ?thesis
       using c integral_spike  by (metis(no_types,lifting))
   qed
   have phi_in_int: "(\<And>x. x \<in> {0..1} \<Longrightarrow> \<phi> x \<in> {0..1})" using phi
     by(auto simp add:)
   then have e: "((\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) has_integral
                              integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))){0..1}"
   proof-
     have 0:"negligible s" using s(1) by auto
     have c':"\<forall> x\<in> {0..1} -s. D x * (F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative \<gamma>2 (at (\<phi> x) within {0..1}) \<bullet> b)) = 
                                     F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)"
       using c by auto
     have has_integral_spike_eq': "\<And>s t f g y. negligible s \<Longrightarrow>
                                                \<forall>x\<in>t - s. g x = f x \<Longrightarrow> (f has_integral y) t = (g has_integral y) t"
       using has_integral_spike_eq by metis
     show ?thesis
       using a has_integral_spike_eq'[OF 0 c']  by blast
   qed
   then have f: "((\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative  \<gamma>1 (at x within {0..1}) \<bullet> b)) has_integral
                      integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
                        {0..1}"
   proof-
     assume ass: "((\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) has_integral
                             integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b)))
                              {0..1}"
     have *:"\<forall>x\<in>{0..1} - (s\<union>{0,1}). (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) x =
                                                   (\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative \<gamma>1 (at x within {0..1}) \<bullet> b)) x"
     proof-
       have "\<forall>x\<in>{0<..<1} - s. (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b) = (vector_derivative  (\<gamma>1) (at x within {0..1}) \<bullet> b)"
       proof
         have i: "open ({0<..<1} - s)" using open_diff s open_greaterThanLessThan by blast
         have ii: "\<forall>x\<in>{0<..<1::real} - s. (\<gamma>2 \<circ> \<phi>) x = \<gamma>1 x" using phi(1)
           by auto                              
         fix x::real
         assume ass: " x \<in> {0<..<1::real} - s"
         then have iii: "(\<gamma>2 \<circ> \<phi> has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})) (at x)"
           using has_vector_derivative_at_within gamma2_vec_deriv_within gamma2_vec_diffable
           by auto
             (*Most crucial but annoying step*)
         then have iv:"(\<gamma>1 has_vector_derivative vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})) (at x)"
           using has_derivative_transform_within_open i ii ass
           apply(auto simp add: has_vector_derivative_def)
           by force
         have v: "0 \<le> x" "x \<le> 1" using ass by auto
         have 0: "vector_derivative \<gamma>1 (at x within {0..1}) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
           using vector_derivative_at_within_ivl[OF iv v(1) v(2) zero_less_one]
           by force
         have 1: "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
           using vector_derivative_at_within_ivl[OF iii v(1) v(2) zero_less_one]
           by force
         then have "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) = vector_derivative \<gamma>1 (at x within {0..1})"
           using 0 1 by auto
         then show "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b = vector_derivative \<gamma>1 (at x within {0..1}) \<bullet> b" by auto
       qed
       then have i: "\<forall>x\<in>{0..1} - (s\<union>{0,1}). (vector_derivative  (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b) = (vector_derivative  (\<gamma>1) (at x within {0..1}) \<bullet> b)"
         by auto
       have ii: "\<forall>x\<in>{0..1} - (s\<union>{0,1}).  F (\<gamma>1 x) \<bullet> b =  F (\<gamma>2 (\<phi> x)) \<bullet> b"
         using phi(1)  by auto
       show ?thesis 
         using i ii by auto
     qed
     have **: "negligible (s\<union>{0,1})" using s(1) by auto
     have has_integral_spike_eq': "\<And>s t g f y. negligible s \<Longrightarrow>
                                 \<forall>x\<in>t - s. g x = f x \<Longrightarrow> (f has_integral y) t = (g has_integral y) t"
       using has_integral_spike_eq by metis
     show ?thesis
       using has_integral_spike_eq'[OF ** *] ass
       by blast
   qed
   then show "line_integral_exists F {b} \<gamma>1"
     using phi  by(auto simp add: line_integral_exists_def)
   have "integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) =
                       integral ({0..1}) (\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative \<gamma>1  (at x within {0..1}) \<bullet> b))"
     using integral_unique[OF e] integral_unique[OF f]
     by metis
   moreover have "integral ({0..1}) (\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) =
                          integral ({0..1}) (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))"
     using b d phi  by (auto simp add:)
   ultimately show "line_integral F {b} \<gamma>1 = line_integral F {b} \<gamma>2"
     using phi by(auto simp add: line_integral_def)
 qed
 
 lemma line_integral_sum_basis:
   assumes "finite (basis::('a::euclidean_space) set)"  "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>"
   shows "line_integral F basis \<gamma> = (\<Sum>b\<in>basis. line_integral F {b} \<gamma>)"
         "line_integral_exists F basis \<gamma>"
   using assms
 proof(induction basis)
   show "line_integral F {} \<gamma> = (\<Sum>b\<in>{}. line_integral F {b} \<gamma>)"
     by(auto simp add: line_integral_def)
   show "\<forall>b\<in>{}. line_integral_exists F {b} \<gamma> \<Longrightarrow> line_integral_exists F {} \<gamma>"
     by(simp add: line_integral_exists_def integrable_0)
 next
   fix basis::"('a::euclidean_space) set"
   fix x::"'a::euclidean_space"
   fix  \<gamma>
   assume ind_hyp: "(\<forall>b\<in>basis. line_integral_exists F {b} \<gamma> \<Longrightarrow> line_integral_exists F basis \<gamma>)"
     "(\<forall>b\<in>basis. line_integral_exists F {b} \<gamma> \<Longrightarrow> line_integral F basis \<gamma> = (\<Sum>b\<in>basis. line_integral F {b} \<gamma>))"
   assume step: "finite basis"
     "x \<notin> basis"
     "\<forall>b\<in>insert x basis. line_integral_exists F {b} \<gamma>"
   then have 0: "line_integral_exists F {x} \<gamma>" by auto
   have 1:"line_integral_exists F basis \<gamma>"
     using ind_hyp step by auto
   then show "line_integral_exists F (insert x basis) \<gamma>"
     using step(1) step(2) line_integral_sum_gen(2)[OF _ 0 1] by simp
   have 3: "finite (insert x basis)" using step(1) by auto
   have "line_integral F basis \<gamma> = (\<Sum>b\<in>basis. line_integral F {b} \<gamma>)"
     using ind_hyp step by auto
   then show "line_integral F (insert x basis) \<gamma> = (\<Sum>b\<in>insert x basis. line_integral F {b} \<gamma>)"
     using step(1) step(2) line_integral_sum_gen(1)[OF 3 0 1]
     by force
 qed
 
 lemma reparam_weak_eq_line_integrals_basis:
   assumes "reparam_weak \<gamma>1 \<gamma>2"
     "\<gamma>2 C1_differentiable_on {0..1}" (*To generalise this to valid_path we need a version of has_integral_substitution_strong that allows finite discontinuities*)
     "\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "finite basis"
   shows "line_integral F basis \<gamma>1 = line_integral F basis \<gamma>2"
     "line_integral_exists F basis \<gamma>1"
 proof- 
   show "line_integral_exists F basis \<gamma>1"
     using reparam_weak_eq_line_integrals(2)[OF assms(1) assms(2)] assms(3-4) line_integral_sum_basis(2)[OF assms(4)]
     by(simp add: subset_iff)
   show "line_integral F basis \<gamma>1 = line_integral F basis \<gamma>2"
     using reparam_weak_eq_line_integrals[OF assms(1) assms(2)] assms(3-4) line_integral_sum_basis(1)[OF assms(4)]
       line_integral_exists_smooth_one_base[OF assms(2)]
     by(simp add: subset_iff)
 qed
 
 lemma reparam_eq_line_integrals_basis:
   assumes "reparam \<gamma>1 \<gamma>2"
     "\<gamma>2 piecewise_C1_differentiable_on {0..1}" 
     "\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "finite basis"
     "\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>2" (*We need to remove this and work on special cases like conservative fields and field/line combinations that satisfy the improper integrals theorem assumptions*)
   shows "line_integral F basis \<gamma>1 = line_integral F basis \<gamma>2"
     "line_integral_exists F basis \<gamma>1"
 proof- 
   show "line_integral_exists F basis \<gamma>1"
     using reparam_eq_line_integrals(2)[OF assms(1) assms(2)] assms(3-5) line_integral_sum_basis(2)[OF assms(4)]
     by(simp add: subset_iff)
   show "line_integral F basis \<gamma>1 = line_integral F basis \<gamma>2"
     using reparam_eq_line_integrals[OF assms(1) assms(2)] assms(3-5) line_integral_sum_basis(1)[OF assms(4)]
     by(simp add: subset_iff)
 qed
 
 lemma line_integral_exists_smooth:
   assumes "\<gamma> C1_differentiable_on {0..1}" (*To generalise this to valid_path we need a version of has_integral_substitution_strong that allows finite discontinuities*)
     "\<forall>(b::'a::euclidean_space) \<in>basis. continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
     "finite basis"
   shows "line_integral_exists F basis \<gamma>"
   using reparam_weak_eq_line_integrals_basis(2)[OF reparam_weak_eq_refl[where ?\<gamma>1.0 = \<gamma>]] assms
   by fastforce
 
 lemma smooth_path_imp_reverse:
   assumes "g C1_differentiable_on {0..1}"
   shows "(reversepath g) C1_differentiable_on {0..1}"
   using assms continuous_on_const
   apply (auto simp: reversepath_def)
   apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
     apply (auto simp: C1_differentiable_on_eq)
   apply (simp add: finite_vimageI inj_on_def)
   done
 
 lemma piecewise_smooth_path_imp_reverse:
   assumes "g piecewise_C1_differentiable_on {0..1}"
   shows "(reversepath g) piecewise_C1_differentiable_on {0..1}"
   using assms valid_path_reversepath
   using valid_path_def by blast
 
 definition chain_reparam_weak_chain where
   "chain_reparam_weak_chain one_chain1 one_chain2 \<equiv>
       \<exists>f. bij f \<and> f ` one_chain1 = one_chain2 \<and> (\<forall>(k,\<gamma>)\<in>one_chain1. if k = fst (f(k,\<gamma>)) then reparam_weak \<gamma> (snd (f(k,\<gamma>))) else reparam_weak \<gamma> (reversepath (snd (f(k,\<gamma>)))))"
 
 lemma chain_reparam_weak_chain_line_integral:
   assumes "chain_reparam_weak_chain one_chain1 one_chain2"
     "\<forall>(k2,\<gamma>2)\<in>one_chain2. \<gamma>2 C1_differentiable_on {0..1}"
     "\<forall>(k2,\<gamma>2)\<in>one_chain2.\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "finite basis"                                                                     
   and bound1: "boundary_chain one_chain1"
   and bound2: "boundary_chain one_chain2"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
 proof-
   obtain f where f: "bij f"
     "(\<forall>(k,\<gamma>)\<in>one_chain1. if k = fst (f(k,\<gamma>)) then reparam_weak \<gamma> (snd (f(k,\<gamma>))) else reparam_weak \<gamma> (reversepath (snd (f(k,\<gamma>)))))"
     "f ` one_chain1 = one_chain2"
     using assms(1)
     by (auto simp add: chain_reparam_weak_chain_def)
   have 0:" \<forall>x\<in>one_chain1. (case x of (k, \<gamma>) \<Rightarrow> (real_of_int k * line_integral F basis \<gamma>) = (case f x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma>) \<and>
                                                         line_integral_exists F basis \<gamma>)"
   proof
     {fix k1 \<gamma>1
       assume ass1: "(k1,\<gamma>1) \<in>one_chain1"
       have "real_of_int k1 * line_integral F basis \<gamma>1 = (case (f (k1,\<gamma>1)) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                       line_integral_exists F basis \<gamma>1"
       proof(cases)
         assume ass2: "k1 = 1"
         let ?k2 = "fst (f (k1, \<gamma>1))"
         let ?\<gamma>2 = "snd (f (k1, \<gamma>1))"
         have "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2\<and>
                               line_integral_exists F basis \<gamma>1"
         proof(cases)
           assume ass3: "?k2 = 1"
           then have 0: "reparam_weak \<gamma>1 ?\<gamma>2"
             using ass1 ass2 f(2)
             by auto
           have 1: "?\<gamma>2 C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image ?\<gamma>2) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1
             by force
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_weak_eq_line_integrals_basis[OF 0 1 2 assms(4)]
               ass2 ass3
             by auto
         next
           assume "?k2 \<noteq> 1"
           then have ass3: "?k2 = -1"
             using bound2 ass1 f(3) unfolding boundary_chain_def by force
           then have 0: "reparam_weak \<gamma>1 (reversepath ?\<gamma>2)"
             using ass1 ass2 f(2)
             by auto
           have 1: "(reversepath ?\<gamma>2) C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1 smooth_path_imp_reverse
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image (reversepath ?\<gamma>2)) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1 path_image_reversepath
             by force
           have 3: "line_integral F basis ?\<gamma>2 = - line_integral F basis (reversepath ?\<gamma>2)"
           proof-
             have i:"valid_path (reversepath ?\<gamma>2) "
               using 1 C1_differentiable_imp_piecewise
               by (auto simp add: valid_path_def)
             show ?thesis 
               using line_integral_on_reverse_path(1)[OF i line_integral_exists_smooth[OF 1 2 ]] assms
               by auto
           qed
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_weak_eq_line_integrals_basis[OF 0 1 2 assms(4)]
               ass2 ass3 3
             by auto
         qed
         then show "real_of_int k1 * line_integral F basis \<gamma>1 = (case f (k1, \<gamma>1) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                                    line_integral_exists F basis \<gamma>1"
           by (simp add: case_prod_beta')
       next
         assume "k1 \<noteq> 1"
         then have ass2: "k1 = -1"
           using bound1 ass1 f(3) unfolding boundary_chain_def by force
         let ?k2 = "fst (f (k1, \<gamma>1))"
         let ?\<gamma>2 = "snd (f (k1, \<gamma>1))"
         have "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                               line_integral_exists F basis \<gamma>1"
         proof(cases)
           assume ass3: "?k2 = 1"
           then have 0: "reparam_weak \<gamma>1 (reversepath ?\<gamma>2)"
             using ass1 ass2 f(2)
             by auto
           have 1: "(reversepath ?\<gamma>2) C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1 smooth_path_imp_reverse
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image (reversepath ?\<gamma>2)) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1 path_image_reversepath
             by force
           have 3: "line_integral F basis ?\<gamma>2 = - line_integral F basis (reversepath ?\<gamma>2)"
           proof-
             have i:"valid_path (reversepath ?\<gamma>2) "
               using 1 C1_differentiable_imp_piecewise
               by (auto simp add: valid_path_def)
             show ?thesis 
               using line_integral_on_reverse_path(1)[OF i line_integral_exists_smooth[OF 1 2 assms(4)]]
               by auto
           qed
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_weak_eq_line_integrals_basis[OF 0 1 2 assms(4)]
               ass2 ass3 3
             by auto
         next
           assume "?k2 \<noteq> 1"
           then have ass3: "?k2 = -1"
             using bound2 ass1 f(3) unfolding boundary_chain_def by force
           then have 0: "reparam_weak \<gamma>1 ?\<gamma>2"
             using ass1 ass2 f(2)  by auto
           have 1: "?\<gamma>2 C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1  by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image ?\<gamma>2) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1  by force
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_weak_eq_line_integrals_basis[OF 0 1 2 assms(4)]  ass2 ass3
             by auto
         qed
         then show "real_of_int k1 * line_integral F basis \<gamma>1 = (case f (k1, \<gamma>1) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                                    line_integral_exists F basis \<gamma>1"
           by (simp add: case_prod_beta')
       qed
     }
     then show "\<And>x. x \<in> one_chain1 \<Longrightarrow>
                                 (case x of (k, \<gamma>) \<Rightarrow> (real_of_int k * line_integral F basis \<gamma>) = (case f x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma>) \<and>
                                                         line_integral_exists F basis \<gamma>)"
       by (auto simp add: case_prod_beta')
   qed
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     using 0 by(simp add: one_chain_line_integral_def sum_bij[OF f(1) _ f(3)] split_beta)
   show "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     using 0 by blast
 qed
 
 definition chain_reparam_chain where
   "chain_reparam_chain one_chain1 one_chain2 \<equiv>
       \<exists>f. bij f \<and> f ` one_chain1 = one_chain2 \<and> (\<forall>(k,\<gamma>)\<in>one_chain1. if k = fst (f(k,\<gamma>)) then reparam \<gamma> (snd (f(k,\<gamma>))) else reparam \<gamma> (reversepath (snd (f(k,\<gamma>)))))"
 
 definition chain_reparam_weak_path::"((real) \<Rightarrow> (real * real)) \<Rightarrow> ((int * ((real) \<Rightarrow> (real * real))) set) \<Rightarrow> bool" where
   "chain_reparam_weak_path \<gamma> one_chain
            \<equiv> \<exists>l. set l = one_chain \<and> distinct l \<and> reparam \<gamma> (rec_join l) \<and> valid_chain_list l \<and> l \<noteq> []"
 
 lemma chain_reparam_chain_line_integral:
   assumes "chain_reparam_chain one_chain1 one_chain2"
     "\<forall>(k2,\<gamma>2)\<in>one_chain2. \<gamma>2 piecewise_C1_differentiable_on {0..1}"
     "\<forall>(k2,\<gamma>2)\<in>one_chain2.\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "finite basis"                                                                     
   and bound1: "boundary_chain one_chain1"
   and bound2: "boundary_chain one_chain2"
   and line: "\<forall>(k2,\<gamma>2)\<in>one_chain2. (\<forall>b\<in>basis. line_integral_exists F {b} \<gamma>2)"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
 proof-
   obtain f where f: "bij f"
     "(\<forall>(k,\<gamma>)\<in>one_chain1. if k = fst (f(k,\<gamma>)) then reparam \<gamma> (snd (f(k,\<gamma>))) else reparam \<gamma> (reversepath (snd (f(k,\<gamma>)))))"
     "f ` one_chain1 = one_chain2"
     using assms(1)
     by (auto simp add: chain_reparam_chain_def)
   have integ_exist_b: "\<forall>(k1,\<gamma>1)\<in>one_chain1. \<forall>b\<in>basis. line_integral_exists F {b} (snd (f (k1, \<gamma>1)))"
     using line f by fastforce
   have valid_cubes: "\<forall>(k1,\<gamma>1)\<in>one_chain1. valid_path (snd (f (k1, \<gamma>1)))"
     using assms(2) f(3) valid_path_def by fastforce
   have integ_rev_exist_b: "\<forall>(k1,\<gamma>1)\<in>one_chain1. \<forall>b\<in>basis. line_integral_exists F {b} (reversepath (snd (f (k1, \<gamma>1))))"
     using line_integral_on_reverse_path(2) integ_exist_b valid_cubes
     by blast                             
   have 0:" \<forall>x\<in>one_chain1. (case x of (k, \<gamma>) \<Rightarrow> (real_of_int k * line_integral F basis \<gamma>) = (case f x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma>) \<and>
                                                         line_integral_exists F basis \<gamma>)"
   proof
     {fix k1 \<gamma>1
       assume ass1: "(k1,\<gamma>1) \<in>one_chain1"
       have "real_of_int k1 * line_integral F basis \<gamma>1 = (case (f (k1,\<gamma>1)) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                       line_integral_exists F basis \<gamma>1"
       proof(cases)
         assume ass2: "k1 = 1"
         let ?k2 = "fst (f (k1, \<gamma>1))"
         let ?\<gamma>2 = "snd (f (k1, \<gamma>1))"
         have "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2\<and>
                               line_integral_exists F basis \<gamma>1"
         proof(cases)
           assume ass3: "?k2 = 1"
           then have 0: "reparam \<gamma>1 ?\<gamma>2"
             using ass1 ass2 f(2)
             by auto
           have 1: "?\<gamma>2 piecewise_C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image ?\<gamma>2) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1
             by force
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_eq_line_integrals_basis[OF 0 1 2 assms(4)] integ_exist_b
               ass1 ass2 ass3
             by auto
         next
           assume "?k2 \<noteq> 1"
           then have ass3: "?k2 = -1"
             using bound2 ass1 f(3) unfolding boundary_chain_def by force
           then have 0: "reparam \<gamma>1 (reversepath ?\<gamma>2)"
             using ass1 ass2 f(2)
             by auto
           have 1: "(reversepath ?\<gamma>2) piecewise_C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1 piecewise_smooth_path_imp_reverse
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image (reversepath ?\<gamma>2)) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1 path_image_reversepath
             by force
           have 3: "line_integral F basis ?\<gamma>2 = - line_integral F basis (reversepath ?\<gamma>2)"
           proof-
             have i:"valid_path (reversepath ?\<gamma>2) "
               using 1 C1_differentiable_imp_piecewise
               by (auto simp add: valid_path_def)
             have ii: "line_integral_exists F basis (snd (f (k1, \<gamma>1)))"
               using assms(4) line_integral_sum_basis(2) integ_exist_b ass1
               by fastforce
             show ?thesis                                            
               using i ii line_integral_on_reverse_path(1) valid_path_reversepath by blast
           qed
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_eq_line_integrals_basis[OF 0 1 2 assms(4)] integ_rev_exist_b
               ass1 ass2 ass3 3 
             by auto
         qed
         then show "real_of_int k1 * line_integral F basis \<gamma>1 = (case f (k1, \<gamma>1) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                                    line_integral_exists F basis \<gamma>1"
           by (simp add: case_prod_beta')
       next
         assume "k1 \<noteq> 1"
         then have ass2: "k1 = -1"
           using bound1 ass1 f(3) unfolding boundary_chain_def by force
         let ?k2 = "fst (f (k1, \<gamma>1))"
         let ?\<gamma>2 = "snd (f (k1, \<gamma>1))"
         have "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                               line_integral_exists F basis \<gamma>1"
         proof(cases)
           assume ass3: "?k2 = 1"
           then have 0: "reparam \<gamma>1 (reversepath ?\<gamma>2)"
             using ass1 ass2 f(2)
             by auto
           have 1: "(reversepath ?\<gamma>2) piecewise_C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1 piecewise_smooth_path_imp_reverse
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image (reversepath ?\<gamma>2)) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1 path_image_reversepath
             by force
           have 3: "line_integral F basis ?\<gamma>2 = - line_integral F basis (reversepath ?\<gamma>2)"
           proof-
             have i:"valid_path (reversepath ?\<gamma>2) "
               using 1 C1_differentiable_imp_piecewise
               by (auto simp add: valid_path_def)
             show ?thesis 
               using line_integral_on_reverse_path(1)[OF i] integ_rev_exist_b
               using ass1 assms(4) line_integral_sum_basis(2) by fastforce
           qed
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_eq_line_integrals_basis[OF 0 1 2 assms(4)]
               ass2 ass3 3
             using ass1 integ_rev_exist_b by auto
         next
           assume "?k2 \<noteq> 1"
           then have ass3: "?k2 = -1"
             using bound2 ass1 f(3) unfolding boundary_chain_def by force
           then have 0: "reparam \<gamma>1 ?\<gamma>2"
             using ass1 ass2 f(2) by auto
           have 1: "?\<gamma>2 piecewise_C1_differentiable_on {0..1}"
             using f(3) assms(2) ass1
             by force
           have 2: "\<forall>b\<in>basis. continuous_on (path_image ?\<gamma>2) (\<lambda>x. F x \<bullet> b)"
             using f(3) assms(3) ass1
             by force
           show "real_of_int k1 * line_integral F basis \<gamma>1 = real_of_int ?k2 * line_integral F basis ?\<gamma>2 \<and>
                                      line_integral_exists F basis \<gamma>1"
             using assms reparam_eq_line_integrals_basis[OF 0 1 2 assms(4)]
               ass2 ass3
             using ass1 integ_exist_b by auto
         qed
         then show "real_of_int k1 * line_integral F basis \<gamma>1 = (case f (k1, \<gamma>1) of (k2, \<gamma>2) \<Rightarrow> real_of_int k2 * line_integral F basis \<gamma>2) \<and>
                                    line_integral_exists F basis \<gamma>1"
           by (simp add: case_prod_beta')
       qed
     }
     then show "\<And>x. x \<in> one_chain1 \<Longrightarrow>
                                 (case x of (k, \<gamma>) \<Rightarrow> (real_of_int k * line_integral F basis \<gamma>) = (case f x of (k, \<gamma>) \<Rightarrow> real_of_int k * line_integral F basis \<gamma>) \<and>
                                                         line_integral_exists F basis \<gamma>)"
       by (auto simp add: case_prod_beta')
   qed
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     using 0 by (simp add: one_chain_line_integral_def sum_bij[OF f(1) _ f(3)] prod.case_eq_if)
   show "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     using 0 by blast
 qed
 
 lemma path_image_rec_join:
   fixes \<gamma>::"real \<Rightarrow> (real \<times> real)"
   fixes k::int
   fixes l
   shows "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_chain_list l \<Longrightarrow> path_image \<gamma> \<subseteq> path_image (rec_join l)"
 proof(induction l)
   case Nil
   then show ?case by auto
 next
   case ass: (Cons a l)
   obtain k' \<gamma>' where a: "a = (k',\<gamma>')" by force
   have "path_image \<gamma> \<subseteq> path_image (rec_join ((k',\<gamma>') # l))"
   proof(cases)
     assume "l=[]"
     then show "path_image \<gamma> \<subseteq> path_image (rec_join ((k',\<gamma>') # l))"
       using ass(2-3) a
       by(auto simp add:)
   next
     assume "l\<noteq>[]"
     then obtain b l' where b_l: "l = b # l'"
       by (meson rec_join.elims)
     obtain k'' \<gamma>'' where b: "b = (k'',\<gamma>'')" by force
     show ?thesis
       using ass path_image_reversepath b_l path_image_join
       by(fastforce simp add: a)
   qed
   then show ?case
     using a  by auto
 qed
 
 lemma path_image_rec_join_2:
   fixes l
   shows "l \<noteq> [] \<Longrightarrow> valid_chain_list l \<Longrightarrow> path_image (rec_join l) \<subseteq> (\<Union>(k, \<gamma>) \<in> set l. path_image \<gamma>)"
 proof(induction l)
   case Nil
   then show ?case by auto
 next
   case ass: (Cons a l)
   obtain k' \<gamma>' where a: "a = (k',\<gamma>')" by force
   have "path_image (rec_join (a # l)) \<subseteq> (\<Union>(k, y)\<in>set (a # l). path_image y)"
   proof(cases)
     assume "l=[]"
     then show "path_image (rec_join (a # l)) \<subseteq> (\<Union>(k, y)\<in>set (a # l). path_image y)"
       using step a  by(auto simp add:)
   next
     assume "l\<noteq>[]"
     then obtain b l' where b_l: "l = b # l'"
       by (meson rec_join.elims)
     obtain k'' \<gamma>'' where b: "b = (k'',\<gamma>'')" by force
     show ?thesis
       using ass
         path_image_reversepath b_l path_image_join
       apply(auto simp add: a) (*Excuse the ugliness of the next script*)
        apply blast
       by fastforce
   qed
   then show ?case
     using a by auto
 qed
 
 lemma continuous_on_closed_UN:
   assumes "finite S"
   shows "((\<And>s. s \<in> S \<Longrightarrow> closed s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f)"
   using assms
 proof(induction S)
   case empty
   then show ?case by auto
 next
   case (insert x F)
   then show ?case using continuous_on_closed_Un closed_Union
     by (simp add: closed_Union continuous_on_closed_Un)
 qed
 
 lemma chain_reparam_weak_path_line_integral:
   assumes path_eq_chain: "chain_reparam_weak_path \<gamma> one_chain" and
     boundary_chain: "boundary_chain one_chain" and
     line_integral_exists: "\<forall>b\<in>basis. \<forall>(k::int, \<gamma>) \<in> one_chain. line_integral_exists F {b} \<gamma>" and
     valid_path: "\<forall>(k::int, \<gamma>) \<in> one_chain. valid_path \<gamma>" and
     finite_basis: "finite basis" and 
     cont: "\<forall>b\<in>basis. \<forall>(k,\<gamma>2)\<in>one_chain. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)" and
     finite_one_chain: "finite one_chain"
   shows "line_integral F basis \<gamma> = one_chain_line_integral F basis one_chain"
     "line_integral_exists F basis \<gamma>"
     (*"valid_path \<gamma>" This cannot be proven, see the crappy assumption of derivs/eq_pw_smooth :( *)
 proof -
   obtain l where l_props: "set l = one_chain" "distinct l" "reparam \<gamma> (rec_join l)" "valid_chain_list l" "l \<noteq> []"
     using chain_reparam_weak_path_def assms
     by auto
   have bchain_l: "boundary_chain (set l)"
     using l_props boundary_chain
     by (simp add: boundary_chain_def)
   have cont_forall: "\<forall>b\<in>basis. continuous_on (\<Union>(k, \<gamma>)\<in>one_chain. path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
   proof
     fix b
     assume ass: "b \<in> basis"
     have "continuous_on (\<Union>((path_image \<circ> snd) ` one_chain))  (\<lambda>x. F x \<bullet> b)"
       apply(rule continuous_on_closed_UN[where ?S = "(path_image o snd) ` one_chain" ])
     proof
       show "finite one_chain" using finite_one_chain by auto
       show "\<And>s. s \<in> (path_image \<circ> snd) ` one_chain \<Longrightarrow> closed s"
         using  closed_valid_path_image[OF] valid_path  
         by fastforce
       show "\<And>s. s \<in> (path_image \<circ> snd) ` one_chain \<Longrightarrow> continuous_on s (\<lambda>x. F x \<bullet> b)"
         using cont ass by force
     qed
     then show "continuous_on (\<Union>(k, \<gamma>)\<in>one_chain. path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
       by (auto simp add: Union_eq case_prod_beta)
   qed
   show "line_integral_exists F basis \<gamma>"
   proof (rule reparam_eq_line_integrals_basis[OF l_props(3) _ _  finite_basis])
     let ?\<gamma>2.0="rec_join l"
     show "?\<gamma>2.0 piecewise_C1_differentiable_on {0..1}"
       apply(simp only: valid_path_def[symmetric])
       apply(rule joined_is_valid)
       using assms l_props by auto
     show "\<forall>b\<in>basis. continuous_on (path_image (rec_join l)) (\<lambda>x. F x \<bullet> b)" using path_image_rec_join_2[OF l_props(5) l_props(4)] l_props(1)
       using cont_forall continuous_on_subset by blast
     show "\<forall>b\<in>basis. line_integral_exists F {b} (rec_join l)"
     proof
       fix b
       assume ass: "b\<in>basis"
       show "line_integral_exists F {b} (rec_join l)"
       proof (rule line_integral_exists_on_rec_join)
         show "boundary_chain (set l)"
           using l_props boundary_chain by auto
         show "valid_chain_list l" using l_props by auto
         show "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>" using l_props assms by auto
         show "\<forall>(k, \<gamma>)\<in>set l. line_integral_exists F {b} \<gamma>" using l_props line_integral_exists ass by blast
       qed
     qed
   qed
   show "line_integral F basis \<gamma> = one_chain_line_integral F basis one_chain"
   proof-
     have i: "line_integral F basis (rec_join l) = one_chain_line_integral F basis one_chain"
     proof (rule one_chain_line_integral_rec_join)
       show "set l = one_chain" "distinct l" "valid_chain_list l" using l_props by auto
       show "boundary_chain one_chain" using boundary_chain by auto
       show "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"
         using line_integral_sum_basis(2)[OF finite_basis] line_integral_exists by blast
       show "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>" using valid_path by auto
       show "finite basis" using finite_basis by auto
     qed
     have ii: "line_integral F basis \<gamma> = line_integral F basis (rec_join l)"
     proof (rule reparam_eq_line_integrals_basis[OF l_props(3) _ _  finite_basis])
       let ?\<gamma>2.0="rec_join l"
       show "?\<gamma>2.0 piecewise_C1_differentiable_on {0..1}"
         apply(simp only: valid_path_def[symmetric])
         apply(rule joined_is_valid)
         using assms l_props by auto
       show "\<forall>b\<in>basis. continuous_on (path_image (rec_join l)) (\<lambda>x. F x \<bullet> b)" using path_image_rec_join_2[OF l_props(5) l_props(4)] l_props(1)
         using cont_forall continuous_on_subset by blast
       show "\<forall>b\<in>basis. line_integral_exists F {b} (rec_join l)"
       proof
         fix b
         assume ass: "b\<in>basis"
         show "line_integral_exists F {b} (rec_join l)"
         proof (rule line_integral_exists_on_rec_join)
           show "boundary_chain (set l)"
             using l_props boundary_chain by auto
           show "valid_chain_list l" using l_props by auto
           show "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>" using l_props assms by auto
           show "\<forall>(k, \<gamma>)\<in>set l. line_integral_exists F {b} \<gamma>" using l_props line_integral_exists ass by blast
         qed
       qed
     qed
     show "line_integral F basis \<gamma> = one_chain_line_integral F basis one_chain" using i ii by auto
   qed
 qed
 
 definition chain_reparam_chain' where
   "chain_reparam_chain' one_chain1 subdiv
        \<equiv> \<exists>f. ((\<Union>(f ` one_chain1)) = subdiv) \<and>
               (\<forall>cube \<in> one_chain1. chain_reparam_weak_path (rec_join [cube]) (f cube)) \<and>
               (\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {}) \<and>
               (\<forall>x \<in> one_chain1. finite (f x))"
 
 lemma chain_reparam_chain'_imp_finite_subdiv:
   assumes "finite one_chain1"
     "chain_reparam_chain' one_chain1 subdiv"
   shows "finite subdiv"
   using assms
   by(auto simp add: chain_reparam_chain'_def)
 
 lemma chain_reparam_chain'_line_integral:
   assumes chain1_eq_chain2: "chain_reparam_chain' one_chain1 subdiv" and
     boundary_chain1: "boundary_chain one_chain1" and
     boundary_chain2: "boundary_chain subdiv" and
     line_integral_exists_on_chain2: "\<forall>b\<in>basis. \<forall>(k::int, \<gamma>) \<in> subdiv. line_integral_exists F {b} \<gamma>" and
     valid_path: "\<forall>(k, \<gamma>) \<in> subdiv. valid_path \<gamma>" and
     valid_path_2: "\<forall>(k, \<gamma>) \<in> one_chain1. valid_path \<gamma>" and
     finite_chain1: "finite one_chain1" and
     finite_basis: "finite basis" and
     cont_field: " \<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>subdiv. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv"
     "\<forall>(k, \<gamma>) \<in> one_chain1. line_integral_exists F basis \<gamma>"
 proof -
   obtain f where f_props:
     "((\<Union>(f ` one_chain1)) = subdiv)"
     "(\<forall>cube \<in> one_chain1. chain_reparam_weak_path (rec_join [cube]) (f cube))"
     "(\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {})"
     "(\<forall>x \<in> one_chain1. finite (f x))"
     using chain1_eq_chain2 
     by (auto simp add: chain_reparam_chain'_def)
   then have 0: "one_chain_line_integral F basis subdiv = one_chain_line_integral F basis (\<Union>(f ` one_chain1))"
     by auto
   {fix k \<gamma>
     assume ass:"(k, \<gamma>) \<in> one_chain1"
     have "line_integral_exists F basis \<gamma> \<and>
                one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
     proof(cases "k = 1")
       assume k_eq_1: "k = 1"
       then have i:"chain_reparam_weak_path \<gamma> (f(k,\<gamma>))"
         using f_props(2) ass by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms  unfolding boundary_chain_def
         by blast
       have iii:"\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). line_integral_exists F {b} \<gamma>"
         using f_props(1) ass assms
         by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms
         by blast
       have v: "\<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>f (k, \<gamma>). continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
         using f_props(1) ass assms
         by blast        
       have "line_integral_exists F basis \<gamma> \<and> one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis \<gamma>"
         using chain_reparam_weak_path_line_integral[OF i ii iii iv finite_basis v] ass f_props(4)
         by (auto)
       then show "line_integral_exists F basis \<gamma> \<and> one_chain_line_integral F basis (f (k, \<gamma>)) = k * line_integral F basis \<gamma>"
         using k_eq_1  by auto
     next
       assume "k \<noteq> 1"
       then have k_eq_neg1: "k = -1"
         using ass boundary_chain1
         by (auto simp add: boundary_chain_def) 
       then have i:"chain_reparam_weak_path (reversepath \<gamma>) (f(k,\<gamma>))"
         using f_props(2) ass by auto
       have ii:"boundary_chain (f(k,\<gamma>))"
         using f_props(1) ass assms unfolding boundary_chain_def
         by blast
       have iii:"\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). line_integral_exists F {b} \<gamma>"
         using f_props(1) ass assms
         by blast
       have "iv": " \<forall>(k, \<gamma>)\<in>f (k, \<gamma>). valid_path \<gamma>"
         using f_props(1) ass assms by blast
       have v: "\<forall>b\<in>basis. \<forall>(k, \<gamma>2)\<in>f (k, \<gamma>). continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
         using f_props(1) ass assms  by blast        
       have x:"line_integral_exists F basis (reversepath \<gamma>) \<and> one_chain_line_integral F basis (f (k, \<gamma>)) = line_integral F basis (reversepath \<gamma>)"
         using chain_reparam_weak_path_line_integral[OF i ii iii iv finite_basis v] ass f_props(4)
         by auto
       have "valid_path (reversepath \<gamma>)"
         using f_props(1) ass assms  by auto
       then have "line_integral_exists F basis \<gamma> \<and> one_chain_line_integral F basis (f (k, \<gamma>)) = - (line_integral F basis \<gamma>)"
         using line_integral_on_reverse_path reversepath_reversepath x ass
         by metis
       then show "line_integral_exists F basis \<gamma> \<and> one_chain_line_integral F basis (f (k::int, \<gamma>)) = k * line_integral F basis \<gamma>"
         using k_eq_neg1  by auto
     qed}
   note cube_line_integ = this
   have finite_chain2: "finite subdiv" 
     using finite_chain1 f_props(1) f_props(4) by auto
   show line_integral_exists_on_chain1: "\<forall>(k, \<gamma>) \<in> one_chain1. line_integral_exists F basis \<gamma>"
     using cube_line_integ by auto
   have 1: "one_chain_line_integral F basis (\<Union>(f ` one_chain1)) = one_chain_line_integral F basis one_chain1"
   proof -
     have 0:"one_chain_line_integral F basis (\<Union>(f ` one_chain1)) = 
                            (\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain)"
     proof -
       have finite: "\<forall>chain \<in> (f ` one_chain1). finite chain"
         using f_props(1) finite_chain2 
         by (meson Sup_upper finite_subset)
       have disj: " \<forall>A\<in>f ` one_chain1. \<forall>B\<in>f ` one_chain1. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
         apply(simp add:image_def)
         using f_props(3)
         by metis
       show "one_chain_line_integral F basis (\<Union>(f ` one_chain1)) = 
                                 (\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain)"
         using Groups_Big.comm_monoid_add_class.sum.Union_disjoint[OF finite disj]
           one_chain_line_integral_def
         by auto
     qed
     have 1:"(\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain) = 
                             one_chain_line_integral F basis one_chain1"
     proof -
       have "(\<Sum>one_chain \<in> (f ` one_chain1). one_chain_line_integral F basis one_chain) = 
                                      (\<Sum>(k,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>))"
       proof -
         have i:"(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) = 
                                        (\<Sum>(k,\<gamma>)\<in>one_chain1 - {p. f p = {}}. k*(line_integral F basis \<gamma>))"
         proof -
           have "inj_on f (one_chain1 - {p. f p = {}})"
             unfolding inj_on_def
             using f_props(3) by force
           then have 0: "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain)
                                                        = (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). one_chain_line_integral F basis (f (k, \<gamma>)))"
             using Groups_Big.comm_monoid_add_class.sum.reindex
             by auto
           have "\<And> k \<gamma>. (k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}) \<Longrightarrow>
                                                      one_chain_line_integral F basis (f(k, \<gamma>)) = k* (line_integral F basis \<gamma>)"
             using cube_line_integ by auto
           then have "(\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). one_chain_line_integral F basis (f (k, \<gamma>)))
                    = (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). k* (line_integral F basis \<gamma>))"
             by (auto intro!: Finite_Cartesian_Product.sum_cong_aux)
           then show "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                      (\<Sum>(k, \<gamma>) \<in> (one_chain1 - {p. f p = {}}). k* (line_integral F basis \<gamma>))"
             using 0  by auto
         qed
         have "\<And> (k::int) \<gamma>. (k, \<gamma>) \<in> one_chain1 \<Longrightarrow> (f (k, \<gamma>) = {}) \<Longrightarrow> (k, \<gamma>) \<in> {(k',\<gamma>'). k'* (line_integral F basis \<gamma>') = 0}"
         proof-
           fix k::"int"
           fix \<gamma>::"real\<Rightarrow>real*real"
           assume ass:"(k, \<gamma>) \<in> one_chain1"
             "(f (k, \<gamma>) = {})"
           then have zero_line_integral:"one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
             using one_chain_line_integral_def
             by auto
           show "(k, \<gamma>) \<in> {(k'::int, \<gamma>'). k' * line_integral F basis \<gamma>' = 0}"
             using zero_line_integral cube_line_integ ass
             by force
         qed
         then have ii:"(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                           (\<Sum>one_chain \<in> (f ` (one_chain1)). one_chain_line_integral F basis one_chain)"
         proof -
           have "\<And>one_chain. one_chain \<in> (f ` (one_chain1)) - (f ` (one_chain1 - {p. f p = {}})) \<Longrightarrow>
                                                          one_chain_line_integral F basis one_chain = 0"
           proof -
             fix one_chain
             assume "one_chain \<in> (f ` (one_chain1)) - (f ` (one_chain1 - {p. f p = {}}))"
             then have "one_chain = {}"
               by auto
             then show "one_chain_line_integral F basis one_chain = 0"
               by (auto simp add: one_chain_line_integral_def)
           qed
           then have 0:"(\<Sum>one_chain \<in> f ` (one_chain1) - (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) 
                                                        = 0"
             using Groups_Big.comm_monoid_add_class.sum.neutral
             by auto
           then have "(\<Sum>one_chain \<in> f ` (one_chain1). one_chain_line_integral F basis one_chain)
                                                       - (\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) 
                                                        = 0"
           proof -
             have finte: "finite (f ` one_chain1)" using finite_chain1 by auto
             show ?thesis
               using Groups_Big.sum_diff[OF finte, of " (f ` (one_chain1 - {p. f p = {}}))"
                   "one_chain_line_integral F basis"]
                 0
               by auto
           qed                                           
           then show "(\<Sum>one_chain \<in> (f ` (one_chain1 - {p. f p = {}})). one_chain_line_integral F basis one_chain) =
                                                           (\<Sum>one_chain \<in> (f ` (one_chain1)). one_chain_line_integral F basis one_chain)"
             by auto
         qed
         have "\<And> (k::int) \<gamma>. (k, \<gamma>) \<in> one_chain1 \<Longrightarrow> (f (k, \<gamma>) = {}) \<Longrightarrow> f (k, \<gamma>) \<in> {chain. one_chain_line_integral F basis chain = 0}"
         proof-
           fix k::"int"
           fix \<gamma>::"real\<Rightarrow>real*real"
           assume ass:"(k, \<gamma>) \<in> one_chain1"  "(f (k, \<gamma>) = {})"
           then have "one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
             using one_chain_line_integral_def
             by auto
           then show "f (k, \<gamma>) \<in> {p'. (one_chain_line_integral F basis p' = 0)}"
             by auto
         qed
         then have iii:"(\<Sum>(k::int,\<gamma>)\<in>one_chain1 - {p. f p = {}}. k*(line_integral F basis \<gamma>))
                                                  = (\<Sum>(k::int,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>))"
         proof-
           have "\<And> k \<gamma>. (k,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}})
                                                     \<Longrightarrow> k* (line_integral F basis \<gamma>) = 0"
           proof-
             fix k \<gamma>
             assume ass: "(k,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}})"
             then have "f(k, \<gamma>) = {}"
               by auto
             then have "one_chain_line_integral F basis (f(k, \<gamma>)) = 0"
               by (auto simp add: one_chain_line_integral_def)
             then have zero_line_integral:"one_chain_line_integral F basis (f (k, \<gamma>)) = 0"
               using one_chain_line_integral_def
               by auto
             then show "k* (line_integral F basis \<gamma>) = 0"
               using zero_line_integral cube_line_integ ass
               by force
           qed
           then have "\<forall>(k::int,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}).
                        k* (line_integral F basis \<gamma>) = 0" by auto
           then have "(\<Sum>(k::int,\<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}). k*(line_integral F basis \<gamma>)) = 0"
             using Groups_Big.comm_monoid_add_class.sum.neutral
               [of "one_chain1 - (one_chain1 - {p. f p = {}})" "(\<lambda>(k::int,\<gamma>). k* (line_integral F basis \<gamma>))"]
             by (simp add: split_beta)
           then have "(\<Sum>(k::int,\<gamma>)\<in>one_chain1. k*(line_integral F basis \<gamma>)) -
                                                     (\<Sum>(k::int,\<gamma>)\<in> (one_chain1 - {p. f p = {}}). k*(line_integral F basis \<gamma>))  = 0"
             using Groups_Big.sum_diff[OF finite_chain1]
             by (metis (no_types) Diff_subset \<open>(\<Sum>(k, \<gamma>)\<in>one_chain1 - (one_chain1 - {p. f p = {}}). k * line_integral F basis \<gamma>) = 0\<close> \<open>\<And>f B. B \<subseteq> one_chain1 \<Longrightarrow> sum f (one_chain1 - B) = sum f one_chain1 - sum f B\<close>)
           then show ?thesis by auto
         qed
         show ?thesis using i ii iii by auto
       qed
       then show ?thesis using one_chain_line_integral_def by auto
     qed                    
     show ?thesis using 0 1 by auto
   qed
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis subdiv" using 0 1 by auto
 qed
 
 lemma chain_reparam_chain'_line_integral_smooth_cubes:
   assumes "chain_reparam_chain' one_chain1 one_chain2"
     "\<forall>(k2,\<gamma>2)\<in>one_chain2. \<gamma>2 C1_differentiable_on {0..1}"
     "\<forall>b\<in>basis.\<forall>(k2,\<gamma>2)\<in>one_chain2. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "finite basis"                                                                     
     "finite one_chain1"
     "boundary_chain one_chain1"
     "boundary_chain one_chain2"
     "\<forall>(k,\<gamma>)\<in>one_chain1. valid_path \<gamma>"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
 proof-
   {fix b
     assume "b \<in> basis"
     fix k \<gamma>
     assume "(k, \<gamma>)\<in>one_chain2"
     have "line_integral_exists F {b} \<gamma>"
       apply(rule line_integral_exists_smooth)
       using \<open>(k, \<gamma>) \<in> one_chain2\<close> assms(2) apply blast
       using assms
       using \<open>(k, \<gamma>) \<in> one_chain2\<close> \<open>b \<in> basis\<close> apply blast
       using \<open>b \<in> basis\<close> by blast}
   then have a: "\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F {b} \<gamma>"
     by auto
   have b: "\<forall>(k2,\<gamma>2)\<in>one_chain2. valid_path \<gamma>2"
     using assms(2)
     by (simp add: C1_differentiable_imp_piecewise case_prod_beta valid_path_def)              
   show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     by (rule chain_reparam_chain'_line_integral[OF assms(1) assms(6) assms(7) a b assms(8) assms(5) assms(4) assms(3)])
   show "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     by (rule chain_reparam_chain'_line_integral[OF assms(1) assms(6) assms(7) a b assms(8) assms(5) assms(4) assms(3)])
 qed
 
 lemma chain_subdiv_path_pathimg_subset:
   assumes "chain_subdiv_path \<gamma> subdiv"
   shows "\<forall>(k',\<gamma>')\<in>subdiv. (path_image \<gamma>') \<subseteq> path_image \<gamma>"
   using assms chain_subdiv_path.simps path_image_rec_join
   by(auto simp add: chain_subdiv_path.simps)
 
 lemma reparam_path_image:
   assumes "reparam \<gamma>1 \<gamma>2"
   shows "path_image \<gamma>1 = path_image \<gamma>2"
   using assms
   apply (auto simp add: reparam_def path_image_def image_def bij_betw_def)
   apply (force dest!: equalityD2)
   done
 
 lemma chain_reparam_weak_path_pathimg_subset:
   assumes "chain_reparam_weak_path \<gamma> subdiv"
   shows "\<forall>(k',\<gamma>')\<in>subdiv. (path_image \<gamma>') \<subseteq> path_image \<gamma>"
   using assms
   apply(auto simp add: chain_reparam_weak_path_def)
   using path_image_rec_join reparam_path_image by blast
 
 lemma chain_subdiv_chain_pathimg_subset':
   assumes "chain_subdiv_chain one_chain subdiv"
   assumes "(k,\<gamma>)\<in> subdiv"
   shows " \<exists>k' \<gamma>'. (k',\<gamma>')\<in> one_chain \<and> path_image \<gamma> \<subseteq> path_image \<gamma>'"
   using assms unfolding chain_subdiv_chain_def  pairwise_def
   apply auto
   by (metis chain_subdiv_path.cases coeff_cube_to_path.simps path_image_rec_join path_image_reversepath)
 
 lemma chain_subdiv_chain_pathimg_subset:
   assumes "chain_subdiv_chain one_chain subdiv"
   shows "\<Union> (path_image ` {\<gamma>. \<exists>k. (k,\<gamma>)\<in> subdiv}) \<subseteq> \<Union> (path_image ` {\<gamma>. \<exists>k. (k,\<gamma>)\<in> one_chain})"
   using assms unfolding chain_subdiv_chain_def pairwise_def
   apply auto
   by (metis UN_iff assms chain_subdiv_chain_pathimg_subset' subsetCE case_prodI2) 
 
 lemma chain_reparam_chain'_pathimg_subset':
   assumes "chain_reparam_chain' one_chain subdiv"
   assumes "(k,\<gamma>)\<in> subdiv"
   shows " \<exists>k' \<gamma>'. (k',\<gamma>')\<in> one_chain \<and> path_image \<gamma> \<subseteq> path_image \<gamma>'"
   using assms chain_reparam_weak_path_pathimg_subset
   apply(auto simp add: chain_reparam_chain'_def set_eq_iff)
   using  path_image_reversepath case_prodE case_prodD old.prod.exhaust
   apply (simp add: list.distinct(1) list.inject rec_join.elims)
   by (smt case_prodD coeff_cube_to_path.simps rec_join.simps(2) reversepath_simps(2) surj_pair)
 
  definition common_reparam_exists:: "(int \<times> (real \<Rightarrow> real \<times> real)) set \<Rightarrow> (int \<times> (real \<Rightarrow> real \<times> real)) set \<Rightarrow> bool" where
     "common_reparam_exists one_chain1 one_chain2 \<equiv>
     (\<exists>subdiv ps1 ps2.
     chain_reparam_chain' (one_chain1 - ps1) subdiv \<and>
     chain_reparam_chain' (one_chain2 - ps2) subdiv \<and>
     (\<forall>(k, \<gamma>)\<in>subdiv.  \<gamma> C1_differentiable_on {0..1}) \<and>
     boundary_chain subdiv \<and>
     (\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>) \<and>
     (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>))"
 
 lemma common_reparam_exists_imp_eq_line_integral:
   assumes finite_basis: "finite basis" and 
     "finite one_chain1"
     "finite one_chain2"
     "boundary_chain (one_chain1::(int \<times> (real \<Rightarrow> real \<times> real)) set)"
     "boundary_chain (one_chain2::(int \<times> (real \<Rightarrow> real \<times> real)) set)"
     " \<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     "(common_reparam_exists one_chain1 one_chain2)"
     "(\<forall>(k, \<gamma>)\<in>one_chain1.  valid_path \<gamma>)"
     "(\<forall>(k, \<gamma>)\<in>one_chain2.  valid_path \<gamma>)"
   shows "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
 proof-
   obtain subdiv ps1 ps2 where props:
     "chain_reparam_chain' (one_chain1 - ps1) subdiv"
     "chain_reparam_chain' (one_chain2 - ps2) subdiv "
     "(\<forall>(k, \<gamma>)\<in>subdiv.  \<gamma> C1_differentiable_on {0..1})"
     "boundary_chain subdiv"
     "(\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>)"
     "(\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>)"
     using assms
     by (auto simp add: common_reparam_exists_def)
   have subdiv_valid: "(\<forall>(k, \<gamma>)\<in>subdiv.  valid_path \<gamma>)"
     apply(simp add: valid_path_def)
     using props(3)
     using C1_differentiable_imp_piecewise by blast
   have onechain_boundary1: "boundary_chain (one_chain1 - ps1)" using assms(4) by (auto simp add: boundary_chain_def)
   have onechain_boundary2: "boundary_chain (one_chain2 - ps1)" using assms(5) by (auto simp add: boundary_chain_def)
   {fix k2 \<gamma>2 b
     assume ass: "(k2, \<gamma>2)\<in>subdiv "" b\<in>basis"
     have "\<And> k \<gamma>. (k, \<gamma>) \<in> subdiv \<Longrightarrow> \<exists>k' \<gamma>'. (k', \<gamma>') \<in> one_chain2 \<and> path_image \<gamma> \<subseteq> path_image \<gamma>'"
       by (meson chain_reparam_chain'_pathimg_subset' props Diff_subset subsetCE)
     then have "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
       using assms(6) continuous_on_subset[where ?f = " (\<lambda>x. F x \<bullet> b)"] ass
       apply(auto simp add:  subset_iff)
       by (metis (mono_tags, lifting) case_prodD)}
   then have cont_field: "\<forall>b\<in>basis. \<forall>(k2, \<gamma>2)\<in>subdiv. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
     by auto
   have one_chain1_ps_valid: "(\<forall>(k, \<gamma>)\<in>one_chain1 - ps1.  valid_path \<gamma>)" using assms by auto
   have one_chain2_ps_valid: "(\<forall>(k, \<gamma>)\<in>one_chain2 - ps1.  valid_path \<gamma>)" using assms by auto
   have 0: "one_chain_line_integral F basis (one_chain1 - ps1) = one_chain_line_integral F basis subdiv"
     apply(rule chain_reparam_chain'_line_integral_smooth_cubes[OF props(1) props(3) cont_field finite_basis])
     using props assms
     apply blast
     using props assms
     using onechain_boundary1 apply blast
     using props assms
     apply blast
     using one_chain1_ps_valid by blast
   have 1:"\<forall>(k, \<gamma>)\<in>(one_chain1 - ps1). line_integral_exists F basis \<gamma>"
     apply(rule chain_reparam_chain'_line_integral_smooth_cubes[OF props(1) props(3) cont_field finite_basis])
     using props assms
     apply blast
     using props assms
     using onechain_boundary1 apply blast
     using props assms
     apply blast
     using one_chain1_ps_valid by blast
   have 2: "one_chain_line_integral F basis (one_chain2 - ps2) = one_chain_line_integral F basis subdiv"
     apply(rule chain_reparam_chain'_line_integral_smooth_cubes[OF props(2) props(3) cont_field finite_basis])
     using props assms
     apply blast
     apply (simp add: assms(5) boundary_chain_diff)
     apply (simp add: props(4))
     by (simp add: assms(9))
   have 3:"\<forall>(k, \<gamma>)\<in>(one_chain2 - ps2). line_integral_exists F basis \<gamma>"
     apply(rule chain_reparam_chain'_line_integral_smooth_cubes[OF props(2) props(3) cont_field finite_basis])
     using props assms
     apply blast
     apply (simp add: assms(5) boundary_chain_diff)
     apply (simp add: props(4))
     by (simp add: assms(9))
   show line_int_ex_chain1: "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"
     using 0 
     using "1" finite_basis line_integral_exists_point_path props(5) by fastforce
   then show "one_chain_line_integral F basis one_chain1 = one_chain_line_integral F basis one_chain2"
     using 0 1 2 3
     using assms(2-3) finite_basis one_chain_line_integral_point_paths props(5) props(6) by auto
 qed
 
 definition subcube :: "real \<Rightarrow> real \<Rightarrow> (int \<times> (real \<Rightarrow> real \<times> real)) \<Rightarrow> (int \<times> (real \<Rightarrow> real \<times> real))" where
   "subcube a b cube = (fst cube, subpath a b (snd cube))"
 
 lemma subcube_valid_path:
   assumes "valid_path (snd cube)" "a \<in> {0..1}" "b \<in> {0..1}"
   shows "valid_path (snd (subcube a b cube))"
   using valid_path_subpath[OF assms] by (auto simp add: subcube_def)
 
 end
diff --git a/thys/Hybrid_Systems_VCs/HS_ODEs.thy b/thys/Hybrid_Systems_VCs/HS_ODEs.thy
--- a/thys/Hybrid_Systems_VCs/HS_ODEs.thy
+++ b/thys/Hybrid_Systems_VCs/HS_ODEs.thy
@@ -1,571 +1,571 @@
 (*  Title:       ODEs and Dynamical Systems for HS verification
     Author:      Jonathan Julián Huerta y Munive, 2019
     Maintainer:  Jonathan Julián Huerta y Munive <jjhuertaymunive1@sheffield.ac.uk>
 *)
 
 section \<open> Ordinary Differential Equations \<close>
 
 text \<open>Vector fields @{text "f::real \<Rightarrow> 'a \<Rightarrow> ('a::real_normed_vector)"} represent systems 
 of ordinary differential equations (ODEs). Picard-Lindeloef's theorem guarantees existence 
 and uniqueness of local solutions to initial value problems involving Lipschitz continuous 
 vector fields. A (local) flow @{text "\<phi>::real \<Rightarrow> 'a \<Rightarrow> ('a::real_normed_vector)"} for such 
 a system is the function that maps initial conditions to their unique solutions. In dynamical 
 systems, the set of all points @{text "\<phi> t s::'a"} for a fixed @{text "s::'a"} is the flow's 
 orbit. If the orbit of each @{text "s \<in> I"} is conatined in @{text I}, then @{text I} is an 
 invariant set of this system. This section formalises these concepts with a focus on hybrid 
 systems (HS) verification.\<close>
 
 theory HS_ODEs
   imports "HS_Preliminaries"
 begin
 
 subsection \<open> Initial value problems and orbits \<close>
 
 notation image ("\<P>")
 
 lemma image_le_pred[simp]: "(\<P> f A \<subseteq> {s. G s}) = (\<forall>x\<in>A. G (f x))"
   unfolding image_def by force
 
 definition ivp_sols :: "(real \<Rightarrow> 'a \<Rightarrow> ('a::real_normed_vector)) \<Rightarrow> real set \<Rightarrow> 'a set \<Rightarrow> 
   real \<Rightarrow> 'a \<Rightarrow> (real \<Rightarrow> 'a) set" ("Sols")
   where "Sols f T S t\<^sub>0 s = {X |X. (D X = (\<lambda>t. f t (X t)) on T) \<and> X t\<^sub>0 = s \<and> X \<in> T \<rightarrow> S}"
 
 lemma ivp_solsI: 
   assumes "D X = (\<lambda>t. f t (X t)) on T" "X t\<^sub>0 = s" "X \<in> T \<rightarrow> S"
   shows "X \<in> Sols f T S t\<^sub>0 s"
   using assms unfolding ivp_sols_def by blast
 
 lemma ivp_solsD:
   assumes "X \<in> Sols f T S t\<^sub>0 s"
   shows "D X = (\<lambda>t. f t (X t)) on T"
     and "X t\<^sub>0 = s" and "X \<in> T \<rightarrow> S"
   using assms unfolding ivp_sols_def by auto
 
 abbreviation "down T t \<equiv> {\<tau>\<in>T. \<tau>\<le> t}"
 
 definition g_orbit :: "(('a::ord) \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set" ("\<gamma>")
   where "\<gamma> X G T = \<Union>{\<P> X (down T t) |t. \<P> X (down T t) \<subseteq> {s. G s}}"
 
 lemma g_orbit_eq: 
   fixes X::"('a::preorder) \<Rightarrow> 'b"
   shows "\<gamma> X G T = {X t |t. t \<in> T \<and> (\<forall>\<tau>\<in>down T t. G (X \<tau>))}"
   unfolding g_orbit_def apply safe
   using le_left_mono by blast auto
 
 definition g_orbital :: "('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> real set \<Rightarrow> 'a set \<Rightarrow> real \<Rightarrow> 
   ('a::real_normed_vector) \<Rightarrow> 'a set" 
   where "g_orbital f G T S t\<^sub>0 s = \<Union>{\<gamma> X G T |X. X \<in> ivp_sols (\<lambda>t. f) T S t\<^sub>0 s}"
 
 lemma g_orbital_eq: "g_orbital f G T S t\<^sub>0 s = 
   {X t |t X. t \<in> T \<and> \<P> X (down T t) \<subseteq> {s. G s} \<and> X \<in> Sols (\<lambda>t. f) T S t\<^sub>0 s }" 
   unfolding g_orbital_def ivp_sols_def g_orbit_eq image_le_pred by auto
 
 lemma g_orbitalI:
   assumes "X \<in> Sols (\<lambda>t. f) T S t\<^sub>0 s"
     and "t \<in> T" and "(\<P> X (down T t) \<subseteq> {s. G s})"
   shows "X t \<in> g_orbital f G T S t\<^sub>0 s"
   using assms unfolding g_orbital_eq(1) by auto
 
 lemma g_orbitalD:
   assumes "s' \<in> g_orbital f G T S t\<^sub>0 s"
   obtains X and t where "X \<in> Sols (\<lambda>t. f) T S t\<^sub>0 s"
   and "X t = s'" and "t \<in> T" and "(\<P> X (down T t) \<subseteq> {s. G s})"
   using assms unfolding g_orbital_def g_orbit_eq by auto
 
 no_notation g_orbit ("\<gamma>")
 
 
 subsection \<open> Differential Invariants \<close>
 
 definition diff_invariant :: "('a \<Rightarrow> bool) \<Rightarrow> (('a::real_normed_vector) \<Rightarrow> 'a) \<Rightarrow> real set \<Rightarrow> 
   'a set \<Rightarrow> real \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" 
   where "diff_invariant I f T S t\<^sub>0 G \<equiv> (\<Union> \<circ> (\<P> (g_orbital f G T S t\<^sub>0))) {s. I s} \<subseteq> {s. I s}"
 
 lemma diff_invariant_eq: "diff_invariant I f T S t\<^sub>0 G = 
   (\<forall>s. I s \<longrightarrow> (\<forall>X\<in>Sols (\<lambda>t. f) T S t\<^sub>0 s. (\<forall>t\<in>T.(\<forall>\<tau>\<in>(down T t). G (X \<tau>)) \<longrightarrow> I (X t))))"
   unfolding diff_invariant_def g_orbital_eq image_le_pred by auto
 
 lemma diff_inv_eq_inv_set:
   "diff_invariant I f T S t\<^sub>0 G = (\<forall>s. I s \<longrightarrow> (g_orbital f G T S t\<^sub>0 s) \<subseteq> {s. I s})"
   unfolding diff_invariant_eq g_orbital_eq image_le_pred by auto
 
 named_theorems diff_invariant_rules "rules for certifying differential invariants."
 
 lemma diff_invariant_eq_rule [diff_invariant_rules]:
   assumes Thyp: "is_interval T" "t\<^sub>0 \<in> T"
     and "\<forall>X. (D X = (\<lambda>\<tau>. f (X \<tau>)) on T) \<longrightarrow> (D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = ((*\<^sub>R) 0) on T)"
   shows "diff_invariant (\<lambda>s. \<mu> s = \<nu> s) f T S t\<^sub>0 G"
 proof(simp add: diff_invariant_eq ivp_sols_def, clarsimp)
   fix X \<tau> assume tHyp:"\<tau> \<in> T" and x_ivp:"D X = (\<lambda>\<tau>. f (X \<tau>)) on T" "\<mu> (X t\<^sub>0) = \<nu> (X t\<^sub>0)" 
   hence obs1: "\<forall>t\<in>T. D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R 0) at t within T"
     using assms by (auto simp: has_vderiv_on_def has_vector_derivative_def)
   have obs2: "{t\<^sub>0--\<tau>} \<subseteq> T"
     using closed_segment_subset_interval tHyp Thyp by blast
   hence "D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<tau> *\<^sub>R 0) on {t\<^sub>0--\<tau>}"
     using obs1 x_ivp by (auto intro!: has_derivative_subset[OF _ obs2]
         simp: has_vderiv_on_def has_vector_derivative_def)
   then obtain t where "t \<in> {t\<^sub>0--\<tau>}" and "\<mu> (X \<tau>) - \<nu> (X \<tau>) - (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0)) = (\<tau> - t\<^sub>0) * t *\<^sub>R 0"
     using mvt_very_simple_closed_segmentE by blast
   thus "\<mu> (X \<tau>) = \<nu> (X \<tau>)" 
     by (simp add: x_ivp(2))
 qed
 
 lemma diff_invariant_leq_rule [diff_invariant_rules]:
   fixes \<mu>::"'a::banach \<Rightarrow> real"
   assumes Thyp: "is_interval T" "t\<^sub>0 \<in> T"
     and "\<forall>X. (D X = (\<lambda>\<tau>. f (X \<tau>)) on T) \<longrightarrow> (\<forall>\<tau>\<in>T. (\<tau> > t\<^sub>0 \<longrightarrow> \<mu>' (X \<tau>) \<ge> \<nu>' (X \<tau>)) \<and> 
   (\<tau> < t\<^sub>0 \<longrightarrow> \<mu>' (X \<tau>) \<le> \<nu>' (X \<tau>))) \<and> (D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on T)"
   shows "diff_invariant (\<lambda>s. \<nu> s \<le> \<mu> s) f T S t\<^sub>0 G"
 proof(simp add: diff_invariant_eq ivp_sols_def, clarsimp)
   fix X \<tau> assume "\<tau> \<in> T" and x_ivp:"D X = (\<lambda>\<tau>. f (X \<tau>)) on T" "\<nu> (X t\<^sub>0) \<le> \<mu> (X t\<^sub>0)"
   {assume "\<tau> \<noteq> t\<^sub>0"
   hence primed: "\<And>\<tau>. \<tau> \<in> T \<Longrightarrow> \<tau> > t\<^sub>0 \<Longrightarrow> \<mu>' (X \<tau>) \<ge> \<nu>' (X \<tau>)" 
     "\<And>\<tau>. \<tau> \<in> T \<Longrightarrow> \<tau> < t\<^sub>0 \<Longrightarrow> \<mu>' (X \<tau>) \<le> \<nu>' (X \<tau>)"
     using x_ivp assms by auto
   have obs1: "\<forall>t\<in>T. D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R (\<mu>' (X t) - \<nu>' (X t))) at t within T"
     using assms x_ivp by (auto simp: has_vderiv_on_def has_vector_derivative_def)
   have obs2: "{t\<^sub>0<--<\<tau>} \<subseteq> T" "{t\<^sub>0--\<tau>} \<subseteq> T"
     using \<open>\<tau> \<in> T\<close> Thyp \<open>\<tau> \<noteq> t\<^sub>0\<close> by (auto simp: convex_contains_open_segment 
         is_interval_convex_1 closed_segment_subset_interval)
   hence "D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on {t\<^sub>0--\<tau>}"
     using obs1 x_ivp by (auto intro!: has_derivative_subset[OF _ obs2(2)]
         simp: has_vderiv_on_def has_vector_derivative_def)
   then obtain t where "t \<in> {t\<^sub>0<--<\<tau>}" and
     "(\<mu> (X \<tau>) - \<nu> (X \<tau>)) - (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0)) = (\<lambda>\<tau>. \<tau> * (\<mu>' (X t) -  \<nu>' (X t))) (\<tau> - t\<^sub>0)"
     using mvt_simple_closed_segmentE \<open>\<tau> \<noteq> t\<^sub>0\<close> by blast
   hence mvt: "\<mu> (X \<tau>) - \<nu> (X \<tau>) = (\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) + (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0))" 
     by force
   have  "\<tau> > t\<^sub>0 \<Longrightarrow> t > t\<^sub>0" "\<not> t\<^sub>0 \<le> \<tau> \<Longrightarrow> t < t\<^sub>0" "t \<in> T"
     using \<open>t \<in> {t\<^sub>0<--<\<tau>}\<close> obs2 unfolding open_segment_eq_real_ivl by auto
   moreover have "t > t\<^sub>0 \<Longrightarrow> (\<mu>' (X t) -  \<nu>' (X t)) \<ge> 0" "t < t\<^sub>0 \<Longrightarrow> (\<mu>' (X t) -  \<nu>' (X t)) \<le> 0"
     using primed(1,2)[OF \<open>t \<in> T\<close>] by auto
   ultimately have "(\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) \<ge> 0"
     apply(case_tac "\<tau> \<ge> t\<^sub>0") 
     by (force, auto simp: split_mult_pos_le)
   hence "(\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) + (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0)) \<ge> 0" 
     using  x_ivp(2) by auto
   hence "\<nu> (X \<tau>) \<le> \<mu> (X \<tau>)" 
     using mvt by simp}
   thus "\<nu> (X \<tau>) \<le> \<mu> (X \<tau>)"
     using x_ivp by blast
 qed
 
 lemma diff_invariant_less_rule [diff_invariant_rules]:
   fixes \<mu>::"'a::banach \<Rightarrow> real"
   assumes Thyp: "is_interval T" "t\<^sub>0 \<in> T"
     and "\<forall> X. (D X = (\<lambda>\<tau>. f (X \<tau>)) on T) \<longrightarrow> (\<forall>\<tau>\<in>T. (\<tau> > t\<^sub>0 \<longrightarrow> \<mu>' (X \<tau>) \<ge> \<nu>' (X \<tau>)) \<and> 
   (\<tau> < t\<^sub>0 \<longrightarrow> \<mu>' (X \<tau>) \<le> \<nu>' (X \<tau>))) \<and> (D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on T)"
   shows "diff_invariant (\<lambda>s. \<nu> s < \<mu> s) f T S t\<^sub>0 G"
 proof(simp add: diff_invariant_eq ivp_sols_def, clarsimp)
   fix X \<tau> assume "\<tau> \<in> T" and x_ivp:"D X = (\<lambda>\<tau>. f (X \<tau>)) on T" "\<nu> (X t\<^sub>0) < \<mu> (X t\<^sub>0)"
   {assume "\<tau> \<noteq> t\<^sub>0"
   hence primed: "\<And>\<tau>. \<tau> \<in> T \<Longrightarrow> \<tau> > t\<^sub>0 \<Longrightarrow> \<mu>' (X \<tau>) \<ge> \<nu>' (X \<tau>)" 
     "\<And>\<tau>. \<tau> \<in> T \<Longrightarrow> \<tau> < t\<^sub>0 \<Longrightarrow> \<mu>' (X \<tau>) \<le> \<nu>' (X \<tau>)"
     using x_ivp assms by auto
   have obs1: "\<forall>t\<in>T. D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R (\<mu>' (X t) - \<nu>' (X t))) at t within T"
     using assms x_ivp by (auto simp: has_vderiv_on_def has_vector_derivative_def)
   have obs2: "{t\<^sub>0<--<\<tau>} \<subseteq> T" "{t\<^sub>0--\<tau>} \<subseteq> T"
     using \<open>\<tau> \<in> T\<close> Thyp \<open>\<tau> \<noteq> t\<^sub>0\<close> by (auto simp: convex_contains_open_segment 
         is_interval_convex_1 closed_segment_subset_interval)
   hence "D (\<lambda>\<tau>. \<mu> (X \<tau>) - \<nu> (X \<tau>)) = (\<lambda>\<tau>. \<mu>' (X \<tau>) - \<nu>' (X \<tau>)) on {t\<^sub>0--\<tau>}"
     using obs1 x_ivp by (auto intro!: has_derivative_subset[OF _ obs2(2)]
         simp: has_vderiv_on_def has_vector_derivative_def)
   then obtain t where "t \<in> {t\<^sub>0<--<\<tau>}" and
     "(\<mu> (X \<tau>) - \<nu> (X \<tau>)) - (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0)) = (\<lambda>\<tau>. \<tau> * (\<mu>' (X t) -  \<nu>' (X t))) (\<tau> - t\<^sub>0)"
     using mvt_simple_closed_segmentE \<open>\<tau> \<noteq> t\<^sub>0\<close> by blast
   hence mvt: "\<mu> (X \<tau>) - \<nu> (X \<tau>) = (\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) + (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0))" 
     by force
   have  "\<tau> > t\<^sub>0 \<Longrightarrow> t > t\<^sub>0" "\<not> t\<^sub>0 \<le> \<tau> \<Longrightarrow> t < t\<^sub>0" "t \<in> T"
     using \<open>t \<in> {t\<^sub>0<--<\<tau>}\<close> obs2 unfolding open_segment_eq_real_ivl by auto
   moreover have "t > t\<^sub>0 \<Longrightarrow> (\<mu>' (X t) -  \<nu>' (X t)) \<ge> 0" "t < t\<^sub>0 \<Longrightarrow> (\<mu>' (X t) -  \<nu>' (X t)) \<le> 0"
     using primed(1,2)[OF \<open>t \<in> T\<close>] by auto
   ultimately have "(\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) \<ge> 0"
     apply(case_tac "\<tau> \<ge> t\<^sub>0") 
     by (force, auto simp: split_mult_pos_le)
   hence "(\<tau> - t\<^sub>0) * (\<mu>' (X t) -  \<nu>' (X t)) + (\<mu> (X t\<^sub>0) - \<nu> (X t\<^sub>0)) > 0" 
     using x_ivp(2) by auto
   hence "\<nu> (X \<tau>) < \<mu> (X \<tau>)" 
     using mvt by simp}
   thus "\<nu> (X \<tau>) < \<mu> (X \<tau>)"
     using x_ivp by blast
 qed
 
 lemma diff_invariant_conj_rule [diff_invariant_rules]:
 assumes "diff_invariant I\<^sub>1 f T S t\<^sub>0 G" 
     and "diff_invariant I\<^sub>2 f T S t\<^sub>0 G"
 shows "diff_invariant (\<lambda>s. I\<^sub>1 s \<and> I\<^sub>2 s) f T S t\<^sub>0 G"
   using assms unfolding diff_invariant_def by auto
 
 lemma diff_invariant_disj_rule [diff_invariant_rules]:
 assumes "diff_invariant I\<^sub>1 f T S t\<^sub>0 G" 
     and "diff_invariant I\<^sub>2 f T S t\<^sub>0 G"
 shows "diff_invariant (\<lambda>s. I\<^sub>1 s \<or> I\<^sub>2 s) f T S t\<^sub>0 G"
   using assms unfolding diff_invariant_def by auto
 
 
 subsection \<open> Picard-Lindeloef \<close>
 
 text\<open> A locale with the assumptions of Picard-Lindeloef theorem. It extends 
 @{term "ll_on_open_it"} by providing an initial time @{term "t\<^sub>0 \<in> T"}.\<close>
 
 locale picard_lindeloef =
   fixes f::"real \<Rightarrow> ('a::{heine_borel,banach}) \<Rightarrow> 'a" and T::"real set" and S::"'a set" and t\<^sub>0::real
   assumes open_domain: "open T" "open S"
     and interval_time: "is_interval T"
     and init_time: "t\<^sub>0 \<in> T"
     and cont_vec_field: "\<forall>s \<in> S. continuous_on T (\<lambda>t. f t s)"
     and lipschitz_vec_field: "local_lipschitz T S f"
 begin
 
 sublocale ll_on_open_it T f S t\<^sub>0
   by (unfold_locales) (auto simp: cont_vec_field lipschitz_vec_field interval_time open_domain) 
 
 lemmas subintervalI = closed_segment_subset_domain
 
 lemma csols_eq: "csols t\<^sub>0 s = {(X, t). t \<in> T \<and>  X \<in> Sols f {t\<^sub>0--t} S t\<^sub>0 s}"
   unfolding ivp_sols_def csols_def solves_ode_def using subintervalI[OF init_time] by auto
 
 abbreviation "ex_ivl s \<equiv> existence_ivl t\<^sub>0 s"
 
 lemma unique_solution:
   assumes xivp: "D X = (\<lambda>t. f t (X t)) on {t\<^sub>0--t}" "X t\<^sub>0 = s" "X \<in> {t\<^sub>0--t} \<rightarrow> S" and "t \<in> T"
     and yivp: "D Y = (\<lambda>t. f t (Y t)) on {t\<^sub>0--t}" "Y t\<^sub>0 = s" "Y \<in> {t\<^sub>0--t} \<rightarrow> S" and "s \<in> S" 
   shows "X t = Y t"
 proof-
   have "(X, t) \<in> csols t\<^sub>0 s"
     using xivp \<open>t \<in> T\<close> unfolding csols_eq ivp_sols_def by auto
   hence ivl_fact: "{t\<^sub>0--t} \<subseteq> ex_ivl s"
     unfolding existence_ivl_def by auto
   have obs: "\<And>z T'. t\<^sub>0 \<in> T' \<and> is_interval T' \<and> T' \<subseteq> ex_ivl s \<and> (z solves_ode f) T' S \<Longrightarrow> 
   z t\<^sub>0 = flow t\<^sub>0 s t\<^sub>0 \<Longrightarrow> (\<forall>t\<in>T'. z t = flow t\<^sub>0 s t)"
     using flow_usolves_ode[OF init_time \<open>s \<in> S\<close>] unfolding usolves_ode_from_def by blast
   have "\<forall>\<tau>\<in>{t\<^sub>0--t}. X \<tau> = flow t\<^sub>0 s \<tau>"
     using obs[of "{t\<^sub>0--t}" X] xivp ivl_fact flow_initial_time[OF init_time  \<open>s \<in> S\<close>] 
     unfolding solves_ode_def by simp
   also have "\<forall>\<tau>\<in>{t\<^sub>0--t}. Y \<tau> = flow t\<^sub>0 s \<tau>"
     using obs[of "{t\<^sub>0--t}" Y] yivp ivl_fact flow_initial_time[OF init_time  \<open>s \<in> S\<close>] 
     unfolding solves_ode_def by simp
   ultimately show "X t = Y t"
     by auto
 qed
 
 lemma solution_eq_flow:
   assumes xivp: "D X = (\<lambda>t. f t (X t)) on ex_ivl s" "X t\<^sub>0 = s" "X \<in> ex_ivl s \<rightarrow> S" 
     and "t \<in> ex_ivl s" and "s \<in> S" 
   shows "X t = flow t\<^sub>0 s t"
 proof-
   have obs: "\<And>z T'. t\<^sub>0 \<in> T' \<and> is_interval T' \<and> T' \<subseteq> ex_ivl s \<and> (z solves_ode f) T' S \<Longrightarrow> 
   z t\<^sub>0 = flow t\<^sub>0 s t\<^sub>0 \<Longrightarrow> (\<forall>t\<in>T'. z t = flow t\<^sub>0 s t)"
     using flow_usolves_ode[OF init_time \<open>s \<in> S\<close>] unfolding usolves_ode_from_def by blast
   have "\<forall>\<tau>\<in>ex_ivl s. X \<tau> = flow t\<^sub>0 s \<tau>"
     using obs[of "ex_ivl s" X] existence_ivl_initial_time[OF init_time \<open>s \<in> S\<close>]
       xivp flow_initial_time[OF init_time \<open>s \<in> S\<close>] unfolding solves_ode_def by simp
   thus "X t = flow t\<^sub>0 s t"
     by (auto simp: \<open>t \<in> ex_ivl s\<close>)
 qed
 
 end
 
 lemma local_lipschitz_add: 
   fixes f1 f2 :: "real \<Rightarrow> 'a::banach \<Rightarrow> 'a"
   assumes "local_lipschitz T S f1"
       and "local_lipschitz T S f2" 
     shows "local_lipschitz T S (\<lambda>t s. f1 t s + f2 t s)"
 proof(unfold local_lipschitz_def, clarsimp)
   fix s and t assume "s \<in> S" and "t \<in> T"
   obtain \<epsilon>\<^sub>1 L1 where "\<epsilon>\<^sub>1 > 0" and L1: "\<And>\<tau>. \<tau>\<in>cball t \<epsilon>\<^sub>1 \<inter> T \<Longrightarrow> L1-lipschitz_on (cball s \<epsilon>\<^sub>1 \<inter> S) (f1 \<tau>)"
     using local_lipschitzE[OF assms(1) \<open>t \<in> T\<close> \<open>s \<in> S\<close>] by blast
   obtain \<epsilon>\<^sub>2 L2 where "\<epsilon>\<^sub>2 > 0" and L2: "\<And>\<tau>. \<tau>\<in>cball t \<epsilon>\<^sub>2 \<inter> T \<Longrightarrow> L2-lipschitz_on (cball s \<epsilon>\<^sub>2 \<inter> S) (f2 \<tau>)"
     using local_lipschitzE[OF assms(2) \<open>t \<in> T\<close> \<open>s \<in> S\<close>] by blast
   have ballH: "cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S \<subseteq> cball s \<epsilon>\<^sub>1 \<inter> S" "cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S \<subseteq> cball s \<epsilon>\<^sub>2 \<inter> S"
     by auto
   have obs1: "\<forall>\<tau>\<in>cball t \<epsilon>\<^sub>1 \<inter> T. L1-lipschitz_on (cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S) (f1 \<tau>)"
     using lipschitz_on_subset[OF L1 ballH(1)] by blast
   also have obs2: "\<forall>\<tau>\<in>cball t \<epsilon>\<^sub>2 \<inter> T. L2-lipschitz_on (cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S) (f2 \<tau>)"
     using lipschitz_on_subset[OF L2 ballH(2)] by blast
   ultimately have "\<forall>\<tau>\<in>cball t (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> T. 
     (L1 + L2)-lipschitz_on (cball s (min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2) \<inter> S) (\<lambda>s. f1 \<tau> s + f2 \<tau> s)"
     using lipschitz_on_add by fastforce
   thus "\<exists>u>0. \<exists>L. \<forall>t\<in>cball t u \<inter> T. L-lipschitz_on (cball s u \<inter> S) (\<lambda>s. f1 t s + f2 t s)"
     apply(rule_tac x="min \<epsilon>\<^sub>1 \<epsilon>\<^sub>2" in exI)
     using \<open>\<epsilon>\<^sub>1 > 0\<close> \<open>\<epsilon>\<^sub>2 > 0\<close> by force
 qed
 
 lemma picard_lindeloef_add: "picard_lindeloef f1 T S t\<^sub>0 \<Longrightarrow> picard_lindeloef f2 T S t\<^sub>0 \<Longrightarrow> 
   picard_lindeloef (\<lambda>t s. f1 t s + f2 t s) T S t\<^sub>0"
   unfolding picard_lindeloef_def apply(clarsimp, rule conjI)
   using continuous_on_add apply fastforce
   using local_lipschitz_add by blast
 
 lemma picard_lindeloef_constant: "picard_lindeloef (\<lambda>t s. c) UNIV UNIV t\<^sub>0"
   apply(unfold_locales, simp_all add: local_lipschitz_def lipschitz_on_def, clarsimp)
   by (rule_tac x=1 in exI, clarsimp, rule_tac x="1/2" in exI, simp)
 
 
 subsection \<open> Flows for ODEs \<close>
 
 text\<open> A locale designed for verification of hybrid systems. The user can select the interval 
 of existence and the defining flow equation via the variables @{term "T"} and @{term "\<phi>"}.\<close>
 
 locale local_flow = picard_lindeloef "(\<lambda> t. f)" T S 0 
   for f::"'a::{heine_borel,banach} \<Rightarrow> 'a" and T S L +
   fixes \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"
   assumes ivp:
     "\<And> t s. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--t}"
     "\<And> s. s \<in> S \<Longrightarrow> \<phi> 0 s = s"
     "\<And> t s. t \<in> T \<Longrightarrow> s \<in> S \<Longrightarrow> (\<lambda>t. \<phi> t s) \<in> {0--t} \<rightarrow> S"
 begin
 
 lemma in_ivp_sols_ivl: 
   assumes "t \<in> T" "s \<in> S"
   shows "(\<lambda>t. \<phi> t s) \<in> Sols (\<lambda>t. f) {0--t} S 0 s"
   apply(rule ivp_solsI)
   using ivp assms by auto
 
 lemma eq_solution_ivl:
   assumes xivp: "D X = (\<lambda>t. f (X t)) on {0--t}" "X 0 = s" "X \<in> {0--t} \<rightarrow> S" 
     and indom: "t \<in> T" "s \<in> S"
   shows "X t = \<phi> t s"
   apply(rule unique_solution[OF xivp \<open>t \<in> T\<close>])
   using \<open>s \<in> S\<close> ivp indom by auto
 
 lemma ex_ivl_eq:
   assumes "s \<in> S"
   shows "ex_ivl s = T"
   using existence_ivl_subset[of s] apply safe
   unfolding existence_ivl_def csols_eq 
   using in_ivp_sols_ivl[OF _ assms] by blast
 
 lemma has_derivative_on_open1: 
   assumes  "t > 0" "t \<in> T" "s \<in> S"
   obtains B where "t \<in> B" and "open B" and "B \<subseteq> T"
     and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B" 
 proof-
   obtain r::real where rHyp: "r > 0" "ball t r \<subseteq> T"
     using open_contains_ball_eq open_domain(1) \<open>t \<in> T\<close> by blast
   moreover have "t + r/2 > 0"
     using \<open>r > 0\<close> \<open>t > 0\<close> by auto
   moreover have "{0--t} \<subseteq> T" 
     using subintervalI[OF init_time \<open>t \<in> T\<close>] .
   ultimately have subs: "{0<--<t + r/2} \<subseteq> T"
     unfolding abs_le_eq abs_le_eq real_ivl_eqs[OF \<open>t > 0\<close>] real_ivl_eqs[OF \<open>t + r/2 > 0\<close>] 
     by clarify (case_tac "t < x", simp_all add: cball_def ball_def dist_norm subset_eq field_simps)
   have "t + r/2 \<in> T"
     using rHyp unfolding real_ivl_eqs[OF rHyp(1)] by (simp add: subset_eq)
   hence "{0--t + r/2} \<subseteq> T"
     using subintervalI[OF init_time] by blast
   hence "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--(t + r/2)})"
     using ivp(1)[OF _ \<open>s \<in> S\<close>] by auto
   hence vderiv: "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0<--<t + r/2})"
     apply(rule has_vderiv_on_subset)
     unfolding real_ivl_eqs[OF \<open>t + r/2 > 0\<close>] by auto
   have "t \<in> {0<--<t + r/2}"
     unfolding real_ivl_eqs[OF \<open>t + r/2 > 0\<close>] using rHyp \<open>t > 0\<close> by simp
   moreover have "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) (at t within {0<--<t + r/2})"
     using vderiv calculation unfolding has_vderiv_on_def has_vector_derivative_def by blast
   moreover have "open {0<--<t + r/2}"
     unfolding real_ivl_eqs[OF \<open>t + r/2 > 0\<close>] by simp
   ultimately show ?thesis
     using subs that by blast
 qed
 
 lemma has_derivative_on_open2: 
   assumes "t < 0" "t \<in> T" "s \<in> S"
   obtains B where "t \<in> B" and "open B" and "B \<subseteq> T"
     and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B" 
 proof-
   obtain r::real where rHyp: "r > 0" "ball t r \<subseteq> T"
     using open_contains_ball_eq open_domain(1) \<open>t \<in> T\<close> by blast
   moreover have "t - r/2 < 0"
     using \<open>r > 0\<close> \<open>t < 0\<close> by auto
   moreover have "{0--t} \<subseteq> T" 
     using subintervalI[OF init_time \<open>t \<in> T\<close>] .
   ultimately have subs: "{0<--<t - r/2} \<subseteq> T"
     unfolding open_segment_eq_real_ivl closed_segment_eq_real_ivl
       real_ivl_eqs[OF rHyp(1)] by(auto simp: subset_eq)
   have "t - r/2 \<in> T"
     using rHyp unfolding real_ivl_eqs by (simp add: subset_eq)
   hence "{0--t - r/2} \<subseteq> T"
     using subintervalI[OF init_time] by blast
   hence "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--(t - r/2)})"
     using ivp(1)[OF _ \<open>s \<in> S\<close>] by auto
   hence vderiv: "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0<--<t - r/2})"
     apply(rule has_vderiv_on_subset)
     unfolding open_segment_eq_real_ivl closed_segment_eq_real_ivl by auto
   have "t \<in> {0<--<t - r/2}"
     unfolding open_segment_eq_real_ivl using rHyp \<open>t < 0\<close> by simp
   moreover have "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) (at t within {0<--<t - r/2})"
     using vderiv calculation unfolding has_vderiv_on_def has_vector_derivative_def by blast
   moreover have "open {0<--<t - r/2}"
     unfolding open_segment_eq_real_ivl by simp
   ultimately show ?thesis
     using subs that by blast
 qed
 
 lemma has_derivative_on_open3: 
   assumes "s \<in> S"
   obtains B where "0 \<in> B" and "open B" and "B \<subseteq> T"
     and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> 0 s)) at 0 within B" 
 proof-
   obtain r::real where rHyp: "r > 0" "ball 0 r \<subseteq> T"
     using open_contains_ball_eq open_domain(1) init_time by blast
   hence "r/2 \<in> T" "-r/2 \<in> T" "r/2 > 0"
     unfolding real_ivl_eqs by auto
   hence subs: "{0--r/2} \<subseteq> T" "{0--(-r/2)} \<subseteq> T"
     using subintervalI[OF init_time] by auto
   hence "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--r/2})"
     "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--(-r/2)})"
     using ivp(1)[OF _ \<open>s \<in> S\<close>] by auto
   also have "{0--r/2} = {0--r/2} \<union> closure {0--r/2} \<inter> closure {0--(-r/2)}"
     "{0--(-r/2)} = {0--(-r/2)} \<union> closure {0--r/2} \<inter> closure {0--(-r/2)}"
     unfolding closed_segment_eq_real_ivl \<open>r/2 > 0\<close> by auto
   ultimately have vderivs:
     "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--r/2} \<union> closure {0--r/2} \<inter> closure {0--(-r/2)})"
     "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {0--(-r/2)} \<union> closure {0--r/2} \<inter> closure {0--(-r/2)})"
     unfolding closed_segment_eq_real_ivl \<open>r/2 > 0\<close> by auto
   have obs: "0 \<in> {-r/2<--<r/2}"
     unfolding open_segment_eq_real_ivl using \<open>r/2 > 0\<close> by auto
   have union: "{-r/2--r/2} = {0--r/2} \<union> {0--(-r/2)}"
     unfolding closed_segment_eq_real_ivl by auto
   hence "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {-r/2--r/2})"
     using has_vderiv_on_union[OF vderivs] by simp
   hence "(D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on {-r/2<--<r/2})"
     using has_vderiv_on_subset[OF _ segment_open_subset_closed[of "-r/2" "r/2"]] by auto
   hence "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> 0 s)) (at 0 within {-r/2<--<r/2})"
     unfolding has_vderiv_on_def has_vector_derivative_def using obs by blast
   moreover have "open {-r/2<--<r/2}"
     unfolding open_segment_eq_real_ivl by simp
   moreover have "{-r/2<--<r/2} \<subseteq> T"
     using subs union segment_open_subset_closed by blast 
   ultimately show ?thesis
     using obs that by blast
 qed
 
 lemma has_derivative_on_open: 
   assumes "t \<in> T" "s \<in> S"
   obtains B where "t \<in> B" and "open B" and "B \<subseteq> T"
     and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B" 
   apply(subgoal_tac "t < 0 \<or> t = 0 \<or> t > 0")
   using has_derivative_on_open1[OF _ assms] has_derivative_on_open2[OF _ assms]
     has_derivative_on_open3[OF \<open>s \<in> S\<close>] by blast force
 
 lemma in_domain:
   assumes "s \<in> S"
   shows "(\<lambda>t. \<phi> t s) \<in> T \<rightarrow> S"
   unfolding ex_ivl_eq[symmetric] existence_ivl_def
   using local.mem_existence_ivl_subset ivp(3)[OF _ assms] by blast
 
 lemma has_vderiv_on_domain:
   assumes "s \<in> S"
   shows "D (\<lambda>t. \<phi> t s) = (\<lambda>t. f (\<phi> t s)) on T"
 proof(unfold has_vderiv_on_def has_vector_derivative_def, clarsimp)
   fix t assume "t \<in> T"
   then obtain B where "t \<in> B" and "open B" and "B \<subseteq> T" 
     and Dhyp: "D (\<lambda>t. \<phi> t s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within B"
     using assms has_derivative_on_open[OF \<open>t \<in> T\<close>] by blast
   hence "t \<in> interior B"
     using interior_eq by auto
   thus "D (\<lambda>t. \<phi> t s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> t s)) at t within T"
     using has_derivative_at_within_mono[OF _ \<open>B \<subseteq> T\<close> Dhyp] by blast
 qed
 
 lemma in_ivp_sols: 
   assumes "s \<in> S"
   shows "(\<lambda>t. \<phi> t s) \<in> Sols (\<lambda>t. f) T S 0 s"
   using has_vderiv_on_domain ivp(2) in_domain apply(rule ivp_solsI)
   using assms by auto
 
 lemma eq_solution:
   assumes "X \<in> Sols (\<lambda>t. f) T S 0 s" and "t \<in> T" and "s \<in> S"
   shows "X t = \<phi> t s"
 proof-
   have "D X = (\<lambda>t. f (X t)) on (ex_ivl s)" and "X 0 = s" and "X \<in> (ex_ivl s) \<rightarrow> S"
     using ivp_solsD[OF assms(1)] unfolding ex_ivl_eq[OF \<open>s \<in> S\<close>] by auto
   note solution_eq_flow[OF this]
   hence "X t = flow 0 s t"
     unfolding ex_ivl_eq[OF \<open>s \<in> S\<close>] using assms by blast
   also have "\<phi> t s = flow 0 s t"
     apply(rule solution_eq_flow ivp)
         apply(simp_all add: assms(2,3) ivp(2)[OF \<open>s \<in> S\<close>])
     unfolding ex_ivl_eq[OF \<open>s \<in> S\<close>] by (auto simp: has_vderiv_on_domain assms in_domain)
   ultimately show "X t = \<phi> t s"
     by simp
 qed
 
 lemma ivp_sols_collapse: 
   assumes "T = UNIV" and "s \<in> S"
   shows "Sols (\<lambda>t. f) T S 0 s = {(\<lambda>t. \<phi> t s)}"
   using in_ivp_sols eq_solution assms by auto
 
 lemma additive_in_ivp_sols:
   assumes "s \<in> S" and "\<P> (\<lambda>\<tau>. \<tau> + t) T \<subseteq> T"
   shows "(\<lambda>\<tau>. \<phi> (\<tau> + t) s) \<in> Sols (\<lambda>t. f) T S 0 (\<phi> (0 + t) s)"
   apply(rule ivp_solsI, rule vderiv_on_compose_add)
   using has_vderiv_on_domain has_vderiv_on_subset assms 
     apply blast
   using in_domain assms by auto
 
 lemma is_monoid_action:
   assumes "s \<in> S" and "T = UNIV"
   shows "\<phi> 0 s = s" and "\<phi> (t\<^sub>1 + t\<^sub>2) s = \<phi> t\<^sub>1 (\<phi> t\<^sub>2 s)"
 proof-
   show "\<phi> 0 s = s"
     using ivp assms by simp
   have "\<phi> (0 + t\<^sub>2) s = \<phi> t\<^sub>2 s" 
     by simp
   also have "\<phi> t\<^sub>2 s \<in> S"
     using in_domain assms by auto
   finally show "\<phi> (t\<^sub>1 + t\<^sub>2) s = \<phi> t\<^sub>1 (\<phi> t\<^sub>2 s)"
     using eq_solution[OF additive_in_ivp_sols] assms by auto
 qed
 
 definition orbit :: "'a \<Rightarrow> 'a set" ("\<gamma>\<^sup>\<phi>")
   where "\<gamma>\<^sup>\<phi> s = g_orbital f (\<lambda>s. True) T S 0 s"
 
 lemma orbit_eq[simp]: 
   assumes "s \<in> S"
   shows "\<gamma>\<^sup>\<phi> s = {\<phi> t s| t. t \<in> T}"
   using eq_solution assms unfolding orbit_def g_orbital_eq ivp_sols_def
   by(auto intro!: has_vderiv_on_domain ivp(2) in_domain)
 
 lemma g_orbital_collapses: 
   assumes "s \<in> S"
   shows "g_orbital f G T S 0 s = {\<phi> t s| t. t \<in> T \<and> (\<forall>\<tau>\<in>down T t. G (\<phi> \<tau> s))}"
 proof(rule subset_antisym, simp_all only: subset_eq)
   let ?gorbit = "{\<phi> t s |t. t \<in> T \<and> (\<forall>\<tau>\<in>down T t. G (\<phi> \<tau> s))}"
   {fix s' assume "s' \<in> g_orbital f G T S 0 s"
     then obtain X and t where x_ivp:"X \<in> Sols (\<lambda>t. f) T S 0 s" 
       and "X t = s'" and "t \<in> T" and guard:"(\<P> X (down T t) \<subseteq> {s. G s})"
       unfolding g_orbital_def g_orbit_eq by auto
     have obs:"\<forall>\<tau>\<in>(down T t). X \<tau> = \<phi> \<tau> s"
       using eq_solution[OF x_ivp _ assms] by blast
     hence "\<P> (\<lambda>t. \<phi> t s) (down T t) \<subseteq> {s. G s}"
       using guard by auto 
     also have "\<phi> t s = X t"
       using eq_solution[OF x_ivp \<open>t \<in> T\<close> assms] by simp
     ultimately have "s' \<in> ?gorbit"
       using \<open>X t = s'\<close> \<open>t \<in> T\<close> by auto}
   thus "\<forall>s'\<in> g_orbital f G T S 0 s. s' \<in> ?gorbit"
     by blast
 next
   let ?gorbit = "{\<phi> t s |t. t \<in> T \<and> (\<forall>\<tau>\<in>down T t. G (\<phi> \<tau> s))}"
   {fix s' assume "s' \<in> ?gorbit"
     then obtain t where "\<P> (\<lambda>t. \<phi> t s) (down T t) \<subseteq> {s. G s}" and "t \<in> T" and "\<phi> t s = s'"
       by blast
     hence "s' \<in> g_orbital f G T S 0 s"
       using assms by(auto intro!: g_orbitalI in_ivp_sols)}
   thus "\<forall>s'\<in>?gorbit. s' \<in> g_orbital f G T S 0 s"
     by blast
 qed
 
 end
 
 lemma line_is_local_flow: 
   "0 \<in> T \<Longrightarrow> is_interval T \<Longrightarrow> open T \<Longrightarrow> local_flow (\<lambda> s. c) T UNIV (\<lambda> t s. s + t *\<^sub>R c)"
   apply(unfold_locales, simp_all add: local_lipschitz_def lipschitz_on_def, clarsimp)
    apply(rule_tac x=1 in exI, clarsimp, rule_tac x="1/2" in exI, simp)
-  apply(rule_tac f'1="\<lambda> s. 0" and g'1="\<lambda> s. c" in derivative_intros(191))
+  apply(rule_tac f'1="\<lambda> s. 0" and g'1="\<lambda> s. c" in has_vderiv_on_add[THEN has_vderiv_on_eq_rhs])
     apply(rule derivative_intros, simp)+
   by simp_all
 
 end
\ No newline at end of file
diff --git a/thys/Hybrid_Systems_VCs/HS_Preliminaries.thy b/thys/Hybrid_Systems_VCs/HS_Preliminaries.thy
--- a/thys/Hybrid_Systems_VCs/HS_Preliminaries.thy
+++ b/thys/Hybrid_Systems_VCs/HS_Preliminaries.thy
@@ -1,315 +1,317 @@
 (*  Title:       Preliminaries for hybrid systems verification
     Author:      Jonathan Julián Huerta y Munive, 2019
     Maintainer:  Jonathan Julián Huerta y Munive <jjhuertaymunive1@sheffield.ac.uk>
 *)
 
 section \<open> Hybrid Systems Preliminaries \<close>
 
 text \<open>Hybrid systems combine continuous dynamics with discrete control. This section contains
 auxiliary lemmas for verification of hybrid systems.\<close>
 
 theory HS_Preliminaries
   imports "Ordinary_Differential_Equations.Picard_Lindeloef_Qualitative"
 begin
 
 
 subsection \<open> Real numbers \<close>
 
 lemma abs_le_eq:
   shows "(r::real) > 0 \<Longrightarrow> (\<bar>x\<bar> < r) = (-r < x \<and> x < r)"
     and "(r::real) > 0 \<Longrightarrow> (\<bar>x\<bar> \<le> r) = (-r \<le> x \<and> x \<le> r)"
   by linarith+
 
 lemma real_ivl_eqs:
   assumes "0 < r"
   shows "ball x r = {x-r<--< x+r}"         and "{x-r<--< x+r} = {x-r<..< x+r}"
     and "ball (r / 2) (r / 2) = {0<--<r}"  and "{0<--<r} = {0<..<r}"
     and "ball 0 r = {-r<--<r}"             and "{-r<--<r} = {-r<..<r}"
     and "cball x r = {x-r--x+r}"           and "{x-r--x+r} = {x-r..x+r}"
     and "cball (r / 2) (r / 2) = {0--r}"   and "{0--r} = {0..r}"
     and "cball 0 r = {-r--r}"              and "{-r--r} = {-r..r}"
   unfolding open_segment_eq_real_ivl closed_segment_eq_real_ivl
   using assms by (auto simp: cball_def ball_def dist_norm field_simps)
 
 lemma norm_rotate_eq[simp]:
   fixes x :: "'a:: {banach,real_normed_field}"
   shows "(x * cos t - y * sin t)\<^sup>2 + (x * sin t + y * cos t)\<^sup>2 = x\<^sup>2 + y\<^sup>2"
     and "(x * cos t + y * sin t)\<^sup>2 + (y * cos t - x * sin t)\<^sup>2 = x\<^sup>2 + y\<^sup>2"
 proof-
   have "(x * cos t - y * sin t)\<^sup>2 = x\<^sup>2 * (cos t)\<^sup>2 + y\<^sup>2 * (sin t)\<^sup>2 - 2 * (x * cos t) * (y * sin t)"
     by(simp add: power2_diff power_mult_distrib)
   also have "(x * sin t + y * cos t)\<^sup>2 = y\<^sup>2 * (cos t)\<^sup>2 + x\<^sup>2 * (sin t)\<^sup>2 + 2 * (x * cos t) * (y * sin t)"
     by(simp add: power2_sum power_mult_distrib)
   ultimately show "(x * cos t - y * sin t)\<^sup>2 + (x * sin t + y * cos t)\<^sup>2 = x\<^sup>2 + y\<^sup>2"
     by (simp add: Groups.mult_ac(2) Groups.mult_ac(3) right_diff_distrib sin_squared_eq)
   thus "(x * cos t + y * sin t)\<^sup>2 + (y * cos t - x * sin t)\<^sup>2 = x\<^sup>2 + y\<^sup>2"
     by (simp add: add.commute add.left_commute power2_diff power2_sum)
 qed
 
 
 subsection \<open> Single variable derivatives \<close>
 
 notation has_derivative ("(1(D _ \<mapsto> (_))/ _)" [65,65] 61)
 notation has_vderiv_on ("(1 D _ = (_)/ on _)" [65,65] 61)
 notation norm ("(1\<parallel>_\<parallel>)" [65] 61)
 
+thy_deps
+
 named_theorems poly_derivatives "compilation of optimised miscellaneous derivative rules."
 
 declare has_vderiv_on_const [poly_derivatives]
     and has_vderiv_on_id [poly_derivatives]
-    and derivative_intros(191) [poly_derivatives]
-    and derivative_intros(192) [poly_derivatives]
-    and derivative_intros(194) [poly_derivatives]
+    and has_vderiv_on_add[THEN has_vderiv_on_eq_rhs, poly_derivatives]
+    and has_vderiv_on_diff[THEN has_vderiv_on_eq_rhs, poly_derivatives]
+    and has_vderiv_on_mult[THEN has_vderiv_on_eq_rhs, poly_derivatives]
 
 lemma has_vderiv_on_compose_eq:
   assumes "D f = f' on g ` T"
     and " D g = g' on T"
     and "h = (\<lambda>x. g' x *\<^sub>R f' (g x))"
   shows "D (\<lambda>t. f (g t)) = h on T"
   apply(subst ssubst[of h], simp)
   using assms has_vderiv_on_compose by auto
 
 lemma vderiv_on_compose_add [derivative_intros]:
   assumes "D x = x' on (\<lambda>\<tau>. \<tau> + t) ` T"
   shows "D (\<lambda>\<tau>. x (\<tau> + t)) = (\<lambda>\<tau>. x' (\<tau> + t)) on T"
   apply(rule has_vderiv_on_compose_eq[OF assms])
   by(auto intro: derivative_intros)
 
 lemma has_vderiv_on_divide_cnst: "a \<noteq> 0 \<Longrightarrow> D (\<lambda>t. t/a) = (\<lambda>t. 1/a) on T"
   unfolding has_vderiv_on_def has_vector_derivative_def
   apply clarify
-  apply(rule_tac f'1="\<lambda>t. t" and g'1="\<lambda> x. 0" in derivative_eq_intros(18))
+  apply(rule_tac f'1="\<lambda>t. t" and g'1="\<lambda> x. 0" in has_derivative_divide[THEN has_derivative_eq_rhs])
   by(auto intro: derivative_eq_intros)
 
 lemma has_vderiv_on_power: "n \<ge> 1 \<Longrightarrow> D (\<lambda>t. t ^ n) = (\<lambda>t. n * (t ^ (n - 1))) on T"
   unfolding has_vderiv_on_def has_vector_derivative_def
   by (auto intro!: derivative_eq_intros)
 
 lemma has_vderiv_on_exp: "D (\<lambda>t. exp t) = (\<lambda>t. exp t) on T"
   unfolding has_vderiv_on_def has_vector_derivative_def by (auto intro: derivative_intros)
 
 lemma has_vderiv_on_cos_comp:
   "D (f::real \<Rightarrow> real) = f' on T \<Longrightarrow> D (\<lambda>t. cos (f t)) = (\<lambda>t. - (f' t) * sin (f t)) on T"
   apply(rule has_vderiv_on_compose_eq[of "\<lambda>t. cos t"])
   unfolding has_vderiv_on_def has_vector_derivative_def
     apply clarify
   by(auto intro!: derivative_eq_intros simp: fun_eq_iff)
 
 lemma has_vderiv_on_sin_comp:
   "D (f::real \<Rightarrow> real) = f' on T \<Longrightarrow> D (\<lambda>t. sin (f t)) = (\<lambda>t. (f' t) * cos (f t)) on T"
   apply(rule has_vderiv_on_compose_eq[of "\<lambda>t. sin t"])
   unfolding has_vderiv_on_def has_vector_derivative_def
     apply clarify
   by(auto intro!: derivative_eq_intros simp: fun_eq_iff)
 
 lemma has_vderiv_on_exp_comp:
   "D (f::real \<Rightarrow> real) = f' on T \<Longrightarrow> D (\<lambda>t. exp (f t)) = (\<lambda>t. (f' t) * exp (f t)) on T"
   apply(rule has_vderiv_on_compose_eq[of "\<lambda>t. exp t"])
   by (rule has_vderiv_on_exp, simp_all add: mult.commute)
 
 lemma vderiv_uminus_intro [poly_derivatives]:
   "D f = f' on T \<Longrightarrow> g = (\<lambda>t. - f' t) \<Longrightarrow> D (\<lambda>t. - f t) = g on T"
   using has_vderiv_on_uminus by auto
 
 lemma vderiv_div_cnst_intro [poly_derivatives]:
   assumes "(a::real) \<noteq> 0" and "D f = f' on T" and "g = (\<lambda>t. (f' t)/a)"
   shows "D (\<lambda>t. (f t)/a) = g on T"
   apply(rule has_vderiv_on_compose_eq[of "\<lambda>t. t/a" "\<lambda>t. 1/a"])
   using assms by(auto intro: has_vderiv_on_divide_cnst)
 
 lemma vderiv_npow_intro [poly_derivatives]:
   fixes f::"real \<Rightarrow> real"
   assumes "n \<ge> 1" and "D f = f' on T" and "g = (\<lambda>t. n * (f' t) * (f t)^(n-1))"
   shows "D (\<lambda>t. (f t)^n) = g on T"
   apply(rule has_vderiv_on_compose_eq[of "\<lambda>t. t^n"])
   using assms(1)
     apply(rule has_vderiv_on_power)
   using assms by auto
 
 lemma vderiv_cos_intro [poly_derivatives]:
   assumes "D (f::real \<Rightarrow> real) = f' on T" and "g = (\<lambda>t. - (f' t) * sin (f t))"
   shows "D (\<lambda>t. cos (f t)) = g on T"
   using assms and has_vderiv_on_cos_comp by auto
 
 lemma vderiv_sin_intro [poly_derivatives]:
   assumes "D (f::real \<Rightarrow> real) = f' on T" and "g = (\<lambda>t. (f' t) * cos (f t))"
   shows "D (\<lambda>t. sin (f t)) = g on T"
   using assms and has_vderiv_on_sin_comp by auto
 
 lemma vderiv_exp_intro [poly_derivatives]:
   assumes "D (f::real \<Rightarrow> real) = f' on T" and "g = (\<lambda>t. (f' t) * exp (f t))"
   shows "D (\<lambda>t. exp (f t)) = g on T"
   using assms and has_vderiv_on_exp_comp by auto
 
 \<comment> \<open>Examples for checking derivatives\<close>
 
 lemma "D (\<lambda>t. a * t\<^sup>2 / 2 + v * t + x) = (\<lambda>t. a * t + v) on T"
   by(auto intro!: poly_derivatives)
 
 lemma "D (\<lambda>t. v * t - a * t\<^sup>2 / 2 + x) = (\<lambda>x. v - a * x) on T"
   by(auto intro!: poly_derivatives)
 
 lemma "c \<noteq> 0 \<Longrightarrow> D (\<lambda>t. a5 * t^5 + a3 * (t^3 / c) - a2 * exp (t^2) + a1 * cos t + a0) =
   (\<lambda>t. 5 * a5 * t^4 + 3 * a3 * (t^2 / c) - 2 * a2 * t * exp (t^2) - a1 * sin t) on T"
   by(auto intro!: poly_derivatives)
 
 lemma "c \<noteq> 0 \<Longrightarrow> D (\<lambda>t. - a3 * exp (t^3 / c) + a1 * sin t + a2 * t^2) =
   (\<lambda>t. a1 * cos t + 2 * a2 * t - 3 * a3 * t^2 / c * exp (t^3 / c)) on T"
   apply(intro poly_derivatives)
   using poly_derivatives(1,2) by force+
 
 lemma "c \<noteq> 0 \<Longrightarrow> D (\<lambda>t. exp (a * sin (cos (t^4) / c))) =
 (\<lambda>t. - 4 * a * t^3 * sin (t^4) / c * cos (cos (t^4) / c) * exp (a * sin (cos (t^4) / c))) on T"
   apply(intro poly_derivatives)
   using poly_derivatives(1,2) by force+
 
 
 subsection \<open> Filters \<close>
 
 lemma eventually_at_within_mono:
   assumes "t \<in> interior T" and "T \<subseteq> S"
     and "eventually P (at t within T)"
   shows "eventually P (at t within S)"
   by (meson assms eventually_within_interior interior_mono subsetD)
 
 lemma netlimit_at_within_mono:
   fixes t::"'a::{perfect_space,t2_space}"
   assumes "t \<in> interior T" and "T \<subseteq> S"
   shows "netlimit (at t within S) = t"
   using assms(1) interior_mono[OF \<open>T \<subseteq> S\<close>] netlimit_within_interior by auto
 
 lemma has_derivative_at_within_mono:
   assumes "(t::real) \<in> interior T" and "T \<subseteq> S"
     and "D f \<mapsto> f' at t within T"
   shows "D f \<mapsto> f' at t within S"
   using assms(3) apply(unfold has_derivative_def tendsto_iff, safe)
   unfolding netlimit_at_within_mono[OF assms(1,2)] netlimit_within_interior[OF assms(1)]
   by (rule eventually_at_within_mono[OF assms(1,2)]) simp
 
 lemma eventually_all_finite2:
   fixes P :: "('a::finite) \<Rightarrow> 'b \<Rightarrow> bool"
   assumes h:"\<forall>i. eventually (P i) F"
   shows "eventually (\<lambda>x. \<forall>i. P i x) F"
 proof(unfold eventually_def)
   let ?F = "Rep_filter F"
   have obs: "\<forall>i. ?F (P i)"
     using h by auto
   have "?F (\<lambda>x. \<forall>i \<in> UNIV. P i x)"
     apply(rule finite_induct)
     by(auto intro: eventually_conj simp: obs h)
   thus "?F (\<lambda>x. \<forall>i. P i x)"
     by simp
 qed
 
 lemma eventually_all_finite_mono:
   fixes P :: "('a::finite) \<Rightarrow> 'b \<Rightarrow> bool"
   assumes h1: "\<forall>i. eventually (P i) F"
       and h2: "\<forall>x. (\<forall>i. (P i x)) \<longrightarrow> Q x"
   shows "eventually Q F"
 proof-
   have "eventually (\<lambda>x. \<forall>i. P i x) F"
     using h1 eventually_all_finite2 by blast
   thus "eventually Q F"
     unfolding eventually_def
     using h2 eventually_mono by auto
 qed
 
 
 subsection \<open> Multivariable derivatives \<close>
 
 lemma frechet_vec_lambda:
   fixes f::"real \<Rightarrow> ('a::banach)^('m::finite)" and x::real and T::"real set"
   defines "x\<^sub>0 \<equiv> netlimit (at x within T)" and "m \<equiv> real CARD('m)"
   assumes "\<forall>i. ((\<lambda>y. (f y $ i - f x\<^sub>0 $ i - (y - x\<^sub>0) *\<^sub>R f' x $ i) /\<^sub>R (\<parallel>y - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at x within T)"
   shows "((\<lambda>y. (f y - f x\<^sub>0 - (y - x\<^sub>0) *\<^sub>R f' x) /\<^sub>R (\<parallel>y - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at x within T)"
 proof(simp add: tendsto_iff, clarify)
   fix \<epsilon>::real assume "0 < \<epsilon>"
   let "?\<Delta>" = "\<lambda>y. y - x\<^sub>0" and "?\<Delta>f" = "\<lambda>y. f y - f x\<^sub>0"
   let "?P" = "\<lambda>i e y. inverse \<bar>?\<Delta> y\<bar> * (\<parallel>f y $ i - f x\<^sub>0 $ i - ?\<Delta> y *\<^sub>R f' x $ i\<parallel>) < e"
     and "?Q" = "\<lambda>y. inverse \<bar>?\<Delta> y\<bar> * (\<parallel>?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x\<parallel>) < \<epsilon>"
   have "0 < \<epsilon> / sqrt m"
     using \<open>0 < \<epsilon>\<close> by (auto simp: assms)
   hence "\<forall>i. eventually (\<lambda>y. ?P i (\<epsilon> / sqrt m) y) (at x within T)"
     using assms unfolding tendsto_iff by simp
   thus "eventually ?Q (at x within T)"
   proof(rule eventually_all_finite_mono, simp add: norm_vec_def L2_set_def, clarify)
     fix t::real
     let ?c = "inverse \<bar>t - x\<^sub>0\<bar>" and "?u t" = "\<lambda>i. f t $ i - f x\<^sub>0 $ i - ?\<Delta> t *\<^sub>R f' x $ i"
     assume hyp:"\<forall>i. ?c * (\<parallel>?u t i\<parallel>) < \<epsilon> / sqrt m"
     hence "\<forall>i. (?c *\<^sub>R (\<parallel>?u t i\<parallel>))\<^sup>2 < (\<epsilon> / sqrt m)\<^sup>2"
       by (simp add: power_strict_mono)
     hence "\<forall>i. ?c\<^sup>2 * ((\<parallel>?u t i\<parallel>))\<^sup>2 < \<epsilon>\<^sup>2 / m"
       by (simp add: power_mult_distrib power_divide assms)
     hence "\<forall>i. ?c\<^sup>2 * ((\<parallel>?u t i\<parallel>))\<^sup>2 < \<epsilon>\<^sup>2 / m"
       by (auto simp: assms)
     also have "({}::'m set) \<noteq> UNIV \<and> finite (UNIV :: 'm set)"
       by simp
     ultimately have "(\<Sum>i\<in>UNIV. ?c\<^sup>2 * ((\<parallel>?u t i\<parallel>))\<^sup>2) < (\<Sum>(i::'m)\<in>UNIV. \<epsilon>\<^sup>2 / m)"
       by (metis (lifting) sum_strict_mono)
     moreover have "?c\<^sup>2 * (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2) = (\<Sum>i\<in>UNIV. ?c\<^sup>2 *  (\<parallel>?u t i\<parallel>)\<^sup>2)"
       using sum_distrib_left by blast
     ultimately have "?c\<^sup>2 * (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2) < \<epsilon>\<^sup>2"
       by (simp add: assms)
     hence "sqrt (?c\<^sup>2 * (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2)) < sqrt (\<epsilon>\<^sup>2)"
       using real_sqrt_less_iff by blast
     also have "... = \<epsilon>"
       using \<open>0 < \<epsilon>\<close> by auto
     moreover have "?c * sqrt (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2) = sqrt (?c\<^sup>2 * (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2))"
       by (simp add: real_sqrt_mult)
     ultimately show "?c * sqrt (\<Sum>i\<in>UNIV. (\<parallel>?u t i\<parallel>)\<^sup>2) < \<epsilon>"
       by simp
   qed
 qed
 
 lemma frechet_vec_nth:
   fixes f::"real \<Rightarrow> ('a::real_normed_vector)^'m" and x::real and T::"real set"
   defines "x\<^sub>0 \<equiv> netlimit (at x within T)"
   assumes "((\<lambda>y. (f y - f x\<^sub>0 - (y - x\<^sub>0) *\<^sub>R f' x) /\<^sub>R (\<parallel>y - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at x within T)"
   shows "((\<lambda>y. (f y $ i - f x\<^sub>0 $ i - (y - x\<^sub>0) *\<^sub>R f' x $ i) /\<^sub>R (\<parallel>y - x\<^sub>0\<parallel>)) \<longlongrightarrow> 0) (at x within T)"
 proof(unfold tendsto_iff dist_norm, clarify)
   let "?\<Delta>" = "\<lambda>y. y - x\<^sub>0" and "?\<Delta>f" = "\<lambda>y. f y - f x\<^sub>0"
   fix \<epsilon>::real assume "0 < \<epsilon>"
   let "?P" = "\<lambda>y. \<parallel>(?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x) /\<^sub>R (\<parallel>?\<Delta> y\<parallel>) - 0\<parallel> < \<epsilon>"
   and "?Q" = "\<lambda>y. \<parallel>(f y $ i - f x\<^sub>0 $ i - ?\<Delta> y *\<^sub>R f' x $ i) /\<^sub>R (\<parallel>?\<Delta> y\<parallel>) - 0\<parallel> < \<epsilon>"
   have "eventually ?P (at x within T)"
     using \<open>0 < \<epsilon>\<close> assms unfolding tendsto_iff by auto
   thus "eventually ?Q (at x within T)"
   proof(rule_tac P="?P" in eventually_mono, simp_all)
     fix y
     let "?u i" = "f y $ i - f x\<^sub>0 $ i - ?\<Delta> y *\<^sub>R f' x $ i"
     assume hyp:"inverse \<bar>?\<Delta> y\<bar> * (\<parallel>?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x\<parallel>) < \<epsilon>"
     have "\<parallel>(?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x) $ i\<parallel> \<le> \<parallel>?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x\<parallel>"
       using Finite_Cartesian_Product.norm_nth_le by blast
     also have "\<parallel>?u i\<parallel> = \<parallel>(?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x) $ i\<parallel>"
       by simp
     ultimately have "\<parallel>?u i\<parallel> \<le> \<parallel>?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x\<parallel>"
       by linarith
     hence "inverse \<bar>?\<Delta> y\<bar> * (\<parallel>?u i\<parallel>) \<le> inverse \<bar>?\<Delta> y\<bar> * (\<parallel>?\<Delta>f y - ?\<Delta> y *\<^sub>R f' x\<parallel>)"
       by (simp add: mult_left_mono)
     thus "inverse \<bar>?\<Delta> y\<bar> * (\<parallel>f y $ i - f x\<^sub>0 $ i - ?\<Delta> y *\<^sub>R f' x $ i\<parallel>) < \<epsilon>"
       using hyp by linarith
   qed
 qed
 
 lemma has_derivative_vec_lambda:
   fixes f::"real \<Rightarrow> ('a::banach)^('n::finite)"
   assumes "\<forall>i. D (\<lambda>t. f t $ i) \<mapsto> (\<lambda> h. h *\<^sub>R f' x $ i) (at x within T)"
   shows "D f \<mapsto> (\<lambda>h. h *\<^sub>R f' x) at x within T"
   apply(unfold has_derivative_def, safe)
    apply(force simp: bounded_linear_def bounded_linear_axioms_def)
   using assms frechet_vec_lambda[of x T ] unfolding has_derivative_def by auto
 
 lemma has_derivative_vec_nth:
   assumes "D f \<mapsto> (\<lambda>h. h *\<^sub>R f' x) at x within T"
   shows "D (\<lambda>t. f t $ i) \<mapsto> (\<lambda>h. h *\<^sub>R f' x $ i) at x within T"
   apply(unfold has_derivative_def, safe)
    apply(force simp: bounded_linear_def bounded_linear_axioms_def)
   using frechet_vec_nth[of x T f] assms unfolding has_derivative_def by auto
 
 lemma has_vderiv_on_vec_eq[simp]:
   fixes x::"real \<Rightarrow> ('a::banach)^('n::finite)"
   shows "(D x = x' on T) = (\<forall>i. D (\<lambda>t. x t $ i) = (\<lambda>t. x' t $ i) on T)"
   unfolding has_vderiv_on_def has_vector_derivative_def apply safe
   using has_derivative_vec_nth has_derivative_vec_lambda by blast+
 
 end
\ No newline at end of file
diff --git a/thys/Stirling_Formula/Ln_Gamma_Asymptotics.thy b/thys/Stirling_Formula/Ln_Gamma_Asymptotics.thy
--- a/thys/Stirling_Formula/Ln_Gamma_Asymptotics.thy
+++ b/thys/Stirling_Formula/Ln_Gamma_Asymptotics.thy
@@ -1,1328 +1,1328 @@
 (*
   File:    Ln_Gamma_Asymptotics.thy
   Author:  Manuel Eberl
 
   The complete asymptotics of the logarithmic Gamma function and the Polygamma functions.
 *)
 section \<open>Complete asymptotics of the logarithmic Gamma function\<close>
 theory Ln_Gamma_Asymptotics
 imports 
-  "HOL-Analysis.Analysis"
+  "HOL-Complex_Analysis.Complex_Analysis"
   Bernoulli.Bernoulli_FPS 
   Bernoulli.Periodic_Bernpoly 
   Stirling_Formula
 begin
   
 subsection \<open>Auxiliary Facts\<close>
   
 (* TODO Move *)
 lemma filterlim_at_infinity_conv_norm_at_top:
   "filterlim f at_infinity G \<longleftrightarrow> filterlim (\<lambda>x. norm (f x)) at_top G"
   by (auto simp: filterlim_at_infinity[OF order.refl] filterlim_at_top_gt[of _ _ 0])
 
 corollary Ln_times_of_nat:
     "\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(of_nat r * z :: complex) = ln (of_nat r) + Ln(z)"
   using Ln_times_of_real[of "of_nat r" z] by simp
 
 lemma tendsto_of_real_0_I: 
   "(f \<longlongrightarrow> 0) G \<Longrightarrow> ((\<lambda>x. (of_real (f x))) \<longlongrightarrow> (0 ::'a::real_normed_div_algebra)) G"
   by (subst (asm) tendsto_of_real_iff [symmetric]) simp  
 
 lemma negligible_atLeastAtMostI: "b \<le> a \<Longrightarrow> negligible {a..(b::real)}"
   by (cases "b < a") auto
     
 lemma integrable_on_negligible: 
  "negligible A \<Longrightarrow> (f :: 'n :: euclidean_space \<Rightarrow> 'a :: banach) integrable_on A"
   by (subst integrable_spike_set_eq[of _ "{}"]) (simp_all add: integrable_on_empty)  
 
 lemma vector_derivative_cong_eq:
   assumes "eventually (\<lambda>x. x \<in> A \<longrightarrow> f x = g x) (nhds x)" "x = y" "A = B" "x \<in> A"
   shows   "vector_derivative f (at x within A) = vector_derivative g (at y within B)"
 proof -
   from eventually_nhds_x_imp_x[OF assms(1)] assms(4) have "f x = g x" by blast
   hence "(\<lambda>D. (f has_vector_derivative D) (at x within A)) = 
            (\<lambda>D. (g has_vector_derivative D) (at x within A))" using assms
     by (intro ext has_vector_derivative_cong_ev refl assms) simp_all
   thus ?thesis by (simp add: vector_derivative_def assms)
 qed
 
 lemma differentiable_of_real [simp]: "of_real differentiable at x within A"
 proof -
   have "(of_real has_vector_derivative 1) (at x within A)"
     by (auto intro!: derivative_eq_intros)
   thus ?thesis by (rule differentiableI_vector)
 qed
   
 lemma higher_deriv_cong_ev:
   assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
   shows   "(deriv ^^ n) f x = (deriv ^^ n) g y"
 proof -
   from assms(1) have "eventually (\<lambda>x. (deriv ^^ n) f x = (deriv ^^ n) g x) (nhds x)"
   proof (induction n arbitrary: f g)
     case (Suc n)
     from Suc.prems have "eventually (\<lambda>y. eventually (\<lambda>z. f z = g z) (nhds y)) (nhds x)"
       by (simp add: eventually_eventually)
     hence "eventually (\<lambda>x. deriv f x = deriv g x) (nhds x)"
       by eventually_elim (rule deriv_cong_ev, simp_all)
     thus ?case by (auto intro!: deriv_cong_ev Suc simp: funpow_Suc_right simp del: funpow.simps)
   qed auto
   from eventually_nhds_x_imp_x[OF this] assms(2) show ?thesis by simp
 qed
 
 lemma deriv_of_real [simp]: 
   "at x within A \<noteq> bot \<Longrightarrow> vector_derivative of_real (at x within A) = 1"
   by (auto intro!: vector_derivative_within derivative_eq_intros)
 
 lemma deriv_Re [simp]: "deriv Re = (\<lambda>_. 1)"
   by (auto intro!: DERIV_imp_deriv simp: fun_eq_iff)
     
 lemma vector_derivative_of_real_left:
   assumes "f differentiable at x"
   shows   "vector_derivative (\<lambda>x. of_real (f x)) (at x) = of_real (deriv f x)"
 proof -
   have "vector_derivative (of_real \<circ> f) (at x) = (of_real (deriv f x))"
     by (subst vector_derivative_chain_at)
        (simp_all add: scaleR_conv_of_real field_derivative_eq_vector_derivative assms)
   thus ?thesis by (simp add: o_def)
 qed
   
 lemma vector_derivative_of_real_right:
   assumes "f field_differentiable at (of_real x)"
   shows   "vector_derivative (\<lambda>x. f (of_real x)) (at x) = deriv f (of_real x)"
 proof -
   have "vector_derivative (f \<circ> of_real) (at x) = deriv f (of_real x)"
     using assms by (subst vector_derivative_chain_at_general) simp_all
   thus ?thesis by (simp add: o_def)
 qed
   
 lemma Ln_holomorphic [holomorphic_intros]:
   assumes "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}"
   shows   "Ln holomorphic_on (A :: complex set)"
 proof (intro holomorphic_onI)
   fix z assume "z \<in> A"
   with assms have "(Ln has_field_derivative inverse z) (at z within A)"
     by (auto intro!: derivative_eq_intros)
   thus "Ln field_differentiable at z within A" by (auto simp: field_differentiable_def)
 qed
 
 lemma ln_Gamma_holomorphic [holomorphic_intros]: 
   assumes "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}"
   shows   "ln_Gamma holomorphic_on (A :: complex set)"
 proof (intro holomorphic_onI)
   fix z assume "z \<in> A"
   with assms have "(ln_Gamma has_field_derivative Digamma z) (at z within A)"
     by (auto intro!: derivative_eq_intros)
   thus "ln_Gamma field_differentiable at z within A" by (auto simp: field_differentiable_def)
 qed
   
 lemma higher_deriv_Polygamma:
   assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
   shows   "(deriv ^^ n) (Polygamma m) z = 
              Polygamma (m + n) (z :: 'a :: {real_normed_field,euclidean_space})"
 proof -
   have "eventually (\<lambda>u. (deriv ^^ n) (Polygamma m) u = Polygamma (m + n) u) (nhds z)"
   proof (induction n)
     case (Suc n)
     from Suc.IH have "eventually (\<lambda>z. eventually (\<lambda>u. (deriv ^^ n) (Polygamma m) u = Polygamma (m + n) u) (nhds z)) (nhds z)"
       by (simp add: eventually_eventually)
     hence "eventually (\<lambda>z. deriv ((deriv ^^ n) (Polygamma m)) z = 
              deriv (Polygamma (m + n)) z) (nhds z)"
       by eventually_elim (intro deriv_cong_ev refl)
     moreover have "eventually (\<lambda>z. z \<in> UNIV - \<int>\<^sub>\<le>\<^sub>0) (nhds z)" using assms
       by (intro eventually_nhds_in_open open_Diff open_UNIV) auto
     ultimately show ?case by eventually_elim (simp_all add: deriv_Polygamma)
   qed simp_all
   thus ?thesis by (rule eventually_nhds_x_imp_x)
 qed
   
 lemma higher_deriv_cmult:
   assumes "f holomorphic_on A" "x \<in> A" "open A"
   shows   "(deriv ^^ j) (\<lambda>x. c * f x) x = c * (deriv ^^ j) f x"
   using assms
 proof (induction j arbitrary: f x)
   case (Suc j f x)
   have "deriv ((deriv ^^ j) (\<lambda>x. c * f x)) x = deriv (\<lambda>x. c * (deriv ^^ j) f x) x"
     using eventually_nhds_in_open[of A x] assms(2,3) Suc.prems
     by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH)
   also have "\<dots> = c * deriv ((deriv ^^ j) f) x" using Suc.prems assms(2,3)
     by (intro deriv_cmult holomorphic_on_imp_differentiable_at holomorphic_higher_deriv) auto
   finally show ?case by simp
 qed simp_all
 
 lemma higher_deriv_ln_Gamma_complex:
   assumes "(x::complex) \<notin> \<real>\<^sub>\<le>\<^sub>0"
   shows   "(deriv ^^ j) ln_Gamma x = (if j = 0 then ln_Gamma x else Polygamma (j - 1) x)"
 proof (cases j)
   case (Suc j')
   have "(deriv ^^ j') (deriv ln_Gamma) x = (deriv ^^ j') Digamma x"
     using eventually_nhds_in_open[of "UNIV - \<real>\<^sub>\<le>\<^sub>0" x] assms
     by (intro higher_deriv_cong_ev refl)
        (auto elim!: eventually_mono simp: open_Diff deriv_ln_Gamma_complex)
   also have "\<dots> = Polygamma j' x" using assms
     by (subst higher_deriv_Polygamma)
        (auto elim!: nonpos_Ints_cases simp: complex_nonpos_Reals_iff)
   finally show ?thesis using Suc by (simp del: funpow.simps add: funpow_Suc_right)
 qed simp_all
 
 lemma higher_deriv_ln_Gamma_real:
   assumes "(x::real) > 0"
   shows   "(deriv ^^ j) ln_Gamma x = (if j = 0 then ln_Gamma x else Polygamma (j - 1) x)"
 proof (cases j)
   case (Suc j')
   have "(deriv ^^ j') (deriv ln_Gamma) x = (deriv ^^ j') Digamma x"
     using eventually_nhds_in_open[of "{0<..}" x] assms
     by (intro higher_deriv_cong_ev refl)
        (auto elim!: eventually_mono simp: open_Diff deriv_ln_Gamma_real)
   also have "\<dots> = Polygamma j' x" using assms
     by (subst higher_deriv_Polygamma)
        (auto elim!: nonpos_Ints_cases simp: complex_nonpos_Reals_iff)
   finally show ?thesis using Suc by (simp del: funpow.simps add: funpow_Suc_right)
 qed simp_all
   
 lemma higher_deriv_ln_Gamma_complex_of_real:
   assumes "(x :: real) > 0"
   shows   "(deriv ^^ j) ln_Gamma (complex_of_real x) = of_real ((deriv ^^ j) ln_Gamma x)"
     using assms
     by (auto simp: higher_deriv_ln_Gamma_real higher_deriv_ln_Gamma_complex
                    ln_Gamma_complex_of_real Polygamma_of_real)
 (* END TODO *)
 
 lemma stirling_limit_aux1: 
   "((\<lambda>y. Ln (1 + z * of_real y) / of_real y) \<longlongrightarrow> z) (at_right 0)" for z :: complex
 proof (cases "z = 0")
   case True
   then show ?thesis by simp
 next
   case False
   have "((\<lambda>y. ln (1 + z * of_real y)) has_vector_derivative 1 * z) (at 0)"
     by (rule has_vector_derivative_real_field) (auto intro!: derivative_eq_intros)
   then have "(\<lambda>y. (Ln (1 + z * of_real y) - of_real y * z) / of_real \<bar>y\<bar>) \<midarrow>0\<rightarrow> 0"
     by (auto simp add: has_vector_derivative_def has_derivative_def netlimit_at 
           scaleR_conv_of_real field_simps)
   then have "((\<lambda>y. (Ln (1 + z * of_real y) - of_real y * z) / of_real \<bar>y\<bar>) \<longlongrightarrow> 0) (at_right 0)"
     by (rule filterlim_mono[OF _ _ at_le]) simp_all
   also have "?this \<longleftrightarrow> ((\<lambda>y. Ln (1 + z * of_real y) / (of_real y) - z) \<longlongrightarrow> 0) (at_right 0)"
     using eventually_at_right_less[of "0::real"]
     by (intro filterlim_cong refl) (auto elim!: eventually_mono simp: field_simps)
   finally show ?thesis by (simp only: LIM_zero_iff)
 qed
   
 lemma stirling_limit_aux2: 
   "((\<lambda>y. y * Ln (1 + z / of_real y)) \<longlongrightarrow> z) at_top" for z :: complex
   using stirling_limit_aux1[of z] by (subst filterlim_at_top_to_right) (simp add: field_simps)
 
 lemma Union_atLeastAtMost: 
   assumes "N > 0" 
   shows   "(\<Union>n\<in>{0..<N}. {real n..real (n + 1)}) = {0..real N}"
 proof (intro equalityI subsetI)
   fix x assume x: "x \<in> {0..real N}"
   thus "x \<in> (\<Union>n\<in>{0..<N}. {real n..real (n + 1)})"
   proof (cases "x = real N")
     case True
     with assms show ?thesis by (auto intro!: bexI[of _ "N - 1"])
   next
     case False
     with x have x: "x \<ge> 0" "x < real N" by simp_all
     hence "x \<ge> real (nat \<lfloor>x\<rfloor>)" "x \<le> real (nat \<lfloor>x\<rfloor> + 1)" by linarith+
     moreover from x have "nat \<lfloor>x\<rfloor> < N" by linarith
     ultimately have "\<exists>n\<in>{0..<N}. x \<in> {real n..real (n + 1)}"
       by (intro bexI[of _ "nat \<lfloor>x\<rfloor>"]) simp_all
     thus ?thesis by blast
   qed
 qed auto
 
 
 subsection \<open>Asymptotics of @{term ln_Gamma}\<close>
 
 text \<open>
   This is the error term that occurs in the expansion of @{term ln_Gamma}. It can be shown to 
   be of order $O(s^{-n})$.
 \<close>
 definition stirling_integral :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra, banach} \<Rightarrow> 'a" where
   "stirling_integral n s = 
      lim (\<lambda>N. integral {0..N} (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n))"
 
 text \<open>
   
 \<close>
 
 context
   fixes s :: complex assumes s: "Re s > 0"
   fixes approx :: "nat \<Rightarrow> complex"
   defines "approx \<equiv> (\<lambda>N. 
     (\<Sum>n = 1..<N. s / of_nat n - ln (1 + s / of_nat n)) - (euler_mascheroni * s + ln s) - \<comment> \<open>\<open>\<longrightarrow> ln_Gamma s\<close>\<close>
     (ln_Gamma (of_nat N) - ln (2 * pi / of_nat N) / 2 - of_nat N * ln (of_nat N) + of_nat N) - \<comment> \<open>\<open>\<longrightarrow> 0\<close>\<close>
     s * (harm (N - 1) - ln (of_nat (N - 1)) - euler_mascheroni) + \<comment> \<open>\<open>\<longrightarrow> 0\<close>\<close>
     s * (ln (of_nat N + s) - ln (of_nat (N - 1))) - \<comment> \<open>\<open>\<longrightarrow> 0\<close>\<close>
     (1/2) * (ln (of_nat N + s) - ln (of_nat N)) +       \<comment> \<open>\<open>\<longrightarrow> 0\<close>\<close>
     of_nat N * (ln (of_nat N + s) - ln (of_nat N)) -  \<comment> \<open>\<open>\<longrightarrow> s\<close>\<close>
     (s - 1/2) * ln s - ln (2 * pi) / 2)"
 begin       
   
 qualified lemma
   assumes N: "N > 0"
   shows   integrable_pbernpoly_1:
             "(\<lambda>x. of_real (-pbernpoly 1 x) / (of_real x + s)) integrable_on {0..real N}"
   and     integral_pbernpoly_1_aux:
             "integral {0..real N} (\<lambda>x. -of_real (pbernpoly 1 x) / (of_real x + s)) = approx N"
   and     has_integral_pbernpoly_1:
             "((\<lambda>x. pbernpoly 1 x /(x + s)) has_integral 
               (\<Sum>m<N. (of_nat m + 1 / 2 + s) * (ln (of_nat m + s) - 
                         ln (of_nat m + 1 + s)) + 1)) {0..real N}"
 proof -
   let ?A = "(\<lambda>n. {of_nat n..of_nat (n+1)}) ` {0..<N}"
   have has_integral: 
     "((\<lambda>x. -pbernpoly 1 x / (x + s)) has_integral 
              (of_nat n + 1/2 + s) * (ln (of_nat (n + 1) + s) - ln (of_nat n + s)) - 1) 
            {of_nat n..of_nat (n + 1)}" for n
   proof (rule has_integral_spike)      
     have "((\<lambda>x. (of_nat n + 1/2 + s) * (1 / (x + s)) - 1) has_integral 
               (of_nat n + 1/2 + s) * (ln (of_real (real (n + 1)) + s) - ln (of_real (real n) + s)) - 1) 
             {of_nat n..of_nat (n + 1)}" 
       using s has_integral_const_real[of 1 "of_nat n" "of_nat (n + 1)"]
       by (intro has_integral_diff has_integral_mult_right fundamental_theorem_of_calculus)
          (auto intro!: derivative_eq_intros has_vector_derivative_real_field
                simp: has_field_derivative_iff_has_vector_derivative [symmetric] field_simps
                      complex_nonpos_Reals_iff)
     thus "((\<lambda>x. (of_nat n + 1/2 + s) * (1 / (x + s)) - 1) has_integral 
               (of_nat n + 1/2 + s) * (ln (of_nat (n + 1) + s) - ln (of_nat n + s)) - 1) 
             {of_nat n..of_nat (n + 1)}" by simp
              
     show "-pbernpoly 1 x / (x + s) = (of_nat n + 1/2 + s) * (1 / (x + s)) - 1"
          if "x \<in> {of_nat n..of_nat (n + 1)} - {of_nat (n + 1)}" for x
     proof -
       have x: "x \<ge> real n" "x < real (n + 1)" using that by simp_all
       hence "floor x = int n" by linarith
       moreover from s x have "complex_of_real x \<noteq> -s" 
         by (auto simp add: complex_eq_iff simp del: of_nat_Suc)
       ultimately show "-pbernpoly 1 x / (x + s) = (of_nat n + 1/2 + s) * (1 / (x + s)) - 1"
         by (auto simp: pbernpoly_def bernpoly_def frac_def divide_simps add_eq_0_iff2)
     qed
   qed simp_all
   hence *: "\<And>I. I\<in>?A \<Longrightarrow> ((\<lambda>x. -pbernpoly 1 x / (x + s)) has_integral 
               (Inf I + 1/2 + s) * (ln (Inf I + 1 + s) - ln (Inf I + s)) - 1) I"
     by (auto simp: add_ac)
   have "((\<lambda>x. - pbernpoly 1 x / (x + s)) has_integral
           (\<Sum>I\<in>?A. (Inf I + 1 / 2 + s) * (ln (Inf I + 1 + s) - ln (Inf I + s)) - 1))
           (\<Union>n\<in>{0..<N}. {real n..real (n + 1)})" (is "(_ has_integral ?i) _")
     apply (intro has_integral_Union * finite_imageI)
       apply (force intro!: negligible_atLeastAtMostI pairwiseI)+
     done
   hence has_integral: "((\<lambda>x. - pbernpoly 1 x / (x + s)) has_integral ?i) {0..real N}"
     by (subst has_integral_spike_set_eq)
        (use Union_atLeastAtMost assms in \<open>auto simp: intro!: empty_imp_negligible\<close>)
   hence "(\<lambda>x. - pbernpoly 1 x / (x + s)) integrable_on {0..real N}"
     and integral:   "integral {0..real N} (\<lambda>x. - pbernpoly 1 x / (x + s)) = ?i"
     by (simp_all add: has_integral_iff)
   show "(\<lambda>x. - pbernpoly 1 x / (x + s)) integrable_on {0..real N}" by fact
 
   note has_integral_neg[OF has_integral]
   also have "-?i = (\<Sum>x<N. (of_nat x + 1 / 2 + s) * (ln (of_nat x + s) - ln (of_nat x + 1 + s)) + 1)" 
     by (subst sum.reindex) 
        (simp_all add: inj_on_def atLeast0LessThan algebra_simps sum_negf [symmetric])
   finally show has_integral: 
     "((\<lambda>x. of_real (pbernpoly 1 x) / (of_real x + s)) has_integral \<dots>) {0..real N}" by simp
       
   note integral
   also have "?i = (\<Sum>n<N. (of_nat n + 1 / 2 + s) * 
                     (ln (of_nat n + 1 + s) - ln (of_nat n + s))) - N" (is "_ = ?S - _")
     by (subst sum.reindex) (simp_all add: inj_on_def sum_subtractf atLeast0LessThan)
   also have "?S = (\<Sum>n<N. of_nat n * (ln (of_nat n + 1 + s) - ln (of_nat n + s))) +
                     (s + 1 / 2) * (\<Sum>n<N. ln (of_nat (Suc n) + s) - ln (of_nat n + s))" 
     (is "_ = ?S1 + _ * ?S2") by (simp add: algebra_simps sum.distrib sum_subtractf sum_distrib_left)
   also have "?S2 = ln (of_nat N + s) - ln s" by (subst sum_lessThan_telescope) simp
   also have "?S1 = (\<Sum>n=1..<N. of_nat n * (ln (of_nat n + 1 + s) - ln (of_nat n + s)))"
     by (intro sum.mono_neutral_right) auto
   also have "\<dots> = (\<Sum>n=1..<N. of_nat n * ln (of_nat n + 1 + s)) - (\<Sum>n=1..<N. of_nat n * ln (of_nat n + s))"
     by (simp add: algebra_simps sum_subtractf)
   also have "(\<Sum>n=1..<N. of_nat n * ln (of_nat n + 1 + s)) = 
                (\<Sum>n=1..<N. (of_nat n - 1) * ln (of_nat n + s)) + (N - 1) * ln (of_nat N + s)"
     by (induction N) (simp_all add: add_ac of_nat_diff)
   also have "\<dots> - (\<Sum>n = 1..<N. of_nat n * ln (of_nat n + s)) =
                -(\<Sum>n=1..<N. ln (of_nat n + s)) + (N - 1) * ln (of_nat N + s)"
     by (induction N) (simp_all add: algebra_simps)
   also from s have neq: "s + of_nat x \<noteq> 0" for x by (auto simp: complex_eq_iff)
   hence "(\<Sum>n=1..<N. ln (of_nat n + s)) = (\<Sum>n=1..<N. ln (of_nat n) + ln (1 + s/n))"
     by (intro sum.cong refl, subst Ln_times_of_nat [symmetric]) (auto simp: divide_simps add_ac)
   also have "\<dots> = ln (fact (N - 1)) + (\<Sum>n=1..<N. ln (1 + s/n))"
     by (induction N) (simp_all add: Ln_times_of_nat fact_reduce add_ac)
   also have "(\<Sum>n=1..<N. ln (1 + s/n)) = -(\<Sum>n=1..<N. s / n - ln (1 + s/n)) + s * (\<Sum>n=1..<N. 1 / of_nat n)"
     by (simp add: sum_distrib_left sum_subtractf) 
   also from N have "ln (fact (N - 1)) = ln_Gamma (of_nat N :: complex)" 
     by (simp add: ln_Gamma_complex_conv_fact)
   also have "{1..<N} = {1..N - 1}" by auto
   hence "(\<Sum>n = 1..<N. 1 / of_nat n) = (harm (N - 1) :: complex)" 
     by (simp add: harm_def divide_simps)
   also have "- (ln_Gamma (of_nat N) + (- (\<Sum>n = 1..<N. s / of_nat n - ln (1 + s / of_nat n)) +
                  s * harm (N - 1))) + of_nat (N - 1) * ln (of_nat N + s) +
                 (s + 1 / 2) * (ln (of_nat N + s) - ln s) - of_nat N = approx N"
     using N by (simp add: field_simps of_nat_diff ln_div approx_def Ln_of_nat 
                           ln_Gamma_complex_of_real [symmetric])
   finally show "integral {0..of_nat N} (\<lambda>x. -of_real (pbernpoly 1 x) / (of_real x + s)) = \<dots>" 
     by simp
 qed
   
 lemma integrable_ln_Gamma_aux:
   shows   "(\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n) integrable_on {0..real N}"
 proof (cases "n = 1")
   case True
   with s show ?thesis using integrable_neg[OF integrable_pbernpoly_1[of N]] 
     by (cases "N = 0") (simp_all add: integrable_on_negligible)
 next
   case False
   from s have "of_real x + s \<noteq> 0" if "x \<ge> 0" for x using that 
     by (auto simp: complex_eq_iff add_eq_0_iff2)
   with False s show ?thesis
     by (auto intro!: integrable_continuous_real continuous_intros)
 qed
   
 text \<open>
   This following proof is based on ``Rudiments of the theory of the gamma function'' 
   by Bruce Berndt~\cite{berndt}.
 \<close>
 qualified lemma integral_pbernpoly_1: 
   "(\<lambda>N. integral {0..real N} (\<lambda>x. pbernpoly 1 x / (x + s)))
      \<longlonglongrightarrow> -ln_Gamma s - s + (s - 1 / 2) * ln s + ln (2 * pi) / 2"
 proof -  
   have neq: "s + of_real x \<noteq> 0" if "x \<ge> 0" for x :: real
     using that s by (auto simp: complex_eq_iff)
   have "(approx \<longlongrightarrow> ln_Gamma s - 0 - 0 + 0 - 0 + s - (s - 1/2) * ln s - ln (2 * pi) / 2) at_top"
     unfolding approx_def
   proof (intro tendsto_add tendsto_diff)
     from s have s': "s \<notin> \<int>\<^sub>\<le>\<^sub>0" by (auto elim!: nonpos_Ints_cases)
     have "(\<lambda>n. \<Sum>i=1..<n. s / of_nat i - ln (1 + s / of_nat i)) \<longlonglongrightarrow> 
              ln_Gamma s + euler_mascheroni * s + ln s" (is "?f \<longlonglongrightarrow> _")
       using ln_Gamma_series'_aux[OF s'] unfolding sums_def 
       by (subst LIMSEQ_Suc_iff [symmetric], subst (asm) sum.atLeast1_atMost_eq [symmetric]) 
          (simp add: atLeastLessThanSuc_atLeastAtMost)
     thus "((\<lambda>n. ?f n - (euler_mascheroni * s + ln s)) \<longlongrightarrow> ln_Gamma s) at_top"
       by (auto intro: tendsto_eq_intros)
   next
     show "(\<lambda>x. complex_of_real (ln_Gamma (real x) - ln (2 * pi / real x) / 2 - 
                  real x * ln (real x) + real x)) \<longlonglongrightarrow> 0"
     proof (intro tendsto_of_real_0_I 
              filterlim_compose[OF tendsto_sandwich filterlim_real_sequentially])
       show "eventually (\<lambda>x::real. ln_Gamma x - ln (2 * pi / x) / 2 - x * ln x + x \<ge> 0) at_top"
         using eventually_ge_at_top[of "1::real"] 
         by eventually_elim (insert ln_Gamma_bounds(1), simp add: algebra_simps)
       show "eventually (\<lambda>x::real. ln_Gamma x - ln (2 * pi / x) / 2 - x * ln x + x \<le> 
               1 / 12 * inverse x) at_top"
         using eventually_ge_at_top[of "1::real"] 
         by eventually_elim (insert ln_Gamma_bounds(2), simp add: field_simps)
       show "((\<lambda>x::real. 1 / 12 * inverse x) \<longlongrightarrow> 0) at_top"
         by (intro tendsto_mult_right_zero tendsto_inverse_0_at_top filterlim_ident)
     qed simp_all
   next
     have "(\<lambda>x. s * of_real (harm (x - 1) - ln (real (x - 1)) - euler_mascheroni)) \<longlonglongrightarrow> 
             s * of_real (euler_mascheroni - euler_mascheroni)"
       by (subst LIMSEQ_Suc_iff [symmetric], intro tendsto_intros) 
          (insert euler_mascheroni_LIMSEQ, simp_all)
     also have "?this \<longleftrightarrow> (\<lambda>x. s * (harm (x - 1) - ln (of_nat (x - 1)) - euler_mascheroni)) \<longlonglongrightarrow> 0"
       by (intro filterlim_cong refl eventually_mono[OF eventually_gt_at_top[of "1::nat"]]) 
          (auto simp: Ln_of_nat of_real_harm)
     finally show "(\<lambda>x. s * (harm (x - 1) - ln (of_nat (x - 1)) - euler_mascheroni)) \<longlonglongrightarrow> 0"  .
   next
     have "((\<lambda>x. ln (1 + (s + 1) / of_real x)) \<longlongrightarrow> ln (1 + 0)) at_top" (is ?P)
       by (intro tendsto_intros tendsto_divide_0[OF tendsto_const]) 
          (simp_all add: filterlim_ident filterlim_at_infinity_conv_norm_at_top filterlim_abs_real)
     also have "ln (of_real (x + 1) + s) - ln (complex_of_real x) = ln (1 + (s + 1) / of_real x)" 
       if "x > 1" for x using that s
       using Ln_divide_of_real[of x "of_real (x + 1) + s", symmetric] neq[of "x+1"]
       by (simp add: field_simps Ln_of_real)
     hence "?P \<longleftrightarrow> ((\<lambda>x. ln (of_real (x + 1) + s) - ln (of_real x)) \<longlongrightarrow> 0) at_top"
       by (intro filterlim_cong refl) 
          (auto intro: eventually_mono[OF eventually_gt_at_top[of "1::real"]])
     finally have "((\<lambda>n. ln (of_real (real n + 1) + s) - ln (of_real (real n))) \<longlongrightarrow> 0) at_top"
       by (rule filterlim_compose[OF _ filterlim_real_sequentially])
     hence "((\<lambda>n. ln (of_nat n + s) - ln (of_nat (n - 1))) \<longlongrightarrow> 0) at_top"
       by (subst LIMSEQ_Suc_iff [symmetric]) (simp add: add_ac)
     thus "(\<lambda>x. s * (ln (of_nat x + s) - ln (of_nat (x - 1)))) \<longlonglongrightarrow> 0"
       by (rule tendsto_mult_right_zero)
   next
     have "((\<lambda>x. ln (1 + s / of_real x)) \<longlongrightarrow> ln (1 + 0)) at_top" (is ?P)
       by (intro tendsto_intros tendsto_divide_0[OF tendsto_const]) 
          (simp_all add: filterlim_ident  filterlim_at_infinity_conv_norm_at_top filterlim_abs_real)
     also have "ln (of_real x + s) - ln (of_real x) = ln (1 + s / of_real x)" if "x > 0" for x
       using Ln_divide_of_real[of x "of_real x + s"] neq[of x] that
       by (auto simp: field_simps Ln_of_real)
     hence "?P \<longleftrightarrow> ((\<lambda>x. ln (of_real x + s) - ln (of_real x)) \<longlongrightarrow> 0) at_top"
       using s by (intro filterlim_cong refl) 
                  (auto intro: eventually_mono [OF eventually_gt_at_top[of "1::real"]])
     finally have "(\<lambda>x. (1/2) * (ln (of_real (real x) + s) - ln (of_real (real x)))) \<longlonglongrightarrow> 0"        
       by (rule tendsto_mult_right_zero[OF filterlim_compose[OF _ filterlim_real_sequentially]])
     thus "(\<lambda>x. (1/2) * (ln (of_nat x + s) - ln (of_nat x))) \<longlonglongrightarrow> 0" by simp
   next
     have "((\<lambda>x. x * (ln (1 + s / of_real x))) \<longlongrightarrow> s) at_top" (is ?P) 
       by (rule stirling_limit_aux2)
     also have "ln (1 + s / of_real x) = ln (of_real x + s) - ln (of_real x)" if "x > 1" for x 
       using that s Ln_divide_of_real [of x "of_real x + s", symmetric] neq[of x]
       by (auto simp: Ln_of_real field_simps)
     hence "?P \<longleftrightarrow> ((\<lambda>x. of_real x * (ln (of_real x + s) - ln (of_real x))) \<longlongrightarrow> s) at_top"
       by (intro filterlim_cong refl) 
          (auto intro: eventually_mono[OF eventually_gt_at_top[of "1::real"]])
     finally have "(\<lambda>n. of_real (real n) * (ln (of_real (real n) + s) - ln (of_real (real n)))) \<longlonglongrightarrow> s"
       by (rule filterlim_compose[OF _ filterlim_real_sequentially])
     thus "(\<lambda>n. of_nat n * (ln (of_nat n + s) - ln (of_nat n))) \<longlonglongrightarrow> s" by simp
   qed simp_all
   also have "?this \<longleftrightarrow> ((\<lambda>N. integral {0..real N} (\<lambda>x. -pbernpoly 1 x / (x + s))) \<longlongrightarrow>
                          ln_Gamma s + s - (s - 1/2) * ln s - ln (2 * pi) / 2) at_top"
     using integral_pbernpoly_1_aux
     by (intro filterlim_cong refl) 
        (auto intro: eventually_mono[OF eventually_gt_at_top[of "0::nat"]])
   also have "(\<lambda>N. integral {0..real N} (\<lambda>x. -pbernpoly 1 x / (x + s))) =
                (\<lambda>N. -integral {0..real N} (\<lambda>x. pbernpoly 1 x / (x + s)))"
     by (simp add: fun_eq_iff)
   finally show ?thesis by (simp add: tendsto_minus_cancel_left [symmetric] algebra_simps)
 qed
 
 
 qualified lemma pbernpoly_integral_conv_pbernpoly_integral_Suc:
   assumes "n \<ge> 1"
   shows   "integral {0..real N} (\<lambda>x. pbernpoly n x / (x + s) ^ n) =
              of_real (pbernpoly (Suc n) (real N)) / (of_nat (Suc n) * (s + of_nat N) ^ n) -
              of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n) + of_nat n / of_nat (Suc n) * 
                integral {0..real N} (\<lambda>x. of_real (pbernpoly (Suc n) x) / (of_real x + s) ^ Suc n)"
 proof - 
   note [derivative_intros] = has_field_derivative_pbernpoly_Suc'
   define I where "I = -of_real (pbernpoly (Suc n) (of_nat N)) / (of_nat (Suc n) * (of_nat N + s) ^ n) +
             of_real (bernoulli (Suc n) / real (Suc n)) / s ^ n +
             integral {0..real N} (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n)"
   have "((\<lambda>x. (-of_nat n * inverse (of_real x + s) ^ Suc n) * 
           (of_real (pbernpoly (Suc n) x) / (of_nat (Suc n))))
           has_integral -I) {0..real N}"
   proof (rule integration_by_parts_interior_strong[OF bounded_bilinear_mult])
     fix x :: real assume x: "x \<in> {0<..<real N} - real ` {0..N}"
     have "x \<notin> \<int>"
     proof
       assume "x \<in> \<int>"
       then obtain n where "x = of_int n" by (auto elim!: Ints_cases)
       with x have x': "x = of_nat (nat n)" by simp
       from x show False by (auto simp: x')
     qed
     hence "((\<lambda>x. of_real (pbernpoly (Suc n) x / of_nat (Suc n))) has_vector_derivative
         complex_of_real (pbernpoly n x)) (at x)"
       by (intro has_vector_derivative_of_real) (auto intro!: derivative_eq_intros)
     thus "((\<lambda>x. of_real (pbernpoly (Suc n) x) / of_nat (Suc n)) has_vector_derivative
             complex_of_real (pbernpoly n x)) (at x)" by simp
     from x s have "complex_of_real x + s \<noteq> 0" by (auto simp: complex_eq_iff)
     thus "((\<lambda>x. inverse (of_real x + s) ^ n) has_vector_derivative 
              - of_nat n * inverse (of_real x + s) ^ Suc n) (at x)" using x s assms
       by (auto intro!: derivative_eq_intros has_vector_derivative_real_field simp: divide_simps power_add [symmetric]
                simp del: power_Suc)
   next
     have "complex_of_real x + s \<noteq> 0" if "x \<ge> 0" for x 
       using that s by (auto simp: complex_eq_iff)
     thus "continuous_on {0..real N} (\<lambda>x. inverse (of_real x + s) ^ n)" 
          "continuous_on {0..real N} (\<lambda>x. complex_of_real (pbernpoly (Suc n) x) / of_nat (Suc n))"
       using assms s by (auto intro!: continuous_intros simp del: of_nat_Suc)
   next
     have "((\<lambda>x. inverse (of_real x + s) ^ n * of_real (pbernpoly n x)) has_integral
             pbernpoly (Suc n) (of_nat N) / (of_nat (Suc n) * (of_nat N + s) ^ n) -
             of_real (bernoulli (Suc n) / real (Suc n)) / s ^ n - -I) {0..real N}" 
       using integrable_ln_Gamma_aux[of n N] assms
       by (auto simp: I_def has_integral_integral divide_simps)
     thus "((\<lambda>x. inverse (of_real x + s) ^ n * of_real (pbernpoly n x)) has_integral
               inverse (of_real (real N) + s) ^ n * (of_real (pbernpoly (Suc n) (real N)) / 
                   of_nat (Suc n)) -
               inverse (of_real 0 + s) ^ n * (of_real (pbernpoly (Suc n) 0) / of_nat (Suc n)) - - I)
             {0..real N}" by (simp_all add: field_simps)
   qed simp_all
   also have "(\<lambda>x. - of_nat n * inverse (of_real x + s) ^ Suc n * (of_real (pbernpoly (Suc n) x) /
                          of_nat (Suc n))) =
              (\<lambda>x. - (of_nat n / of_nat (Suc n) * of_real (pbernpoly (Suc n) x) / 
                          (of_real x + s) ^ Suc n))"
     by (simp add: divide_simps fun_eq_iff)
   finally have "((\<lambda>x. - (of_nat n / of_nat (Suc n) * of_real (pbernpoly (Suc n) x) /
                             (of_real x + s) ^ Suc n)) has_integral - I) {0..real N}" .
   from has_integral_neg[OF this] show ?thesis
     by (auto simp add: I_def has_integral_iff algebra_simps integral_mult_right [symmetric] 
              simp del: power_Suc of_nat_Suc )
 qed
 
 lemma pbernpoly_over_power_tendsto_0: 
   assumes "n > 0"
   shows   "(\<lambda>x. of_real (pbernpoly (Suc n) (real x)) / (of_nat (Suc n) * (s + of_nat x) ^ n)) \<longlonglongrightarrow> 0"
 proof -
   from s have neq: "s + of_nat n \<noteq> 0" for n by (auto simp: complex_eq_iff)
   from bounded_pbernpoly[of "Suc n"] guess c . note c = this
   have "eventually (\<lambda>x. norm (of_real (pbernpoly (Suc n) (real x)) / 
                                     (of_nat (Suc n) * (s + of_nat x) ^ n)) \<le>
                           (c / real (Suc n)) / real x ^ n) at_top"
     using eventually_gt_at_top[of "0::nat"]
   proof eventually_elim
     case (elim x)
     have "norm (of_real (pbernpoly (Suc n) (real x)) / 
                                     (of_nat (Suc n) * (s + of_nat x) ^ n)) \<le>
             (c / real (Suc n)) / norm (s + of_nat x) ^ n" (is "_ \<le> ?rhs") using c[of x]
       by (auto simp: norm_divide norm_mult norm_power neq field_simps simp del: of_nat_Suc)
     also have "real x \<le> cmod (s + of_nat x)"
       using complex_Re_le_cmod[of "s + of_nat x"] s by simp
     hence "?rhs \<le> (c / real (Suc n)) / real x ^ n" using s elim c[of 0] neq[of x]
       by (intro divide_left_mono power_mono mult_pos_pos divide_nonneg_pos zero_less_power) auto
     finally show ?case .
   qed 
   moreover have "(\<lambda>x. (c / real (Suc n)) / real x ^ n) \<longlonglongrightarrow> 0"
     by (intro real_tendsto_divide_at_top[OF tendsto_const] filterlim_pow_at_top assms
               filterlim_real_sequentially)
   ultimately show ?thesis by (rule Lim_null_comparison)
 qed
 
 lemma convergent_stirling_integral:
   assumes "n > 0"
   shows   "convergent (\<lambda>N. integral {0..real N} 
              (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n))" (is "convergent (?f n)")
 proof -
   have "convergent (?f (Suc n))" for n
   proof (induction n)
     case 0
     thus ?case using integral_pbernpoly_1 by (auto intro!: convergentI)
   next
     case (Suc n)
     have "convergent (\<lambda>N. ?f (Suc n) N -
             of_real (pbernpoly (Suc (Suc n)) (real N)) / 
                 (of_nat (Suc (Suc n)) * (s + of_nat N) ^ Suc n) +
             of_real (bernoulli (Suc (Suc n)) / (real (Suc (Suc n)))) / s ^ Suc n)" 
       (is "convergent ?g")
       by (intro convergent_add convergent_diff Suc 
             convergent_const convergentI[OF pbernpoly_over_power_tendsto_0]) simp_all
     also have "?g = (\<lambda>N. of_nat (Suc n) / of_nat (Suc (Suc n)) * ?f (Suc (Suc n)) N)" using s
       by (subst pbernpoly_integral_conv_pbernpoly_integral_Suc) 
          (auto simp: fun_eq_iff field_simps simp del: of_nat_Suc power_Suc)
     also have "convergent \<dots> \<longleftrightarrow> convergent (?f (Suc (Suc n)))"
       by (intro convergent_mult_const_iff) (simp_all del: of_nat_Suc)
     finally show ?case .
   qed
   from this[of "n - 1"] assms show ?thesis by simp
 qed
 
 lemma stirling_integral_conv_stirling_integral_Suc:
   assumes "n > 0"
   shows   "stirling_integral n s =
              of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s -
              of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n)"
 proof -
   have "(\<lambda>N. of_real (pbernpoly (Suc n) (real N)) / (of_nat (Suc n) * (s + of_nat N) ^ n) -
              of_real (bernoulli (Suc n)) / (real (Suc n) * s ^ n) +
              integral {0..real N} (\<lambda>x. of_nat n / of_nat (Suc n) * 
                 (of_real (pbernpoly (Suc n) x) / (of_real x + s) ^ Suc n)))
            \<longlonglongrightarrow> 0 - of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n) +
                    of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s" (is "?f \<longlonglongrightarrow> _")
     unfolding stirling_integral_def integral_mult_right
     using convergent_stirling_integral[of "Suc n"] assms s
     by (intro tendsto_intros pbernpoly_over_power_tendsto_0)
        (auto simp: convergent_LIMSEQ_iff simp del: of_nat_Suc)
   also have "?this \<longleftrightarrow> (\<lambda>N. integral {0..real N} 
                (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n)) \<longlonglongrightarrow>
                of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s -
                  of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n)" 
     using eventually_gt_at_top[of "0::nat"] pbernpoly_integral_conv_pbernpoly_integral_Suc[of n] 
           assms unfolding integral_mult_right
     by (intro filterlim_cong refl) (auto elim!: eventually_mono simp del: power_Suc)
   finally show ?thesis unfolding stirling_integral_def[of n] by (rule limI)
 qed
 
 lemma stirling_integral_1_unfold:
   assumes "m > 0"
   shows   "stirling_integral 1 s = stirling_integral m s / of_nat m - 
              (\<Sum>k=1..<m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k))"
 proof -
   have "stirling_integral 1 s = stirling_integral (Suc m) s / of_nat (Suc m) -
           (\<Sum>k=1..<Suc m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k))" for m
   proof (induction m)
     case (Suc m)
     let ?C = "(\<Sum>k = 1..<Suc m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k))"
     note Suc.IH
     also have "stirling_integral (Suc m) s / of_nat (Suc m) = 
                  stirling_integral (Suc (Suc m)) s / of_nat (Suc (Suc m)) -
                  of_real (bernoulli (Suc (Suc m))) / 
                    (of_nat (Suc m) * of_nat (Suc (Suc m)) * s ^ Suc m)"
       (is "_ = ?A - ?B") by (subst stirling_integral_conv_stirling_integral_Suc)
                             (simp_all del: of_nat_Suc power_Suc add: divide_simps)
     also have "?A - ?B - ?C = ?A - (?B + ?C)" by (rule diff_diff_eq)
     also have "?B + ?C = (\<Sum>k = 1..<Suc (Suc m). of_real (bernoulli (Suc k)) /
                              (of_nat k * of_nat (Suc k) * s ^ k))" 
       using s by (simp add: divide_simps)
     finally show ?case .
   qed simp_all
   note this[of "m - 1"]
   also from assms have "Suc (m - 1) = m" by simp
   finally show ?thesis .
 qed
   
 lemma ln_Gamma_stirling_complex:
   assumes "m > 0"
   shows   "ln_Gamma s = (s - 1 / 2) * ln s - s + ln (2 * pi) / 2 +
              (\<Sum>k=1..<m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k)) - 
              stirling_integral m s / of_nat m"
 proof -
   have "ln_Gamma s = (s - 1 / 2) * ln s - s + ln (2 * pi) / 2 - stirling_integral 1 s"
     using limI[OF integral_pbernpoly_1] by (simp add: stirling_integral_def algebra_simps)
   also have "stirling_integral 1 s = stirling_integral m s / of_nat m -
                (\<Sum>k = 1..<m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k))"
     using assms by (rule stirling_integral_1_unfold)
   finally show ?thesis by simp
 qed
 
 lemma LIMSEQ_stirling_integral:
   "n > 0 \<Longrightarrow> (\<lambda>x. integral {0..real x} (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n))
      \<longlonglongrightarrow> stirling_integral n s" unfolding stirling_integral_def 
   using convergent_stirling_integral[of n] by (simp only: convergent_LIMSEQ_iff)
 
 end
 
 lemmas has_integral_of_real = has_integral_linear[OF _ bounded_linear_of_real, unfolded o_def]
 lemmas integral_of_real = integral_linear[OF _ bounded_linear_of_real, unfolded o_def]
 
 lemma integrable_ln_Gamma_aux_real:
   assumes "0 < s"
   shows   "(\<lambda>x. pbernpoly n x / (x + s) ^ n) integrable_on {0..real N}"
 proof -
   have "(\<lambda>x. complex_of_real (pbernpoly n x / (x + s) ^ n)) integrable_on {0..real N}"
     using integrable_ln_Gamma_aux[of "of_real s" n N] assms by simp
   from integrable_linear[OF this bounded_linear_Re] show ?thesis 
     by (simp only: o_def Re_complex_of_real) 
 qed
   
 lemma  
   assumes "x > 0" "n > 0"
   shows   stirling_integral_complex_of_real:
             "stirling_integral n (complex_of_real x) = of_real (stirling_integral n x)"
     and   LIMSEQ_stirling_integral_real:
             "(\<lambda>N. integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n))
             \<longlonglongrightarrow> stirling_integral n x"
     and   stirling_integral_real_convergent:
             "convergent (\<lambda>N. integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n))"
 proof -
   have "(\<lambda>N. integral {0..real N} (\<lambda>t. of_real (pbernpoly n t / (t + x) ^ n)))
            \<longlonglongrightarrow> stirling_integral n (complex_of_real x)"
     using LIMSEQ_stirling_integral[of "complex_of_real x" n] assms by simp
   hence "(\<lambda>N. of_real (integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n)))
            \<longlonglongrightarrow> stirling_integral n (complex_of_real x)"
     using integrable_ln_Gamma_aux_real[OF assms(1), of n] 
     by (subst (asm) integral_of_real) simp
   from tendsto_Re[OF this] 
     have "(\<lambda>N. integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n))
            \<longlonglongrightarrow> Re (stirling_integral n (complex_of_real x))" by simp
   thus "convergent (\<lambda>N. integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n))"
     by (rule convergentI)
   thus "(\<lambda>N. integral {0..real N} (\<lambda>t. pbernpoly n t / (t + x) ^ n))
           \<longlonglongrightarrow> stirling_integral n x" unfolding stirling_integral_def
     by (simp add: convergent_LIMSEQ_iff)
   from tendsto_of_real[OF this, where 'a = complex] 
        integrable_ln_Gamma_aux_real[OF assms(1), of n]
     have "(\<lambda>xa. integral {0..real xa} 
                     (\<lambda>xa. complex_of_real (pbernpoly n xa) / (complex_of_real xa + x) ^ n))
              \<longlonglongrightarrow> complex_of_real (stirling_integral n x)"
     by (subst (asm) integral_of_real [symmetric]) simp_all
   from LIMSEQ_unique[OF this LIMSEQ_stirling_integral[of "complex_of_real x" n]] assms
     show "stirling_integral n (complex_of_real x) = of_real (stirling_integral n x)" by simp
 qed
 
 lemma ln_Gamma_stirling_real:
   assumes "x > (0 :: real)" "m > (0::nat)"
   shows   "ln_Gamma x = (x - 1 / 2) * ln x - x + ln (2 * pi) / 2 +
               (\<Sum>k = 1..<m. bernoulli (Suc k) / (of_nat k * of_nat (Suc k) * x ^ k)) -
               stirling_integral m x / of_nat m"
 proof -
   from assms have "complex_of_real (ln_Gamma x) = ln_Gamma (complex_of_real x)"
     by (simp add: ln_Gamma_complex_of_real)
   also have "ln_Gamma (complex_of_real x) = complex_of_real (
                 (x - 1 / 2) * ln x - x + ln (2 * pi) / 2 +
                 (\<Sum>k = 1..<m. bernoulli (Suc k) / (of_nat k * of_nat (Suc k) * x ^ k)) -
                 stirling_integral m x / of_nat m)" using assms
     by (subst ln_Gamma_stirling_complex[of _ m])
        (simp_all add: Ln_of_real stirling_integral_complex_of_real)
   finally show ?thesis by (subst (asm) of_real_eq_iff)
 qed
 
 
 context
 begin
 
 private lemma stirling_integral_bound_aux:
   assumes n: "n > (1::nat)"
   obtains c where "\<And>s. Re s > 0 \<Longrightarrow> norm (stirling_integral n s) \<le>  c / Re s ^ (n - 1)"
 proof -
   obtain c where c: "norm (pbernpoly n x) \<le> c" for x by (rule bounded_pbernpoly[of n]) blast
   have c': "pbernpoly n x \<le> c" for x using c[of x] by (simp add: abs_real_def split: if_splits)
   from c[of 0] have c_nonneg: "c \<ge> 0" by simp
   have "norm (stirling_integral n s) \<le> c / (real n - 1) / Re s ^ (n - 1)" if s: "Re s > 0" for s
   proof (rule Lim_norm_ubound[OF _ LIMSEQ_stirling_integral])
     have pos: "x + norm s > 0" if "x \<ge> 0" for x using s that by (intro add_nonneg_pos) auto
     have nz: "of_real x + s \<noteq> 0" if "x \<ge> 0" for x using s that by (auto simp: complex_eq_iff)
     let ?bound = "\<lambda>N. c / (Re s ^ (n - 1) * (real n - 1)) - 
                         c / ((real N + Re s) ^ (n - 1) * (real n - 1))"
     show "eventually (\<lambda>N. norm (integral {0..real N} 
               (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n)) \<le> 
             c / (real n - 1) / Re s ^ (n - 1)) at_top"
       using eventually_gt_at_top[of "0::nat"]
     proof eventually_elim
       case (elim N)
       let ?F = "\<lambda>x. -c / ((x + Re s) ^ (n - 1) * (real n - 1))"
       from n s have "((\<lambda>x. c / (x + Re s) ^ n) has_integral (?F (real N) - ?F 0)) {0..real N}"
         by (intro fundamental_theorem_of_calculus)
            (auto intro!: derivative_eq_intros simp: divide_simps power_diff add_eq_0_iff2
                    has_field_derivative_iff_has_vector_derivative [symmetric])      
       also have "?F (real N) - ?F 0 = ?bound N" by simp
       finally have *: "((\<lambda>x. c / (x + Re s) ^ n) has_integral ?bound N) {0..real N}" .
       have "norm (integral {0..real N} (\<lambda>x. of_real (pbernpoly n x) / (of_real x + s) ^ n)) \<le>
               integral {0..real N} (\<lambda>x. c / (x + Re s) ^ n)"
       proof (intro integral_norm_bound_integral integrable_ln_Gamma_aux s ballI)
         fix x assume x: "x \<in> {0..real N}"
         have "norm (of_real (pbernpoly n x) / (of_real x + s) ^ n) \<le> c / norm (of_real x + s) ^ n"
           unfolding norm_divide norm_power using c by (intro divide_right_mono) simp_all
         also have "\<dots> \<le> c / (x + Re s) ^ n" 
           using x c c_nonneg s nz[of x] complex_Re_le_cmod[of "of_real x + s"]
           by (intro divide_left_mono power_mono mult_pos_pos zero_less_power add_nonneg_pos) auto
         finally show "norm (of_real (pbernpoly n x) / (of_real x + s) ^ n) \<le> \<dots>" .
       qed (insert n s * pos nz c, auto)
       also have "\<dots> = ?bound N" using * by (simp add: has_integral_iff)
       also have "\<dots> \<le> c / (Re s ^ (n - 1) * (real n - 1))" using c_nonneg elim s n by simp
       also have "\<dots> = c / (real n - 1) / (Re s ^ (n - 1))" by simp
       finally show "norm (integral {0..real N} (\<lambda>x. of_real (pbernpoly n x) /
                       (of_real x + s) ^ n)) \<le> c / (real n - 1) / Re s ^ (n - 1)" .
     qed
   qed (insert s n, simp_all)
   thus ?thesis by (rule that)
 qed
 
 lemma stirling_integral_bound:
   assumes "n > 0"
   obtains c where 
     "\<And>s. Re s > 0 \<Longrightarrow> norm (stirling_integral n s) \<le> c / Re s ^ n"
 proof -
   let ?f = "\<lambda>s. of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s -
                   of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n)"
   from stirling_integral_bound_aux[of "Suc n"] assms obtain c where 
     c: "\<And>s. Re s > 0 \<Longrightarrow> norm (stirling_integral (Suc n) s) \<le> c / Re s ^ n" by auto
   define c1 where "c1 = real n / real (Suc n) * c"
   define c2 where "c2 = \<bar>bernoulli (Suc n)\<bar> / real (Suc n)"
   have c2_nonneg: "c2 \<ge> 0" by (simp add: c2_def)
   show ?thesis
   proof (rule that)
     fix s :: complex assume s: "Re s > 0"
     have "stirling_integral n s = ?f s" using s assms 
       by (rule stirling_integral_conv_stirling_integral_Suc)
     also have "norm \<dots> \<le> norm (of_nat n / of_nat (Suc n) * stirling_integral (Suc n) s) +
                            norm (of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n))"
       by (rule norm_triangle_ineq4)
     also have "\<dots> = real n / real (Suc n) * norm (stirling_integral (Suc n) s) +
                       c2 / norm s ^ n" (is "_ = ?A + ?B")
       by (simp add: norm_divide norm_mult norm_power c2_def field_simps del: of_nat_Suc)
     also have "?A \<le> real n / real (Suc n) * (c / Re s ^ n)"
       by (intro mult_left_mono c s) simp_all
     also have "\<dots> = c1 / Re s ^ n" by (simp add: c1_def)
     also have "c2 / norm s ^ n \<le> c2 / Re s ^ n" using s c2_nonneg
       by (intro divide_left_mono power_mono complex_Re_le_cmod mult_pos_pos zero_less_power) auto
     also have "c1 / Re s ^ n + c2 / Re s ^ n = (c1 + c2) / Re s ^ n" 
       using s by (simp add: field_simps)
     finally show "norm (stirling_integral n s) \<le> (c1 + c2) / Re s ^ n" by - simp_all
   qed
 qed
 
 end
 
 
 lemma stirling_integral_holomorphic [holomorphic_intros]:
   assumes m: "m > 0" and "\<forall>s\<in>A. Re s > 0"
   shows   "stirling_integral m holomorphic_on A"  
 proof -
   let ?f = "\<lambda>s::complex. of_nat m * ((s - 1 / 2) * Ln s - s + of_real (ln (2 * pi) / 2) +
           (\<Sum>k=1..<m. of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k)) - 
           ln_Gamma s)"
   have "?f holomorphic_on A" using assms
     by (auto intro!: holomorphic_intros simp del: of_nat_Suc elim!: nonpos_Reals_cases)
   also have "?this \<longleftrightarrow> stirling_integral m holomorphic_on A" 
     using assms by (intro holomorphic_cong refl) 
                    (simp_all add: field_simps ln_Gamma_stirling_complex)
   finally show "stirling_integral m holomorphic_on A" .
 qed
 
 lemma stirling_integral_continuous_on [continuous_intros]:
   assumes m: "m > 0" and "\<forall>s\<in>A. Re s > 0"
   shows   "continuous_on A (stirling_integral m)"
   by (intro holomorphic_on_imp_continuous_on stirling_integral_holomorphic assms)
     
 lemma has_field_derivative_stirling_integral:
   assumes "Re x > 0" "n > 0"
   shows   "(stirling_integral n has_field_derivative deriv (stirling_integral n) x) (at x)"
   using assms 
   by (intro holomorphic_derivI[OF stirling_integral_holomorphic, of n  "{s. Re s > 0}"] 
         open_halfspace_Re_gt) auto
 
 
  
 lemma
   assumes n: "n > 0" and "x > 0"
   shows   deriv_stirling_integral_complex_of_real:
             "(deriv ^^ j) (stirling_integral n) (complex_of_real x) =
                complex_of_real ((deriv ^^ j) (stirling_integral n) x)" (is "?lhs x = ?rhs x")
   and     differentiable_stirling_integral_real:
             "(deriv ^^ j) (stirling_integral n) field_differentiable at x" (is ?thesis2)
 proof -
   let ?A = "{s. Re s > 0}"
   let ?f = "\<lambda>j x. (deriv ^^ j) (stirling_integral n) (complex_of_real x)"
   let ?f' = "\<lambda>j x. complex_of_real ((deriv ^^ j) (stirling_integral n) x)"
     
   have [simp]: "open ?A" by (simp add: open_halfspace_Re_gt)      
 
   have "?lhs x = ?rhs x \<and> (deriv ^^ j) (stirling_integral n) field_differentiable at x" 
     if "x > 0" for x using that
   proof (induction j arbitrary: x)
     case 0
     have "((\<lambda>x. Re (stirling_integral n (of_real x))) has_field_derivative 
                   Re (deriv (\<lambda>x. stirling_integral n x) (of_real x))) (at x)" using 0 n
       by (auto intro!: derivative_intros has_vector_derivative_real_field
                  field_differentiable_derivI holomorphic_on_imp_differentiable_at[of _ ?A]
                  stirling_integral_holomorphic)
     also have "?this \<longleftrightarrow> (stirling_integral n has_field_derivative 
              Re (deriv (\<lambda>x. stirling_integral n x) (of_real x))) (at x)"
       using eventually_nhds_in_open[of "{0<..}" x] 0 n
       by (intro has_field_derivative_cong_ev refl) 
          (auto elim!: eventually_mono simp: stirling_integral_complex_of_real)
     finally have "stirling_integral n field_differentiable at x"
       by (auto simp: field_differentiable_def)
     with 0 n show ?case by (auto simp: stirling_integral_complex_of_real)
   next
     case (Suc j x)
     note IH = conjunct1[OF Suc.IH] conjunct2[OF Suc.IH]
     have *: "(deriv ^^ Suc j) (stirling_integral n) (complex_of_real x) =
                  of_real ((deriv ^^ Suc j) (stirling_integral n) x)" if x: "x > 0" for x
     proof -
       have "deriv ((deriv ^^ j) (stirling_integral n)) (complex_of_real x) =
               vector_derivative (\<lambda>x. (deriv ^^ j) (stirling_integral n) (of_real x)) (at x)"
         using n x
         by (intro vector_derivative_of_real_right [symmetric] 
                    holomorphic_on_imp_differentiable_at[of _ ?A] holomorphic_higher_deriv
                    stirling_integral_holomorphic) auto
       also have "\<dots> = vector_derivative (\<lambda>x. of_real ((deriv ^^ j) (stirling_integral n) x)) (at x)"
         using eventually_nhds_in_open[of "{0<..}" x] x
         by (intro vector_derivative_cong_eq) (auto elim!: eventually_mono simp: IH(1))
       also have "\<dots> = of_real (deriv ((deriv ^^ j) (stirling_integral n)) x)"
         by (intro vector_derivative_of_real_left holomorphic_on_imp_differentiable_at[of _ ?A]
               field_differentiable_imp_differentiable IH(2) x)
       finally show ?thesis by simp
     qed
     have "((\<lambda>x. Re ((deriv ^^ Suc j) (stirling_integral n) (of_real x))) has_field_derivative 
              Re (deriv ((deriv ^^ Suc j) (stirling_integral n)) (of_real x))) (at x)"
       using Suc.prems n
       by (intro derivative_intros has_vector_derivative_real_field field_differentiable_derivI
                 holomorphic_on_imp_differentiable_at[of _ ?A] stirling_integral_holomorphic
                 holomorphic_higher_deriv) auto
     also have "?this \<longleftrightarrow> ((deriv ^^ Suc j) (stirling_integral n) has_field_derivative 
                   Re (deriv ((deriv ^^ Suc j) (stirling_integral n)) (of_real x))) (at x)"  
       using eventually_nhds_in_open[of "{0<..}" x] Suc.prems *
       by (intro has_field_derivative_cong_ev refl) (auto elim!: eventually_mono)
     finally have "(deriv ^^ Suc j) (stirling_integral n) field_differentiable at x"
       by (auto simp: field_differentiable_def)
     with *[OF Suc.prems] show ?case by blast
   qed
   from this[OF assms(2)] show "?lhs x = ?rhs x" ?thesis2 by blast+
 qed
 
 text \<open>
   Unfortunately, asymptotic power series cannot, in general, be differentiated. However, since 
   @{term ln_Gamma} is holomorphic on the entire positive real half-space, we can differentiate 
   its asymptotic expansion after all.
 
   To do this, we use an ad-hoc version of the more general approach outlined in Erdelyi's
   ``Asymptotic Expansions'' for holomorphic functions: We bound the value of the $j$-th derivative 
   of the remainder term at some value $x$ by applying Cauchy's integral formula along a circle 
   centred at $x$ with radius $\frac{1}{2} x$.
 \<close>
 lemma deriv_stirling_integral_real_bound:
   assumes m: "m > 0"
   shows   "(deriv ^^ j) (stirling_integral m) \<in> O(\<lambda>x::real. 1 / x ^ (m + j))"
 proof -
   from stirling_integral_bound[OF m] guess c . note c = this
   have "0 \<le> cmod (stirling_integral m 1)" by simp
   also have "\<dots> \<le> c" using c[of 1] by simp
   finally have c_nonneg: "c \<ge> 0" .
   define B where "B = c * 2 ^ (m + Suc j)"
   define B' where "B' = B * fact j / 2"
 
   have "eventually (\<lambda>x::real. norm ((deriv ^^ j) (stirling_integral m) x) \<le> 
           B' * norm (1 / x ^ (m+ j))) at_top"
     using eventually_gt_at_top[of "0::real"]
   proof eventually_elim
     case (elim x)
     have "Re s > 0" if "s \<in> cball (of_real x) (x/2)" for s :: complex
     proof -
       have "x - Re s \<le> norm (of_real x - s)" using complex_Re_le_cmod[of "of_real x - s"] by simp
       also from that have "\<dots> \<le> x/2" by (simp add: dist_complex_def)
       finally show ?thesis using elim by simp
     qed
     hence "((\<lambda>u. stirling_integral m u / (u - of_real x) ^ Suc j) has_contour_integral
             complex_of_real (2 * pi) * \<i> / fact j * 
               (deriv ^^ j) (stirling_integral m) (of_real x)) (circlepath (of_real x) (x/2))"
       using m elim
       by (intro Cauchy_has_contour_integral_higher_derivative_circlepath 
                 stirling_integral_continuous_on stirling_integral_holomorphic) auto
     hence "norm (of_real (2 * pi) * \<i> / fact j * (deriv ^^ j) (stirling_integral m) (of_real x)) \<le>
             B / x ^ (m + Suc j) * (2 * pi * (x / 2))"
     proof (rule has_contour_integral_bound_circlepath)
       fix u :: complex assume dist: "norm (u - of_real x) = x / 2"
       have "Re (of_real x - u) \<le> norm (of_real x - u)" by (rule complex_Re_le_cmod)
       also have "\<dots> = x / 2" using dist by (simp add: norm_minus_commute)
       finally have Re_u: "Re u \<ge> x/2" using elim by simp
       have "norm (stirling_integral m u / (u - of_real x) ^ Suc j) \<le> 
               c / Re u ^ m / (x / 2) ^ Suc j" using Re_u elim
         unfolding norm_divide norm_power dist
         by (intro divide_right_mono zero_le_power c) simp_all
       also have "\<dots> \<le> c / (x/2) ^ m / (x / 2) ^ Suc j" using c_nonneg elim Re_u
         by (intro divide_right_mono divide_left_mono power_mono) simp_all
       also have "\<dots> = B / x ^ (m + Suc j)" using elim by (simp add: B_def field_simps power_add)
       finally show "norm (stirling_integral m u / (u - of_real x) ^ Suc j) \<le> B / x ^ (m + Suc j)" .
     qed (insert elim c_nonneg, auto simp: B_def simp del: power_Suc)
     hence "cmod ((deriv ^^ j) (stirling_integral m) (of_real x)) \<le> B' / x ^ (j + m)"
       using elim by (simp add: field_simps norm_divide norm_mult norm_power B'_def)
     with elim m show ?case by (simp_all add: add_ac deriv_stirling_integral_complex_of_real)
   qed
   thus ?thesis by (rule bigoI)
 qed
 
 definition stirling_sum where
   "stirling_sum j m x = 
      (-1) ^ j * (\<Sum>k = 1..<m. (of_real (bernoulli (Suc k)) * pochhammer (of_nat k) j / (of_nat k *
                                  of_nat (Suc k))) * inverse x ^ (k + j))"
   
 definition stirling_sum' where
   "stirling_sum' j m x = 
      (-1) ^ (Suc j) * (\<Sum>k\<le>m. (of_real (bernoulli' k) * 
        pochhammer (of_nat (Suc k)) (j - 1) * inverse x ^ (k + j)))"
 
 lemma stirling_sum_complex_of_real:
   "stirling_sum j m (complex_of_real x) = complex_of_real (stirling_sum j m x)"
   by (simp add: stirling_sum_def pochhammer_of_real [symmetric] del: of_nat_Suc)
 
 lemma stirling_sum'_complex_of_real:
   "stirling_sum' j m (complex_of_real x) = complex_of_real (stirling_sum' j m x)"
   by (simp add: stirling_sum'_def pochhammer_of_real [symmetric] del: of_nat_Suc)
 
 lemma has_field_derivative_stirling_sum_complex [derivative_intros]:
   "Re x > 0 \<Longrightarrow> (stirling_sum j m has_field_derivative stirling_sum (Suc j) m x) (at x)"
     unfolding stirling_sum_def [abs_def] sum_distrib_left
     by (rule DERIV_sum) (auto intro!: derivative_eq_intros simp del: of_nat_Suc 
                               simp: pochhammer_Suc power_diff)
 
 lemma has_field_derivative_stirling_sum_real [derivative_intros]:
   "x > (0::real) \<Longrightarrow> (stirling_sum j m has_field_derivative stirling_sum (Suc j) m x) (at x)"
     unfolding stirling_sum_def [abs_def] sum_distrib_left
     by (rule DERIV_sum) (auto intro!: derivative_eq_intros simp del: of_nat_Suc 
                               simp: pochhammer_Suc power_diff)
 
 lemma has_field_derivative_stirling_sum'_complex [derivative_intros]:
   assumes "j > 0" "Re x > 0"
   shows   "(stirling_sum' j m has_field_derivative stirling_sum' (Suc j) m x) (at x)"
 proof (cases j)
   case (Suc j')
   from assms have [simp]: "x \<noteq> 0" by auto
   define c where "c = (\<lambda>n. (-1) ^ Suc j * complex_of_real (bernoulli' n) * 
                           pochhammer (of_nat (Suc n)) j')"
   define T where "T = (\<lambda>n x. c n * inverse x ^ (j + n))"
   define T' where "T' = (\<lambda>n x. - (of_nat (j + n)) * c n * inverse x ^ (Suc (j + n)))"
   have "((\<lambda>x. \<Sum>k\<le>m. T k x) has_field_derivative (\<Sum>k\<le>m. T' k x)) (at x)" using assms Suc
     by (intro DERIV_sum)
        (auto simp: T_def T'_def intro!: derivative_eq_intros 
              simp: field_simps power_add [symmetric]  simp del: of_nat_Suc power_Suc of_nat_add)
   also have "(\<lambda>x. (\<Sum>k\<le>m. T k x)) = stirling_sum' j m"
     by (simp add: Suc T_def c_def stirling_sum'_def fun_eq_iff add_ac mult.assoc sum_distrib_left)
   also have "(\<Sum>k\<le>m. T' k x) = stirling_sum' (Suc j) m x"
     by (simp add: T'_def c_def Suc stirling_sum'_def sum_distrib_left 
           sum_distrib_right algebra_simps pochhammer_Suc)
   finally show ?thesis .
 qed (insert assms, simp_all)
   
 lemma has_field_derivative_stirling_sum'_real [derivative_intros]:
   assumes "j > 0" "x > (0::real)"
   shows   "(stirling_sum' j m has_field_derivative stirling_sum' (Suc j) m x) (at x)"
 proof (cases j)
   case (Suc j')
   from assms have [simp]: "x \<noteq> 0" by auto
   define c where "c = (\<lambda>n. (-1) ^ Suc j * (bernoulli' n) * pochhammer (of_nat (Suc n)) j')"
   define T where "T = (\<lambda>n x. c n * inverse x ^ (j + n))"
   define T' where "T' = (\<lambda>n x. - (of_nat (j + n)) * c n * inverse x ^ (Suc (j + n)))"
   have "((\<lambda>x. \<Sum>k\<le>m. T k x) has_field_derivative (\<Sum>k\<le>m. T' k x)) (at x)" using assms Suc
     by (intro DERIV_sum)
        (auto simp: T_def T'_def intro!: derivative_eq_intros 
              simp: field_simps power_add [symmetric]  simp del: of_nat_Suc power_Suc of_nat_add)
   also have "(\<lambda>x. (\<Sum>k\<le>m. T k x)) = stirling_sum' j m"
     by (simp add: Suc T_def c_def stirling_sum'_def fun_eq_iff add_ac mult.assoc sum_distrib_left)
   also have "(\<Sum>k\<le>m. T' k x) = stirling_sum' (Suc j) m x"
     by (simp add: T'_def c_def Suc stirling_sum'_def sum_distrib_left 
           sum_distrib_right algebra_simps pochhammer_Suc)
   finally show ?thesis .
 qed (insert assms, simp_all)
 
 lemma higher_deriv_stirling_sum_complex:
   "Re x > 0 \<Longrightarrow> (deriv ^^ i) (stirling_sum j m) x = stirling_sum (i + j) m x"
 proof (induction i arbitrary: x)
   case (Suc i)
   have "deriv ((deriv ^^ i) (stirling_sum j m)) x = deriv (stirling_sum (i + j) m) x"
     using eventually_nhds_in_open[of "{x. Re x > 0}" x] Suc.prems
     by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: open_halfspace_Re_gt Suc.IH)
   also from Suc.prems have "\<dots> = stirling_sum (Suc (i + j)) m x"
     by (intro DERIV_imp_deriv has_field_derivative_stirling_sum_complex)
   finally show ?case by simp
 qed simp_all
   
 
 definition Polygamma_approx :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a :: {real_normed_field, ln}" where
   "Polygamma_approx j m = 
      (deriv ^^ j) (\<lambda>x. (x - 1 / 2) * ln x - x + of_real (ln (2 * pi)) / 2 + stirling_sum 0 m x)"
   
 lemma Polygamma_approx_Suc: "Polygamma_approx (Suc j) m = deriv (Polygamma_approx j m)"
   by (simp add: Polygamma_approx_def)  
 
 lemma Polygamma_approx_0: 
   "Polygamma_approx 0 m x = (x - 1/2) * ln x - x + of_real (ln (2*pi)) / 2 + stirling_sum 0 m x"
   by (simp add: Polygamma_approx_def)
     
 lemma Polygamma_approx_1_complex: 
   "Re x > 0 \<Longrightarrow> 
      Polygamma_approx (Suc 0) m x = ln x - 1 / (2*x) + stirling_sum (Suc 0) m x"
   unfolding Polygamma_approx_Suc Polygamma_approx_0
   by (intro DERIV_imp_deriv) 
      (auto intro!: derivative_eq_intros elim!: nonpos_Reals_cases simp: field_simps)
      
 lemma Polygamma_approx_1_real: 
   "x > (0 :: real) \<Longrightarrow> 
      Polygamma_approx (Suc 0) m x = ln x - 1 / (2*x) + stirling_sum (Suc 0) m x"
   unfolding Polygamma_approx_Suc Polygamma_approx_0
   by (intro DERIV_imp_deriv) 
      (auto intro!: derivative_eq_intros elim!: nonpos_Reals_cases simp: field_simps)
  
 lemma stirling_sum_2_conv_stirling_sum'_1:
   fixes x :: "'a :: {real_div_algebra, field_char_0}"
   assumes "m > 0" "x \<noteq> 0"
   shows   "stirling_sum' 1 m x = 1 / x + 1 / (2 * x^2) + stirling_sum 2 m x"
 proof -
   have pochhammer_2: "pochhammer (of_nat k) 2 = of_nat k * of_nat (Suc k)" for k 
     by (simp add: pochhammer_Suc eval_nat_numeral add_ac)
   have "stirling_sum 2 m x = 
           (\<Sum>k = Suc 0..<m. of_real (bernoulli' (Suc k)) * inverse x ^ Suc (Suc k))"
     unfolding stirling_sum_def pochhammer_2 power2_minus power_one mult_1_left
     by (intro sum.cong refl)
        (simp_all add: stirling_sum_def pochhammer_2 power2_eq_square divide_simps bernoulli'_def
                  del: of_nat_Suc power_Suc)
   also have "1 / (2 * x^2) + \<dots> = 
                (\<Sum>k=0..<m. of_real (bernoulli' (Suc k)) * inverse x ^ Suc (Suc k))" using assms
     by (subst (2) sum.atLeast_Suc_lessThan) (simp_all add: power2_eq_square field_simps)
   also have "1 / x + \<dots> = (\<Sum>k=0..<Suc m. of_real (bernoulli' k) * inverse x ^ Suc k)"
     by (subst sum.atLeast0_lessThan_Suc_shift) (simp_all add: bernoulli'_def divide_simps)
   also have "\<dots> = (\<Sum>k\<le>m. of_real (bernoulli' k) * inverse x ^ Suc k)"
     by (intro sum.cong) auto
   also have "\<dots> = stirling_sum' 1 m x" by (simp add: stirling_sum'_def)
   finally show ?thesis by (simp add: add_ac)
 qed
 
 lemma Polygamma_approx_2_real: 
   assumes "x > (0::real)" "m > 0"
   shows   "Polygamma_approx (Suc (Suc 0)) m x = stirling_sum' 1 m x"
 proof -
   have "Polygamma_approx (Suc (Suc 0)) m x = deriv (Polygamma_approx (Suc 0) m) x" 
     by (simp add: Polygamma_approx_Suc)
   also have "\<dots> = deriv (\<lambda>x. ln x - 1 / (2*x) + stirling_sum (Suc 0) m x) x"
     using eventually_nhds_in_open[of "{0<..}" x] assms
     by (intro deriv_cong_ev) (auto elim!: eventually_mono simp: Polygamma_approx_1_real)
   also have "\<dots> = 1 / x + 1 / (2*x^2) + stirling_sum (Suc (Suc 0)) m x" using assms
     by (intro DERIV_imp_deriv) (auto intro!: derivative_eq_intros 
            elim!: nonpos_Reals_cases simp: field_simps power2_eq_square)
   also have "\<dots> = stirling_sum' 1 m x" using stirling_sum_2_conv_stirling_sum'_1[of m x] assms
     by (simp add: eval_nat_numeral)
   finally show ?thesis .
 qed
   
 lemma Polygamma_approx_2_complex: 
   assumes "Re x > 0" "m > 0"
   shows   "Polygamma_approx (Suc (Suc 0)) m x = stirling_sum' 1 m x"
 proof -
   have "Polygamma_approx (Suc (Suc 0)) m x = deriv (Polygamma_approx (Suc 0) m) x" 
     by (simp add: Polygamma_approx_Suc)
   also have "\<dots> = deriv (\<lambda>x. ln x - 1 / (2*x) + stirling_sum (Suc 0) m x) x"
     using eventually_nhds_in_open[of "{s. Re s > 0}" x] assms
     by (intro deriv_cong_ev)
        (auto simp: open_halfspace_Re_gt elim!: eventually_mono simp: Polygamma_approx_1_complex)
   also have "\<dots> = 1 / x + 1 / (2*x^2) + stirling_sum (Suc (Suc 0)) m x" using assms
     by (intro DERIV_imp_deriv) (auto intro!: derivative_eq_intros 
            elim!: nonpos_Reals_cases simp: field_simps power2_eq_square)
   also have "\<dots> = stirling_sum' 1 m x" using stirling_sum_2_conv_stirling_sum'_1[of m x] assms
       by (subst stirling_sum_2_conv_stirling_sum'_1) (auto simp: eval_nat_numeral)
   finally show ?thesis .
 qed
   
 lemma Polygamma_approx_ge_2_real: 
   assumes "x > (0::real)" "m > 0"
   shows   "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
 using assms(1)
 proof (induction j arbitrary: x)
   case (0 x)
   with assms show ?case by (simp add: Polygamma_approx_2_real)
 next
   case (Suc j x)
   have "Polygamma_approx (Suc (Suc (Suc j))) m x = deriv (Polygamma_approx (Suc (Suc j)) m) x"
     by (simp add: Polygamma_approx_Suc)
   also have "\<dots> = deriv (stirling_sum' (Suc j) m) x"
     using eventually_nhds_in_open[of "{0<..}" x] Suc.prems
     by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH)
   also have "\<dots> = stirling_sum' (Suc (Suc j)) m x" using Suc.prems
     by (intro DERIV_imp_deriv derivative_intros) simp_all
   finally show ?case .
 qed
   
 lemma Polygamma_approx_ge_2_complex:
   assumes "Re x > 0" "m > 0"
   shows   "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
 using assms(1)
 proof (induction j arbitrary: x)
   case (0 x)
   with assms show ?case by (simp add: Polygamma_approx_2_complex)
 next
   case (Suc j x)
   have "Polygamma_approx (Suc (Suc (Suc j))) m x = deriv (Polygamma_approx (Suc (Suc j)) m) x"
     by (simp add: Polygamma_approx_Suc)
   also have "\<dots> = deriv (stirling_sum' (Suc j) m) x"
     using eventually_nhds_in_open[of "{x. Re x > 0}" x] Suc.prems
     by (intro deriv_cong_ev refl) (auto elim!: eventually_mono simp: Suc.IH open_halfspace_Re_gt)
   also have "\<dots> = stirling_sum' (Suc (Suc j)) m x" using Suc.prems
     by (intro DERIV_imp_deriv derivative_intros) simp_all
   finally show ?case .
 qed
 
 lemma Polygamma_approx_complex_of_real:
   assumes "x > 0" "m > 0"
   shows   "Polygamma_approx j m (complex_of_real x) = of_real (Polygamma_approx j m x)"
 proof (cases j)
   case 0
   with assms show ?thesis by (simp add: Polygamma_approx_0 Ln_of_real stirling_sum_complex_of_real)
 next
   case [simp]: (Suc j')
   thus ?thesis
   proof (cases j')
     case 0
     with assms show ?thesis 
       by (simp add: Polygamma_approx_1_complex 
                     Polygamma_approx_1_real stirling_sum_complex_of_real Ln_of_real)
   next
     case (Suc j'')
     with assms show ?thesis
       by (simp add: Polygamma_approx_ge_2_complex Polygamma_approx_ge_2_real 
                     stirling_sum'_complex_of_real)
   qed
 qed
   
 lemma higher_deriv_Polygamma_approx [simp]: 
   "(deriv ^^ j) (Polygamma_approx i m) = Polygamma_approx (j + i) m"
   by (simp add: Polygamma_approx_def funpow_add)
   
 lemma stirling_sum_holomorphic [holomorphic_intros]:
   "0 \<notin> A \<Longrightarrow> stirling_sum j m holomorphic_on A"
   unfolding stirling_sum_def by (intro holomorphic_intros) auto
   
 lemma Polygamma_approx_holomorphic [holomorphic_intros]:
   "Polygamma_approx j m holomorphic_on {s. Re s > 0}"
   unfolding Polygamma_approx_def
   by (intro holomorphic_intros) (auto simp: open_halfspace_Re_gt elim!: nonpos_Reals_cases)
   
 lemma higher_deriv_lnGamma_stirling:
   assumes m: "m > 0"
   shows   "(\<lambda>x::real. (deriv ^^ j) ln_Gamma x - Polygamma_approx j m x) \<in> O(\<lambda>x. 1 / x ^ (m + j))"
 proof -
   have "eventually (\<lambda>x. \<bar>(deriv ^^ j) ln_Gamma x - Polygamma_approx j m x\<bar> =
                           inverse (real m) * \<bar>(deriv ^^ j) (stirling_integral m) x\<bar>) at_top"
     using eventually_gt_at_top[of "0::real"]
   proof eventually_elim
     case (elim x)
     note x = this
     have "(deriv ^^ j) (\<lambda>x. ln_Gamma x - Polygamma_approx 0 m x) (complex_of_real x) =
             (deriv ^^ j) (\<lambda>x. (-inverse (of_nat m)) * stirling_integral m x) (complex_of_real x)"
       using eventually_nhds_in_open[of "{s. Re s > 0}" x] x m
       by (intro higher_deriv_cong_ev refl)
          (auto elim!: eventually_mono simp: ln_Gamma_stirling_complex Polygamma_approx_def 
             field_simps open_halfspace_Re_gt stirling_sum_def)
     also have "\<dots> = - inverse (of_nat m) * (deriv ^^ j) (stirling_integral m) (of_real x)" using x m
       by (intro higher_deriv_cmult[of _ "{s. Re s > 0}"] stirling_integral_holomorphic)
          (auto simp: open_halfspace_Re_gt)
     also have "(deriv ^^ j) (\<lambda>x. ln_Gamma x - Polygamma_approx 0 m x) (complex_of_real x) = 
                  (deriv ^^ j) ln_Gamma (of_real x) - (deriv ^^ j) (Polygamma_approx 0 m) (of_real x)"
       using x 
       by (intro higher_deriv_diff[of _ "{s. Re s > 0}"])
          (auto intro!: holomorphic_intros elim!: nonpos_Reals_cases simp: open_halfspace_Re_gt)
     also have "(deriv ^^ j) (Polygamma_approx 0 m) (complex_of_real x) =
                  of_real (Polygamma_approx j m x)" using x m
       by (simp add: Polygamma_approx_complex_of_real)
     also have "norm (- inverse (of_nat m) * (deriv ^^ j) (stirling_integral m) (complex_of_real x)) = 
                  inverse (real m) * \<bar>(deriv ^^ j) (stirling_integral m) x\<bar>"
       using x m by (simp add: norm_mult norm_inverse deriv_stirling_integral_complex_of_real)
     also have "(deriv ^^ j) ln_Gamma (complex_of_real x) = of_real ((deriv ^^ j) ln_Gamma x)" using x
       by (simp add: higher_deriv_ln_Gamma_complex_of_real)
     also have "norm (\<dots> - of_real (Polygamma_approx j m x)) = 
                  \<bar>(deriv ^^ j) ln_Gamma x - Polygamma_approx j m x\<bar>"
       by (simp only: of_real_diff [symmetric] norm_of_real)
     finally show ?case .
   qed
   from bigthetaI_cong[OF this] m
     have "(\<lambda>x::real. (deriv ^^ j) ln_Gamma x - Polygamma_approx j m x) \<in> 
              \<Theta>(\<lambda>x. (deriv ^^ j) (stirling_integral m) x)" by simp
   also have "(\<lambda>x::real. (deriv ^^ j) (stirling_integral m) x) \<in> O(\<lambda>x. 1 / x ^ (m + j))" using m
       by (rule deriv_stirling_integral_real_bound)
   finally show ?thesis .
 qed
 
 lemma Polygamma_approx_1_real':
   assumes x: "(x::real) > 0" and m: "m > 0"
   shows   "Polygamma_approx 1 m x = ln x - (\<Sum>k = Suc 0..m. bernoulli' k * inverse x ^ k / real k)"
 proof -
   have "Polygamma_approx 1 m x = ln x - (1 / (2 * x) + 
           (\<Sum>k=Suc 0..<m. bernoulli (Suc k) * inverse x ^ Suc k / real (Suc k)))"
     (is "_ = _ - (_ + ?S)") using x by (simp add: Polygamma_approx_1_real stirling_sum_def)
   also have "?S = (\<Sum>k=Suc 0..<m. bernoulli' (Suc k) * inverse x ^ Suc k / real (Suc k))"
     by (intro sum.cong refl) (simp_all add: bernoulli'_def)
   also have "1 / (2 * x) + \<dots> = 
                (\<Sum>k=0..<m. bernoulli' (Suc k) * inverse x ^ Suc k / real (Suc k))" using m
     by (subst (2) sum.atLeast_Suc_lessThan) (simp_all add: field_simps)
   also have "\<dots> = (\<Sum>k = Suc 0..m. bernoulli' k * inverse x ^ k / real k)" using assms
     by (subst sum.shift_bounds_Suc_ivl [symmetric]) (simp add: atLeastLessThanSuc_atLeastAtMost)
   finally show ?thesis .
 qed
 
 theorem
   assumes m: "m > 0"
   shows   ln_Gamma_real_asymptotics:
             "(\<lambda>x. ln_Gamma x - ((x - 1 / 2) * ln x - x + ln (2 * pi) / 2 +
                      (\<Sum>k = 1..<m. bernoulli (Suc k) / (real k * real (Suc k)) / x^k)))
                 \<in> O(\<lambda>x. 1 / x ^ m)" (is ?th1)
     and   Digamma_real_asymptotics:
             "(\<lambda>x. Digamma x - (ln x - (\<Sum>k=1..m. bernoulli' k / real k / x ^ k)))
                 \<in> O(\<lambda>x. 1 / (x ^ Suc m))" (is ?th2)
     and   Polygamma_real_asymptotics: "j > 0 \<Longrightarrow> 
              (\<lambda>x. Polygamma j x - (- 1) ^ Suc j * (\<Sum>k\<le>m. bernoulli' k *
                      pochhammer (real (Suc k)) (j - 1) / x ^ (k + j))) 
                 \<in> O(\<lambda>x. 1 / x ^ (m+j+1))" (is "_ \<Longrightarrow> ?th3")
 proof -
   define G :: "nat \<Rightarrow> real \<Rightarrow> real" where 
     "G = (\<lambda>m. if m = 0 then ln_Gamma else Polygamma (m - 1))"
   have *: "(\<lambda>x. G j x - h x) \<in> O(\<lambda>x. 1 / x ^ (m + j))"
     if "\<And>x::real. x > 0 \<Longrightarrow> Polygamma_approx j m x = h x" for j h
   proof -
     have "(\<lambda>x. G j x - h x) \<in> 
             \<Theta>(\<lambda>x. (deriv ^^ j) ln_Gamma x - Polygamma_approx j m x)" (is "_ \<in> \<Theta>(?f)")
       using that
       by (intro bigthetaI_cong) (auto intro: eventually_mono[OF eventually_gt_at_top[of "0::real"]]
             simp del: funpow.simps simp: higher_deriv_ln_Gamma_real G_def)
     also have "?f \<in> O(\<lambda>x::real. 1 / x ^ (m + j))" using m
       by (rule higher_deriv_lnGamma_stirling)
     finally show ?thesis .
   qed  
     
   note [[simproc del: simplify_landau_sum]]
   from *[OF Polygamma_approx_0] assms show ?th1 
     by (simp add: G_def Polygamma_approx_0 stirling_sum_def field_simps)
   from *[OF Polygamma_approx_1_real'] assms show ?th2 by (simp add: G_def field_simps)
       
   assume j: "j > 0"
   from *[OF Polygamma_approx_ge_2_real, of "j - 1"] assms j show ?th3
     by (simp add: G_def stirling_sum'_def power_add power_diff field_simps)
 qed
 
 end
diff --git a/thys/Stirling_Formula/ROOT b/thys/Stirling_Formula/ROOT
--- a/thys/Stirling_Formula/ROOT
+++ b/thys/Stirling_Formula/ROOT
@@ -1,13 +1,13 @@
 chapter AFP
 
-session Stirling_Formula (AFP) = "HOL-Analysis" +
+session Stirling_Formula (AFP) = Bernoulli +
   options [timeout = 600]
   sessions
     Bernoulli
     Landau_Symbols
   theories
     Stirling_Formula
     Ln_Gamma_Asymptotics
   document_files
     "root.tex"
     "root.bib"
diff --git a/thys/Transcendence_Series_Hancl_Rucki/ROOT b/thys/Transcendence_Series_Hancl_Rucki/ROOT
--- a/thys/Transcendence_Series_Hancl_Rucki/ROOT
+++ b/thys/Transcendence_Series_Hancl_Rucki/ROOT
@@ -1,12 +1,12 @@
 chapter AFP
 
-session Transcendence_Series_Hancl_Rucki (AFP) = "HOL-Analysis" +
+session Transcendence_Series_Hancl_Rucki (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 1200]
   sessions
     Prime_Number_Theorem
   theories
     Transcendence_Series
   document_files
     "root.tex"
     "root.bib"
 
diff --git a/thys/Winding_Number_Eval/Cauchy_Index_Theorem.thy b/thys/Winding_Number_Eval/Cauchy_Index_Theorem.thy
--- a/thys/Winding_Number_Eval/Cauchy_Index_Theorem.thy
+++ b/thys/Winding_Number_Eval/Cauchy_Index_Theorem.thy
@@ -1,4682 +1,4682 @@
 (*
     Author:     Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
 *)
 
 section \<open>Cauchy's index theorem\<close>
 
 theory Cauchy_Index_Theorem imports 
-  "HOL-Analysis.Multivariate_Analysis"
+  "HOL-Complex_Analysis.Complex_Analysis"
   "Sturm_Tarski.Sturm_Tarski"
   "HOL-Computational_Algebra.Fundamental_Theorem_Algebra"
   Missing_Transcendental
   Missing_Algebraic
   Missing_Analysis
 begin
 
 text \<open>This theory formalises Cauchy indices on the complex plane and relate them to winding numbers\<close>
 
 subsection \<open>Misc\<close>
   
 (*refined version of the library one with the same name by dropping unnecessary assumptions*) 
 lemma atMostAtLeast_subset_convex:
   fixes C :: "real set"
   assumes "convex C"
     and "x \<in> C" "y \<in> C" 
   shows "{x .. y} \<subseteq> C"
 proof safe
   fix z assume z: "z \<in> {x .. y}"
   have "z \<in> C" if *: "x < z" "z < y"
   proof -
     let ?\<mu> = "(y - z) / (y - x)"
     have "0 \<le> ?\<mu>" "?\<mu> \<le> 1"
       using assms * by (auto simp: field_simps)
     then have comb: "?\<mu> * x + (1 - ?\<mu>) * y \<in> C"
       using assms iffD1[OF convex_alt, rule_format, of C y x ?\<mu>]
       by (simp add: algebra_simps)
     have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
       by (auto simp: field_simps)
     also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
       using * by (simp only: add_divide_distrib) (auto simp: field_simps)
     also have "\<dots> = z"
       using assms * by (auto simp: field_simps)
     finally show ?thesis
       using comb by auto
   qed
   then show "z \<in> C"
     using z assms by (auto simp: le_less)
 qed
   
 lemma arg_elim:
   "f x \<Longrightarrow> x= y \<Longrightarrow> f y"
   by auto
  
 lemma arg_elim2:
   "f x1 x2 \<Longrightarrow> x1= y1 \<Longrightarrow>x2=y2 \<Longrightarrow> f y1 y2"
   by auto
  
 lemma arg_elim3:
   "\<lbrakk>f x1 x2 x3;x1= y1;x2=y2;x3=y3 \<rbrakk> \<Longrightarrow> f y1 y2 y3"
   by auto  
     
 lemma IVT_strict:
   fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b::linorder_topology"
   assumes "(f a > y \<and> y > f b) \<or> (f a < y \<and> y < f b)" "a<b" "continuous_on {a .. b} f"
   shows "\<exists>x. a < x \<and> x < b \<and> f x = y"
 by (metis IVT' IVT2' assms(1) assms(2) assms(3) linorder_neq_iff order_le_less order_less_imp_le)    
   
 lemma (in dense_linorder) atLeastAtMost_subseteq_greaterThanLessThan_iff:
   "{a .. b} \<subseteq> { c <..< d } \<longleftrightarrow> (a \<le> b \<longrightarrow> c < a \<and> b < d)"
   using dense[of "a" "min c b"] dense[of "max a d" "b"]
   by (force simp: subset_eq Ball_def not_less[symmetric])    
 
 lemma Re_winding_number_half_right:
   assumes "\<forall>p\<in>path_image \<gamma>. Re p \<ge> Re z" and "valid_path \<gamma>" and  "z\<notin>path_image \<gamma>"
   shows "Re(winding_number \<gamma> z) = (Im (Ln (pathfinish \<gamma> - z)) - Im (Ln (pathstart \<gamma> - z)))/(2*pi)"
 proof -
   define g where "g=(\<lambda>t. \<gamma> t - z)"
   define st fi where "st\<equiv>pathstart g" and "fi\<equiv>pathfinish g"
   have "valid_path g" "0\<notin>path_image g" and pos_img:"\<forall>p\<in>path_image g. Re p \<ge> 0" unfolding g_def 
     subgoal using assms(2) by auto
     subgoal using assms(3) by auto
     subgoal using assms(1) by fastforce
     done
   have "(inverse has_contour_integral Ln fi - Ln st) g"
     unfolding fi_def st_def
   proof (rule contour_integral_primitive[OF _ \<open>valid_path g\<close>,of " - \<real>\<^sub>\<le>\<^sub>0"])
     fix x::complex assume "x \<in> - \<real>\<^sub>\<le>\<^sub>0"
     then have "(Ln has_field_derivative inverse x) (at x)" using has_field_derivative_Ln
       by auto
     then show "(Ln has_field_derivative inverse x) (at x within - \<real>\<^sub>\<le>\<^sub>0)"
       using has_field_derivative_at_within by auto
   next
     show "path_image g \<subseteq> - \<real>\<^sub>\<le>\<^sub>0" using pos_img \<open>0\<notin>path_image g\<close>
       by (metis ComplI antisym assms(3) complex_nonpos_Reals_iff complex_surj
           subsetI zero_complex.code)
   qed
   then have winding_eq:"2*pi*\<i>*winding_number g 0 = (Ln fi - Ln st)"
     using has_contour_integral_winding_number[OF \<open>valid_path g\<close> \<open>0\<notin>path_image g\<close>
         ,simplified,folded inverse_eq_divide] has_contour_integral_unique 
     by auto
   have "Re(winding_number g 0) 
       = (Im (Ln fi) - Im (Ln st))/(2*pi)"
     (is "?L=?R")
   proof -
     have "?L = Re((Ln fi - Ln st)/(2*pi*\<i>))"
       using winding_eq[symmetric] by auto
     also have "... = ?R"
       by (metis Im_divide_of_real Im_i_times complex_i_not_zero minus_complex.simps(2) 
           mult.commute mult_divide_mult_cancel_left_if times_divide_eq_right)
     finally show ?thesis .
   qed
   then show ?thesis unfolding g_def fi_def st_def using winding_number_offset by simp
 qed      
   
 lemma Re_winding_number_half_upper:
   assumes pimage:"\<forall>p\<in>path_image \<gamma>. Im p \<ge> Im z" and "valid_path \<gamma>" and "z\<notin>path_image \<gamma>"
   shows "Re(winding_number \<gamma> z) = 
             (Im (Ln (\<i>*z - \<i>*pathfinish \<gamma>)) - Im (Ln (\<i>*z - \<i>*pathstart \<gamma> )))/(2*pi)"
 proof -
   define \<gamma>' where "\<gamma>'=(\<lambda>t. - \<i> * (\<gamma> t - z) + z)"
   have "Re (winding_number \<gamma>' z) = (Im (Ln (pathfinish \<gamma>' - z)) - Im (Ln (pathstart \<gamma>' - z))) / (2 * pi)"
     unfolding \<gamma>'_def
     apply (rule Re_winding_number_half_right)
     subgoal using pimage unfolding path_image_def by auto
     subgoal 
       apply (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close>,of "\<lambda>x. -\<i> * (x-z) + z" UNIV
             , unfolded comp_def])
       by (auto intro!:holomorphic_intros)
     subgoal using \<open>z\<notin>path_image \<gamma>\<close> unfolding path_image_def by auto
     done
   moreover have "winding_number \<gamma>' z = winding_number \<gamma> z"
   proof -
     define f where "f=(\<lambda>x. -\<i> * (x-z) + z)"
     define c where "c= 1 / (complex_of_real (2 * pi) * \<i>)"
     have "winding_number \<gamma>' z = winding_number (f o \<gamma>) z" 
       unfolding \<gamma>'_def comp_def f_def by auto
     also have "... = c * contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))" unfolding c_def
     proof (rule winding_number_comp[of UNIV])
       show "z \<notin> path_image (f \<circ> \<gamma>)" using \<open>z\<notin>path_image \<gamma>\<close> unfolding f_def path_image_def by auto
     qed (auto simp add:f_def \<open>valid_path \<gamma>\<close> intro!:holomorphic_intros)
     also have "... = c * contour_integral \<gamma> (\<lambda>w.  1 / (w - z))"
     proof -
       have "deriv f x = -\<i>" for x
         unfolding f_def
         by (auto intro!:derivative_eq_intros DERIV_imp_deriv)
       then show ?thesis    
         unfolding f_def c_def 
         by (auto simp add:field_simps divide_simps intro!:arg_cong2[where f=contour_integral])
     qed
     also have "... = winding_number \<gamma> z"
       using winding_number_valid_path[OF \<open>valid_path \<gamma>\<close> \<open>z\<notin>path_image \<gamma>\<close>,folded c_def] by simp
     finally show ?thesis .
   qed    
   moreover have "pathfinish \<gamma>' = z+ \<i>*z -\<i>* pathfinish \<gamma>" "pathstart \<gamma>' = z+ \<i>*z -\<i>*pathstart \<gamma>"  
     unfolding \<gamma>'_def path_defs by (auto simp add:algebra_simps)
   ultimately show ?thesis by auto
 qed
     
 lemma Re_winding_number_half_lower:
   assumes pimage:"\<forall>p\<in>path_image \<gamma>. Im p \<le> Im z" and "valid_path \<gamma>" and "z\<notin>path_image \<gamma>"
   shows "Re(winding_number \<gamma> z) = 
              (Im (Ln ( \<i>*pathfinish \<gamma> - \<i>*z)) - Im (Ln (\<i>*pathstart \<gamma> - \<i>*z)))/(2*pi)"
 proof -
   define \<gamma>' where "\<gamma>'=(\<lambda>t. \<i> * (\<gamma> t - z) + z)"
   have "Re (winding_number \<gamma>' z) = (Im (Ln (pathfinish \<gamma>' - z)) - Im (Ln (pathstart \<gamma>' - z))) / (2 * pi)"
     unfolding \<gamma>'_def
     apply (rule Re_winding_number_half_right)
     subgoal using pimage unfolding path_image_def by auto
     subgoal 
       apply (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close>,of "\<lambda>x. \<i> * (x-z) + z" UNIV
             , unfolded comp_def])
       by (auto intro!:holomorphic_intros)
     subgoal using \<open>z\<notin>path_image \<gamma>\<close> unfolding path_image_def by auto
     done
   moreover have "winding_number \<gamma>' z = winding_number \<gamma> z"
   proof -
     define f where "f=(\<lambda>x. \<i> * (x-z) + z)"
     define c where "c= 1 / (complex_of_real (2 * pi) * \<i>)"
     have "winding_number \<gamma>' z = winding_number (f o \<gamma>) z" 
       unfolding \<gamma>'_def comp_def f_def by auto
     also have "... = c * contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))" unfolding c_def
     proof (rule winding_number_comp[of UNIV])
       show "z \<notin> path_image (f \<circ> \<gamma>)" using \<open>z\<notin>path_image \<gamma>\<close> unfolding f_def path_image_def by auto
     qed (auto simp add:f_def \<open>valid_path \<gamma>\<close> intro!:holomorphic_intros)
     also have "... = c * contour_integral \<gamma> (\<lambda>w.  1 / (w - z))"
     proof -
       have "deriv f x = \<i>" for x
         unfolding f_def
         by (auto intro!:derivative_eq_intros DERIV_imp_deriv)
       then show ?thesis    
         unfolding f_def c_def 
         by (auto simp add:field_simps divide_simps intro!:arg_cong2[where f=contour_integral])
     qed
     also have "... = winding_number \<gamma> z"
       using winding_number_valid_path[OF \<open>valid_path \<gamma>\<close> \<open>z\<notin>path_image \<gamma>\<close>,folded c_def] by simp
     finally show ?thesis .
   qed    
   moreover have "pathfinish \<gamma>' = z+ \<i>* pathfinish \<gamma> - \<i>*z" "pathstart \<gamma>' = z+ \<i>*pathstart \<gamma> - \<i>*z"  
     unfolding \<gamma>'_def path_defs by (auto simp add:algebra_simps)
   ultimately show ?thesis by auto
 qed    
     
       
 lemma Re_winding_number_half_left:
   assumes neg_img:"\<forall>p\<in>path_image \<gamma>. Re p \<le> Re z" and "valid_path \<gamma>" and "z\<notin>path_image \<gamma>"
   shows "Re(winding_number \<gamma> z) = (Im (Ln (z - pathfinish \<gamma>)) - Im (Ln (z - pathstart \<gamma> )))/(2*pi)"
 proof -
   define \<gamma>' where "\<gamma>'\<equiv>(\<lambda>t. 2*z - \<gamma> t)"
   have "Re (winding_number \<gamma>' z) = (Im (Ln (pathfinish \<gamma>' - z)) - Im (Ln (pathstart \<gamma>' - z))) / (2 * pi)"
     unfolding \<gamma>'_def
     apply (rule Re_winding_number_half_right)
     subgoal using neg_img unfolding path_image_def by auto
     subgoal 
       apply (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close>,of "\<lambda>t. 2*z-t" UNIV,
             unfolded comp_def])
       by (auto intro:holomorphic_intros)
     subgoal using \<open>z\<notin>path_image \<gamma>\<close> unfolding path_image_def by auto
     done
   moreover have "winding_number \<gamma>' z = winding_number \<gamma> z"
   proof -
     define f where "f=(\<lambda>t. 2*z-t)"
     define c where "c= 1 / (complex_of_real (2 * pi) * \<i>)"
     have "winding_number \<gamma>' z = winding_number (f o \<gamma>) z" 
       unfolding \<gamma>'_def comp_def f_def by auto
     also have "... = c * contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))" unfolding c_def
     proof (rule winding_number_comp[of UNIV])
       show "z \<notin> path_image (f \<circ> \<gamma>)" using \<open>z\<notin>path_image \<gamma>\<close> unfolding f_def path_image_def by auto
     qed (auto simp add:f_def \<open>valid_path \<gamma>\<close> intro:holomorphic_intros)
     also have "... = c * contour_integral \<gamma> (\<lambda>w.  1 / (w - z))" 
       unfolding f_def c_def 
       by (auto simp add:field_simps divide_simps intro!:arg_cong2[where f=contour_integral])
     also have "... = winding_number \<gamma> z"
       using winding_number_valid_path[OF \<open>valid_path \<gamma>\<close> \<open>z\<notin>path_image \<gamma>\<close>,folded c_def] by simp
     finally show ?thesis .
   qed
   moreover have "pathfinish \<gamma>' = 2*z - pathfinish \<gamma>" "pathstart \<gamma>' = 2*z - pathstart \<gamma>"  
     unfolding \<gamma>'_def path_defs by auto 
   ultimately show ?thesis by auto    
 qed 
   
 lemma continuous_on_open_Collect_neq:
   fixes f g :: "'a::topological_space \<Rightarrow> 'b::t2_space"
   assumes f: "continuous_on S f" and g: "continuous_on S g" and "open S"
   shows "open {x\<in>S. f x \<noteq> g x}"
 proof (rule topological_space_class.openI)
   fix t
   assume "t \<in> {x\<in>S. f x \<noteq> g x}"
   then obtain U0 V0 where "open U0" "open V0" "f t \<in> U0" "g t \<in> V0" "U0 \<inter> V0 = {}" "t\<in>S"
     by (auto simp add: separation_t2)
   obtain U1 where "open U1" "t \<in> U1" "\<forall>y\<in>(S \<inter> U1). f y \<in> U0"
     using f[unfolded continuous_on_topological,rule_format,OF \<open>t\<in>S\<close> \<open>open U0\<close> \<open>f t \<in>U0\<close>] by auto
   obtain V1 where "open V1" "t \<in> V1" "\<forall>y\<in>(S \<inter> V1). g y \<in> V0"
     using g[unfolded continuous_on_topological,rule_format,OF \<open>t\<in>S\<close> \<open>open V0\<close> \<open>g t \<in>V0\<close>] by auto 
   define T where "T=V1 \<inter> U1 \<inter> S"
   have "open T" unfolding T_def using \<open>open U1\<close> \<open>open V1\<close> \<open>open S\<close> by auto
   moreover have "t \<in> T" unfolding T_def using \<open>t\<in>U1\<close> \<open>t\<in>V1\<close> \<open>t\<in>S\<close> by auto
   moreover have "T \<subseteq> {x \<in> S. f x \<noteq> g x}" unfolding T_def
     using \<open>U0 \<inter> V0 = {}\<close> \<open>\<forall>y\<in>S \<inter> U1. f y \<in> U0\<close> \<open>\<forall>y\<in>S \<inter> V1. g y \<in> V0\<close> by auto
   ultimately show "\<exists>T. open T \<and> t \<in> T \<and> T \<subseteq> {x \<in> S. f x \<noteq> g x}" by auto
 qed  
     
 subsection \<open>Sign at a filter\<close>    
  
 (*Relaxation in types could be done in the future.*)
 definition has_sgnx::"(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool" 
     (infixr "has'_sgnx" 55) where
   "(f has_sgnx c) F= (eventually (\<lambda>x. sgn(f x) = c) F)"    
   
 definition sgnx_able (infixr "sgnx'_able" 55) where
   "(f sgnx_able F) = (\<exists>c. (f has_sgnx c) F)"
   
 definition sgnx where
   "sgnx f F = (SOME c. (f has_sgnx c) F)"
   
 lemma has_sgnx_eq_rhs: "(f has_sgnx x) F \<Longrightarrow> x = y \<Longrightarrow> (f has_sgnx y) F"
   by simp
 
 named_theorems sgnx_intros "introduction rules for has_sgnx"
 setup \<open>
   Global_Theory.add_thms_dynamic (@{binding sgnx_eq_intros},
     fn context =>
       Named_Theorems.get (Context.proof_of context) @{named_theorems sgnx_intros}
       |> map_filter (try (fn thm => @{thm has_sgnx_eq_rhs} OF [thm])))
 \<close>  
 
 lemma sgnx_able_sgnx:"f sgnx_able F \<Longrightarrow> (f has_sgnx (sgnx f F)) F"
   unfolding sgnx_able_def sgnx_def using someI_ex by metis    
 
 lemma has_sgnx_imp_sgnx_able[elim]:
   "(f has_sgnx c) F \<Longrightarrow> f sgnx_able F"
 unfolding sgnx_able_def by auto
 
 lemma has_sgnx_unique:
   assumes "F\<noteq>bot" "(f has_sgnx c1) F" "(f has_sgnx c2) F" 
   shows "c1=c2"
 proof (rule ccontr)
   assume "c1 \<noteq> c2 "
   have "eventually (\<lambda>x. sgn(f x) = c1 \<and> sgn(f x) = c2) F" 
     using assms  unfolding has_sgnx_def eventually_conj_iff by simp
   then have "eventually (\<lambda>_. c1 = c2) F" by (elim eventually_mono,auto)
   then have "eventually (\<lambda>_. False) F" using \<open>c1 \<noteq> c2\<close> by auto
   then show False using \<open>F \<noteq> bot\<close> eventually_False by auto
 qed
 
 lemma has_sgnx_imp_sgnx[elim]:
   "(f has_sgnx c) F \<Longrightarrow>F\<noteq>bot \<Longrightarrow> sgnx f F = c"
   using has_sgnx_unique sgnx_def by auto
 
 lemma has_sgnx_const[simp,sgnx_intros]:
   "((\<lambda>_. c) has_sgnx sgn c) F"
 by (simp add: has_sgnx_def)
 
 lemma finite_sgnx_at_left_at_right:
   assumes "finite {t. f t=0 \<and> a<t \<and> t<b}" "continuous_on ({a<..<b} - s) f" "finite s" 
       and x:"x\<in>{a<..<b}"
   shows "f sgnx_able (at_left x)" "sgnx f (at_left x)\<noteq>0"
         "f sgnx_able (at_right x)" "sgnx f (at_right x)\<noteq>0"
 proof -
   define ls where "ls \<equiv> {t. (f t=0 \<or> t\<in>s) \<and> a<t \<and>t<x }"
   define l where "l\<equiv>(if ls = {} then (a+x)/2 else (Max ls + x)/2)" 
   have "finite ls"
   proof -
     have "{t. f t=0 \<and> a<t \<and> t<x} \<subseteq> {t. f t=0 \<and> a<t \<and> t<b}" using x by auto
     then have "finite {t. f t=0 \<and> a<t \<and> t<x}" using assms(1) 
       using finite_subset by blast
     moreover have "finite {t. t\<in>s \<and> a<t \<and> t<x}" using assms(3) by auto
     moreover have "ls = {t. f t=0 \<and> a<t \<and> t<x} \<union> {t. t\<in>s \<and> a<t \<and> t<x}"
       unfolding ls_def by auto
     ultimately show ?thesis by auto
   qed
   have [simp]: "l<x" "a<l" "l<b"
   proof -
     have "l<x \<and> a<l \<and> l<b" when "ls = {}" 
       using that x unfolding l_def by auto
     moreover have "l<x \<and> a<l \<and> l<b" when "ls \<noteq>{}"
     proof -
       have "Max ls \<in> ls" using assms(1,3) that \<open>finite ls\<close>
         apply (intro linorder_class.Max_in)
         by auto
       then have "a<Max ls \<and> Max ls < x" unfolding ls_def by auto
       then show ?thesis unfolding l_def using that x by auto
     qed
     ultimately show "l<x" "a<l" "l<b" by auto
   qed  
   have noroot:"f t\<noteq>0" when t:"t\<in>{l..<x}" for t
   proof (cases "ls = {}")
     case True
     have False when "f t=0"
     proof -
       have "t>a" using t \<open>l>a\<close> by (meson atLeastLessThan_iff less_le_trans)
       then have "t\<in>ls" using that t unfolding ls_def by auto
       then show False using True by auto
     qed
     then show ?thesis by auto 
   next
     case False
     have "t>Max ls" using that False \<open>l<x\<close> unfolding l_def by auto
     have False when "f t=0" 
     proof -
       have "t>a" using t \<open>l>a\<close> by (meson atLeastLessThan_iff less_le_trans)
       then have "t\<in>ls" using that t unfolding ls_def by auto
       then have "t\<le>Max ls" using \<open>finite ls\<close> by auto
       then show False using \<open>t>Max ls\<close> by auto
     qed
     then show ?thesis by auto
   qed
   have "(f has_sgnx sgn (f l)) (at_left x)" unfolding has_sgnx_def
   proof (rule eventually_at_leftI[OF _ \<open>l<x\<close>])
     fix t assume t:"t\<in>{l<..<x}"
     then have [simp]:"t>a" "t<b" using \<open>l>a\<close> x 
       by (meson greaterThanLessThan_iff less_trans)+
     have False when "f t = 0" 
       using noroot t that by auto
     moreover have False when "f l=0" 
       using noroot t that by auto
     moreover have False when "f l>0 \<and> f t<0 \<or> f l <0 \<and> f t >0" 
     proof -
       have False when "{l..t} \<inter> s \<noteq>{}" 
       proof -
         obtain t' where t':"t'\<in>{l..t}" "t'\<in>s" 
           using \<open>{l..t} \<inter> s \<noteq> {}\<close> by blast
         then have "a<t' \<and> t'<x" 
           by (metis \<open>a < l\<close> atLeastAtMost_iff greaterThanLessThan_iff le_less less_trans t)
         then have "t'\<in>ls" unfolding ls_def using \<open>t'\<in>s\<close> by auto
         then have "t'\<le>Max ls" using \<open>finite ls\<close> by auto
         moreover have "Max ls<l" 
           using \<open>l<x\<close> \<open>t'\<in>ls\<close> \<open>finite ls\<close> unfolding l_def by (auto simp add:ls_def )
         ultimately show False using t'(1) by auto
       qed
       moreover have "{l..t} \<subseteq> {a<..<b}" 
         by (intro atMostAtLeast_subset_convex,auto)
       ultimately have "continuous_on {l..t} f" using assms(2) 
         by (elim continuous_on_subset,auto) 
       then have "\<exists>x>l. x < t \<and> f x = 0"
         apply (intro IVT_strict)
         using that t assms(2) by auto 
       then obtain t' where "l<t'" "t'<t" "f t'=0" by auto
       then have "t'\<in>{l..<x}" unfolding ls_def using t by auto
       then show False using noroot \<open>f t'=0\<close> by auto
     qed
     ultimately show "sgn (f t) = sgn (f l)" 
       by (metis le_less not_less sgn_if)
   qed
   then show "f sgnx_able (at_left x)" by auto   
   show "sgnx f (at_left x)\<noteq>0" 
     using noroot[of l,simplified] \<open>(f has_sgnx sgn (f l)) (at_left x)\<close>
     by (simp add: has_sgnx_imp_sgnx sgn_if)
 next
   define rs where "rs \<equiv> {t. (f t=0 \<or> t\<in>s) \<and> x<t \<and> t<b}"
   define r where "r\<equiv>(if rs = {} then (x+b)/2 else (Min rs + x)/2)" 
   have "finite rs" 
   proof -
     have "{t. f t=0 \<and> x<t \<and> t<b} \<subseteq> {t. f t=0 \<and> a<t \<and> t<b}" using x by auto
     then have "finite {t. f t=0 \<and> x<t \<and> t<b}" using assms(1) 
       using finite_subset by blast
     moreover have "finite {t. t\<in>s \<and> x<t \<and> t<b}" using assms(3) by auto
     moreover have "rs = {t. f t=0 \<and> x<t \<and> t<b} \<union> {t. t\<in>s \<and> x<t \<and> t<b}"
       unfolding rs_def by auto
     ultimately show ?thesis by auto
   qed  
   
   have [simp]: "r>x" "a<r" "r<b"
   proof -
     have "r>x \<and> a<r \<and> r<b" when "rs = {}" 
       using that x unfolding r_def by auto
     moreover have "r>x \<and> a<r \<and> r<b" when "rs \<noteq>{}"
     proof -
       have "Min rs \<in> rs" using assms(1,3) that \<open>finite rs\<close>
         apply (intro linorder_class.Min_in)
         by auto
       then have "x<Min rs \<and> Min rs < b" unfolding rs_def by auto
       then show ?thesis unfolding r_def using that x by auto
     qed
     ultimately show "r>x" "a<r" "r<b" by auto
   qed   
   have noroot:"f t\<noteq>0" when t:"t\<in>{x<..r}" for t
   proof (cases "rs = {}")
     case True
     have False when "f t=0"
     proof -
       have "t<b" using t \<open>r<b\<close> 
         using greaterThanAtMost_iff by fastforce 
       then have "t\<in>rs" using that t unfolding rs_def by auto
       then show False using True by auto
     qed
     then show ?thesis by auto
   next
     case False
     have "t<Min rs" using that False \<open>r>x\<close> unfolding r_def by auto
     have False when "f t=0" 
     proof -
       have "t<b" using t \<open>r<b\<close> by (metis greaterThanAtMost_iff le_less less_trans)
       then have "t\<in>rs" using that t unfolding rs_def by auto
       then have "t\<ge>Min rs" using \<open>finite rs\<close> by auto
       then show False using \<open>t<Min rs\<close> by auto
     qed
     then show ?thesis by auto
   qed
   have "(f has_sgnx sgn (f r)) (at_right x)" unfolding has_sgnx_def
   proof (rule eventually_at_rightI[OF _ \<open>r>x\<close>])
     fix t assume t:"t\<in>{x<..<r}"
     then have [simp]:"t>a" "t<b" using \<open>r<b\<close> x 
       by (meson greaterThanLessThan_iff less_trans)+
     have False when "f t = 0" 
       using noroot t that by auto
     moreover have False when "f r=0" 
       using noroot t that by auto
     moreover have False when "f r>0 \<and> f t<0 \<or> f r <0 \<and> f t >0" 
     proof -
       have False when "{t..r} \<inter> s \<noteq>{}" 
       proof -
         obtain t' where t':"t'\<in>{t..r}" "t'\<in>s" 
           using \<open>{t..r} \<inter> s \<noteq> {}\<close> by blast
         then have "x<t' \<and> t'<b" 
           by (meson \<open>r < b\<close> atLeastAtMost_iff greaterThanLessThan_iff less_le_trans not_le t)
         then have "t'\<in>rs" unfolding rs_def using t \<open>t'\<in>s\<close> by auto
         then have "t'\<ge>Min rs" using \<open>finite rs\<close> by auto
         moreover have "Min rs>r" 
           using \<open>r>x\<close> \<open>t'\<in>rs\<close> \<open>finite rs\<close> unfolding r_def by (auto simp add:rs_def )
         ultimately show False using t'(1) by auto
       qed
       moreover have "{t..r} \<subseteq> {a<..<b}"
         by (intro atMostAtLeast_subset_convex,auto)
       ultimately have "continuous_on {t..r} f" using assms(2) by (elim continuous_on_subset,auto) 
       then have "\<exists>x>t. x < r \<and> f x = 0"
         apply (intro IVT_strict)
         using that t assms(2) by auto
       then obtain t' where "t<t'" "t'<r" "f t'=0" by auto
       then have "t'\<in>{x<..r}" unfolding rs_def using t by auto
       then show False using noroot \<open>f t'=0\<close> by auto
     qed
     ultimately show "sgn (f t) = sgn (f r)" 
       by (metis le_less not_less sgn_if)
   qed
   then show "f sgnx_able (at_right x)" by auto   
   show "sgnx f (at_right x)\<noteq>0" 
     using noroot[of r,simplified] \<open>(f has_sgnx sgn (f r)) (at_right x)\<close>
     by (simp add: has_sgnx_imp_sgnx sgn_if)    
 qed
 
 lemma sgnx_able_poly[simp]:
   "(poly p) sgnx_able (at_right a)"
   "(poly p) sgnx_able (at_left a)"
   "(poly p) sgnx_able at_top"
   "(poly p) sgnx_able at_bot"
 proof -
   show "(poly p) sgnx_able at_top"
     using has_sgnx_def poly_sgn_eventually_at_top sgnx_able_def by blast
   show "(poly p) sgnx_able at_bot"
     using has_sgnx_def poly_sgn_eventually_at_bot sgnx_able_def by blast
   show "(poly p) sgnx_able (at_right a)"
   proof (cases "p=0")
     case True
     then show ?thesis unfolding sgnx_able_def has_sgnx_def eventually_at_right
       using linordered_field_no_ub by force
   next
     case False
     obtain ub where "ub>a" and ub:"\<forall>z. a<z\<and>z\<le>ub\<longrightarrow>poly p z\<noteq>0" 
       using next_non_root_interval[OF False] by auto 
     have "\<forall>z. a<z\<and>z\<le>ub\<longrightarrow>sgn(poly p z) = sgn (poly p ub)"
     proof (rule ccontr)
       assume "\<not> (\<forall>z. a < z \<and> z \<le> ub \<longrightarrow> sgn (poly p z) = sgn (poly p ub))"
       then obtain z where "a<z" "z\<le>ub" "sgn(poly p z) \<noteq> sgn (poly p ub)" by auto
       moreover then have "poly p z\<noteq>0" "poly p ub\<noteq>0" "z\<noteq>ub" using ub \<open>ub>a\<close> by blast+
       ultimately have "(poly p z>0 \<and> poly p ub<0) \<or> (poly p z<0 \<and> poly p ub>0)"
         by (metis linorder_neqE_linordered_idom sgn_neg sgn_pos)
       then have "\<exists>x>z. x < ub \<and> poly p x = 0" 
         using poly_IVT_neg[of z ub p] poly_IVT_pos[of z ub p] \<open>z\<le>ub\<close> \<open>z\<noteq>ub\<close> by argo  
       then show False using ub \<open>a < z\<close> by auto  
     qed
     then show ?thesis unfolding sgnx_able_def has_sgnx_def eventually_at_right
       apply (rule_tac exI[where x="sgn(poly p ub)"])
       apply (rule_tac exI[where x="ub"])
       using less_eq_real_def \<open>ub>a\<close> by blast
   qed
   show "(poly p) sgnx_able (at_left a)"
   proof (cases "p=0")
     case True
     then show ?thesis unfolding sgnx_able_def has_sgnx_def eventually_at_right
       using linordered_field_no_ub by force
   next
     case False
     obtain lb where "lb<a" and ub:"\<forall>z. lb\<le>z\<and>z<a\<longrightarrow>poly p z\<noteq>0" 
       using last_non_root_interval[OF False] by auto 
     have "\<forall>z. lb\<le>z\<and>z<a\<longrightarrow>sgn(poly p z) = sgn (poly p lb)"
     proof (rule ccontr)
       assume "\<not> (\<forall>z. lb\<le>z\<and>z<a \<longrightarrow> sgn (poly p z) = sgn (poly p lb))"
       then obtain z where "lb\<le>z" "z<a" "sgn(poly p z) \<noteq> sgn (poly p lb)" by auto
       moreover then have "poly p z\<noteq>0" "poly p lb\<noteq>0" "z\<noteq>lb" using ub \<open>lb<a\<close> by blast+
       ultimately have "(poly p z>0 \<and> poly p lb<0) \<or> (poly p z<0 \<and> poly p lb>0)"
         by (metis linorder_neqE_linordered_idom sgn_neg sgn_pos)
       then have "\<exists>x>lb. x < z \<and> poly p x = 0" 
         using poly_IVT_neg[of lb z p] poly_IVT_pos[of lb z p] \<open>lb\<le>z\<close> \<open>z\<noteq>lb\<close> by argo  
       then show False using ub \<open>z < a\<close> by auto  
     qed
     then show ?thesis unfolding sgnx_able_def has_sgnx_def eventually_at_left
       apply (rule_tac exI[where x="sgn(poly p lb)"])
       apply (rule_tac exI[where x="lb"])
       using less_eq_real_def \<open>lb<a\<close> by blast
   qed
 qed
 
 lemma has_sgnx_identity[intro,sgnx_intros]:
   shows "x\<ge>0 \<Longrightarrow>((\<lambda>x. x) has_sgnx 1) (at_right x)" 
         "x\<le>0 \<Longrightarrow> ((\<lambda>x. x) has_sgnx -1) (at_left x)"  
 proof -
   show "x\<ge>0 \<Longrightarrow> ((\<lambda>x. x) has_sgnx 1) (at_right x)"
     unfolding has_sgnx_def eventually_at_right
     apply (intro exI[where x="x+1"])
     by auto
   show "x\<le>0 \<Longrightarrow> ((\<lambda>x. x) has_sgnx -1) (at_left x)"
     unfolding has_sgnx_def eventually_at_left
     apply (intro exI[where x="x-1"])
     by auto
 qed
     
 lemma has_sgnx_divide[sgnx_intros]:
   assumes "(f has_sgnx c1) F" "(g has_sgnx c2) F"
   shows "((\<lambda>x. f x / g x) has_sgnx c1 / c2) F"
 proof -
   have "\<forall>\<^sub>F x in F. sgn (f x) = c1 \<and> sgn (g x) = c2" 
     using assms unfolding has_sgnx_def by (intro eventually_conj,auto)
   then have "\<forall>\<^sub>F x in F. sgn (f x / g x) = c1 / c2" 
     apply (elim eventually_mono)
     by (simp add: sgn_mult sgn_divide)
   then show "((\<lambda>x. f x / g x) has_sgnx c1 / c2) F" unfolding has_sgnx_def by auto
 qed
 
 lemma sgnx_able_divide[sgnx_intros]:
   assumes "f sgnx_able F" "g sgnx_able F"
   shows "(\<lambda>x. f x / g x) sgnx_able F"  
 using has_sgnx_divide by (meson assms(1) assms(2) sgnx_able_def)   
 
 lemma sgnx_divide:
   assumes "F\<noteq>bot" "f sgnx_able F" "g sgnx_able F"
   shows "sgnx (\<lambda>x. f x / g x) F =sgnx f F / sgnx g F"
 proof -
   obtain c1 c2 where c1:"(f has_sgnx c1) F" and c2:"(g has_sgnx c2) F"
     using assms unfolding sgnx_able_def by auto
   have "sgnx f F=c1" "sgnx g F=c2" using c1 c2 \<open>F\<noteq>bot\<close> by auto
   moreover have "((\<lambda>x. f x / g x) has_sgnx c1 / c2) F" 
     using has_sgnx_divide[OF c1 c2] .
   ultimately show ?thesis using assms(1) has_sgnx_imp_sgnx by blast
 qed
   
 lemma has_sgnx_times[sgnx_intros]:
   assumes "(f has_sgnx c1) F" "(g has_sgnx c2) F"
   shows "((\<lambda>x. f x* g x) has_sgnx c1 * c2) F"
 proof -
   have "\<forall>\<^sub>F x in F. sgn (f x) = c1 \<and> sgn (g x) = c2" 
     using assms unfolding has_sgnx_def by (intro eventually_conj,auto)
   then have "\<forall>\<^sub>F x in F. sgn (f x * g x) = c1 * c2" 
     apply (elim eventually_mono)
     by (simp add: sgn_mult)
   then show "((\<lambda>x. f x* g x) has_sgnx c1 * c2) F" unfolding has_sgnx_def by auto
 qed
 
 lemma sgnx_able_times[sgnx_intros]:
   assumes "f sgnx_able F" "g sgnx_able F"
   shows "(\<lambda>x. f x * g x) sgnx_able F"  
 using has_sgnx_times by (meson assms(1) assms(2) sgnx_able_def)   
 
 lemma sgnx_times:
   assumes "F\<noteq>bot" "f sgnx_able F" "g sgnx_able F"
   shows "sgnx (\<lambda>x. f x * g x) F =sgnx f F * sgnx g F"
 proof -
   obtain c1 c2 where c1:"(f has_sgnx c1) F" and c2:"(g has_sgnx c2) F"
     using assms unfolding sgnx_able_def by auto
   have "sgnx f F=c1" "sgnx g F=c2" using c1 c2 \<open>F\<noteq>bot\<close> by auto
   moreover have "((\<lambda>x. f x* g x) has_sgnx c1 * c2) F" 
     using has_sgnx_times[OF c1 c2] .
   ultimately show ?thesis using assms(1) has_sgnx_imp_sgnx by blast
 qed
 
 lemma tendsto_nonzero_has_sgnx:
   assumes "(f \<longlongrightarrow> c) F" "c\<noteq>0"
   shows "(f has_sgnx sgn c) F"
 proof (cases rule:linorder_cases[of c 0])
   case less
   then have "\<forall>\<^sub>F x in F. f x<0"
     using order_topology_class.order_tendstoD[OF assms(1),of 0] by auto
   then show ?thesis 
     unfolding has_sgnx_def 
     apply (elim eventually_mono)
     using less by auto
 next
   case equal
   then show ?thesis using \<open>c\<noteq>0\<close> by auto
 next
   case greater
   then have "\<forall>\<^sub>F x in F. f x>0"
     using order_topology_class.order_tendstoD[OF assms(1),of 0] by auto
   then show ?thesis 
     unfolding has_sgnx_def 
     apply (elim eventually_mono)
     using greater by auto
 qed
 
 lemma tendsto_nonzero_sgnx:
   assumes "(f \<longlongrightarrow> c) F" "F\<noteq>bot" "c\<noteq>0"
   shows "sgnx f F = sgn c"
   using tendsto_nonzero_has_sgnx
 by (simp add: assms has_sgnx_imp_sgnx)
 
 
 lemma filterlim_divide_at_bot_at_top_iff:
   assumes "(f \<longlongrightarrow> c) F" "c\<noteq>0" 
   shows 
     "(LIM x F. f x / g x :> at_bot) \<longleftrightarrow> (g \<longlongrightarrow> 0) F 
       \<and> ((\<lambda>x. g x) has_sgnx - sgn c) F"
     "(LIM x F. f x / g x :> at_top) \<longleftrightarrow> (g \<longlongrightarrow> 0) F 
       \<and> ((\<lambda>x. g x) has_sgnx sgn c) F"
 proof -
   show "(LIM x F. f x / g x :> at_bot) \<longleftrightarrow> ((g \<longlongrightarrow> 0) F )  
     \<and> ((\<lambda>x. g x) has_sgnx - sgn c) F"
   proof 
     assume asm:"LIM x F. f x / g x :> at_bot"
     then have "filterlim g (at 0) F" 
       using filterlim_at_infinity_divide_iff[OF assms(1,2),of g] 
       at_bot_le_at_infinity filterlim_mono by blast
     then have "(g \<longlongrightarrow> 0) F" using filterlim_at by blast
     moreover have "(g has_sgnx - sgn c) F"
     proof -
       have "((\<lambda>x. sgn c * inverse (f x)) \<longlongrightarrow> sgn c * inverse c) F"
         using assms(1,2) by (auto intro:tendsto_intros)
       then have "LIM x F. sgn c * inverse (f x) * (f x / g x) :> at_bot"
         apply (elim filterlim_tendsto_pos_mult_at_bot[OF _ _ asm])
         using \<open>c\<noteq>0\<close> sgn_real_def by auto
       then have "LIM x F. sgn c / g x :> at_bot"
         apply (elim filterlim_mono_eventually)
         using eventually_times_inverse_1[OF assms] by (auto elim:eventually_mono)
       then have "\<forall>\<^sub>F x in F. sgn c / g x < 0" 
         using filterlim_at_bot_dense[of "\<lambda>x. sgn c/g x" F] by auto
       then show ?thesis unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (metis add.inverse_inverse divide_less_0_iff sgn_neg sgn_pos sgn_sgn)
     qed  
     ultimately show "(g \<longlongrightarrow> 0) F \<and> (g has_sgnx - sgn c) F" by auto
   next
     assume "(g \<longlongrightarrow> 0) F \<and> (g has_sgnx - sgn c) F"
     then have asm:"(g \<longlongrightarrow> 0) F" "(g has_sgnx - sgn c) F" by auto
     have "LIM x F. inverse (g x * sgn c) :> at_bot"
     proof (rule filterlim_inverse_at_bot)
       show "((\<lambda>x. g x * sgn c) \<longlongrightarrow> 0) F"  
         apply (rule tendsto_mult_left_zero)
         using asm(1) by blast
     next
       show "\<forall>\<^sub>F x in F. g x * sgn c < 0" using asm(2) unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (metis add.inverse_inverse assms(2) linorder_neqE_linordered_idom mult_less_0_iff 
             neg_0_less_iff_less sgn_greater sgn_zero_iff)
     qed
     moreover have "((\<lambda>x. f x * sgn c) \<longlongrightarrow> c * sgn c) F"
       using \<open>(f \<longlongrightarrow> c) F\<close> \<open>c\<noteq>0\<close>
       apply (intro tendsto_intros)
       by (auto simp add:sgn_zero_iff)
     moreover have "c * sgn c >0" using \<open>c\<noteq>0\<close> by (simp add: sgn_real_def)    
     ultimately have "LIM x F. (f x * sgn c) *inverse (g x * sgn c) :> at_bot"
       using filterlim_tendsto_pos_mult_at_bot by blast
     then show "LIM x F. f x / g x :> at_bot" 
       using \<open>c\<noteq>0\<close> by (auto simp add:field_simps sgn_zero_iff)
   qed
   show "(LIM x F. f x / g x :> at_top) \<longleftrightarrow> ((g \<longlongrightarrow> 0) F )  
     \<and> ((\<lambda>x. g x) has_sgnx sgn c) F"
   proof 
     assume asm:"LIM x F. f x / g x :> at_top"
     then have "filterlim g (at 0) F" 
       using filterlim_at_infinity_divide_iff[OF assms(1,2),of g] 
       at_top_le_at_infinity filterlim_mono by blast
     then have "(g \<longlongrightarrow> 0) F" using filterlim_at by blast
     moreover have "(g has_sgnx sgn c) F"
     proof -
       have "((\<lambda>x. sgn c * inverse (f x)) \<longlongrightarrow> sgn c * inverse c) F"
         using assms(1,2) by (auto intro:tendsto_intros)
       then have "LIM x F. sgn c * inverse (f x) * (f x / g x) :> at_top"
         apply (elim filterlim_tendsto_pos_mult_at_top[OF _ _ asm])
         using \<open>c\<noteq>0\<close> sgn_real_def by auto
       then have "LIM x F. sgn c / g x :> at_top"
         apply (elim filterlim_mono_eventually)
         using eventually_times_inverse_1[OF assms] by (auto elim:eventually_mono)
       then have "\<forall>\<^sub>F x in F. sgn c / g x > 0" 
         using filterlim_at_top_dense[of "\<lambda>x. sgn c/g x" F] by auto
       then show ?thesis unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (metis sgn_greater sgn_less sgn_neg sgn_pos zero_less_divide_iff)
     qed  
     ultimately show "(g \<longlongrightarrow> 0) F \<and> (g has_sgnx sgn c) F" by auto
   next
     assume "(g \<longlongrightarrow> 0) F \<and> (g has_sgnx sgn c) F"
     then have asm:"(g \<longlongrightarrow> 0) F" "(g has_sgnx sgn c) F" by auto
     have "LIM x F. inverse (g x * sgn c) :> at_top"
     proof (rule filterlim_inverse_at_top)
       show "((\<lambda>x. g x * sgn c) \<longlongrightarrow> 0) F"  
         apply (rule tendsto_mult_left_zero)
         using asm(1) by blast
     next
       show "\<forall>\<^sub>F x in F. g x * sgn c > 0" using asm(2) unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (metis assms(2) sgn_1_neg sgn_greater sgn_if zero_less_mult_iff)
     qed
     moreover have "((\<lambda>x. f x * sgn c) \<longlongrightarrow> c * sgn c) F"
       using \<open>(f \<longlongrightarrow> c) F\<close> \<open>c\<noteq>0\<close>
       apply (intro tendsto_intros)
       by (auto simp add:sgn_zero_iff)
     moreover have "c * sgn c >0" using \<open>c\<noteq>0\<close> by (simp add: sgn_real_def)    
     ultimately have "LIM x F. (f x * sgn c) *inverse (g x * sgn c) :> at_top"
       using filterlim_tendsto_pos_mult_at_top by blast
     then show "LIM x F. f x / g x :> at_top" 
       using \<open>c\<noteq>0\<close> by (auto simp add:field_simps sgn_zero_iff)
   qed    
 qed 
 
 
 lemma poly_sgnx_left_right:
   fixes c a::real and p::"real poly"
   assumes "p\<noteq>0"
   shows "sgnx (poly p) (at_left a) = (if even (order a p) 
             then sgnx (poly p) (at_right a)
             else -sgnx (poly p) (at_right a))"
   using assms
 proof (induction "degree p" arbitrary: p rule: less_induct)
   case less
   have ?case when "poly p a\<noteq>0" 
   proof -
     have "sgnx (poly p) (at_left a) = sgn (poly p a)" 
       by (simp add: has_sgnx_imp_sgnx tendsto_nonzero_has_sgnx that)
     moreover have "sgnx (poly p) (at_right a) = sgn (poly p a)"
       by (simp add: has_sgnx_imp_sgnx tendsto_nonzero_has_sgnx that)
     moreover have "order a p = 0" using that by (simp add: order_0I)
     ultimately show ?thesis by auto
   qed
   moreover have ?case when "poly p a=0"
   proof -
     obtain q where pq:"p= [:-a,1:] * q" 
       using \<open>poly p a=0\<close> by (meson dvdE poly_eq_0_iff_dvd)
     then have "q\<noteq>0" using \<open>p\<noteq>0\<close> by auto
     then have "degree q < degree p" unfolding pq by (subst degree_mult_eq,auto)
     have "sgnx (poly p) (at_left a) = - sgnx (poly q) (at_left a)" 
     proof -
       have "sgnx (\<lambda>x. poly p x) (at_left a) 
           = sgnx (poly q) (at_left a) * sgnx (poly [:-a,1:]) (at_left a)"
         unfolding pq
         apply (subst poly_mult)
         apply (subst sgnx_times)
         by auto
       moreover have "sgnx (\<lambda>x. poly [:-a,1:] x) (at_left a) = -1"
         apply (intro has_sgnx_imp_sgnx)
         unfolding has_sgnx_def eventually_at_left
         by (auto simp add: linordered_field_no_lb)
       ultimately show ?thesis by auto
     qed
     moreover have "sgnx (poly p) (at_right a) = sgnx (poly q) (at_right a)" 
     proof -
       have "sgnx (\<lambda>x. poly p x) (at_right a) 
           = sgnx (poly q) (at_right a) * sgnx (poly [:-a,1:]) (at_right a)"
         unfolding pq
         apply (subst poly_mult)
         apply (subst sgnx_times)
         by auto
       moreover have "sgnx (\<lambda>x. poly [:-a,1:] x) (at_right a) = 1"
         apply (intro has_sgnx_imp_sgnx)
         unfolding has_sgnx_def eventually_at_right
         by (auto simp add: linordered_field_no_ub)
       ultimately show ?thesis by auto
     qed
     moreover have "even (order a p) \<longleftrightarrow> odd (order a q)"
       unfolding pq 
       apply (subst order_mult[OF \<open>p \<noteq> 0\<close>[unfolded pq]])
       using  \<open>q\<noteq>0\<close> by (auto simp add:order_power_n_n[of _ 1, simplified])
     moreover note less.hyps[OF \<open>degree q < degree p\<close> \<open>q\<noteq>0\<close>]
     ultimately show ?thesis by auto
   qed
   ultimately show ?case by blast
 qed
 
 lemma poly_has_sgnx_left_right:
   fixes c a::real and p::"real poly"
   assumes "p\<noteq>0"
   shows "(poly p has_sgnx c) (at_left a) \<longleftrightarrow> (if even (order a p) 
             then (poly p has_sgnx c) (at_right a)
             else (poly p has_sgnx -c) (at_right a))"
 using poly_sgnx_left_right 
 by (metis (no_types, hide_lams) add.inverse_inverse assms has_sgnx_unique 
      sgnx_able_poly sgnx_able_sgnx trivial_limit_at_left_real trivial_limit_at_right_real)    
     
     
 lemma sign_r_pos_sgnx_iff:
   "sign_r_pos p a \<longleftrightarrow> sgnx (poly p) (at_right a) > 0"
 proof 
   assume asm:"0 < sgnx (poly p) (at_right a)"
   obtain c where c_def:"(poly p has_sgnx c) (at_right a)"
     using sgnx_able_poly(1) sgnx_able_sgnx by blast
   then have "c>0" using asm 
     using has_sgnx_imp_sgnx trivial_limit_at_right_real by blast
   then show "sign_r_pos p a" using c_def unfolding sign_r_pos_def has_sgnx_def
     apply (elim eventually_mono)
     by force
 next
   assume asm:"sign_r_pos p a"
   define c where "c = sgnx (poly p) (at_right a)"
   then have "(poly p has_sgnx c) (at_right a)" 
     by (simp add: sgnx_able_sgnx)
   then have "(\<forall>\<^sub>F x in (at_right a). poly p x>0 \<and> sgn (poly p x) = c)"
     using asm unfolding has_sgnx_def sign_r_pos_def
     by (simp add:eventually_conj_iff)
   then have "\<forall>\<^sub>F x in (at_right a). c > 0"
     apply (elim eventually_mono)
     by fastforce
   then show "c>0" by auto
 qed
 
 lemma sgnx_values:
   assumes "f sgnx_able F" "F \<noteq> bot"
   shows "sgnx f F = -1 \<or> sgnx f F = 0 \<or> sgnx f F = 1"
 proof -
   obtain c where c_def:"(f has_sgnx c) F" 
     using assms(1) unfolding sgnx_able_def by auto
   then obtain x where "sgn(f x) = c" 
     unfolding has_sgnx_def using assms(2) eventually_happens 
     by blast
   then have "c=-1 \<or> c=0 \<or> c=1" using sgn_if by metis
   moreover have "sgnx f F = c" using c_def by (simp add: assms(2) has_sgnx_imp_sgnx)
   ultimately show ?thesis by auto
 qed
   
 lemma has_sgnx_poly_at_top:
     "(poly p has_sgnx  sgn_pos_inf p) at_top"
   using has_sgnx_def poly_sgn_eventually_at_top by blast
 
 lemma has_sgnx_poly_at_bot:
     "(poly p has_sgnx  sgn_neg_inf p) at_bot"
   using has_sgnx_def poly_sgn_eventually_at_bot by blast
     
 lemma sgnx_poly_at_top:
   "sgnx (poly p) at_top = sgn_pos_inf p"
 by (simp add: has_sgnx_def has_sgnx_imp_sgnx poly_sgn_eventually_at_top)
     
 lemma sgnx_poly_at_bot:
   "sgnx (poly p) at_bot = sgn_neg_inf p"
 by (simp add: has_sgnx_def has_sgnx_imp_sgnx poly_sgn_eventually_at_bot)  
   
 lemma poly_has_sgnx_values:
   assumes "p\<noteq>0"
   shows 
     "(poly p has_sgnx 1) (at_left a) \<or> (poly p has_sgnx - 1) (at_left a)"
     "(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
     "(poly p has_sgnx 1) at_top \<or> (poly p has_sgnx - 1) at_top"
     "(poly p has_sgnx 1) at_bot \<or> (poly p has_sgnx - 1) at_bot"
 proof -
   have "sgn_pos_inf p = 1 \<or> sgn_pos_inf p = -1"
     unfolding sgn_pos_inf_def by (simp add: assms sgn_if)
   then show "(poly p has_sgnx 1) at_top \<or> (poly p has_sgnx - 1) at_top"
     using has_sgnx_poly_at_top by metis
 next
   have "sgn_neg_inf p = 1 \<or> sgn_neg_inf p = -1"
     unfolding sgn_neg_inf_def by (simp add: assms sgn_if)
   then show "(poly p has_sgnx 1) at_bot \<or> (poly p has_sgnx - 1) at_bot"
     using has_sgnx_poly_at_bot by metis
 next
   obtain c where c_def:"(poly p has_sgnx c) (at_left a)" 
     using sgnx_able_poly(2) sgnx_able_sgnx by blast
   then have "sgnx (poly p) (at_left a) = c" using assms by auto
   then have "c=-1 \<or> c=0 \<or> c=1"
     using sgnx_values sgnx_able_poly(2) trivial_limit_at_left_real by blast
   moreover have False when "c=0" 
   proof -
     have "(poly p has_sgnx 0) (at_left a)" using c_def that by auto
     then obtain lb where "lb<a" "\<forall>y. (lb<y \<and> y < a) \<longrightarrow> poly p y = 0"
       unfolding has_sgnx_def eventually_at_left sgn_if
       by (metis one_neq_zero zero_neq_neg_one)
     then have "{lb<..<a} \<subseteq> proots p" unfolding proots_within_def by auto
     then have "infinite (proots p)"
       apply (elim infinite_super)
       using \<open>lb<a\<close> by auto
     moreover have "finite (proots p)" using finite_proots[OF \<open>p\<noteq>0\<close>] by auto
     ultimately show False by auto
   qed
   ultimately have "c=-1 \<or> c=1" by auto
   then show "(poly p has_sgnx 1) (at_left a) \<or> (poly p has_sgnx - 1) (at_left a)"
     using c_def by auto
 next
   obtain c where c_def:"(poly p has_sgnx c) (at_right a)" 
     using sgnx_able_poly(1) sgnx_able_sgnx by blast
   then have "sgnx (poly p) (at_right a) = c" using assms by auto
   then have "c=-1 \<or> c=0 \<or> c=1"
     using sgnx_values sgnx_able_poly(1) trivial_limit_at_right_real by blast
   moreover have False when "c=0" 
   proof -
     have "(poly p has_sgnx 0) (at_right a)" using c_def that by auto
     then obtain ub where "ub>a" "\<forall>y. (a<y \<and> y < ub) \<longrightarrow> poly p y = 0"
       unfolding has_sgnx_def eventually_at_right sgn_if
       by (metis one_neq_zero zero_neq_neg_one)
     then have "{a<..<ub} \<subseteq> proots p" unfolding proots_within_def by auto
     then have "infinite (proots p)"
       apply (elim infinite_super)
       using \<open>ub>a\<close> by auto
     moreover have "finite (proots p)" using finite_proots[OF \<open>p\<noteq>0\<close>] by auto
     ultimately show False by auto
   qed
   ultimately have "c=-1 \<or> c=1" by auto
   then show "(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
     using c_def by auto
 qed
 
 lemma poly_sgnx_values:
   assumes "p\<noteq>0"
   shows "sgnx (poly p) (at_left a) = 1 \<or> sgnx (poly p) (at_left a) = -1"
         "sgnx (poly p) (at_right a) = 1 \<or> sgnx (poly p) (at_right a) = -1" 
   using poly_has_sgnx_values[OF \<open>p\<noteq>0\<close>] has_sgnx_imp_sgnx trivial_limit_at_left_real 
     trivial_limit_at_right_real by blast+
  
 lemma has_sgnx_inverse: "(f has_sgnx c) F \<longleftrightarrow> ((inverse o f) has_sgnx (inverse c)) F"
   unfolding has_sgnx_def comp_def
   apply (rule eventually_subst)  
   apply (rule always_eventually)
   by (metis inverse_inverse_eq sgn_inverse)
 
 
 lemma has_sgnx_derivative_at_left:
   assumes g_deriv:"(g has_field_derivative c) (at x)" and "g x=0" and "c\<noteq>0" 
   shows "(g has_sgnx - sgn c) (at_left x)" 
 proof -
   have "(g has_sgnx -1) (at_left x)" when "c>0"
   proof -
     obtain d1 where "d1>0" and d1_def:"\<forall>h>0. h < d1 \<longrightarrow> g (x - h) < g x"
       using DERIV_pos_inc_left[OF g_deriv \<open>c>0\<close>] \<open>g x=0\<close> by auto
     have "(g has_sgnx -1) (at_left x)" 
       unfolding has_sgnx_def eventually_at_left
       apply (intro exI[where x="x-d1"])
       using \<open>d1>0\<close> d1_def 
       by (metis (no_types, hide_lams) add.commute add_uminus_conv_diff assms(2) diff_add_cancel 
           diff_strict_left_mono diff_zero minus_diff_eq sgn_neg)  
     thus ?thesis by auto 
   qed
   moreover have "(g has_sgnx 1) (at_left x)" when "c<0"
   proof -
     obtain d1 where "d1>0" and d1_def:"\<forall>h>0. h < d1 \<longrightarrow> g (x - h) > g x"
         using DERIV_neg_dec_left[OF g_deriv \<open>c<0\<close>] \<open>g x=0\<close> by auto
     have "(g has_sgnx 1) (at_left x)" 
         unfolding has_sgnx_def eventually_at_left
         apply (intro exI[where x="x-d1"])
         using \<open>d1>0\<close> d1_def 
         by (metis (no_types, hide_lams) add.commute add_uminus_conv_diff assms(2) diff_add_cancel
             diff_zero less_diff_eq minus_diff_eq sgn_pos)
     thus ?thesis using \<open>c<0\<close>  by auto
   qed
   ultimately show ?thesis using \<open>c\<noteq>0\<close> using sgn_real_def by auto
 qed
 
 lemma has_sgnx_derivative_at_right:
   assumes g_deriv:"(g has_field_derivative c) (at x)" and "g x=0" and "c\<noteq>0"
   shows "(g has_sgnx sgn c) (at_right x)"  
 proof -
   have "(g has_sgnx 1) (at_right x)" when "c>0"
   proof -
     obtain d2 where "d2>0" and d2_def:"\<forall>h>0. h < d2 \<longrightarrow> g x < g (x + h)"
         using DERIV_pos_inc_right[OF g_deriv \<open>c>0\<close>] \<open>g x=0\<close> by auto 
     have "(g has_sgnx 1) (at_right x)" 
       unfolding has_sgnx_def eventually_at_right
       apply (intro exI[where x="x+d2"])
       using \<open>d2>0\<close> d2_def 
       by (metis add.commute assms(2) diff_add_cancel diff_less_eq less_add_same_cancel1 sgn_pos)
     thus ?thesis using \<open>c>0\<close> by auto
   qed
   moreover have "(g has_sgnx -1) (at_right x)" when "c<0"
   proof -
     obtain d2 where "d2>0" and d2_def:"\<forall>h>0. h < d2 \<longrightarrow> g x > g (x + h)"
       using DERIV_neg_dec_right[OF g_deriv \<open>c<0\<close>] \<open>g x=0\<close> by auto 
     have "(g has_sgnx -1) (at_right x)" 
       unfolding has_sgnx_def eventually_at_right
       apply (intro exI[where x="x+d2"])
       using \<open>d2>0\<close> d2_def 
       by (metis (no_types, hide_lams) add.commute add.right_inverse add_uminus_conv_diff assms(2) 
           diff_add_cancel diff_less_eq sgn_neg)
     thus ?thesis using \<open>c<0\<close> by auto
   qed
   ultimately show ?thesis using \<open>c\<noteq>0\<close> using sgn_real_def by auto
 qed  
 
 lemma has_sgnx_split:
   "(f has_sgnx c) (at x) \<longleftrightarrow> (f has_sgnx c) (at_left x) \<and> (f has_sgnx c) (at_right x)"
 unfolding has_sgnx_def using eventually_at_split by auto    
         
 lemma sgnx_at_top_IVT:
   assumes "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) at_top"
   shows "\<exists>x>a. poly p x=0"
 proof (cases "p=0")
   case True
   then show ?thesis using gt_ex[of a] by simp
 next
   case False
   from poly_has_sgnx_values[OF this] 
   have "(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
     "(poly p has_sgnx 1) at_top \<or> (poly p has_sgnx - 1) at_top"
     by auto
   moreover have ?thesis when has_r:"(poly p has_sgnx 1) (at_right a)" 
       and has_top:"(poly p has_sgnx -1) at_top"
   proof -
     obtain b where "b>a" "poly p b>0"
     proof -
       obtain a' where "a'>a" and a'_def:"\<forall>y>a. y < a' \<longrightarrow> sgn (poly p y) = 1"
         using has_r[unfolded has_sgnx_def eventually_at_right] by auto
       define b where "b=(a+a')/2"
       have "a<b" "b<a'" unfolding b_def using \<open>a'>a\<close> by auto
       moreover have "poly p b>0"
         using a'_def[rule_format,OF \<open>b>a\<close> \<open>b<a'\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain c where "c>b" "poly p c<0"
     proof -
       obtain b' where b'_def:"\<forall>n\<ge>b'. sgn (poly p n) = - 1"
         using has_top[unfolded has_sgnx_def eventually_at_top_linorder] by auto
       define c where "c=1+max b b'"
       have "c>b" "c\<ge>b'" unfolding c_def using \<open>b>a\<close> by auto
       moreover have "poly p c<0"
         using b'_def[rule_format,OF \<open>b'\<le>c\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately show ?thesis using poly_IVT_neg[of b c p] not_less by fastforce
   qed
   moreover have ?thesis when has_r:"(poly p has_sgnx -1) (at_right a)" 
       and has_top:"(poly p has_sgnx 1) at_top"
   proof -
     obtain b where "b>a" "poly p b<0"
     proof -
       obtain a' where "a'>a" and a'_def:"\<forall>y>a. y < a' \<longrightarrow> sgn (poly p y) = -1"
         using has_r[unfolded has_sgnx_def eventually_at_right] by auto
       define b where "b=(a+a')/2"
       have "a<b" "b<a'" unfolding b_def using \<open>a'>a\<close> by auto
       moreover have "poly p b<0"
         using a'_def[rule_format,OF \<open>b>a\<close> \<open>b<a'\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain c where "c>b" "poly p c>0"
     proof -
       obtain b' where b'_def:"\<forall>n\<ge>b'. sgn (poly p n) = 1"
         using has_top[unfolded has_sgnx_def eventually_at_top_linorder] by auto
       define c where "c=1+max b b'"
       have "c>b" "c\<ge>b'" unfolding c_def using \<open>b>a\<close> by auto
       moreover have "poly p c>0"
         using b'_def[rule_format,OF \<open>b'\<le>c\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately show ?thesis using poly_IVT_pos[of b c p] not_less by fastforce
   qed
   moreover have ?thesis when 
     "(poly p has_sgnx 1) (at_right a) \<and> (poly p has_sgnx 1) at_top
      \<or> (poly p has_sgnx - 1) (at_right a) \<and> (poly p has_sgnx -1) at_top" 
   proof -
     have "sgnx (poly p) (at_right a) = sgnx (poly p) at_top"
       using that has_sgnx_imp_sgnx by auto
     then have False using assms by simp
     then show ?thesis by auto
   qed
   ultimately show ?thesis by blast
 qed
   
 lemma sgnx_at_left_at_right_IVT:
   assumes "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) (at_left b)" "a<b"
   shows "\<exists>x. a<x \<and> x<b \<and> poly p x=0"
 proof (cases "p=0")
   case True
   then show ?thesis using \<open>a<b\<close> by (auto intro:exI[where x="(a+b)/2"])
 next
   case False
   from poly_has_sgnx_values[OF this]
   have "(poly p has_sgnx 1) (at_right a) \<or> (poly p has_sgnx - 1) (at_right a)"
     "(poly p has_sgnx 1) (at_left b) \<or> (poly p has_sgnx - 1) (at_left b)"
     by auto
   moreover have ?thesis when has_r:"(poly p has_sgnx 1) (at_right a)" 
       and has_l:"(poly p has_sgnx -1) (at_left b)"
   proof -
     obtain c where "a<c" "c<b" "poly p c>0"
     proof -
       obtain a' where "a'>a" and a'_def:"\<forall>y>a. y < a' \<longrightarrow> sgn (poly p y) = 1"
         using has_r[unfolded has_sgnx_def eventually_at_right] by auto
       define c where "c=(a+min a' b)/2"
       have "a<c" "c<a'" "c<b" unfolding c_def using \<open>a'>a\<close> \<open>b>a\<close> by auto
       moreover have "poly p c>0"
         using a'_def[rule_format,OF \<open>c>a\<close> \<open>c<a'\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain d where "c<d""d<b" "poly p d<0"
     proof -
       obtain b' where "b'<b" and b'_def:"\<forall>y>b'. y < b \<longrightarrow> sgn (poly p y) = - 1"
         using has_l[unfolded has_sgnx_def eventually_at_left] by auto
       define d where "d=(b+max b' c)/2"
       have "b'<d" "d<b" "d>c" 
         unfolding d_def using \<open>b>b'\<close> \<open>b>c\<close> by auto
       moreover have "poly p d<0"
         using b'_def[rule_format, OF \<open>b'<d\<close> \<open>d<b\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately obtain x where "c<x" "x<d" "poly p x=0"
       using poly_IVT_neg[of c d p] by auto
     then show ?thesis using \<open>c>a\<close> \<open>d<b\<close> by (auto intro: exI[where x=x])
   qed
   moreover have ?thesis when has_r:"(poly p has_sgnx -1) (at_right a)" 
       and has_l:"(poly p has_sgnx 1) (at_left b)"
   proof -
     obtain c where "a<c" "c<b" "poly p c<0"
     proof -
       obtain a' where "a'>a" and a'_def:"\<forall>y>a. y < a' \<longrightarrow> sgn (poly p y) = -1"
         using has_r[unfolded has_sgnx_def eventually_at_right] by auto
       define c where "c=(a+min a' b)/2"
       have "a<c" "c<a'" "c<b" unfolding c_def using \<open>a'>a\<close> \<open>b>a\<close> by auto
       moreover have "poly p c<0"
         using a'_def[rule_format,OF \<open>c>a\<close> \<open>c<a'\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain d where "c<d""d<b" "poly p d>0"
     proof -
       obtain b' where "b'<b" and b'_def:"\<forall>y>b'. y < b \<longrightarrow> sgn (poly p y) = 1"
         using has_l[unfolded has_sgnx_def eventually_at_left] by auto
       define d where "d=(b+max b' c)/2"
       have "b'<d" "d<b" "d>c" 
         unfolding d_def using \<open>b>b'\<close> \<open>b>c\<close> by auto
       moreover have "poly p d>0"
         using b'_def[rule_format, OF \<open>b'<d\<close> \<open>d<b\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately obtain x where "c<x" "x<d" "poly p x=0"
       using poly_IVT_pos[of c d p] by auto
     then show ?thesis using \<open>c>a\<close> \<open>d<b\<close> by (auto intro: exI[where x=x])
   qed
   moreover have ?thesis when 
     "(poly p has_sgnx 1) (at_right a) \<and> (poly p has_sgnx 1) (at_left b)
      \<or> (poly p has_sgnx - 1) (at_right a) \<and> (poly p has_sgnx -1) (at_left b)" 
   proof -
     have "sgnx (poly p) (at_right a) = sgnx (poly p) (at_left b)"
       using that has_sgnx_imp_sgnx by auto
     then have False using assms by simp
     then show ?thesis by auto
   qed
   ultimately show ?thesis by blast
 qed
    
 lemma sgnx_at_bot_IVT:
   assumes "sgnx (poly p) (at_left a) \<noteq> sgnx (poly p) at_bot"
   shows "\<exists>x<a. poly p x=0"
 proof (cases "p=0")
   case True
   then show ?thesis using lt_ex[of a] by simp
 next
   case False
   from poly_has_sgnx_values[OF this] 
   have "(poly p has_sgnx 1) (at_left a) \<or> (poly p has_sgnx - 1) (at_left a)"
     "(poly p has_sgnx 1) at_bot \<or> (poly p has_sgnx - 1) at_bot"
     by auto
   moreover have ?thesis when has_l:"(poly p has_sgnx 1) (at_left a)" 
       and has_bot:"(poly p has_sgnx -1) at_bot"
   proof -
     obtain b where "b<a" "poly p b>0"
     proof -
       obtain a' where "a'<a" and a'_def:"\<forall>y>a'. y < a \<longrightarrow> sgn (poly p y) = 1"
         using has_l[unfolded has_sgnx_def eventually_at_left] by auto
       define b where "b=(a+a')/2"
       have "a>b" "b>a'" unfolding b_def using \<open>a'<a\<close> by auto
       moreover have "poly p b>0"
         using a'_def[rule_format,OF \<open>b>a'\<close> \<open>b<a\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain c where "c<b" "poly p c<0"
     proof -
       obtain b' where b'_def:"\<forall>n\<le>b'. sgn (poly p n) = - 1"
         using has_bot[unfolded has_sgnx_def eventually_at_bot_linorder] by auto
       define c where "c=min b b'- 1"
       have "c<b" "c\<le>b'" unfolding c_def using \<open>b<a\<close> by auto
       moreover have "poly p c<0"
         using b'_def[rule_format,OF \<open>b'\<ge>c\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately show ?thesis using poly_IVT_pos[of c b p] using not_less by fastforce
   qed
   moreover have ?thesis when has_l:"(poly p has_sgnx -1) (at_left a)" 
       and has_bot:"(poly p has_sgnx 1) at_bot"
   proof -
     obtain b where "b<a" "poly p b<0"
     proof -
       obtain a' where "a'<a" and a'_def:"\<forall>y>a'. y < a \<longrightarrow> sgn (poly p y) = -1"
         using has_l[unfolded has_sgnx_def eventually_at_left] by auto
       define b where "b=(a+a')/2"
       have "a>b" "b>a'" unfolding b_def using \<open>a'<a\<close> by auto
       moreover have "poly p b<0"
         using a'_def[rule_format,OF \<open>b>a'\<close> \<open>b<a\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     moreover obtain c where "c<b" "poly p c>0"
     proof -
       obtain b' where b'_def:"\<forall>n\<le>b'. sgn (poly p n) = 1"
         using has_bot[unfolded has_sgnx_def eventually_at_bot_linorder] by auto
       define c where "c=min b b'- 1"
       have "c<b" "c\<le>b'" unfolding c_def using \<open>b<a\<close> by auto
       moreover have "poly p c>0"
         using b'_def[rule_format,OF \<open>b'\<ge>c\<close>] unfolding sgn_if by argo
       ultimately show ?thesis using that by auto
     qed
     ultimately show ?thesis using poly_IVT_neg[of c b p] using not_less by fastforce
   qed
   moreover have ?thesis when 
     "(poly p has_sgnx 1) (at_left a) \<and> (poly p has_sgnx 1) at_bot
      \<or> (poly p has_sgnx - 1) (at_left a) \<and> (poly p has_sgnx -1) at_bot" 
   proof -
     have "sgnx (poly p) (at_left a) = sgnx (poly p) at_bot"
       using that has_sgnx_imp_sgnx by auto
     then have False using assms by simp
     then show ?thesis by auto
   qed
   ultimately show ?thesis by blast
 qed    
 
 lemma sgnx_poly_nz:
   assumes "poly p x\<noteq>0"
   shows "sgnx (poly p) (at_left x) = sgn (poly p x)"
         "sgnx (poly p) (at_right x) = sgn (poly p x)"
 proof -
   have "(poly p has_sgnx sgn(poly p x)) (at x)"
     apply (rule tendsto_nonzero_has_sgnx)
     using assms by auto
   then show "sgnx (poly p) (at_left x) = sgn (poly p x)"
         "sgnx (poly p) (at_right x) = sgn (poly p x)"
     unfolding has_sgnx_split by auto
 qed
   
 subsection \<open>Finite predicate segments over an interval\<close>
 
 inductive finite_Psegments::"(real \<Rightarrow> bool) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool" for P where
   emptyI: "a\<ge>b \<Longrightarrow> finite_Psegments P a b"|
   insertI_1: "\<lbrakk>s\<in>{a..<b};s=a\<or>P s;\<forall>t\<in>{s<..<b}. P t; finite_Psegments P a s\<rbrakk> 
         \<Longrightarrow> finite_Psegments P a b"|
   insertI_2: "\<lbrakk>s\<in>{a..<b};s=a\<or>P s;(\<forall>t\<in>{s<..<b}. \<not>P t);finite_Psegments P a s\<rbrakk> 
         \<Longrightarrow> finite_Psegments P a b" 
 
 lemma finite_Psegments_pos_linear:
   assumes "finite_Psegments P (b*lb+c) (b*ub+c) " and "b>0"
   shows "finite_Psegments (P o (\<lambda>t. b*t+c)) lb ub" 
 proof -    
   have [simp]:"b\<noteq>0" using \<open>b>0\<close> by auto
   show ?thesis 
   proof (rule finite_Psegments.induct[OF assms(1),
         of "\<lambda>lb' ub'. finite_Psegments (P o (\<lambda>t. b*t+c)) ((lb'-c)/b) ((ub'-c)/b)",simplified])
     (*really weird application of the induction rule, is there an alternative?*)
     fix lb ub f assume "(lb::real)\<le>ub"  
     then have "(lb - c) / b \<le> (ub - c) / b" 
       using \<open>b>0\<close> by (auto simp add:field_simps)
     then show "finite_Psegments (f \<circ> (\<lambda>t. b * t + c)) ((ub - c) / b) ((lb - c) / b)"
       by (rule finite_Psegments.emptyI)
   next
     fix s lb ub P
     assume asm: "lb \<le> s \<and> s < ub" 
        "\<forall>t\<in>{s<..<ub}. P t" 
        "finite_Psegments (P \<circ> (\<lambda>t. b * t + c)) ((lb - c) / b) ((s - c) / b)"
        "s = lb \<or> P s"
     show "finite_Psegments (P \<circ> (\<lambda>t. b * t + c)) ((lb - c) / b) ((ub - c) / b)"
       apply (rule finite_Psegments.insertI_1[of "(s-c)/b"])
       using asm \<open>b>0\<close>  by (auto simp add:field_simps)
   next
     fix s lb ub P
     assume asm: "lb \<le> s \<and> s < ub" 
        "\<forall>t\<in>{s<..<ub}. \<not> P t" 
        "finite_Psegments (P \<circ> (\<lambda>t. b * t + c)) ((lb - c) / b) ((s - c) / b)" 
        "s=lb \<or> P s"
     show "finite_Psegments (P \<circ> (\<lambda>t. b * t + c)) ((lb - c) / b) ((ub - c) / b)"
       apply (rule finite_Psegments.insertI_2[of "(s-c)/b"])
       using asm \<open>b>0\<close>  by (auto simp add:field_simps)
   qed
 qed  
 
 lemma finite_Psegments_congE:
   assumes "finite_Psegments Q lb ub" 
     "\<And>t. \<lbrakk>lb<t;t<ub\<rbrakk> \<Longrightarrow> Q t \<longleftrightarrow> P t "
   shows "finite_Psegments P lb ub" using assms
 proof (induct rule:finite_Psegments.induct)
   case (emptyI a b)
   then show ?case using finite_Psegments.emptyI by auto
 next
   case (insertI_1 s a b)
   show ?case 
   proof (rule finite_Psegments.insertI_1[of s])
     have "P s" when "s\<noteq>a"
     proof -
       have "s\<in>{a<..<b}" using \<open>s \<in> {a..<b}\<close> that by auto 
       then show ?thesis using insertI_1 by auto
     qed
     then show "s = a \<or> P s" by auto
   next
     show "s \<in> {a..<b}" " \<forall>t\<in>{s<..<b}. P t" "finite_Psegments P a s" using insertI_1 by auto
   qed  
 next
   case (insertI_2 s a b)
   show ?case 
   proof (rule finite_Psegments.insertI_2[of s]) 
     have "P s" when "s\<noteq>a"
     proof -
       have "s\<in>{a<..<b}" using \<open>s \<in> {a..<b}\<close> that by auto 
       then show ?thesis using insertI_2 by auto
     qed
     then show "s = a \<or> P s" by auto
   next
     show "s \<in> {a..<b}" " \<forall>t\<in>{s<..<b}. \<not> P t" "finite_Psegments P a s" using insertI_2 by auto
   qed   
 qed
   
 lemma finite_Psegments_constI:
   assumes "\<And>t. \<lbrakk>a<t;t<b\<rbrakk> \<Longrightarrow> P t = c"
   shows "finite_Psegments P a b"
 proof -
   have "finite_Psegments (\<lambda>_. c) a b"
   proof -
     have ?thesis when "a\<ge>b"
       using that finite_Psegments.emptyI by auto
     moreover have ?thesis when "a<b" "c"
       apply (rule finite_Psegments.insertI_1[of a])
       using that by (auto intro: finite_Psegments.emptyI)
     moreover have ?thesis when "a<b" "\<not>c"
       apply (rule finite_Psegments.insertI_2[of a])
       using that by (auto intro: finite_Psegments.emptyI)
     ultimately show ?thesis by argo
   qed
   then show ?thesis
     apply (elim finite_Psegments_congE)
     using assms by auto
 qed    
 
 context 
 begin  
 
 private lemma finite_Psegments_less_eq1:
   assumes "finite_Psegments P a c" "b\<le>c"
   shows "finite_Psegments P a b" using assms
 proof (induct arbitrary: b rule:finite_Psegments.induct)
   case (emptyI a c)
   then show ?case using finite_Psegments.emptyI by auto
 next
   case (insertI_1 s a c)  
   have ?case when "b\<le>s" using insertI_1 that by auto
   moreover have ?case when "b>s" 
   proof -
     have "s \<in> {a..<b}" using that \<open>s \<in> {a..<c}\<close> \<open>b \<le> c\<close> by auto 
     moreover have "\<forall>t\<in>{s<..<b}. P t" using \<open>\<forall>t\<in>{s<..<c}. P t\<close> that \<open>b \<le> c\<close> by auto
     ultimately show ?case 
       using finite_Psegments.insertI_1[OF _ _ _ \<open>finite_Psegments P a s\<close>] \<open> s = a \<or> P s\<close> by auto
   qed
   ultimately show ?case by fastforce
 next
   case (insertI_2 s a c)
   have ?case when "b\<le>s" using insertI_2 that by auto
   moreover have ?case when "b>s" 
   proof -
     have "s \<in> {a..<b}" using that \<open>s \<in> {a..<c}\<close> \<open>b \<le> c\<close> by auto 
     moreover have "\<forall>t\<in>{s<..<b}. \<not> P t" using \<open>\<forall>t\<in>{s<..<c}. \<not> P t\<close> that \<open>b \<le> c\<close> by auto
     ultimately show ?case 
       using finite_Psegments.insertI_2[OF _ _ _ \<open>finite_Psegments P a s\<close>] \<open> s = a \<or> P s\<close> by auto
   qed
   ultimately show ?case by fastforce
 qed  
  
   
 private lemma finite_Psegments_less_eq2:
   assumes "finite_Psegments P a c" "a\<le>b"
   shows "finite_Psegments P b c" using assms  
 proof (induct arbitrary:  rule:finite_Psegments.induct)
   case (emptyI a c)
   then show ?case using finite_Psegments.emptyI by auto 
 next
   case (insertI_1 s a c)
   have ?case when "s\<le>b"
   proof -
     have "\<forall>t\<in>{b<..<c}. P t" using insertI_1 that by auto
     then show ?thesis by (simp add: finite_Psegments_constI)
   qed
   moreover have ?case when "s>b"
     apply (rule finite_Psegments.insertI_1[where s=s])
     using insertI_1 that by auto
   ultimately show ?case by linarith
 next
   case (insertI_2 s a c)
   have ?case when "s\<le>b"
   proof -
     have "\<forall>t\<in>{b<..<c}. \<not> P t" using insertI_2 that by auto
     then show ?thesis by (metis finite_Psegments_constI greaterThanLessThan_iff)
   qed
   moreover have ?case when "s>b"
     apply (rule finite_Psegments.insertI_2[where s=s])
     using insertI_2 that by auto
   ultimately show ?case by linarith
 qed  
 
 
 lemma finite_Psegments_included:
   assumes "finite_Psegments P a d" "a\<le>b" "c\<le>d"
   shows "finite_Psegments P b c"  
   using finite_Psegments_less_eq2 finite_Psegments_less_eq1 assms by blast      
 end      
 
 lemma finite_Psegments_combine:
   assumes "finite_Psegments P a b" "finite_Psegments P b c" "b\<in>{a..c}" "closed ({x. P x} \<inter> {a..c})" 
   shows "finite_Psegments P a c"  using assms(2,1,3,4)
 proof (induct rule:finite_Psegments.induct)
   case (emptyI b c)
   then show ?case using finite_Psegments_included by auto
 next
   case (insertI_1 s b c)
   have "P s" 
   proof -
     have "s<c" using insertI_1 by auto
     define S where "S = {x. P x} \<inter> {s..(s+c)/2}"
     have "closed S"
     proof -
       have "closed ({a. P a} \<inter> {a..c})" using insertI_1(8) .
       moreover have "S = ({a. P a} \<inter> {a..c}) \<inter> {s..(s+c)/2}"
         using insertI_1(1,7) unfolding S_def by (auto simp add:field_simps)
       ultimately show ?thesis
         using closed_Int[of "{a. P a} \<inter> {a..c}" "{s..(s+c)/2}"] by blast
     qed
     moreover have "\<exists>y\<in>S. dist y s < e" when "e>0" for e 
     proof -
       define y where "y = min ((s+c)/2) (e/2+s)"
       have "y\<in>S"
       proof -
         have "y\<in>{s..(s+c)/2}" unfolding y_def
           using \<open>e>0\<close> \<open>s<c\<close> by (auto simp add:min_mult_distrib_left algebra_simps)
         moreover have "P y" 
           apply (rule insertI_1(3)[rule_format])
           unfolding y_def
           using \<open>e>0\<close> \<open>s<c\<close> 
           by (auto simp add:algebra_simps min_mult_distrib_left min_less_iff_disj)
         ultimately show ?thesis unfolding S_def by auto   
       qed
       moreover have "dist y s <e"
         unfolding y_def using \<open>e>0\<close> \<open>s<c\<close> 
         by (auto simp add:algebra_simps min_mult_distrib_left min_less_iff_disj dist_real_def)
       ultimately show ?thesis by auto
     qed
     ultimately have "s\<in>S" using closed_approachable by auto
     then show ?thesis unfolding S_def by auto
   qed
   show ?case 
   proof (rule finite_Psegments.insertI_1[of s])
     show " s \<in> {a..<c}" "s = a \<or> P s" "\<forall>t\<in>{s<..<c}. P t"
       using insertI_1 \<open>P s\<close> by auto
   next
     have "closed ({a. P a} \<inter> {a..s})" 
       using closed_Int[OF \<open>closed ({a. P a} \<inter> {a..c})\<close>,of "{a..s}",simplified]
       apply (elim arg_elim[of closed])
       using \<open>s \<in> {b..<c}\<close> \<open>b \<in> {a..c}\<close> by auto
     then show "finite_Psegments P a s" using insertI_1 by auto
   qed
 next
   case (insertI_2 s b c)
   have ?case when "P s"
   proof (rule finite_Psegments.insertI_2[of s])
     show "s \<in> {a..<c}" "s = a \<or> P s" "\<forall>t\<in>{s<..<c}. \<not> P t" using that insertI_2 by auto
   next
     have "closed ({a. P a} \<inter> {a..s})" 
       using closed_Int[OF \<open>closed ({a. P a} \<inter> {a..c})\<close>,of "{a..s}",simplified]
       apply (elim arg_elim[of closed])
       using  \<open>s \<in> {b..<c}\<close> \<open>b \<in> {a..c}\<close> by auto
     then show "finite_Psegments P a s" using insertI_2 by auto
   qed  
   moreover have ?case when "\<not> P s" "s=b" using \<open>finite_Psegments P a b\<close>  
   proof (cases rule:finite_Psegments.cases)
     case emptyI
     then show ?thesis using insertI_2 that 
       by (metis antisym_conv atLeastAtMost_iff finite_Psegments.insertI_2)
   next
     case (insertI_1 s0)
     have "P s" 
     proof -
       have "s0<s" using insertI_1 atLeastLessThan_iff that(2) by blast    
       define S where "S = {x. P x} \<inter> {(s0+s)/2..s}"
       have "closed S"
         using closed_Int[OF \<open>closed ({a. P a} \<inter> {a..c})\<close>,of "{(s0+s)/2..s}",simplified]  
         apply (elim arg_elim[of closed])
         unfolding S_def using \<open>s0 \<in> {a..<b}\<close> \<open> s \<in> {b..<c}\<close> \<open>b \<in> {a..c}\<close> by auto    
       moreover have "\<exists>y\<in>S. dist y s < e" when "e>0" for e 
       proof -
         define y where "y = max ((s+s0)/2) (s-e/2)"
         have "y\<in>S"
         proof -
           have "y\<in>{(s0+s)/2..s}" unfolding y_def
             using \<open>e>0\<close> \<open>s0<s\<close> by (auto simp add:field_simps min_mult_distrib_left)
           moreover have "P y" 
             apply (rule insertI_1(3)[rule_format])
             unfolding y_def
             using \<open>e>0\<close> \<open>s0<s\<close> \<open>s=b\<close>
             by (auto simp add:field_simps max_mult_distrib_left less_max_iff_disj)
           ultimately show ?thesis unfolding S_def by auto   
         qed
         moreover have "dist y s <e"
           unfolding y_def using \<open>e>0\<close> \<open>s0<s\<close> 
           by (auto simp add:algebra_simps max_mult_distrib_left less_max_iff_disj dist_real_def 
               max_add_distrib_right)
         ultimately show ?thesis by auto
       qed
       ultimately have "s\<in>S" using closed_approachable by auto
       then show ?thesis unfolding S_def by auto
     qed  
     then have False using \<open>\<not> P s\<close> by auto
     then show ?thesis by simp
   next
     case (insertI_2 s0)
     have *: "\<forall>t\<in>{s0<..<c}. \<not> P t" 
       using \<open> \<forall>t\<in>{s<..<c}. \<not> P t\<close> that \<open>\<forall>t\<in>{s0<..<b}. \<not> P t\<close> 
       by force
     show ?thesis 
       apply (rule finite_Psegments.insertI_2[of s0])
       subgoal using insertI_2.prems(2) local.insertI_2(1) by auto   
       subgoal using \<open>s0 = a \<or> P s0\<close> .
       subgoal using * .
       subgoal using \<open>finite_Psegments P a s0\<close> .
       done
   qed
   moreover note \<open>s = b \<or> P s\<close>  
   ultimately show ?case by auto
 qed
  
 subsection \<open>Finite segment intersection of a path with the imaginary axis\<close>
 
 definition finite_ReZ_segments::"(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> bool" where
   "finite_ReZ_segments g z = finite_Psegments (\<lambda>t. Re (g t - z) = 0) 0 1"    
 
 lemma finite_ReZ_segments_joinpaths:
   assumes g1:"finite_ReZ_segments g1 z" and g2: "finite_ReZ_segments g2 z" and
     "path g1" "path g2" "pathfinish g1=pathstart g2"
   shows "finite_ReZ_segments (g1+++g2) z" 
 proof -
   define P where "P = (\<lambda>t. (Re ((g1 +++ g2) t - z) = 0 \<and> 0<t \<and> t<1) \<or> t=0 \<or> t=1)"
   have "finite_Psegments P 0 (1/2)" 
   proof -
     have "finite_Psegments (\<lambda>t. Re (g1 t - z) = 0) 0 1" 
       using g1 unfolding finite_ReZ_segments_def .
     then have "finite_Psegments (\<lambda>t. Re (g1 (2 * t) - z) = 0) 0 (1/2)" 
       apply (drule_tac finite_Psegments_pos_linear[of _ 2 0 0 "1/2",simplified])
       by (auto simp add:comp_def)
     then show ?thesis
       unfolding P_def joinpaths_def
       by (elim finite_Psegments_congE,auto)
   qed
   moreover have "finite_Psegments P (1/2) 1" 
   proof -
     have "finite_Psegments (\<lambda>t. Re (g2 t - z) = 0) 0 1" 
       using g2 unfolding finite_ReZ_segments_def .
     then have "finite_Psegments (\<lambda>t. Re (g2 (2 * t-1) - z) = 0) (1/2) 1"  
       apply (drule_tac finite_Psegments_pos_linear[of _ 2 "1/2" "-1" 1,simplified])
       by (auto simp add:comp_def)
     then show ?thesis
       unfolding P_def joinpaths_def
       apply (elim finite_Psegments_congE)
       by auto
   qed  
   moreover have "closed {x. P x}"
   proof -
     define Q where "Q=(\<lambda>t. Re ((g1 +++ g2) t - z) = 0)"
     have "continuous_on {0<..<1} (g1+++g2)"
       using path_join_imp[OF \<open>path g1\<close> \<open>path g2\<close> \<open>pathfinish g1=pathstart g2\<close>] 
       unfolding path_def by (auto elim:continuous_on_subset)
     from continuous_on_Re[OF this] have "continuous_on {0<..<1} (\<lambda>x. Re ((g1 +++ g2) x))" .
     from continuous_on_open_Collect_neq[OF this,of "\<lambda>_. Re z",OF continuous_on_const,simplified]
     have "open {t. Re ((g1 +++ g2) t - z) \<noteq> 0 \<and> 0<t \<and> t<1}" 
        by (elim arg_elim[where f="open"],auto)
     from closed_Diff[of "{0::real..1}",OF _ this,simplified]
     show "closed {x. P x}" 
       apply (elim arg_elim[where f=closed])
       by (auto simp add:P_def)
   qed
   ultimately have "finite_Psegments P 0 1"
     using finite_Psegments_combine[of _ 0 "1/2" 1] by auto
   then show ?thesis 
     unfolding finite_ReZ_segments_def P_def
     by (elim finite_Psegments_congE,auto)
 qed
       
 lemma finite_ReZ_segments_congE:
   assumes "finite_ReZ_segments p1 z1" 
     "\<And>t. \<lbrakk>0<t;t<1\<rbrakk> \<Longrightarrow>  Re(p1 t- z1) = Re(p2 t - z2)"
   shows "finite_ReZ_segments p2 z2"      
   using assms unfolding finite_ReZ_segments_def
   apply (elim finite_Psegments_congE)
   by auto 
        
 lemma finite_ReZ_segments_constI:
   assumes "\<forall>t. 0<t\<and>t<1 \<longrightarrow> g t = c"
   shows "finite_ReZ_segments g z" 
 proof -
   have "finite_ReZ_segments (\<lambda>_. c) z" 
     unfolding finite_ReZ_segments_def 
     by (rule finite_Psegments_constI,auto)
   then show ?thesis using assms
     by (elim finite_ReZ_segments_congE,auto)
 qed 
 
 lemma finite_ReZ_segment_cases [consumes 1, case_names subEq subNEq,cases pred:finite_ReZ_segments]:
   assumes "finite_ReZ_segments g z"
     and subEq:"(\<And>s. \<lbrakk>s \<in> {0..<1};s=0\<or>Re (g s) = Re z;
           \<forall>t\<in>{s<..<1}. Re (g t) = Re z;finite_ReZ_segments (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P)" 
     and subNEq:"(\<And>s. \<lbrakk>s \<in> {0..<1};s=0\<or>Re (g s) = Re z;
           \<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z;finite_ReZ_segments (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P)"
   shows "P"  
 using assms(1) unfolding finite_ReZ_segments_def
 proof (cases rule:finite_Psegments.cases)
   case emptyI
   then show ?thesis by auto
 next
   case (insertI_1 s)
   have "finite_ReZ_segments (subpath 0 s g) z" 
   proof (cases "s=0")
     case True
     show ?thesis
       apply (rule finite_ReZ_segments_constI)
       using True unfolding subpath_def by auto
   next
     case False
     then have "s>0" using \<open>s\<in>{0..<1}\<close> by auto
     from finite_Psegments_pos_linear[OF _ this,of _ 0 0 1] insertI_1(4)
     show "finite_ReZ_segments (subpath 0 s g) z"
       unfolding finite_ReZ_segments_def comp_def subpath_def by auto
   qed  
   then show ?thesis using subEq insertI_1 by force
 next
   case (insertI_2 s)
   have "finite_ReZ_segments (subpath 0 s g) z" 
   proof (cases "s=0")
     case True
     show ?thesis
       apply (rule finite_ReZ_segments_constI)
       using True unfolding subpath_def by auto
   next
     case False
     then have "s>0" using \<open>s\<in>{0..<1}\<close> by auto
     from finite_Psegments_pos_linear[OF _ this,of _ 0 0 1] insertI_2(4)
     show "finite_ReZ_segments (subpath 0 s g) z"
       unfolding finite_ReZ_segments_def comp_def subpath_def by auto
   qed
   then show ?thesis using subNEq insertI_2 by force
 qed         
   
 lemma finite_ReZ_segments_induct [case_names sub0 subEq subNEq, induct pred:finite_ReZ_segments]:
   assumes "finite_ReZ_segments g z"
   assumes  sub0:"\<And>g z. (P (subpath 0 0 g) z)" 
     and subEq:"(\<And>s g z. \<lbrakk>s \<in> {0..<1};s=0\<or>Re (g s) = Re z;
           \<forall>t\<in>{s<..<1}. Re (g t) = Re z;finite_ReZ_segments (subpath 0 s g) z; 
           P (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P g z)" 
     and subNEq:"(\<And>s g z. \<lbrakk>s \<in> {0..<1};s=0\<or>Re (g s) = Re z;
           \<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z;finite_ReZ_segments (subpath 0 s g) z;
           P (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P g z)"
   shows "P g z" 
 proof -
   have "finite_Psegments (\<lambda>t. Re (g t - z) = 0) 0 1" 
     using assms(1) unfolding finite_ReZ_segments_def by auto
   then have "(0::real)\<le>1 \<longrightarrow> P (subpath 0 1 g) z"
   proof (induct rule: finite_Psegments.induct[of _ 0 1 "\<lambda>a b. b\<ge>a \<longrightarrow> P (subpath a b g) z"] )
     case (emptyI a b)
     then show ?case using sub0[of "subpath a b g"] unfolding subpath_def by auto 
   next
     case (insertI_1 s a b)
     have ?case when "a=b"
       using sub0[of "subpath a b g"] that unfolding subpath_def by auto        
     moreover have ?case when "a\<noteq>b"
     proof -
       have "b>a" using that \<open>s \<in> {a..<b}\<close> by auto
       define s'::real where "s'=(s-a)/(b-a)"  
       have "P (subpath a b g) z" 
       proof (rule subEq[of s' "subpath a b g"])
         show "\<forall>t\<in>{s'<..<1}. Re (subpath a b g t) = Re z" 
         proof 
           fix t assume "t \<in> {s'<..<1}"
           then have "(b - a) * t + a\<in>{s<..<b}" 
             unfolding s'_def using \<open>b>a\<close> \<open>s \<in> {a..<b}\<close>
             apply (auto simp add:field_simps)
             by (sos "((((A<0 * (A<1 * A<2)) * R<1) + (((A<=1 * (A<0 * R<1)) * (R<1 * [1]^2))
                + ((A<=0 * (A<0 * (A<1 * R<1))) * (R<1 * [1]^2)))))")  
           then have "Re (g ((b - a) * t + a) - z) = 0"
             using insertI_1(3)[rule_format,of "(b - a) * t + a"] by auto
           then show "Re (subpath a b g t) = Re z" 
             unfolding subpath_def by auto
         qed
         show "finite_ReZ_segments (subpath 0 s' (subpath a b g)) z"
         proof (cases "s=a")
           case True
           then show ?thesis unfolding s'_def subpath_def 
             by (auto intro:finite_ReZ_segments_constI)
         next
           case False
           have "finite_Psegments (\<lambda>t. Re (g t - z) = 0) a s"
             using insertI_1(4) unfolding finite_ReZ_segments_def by auto
           then have "finite_Psegments ((\<lambda>t. Re (g t - z) = 0) \<circ> (\<lambda>t. (s - a) * t + a)) 0 1" 
             apply (elim finite_Psegments_pos_linear[of _ "s-a" 0 a 1,simplified])
             using False \<open>s\<in>{a..<b}\<close> by auto
           then show ?thesis 
             using \<open>b>a\<close> unfolding finite_ReZ_segments_def subpath_def s'_def comp_def
             by auto
         qed
         show "s' \<in> {0..<1}"
           using \<open>b>a\<close> \<open>s\<in>{a..<b}\<close> unfolding s'_def 
           by (auto simp add:field_simps)
         show "P (subpath 0 s' (subpath a b g)) z"
         proof -
           have "P (subpath a s g) z" using insertI_1(1,5) by auto
           then show ?thesis 
             using \<open>b>a\<close> unfolding s'_def subpath_def  by simp  
         qed
         show "s' = 0 \<or> Re (subpath a b g s') = Re z"
         proof -
           have ?thesis when "s=a"
             using that unfolding s'_def by auto
           moreover have ?thesis when "Re (g s - z) = 0"
             using that unfolding s'_def subpath_def by auto
           ultimately show ?thesis using \<open>s = a \<or> Re (g s - z) = 0\<close> by auto
         qed
       qed
       then show ?thesis using \<open>b>a\<close> by auto
     qed
     ultimately show ?case by auto
   next
     case (insertI_2 s a b)
     have ?case when "a=b"
       using sub0[of "subpath a b g"] that unfolding subpath_def by auto        
     moreover have ?case when "a\<noteq>b"
     proof -
       have "b>a" using that \<open>s \<in> {a..<b}\<close> by auto
       define s'::real where "s'=(s-a)/(b-a)"  
       have "P (subpath a b g) z" 
       proof (rule subNEq[of s' "subpath a b g"])
         show "\<forall>t\<in>{s'<..<1}. Re (subpath a b g t) \<noteq> Re z" 
         proof 
           fix t assume "t \<in> {s'<..<1}"
           then have "(b - a) * t + a\<in>{s<..<b}" 
             unfolding s'_def using \<open>b>a\<close> \<open>s \<in> {a..<b}\<close>
             apply (auto simp add:field_simps)
             by (sos "((((A<0 * (A<1 * A<2)) * R<1) + (((A<=1 * (A<0 * R<1)) * (R<1 * [1]^2)) + 
               ((A<=0 * (A<0 * (A<1 * R<1))) * (R<1 * [1]^2)))))")  
           then have "Re (g ((b - a) * t + a) - z) \<noteq> 0"
             using insertI_2(3)[rule_format,of "(b - a) * t + a"] by auto
           then show "Re (subpath a b g t) \<noteq> Re z" 
             unfolding subpath_def by auto
         qed
         show "finite_ReZ_segments (subpath 0 s' (subpath a b g)) z"
         proof (cases "s=a")
           case True
           then show ?thesis unfolding s'_def subpath_def 
             by (auto intro:finite_ReZ_segments_constI)
         next
           case False
           have "finite_Psegments (\<lambda>t. Re (g t - z) = 0) a s"
             using insertI_2(4) unfolding finite_ReZ_segments_def by auto
           then have "finite_Psegments ((\<lambda>t. Re (g t - z) = 0) \<circ> (\<lambda>t. (s - a) * t + a)) 0 1" 
             apply (elim finite_Psegments_pos_linear[of _ "s-a" 0 a 1,simplified])
             using False \<open>s\<in>{a..<b}\<close> by auto
           then show ?thesis 
             using \<open>b>a\<close> unfolding finite_ReZ_segments_def subpath_def s'_def comp_def
             by auto
         qed
         show "s' \<in> {0..<1}"
           using \<open>b>a\<close> \<open>s\<in>{a..<b}\<close> unfolding s'_def 
           by (auto simp add:field_simps)
         show "P (subpath 0 s' (subpath a b g)) z"
         proof -
           have "P (subpath a s g) z" using insertI_2(1,5) by auto
           then show ?thesis 
             using \<open>b>a\<close> unfolding s'_def subpath_def  by simp  
         qed
         show "s' = 0 \<or> Re (subpath a b g s') = Re z"
         proof -
           have ?thesis when "s=a"
             using that unfolding s'_def by auto
           moreover have ?thesis when "Re (g s - z) = 0"
             using that unfolding s'_def subpath_def by auto
           ultimately show ?thesis using \<open>s = a \<or> Re (g s - z) = 0\<close> by auto
         qed
       qed
       then show ?thesis using \<open>b>a\<close> by auto
     qed
     ultimately show ?case by auto
   qed
   then show ?thesis by auto
 qed
 
 lemma finite_ReZ_segments_shiftpah:
   assumes "finite_ReZ_segments g z" "s\<in>{0..1}" "path g" and loop:"pathfinish g = pathstart g" 
   shows "finite_ReZ_segments (shiftpath s g) z"
 proof -
   have "finite_Psegments (\<lambda>t. Re (shiftpath s g t - z) = 0) 0 (1-s)"
   proof -
     have "finite_Psegments (\<lambda>t. Re (g t) = Re z) s 1" 
       using assms finite_Psegments_included[of _ 0 1 s] unfolding finite_ReZ_segments_def  
       by force   
     then have "finite_Psegments (\<lambda>t. Re (g (s + t) - z) = 0) 0 (1-s)" 
       using finite_Psegments_pos_linear[of "\<lambda>t. Re (g t - z) =0" 1 0 s "1-s",simplified]
       unfolding comp_def by (auto simp add:algebra_simps)
     then show ?thesis unfolding shiftpath_def
       apply (elim finite_Psegments_congE)
       using \<open>s\<in>{0..1}\<close> by auto  
   qed
   moreover have "finite_Psegments (\<lambda>t. Re (shiftpath s g t - z) = 0) (1-s) 1"  
   proof -
     have "finite_Psegments (\<lambda>t. Re (g t) = Re z) 0 s " 
       using assms finite_Psegments_included unfolding finite_ReZ_segments_def  
       by force   
     then have "finite_Psegments (\<lambda>t. Re (g (s + t - 1) - z) = 0) (1-s) 1" 
       using finite_Psegments_pos_linear[of "\<lambda>t. Re (g t - z) =0" 1 "1-s" "s-1" 1,simplified]
       unfolding comp_def by (auto simp add:algebra_simps)
     then show ?thesis unfolding shiftpath_def
       apply (elim finite_Psegments_congE)
       using \<open>s\<in>{0..1}\<close> by auto  
   qed
   moreover have "1 - s \<in> {0..1}" using \<open>s\<in>{0..1}\<close> by auto
   moreover have "closed ({x. Re (shiftpath s g x - z) = 0} \<inter> {0..1})"
   proof -
     let ?f = "\<lambda>x. Re (shiftpath s g x - z)"  
     have "continuous_on {0..1} ?f"
       using path_shiftpath[OF \<open>path g\<close> loop \<open>s\<in>{0..1}\<close>] unfolding path_def
       by (auto intro: continuous_intros)
     from continuous_closed_preimage_constant[OF this,of 0,simplified] 
     show ?thesis 
       apply (elim arg_elim[of closed])
       by force
   qed
   ultimately show ?thesis unfolding finite_ReZ_segments_def 
     by (rule finite_Psegments_combine[where b="1-s"])  
 qed
      
 lemma finite_imp_finite_ReZ_segments:
   assumes "finite {t. Re (g t - z) = 0 \<and> 0 \<le> t \<and> t\<le>1}"
   shows "finite_ReZ_segments g z"
 proof -
   define P where "P = (\<lambda>t. Re (g t - z) = 0)"
   define rs where "rs=(\<lambda>b. {t. P t \<and> 0 < t \<and> t<b})"
   have "finite_Psegments P 0 b" when "finite (rs b)" "b>0" for b
   using that
   proof (induct "card (rs b)" arbitrary:b rule:nat_less_induct)
     case ind:1
     have ?case when "rs b= {}"
       apply (rule finite_Psegments.intros(3)[of 0])
       using that \<open>0 < b\<close> unfolding rs_def by (auto intro:finite_Psegments.intros)  
     moreover have ?case when "rs b\<noteq>{}"
     proof -
       define lj where "lj = Max (rs b)" 
       have "0<lj" "lj<b" "P lj"
         using Max_in[OF \<open>finite (rs b)\<close> \<open>rs b\<noteq>{}\<close>,folded lj_def]
         unfolding rs_def by auto
       show ?thesis 
       proof (rule finite_Psegments.intros(3)[of lj])
         show "lj \<in> {0..<b}" " lj = 0 \<or> P lj" 
           using \<open>0<lj\<close> \<open>lj<b\<close> \<open>P lj\<close> by auto
         show "\<forall>t\<in>{lj<..<b}. \<not> P t"
         proof (rule ccontr)
           assume " \<not> (\<forall>t\<in>{lj<..<b}. \<not> P t) "
           then obtain t where t:"P t" "lj < t" "t < b" by auto
           then have "t\<in>rs b" unfolding rs_def using \<open>lj>0\<close> by auto
           then have "t\<le>lj" using Max_ge[OF \<open>finite (rs b)\<close>,of t] unfolding lj_def by auto
           then show False using \<open>t>lj\<close> by auto
         qed
         show "finite_Psegments P 0 lj"
         proof (rule ind.hyps[rule_format,of "card (rs lj)" lj,simplified])
           show "finite (rs lj)"
             using \<open>finite (rs b)\<close> unfolding rs_def using \<open>lj<b\<close> 
             by (auto elim!:rev_finite_subset )
           show "card (rs lj) < card (rs b)"
             apply (rule psubset_card_mono[OF \<open>finite (rs b)\<close>])
             using Max_in \<open>finite (rs lj)\<close> \<open>lj < b\<close> lj_def rs_def that by fastforce
           show "0 < lj" using \<open>0<lj\<close> .
         qed
       qed
     qed
     ultimately show ?case by auto 
   qed
   moreover have "finite (rs 1)"
     using assms unfolding rs_def P_def
     by (auto elim:rev_finite_subset)
   ultimately have "finite_Psegments P 0 1" by auto
   then show ?thesis unfolding P_def finite_ReZ_segments_def .
 qed
   
 lemma finite_ReZ_segments_poly_linepath:
   shows "finite_ReZ_segments (poly p o linepath a b) z" 
 proof -
   define P where "P=map_poly Re (pcompose (p-[:z:]) [:a,b-a:])"
   have *:"Re ((poly p \<circ> linepath a b) t - z) = 0 \<longleftrightarrow> poly P t=0" for t
     unfolding inner_complex_def P_def linepath_def comp_def
     apply (subst Re_poly_of_real[symmetric])
     by (auto simp add: algebra_simps poly_pcompose scaleR_conv_of_real)
   have ?thesis when "P\<noteq>0"
   proof -
     have "finite {t. poly P t=0}" using that poly_roots_finite by auto
     then have "finite {t. Re ((poly p \<circ> linepath a b) t - z) = 0 \<and> 0 \<le> t \<and> t \<le> 1}"
       using *
       by auto
     then show ?thesis   
       using finite_imp_finite_ReZ_segments[of "poly p o linepath a b" z] by auto
   qed
   moreover have ?thesis when "P=0"
     unfolding finite_ReZ_segments_def
     apply (rule finite_Psegments_constI[where c=True])  
     apply (subst *)
     using that by auto 
   ultimately show ?thesis by auto
 qed  
   
 lemma part_circlepath_half_finite_inter:
   assumes "st\<noteq>tt" "r\<noteq>0" "c\<noteq>0"
   shows "finite {t. part_circlepath z0 r st tt t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}" (is "finite ?T")
 proof -
   let ?S = "{\<theta>. (z0+r*exp (\<i> * \<theta> )) \<bullet> c = d \<and> \<theta> \<in> closed_segment st tt}"
   define S where "S \<equiv> {\<theta>. (z0+r*exp (\<i> * \<theta> )) \<bullet> c = d \<and> \<theta> \<in> closed_segment st tt}"
   have "S = linepath st tt ` ?T"
   proof   
     define g where "g\<equiv>(\<lambda>t. (t-st)/(tt -st))"
     have "0\<le>g t" "g t\<le>1" when "t \<in> closed_segment st tt " for t
       using that \<open>st\<noteq>tt\<close> closed_segment_eq_real_ivl unfolding  g_def real_scaleR_def
       by (auto simp add:divide_simps)
     moreover have "linepath st tt (g t) =t" "g (linepath st tt t) = t" for t
       unfolding linepath_def g_def real_scaleR_def using \<open>st\<noteq>tt\<close>
       apply (simp_all add:divide_simps)
       by (auto simp add:algebra_simps )
     ultimately have "x\<in>linepath st tt ` ?T" when "x\<in>S" for x
       using that unfolding S_def 
       by (auto intro!:image_eqI[where x="g x"] simp add:part_circlepath_def)
     then show "S \<subseteq> linepath st tt ` ?T" by auto
   next
     have "x\<in>S" when "x\<in>linepath st tt ` ?T" for x
       using that unfolding part_circlepath_def S_def
       by (auto simp add: linepath_in_path)
     then show "linepath st tt ` ?T \<subseteq> S" by auto
   qed
   moreover have "finite S" 
   proof -
     define a' b' c' where "a'=r * Re c" and "b' = r* Im c" and "c'=Im c * Im z0 + Re z0 * Re c - d"
     define f where "f \<theta>= a' * cos \<theta> + b' * sin \<theta> + c'" for \<theta>
     have "(z0+r*exp (\<i> * \<theta> )) \<bullet> c = d \<longleftrightarrow> f \<theta> = 0" for \<theta>
       unfolding exp_Euler inner_complex_def f_def a'_def b'_def c'_def
       by (auto simp add:algebra_simps cos_of_real sin_of_real)
     then have *:"S = roots f \<inter> closed_segment st tt"
       unfolding S_def roots_within_def by auto
     have "uniform_discrete S"
     proof -
       have "a' \<noteq> 0 \<or> b' \<noteq> 0 \<or> c' \<noteq> 0" 
         using assms complex_eq_iff unfolding a'_def b'_def c'_def 
         by auto
       then have "periodic_set (roots f) (4 * pi)"
         using periodic_set_sin_cos_linear[of a' b' c',folded f_def] by auto
       then have "uniform_discrete (roots f)" using periodic_imp_uniform_discrete by auto
       then show ?thesis unfolding * by auto
     qed
     moreover have "bounded S" unfolding * 
       by (simp add: bounded_Int bounded_closed_segment)
     ultimately show ?thesis using uniform_discrete_finite_iff by auto
   qed
   moreover have "inj_on (linepath st tt) ?T" 
   proof -
     have "inj (linepath st tt)"
       unfolding linepath_def using assms inj_segment by blast
     then show ?thesis by (auto elim:subset_inj_on)
   qed
   ultimately show ?thesis by (auto elim!: finite_imageD)
 qed
 
 lemma linepath_half_finite_inter:
   assumes "a \<bullet> c \<noteq> d \<or> b \<bullet> c \<noteq> d"
   shows "finite {t. linepath a b t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}" (is "finite ?S")
 proof (rule ccontr)
   assume asm:"infinite ?S"
   obtain t1 t2 where u1u2:"t1\<noteq>t2" "t1\<in>?S" "t2\<in>?S"
   proof -
     obtain t1 where "t1\<in>?S" using not_finite_existsD asm by blast
     moreover have "\<exists>u2. u2\<in>?S-{t1}" 
       using infinite_remove[OF asm,of t1] 
       by (meson finite.emptyI rev_finite_subset subsetI)
     ultimately show ?thesis using that by auto
   qed
   have t1:"(1-t1)*(a \<bullet> c) + t1 * (b \<bullet> c) = d"
     using \<open>t1\<in>?S\<close> unfolding linepath_def by (simp add: inner_left_distrib)
   have t2:"(1-t2)*(a \<bullet> c) + t2 * (b \<bullet> c) = d"
     using \<open>t2\<in>?S\<close> unfolding linepath_def by (simp add: inner_left_distrib)
   have "a \<bullet> c = d"
   proof -
     have "t2*((1-t1)*(a \<bullet> c) + t1 * (b \<bullet> c)) = t2*d" using t1 by auto
     then have *:"(t2-t1*t2)*(a \<bullet> c) + t1*t2 * (b \<bullet> c) = t2*d" by (auto simp add:algebra_simps)
     have "t1*((1-t2)*(a \<bullet> c) + t2 * (b \<bullet> c)) = t1*d" using t2 by auto
     then have **:"(t1-t1*t2)*(a \<bullet> c) + t1*t2 * (b \<bullet> c) = t1*d" by (auto simp add:algebra_simps)
     have "(t2-t1)*(a \<bullet> c) = (t2-t1)*d" using arg_cong2[OF * **,of minus]
       by (auto simp add:algebra_simps)
     then show ?thesis using \<open>t1\<noteq>t2\<close> by auto
   qed
   moreover have "b \<bullet> c = d"
   proof -
     have "(1-t2)*((1-t1)*(a \<bullet> c) + t1 * (b \<bullet> c)) = (1-t2)*d" using t1 by auto
     then have *:"(1-t1)*(1-t2)*(a \<bullet> c) + (t1-t1*t2) * (b \<bullet> c) = (1-t2)*d" by (auto simp add:algebra_simps)
     have "(1-t1)*((1-t2)*(a \<bullet> c) + t2 * (b \<bullet> c)) = (1-t1)*d" using t2 by auto
     then have **:"(1-t1)*(1-t2)*(a \<bullet> c) + (t2-t1*t2) * (b \<bullet> c) = (1-t1)*d" by (auto simp add:algebra_simps)
     have "(t2-t1)*(b \<bullet> c) = (t2-t1)*d" using arg_cong2[OF ** *,of minus]
       by (auto simp add:algebra_simps)
     then show ?thesis using \<open>t1\<noteq>t2\<close> by auto
   qed
   ultimately show False using assms by auto  
 qed
   
 lemma finite_half_joinpaths_inter:
   assumes "finite {t. l1 t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}" "finite {t. l2 t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}"
   shows "finite {t. (l1+++l2) t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}"
 proof -
   let ?l1s = "{t. l1 (2*t) \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1/2}"
   let ?l2s = "{t. l2 (2 * t - 1) \<bullet> c = d \<and> 1/2< t \<and> t\<le>1}"
   let ?ls = "\<lambda>l. {t. l t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1}"
   have "{t. (l1+++l2) t \<bullet> c = d \<and> 0\<le> t \<and> t\<le>1} = ?l1s \<union> ?l2s"
     unfolding joinpaths_def by auto
   moreover have "finite ?l1s"
   proof -
     have "?l1s = ((*) (1/2)) ` ?ls l1" by (auto intro:rev_image_eqI)
     thus ?thesis using assms by simp 
   qed
   moreover have "finite ?l2s"
   proof -
     have "?l2s \<subseteq> (\<lambda>x. x/2 + 1/2) ` ?ls l2" by (auto intro:rev_image_eqI simp add:field_simps)
     thus ?thesis using assms 
       by (auto elim:finite_subset)
   qed
   ultimately show ?thesis by simp
 qed  
   
 lemma finite_ReZ_segments_linepath:
   "finite_ReZ_segments (linepath a b) z"
 proof -
   have ?thesis when "Re a\<noteq>Re z \<or> Re b \<noteq>Re z"
   proof -
     let ?S1="{t. Re (linepath a b t-z) = 0 \<and> 0 \<le> t \<and> t \<le> 1}"
     have "finite ?S1" 
       using linepath_half_finite_inter[of a "Complex 1 0" "Re z" b] that 
       using one_complex.code by auto
     from finite_imp_finite_ReZ_segments[OF this] show ?thesis .
   qed
   moreover have ?thesis when "Re a=Re z" "Re b=Re z"
     unfolding finite_ReZ_segments_def
     apply (rule finite_Psegments.intros(2)[of 0])
     using that unfolding linepath_def by (auto simp add:algebra_simps intro:finite_Psegments.intros)
   ultimately show ?thesis by blast
 qed
   
 lemma finite_ReZ_segments_part_circlepath:
   "finite_ReZ_segments (part_circlepath z0 r st tt) z"
 proof -
   have ?thesis when "st \<noteq> tt" "r \<noteq> 0"
   proof -
     let ?S1="{t. Re (part_circlepath z0 r st tt t-z) = 0 \<and> 0 \<le> t \<and> t \<le> 1}"  
     have "finite ?S1"
       using part_circlepath_half_finite_inter[of st tt r "Complex 1 0" z0 "Re z"] that one_complex.code
       by (auto simp add:inner_complex_def )
     from finite_imp_finite_ReZ_segments[OF this] show ?thesis .
   qed
   moreover have ?thesis when "st =tt \<or> r=0"
   proof -
     define c where "c = z0 + r * exp (\<i> *  tt)"
     have "part_circlepath z0 r st tt = (\<lambda>t. c)"
       unfolding part_circlepath_def c_def using that linepath_refl by auto
     then show ?thesis 
       using finite_ReZ_segments_linepath[of c c z] linepath_refl[of c]
       by auto
   qed
   ultimately show ?thesis by blast
 qed
    
 lemma finite_ReZ_segments_poly_of_real:
   shows "finite_ReZ_segments (poly p o of_real) z" 
   using finite_ReZ_segments_poly_linepath[of p 0 1 z] unfolding linepath_def
   by (auto simp add:scaleR_conv_of_real)
  
 lemma finite_ReZ_segments_subpath:
   assumes "finite_ReZ_segments g z"
     "0\<le>u" "u\<le>v" "v\<le>1"
   shows "finite_ReZ_segments (subpath u v g) z" 
 proof (cases "u=v")
   case True  
   then show ?thesis 
     unfolding subpath_def by (auto intro:finite_ReZ_segments_constI)
 next
   case False
   then have "u<v" using \<open>u\<le>v\<close> by auto
   define P where "P=(\<lambda>t. Re (g t - z) = 0)"
   have "finite_ReZ_segments (subpath u v g) z 
       = finite_Psegments (P o (\<lambda>t. (v - u) * t + u)) 0 1"
     unfolding finite_ReZ_segments_def subpath_def P_def comp_def by auto
   also have "..."
     apply (rule finite_Psegments_pos_linear)
     using assms False unfolding finite_ReZ_segments_def
     by (fold P_def,auto elim:finite_Psegments_included)        
   finally show ?thesis .
 qed   
   
 subsection \<open>jump and jumpF\<close> 
   
 definition jump::"(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> int" where
   "jump f a = (if 
         (LIM x (at_left a). f x :> at_bot) \<and> (LIM x (at_right a). f x :> at_top) 
       then 1 else if 
         (LIM x (at_left a). f x :> at_top) \<and> (LIM x (at_right a). f x :> at_bot) 
       then -1 else 0)"
   
 definition jumpF::"(real \<Rightarrow> real) \<Rightarrow> real filter \<Rightarrow> real" where 
   "jumpF f F \<equiv> (if filterlim f at_top F then 1/2 else 
     if filterlim f at_bot F then -1/2 else (0::real))"
    
 lemma jumpF_const[simp]:
   assumes "F\<noteq>bot"
   shows "jumpF (\<lambda>_. c) F = 0" 
 proof -
   have False when "LIM x F. c :> at_bot"
     using filterlim_at_bot_nhds[OF that _ \<open>F\<noteq>bot\<close>] by auto
   moreover have False when "LIM x F. c :> at_top"
     using filterlim_at_top_nhds[OF that _ \<open>F\<noteq>bot\<close>] by auto
   ultimately show ?thesis unfolding jumpF_def by auto
 qed
 
 lemma jumpF_not_infinity:
   assumes "continuous F g" "F\<noteq>bot"
   shows "jumpF g F = 0"  
 proof -
   have "\<not> filterlim g at_infinity F"
     using not_tendsto_and_filterlim_at_infinity[OF \<open>F \<noteq>bot\<close> assms(1)[unfolded continuous_def]] 
     by auto
   then have "\<not> filterlim g at_bot F" "\<not> filterlim g at_top F"
     using at_bot_le_at_infinity at_top_le_at_infinity filterlim_mono by blast+
   then show ?thesis unfolding jumpF_def by auto
 qed
   
 lemma jumpF_linear_comp:
   assumes "c\<noteq>0"
   shows 
     "jumpF (f o (\<lambda>x. c*x+b)) (at_left x) = 
             (if c>0 then jumpF f (at_left (c*x+b)) else jumpF f (at_right (c*x+b)))"
     (is ?case1)
     "jumpF (f o (\<lambda>x. c*x+b)) (at_right x) = 
             (if c>0 then jumpF f (at_right (c*x+b)) else jumpF f (at_left (c*x+b)))"
     (is ?case2)
 proof -
   let ?g = "\<lambda>x. c*x+b"  
   have ?case1 ?case2 when "\<not> c>0"
   proof -
     have "c<0" using \<open>c\<noteq>0\<close> that by auto
     have "filtermap ?g (at_left x) = at_right (?g x)"
          "filtermap ?g (at_right x) = at_left (?g x)"
       using \<open>c<0\<close> 
       filtermap_linear_at_left[OF \<open>c\<noteq>0\<close>, of b x] 
       filtermap_linear_at_right[OF \<open>c\<noteq>0\<close>, of b x] by auto
     then have 
         "jumpF (f \<circ> ?g) (at_left x) = jumpF f (at_right (?g x))"
         "jumpF (f \<circ> ?g) (at_right x) = jumpF f (at_left (?g x))"
       unfolding jumpF_def filterlim_def comp_def
       by (auto simp add: filtermap_filtermap[of f ?g,symmetric])
     then show ?case1 ?case2 using \<open>c<0\<close> by auto
   qed
   moreover have ?case1 ?case2 when "c>0"
   proof -
     have "filtermap ?g (at_left x) = at_left (?g x)"
          "filtermap ?g (at_right x) = at_right (?g x)"
       using that 
       filtermap_linear_at_left[OF \<open>c\<noteq>0\<close>, of b x] 
       filtermap_linear_at_right[OF \<open>c\<noteq>0\<close>, of b x] by auto
     then have 
         "jumpF (f \<circ> ?g) (at_left x) = jumpF f (at_left (?g x))"
         "jumpF (f \<circ> ?g) (at_right x) = jumpF f (at_right (?g x))"
       unfolding jumpF_def filterlim_def comp_def
       by (auto simp add: filtermap_filtermap[of f ?g,symmetric])
     then show ?case1 ?case2 using that by auto
   qed
   ultimately show ?case1 ?case2 by auto
 qed  
   
 lemma jump_const[simp]:"jump (\<lambda>_. c) a = 0"
 proof -
   have False when "LIM x (at_left a). c :> at_bot"
     apply (rule not_tendsto_and_filterlim_at_infinity[of "at_left a" "\<lambda>_. c" c])
       apply auto
     using at_bot_le_at_infinity filterlim_mono that by blast
   moreover have False when "LIM x (at_left a). c :> at_top"
     apply (rule not_tendsto_and_filterlim_at_infinity[of "at_left a" "\<lambda>_. c" c])
       apply auto
     using at_top_le_at_infinity filterlim_mono that by blast  
   ultimately show ?thesis unfolding jump_def by auto
 qed  
 
 lemma jump_not_infinity:
   "isCont f a \<Longrightarrow> jump f a =0"
   by (meson at_bot_le_at_infinity at_top_le_at_infinity filterlim_at_split 
       filterlim_def isCont_def jump_def not_tendsto_and_filterlim_at_infinity 
       order_trans trivial_limit_at_left_real) 
   
 lemma jump_jump_poly_aux:
   assumes "p\<noteq>0" "coprime p q"
   shows "jump (\<lambda>x. poly q x / poly p x) a = jump_poly q p a"
 proof (cases "q=0")
   case True
   then show ?thesis by auto
 next
   case False
   define f where "f \<equiv> (\<lambda>x. poly q x / poly p x)"
   have ?thesis when "poly q a = 0"
   proof -
     have "poly p a\<noteq>0" using coprime_poly_0[OF \<open>coprime p q\<close>] that by blast
     then have "isCont f a" unfolding f_def by simp
     then have "jump f a=0" using jump_not_infinity by auto
     moreover have "jump_poly q p a=0"
       using jump_poly_not_root[OF \<open>poly p a\<noteq>0\<close>] by auto
     ultimately show ?thesis unfolding f_def by auto
   qed
   moreover have ?thesis when "poly q a\<noteq>0"
   proof (cases "even(order a p)")
     case True
     define c where "c\<equiv>sgn (poly q a)"
     note 
       filterlim_divide_at_bot_at_top_iff
         [OF _ that,of "poly q"  "at_left a" "poly p",folded f_def c_def,simplified]
       filterlim_divide_at_bot_at_top_iff
         [OF _ that,of "poly q"  "at_right a" "poly p",folded f_def c_def,simplified]
     moreover have "(poly p has_sgnx - c) (at_left a) = (poly p has_sgnx - c) (at_right a)"
          "(poly p has_sgnx c) (at_left a) = (poly p has_sgnx c) (at_right a)"
       using poly_has_sgnx_left_right[OF \<open>p\<noteq>0\<close>] True by auto
     moreover have "c\<noteq>0" by (simp add: c_def sgn_if that)
     then have False when 
         "(poly p has_sgnx - c) (at_right a)" 
         "(poly p has_sgnx c) (at_right a)"
       using has_sgnx_unique[OF _ that] by auto
     ultimately have "jump f a = 0"
       unfolding jump_def by auto
     moreover have "jump_poly q p a = 0" unfolding jump_poly_def
       using True by (simp add: order_0I that)
     ultimately show ?thesis unfolding f_def by auto
   next
     case False
     define c where "c\<equiv>sgn (poly q a)"
     have "(poly p \<longlongrightarrow> 0) (at a)" using False 
       by (metis even_zero order_0I poly_tendsto(1))
     then have "(poly p \<longlongrightarrow> 0) (at_left a)" and "(poly p \<longlongrightarrow> 0) (at_right a)"
       by (auto simp add: filterlim_at_split)
     moreover note 
       filterlim_divide_at_bot_at_top_iff
         [OF _ that,of "poly q" _ "poly p",folded f_def c_def]
     moreover have "(poly p has_sgnx c) (at_left a) = (poly p has_sgnx - c) (at_right a)"
          "(poly p has_sgnx - c) (at_left a) = (poly p has_sgnx c) (at_right a)"
       using poly_has_sgnx_left_right[OF \<open>p\<noteq>0\<close>] False by auto
     ultimately have "jump f a = (if (poly p has_sgnx c) (at_right a) then 1
         else if (poly p has_sgnx - c) (at_right a) then -1 else 0)"
       unfolding jump_def by auto
     also have "... = (if sign_r_pos (q * p) a then 1 else - 1)"
     proof -
       have "(poly p has_sgnx c) (at_right a) \<longleftrightarrow>  sign_r_pos (q * p) a " 
       proof 
         assume "(poly p has_sgnx c) (at_right a)"
         then have "sgnx (poly p) (at_right a) = c" by auto
         moreover have "sgnx (poly q) (at_right a) = c"
           unfolding c_def using that by (auto intro!: tendsto_nonzero_sgnx)
         ultimately have "sgnx (\<lambda>x. poly (q*p) x) (at_right a) = c * c"
           by (simp add:sgnx_times)
         moreover have "c\<noteq>0" by (simp add: c_def sgn_if that) 
         ultimately have "sgnx (\<lambda>x. poly (q*p) x) (at_right a) > 0"
           using not_real_square_gt_zero by fastforce
         then show "sign_r_pos (q * p) a" using sign_r_pos_sgnx_iff 
           by blast
       next
         assume asm:"sign_r_pos (q * p) a"
         let ?c1 = "sgnx (poly p) (at_right a)"
         let ?c2 = "sgnx (poly q) (at_right a)"
         have "0 < sgnx (\<lambda>x. poly (q * p) x) (at_right a)"
           using asm sign_r_pos_sgnx_iff by blast
         then have "?c2 * ?c1 >0" 
           apply (subst (asm) poly_mult)
           apply (subst (asm) sgnx_times)
           by auto
         then have "?c2>0 \<and> ?c1>0 \<or> ?c2<0 \<and> ?c1<0"
           by (simp add: zero_less_mult_iff)
         then have "?c1=?c2"
           using sgnx_values[OF sgnx_able_poly(1), of a,simplified]
           by (metis add.inverse_neutral less_minus_iff less_not_sym)    
         moreover have "sgnx (poly q) (at_right a) = c"
           unfolding c_def using that by (auto intro!: tendsto_nonzero_sgnx)
         ultimately have "?c1 = c" by auto
         then show "(poly p has_sgnx c) (at_right a)"
           using sgnx_able_poly(1) sgnx_able_sgnx by blast
       qed  
       then show ?thesis 
         unfolding jump_poly_def using poly_has_sgnx_values[OF \<open>p\<noteq>0\<close>]
         by (metis add.inverse_inverse c_def sgn_if that)
     qed
     also have "... = jump_poly q p a"
       unfolding jump_poly_def using False order_root that by (simp add: order_root assms(1))
     finally show ?thesis unfolding f_def by auto
   qed
   ultimately show ?thesis by auto
 qed
  
 lemma jump_jumpF:
   assumes cont:"isCont (inverse o f) a" and 
       sgnxl:"(f has_sgnx l) (at_left a)" and sgnxr:"(f has_sgnx r) (at_right a)" and
       "l\<noteq>0 " "r\<noteq>0"
   shows "jump f a = jumpF f (at_right a) - jumpF f (at_left a)"
 proof -
   have ?thesis when "filterlim f at_bot (at_left a)" "filterlim f at_top (at_right a)"
     unfolding jump_def jumpF_def 
     using that filterlim_at_top_at_bot[OF _ _ trivial_limit_at_left_real]
     by auto
   moreover have ?thesis when "filterlim f at_top (at_left a)" "filterlim f at_bot (at_right a)"
     unfolding jump_def jumpF_def 
     using that filterlim_at_top_at_bot[OF _ _ trivial_limit_at_right_real]
     by auto
   moreover have ?thesis when 
           "\<not> filterlim f at_bot (at_left a) \<or> \<not> filterlim f at_top (at_right a)"
           "\<not> filterlim f at_top (at_left a) \<or> \<not> filterlim f at_bot (at_right a)"
   proof (cases "f a=0")
     case False
     have "jumpF f (at_right a) = 0" "jumpF f (at_left a) = 0"
     proof -
       have "isCont (inverse o inverse o f) a" using cont False unfolding comp_def
         by (rule_tac continuous_at_within_inverse, auto)
       then have "isCont f a" unfolding comp_def by auto
       then have "(f \<longlongrightarrow> f a) (at_right a)" "(f \<longlongrightarrow> f a) (at_left a)"
         unfolding continuous_at_split by (auto simp add:continuous_within)
       moreover note trivial_limit_at_left_real trivial_limit_at_right_real
       ultimately show "jumpF f (at_right a) = 0" "jumpF f (at_left a) = 0"
         unfolding jumpF_def using filterlim_at_bot_nhds filterlim_at_top_nhds
         by metis+
     qed
     then show ?thesis unfolding jump_def using that by auto
   next
     case True
     then have tends0:"((\<lambda>x. inverse (f x)) \<longlongrightarrow> 0) (at a)" 
       using cont unfolding isCont_def comp_def by auto
     have "jump f a = 0" using that unfolding jump_def by auto
     have r_lim:"if r>0 then filterlim f at_top (at_right a) else filterlim f at_bot (at_right a)"
     proof (cases "r>0")
       case True 
       then have "\<forall>\<^sub>F x in (at_right a). 0 < f x" 
           using sgnxr unfolding has_sgnx_def
           by (auto elim:eventually_mono)
       then have "filterlim f at_top (at_right a)" 
         using filterlim_inverse_at_top[of "\<lambda>x. inverse (f x)", simplified] tends0
         unfolding filterlim_at_split by auto
       then show ?thesis using True by presburger
     next
       case False
       then have "\<forall>\<^sub>F x in (at_right a). 0 > f x" 
         using sgnxr \<open>r\<noteq>0\<close> False unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (meson linorder_neqE_linordered_idom sgn_less)
       then have "filterlim f at_bot (at_right a)" 
         using filterlim_inverse_at_bot[of "\<lambda>x. inverse (f x)", simplified] tends0
         unfolding filterlim_at_split by auto
       then show ?thesis using False by simp
     qed
     have l_lim:"if l>0 then filterlim f at_top (at_left a) else filterlim f at_bot (at_left a)"
     proof (cases "l>0")
       case True 
       then have "\<forall>\<^sub>F x in (at_left a). 0 < f x" 
           using sgnxl unfolding has_sgnx_def
           by (auto elim:eventually_mono)
       then have "filterlim f at_top (at_left a)" 
         using filterlim_inverse_at_top[of "\<lambda>x. inverse (f x)", simplified] tends0
         unfolding filterlim_at_split by auto
       then show ?thesis using True by presburger
     next
       case False
       then have "\<forall>\<^sub>F x in (at_left a). 0 > f x" 
         using sgnxl \<open>l\<noteq>0\<close> False unfolding has_sgnx_def
         apply (elim eventually_mono)
         by (meson linorder_neqE_linordered_idom sgn_less)
       then have "filterlim f at_bot (at_left a)" 
         using filterlim_inverse_at_bot[of "\<lambda>x. inverse (f x)", simplified] tends0
         unfolding filterlim_at_split by auto
       then show ?thesis using False by simp
     qed
       
     have ?thesis when "l>0" "r>0"
       using that l_lim r_lim \<open>jump f a=0\<close> unfolding jumpF_def by auto
     moreover have ?thesis when "\<not> l>0" "\<not> r>0"
     proof -
       have "filterlim f at_bot (at_right a)" "filterlim f at_bot (at_left a)" 
         using r_lim l_lim that by auto
       moreover then have "\<not> filterlim f at_top (at_right a)" "\<not> filterlim f at_top (at_left a)"
         by (auto elim: filterlim_at_top_at_bot)
       ultimately have "jumpF f (at_right a) = -1/2 "  "jumpF f (at_left a) = -1/2"
         unfolding jumpF_def by auto
       then show ?thesis using \<open>jump f a=0\<close> by auto
     qed
     moreover have ?thesis when "l>0" "\<not> r>0"
     proof -
       note \<open>\<not> filterlim f at_top (at_left a) \<or> \<not> filterlim f at_bot (at_right a)\<close>
       moreover have "filterlim f at_bot (at_right a)" "filterlim f at_top (at_left a)" 
         using r_lim l_lim that by auto
       ultimately have False by auto
       then show ?thesis by auto
     qed
     moreover have ?thesis when "\<not> l>0" "r>0"
     proof -
       note \<open>\<not> filterlim f at_bot (at_left a) \<or> \<not> filterlim f at_top (at_right a)\<close>
       moreover have "filterlim f at_bot (at_left a)" "filterlim f at_top (at_right a)" 
         using r_lim l_lim that by auto
       ultimately have False by auto
       then show ?thesis by auto
     qed
     ultimately show ?thesis by auto
   qed
   ultimately show ?thesis by auto
 qed
   
 lemma jump_linear_comp:
   assumes "c\<noteq>0"
   shows "jump (f o (\<lambda>x. c*x+b)) x = (if c>0 then jump f (c*x+b) else -jump f (c*x+b))"
 proof (cases "c>0")
   case False
   then have "c<0" using \<open>c\<noteq>0\<close> by auto
   let ?g = "\<lambda>x. c*x+b"  
   have "filtermap ?g (at_left x) = at_right (?g x)"
        "filtermap ?g (at_right x) = at_left (?g x)"
     using \<open>c<0\<close> 
       filtermap_linear_at_left[OF \<open>c\<noteq>0\<close>, of b x] 
       filtermap_linear_at_right[OF \<open>c\<noteq>0\<close>, of b x] by auto
   then have "jump (f \<circ> ?g) x = - jump f (c * x + b)"
     unfolding jump_def filterlim_def comp_def
     apply (auto simp add: filtermap_filtermap[of f ?g,symmetric])
     apply (fold filterlim_def)
     by (auto elim:filterlim_at_top_at_bot)
   then show ?thesis using \<open>c<0\<close> by auto
 next
   case True
   let ?g = "\<lambda>x. c*x+b"  
   have "filtermap ?g (at_left x) = at_left (?g x)"
        "filtermap ?g (at_right x) = at_right (?g x)"
     using True 
       filtermap_linear_at_left[OF \<open>c\<noteq>0\<close>, of b x] 
       filtermap_linear_at_right[OF \<open>c\<noteq>0\<close>, of b x] by auto
   then have "jump (f \<circ> ?g) x = jump f (c * x + b)"
     unfolding jump_def filterlim_def comp_def
     by (auto simp add: filtermap_filtermap[of f ?g,symmetric])
   then show ?thesis using True by auto
 qed 
   
 lemma jump_divide_derivative:
   assumes "isCont f x" "g x = 0" "f x\<noteq>0" 
     and g_deriv:"(g has_field_derivative c) (at x)" and "c\<noteq>0"
   shows "jump (\<lambda>t. f t/g t) x = (if sgn c = sgn ( f x) then 1 else -1)"  
 proof -
   have g_tendsto:"(g \<longlongrightarrow> 0) (at_left x)" "(g \<longlongrightarrow> 0) (at_right x)"
     by (metis DERIV_isCont Lim_at_imp_Lim_at_within assms(2) assms(4) continuous_at)+
   have f_tendsto:"(f \<longlongrightarrow> f x) (at_left x)" "(f \<longlongrightarrow> f x) (at_right x)"
     using Lim_at_imp_Lim_at_within assms(1) continuous_at by blast+
   
   have ?thesis when "c>0" "f x>0"  
   proof -    
     have "(g has_sgnx - sgn (f x)) (at_left x)" 
       using has_sgnx_derivative_at_left[OF g_deriv \<open>g x=0\<close>] that by auto
     moreover have "(g has_sgnx sgn (f x)) (at_right x)" 
       using has_sgnx_derivative_at_right[OF g_deriv \<open>g x=0\<close>] that by auto
     ultimately have "(LIM t at_left x. f t / g t :> at_bot) \<and> (LIM t at_right x. f t / g t :> at_top)"
       using filterlim_divide_at_bot_at_top_iff[OF _ \<open>f x\<noteq>0\<close>, of f] 
       using f_tendsto(1) f_tendsto(2) g_tendsto(1) g_tendsto(2) by blast
     moreover have "sgn c = sgn (f x)" using that by auto
     ultimately show ?thesis unfolding jump_def by auto
   qed
   moreover have ?thesis when "c>0" "f x<0"  
   proof -    
     have "(g has_sgnx sgn (f x)) (at_left x)" 
       using has_sgnx_derivative_at_left[OF g_deriv \<open>g x=0\<close>] that by auto
     moreover have "(g has_sgnx - sgn (f x)) (at_right x)" 
       using has_sgnx_derivative_at_right[OF g_deriv \<open>g x=0\<close>] that by auto
     ultimately have "(LIM t at_left x. f t / g t :> at_top) \<and> (LIM t at_right x. f t / g t :> at_bot)"
       using filterlim_divide_at_bot_at_top_iff[OF _ \<open>f x\<noteq>0\<close>, of f] 
       using f_tendsto(1) f_tendsto(2) g_tendsto(1) g_tendsto(2) by blast
     moreover from this have "\<not> (LIM t at_left x. f t / g t :> at_bot)"
       using filterlim_at_top_at_bot by fastforce
     moreover have "sgn c \<noteq> sgn (f x)" using that by auto
     ultimately show ?thesis unfolding jump_def by auto
   qed
   moreover have ?thesis when "c<0" "f x>0"  
   proof -    
     have "(g has_sgnx sgn (f x)) (at_left x)" 
       using has_sgnx_derivative_at_left[OF g_deriv \<open>g x=0\<close>] that by auto
     moreover have "(g has_sgnx - sgn (f x)) (at_right x)" 
       using has_sgnx_derivative_at_right[OF g_deriv \<open>g x=0\<close>] that by auto
     ultimately have "(LIM t at_left x. f t / g t :> at_top) \<and> (LIM t at_right x. f t / g t :> at_bot)"
       using filterlim_divide_at_bot_at_top_iff[OF _ \<open>f x\<noteq>0\<close>, of f] 
       using f_tendsto(1) f_tendsto(2) g_tendsto(1) g_tendsto(2) by blast
     moreover from this have "\<not> (LIM t at_left x. f t / g t :> at_bot)"
       using filterlim_at_top_at_bot by fastforce
     moreover have "sgn c \<noteq> sgn (f x)" using that by auto
     ultimately show ?thesis unfolding jump_def by auto
   qed
   moreover have ?thesis when "c<0" "f x<0"  
   proof -    
     have "(g has_sgnx - sgn (f x)) (at_left x)" 
       using has_sgnx_derivative_at_left[OF g_deriv \<open>g x=0\<close>] that by auto
     moreover have "(g has_sgnx sgn (f x)) (at_right x)" 
       using has_sgnx_derivative_at_right[OF g_deriv \<open>g x=0\<close>] that by auto
     ultimately have "(LIM t at_left x. f t / g t :> at_bot) \<and> (LIM t at_right x. f t / g t :> at_top)"
       using filterlim_divide_at_bot_at_top_iff[OF _ \<open>f x\<noteq>0\<close>, of f] 
       using f_tendsto(1) f_tendsto(2) g_tendsto(1) g_tendsto(2) by blast
     moreover have "sgn c =sgn (f x)" using that by auto
     ultimately show ?thesis unfolding jump_def by auto
   qed
   ultimately show ?thesis using \<open>c\<noteq>0\<close> \<open>f x\<noteq>0\<close> by argo   
 qed
     
 lemma jump_jump_poly: "jump (\<lambda>x. poly q x / poly p x) a = jump_poly q p a"
 proof (cases "p=0")
   case True
   then show ?thesis by auto
 next
   case False
   obtain p' q' where p':"p= p'*gcd p q" and q':"q=q'*gcd p q" 
     using gcd_dvd1 gcd_dvd2 dvd_def[of "gcd p q", simplified mult.commute]  by metis
   then have "coprime p' q'" "p'\<noteq>0" "gcd p q\<noteq>0" using gcd_coprime \<open>p\<noteq>0\<close>  by auto
   
   define f where "f \<equiv> (\<lambda>x. poly q' x / poly p' x)"
   define g where "g \<equiv> (\<lambda>x. if poly (gcd p q) x = 0 then 0::real else 1)"
   
   have g_tendsto:"(g \<longlongrightarrow> 1) (at_left a)" "(g \<longlongrightarrow> 1) (at_right a)"
   proof -
     have
       "(poly (gcd p q) has_sgnx 1) (at_left a) 
           \<or> (poly (gcd p q) has_sgnx - 1) (at_left a)"
       "(poly (gcd p q) has_sgnx 1) (at_right a) 
           \<or> (poly (gcd p q) has_sgnx - 1) (at_right a)"
       using \<open>p\<noteq>0\<close> poly_has_sgnx_values by auto
     then have " \<forall>\<^sub>F x in at_left a. g x = 1" " \<forall>\<^sub>F x in at_right a. g x = 1"
       unfolding has_sgnx_def g_def by (auto elim:eventually_mono)
     then show "(g \<longlongrightarrow> 1) (at_left a)" "(g \<longlongrightarrow> 1) (at_right a)"
       using tendsto_eventually by auto
   qed
     
   have "poly q x / poly p x = g x * f x" for x
     unfolding f_def g_def by (subst p',subst q',auto)
   then have "jump (\<lambda>x. poly q x / poly p x) a = jump (\<lambda>x. g x * f x) a"
     by auto
   also have "... = jump f a"
     unfolding jump_def
     apply (subst (1 2) filterlim_tendsto_pos_mult_at_top_iff)
         prefer 5
         apply (subst (1 2) filterlim_tendsto_pos_mult_at_bot_iff)
     using g_tendsto by auto
   also have "... = jump_poly q' p' a"
     using jump_jump_poly_aux[OF \<open>p'\<noteq>0\<close> \<open>coprime p' q'\<close>] unfolding f_def by auto
   also have "... = jump_poly q p a"
     using jump_poly_mult[OF \<open>gcd p q \<noteq> 0\<close>, of q'] p' q'
     by (metis mult.commute)
   finally show ?thesis .
 qed  
   
  
 lemma jump_Im_divide_Re_0:
   assumes "path g" "Re (g x)\<noteq>0" "0<x" "x<1"
   shows "jump (\<lambda>t. Im (g t) / Re (g t)) x = 0"  
 proof -
   have "isCont g x"
     using \<open>path g\<close>[unfolded path_def] \<open>0<x\<close> \<open>x<1\<close>
     apply (elim continuous_on_interior)
     by auto
   then have "isCont (\<lambda>t. Im(g t)/Re(g t)) x" using \<open>Re (g x)\<noteq>0\<close>  
     by (auto intro:continuous_intros isCont_Re isCont_Im)
   then show "jump (\<lambda>t. Im(g t)/Re(g t)) x=0"
     using jump_not_infinity by auto 
 qed 
   
 lemma jumpF_im_divide_Re_0:
   assumes "path g" "Re (g x)\<noteq>0" 
   shows "\<lbrakk>0\<le>x;x<1\<rbrakk> \<Longrightarrow> jumpF (\<lambda>t. Im (g t) / Re (g t)) (at_right x) = 0"
         "\<lbrakk>0<x;x\<le>1\<rbrakk> \<Longrightarrow> jumpF (\<lambda>t. Im (g t) / Re (g t)) (at_left x) = 0"
 proof -
   define g' where "g' = (\<lambda>t. Im (g t) / Re (g t))"
 
   show "jumpF g' (at_right x) = 0" when "0\<le>x" "x<1"
   proof -
     have "(g' \<longlongrightarrow> g' x) (at_right x)"
     proof (cases "x=0")
       case True
       have "continuous (at_right 0) g" 
         using \<open>path g\<close> unfolding path_def 
         by (auto elim:continuous_on_at_right)
       then have "continuous (at_right x) (\<lambda>t. Im(g t))" "continuous (at_right x) (\<lambda>t. Re(g t))"
         using continuous_Im continuous_Re True by auto
       moreover have "Re (g (netlimit (at_right x))) \<noteq> 0" 
         using assms(2) by (simp add: Lim_ident_at)
       ultimately have "continuous (at_right x) (\<lambda>t. Im (g t)/Re(g t))"  
         by (auto intro:continuous_divide)
       then show ?thesis unfolding g'_def continuous_def 
         by (simp add: Lim_ident_at)
     next
       case False
       have "isCont (\<lambda>x. Im (g x)) x" "isCont (\<lambda>x. Re (g x)) x" 
         using \<open>path g\<close> unfolding path_def
         by (metis False atLeastAtMost_iff at_within_Icc_at continuous_Im continuous_Re
           continuous_on_eq_continuous_within less_le that)+      
       then have "isCont g' x" 
         using assms(2) unfolding g'_def
         by (auto intro:continuous_intros)
       then show ?thesis unfolding isCont_def using filterlim_at_split by blast
     qed
     then have "\<not> filterlim g' at_top (at_right x)" "\<not> filterlim g' at_bot (at_right x)"
       using filterlim_at_top_nhds[of g' "at_right x"] filterlim_at_bot_nhds[of g' "at_right x"]
       by auto
     then show ?thesis unfolding jumpF_def by auto
   qed
     
   show "jumpF g' (at_left x) = 0" when "0<x" "x\<le>1"
   proof -
     have "(g' \<longlongrightarrow> g' x) (at_left x)"
     proof (cases "x=1")
       case True
       have "continuous (at_left 1) g" 
         using \<open>path g\<close> unfolding path_def 
         by (auto elim:continuous_on_at_left)
       then have "continuous (at_left x) (\<lambda>t. Im(g t))" "continuous (at_left x) (\<lambda>t. Re(g t))"
         using continuous_Im continuous_Re True by auto
       moreover have "Re (g (netlimit (at_left x))) \<noteq> 0" 
         using assms(2) by (simp add: Lim_ident_at)
       ultimately have "continuous (at_left x) (\<lambda>t. Im (g t)/Re(g t))"  
         by (auto intro:continuous_divide)
       then show ?thesis unfolding g'_def continuous_def 
         by (simp add: Lim_ident_at)
     next
       case False
       have "isCont (\<lambda>x. Im (g x)) x" "isCont (\<lambda>x. Re (g x)) x" 
         using \<open>path g\<close> unfolding path_def
         by (metis False atLeastAtMost_iff at_within_Icc_at continuous_Im continuous_Re
           continuous_on_eq_continuous_within less_le that)+      
       then have "isCont g' x" 
         using assms(2) unfolding g'_def
         by (auto)
       then show ?thesis unfolding isCont_def using filterlim_at_split by blast
     qed
     then have "\<not> filterlim g' at_top (at_left x)" "\<not> filterlim g' at_bot (at_left x)"
       using filterlim_at_top_nhds[of g' "at_left x"] filterlim_at_bot_nhds[of g' "at_left x"]
       by auto
     then show ?thesis unfolding jumpF_def by auto
   qed 
 qed
   
 lemma jump_cong:
   assumes "x=y" and "eventually (\<lambda>x. f x=g x) (at x)"
   shows "jump f x = jump g y"
 proof -
   have left:"eventually (\<lambda>x. f x=g x) (at_left x)" 
     and right:"eventually (\<lambda>x. f x=g x) (at_right x)"
     using assms(2) eventually_at_split by blast+
   from filterlim_cong[OF _ _ this(1)] filterlim_cong[OF _ _ this(2)] 
   show ?thesis unfolding jump_def using assms(1) by fastforce
 qed 
     
 lemma jumpF_cong:
   assumes "F=G" and "eventually (\<lambda>x. f x=g x) F"
   shows "jumpF f F = jumpF g G"
 proof -
   have "\<forall>\<^sub>F r in G. f r = g r"
     using assms(1) assms(2) by force
   then show ?thesis
     by (simp add: assms(1) filterlim_cong jumpF_def)
 qed
   
 lemma jump_at_left_at_right_eq:
   assumes "isCont f x" and "f x \<noteq> 0" and sgnx_eq:"sgnx g (at_left x) = sgnx g (at_right x)"
   shows "jump (\<lambda>t. f t/g t) x = 0"  
 proof -
   define c where "c = sgn (f x)"
   then have "c\<noteq>0" using \<open>f x\<noteq>0\<close> by (simp add: sgn_zero_iff)
   have f_tendsto:"(f \<longlongrightarrow> f x) (at_left x)" " (f \<longlongrightarrow> f x) (at_right x)"
     using \<open>isCont f x\<close> Lim_at_imp_Lim_at_within isCont_def by blast+
   have False when "(g has_sgnx - c) (at_left x)" "(g has_sgnx c) (at_right x)" 
   proof -
     have "sgnx g (at_left x) = -c" using that(1) by auto
     moreover have "sgnx g (at_right x) = c" using that(2) by auto
     ultimately show False using sgnx_eq \<open>c\<noteq>0\<close> by force
   qed
   moreover have False when "(g has_sgnx c) (at_left x)" "(g has_sgnx - c) (at_right x)" 
   proof -
     have "sgnx g (at_left x) = c" using that(1) by auto
     moreover have "sgnx g (at_right x) = - c" using that(2) by auto
     ultimately show False using sgnx_eq \<open>c\<noteq>0\<close> by force
   qed  
   ultimately show ?thesis
     unfolding jump_def
     by (auto simp add:f_tendsto filterlim_divide_at_bot_at_top_iff[OF _ \<open>f x \<noteq> 0\<close>] c_def)  
 qed  
   
 lemma jumpF_pos_has_sgnx:
   assumes "jumpF f F > 0"
   shows "(f has_sgnx 1) F"
 proof -
   have "filterlim f at_top F" using assms unfolding jumpF_def by argo
   then have "eventually (\<lambda>x. f x>0) F" using filterlim_at_top_dense[of f F] by blast
   then show ?thesis unfolding has_sgnx_def 
     apply (elim eventually_mono)
     by auto
 qed  
 
 lemma jumpF_neg_has_sgnx:
   assumes "jumpF f F < 0"
   shows "(f has_sgnx -1) F"
 proof -
   have "filterlim f at_bot F" using assms unfolding jumpF_def by argo
   then have "eventually (\<lambda>x. f x<0) F" using filterlim_at_bot_dense[of f F] by blast
   then show ?thesis unfolding has_sgnx_def 
     apply (elim eventually_mono)
     by auto
 qed
 
   
 lemma jumpF_IVT:
   fixes f::"real \<Rightarrow> real" and a b::real
   defines "right\<equiv>(\<lambda>(R::real \<Rightarrow> real \<Rightarrow> bool). R (jumpF f (at_right a)) 0 
                       \<or> (continuous (at_right a) f \<and> R (f a) 0))"
     and
           "left\<equiv>(\<lambda>(R::real \<Rightarrow> real \<Rightarrow> bool). R (jumpF f (at_left b)) 0 
                       \<or> (continuous (at_left b) f \<and> R (f b) 0))"
   assumes "a<b" and cont:"continuous_on {a<..<b} f" and
     right_left:"right greater \<and> left less \<or> right less \<and> left greater" 
   shows "\<exists>x. a<x \<and> x<b \<and> f x =0"
 proof -
   have ?thesis when "right greater" "left less" 
   proof -
     have "(f has_sgnx 1) (at_right a)"
     proof -
       have ?thesis when "jumpF f (at_right a)>0" using jumpF_pos_has_sgnx[OF that] .
       moreover have ?thesis when "f a > 0" "continuous (at_right a) f"
       proof -
         have "(f \<longlongrightarrow> f a) (at_right a)" using that(2) by (simp add: continuous_within)
         then show ?thesis
           using tendsto_nonzero_has_sgnx[of f "f a" "at_right a"] that by auto
       qed
       ultimately show ?thesis using that(1) unfolding right_def by auto
     qed
     then obtain a' where "a<a'" and a'_def:"\<forall>y. a<y \<and> y < a' \<longrightarrow> f y > 0" 
       unfolding has_sgnx_def eventually_at_right using sgn_1_pos by auto
     have "(f has_sgnx - 1) (at_left b)" 
     proof -
       have ?thesis when "jumpF f (at_left b)<0" using jumpF_neg_has_sgnx[OF that] .
       moreover have ?thesis when "f b < 0" "continuous (at_left b) f" 
       proof -
         have "(f \<longlongrightarrow> f b) (at_left b)" 
           using that(2) by (simp add: continuous_within)
         then show ?thesis
           using tendsto_nonzero_has_sgnx[of f "f b" "at_left b"] that by auto
       qed
       ultimately show ?thesis using that(2) unfolding left_def by auto
     qed
     then obtain b' where "b'<b" and b'_def:"\<forall>y. b'<y \<and> y < b \<longrightarrow> f y < 0" 
       unfolding has_sgnx_def eventually_at_left using sgn_1_neg by auto  
     have "a' \<le> b'"
     proof (rule ccontr)
       assume "\<not> a' \<le> b'"
       then have "{a<..<a'} \<inter> {b'<..<b} \<noteq> {}"
         using \<open>a<a'\<close> \<open>b'<b\<close> \<open>a<b\<close> by auto
       then obtain c where "c\<in>{a<..<a'}" "c\<in>{b'<..<b}" by blast
       then have "f c>0" "f c<0"
         using a'_def b'_def by auto
       then show False by auto
     qed
     define a0 where "a0=(a+a')/2"
     define b0 where "b0=(b+b')/2"
     have [simp]:"a<a0" "a0<a'" "a0<b0" "b'<b0" "b0<b" 
       unfolding a0_def b0_def using \<open>a<a'\<close> \<open>b'<b\<close> \<open>a'\<le>b'\<close> by auto   
     have "f a0>0" "f b0<0" using a'_def[rule_format,of a0] b'_def[rule_format,of b0] by auto
     moreover have "continuous_on {a0..b0} f"
       using cont \<open>a < a0\<close> \<open>b0 < b\<close>
       by (meson atLeastAtMost_subseteq_greaterThanLessThan_iff continuous_on_subset)
     ultimately have "\<exists>x>a0. x < b0 \<and> f x = 0"
       using IVT_strict[of 0 f a0 b0] by auto
     then show ?thesis using \<open>a < a0\<close> \<open>b0 < b\<close> 
       by (meson lessThan_strict_subset_iff psubsetE subset_psubset_trans)
   qed
   moreover have ?thesis when "right less" "left greater" 
   proof -
     have "(f has_sgnx -1) (at_right a)"
     proof -
       have ?thesis when "jumpF f (at_right a)<0" using jumpF_neg_has_sgnx[OF that] .
       moreover have ?thesis when "f a < 0" "continuous (at_right a) f"
       proof -
         have "(f \<longlongrightarrow> f a) (at_right a)" 
           using that(2) by (simp add: continuous_within)
         then show ?thesis
           using tendsto_nonzero_has_sgnx[of f "f a" "at_right a"] that by auto
       qed
       ultimately show ?thesis using that(1) unfolding right_def by auto
     qed
     then obtain a' where "a<a'" and a'_def:"\<forall>y. a<y \<and> y < a' \<longrightarrow> f y < 0" 
       unfolding has_sgnx_def eventually_at_right using sgn_1_neg by auto
     have "(f has_sgnx  1) (at_left b)" 
     proof -
       have ?thesis when "jumpF f (at_left b)>0" using jumpF_pos_has_sgnx[OF that] .
       moreover have ?thesis when "f b > 0" "continuous (at_left b) f"
       proof -
         have "(f \<longlongrightarrow> f b) (at_left b)" 
           using that(2) by (simp add: continuous_within)
         then show ?thesis
           using tendsto_nonzero_has_sgnx[of f "f b" "at_left b"] that by auto
       qed
       ultimately show ?thesis using that(2) unfolding left_def by auto
     qed
     then obtain b' where "b'<b" and b'_def:"\<forall>y. b'<y \<and> y < b \<longrightarrow> f y > 0" 
       unfolding has_sgnx_def eventually_at_left using sgn_1_pos by auto  
     have "a' \<le> b'"
     proof (rule ccontr)
       assume "\<not> a' \<le> b'"
       then have "{a<..<a'} \<inter> {b'<..<b} \<noteq> {}"
         using \<open>a<a'\<close> \<open>b'<b\<close> \<open>a<b\<close> by auto
       then obtain c where "c\<in>{a<..<a'}" "c\<in>{b'<..<b}" by blast
       then have "f c>0" "f c<0"
         using a'_def b'_def by auto
       then show False by auto
     qed
     define a0 where "a0=(a+a')/2"
     define b0 where "b0=(b+b')/2"
     have [simp]:"a<a0" "a0<a'" "a0<b0" "b'<b0" "b0<b" 
       unfolding a0_def b0_def using \<open>a<a'\<close> \<open>b'<b\<close> \<open>a'\<le>b'\<close> by auto   
     have "f a0<0" "f b0>0" using a'_def[rule_format,of a0] b'_def[rule_format,of b0] by auto
     moreover have "continuous_on {a0..b0} f"
       using cont  \<open>a < a0\<close> \<open>b0 < b\<close> 
       by (meson atLeastAtMost_subseteq_greaterThanLessThan_iff continuous_on_subset)  
     ultimately have "\<exists>x>a0. x < b0 \<and> f x = 0"
       using IVT_strict[of 0 f a0 b0] by auto
     then show ?thesis using \<open>a < a0\<close> \<open>b0 < b\<close> 
       by (meson lessThan_strict_subset_iff psubsetE subset_psubset_trans)
   qed  
   ultimately show ?thesis using right_left by auto
 qed
    
 lemma jumpF_eventually_const:
   assumes "eventually (\<lambda>x. f x=c) F" "F\<noteq>bot"
   shows "jumpF f F = 0"
 proof -
   have "jumpF f F = jumpF (\<lambda>_. c) F"
     apply (rule jumpF_cong)
     using assms(1) by auto
   also have "... = 0" using jumpF_const[OF \<open>F\<noteq>bot\<close>] by simp
   finally show ?thesis .
 qed
 
 lemma jumpF_tan_comp:
   "jumpF (f o tan) (at_right x) = (if cos x = 0 
       then jumpF f at_bot else jumpF f (at_right (tan x)))"
   "jumpF (f o tan) (at_left x) = (if cos x =0 
       then jumpF f at_top else jumpF f (at_left (tan x)))"
 proof -
   have "filtermap (f \<circ> tan) (at_right x) = 
       (if cos x = 0 then filtermap f at_bot else filtermap f (at_right (tan x)))"
     unfolding comp_def
     apply (subst filtermap_filtermap[of f tan,symmetric])
     using filtermap_tan_at_right_inf filtermap_tan_at_right by auto
   then show "jumpF (f o tan) (at_right x) = (if cos x = 0 
           then jumpF f at_bot else jumpF f (at_right (tan x)))"
     unfolding jumpF_def filterlim_def by auto
 next
   have "filtermap (f \<circ> tan) (at_left x) = 
       (if cos x = 0 then filtermap f at_top else filtermap f (at_left (tan x)))"
     unfolding comp_def
     apply (subst filtermap_filtermap[of f tan,symmetric])
     using filtermap_tan_at_left_inf filtermap_tan_at_left by auto
   then show "jumpF (f o tan) (at_left x) = (if cos x = 0 
           then jumpF f at_top else jumpF f (at_left (tan x)))"
     unfolding jumpF_def filterlim_def by auto
 qed
 
 subsection \<open>Finite jumpFs over an interval\<close>
 
 definition finite_jumpFs::"(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow>  bool" where
   "finite_jumpFs f a b = finite {x. (jumpF f (at_left x) \<noteq>0 \<or> jumpF f (at_right x) \<noteq>0) \<and> a\<le>x \<and> x\<le>b}" 
   
 lemma finite_jumpFs_linear_pos:
   assumes "c>0"
   shows "finite_jumpFs (f o (\<lambda>x. c * x + b)) lb ub \<longleftrightarrow> finite_jumpFs f (c * lb +b) (c * ub + b)" 
 proof -
   define left where "left = (\<lambda>f lb ub. {x. jumpF f (at_left x) \<noteq> 0 \<and> lb \<le> x \<and> x \<le> ub})"
   define right where "right = (\<lambda>f lb ub. {x. jumpF f (at_right x) \<noteq> 0 \<and> lb \<le> x \<and> x \<le> ub})"
   define g where "g=(\<lambda>x. c*x+b)"
   define gi where "gi = (\<lambda>x. (x-b)/c)"
   have "finite_jumpFs (f o (\<lambda>x. c * x + b)) lb ub 
       = finite (left (f o g) lb ub \<union> right (f o g) lb ub)"
     unfolding finite_jumpFs_def  
     apply (rule arg_cong[where f=finite]) 
     by (auto simp add:left_def right_def g_def)
   also have "... = finite (gi ` (left f (g lb) (g ub) \<union> right f (g lb) (g ub)))"
   proof -
     have j_rw:
       "jumpF (f o g) (at_left x) = jumpF f (at_left (g x))" 
       "jumpF (f o g) (at_right x) = jumpF f (at_right (g x))"
         for x 
       using jumpF_linear_comp[of c f b x] \<open>c>0\<close> unfolding g_def by auto
     then have 
         "left (f o g) lb ub = gi ` left f (g lb) (g ub)"
         "right (f o g) lb ub = gi ` right f (g lb) (g ub)"
       unfolding left_def right_def gi_def
       using \<open>c>0\<close> by (auto simp add:g_def field_simps) 
     then have "left (f o g) lb ub \<union> right (f o g) lb ub 
         = gi ` (left f (g lb) (g ub) \<union> right f (g lb) (g ub))"
       by auto
     then show ?thesis by auto
   qed
   also have "... = finite (left f (g lb) (g ub) \<union> right f (g lb) (g ub))"
     apply (rule finite_image_iff)
     unfolding gi_def using \<open>c >0\<close>  inj_on_def by fastforce
   also have "... =  finite_jumpFs f (c * lb +b) (c * ub + b)"   
     unfolding finite_jumpFs_def
     apply (rule arg_cong[where f=finite]) 
     by (auto simp add:left_def right_def g_def)
   finally show ?thesis .
 qed
   
 lemma finite_jumpFs_consts:
   "finite_jumpFs (\<lambda>_ . c) lb ub"
   unfolding finite_jumpFs_def using jumpF_const by auto
     
 lemma finite_jumpFs_combine:
   assumes "finite_jumpFs f a b" "finite_jumpFs f b c" 
   shows "finite_jumpFs f a c"
 proof -
   define P where "P=(\<lambda>x. jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0)"
   have "{x. P x \<and> a \<le> x \<and> x \<le> c} \<subseteq> {x. P x \<and> a \<le> x \<and> x\<le>b} \<union> {x. P x \<and> b \<le>x \<and> x\<le>c}"
     by auto
   moreover have "finite ({x. P x \<and> a \<le> x \<and> x\<le>b} \<union> {x. P x \<and> b \<le>x \<and> x\<le>c})"
     using assms unfolding finite_jumpFs_def P_def by auto
   ultimately have "finite {x. P x \<and> a \<le> x \<and> x \<le> c}" 
     using finite_subset by auto    
   then show ?thesis unfolding finite_jumpFs_def P_def by auto
 qed
     
 lemma finite_jumpFs_subE:
   assumes "finite_jumpFs f a b" "a\<le>a'" "b'\<le>b" 
   shows "finite_jumpFs f a' b'"  
 using assms unfolding finite_jumpFs_def
   apply (elim rev_finite_subset)
   by auto
    
 lemma finite_Psegments_Re_imp_jumpFs:
   assumes "finite_Psegments (\<lambda>t. Re (g t - z) = 0) a b" "continuous_on {a..b} g" 
   shows "finite_jumpFs (\<lambda>t. Im (g t - z)/Re (g t - z)) a b"
     using assms
 proof (induct rule:finite_Psegments.induct)
   case (emptyI a b) 
   then show ?case unfolding finite_jumpFs_def 
     by (auto intro:rev_finite_subset[of "{a}"])    
 next
   case (insertI_1 s a b)
   define f where "f=(\<lambda>t. Im (g t - z) / Re (g t - z))"
   have "finite_jumpFs f a s" 
   proof -
     have "continuous_on {a..s} g" using \<open>continuous_on {a..b} g\<close> \<open>s \<in> {a..<b}\<close> 
       by (auto elim:continuous_on_subset)
     then show ?thesis using insertI_1 unfolding f_def by auto
   qed
   moreover have "finite_jumpFs f s b" 
   proof -
     have "jumpF f (at_left x) =0" "jumpF f (at_right x) = 0" when "x\<in>{s<..<b}" for x
     proof - 
       show "jumpF f (at_left x) =0"
         apply (rule jumpF_eventually_const[of _ 0])
         unfolding eventually_at_left
          apply (rule exI[where x=s])
         using that insertI_1 unfolding f_def by auto
       show "jumpF f (at_right x) = 0"
         apply (rule jumpF_eventually_const[of _ 0])
         unfolding eventually_at_right
          apply (rule exI[where x=b])
         using that insertI_1 unfolding f_def by auto 
     qed
     then have "{x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> s \<le> x \<and> x \<le> b}
           = {x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> (x=s \<or> x = b)}"
       using  \<open>s\<in>{a..<b}\<close> by force
     then show ?thesis unfolding finite_jumpFs_def by auto
   qed
   ultimately show ?case using finite_jumpFs_combine[of _ a s b] unfolding f_def by auto
 next
   case (insertI_2 s a b)
   define f where "f=(\<lambda>t. Im (g t - z) / Re (g t - z))"
   have "finite_jumpFs f a s" 
   proof -
     have "continuous_on {a..s} g" using \<open>continuous_on {a..b} g\<close> \<open>s \<in> {a..<b}\<close> 
       by (auto elim:continuous_on_subset)
     then show ?thesis using insertI_2 unfolding f_def by auto
   qed
   moreover have "finite_jumpFs f s b" 
   proof -
     have "jumpF f (at_left x) =0" "jumpF f (at_right x) = 0" when "x\<in>{s<..<b}" for x
     proof - 
       have "isCont f x" 
         unfolding f_def
         apply (intro continuous_intros isCont_Im isCont_Re 
             continuous_on_interior[OF \<open>continuous_on {a..b} g\<close>])
         using insertI_2.hyps(1) that 
           apply auto[2]
         using insertI_2.hyps(3) that by blast
       then show "jumpF f (at_left x) =0" "jumpF f (at_right x) = 0"
         by (simp_all add: continuous_at_split jumpF_not_infinity)
     qed
     then have "{x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> s \<le> x \<and> x \<le> b}
           = {x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> (x=s \<or> x = b)}"
       using  \<open>s\<in>{a..<b}\<close> by force
     then show ?thesis unfolding finite_jumpFs_def by auto
   qed
   ultimately show ?case using finite_jumpFs_combine[of _ a s b] unfolding f_def by auto
 qed
   
 lemma finite_ReZ_segments_imp_jumpFs:
   assumes "finite_ReZ_segments g z" "path g" 
   shows "finite_jumpFs (\<lambda>t. Im (g t - z)/Re (g t - z)) 0 1"
   using assms unfolding finite_ReZ_segments_def path_def
   by (rule finite_Psegments_Re_imp_jumpFs)      
     
 subsection \<open>@{term jumpF} at path ends\<close>  
 
 definition jumpF_pathstart::"(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> real" where
   "jumpF_pathstart g z= jumpF (\<lambda>t. Im(g t- z)/Re(g t - z)) (at_right 0)"
  
 definition jumpF_pathfinish::"(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> real" where
   "jumpF_pathfinish g z= jumpF (\<lambda>t. Im(g t - z)/Re(g t -z)) (at_left 1)"
   
 lemma jumpF_pathstart_eq_0:
   assumes "path g" "Re(pathstart g)\<noteq>Re z"
   shows "jumpF_pathstart g z = 0"
 unfolding jumpF_pathstart_def 
   apply (rule jumpF_im_divide_Re_0)
   using assms[unfolded pathstart_def] by auto
 
 lemma jumpF_pathfinish_eq_0:
   assumes "path g" "Re(pathfinish g)\<noteq>Re z"
   shows "jumpF_pathfinish g z = 0"
 unfolding jumpF_pathfinish_def 
   apply (rule jumpF_im_divide_Re_0)
   using assms[unfolded pathfinish_def] by auto  
    
 lemma 
   shows jumpF_pathfinish_reversepath: "jumpF_pathfinish (reversepath g) z = jumpF_pathstart g z"
     and jumpF_pathstart_reversepath: "jumpF_pathstart (reversepath g) z = jumpF_pathfinish g z"
 proof -
   define f where "f=(\<lambda>t. Im (g t - z) / Re (g t - z))"
   define f' where "f'=(\<lambda>t. Im (reversepath g t - z) / Re (reversepath g t - z))"
   have "f o (\<lambda>t. 1 - t) = f'"
     unfolding f_def f'_def comp_def reversepath_def by auto
   then show "jumpF_pathfinish (reversepath g) z = jumpF_pathstart g z" 
         "jumpF_pathstart (reversepath g) z = jumpF_pathfinish g z" 
     unfolding jumpF_pathstart_def jumpF_pathfinish_def
     using jumpF_linear_comp(2)[of "-1" f 1 0,simplified] jumpF_linear_comp(1)[of "-1" f 1 1,simplified]
     apply (fold f_def f'_def)    
     by auto
 qed
      
 lemma jumpF_pathstart_joinpaths[simp]:
   "jumpF_pathstart (g1+++g2) z = jumpF_pathstart g1 z"
 proof -
   let ?h="(\<lambda>t. Im (g1 t - z) / Re (g1 t - z))"
   let ?f="\<lambda>t. Im ((g1 +++ g2) t - z) / Re ((g1 +++ g2) t - z)"
   have "jumpF_pathstart g1 z = jumpF ?h (at_right 0)"
     unfolding jumpF_pathstart_def by simp
   also have "... = jumpF (?h o (\<lambda>t. 2*t)) (at_right 0)"
     using jumpF_linear_comp[of 2 ?h 0 0,simplified] by auto
   also have "... = jumpF ?f (at_right 0)"
   proof (rule jumpF_cong)
     show " \<forall>\<^sub>F x in at_right 0. (?h \<circ> (*) 2) x =?f x"
       unfolding eventually_at_right
       apply (intro exI[where x="1/2"])
       by (auto simp add:joinpaths_def)
   qed simp
   also have "... =jumpF_pathstart (g1+++g2) z"
     unfolding jumpF_pathstart_def by simp
   finally show ?thesis by simp
 qed
   
 lemma jumpF_pathfinish_joinpaths[simp]:
   "jumpF_pathfinish (g1+++g2) z = jumpF_pathfinish g2 z"
 proof -
   let ?h="(\<lambda>t. Im (g2 t - z) / Re (g2 t - z))"
   let ?f="\<lambda>t. Im ((g1 +++ g2) t - z) / Re ((g1 +++ g2) t - z)"
   have "jumpF_pathfinish g2 z = jumpF ?h (at_left 1)"
     unfolding jumpF_pathfinish_def by simp
   also have "... = jumpF (?h o (\<lambda>t. 2*t-1)) (at_left 1)"
     using jumpF_linear_comp[of 2 _ "-1" 1,simplified] by auto
   also have "... = jumpF ?f (at_left 1)"
   proof (rule jumpF_cong)
     show " \<forall>\<^sub>F x in at_left 1. (?h \<circ> (\<lambda>t. 2 * t - 1)) x =?f x"
       unfolding eventually_at_left
       apply (intro exI[where x="1/2"])
       by (auto simp add:joinpaths_def)
   qed simp
   also have "... =jumpF_pathfinish (g1+++g2) z"
     unfolding jumpF_pathfinish_def by simp
   finally show ?thesis by simp
 qed  
   
 subsection \<open>Cauchy index\<close>  
 
 \<comment>\<open>Deprecated, use "cindexE" if possible\<close>
 definition cindex::"real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> int" where
   "cindex a b f = (\<Sum>x\<in>{x. jump f x\<noteq>0 \<and> a<x \<and> x<b}. jump f x )"
 
 definition cindexE::"real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> real" where
   "cindexE a b f = (\<Sum>x\<in>{x. jumpF f (at_right x) \<noteq>0 \<and> a\<le>x \<and> x<b}. jumpF f (at_right x)) 
                     - (\<Sum>x\<in>{x. jumpF f (at_left x) \<noteq>0 \<and> a<x \<and> x\<le>b}. jumpF f (at_left x))"     
 
 definition cindexE_ubd::"(real \<Rightarrow> real) \<Rightarrow> real" where
   "cindexE_ubd f = (\<Sum>x\<in>{x. jumpF f (at_right x) \<noteq>0 }. jumpF f (at_right x)) 
                       - (\<Sum>x\<in>{x. jumpF f (at_left x) \<noteq>0}. jumpF f (at_left x))"
 
   
 lemma cindexE_empty:
   "cindexE a a f = 0"
   unfolding cindexE_def by (simp add: sum.neutral)     
   
 lemma cindex_const: "cindex a b (\<lambda>_. c) = 0"
   unfolding cindex_def
   apply (rule sum.neutral)
   by auto
       
 lemma cindex_eq_cindex_poly: "cindex a b (\<lambda>x. poly q x/poly p x) = cindex_poly a b q p"  
 proof (cases "p=0")
   case True
   then show ?thesis using cindex_const by auto
 next
   case False
   have "cindex_poly a b q p = 
       (\<Sum>x |jump_poly q p x \<noteq>0 \<and> a < x \<and> x < b. jump_poly q p x)"
     unfolding cindex_poly_def 
     apply (rule sum.mono_neutral_cong_right)
     using jump_poly_not_root by (auto simp add: \<open>p\<noteq>0\<close> poly_roots_finite)
   also have "... = cindex a b (\<lambda>x. poly q x/poly p x)"
     unfolding cindex_def
     apply (rule sum.cong)
     using jump_jump_poly[of q] by auto
   finally show ?thesis by auto
 qed
   
 lemma cindex_combine:
   assumes finite:"finite {x. jump f x\<noteq>0 \<and> a<x \<and> x<c}" and "a<b" "b<c"
   shows "cindex a c f = cindex a b f  + jump f b + cindex b c f"
 proof -
   define ssum where "ssum = (\<lambda>s. sum (jump f) ({x. jump f x\<noteq>0 \<and> a<x \<and> x<c} \<inter> s))"  
   have ssum_union:"ssum (A \<union> B) = ssum A + ssum B" when "A \<inter> B ={}" for A B
   proof -
     define C where "C={x. jump f x \<noteq> 0 \<and> a<x \<and> x<c}"
     have "finite C" using finite unfolding C_def .
     then show ?thesis
       unfolding ssum_def
       apply (fold C_def)
       using sum_Un[of "C \<inter> A" "C \<inter> B"] that
       by (simp add: inf_assoc inf_sup_aci(3) inf_sup_distrib1 sum.union_disjoint)
   qed
   have "cindex a c f = ssum ({a<..<b} \<union> {b} \<union> {b<..<c})"
     unfolding ssum_def cindex_def 
     apply (rule sum.cong[of _ _ "jump f" "jump f",simplified])
     using \<open>a<b\<close> \<open>b<c\<close> by fastforce
   moreover have "cindex a b f = ssum {a<..<b}" 
     unfolding cindex_def ssum_def using \<open>a<b\<close> \<open>b<c\<close> 
     by (intro sum.cong,auto)
   moreover have "jump f b = ssum {b}" 
     unfolding ssum_def using \<open>a<b\<close> \<open>b<c\<close>  by (cases "jump f b=0",auto)
   moreover have "cindex b c f = ssum {b<..<c}" 
     unfolding cindex_def ssum_def using \<open>a<b\<close> \<open>b<c\<close>  by (intro sum.cong,auto)
   ultimately show ?thesis 
     apply (subst (asm) ssum_union,simp)
     by (subst (asm) ssum_union,auto)
 qed
  
 lemma cindexE_combine:
   assumes finite:"finite_jumpFs f a c" and "a\<le>b" "b\<le>c"
   shows "cindexE a c f = cindexE a b f + cindexE b c f" 
 proof -
   define S where "S={x. (jumpF f (at_left x) \<noteq> 0 \<or> jumpF f (at_right x) \<noteq> 0) \<and> a \<le> x \<and> x \<le> c}"
   define A0 where "A0={x. jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < c}"
   define A1 where "A1={x. jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b}" 
   define A2 where "A2={x. jumpF f (at_right x) \<noteq> 0 \<and> b \<le> x \<and> x < c}"
   define B0 where "B0={x. jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> c}"
   define B1 where "B1={x. jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b}" 
   define B2 where "B2={x. jumpF f (at_left x) \<noteq> 0 \<and> b < x \<and> x \<le> c}"
   have [simp]:"finite A1" "finite A2" "finite B1" "finite B2" 
   proof -
     have "finite S" using finite unfolding finite_jumpFs_def S_def by auto
     moreover have "A1 \<subseteq> S" "A2 \<subseteq> S" "B1 \<subseteq> S" "B2 \<subseteq> S"
       unfolding A1_def A2_def B1_def B2_def S_def using \<open>a\<le>b\<close> \<open>b\<le>c\<close> by auto
     ultimately show "finite A1" "finite A2" "finite B1" "finite B2" by (auto elim:finite_subset)
   qed 
   have "cindexE a c f = sum (\<lambda>x. jumpF f (at_right x)) A0 
         - sum (\<lambda>x. jumpF f (at_left x)) B0"
     unfolding cindexE_def A0_def B0_def by auto
   also have "... = sum (\<lambda>x. jumpF f (at_right x)) (A1 \<union> A2) 
         - sum (\<lambda>x. jumpF f (at_left x)) (B1 \<union> B2)"
   proof -
     have "A0=A1\<union>A2" unfolding A0_def A1_def A2_def using assms by auto
     moreover have "B0=B1\<union>B2" unfolding B0_def B1_def B2_def using assms by auto
     ultimately show ?thesis by auto
   qed
   also have "... = cindexE a b f + cindexE b c f"
   proof -
     have "A1 \<inter> A2 = {}" unfolding A1_def A2_def by auto
     moreover have "B1 \<inter> B2 = {}" unfolding B1_def B2_def by auto
     ultimately show ?thesis
       unfolding cindexE_def  
       apply (fold A1_def A2_def B1_def B2_def)
       by (auto simp add:sum.union_disjoint) 
   qed
   finally show ?thesis .
 qed
 
 lemma cindex_linear_comp:
   assumes "c\<noteq>0"
   shows "cindex lb ub (f o (\<lambda>x. c*x+b)) = (if c>0 
     then cindex (c*lb+b) (c*ub+b) f 
     else - cindex (c*ub+b) (c*lb+b) f)"
 proof (cases "c>0")
   case False
   then have "c<0" using \<open>c\<noteq>0\<close> by auto
   have "cindex lb ub (f o (\<lambda>x. c*x+b)) = - cindex (c*ub+b) (c*lb+b) f"
     unfolding cindex_def
     apply (subst sum_negf[symmetric])
     apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
     subgoal by (simp add: inj_on_def)
     subgoal using False
       apply (subst jump_linear_comp[OF \<open>c\<noteq>0\<close>])
       by (auto simp add:\<open>c<0\<close> \<open>c\<noteq>0\<close> field_simps)
     subgoal for x
       apply (subst jump_linear_comp[OF \<open>c\<noteq>0\<close>])
       by (auto simp add:\<open>c<0\<close> \<open>c\<noteq>0\<close> False field_simps)
     done 
   then show ?thesis using False by auto
 next
   case True
   have "cindex lb ub (f o (\<lambda>x. c*x+b)) = cindex (c*lb+b) (c*ub+b) f"
     unfolding cindex_def
     apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
     subgoal by (simp add: inj_on_def)
     subgoal 
       apply (subst jump_linear_comp[OF \<open>c\<noteq>0\<close>])
       by (auto simp add: True \<open>c\<noteq>0\<close> field_simps)
     subgoal for x
       apply (subst jump_linear_comp[OF \<open>c\<noteq>0\<close>])
       by (auto simp add: \<open>c\<noteq>0\<close> True field_simps)
     done 
   then show ?thesis using True by auto
 qed     
 
 lemma cindexE_linear_comp: 
   assumes "c\<noteq>0"
   shows "cindexE lb ub (f o (\<lambda>x. c*x+b)) = (if c>0 
     then cindexE (c*lb+b) (c*ub+b) f 
     else - cindexE (c*ub+b) (c*lb+b) f)"  
 proof -
   define cright where "cright = (\<lambda>lb ub f. (\<Sum>x | jumpF f (at_right x) \<noteq> 0 \<and> lb \<le> x \<and> x < ub. 
                         jumpF f (at_right x)))"
   define cleft where "cleft = (\<lambda>lb ub f. (\<Sum>x | jumpF f (at_left x) \<noteq> 0 \<and> lb < x \<and> x \<le> ub. 
                         jumpF f (at_left x)))"
   have cindexE_unfold:"cindexE lb ub f = cright lb ub f - cleft lb ub f"
     for lb ub f unfolding cindexE_def cright_def cleft_def by auto
   have ?thesis when "c<0"
   proof -
     have "cright lb ub (f \<circ> (\<lambda>x. c * x + b)) = cleft (c * ub + b) (c * lb + b) f" 
       unfolding cright_def cleft_def
       apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
       subgoal by (simp add: inj_on_def)
       subgoal using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add:field_simps)
       subgoal for x using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add: field_simps)
       done  
     moreover have "cleft lb ub (f \<circ> (\<lambda>x. c * x + b)) = cright (c*ub+b) (c*lb + b) f" 
       unfolding cright_def cleft_def
       apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
       subgoal by (simp add: inj_on_def)
       subgoal using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add:field_simps)
       subgoal for x using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add: field_simps)
       done 
     ultimately show ?thesis unfolding cindexE_unfold using that by auto 
   qed
   moreover have ?thesis when "c>0"
   proof -
     have "cright lb ub (f \<circ> (\<lambda>x. c * x + b)) = cright (c * lb + b) (c * ub + b) f" 
       unfolding cright_def cleft_def
       apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
       subgoal by (simp add: inj_on_def)
       subgoal using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add:field_simps)
       subgoal for x using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add: field_simps)
       done   
     moreover have "cleft lb ub (f \<circ> (\<lambda>x. c * x + b)) = cleft (c*lb+b) (c*ub + b) f" 
       unfolding cright_def cleft_def
       apply (intro sum.reindex_cong[of "\<lambda>x. (x-b)/c"])
       subgoal by (simp add: inj_on_def)
       subgoal using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add:field_simps)
       subgoal for x using that
         by (subst jumpF_linear_comp[OF \<open>c\<noteq>0\<close>],auto simp add: field_simps)
       done 
     ultimately show ?thesis unfolding cindexE_unfold using that by auto
   qed
   ultimately show ?thesis using \<open>c\<noteq>0\<close> by auto
 qed
 
 lemma cindexE_cong:
   assumes "finite s" and fg_eq:"\<And>x. \<lbrakk>a<x;x<b;x\<notin>s\<rbrakk> \<Longrightarrow> f x = g x"
   shows "cindexE a b f = cindexE a b g"
 proof -
   define left where 
       "left=(\<lambda>f. (\<Sum>x | jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b. jumpF f (at_left x)))"
   define right where 
       "right=(\<lambda>f. (\<Sum>x | jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b. jumpF f (at_right x)))"
   have "left f = left g"
   proof -
     have "jumpF f (at_left x) = jumpF g (at_left x)" when "a<x" "x\<le>b" for x 
     proof (rule jumpF_cong)
       define cs where "cs \<equiv> {y\<in>s. a<y \<and> y<x}"
       define c where "c\<equiv> (if cs = {} then (x+a)/2 else Max cs)"
       have "finite cs" unfolding cs_def using assms(1) by auto
       have "c<x \<and> (\<forall>y. c<y \<and> y<x \<longrightarrow> f y=g y)" 
       proof (cases "cs={}")
         case True
         then have "\<forall>y. c<y \<and> y<x \<longrightarrow> y \<notin> s" unfolding cs_def c_def by force
         moreover have "c=(x+a)/2" using True unfolding c_def by auto
         ultimately show ?thesis using fg_eq using that by auto
       next
         case False
         then have "c\<in>cs" unfolding c_def using False \<open>finite cs\<close> by auto
         moreover have "\<forall>y. c<y \<and> y<x \<longrightarrow> y \<notin> s" 
         proof (rule ccontr)
           assume "\<not> (\<forall>y. c < y \<and> y < x \<longrightarrow> y \<notin> s) "
           then obtain y' where "c<y'" "y'<x" "y'\<in>s" by auto
           then have "y'\<in>cs" using \<open>c\<in>cs\<close> unfolding cs_def by auto
           then have "y'\<le>c" unfolding c_def using False \<open>finite cs\<close> by auto
           then show False using \<open>c<y'\<close> by auto
         qed
         ultimately show ?thesis unfolding cs_def using that by (auto intro!:fg_eq)
       qed  
       then show "\<forall>\<^sub>F x in at_left x. f x = g x"
         unfolding eventually_at_left by auto
     qed simp
     then show ?thesis
       unfolding left_def
       by (auto intro: sum.cong)
   qed
   moreover have "right f = right g"
   proof -
     have "jumpF f (at_right x) = jumpF g (at_right x)" when "a\<le>x" "x<b" for x 
     proof (rule jumpF_cong)
       define cs where "cs \<equiv> {y\<in>s. x<y \<and> y<b}"
       define c where "c\<equiv> (if cs = {} then (x+b)/2 else Min cs)"
       have "finite cs" unfolding cs_def using assms(1) by auto
       have "x<c \<and> (\<forall>y. x<y \<and> y<c \<longrightarrow> f y=g y)" 
       proof (cases "cs={}")
         case True
         then have "\<forall>y. x<y \<and> y<c \<longrightarrow> y \<notin> s" unfolding cs_def c_def by force
         moreover have "c=(x+b)/2" using True unfolding c_def by auto
         ultimately show ?thesis using fg_eq using that by auto
       next
         case False
         then have "c\<in>cs" unfolding c_def using False \<open>finite cs\<close> by auto
         moreover have "\<forall>y. x<y \<and> y<c \<longrightarrow> y \<notin> s" 
         proof (rule ccontr)
           assume "\<not> (\<forall>y. x < y \<and> y < c \<longrightarrow> y \<notin> s) "
           then obtain y' where "x<y'" "y'<c" "y'\<in>s" by auto
           then have "y'\<in>cs" using \<open>c\<in>cs\<close> unfolding cs_def by auto
           then have "y'\<ge>c" unfolding c_def using False \<open>finite cs\<close> by auto
           then show False using \<open>c>y'\<close> by auto
         qed
         ultimately show ?thesis unfolding cs_def using that by (auto intro!:fg_eq)
       qed  
       then show "\<forall>\<^sub>F x in at_right x. f x = g x"
         unfolding eventually_at_right by auto
     qed simp
     then show ?thesis
       unfolding right_def
       by (auto intro: sum.cong)
   qed
   ultimately show ?thesis unfolding cindexE_def left_def right_def by presburger
 qed
       
 lemma cindexE_constI:
   assumes "\<And>t. \<lbrakk>a<t;t<b\<rbrakk> \<Longrightarrow> f t=c"
   shows "cindexE a b f = 0"
 proof -
   define left where 
       "left=(\<lambda>f. (\<Sum>x | jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b. jumpF f (at_left x)))"
   define right where 
       "right=(\<lambda>f. (\<Sum>x | jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b. jumpF f (at_right x)))"
   have "left f = 0"  
   proof -
     have "jumpF f (at_left x) = 0" when "a<x" "x\<le>b" for x 
       apply (rule jumpF_eventually_const[of _ c])
       unfolding eventually_at_left using assms that by auto
     then show ?thesis unfolding left_def by auto
   qed
   moreover have "right f = 0"  
   proof -
     have "jumpF f (at_right x) = 0" when "a\<le>x" "x<b" for x 
       apply (rule jumpF_eventually_const[of _ c])
       unfolding eventually_at_right using assms that by auto
     then show ?thesis unfolding right_def by auto
   qed
   ultimately show ?thesis unfolding cindexE_def left_def right_def by auto
 qed
 
 lemma cindex_eq_cindexE_divide:
   fixes f g::"real \<Rightarrow> real"
   defines "h \<equiv> (\<lambda>x. f x/g x)"
   assumes "a<b" and
     finite_fg: "finite {x. (f x=0\<or>g x=0) \<and> a\<le>x\<and>x\<le>b}" and 
     g_imp_f:"\<forall>x\<in>{a..b}. g x=0 \<longrightarrow> f x\<noteq>0" and
     f_cont:"continuous_on {a..b} f" and
     g_cont:"continuous_on {a..b} g"
   shows "cindexE a b h = jumpF h (at_right a) + cindex a b h - jumpF h (at_left b)"  
 proof -
   define R where "R=(\<lambda>S.{x. jumpF h (at_right x) \<noteq> 0 \<and> x\<in>S})"
   define L where "L=(\<lambda>S.{x. jumpF h (at_left x) \<noteq> 0 \<and> x\<in>S})"
   define right where "right = (\<lambda>S. (\<Sum>x\<in>R S. jumpF h (at_right x)))"
   define left where "left = (\<lambda>S. (\<Sum>x\<in>L S. jumpF h (at_left x)))"
   have jump_gnz:"jumpF h (at_left x) = 0" "jumpF h (at_right x) = 0" "jump h x=0" 
       when "a<x" "x<b" "g x\<noteq>0" for x
     proof -
       have "isCont h x" unfolding h_def using f_cont g_cont that
         by (auto intro!:continuous_intros elim:continuous_on_interior)
       then show "jumpF h (at_left x) = 0" "jumpF h (at_right x) = 0" "jump h x=0" 
         using jumpF_not_infinity jump_not_infinity unfolding continuous_at_split by auto
     qed  
     
   have finite_jFs:"finite_jumpFs h a b" 
   proof -
     define S where "S=(\<lambda>s. {x. (jumpF h (at_left x) \<noteq> 0 \<or> jumpF h (at_right x) \<noteq> 0) \<and> x\<in>s})"
     note jump_gnz
     then have "S {a<..<b} \<subseteq> {x. (f x=0\<or>g x=0) \<and> a\<le>x\<and>x\<le>b}"
       unfolding S_def by auto
     then have "finite (S {a<..<b})" 
       using rev_finite_subset[OF finite_fg] by auto
     moreover have "finite (S {a,b})" unfolding S_def by auto
     moreover have "S {a..b} = S {a<..<b} \<union> S {a,b}" 
       unfolding S_def using \<open>a<b\<close> by auto
     ultimately have "finite (S {a..b})" by auto
     then show ?thesis unfolding S_def finite_jumpFs_def by auto
   qed
   have "cindexE a b h = right {a..<b} - left {a<..b}"
     unfolding cindexE_def right_def left_def R_def L_def by auto
   also have "... = jumpF h (at_right a) +  right {a<..<b} - left {a<..<b} - jumpF h (at_left b)"
   proof -
     have "right {a..<b} = jumpF h (at_right a) +  right {a<..<b}"
     proof (cases "jumpF h (at_right a) =0")
       case True
       then have "R {a..<b} = R {a<..<b}"
         unfolding R_def using less_eq_real_def by auto
       then have "right {a..<b} = right {a<..<b}"
         unfolding right_def by auto
       then show ?thesis using True by auto
     next
       case False
       have "finite (R {a..<b})"
         using finite_jFs unfolding R_def finite_jumpFs_def 
         by (auto elim:rev_finite_subset)
       moreover have "a \<in> R {a..<b}" using False \<open>a<b\<close> unfolding R_def by auto
       moreover have "R {a..<b} - {a} = R {a<..<b}" unfolding R_def by auto
       ultimately show "right {a..<b}= jumpF h (at_right a) 
             + right {a<..<b}"
         using sum.remove[of "R {a..<b}" a "\<lambda>x. jumpF h (at_right x)"]
         unfolding right_def by simp
     qed
     moreover have "left {a<..b} = jumpF h (at_left b) +  left {a<..<b}"
     proof (cases "jumpF h (at_left b) =0")
       case True
       then have "L {a<..b} = L {a<..<b}"
         unfolding L_def using less_eq_real_def by auto
       then have "left {a<..b} = left {a<..<b}"
         unfolding left_def by auto
       then show ?thesis using True by auto
     next
       case False
       have "finite (L {a<..b})"
         using finite_jFs unfolding L_def finite_jumpFs_def 
         by (auto elim:rev_finite_subset)
       moreover have "b \<in> L {a<..b}" using False \<open>a<b\<close> unfolding L_def by auto
       moreover have "L {a<..b} - {b} = L {a<..<b}" unfolding L_def by auto
       ultimately show "left {a<..b}= jumpF h (at_left b) + left {a<..<b}"
         using sum.remove[of "L {a<..b}" b "\<lambda>x. jumpF h (at_left x)"]
         unfolding left_def by simp
     qed  
     ultimately show ?thesis by simp
   qed
   also have "... = jumpF h (at_right a) + cindex a b h - jumpF h (at_left b)"
   proof -
     define S where "S={x. g x=0 \<and> a < x \<and> x < b}"
     have "right {a<..<b} = sum (\<lambda>x. jumpF h (at_right x)) S"
       unfolding right_def S_def R_def
       apply (rule sum.mono_neutral_left)
       subgoal using finite_fg by (auto elim:rev_finite_subset)
       subgoal using jump_gnz by auto
       subgoal by auto
       done
     moreover have "left {a<..<b} = sum (\<lambda>x. jumpF h (at_left x)) S"
       unfolding left_def S_def L_def
       apply (rule sum.mono_neutral_left)
       subgoal using finite_fg by (auto elim:rev_finite_subset)
       subgoal using jump_gnz by auto
       subgoal by auto
       done   
     ultimately have "right {a<..<b} - left {a<..<b} 
         = sum (\<lambda>x. jumpF h (at_right x) - jumpF h (at_left x)) S"
       by (simp add: sum_subtractf)
     also have "... = sum (\<lambda>x. of_int(jump h x)) S"
     proof (rule sum.cong)
       fix x assume "x\<in>S"
       define hr where "hr = sgnx h (at_right x)"
       define hl where "hl = sgnx h (at_left x)"
       have "h sgnx_able (at_left x)" "hr\<noteq>0" "h sgnx_able (at_right x)" "hl\<noteq>0" 
       proof -
         have "finite {t. h t = 0 \<and> a < t \<and> t < b}" 
           using finite_fg unfolding h_def by (auto elim!:rev_finite_subset)
         moreover have "continuous_on ({a<..<b} - {x. g x = 0 \<and> a < x \<and> x < b}) h" 
           unfolding h_def using f_cont g_cont
           by (auto intro!: continuous_intros elim:continuous_on_subset) 
         moreover have "finite {x. g x = 0 \<and> a < x \<and> x < b}" 
           using finite_fg by (auto elim!:rev_finite_subset)
         moreover have  " x \<in> {a<..<b}" 
           using \<open>x\<in>S\<close> unfolding S_def by auto
         ultimately show "h sgnx_able (at_left x)"  "hl\<noteq>0" "h sgnx_able (at_right x)"  "hr\<noteq>0" 
           using  finite_sgnx_at_left_at_right[of h a b "{x. g x=0 \<and> a<x\<and>x<b}" x] 
           unfolding hl_def hr_def by blast+
       qed
       then have "(h has_sgnx hl) (at_left x)" "(h has_sgnx hr) (at_right x)"
         unfolding hl_def hr_def using sgnx_able_sgnx by blast+
       moreover have "isCont (inverse \<circ> h) x" 
       proof -
         have "f x\<noteq>0" using \<open>x\<in>S\<close> g_imp_f unfolding S_def by auto
         then show ?thesis using f_cont g_cont \<open>x\<in>S\<close> unfolding h_def S_def
           by (auto simp add:comp_def intro!:continuous_intros elim:continuous_on_interior)
       qed
       ultimately show "jumpF h (at_right x) - jumpF h (at_left x) = real_of_int (jump h x)" 
         using jump_jumpF[of x h] \<open>hr\<noteq>0\<close> \<open>hl\<noteq>0\<close> by auto
     qed auto
     also have "... = cindex a b h"
       unfolding cindex_def of_int_sum S_def
       apply (rule sum.mono_neutral_cong_right)
       using jump_gnz finite_fg by (auto elim:rev_finite_subset)  
     finally show ?thesis by simp
   qed
   finally show ?thesis .
 qed  
 
 subsection \<open>Cauchy index along a path\<close>
 
 \<comment>\<open>Deprecated, use "cindex\_pathE" if possible\<close>
 definition cindex_path::"(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
   "cindex_path g z = cindex 0 1 (\<lambda>t. Im (g t - z) / Re (g t - z))" 
    
 definition cindex_pathE::"(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> real" where
   "cindex_pathE g z = cindexE 0 1 (\<lambda>t. Im (g t - z) / Re (g t - z))"
 
 lemma cindex_pathE_point: "cindex_pathE (linepath a a) b = 0"
   unfolding cindex_pathE_def by (simp add:cindexE_constI)
 
 lemma cindex_path_reversepath:
   "cindex_path (reversepath g) z = - cindex_path g z"
 proof -
   define f where "f=(\<lambda>t. Im (g t - z) / Re (g t - z))"
   define f' where "f'=(\<lambda>t. Im (reversepath g t - z) / Re (reversepath g t - z))"
   have "f o (\<lambda>t. 1 - t) = f'"
     unfolding f_def f'_def comp_def reversepath_def by auto
   then have "cindex 0 1 f' = - cindex 0 1 f"   
     using cindex_linear_comp[of "-1" 0 1 f 1,simplified] by simp
   then show ?thesis 
     unfolding cindex_path_def 
     apply (fold f_def f'_def)
     by simp
 qed 
 
 lemma cindex_pathE_reversepath: "cindex_pathE (reversepath g) z = -cindex_pathE g z"
   using cindexE_linear_comp[of "-1" 0 1 "\<lambda>t. (Im (g t) - Im z) / (Re (g t) - Re z)" 1]  
   by (simp add: cindex_pathE_def reversepath_def o_def)
 
 lemma cindex_pathE_reversepath': "cindex_pathE g z = -cindex_pathE (reversepath g) z"
   using cindexE_linear_comp[of "-1" 0 1 "\<lambda>t. (Im (g t) - Im z) / (Re (g t) - Re z)" 1]  
   by (simp add: cindex_pathE_def reversepath_def o_def)
 
 lemma cindex_pathE_joinpaths:
   assumes g1:"finite_ReZ_segments g1 z" and g2: "finite_ReZ_segments g2 z" and
     "path g1" "path g2" "pathfinish g1 = pathstart g2"
   shows "cindex_pathE (g1+++g2) z = cindex_pathE g1 z + cindex_pathE g2 z"
 proof -
   define f where "f = (\<lambda>g (t::real). Im (g t - z) / Re (g t - z))"
   have "cindex_pathE (g1 +++ g2) z =  cindexE 0 1 (f (g1+++g2))"
     unfolding cindex_pathE_def f_def by auto
   also have "... = cindexE 0 (1/2) (f (g1+++g2)) + cindexE (1/2) 1 (f (g1+++g2))"
   proof (rule cindexE_combine)
     show "finite_jumpFs (f (g1 +++ g2)) 0 1" 
       unfolding f_def
       apply (rule finite_ReZ_segments_imp_jumpFs)
       subgoal using finite_ReZ_segments_joinpaths[OF g1 g2] assms(3-5) .
       subgoal using path_join_imp[OF \<open>path g1\<close> \<open>path g2\<close> \<open>pathfinish g1=pathstart g2\<close>] .
       done
   qed auto
   also have "... = cindex_pathE g1 z + cindex_pathE g2 z"
   proof -
     have "cindexE 0 (1/2) (f (g1+++g2)) = cindex_pathE g1 z"
     proof -
       have "cindexE 0 (1/2) (f (g1+++g2)) = cindexE 0 (1/2) (f g1 o ((*) 2))"
         apply (rule cindexE_cong)
         unfolding comp_def joinpaths_def f_def by auto
       also have "... = cindexE 0 1 (f g1)"
         using cindexE_linear_comp[of 2 0 "1/2" _ 0,simplified] by simp
       also have "... = cindex_pathE g1 z"
         unfolding cindex_pathE_def f_def by auto
       finally show ?thesis .
     qed
     moreover have "cindexE (1/2) 1 (f (g1+++g2)) = cindex_pathE g2 z"
     proof -
       have "cindexE (1/2) 1 (f (g1+++g2)) = cindexE (1/2) 1 (f g2 o (\<lambda>x. 2*x  - 1))"
         apply (rule cindexE_cong)
         unfolding comp_def joinpaths_def f_def by auto
       also have "... = cindexE 0 1 (f g2)"
         using cindexE_linear_comp[of 2 "1/2" 1 _ "-1",simplified] by simp
       also have "... = cindex_pathE g2 z"
         unfolding cindex_pathE_def f_def by auto
       finally show ?thesis .
     qed  
     ultimately show ?thesis by simp
   qed
   finally show ?thesis .
 qed
 
 lemma cindex_pathE_constI:
   assumes "\<And>t. \<lbrakk>0<t;t<1\<rbrakk> \<Longrightarrow> g t=c"
   shows "cindex_pathE g z = 0"
   unfolding cindex_pathE_def
   apply (rule cindexE_constI)
   using assms by auto
   
 lemma cindex_pathE_subpath_combine:
   assumes g:"finite_ReZ_segments g z"and "path g" and
      "0\<le>a" "a\<le>b" "b\<le>c" "c\<le>1"
   shows "cindex_pathE (subpath a b g) z + cindex_pathE (subpath b c g) z 
           = cindex_pathE (subpath a c g) z"
 proof -
   define f where "f = (\<lambda>t. Im (g t - z) / Re (g t - z))"
   have ?thesis when "a=b"
   proof -
     have "cindex_pathE (subpath a b g) z = 0"
       apply (rule cindex_pathE_constI)
       using that unfolding subpath_def by auto
     then show ?thesis using that by auto
   qed
   moreover have ?thesis when "b=c" 
   proof -
     have "cindex_pathE (subpath b c g) z = 0"
       apply (rule cindex_pathE_constI)
       using that unfolding subpath_def by auto
     then show ?thesis using that by auto
   qed
   moreover have ?thesis when "a\<noteq>b" "b\<noteq>c"
   proof -
     have  [simp]:"a<b" "b<c" "a<c" 
       using that \<open>a\<le>b\<close> \<open>b\<le>c\<close> by auto
     have "cindex_pathE (subpath a b g) z = cindexE a b f"
     proof -
       have "cindex_pathE (subpath a b g) z = cindexE 0 1 (f \<circ> (\<lambda>x. (b - a) * x + a))"
         unfolding cindex_pathE_def f_def comp_def subpath_def by auto
       also have "... =  cindexE a b f"
         using cindexE_linear_comp[of "b-a" 0 1 f a,simplified] that(1) by auto
       finally show ?thesis .
     qed
     moreover have "cindex_pathE (subpath b c g) z = cindexE b c f"
     proof -
       have "cindex_pathE (subpath b c g) z = cindexE 0 1 (f \<circ> (\<lambda>x. (c - b) * x + b))"
         unfolding cindex_pathE_def f_def comp_def subpath_def by auto
       also have "... =  cindexE b c f"
         using cindexE_linear_comp[of "c-b" 0 1 f b,simplified] that(2) by auto
       finally show ?thesis .
     qed
     moreover have "cindex_pathE (subpath a c g) z = cindexE a c f" 
     proof -
       have "cindex_pathE (subpath a c g) z = cindexE 0 1 (f \<circ> (\<lambda>x. (c - a) * x + a))"
         unfolding cindex_pathE_def f_def comp_def subpath_def by auto
       also have "... =  cindexE a c f"
         using cindexE_linear_comp[of "c-a" 0 1 f a,simplified] \<open>a<c\<close> by auto
       finally show ?thesis .
     qed
     moreover have "cindexE a b f + cindexE b c f = cindexE a c f "
     proof -
       have "finite_jumpFs f a c" 
         using finite_ReZ_segments_imp_jumpFs[OF g \<open>path g\<close>] \<open>0\<le>a\<close> \<open>c\<le>1\<close> unfolding f_def
         by (elim finite_jumpFs_subE,auto)
       then show ?thesis using cindexE_linear_comp cindexE_combine[OF _ \<open>a\<le>b\<close> \<open>b\<le>c\<close>] by auto
     qed
     ultimately show ?thesis by auto  
   qed
   ultimately show ?thesis by blast 
 qed
   
 lemma cindex_pathE_shiftpath:
   assumes "finite_ReZ_segments g z" "s\<in>{0..1}" "path g" and loop:"pathfinish g = pathstart g"
   shows "cindex_pathE (shiftpath s g) z = cindex_pathE g z" 
 proof -
   define f where "f=(\<lambda>g t. Im (g (t::real) - z) / Re (g t - z))"
   have "cindex_pathE (shiftpath s g) z = cindexE 0 1 (f (shiftpath s g))"
     unfolding cindex_pathE_def f_def by simp
   also have "... = cindexE 0 (1-s) (f (shiftpath s g)) + cindexE (1-s) 1 (f (shiftpath s g))"
   proof (rule cindexE_combine)
     have "finite_ReZ_segments (shiftpath s g) z"
       using finite_ReZ_segments_shiftpah[OF assms] .
     from finite_ReZ_segments_imp_jumpFs[OF this] path_shiftpath[OF \<open>path g\<close> loop \<open>s\<in>{0..1}\<close>]    
     show "finite_jumpFs (f (shiftpath s g)) 0 1" unfolding f_def by simp
     show "0 \<le> 1 - s" "1 - s \<le> 1" using \<open>s\<in>{0..1}\<close> by auto
   qed 
   also have "... = cindexE 0 s (f g) + cindexE s 1 (f g)"
   proof -
     have "cindexE 0 (1-s) (f (shiftpath s g)) = cindexE s 1 (f g)"
     proof -
       have "cindexE 0 (1-s) (f (shiftpath s g)) = cindexE 0 (1-s) ((f g) o (\<lambda>t. t+s))"
         apply (rule cindexE_cong)
         unfolding shiftpath_def f_def using \<open>s\<in>{0..1}\<close> by (auto simp add:algebra_simps)
       also have "...= cindexE s 1 (f g)"
         using cindexE_linear_comp[of 1 0 "1-s" "f g" s,simplified] .
       finally show ?thesis .
     qed
     moreover have "cindexE (1 - s) 1 (f (shiftpath s g)) = cindexE 0 s (f g)"  
     proof -
       have "cindexE (1 - s) 1 (f (shiftpath s g)) = cindexE (1-s) 1 ((f g) o (\<lambda>t. t+s-1))"
         apply (rule cindexE_cong)
         unfolding shiftpath_def f_def using \<open>s\<in>{0..1}\<close> by (auto simp add:algebra_simps)
       also have "... = cindexE 0 s (f g)"
         using cindexE_linear_comp[of 1 "1-s" 1 "f g" "s-1",simplified] 
         by (simp add:algebra_simps)
       finally show ?thesis .
     qed
     ultimately show ?thesis by auto
   qed
   also have "... = cindexE 0 1 (f g)"
   proof (rule cindexE_combine[symmetric])
     show "finite_jumpFs (f g) 0 1"  
       using finite_ReZ_segments_imp_jumpFs[OF assms(1,3)] unfolding f_def by simp
     show "0 \<le> s" "s\<le>1" using \<open>s\<in>{0..1}\<close> by auto
   qed
   also have "... = cindex_pathE g z"
     unfolding cindex_pathE_def f_def by simp
   finally show ?thesis .
 qed 
 
 subsection \<open>Cauchy's Index Theorem\<close>  
     
 theorem winding_number_cindex_pathE_aux:
   fixes g::"real \<Rightarrow> complex"
   assumes "finite_ReZ_segments g z" and "valid_path g" "z \<notin> path_image g" and
     Re_ends:"Re (g 1) = Re z" "Re (g 0) = Re z"
   shows "2 * Re(winding_number g z) = - cindex_pathE g z" 
   using assms  
 proof (induct rule:finite_ReZ_segments_induct)
   case (sub0 g z)
   have "winding_number (subpath 0 0 g) z = 0" 
     using \<open>z \<notin> path_image (subpath 0 0 g)\<close> unfolding subpath_refl
     by (auto intro!: winding_number_trivial)
   moreover have "cindex_pathE (subpath 0 0 g) z = 0"
     unfolding subpath_def by (auto intro:cindex_pathE_constI)
   ultimately show ?case by auto
 next
   case (subEq s g z)
   have Re_winding_0:"Re(winding_number h z) = 0" 
     when Re_const:"\<forall>t\<in>{0..1}. Re (h t) = Re z" and "valid_path h" "z\<notin>path_image h" for h
   proof -
     have "Re (winding_number (\<lambda>t. h t - z) 0) = (Im (Ln (pathfinish (\<lambda>t. h t - z))) 
               - Im (Ln (pathstart (\<lambda>t. h t - z)))) / (2 * pi)"
       apply (rule Re_winding_number_half_right[of _ 0,simplified])
       using Re_const \<open>valid_path h\<close> \<open>z \<notin> path_image h\<close> 
         apply auto
       by (metis (no_types, hide_lams) add.commute imageE le_add_same_cancel1 order_refl 
             path_image_def plus_complex.simps(1))
     moreover have "Im (Ln (h 1 - z)) = Im (Ln (h 0 - z))"
     proof -
       define z0 where "z0 = h 0 - z"
       define z1 where "z1 = h 1 - z"
       have [simp]: "z0\<noteq>0" "z1\<noteq>0" "Re z0=0" "Re z1=0"
         using \<open>z \<notin> path_image h\<close> that(1) unfolding z1_def z0_def path_image_def by auto
       have ?thesis when [simp]: "Im z0>0" "Im z1>0"
         apply (fold z1_def z0_def)
         using Im_Ln_eq_pi_half[of z1] Im_Ln_eq_pi_half[of z0] by auto
       moreover have ?thesis when [simp]: "Im z0<0" "Im z1<0"
         apply (fold z1_def z0_def)
         using Im_Ln_eq_pi_half[of z1] Im_Ln_eq_pi_half[of z0] by auto
       moreover have False when "Im z0\<ge>0" "Im z1\<le>0" 
       proof -
         define f where "f=(\<lambda>t. Im (h t - z))"
         have "\<exists>x\<ge>0. x \<le> 1 \<and> f x = 0"
           apply (rule IVT2'[of f 1 0 0])
           using that valid_path_imp_path[OF \<open>valid_path h\<close>] 
           unfolding f_def z0_def z1_def path_def 
           by (auto intro:continuous_intros)
         then show False using Re_const  \<open>z \<notin> path_image h\<close> unfolding f_def  
           by (metis atLeastAtMost_iff complex_surj image_eqI minus_complex.simps(2) 
                 path_defs(4) right_minus_eq)
       qed
       moreover have False when "Im z0\<le>0" "Im z1\<ge>0" 
       proof -
         define f where "f=(\<lambda>t. Im (h t - z))"
         have "\<exists>x\<ge>0. x \<le> 1 \<and> f x = 0"
           apply (rule IVT')
           using that valid_path_imp_path[OF \<open>valid_path h\<close>] 
           unfolding f_def z0_def z1_def path_def 
           by (auto intro:continuous_intros)
         then show False using Re_const \<open>z \<notin> path_image h\<close> unfolding f_def  
           by (metis atLeastAtMost_iff complex_surj image_eqI minus_complex.simps(2) 
                 path_defs(4) right_minus_eq)
       qed
       ultimately show ?thesis by argo
     qed
     ultimately have "Re (winding_number (\<lambda>t. h t - z) 0) = 0"
       unfolding pathfinish_def pathstart_def by auto
     then show ?thesis using winding_number_offset by auto
   qed
   have ?case when "s = 0"
   proof -
     have *: "\<forall>t\<in>{0..1}. Re (g t) = Re z" 
       using \<open>\<forall>t\<in>{s<..<1}. Re (g t) = Re z\<close> \<open>Re (g 1) = Re z\<close> \<open>Re (g 0) = Re z\<close> \<open>s=0\<close>
       by force
     have "Re(winding_number g z) = 0"
       by (rule Re_winding_0[OF * \<open>valid_path g\<close> \<open>z \<notin> path_image g\<close>])
     moreover have "cindex_pathE g z = 0"
       unfolding cindex_pathE_def
       apply (rule cindexE_constI)
       using * by auto
     ultimately show ?thesis by auto
   qed
   moreover have ?case when "s\<noteq>0"
   proof -
     define g1 where "g1 = subpath 0 s g"
     define g2 where "g2 = subpath s 1 g"
     have "path g" "s>0" 
       using valid_path_imp_path[OF \<open>valid_path g\<close>] that \<open>s\<in>{0..<1}\<close> by auto
     have "2 * Re (winding_number g z) = 2*Re (winding_number g1 z) + 2*Re (winding_number g2 z)"
       apply (subst winding_number_subpath_combine[OF \<open>path g\<close> \<open>z\<notin>path_image g\<close>,of 0 s 1
             ,simplified,symmetric])
       using \<open>s\<in>{0..<1}\<close> unfolding g1_def g2_def by auto
     also have "... = - cindex_pathE g1 z - cindex_pathE g2 z"
     proof -
       have "2*Re (winding_number g1 z) = - cindex_pathE g1 z" 
         unfolding g1_def
         apply (rule subEq.hyps(5))
         subgoal using subEq.hyps(1) subEq.prems(1) valid_path_subpath by fastforce  
         subgoal by (meson Path_Connected.path_image_subpath_subset atLeastAtMost_iff 
             atLeastLessThan_iff less_eq_real_def subEq(7) subEq.hyps(1) subEq.prems(1) 
             subsetCE valid_path_imp_path zero_le_one) 
         subgoal by (metis Groups.add_ac(2) add_0_left diff_zero mult.right_neutral subEq(2) 
             subEq(9) subpath_def) 
         subgoal by (simp add: subEq.prems(4) subpath_def)
         done
       moreover have "2*Re (winding_number g2 z) = - cindex_pathE g2 z"  
       proof -
         have *: "\<forall>t\<in>{0..1}. Re (g2 t) = Re z"
         proof 
           fix t::real assume "t\<in>{0..1}"
           have "Re (g2 t) = Re z" when "t=0 \<or> t=1"
             using that unfolding g2_def 
             by (metis \<open>s \<noteq> 0\<close> add.left_neutral diff_add_cancel mult.commute mult.left_neutral 
                 mult_zero_left subEq.hyps(2) subEq.prems(3) subpath_def)
           moreover have "Re (g2 t) = Re z" when "t\<in>{0<..<1}"
           proof -
             define t' where "t'=(1 - s) * t + s"
             then have "t'\<in>{s<..<1}" 
               using that \<open>s\<in>{0..<1}\<close> unfolding t'_def 
               apply auto
               by (sos "((((A<0 * (A<1 * A<2)) * R<1) + ((A<=1 * (A<0 * R<1)) * (R<1 * [1]^2))))")
             then have "Re (g t') = Re z" 
               using \<open>\<forall>t\<in>{s<..<1}. Re (g t) = Re z\<close> by auto
             then show ?thesis
               unfolding g2_def subpath_def t'_def .
           qed
           ultimately show "Re (g2 t) = Re z" using \<open>t\<in>{0..1}\<close> by fastforce
         qed
         have "Re(winding_number g2 z) = 0"
           apply (rule Re_winding_0[OF *]) 
           subgoal using g2_def subEq.hyps(1) subEq.prems(1) valid_path_subpath by fastforce
           subgoal by (metis (no_types, hide_lams) Path_Connected.path_image_subpath_subset 
                 atLeastAtMost_iff atLeastLessThan_iff g2_def less_eq_real_def subEq.hyps(1) 
                 subEq.prems(1) subEq.prems(2) subsetCE valid_path_imp_path zero_le_one)  
           done
         moreover have "cindex_pathE g2 z = 0"
           unfolding cindex_pathE_def
           apply (rule cindexE_constI)
           using * by auto
         ultimately show ?thesis by auto
       qed
       ultimately show ?thesis by auto
     qed
     also have "... = - cindex_pathE g z"
     proof -
       have "finite_ReZ_segments g z"
         unfolding finite_ReZ_segments_def
         apply (rule finite_Psegments.insertI_1[of s])
         subgoal using \<open>s \<in> {0..<1}\<close> by auto
         subgoal using \<open>s = 0 \<or> Re (g s) = Re z\<close> by auto
         subgoal using \<open>\<forall>t\<in>{s<..<1}. Re (g t) = Re z\<close> by auto
         subgoal 
         proof -
           have "finite_Psegments (\<lambda>t. Re (g (s * t)) = Re z) 0 1"
             using \<open>finite_ReZ_segments (subpath 0 s g) z\<close>
             unfolding subpath_def finite_ReZ_segments_def by auto
           from finite_Psegments_pos_linear[of _ "1/s" 0 0 s,simplified,OF this] 
           show "finite_Psegments (\<lambda>t. Re (g t - z) = 0) 0 s" 
             using \<open>s>0\<close> unfolding comp_def by auto
         qed
         done
       then show ?thesis    
         using cindex_pathE_subpath_combine[OF _ \<open>path g\<close>,of z 0 s 1,folded g1_def g2_def,simplified]
           \<open>s\<in>{0..<1}\<close> by auto
     qed
     finally show ?thesis .  
   qed
   ultimately show ?case by auto
 next
   case (subNEq s g z)
   have Re_winding:"2*Re(winding_number h z) = jumpF_pathfinish h z - jumpF_pathstart h z" 
     when Re_neq:"\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z" and "Re (h 0) = Re z" "Re (h 1) = Re z"
           and "valid_path h" "z\<notin>path_image h" for h
   proof -
     have Re_winding_pos:
         "2*Re(winding_number h0 0) = jumpF_pathfinish h0 0 - jumpF_pathstart h0 0" 
       when Re_gt:"\<forall>t\<in>{0<..<1}. Re (h0 t) > 0" and "Re (h0 0) = 0" "Re (h0 1) = 0"
           and "valid_path h0" "0\<notin>path_image h0" for h0
     proof -
       define f where "f \<equiv> (\<lambda>(t::real). Im(h0 t) / Re (h0 t))"
       define ln0 where "ln0 = Im (Ln (h0 0)) / pi"
       define ln1 where "ln1 = Im (Ln (h0 1)) / pi"
       have "path h0" using \<open>valid_path h0\<close> valid_path_imp_path by auto
       have "h0 0\<noteq>0" "h0 1\<noteq>0"
         using path_defs(4) that(5) by fastforce+ 
       have "ln1 = jumpF_pathfinish h0 0"  
       proof -
         have sgnx_at_left:"((\<lambda>x. Re (h0 x)) has_sgnx 1) (at_left 1)"
           unfolding has_sgnx_def eventually_at_left using \<open>\<forall>p\<in>{0<..<1}. Re (h0 p) > 0\<close>
           by (intro exI[where x=0],auto)
         have cont:"continuous (at_left 1) (\<lambda>t. Im (h0 t))" 
                 "continuous (at_left 1) (\<lambda>t. Re (h0 t))"
           using \<open>path h0\<close> unfolding path_def
           by (auto intro:continuous_on_at_left[of 0 1] continuous_intros)
         have ?thesis when "Im (h0 1) > 0"
         proof -
           have "ln1 = 1/2"
             using Im_Ln_eq_pi_half[OF \<open>h0 1\<noteq>0\<close>] that \<open>Re (h0 1) = 0\<close> unfolding ln1_def by auto 
           moreover have "jumpF_pathfinish h0 0 = 1/2"
           proof -
             have "filterlim f at_top (at_left 1)" unfolding f_def
               apply (subst filterlim_divide_at_bot_at_top_iff[of _ " Im (h0 1)"])  
               using \<open>Re(h0 1) = 0\<close> sgnx_at_left cont that unfolding continuous_within by auto
             then show ?thesis unfolding jumpF_pathfinish_def jumpF_def f_def by auto
           qed
           ultimately show ?thesis by auto
         qed
         moreover have ?thesis when "Im (h0 1) < 0"
         proof -
           have "ln1 = - 1/2"
             using Im_Ln_eq_pi_half[OF \<open>h0 1\<noteq>0\<close>] that \<open>Re (h0 1) = 0\<close> unfolding ln1_def by auto
           moreover have "jumpF_pathfinish h0 0 = - 1/2"
           proof -
             have "((\<lambda>x. Re (h0 x)) has_sgnx - sgn (Im (h0 1))) (at_left 1)" 
               using sgnx_at_left that by auto
             then have "filterlim f at_bot (at_left 1)" 
               unfolding f_def using cont that
               apply (subst filterlim_divide_at_bot_at_top_iff[of _ " Im (h0 1)"])
               unfolding continuous_within using \<open>Re(h0 1) = 0\<close> by auto
             then show ?thesis unfolding jumpF_pathfinish_def jumpF_def f_def by auto 
           qed
           ultimately show ?thesis by auto
         qed
         moreover have "Im (h0 1)\<noteq>0" using \<open>h0 1\<noteq>0\<close> \<open>Re (h0 1) = 0\<close> 
           using complex.expand by auto   
         ultimately show ?thesis by linarith
       qed
       moreover have "ln0 = jumpF_pathstart h0 0" 
       proof -
         have sgnx_at_right:"((\<lambda>x. Re (h0 x)) has_sgnx 1) (at_right 0)"
           unfolding has_sgnx_def eventually_at_right using \<open>\<forall>p\<in>{0<..<1}. Re (h0 p) > 0\<close>
           by (intro exI[where x=1],auto)
         have cont:"continuous (at_right 0) (\<lambda>t. Im (h0 t))" 
           "continuous (at_right 0) (\<lambda>t. Re (h0 t))"
           using \<open>path h0\<close> unfolding path_def      
           by (auto intro:continuous_on_at_right[of 0 1] continuous_intros)
         have ?thesis when "Im (h0 0) > 0"   
         proof -
           have "ln0 = 1/2" 
             using Im_Ln_eq_pi_half[OF \<open>h0 0\<noteq>0\<close>] that \<open>Re (h0 0) = 0\<close> unfolding ln0_def by auto 
           moreover have "jumpF_pathstart h0 0 = 1/2"
           proof -
             have "filterlim f at_top (at_right 0)" unfolding f_def
               apply (subst filterlim_divide_at_bot_at_top_iff[of _ " Im (h0 0)"])  
               using \<open>Re(h0 0) = 0\<close> sgnx_at_right cont that unfolding continuous_within by auto
             then show ?thesis unfolding jumpF_pathstart_def jumpF_def f_def by auto
           qed
           ultimately show ?thesis by auto
         qed
         moreover have ?thesis when "Im (h0 0) < 0"   
         proof -
           have "ln0 = - 1/2" 
             using Im_Ln_eq_pi_half[OF \<open>h0 0\<noteq>0\<close>] that \<open>Re (h0 0) = 0\<close> unfolding ln0_def by auto 
           moreover have "jumpF_pathstart h0 0 = - 1/2"
           proof -
             have "filterlim f at_bot (at_right 0)" unfolding f_def
               apply (subst filterlim_divide_at_bot_at_top_iff[of _ " Im (h0 0)"])  
               using \<open>Re(h0 0) = 0\<close> sgnx_at_right cont that unfolding continuous_within by auto
             then show ?thesis unfolding jumpF_pathstart_def jumpF_def f_def by auto
           qed
           ultimately show ?thesis by auto
         qed
         moreover have "Im (h0 0)\<noteq>0" using \<open>h0 0\<noteq>0\<close> \<open>Re (h0 0) = 0\<close> 
           using complex.expand by auto   
         ultimately show ?thesis by linarith
       qed
       moreover have "2*Re(winding_number h0 0) = ln1 - ln0"
       proof -
         have "\<forall>p\<in>path_image h0. 0 \<le> Re p" 
         proof 
           fix p assume "p \<in> path_image h0"
           then obtain t where t:"t\<in>{0..1}" "p = h0 t" unfolding path_image_def by auto
           have "0 \<le> Re p" when "t=0 \<or> t=1"  
             using that t \<open>Re (h0 0) = 0\<close> \<open>Re (h0 1) = 0\<close> by auto
           moreover have "0 \<le> Re p" when "t\<in>{0<..<1}"
             using that t Re_gt[rule_format, of t] by fastforce 
           ultimately show "0 \<le> Re p" using t(1) by fastforce    
         qed
         from Re_winding_number_half_right[of _ 0,simplified,OF this \<open>valid_path h0\<close> \<open>0 \<notin> path_image h0\<close>]
         show ?thesis unfolding ln1_def ln0_def pathfinish_def pathstart_def 
           by (auto simp add:field_simps)
       qed
       ultimately show ?thesis by auto
     qed
    
     have ?thesis when "\<forall>t\<in>{0<..<1}. Re (h t) < Re z"
     proof -
       let ?hu= "\<lambda>t. z - h t"
       have "2*Re(winding_number ?hu 0) = jumpF_pathfinish ?hu 0 - jumpF_pathstart ?hu 0"
         apply(rule Re_winding_pos)
         subgoal using that by auto
         subgoal using \<open>Re (h 0) = Re z\<close> by auto
         subgoal using \<open>Re (h 1) = Re z\<close> by auto
         subgoal using \<open>valid_path h\<close> valid_path_offset valid_path_uminus_comp 
             unfolding comp_def by fastforce
         subgoal using \<open>z\<notin>path_image h\<close> by (simp add: image_iff path_defs(4)) 
         done
       moreover have "winding_number ?hu 0 = winding_number h z"
         using winding_number_offset[of h z] 
               winding_number_uminus_comp[of "\<lambda>t. h t- z" 0,unfolded comp_def,simplified] 
               \<open>valid_path h\<close> \<open>z\<notin>path_image h\<close> by auto
       moreover have "jumpF_pathfinish ?hu 0 =  jumpF_pathfinish h z"
         unfolding jumpF_pathfinish_def
         apply (auto intro!:jumpF_cong eventuallyI)
         by (auto simp add:divide_simps algebra_simps)
       moreover have "jumpF_pathstart ?hu 0 =  jumpF_pathstart h z"
         unfolding jumpF_pathstart_def
         apply (auto intro!:jumpF_cong eventuallyI)
         by (auto simp add:divide_simps algebra_simps)
       ultimately show ?thesis by auto
     qed
     moreover have ?thesis when "\<forall>t\<in>{0<..<1}. Re (h t) > Re z"
     proof -
       let ?hu= "\<lambda>t. h t - z"
       have "2*Re(winding_number ?hu 0) = jumpF_pathfinish ?hu 0 - jumpF_pathstart ?hu 0"
         apply(rule Re_winding_pos)
         subgoal using that by auto
         subgoal using \<open>Re (h 0) = Re z\<close> by auto
         subgoal using \<open>Re (h 1) = Re z\<close> by auto
         subgoal using \<open>valid_path h\<close> valid_path_offset valid_path_uminus_comp 
             unfolding comp_def by fastforce
         subgoal using \<open>z\<notin>path_image h\<close> by simp 
         done
       moreover have "winding_number ?hu 0 = winding_number h z"
         using winding_number_offset[of h z] \<open>valid_path h\<close> \<open>z\<notin>path_image h\<close> by auto
       moreover have "jumpF_pathfinish ?hu 0 =  jumpF_pathfinish h z"
         unfolding jumpF_pathfinish_def by auto
       moreover have "jumpF_pathstart ?hu 0 =  jumpF_pathstart h z"
         unfolding jumpF_pathstart_def by auto
       ultimately show ?thesis by auto
     qed  
     moreover have "(\<forall>t\<in>{0<..<1}. Re (h t) > Re z) \<or> (\<forall>t\<in>{0<..<1}. Re (h t) < Re z)" 
     proof (rule ccontr)
       assume " \<not> ((\<forall>t\<in>{0<..<1}. Re z < Re (h t)) \<or> (\<forall>t\<in>{0<..<1}. Re (h t) < Re z))"
       then obtain t1 t2 where t:"t1\<in>{0<..<1}" "t2\<in>{0<..<1}" "Re (h t1)\<le>Re z" "Re (h t2)\<ge>Re z"
         unfolding path_image_def by auto 
       have False when "t1\<le>t2"
       proof -
         have "continuous_on {t1..t2} (\<lambda>t. Re (h t))" 
           using valid_path_imp_path[OF \<open>valid_path h\<close>] t unfolding path_def 
           by (metis (full_types) atLeastatMost_subset_iff continuous_on_Re continuous_on_subset 
             eucl_less_le_not_le greaterThanLessThan_iff)
         then obtain t' where t':"t'\<ge>t1" "t'\<le>t2" "Re (h t') = Re z"
           using  IVT'[of "\<lambda>t. Re (h t)" t1 _ t2] t \<open>t1\<le>t2\<close> by auto
         then have "t'\<in>{0<..<1}" using t by auto
         then have "Re (h t') \<noteq> Re z" using Re_neq by auto
         then show False using \<open>Re (h t') = Re z\<close> by simp 
       qed
       moreover have False when "t1\<ge>t2"
       proof -
         have "continuous_on {t2..t1} (\<lambda>t. Re (h t))" 
           using valid_path_imp_path[OF \<open>valid_path h\<close>] t unfolding path_def 
           by (metis (full_types) atLeastatMost_subset_iff continuous_on_Re continuous_on_subset 
             eucl_less_le_not_le greaterThanLessThan_iff)
         then obtain t' where t':"t'\<le>t1" "t'\<ge>t2" "Re (h t') = Re z"
           using  IVT2'[of "\<lambda>t. Re (h t)" t1 _ t2] t \<open>t1\<ge>t2\<close> by auto
         then have "t'\<in>{0<..<1}" using t by auto
         then have "Re (h t') \<noteq> Re z" using Re_neq by auto
         then show False using \<open>Re (h t') = Re z\<close> by simp 
       qed
       ultimately show False by linarith
     qed
     ultimately show ?thesis by blast
   qed
   
   have index_ends:"cindex_pathE h z = jumpF_pathstart h z - jumpF_pathfinish h z"
     when Re_neq:"\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z" and "valid_path h" for h
   proof -
     define f where "f = (\<lambda>t. Im (h t - z) / Re (h t - z))"
     define Ri where "Ri = {x. jumpF f (at_right x) \<noteq> 0 \<and> 0 \<le> x \<and> x < 1}"
     define Le where "Le = {x. jumpF f (at_left x) \<noteq> 0 \<and> 0 < x \<and> x \<le> 1}"
     have "path h" using \<open>valid_path h\<close> valid_path_imp_path by auto
     have jumpF_eq0: "jumpF f (at_left x) = 0" "jumpF f (at_right x) = 0" when "x\<in>{0<..<1}" for x
     proof -
       have "Re (h x) \<noteq> Re z"
         using  \<open>\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z\<close> that by blast
       then have "isCont f x"
         unfolding f_def using continuous_on_interior[OF \<open>path h\<close>[unfolded path_def]] that
         by (auto intro!: continuous_intros isCont_Im isCont_Re)
       then show "jumpF f (at_left x) = 0" "jumpF f (at_right x) = 0"
         unfolding continuous_at_split by (auto intro: jumpF_not_infinity)
     qed
     have "cindex_pathE h z  = cindexE 0 1 f" 
       unfolding cindex_pathE_def f_def by simp
     also have "... = sum (\<lambda>x. jumpF f (at_right x)) Ri - sum (\<lambda>x. jumpF f (at_left x)) Le"
       unfolding cindexE_def Ri_def Le_def by auto
     also have "... = jumpF f (at_right 0) -  jumpF f (at_left 1)"
     proof -
       have "sum (\<lambda>x. jumpF f (at_right x)) Ri = jumpF f (at_right 0)"
       proof (cases "jumpF f (at_right 0) = 0")
         case True
         hence False if "x \<in> Ri" for x using that
           by (cases "x = 0") (auto simp: jumpF_eq0 Ri_def)
         hence "Ri = {}" by blast
         then show ?thesis using True by auto
       next
         case False
         hence "x \<in> Ri \<longleftrightarrow> x = 0" for x using that
           by (cases "x = 0") (auto simp: jumpF_eq0 Ri_def)
         hence "Ri = {0}" by blast
         then show ?thesis by auto
       qed
       moreover have "sum (\<lambda>x. jumpF f (at_left x)) Le = jumpF f (at_left 1)"
       proof (cases "jumpF f (at_left 1) = 0")
         case True
         then have "Le = {}"
           unfolding Le_def using jumpF_eq0(1) greaterThanLessThan_iff by fastforce
         then show ?thesis using True by auto
       next
         case False
         then have "Le = {1}"
           unfolding Le_def using jumpF_eq0(1) greaterThanLessThan_iff by fastforce
         then show ?thesis by auto
       qed  
       ultimately show ?thesis by auto
     qed
     also have "... = jumpF_pathstart h z - jumpF_pathfinish h z"
       unfolding jumpF_pathstart_def jumpF_pathfinish_def f_def by simp
     finally show ?thesis .
   qed
 
   have ?case when "s=0"
   proof -
     have "2 * Re (winding_number g z) = jumpF_pathfinish g z - jumpF_pathstart g z"
       apply (rule Re_winding)
       using subNEq that by auto
     moreover have "cindex_pathE g z = jumpF_pathstart g z - jumpF_pathfinish g z"
       apply (rule index_ends)
       using subNEq that by auto
     ultimately show ?thesis by auto
   qed
   moreover have ?case when "s\<noteq>0"
   proof -
     define g1 where "g1 = subpath 0 s g"
     define g2 where "g2 = subpath s 1 g"
     have "path g" "s>0" 
       using valid_path_imp_path[OF \<open>valid_path g\<close>] that \<open>s\<in>{0..<1}\<close> by auto
     have "2 * Re (winding_number g z) = 2*Re (winding_number g1 z) + 2*Re (winding_number g2 z)"
       apply (subst winding_number_subpath_combine[OF \<open>path g\<close> \<open>z\<notin>path_image g\<close>,of 0 s 1
             ,simplified,symmetric])
       using \<open>s\<in>{0..<1}\<close> unfolding g1_def g2_def by auto  
     also have "... = - cindex_pathE g1 z - cindex_pathE g2 z"
     proof -
       have "2*Re (winding_number g1 z) = - cindex_pathE g1 z" 
         unfolding g1_def
         apply (rule subNEq.hyps(5))
         subgoal using subNEq.hyps(1) subNEq.prems(1) valid_path_subpath by fastforce  
         subgoal by (meson Path_Connected.path_image_subpath_subset atLeastAtMost_iff 
             atLeastLessThan_iff less_eq_real_def subNEq(7) subNEq.hyps(1) subNEq.prems(1) 
             subsetCE valid_path_imp_path zero_le_one) 
         subgoal by (metis Groups.add_ac(2) add_0_left diff_zero mult.right_neutral subNEq(2) 
             subNEq(9) subpath_def) 
         subgoal by (simp add: subNEq.prems(4) subpath_def)
         done
       moreover have "2*Re (winding_number g2 z) = - cindex_pathE g2 z"  
       proof -
         have *:"\<forall>t\<in>{0<..<1}. Re (g2 t) \<noteq> Re z"
         proof 
           fix t::real assume "t \<in> {0<..<1}"
           define t' where "t'=(1 - s) * t + s"
           have "t'\<in>{s<..<1}" unfolding t'_def using \<open>s\<in>{0..<1}\<close> \<open>t \<in> {0<..<1}\<close> 
             apply (auto simp add:algebra_simps)
             by (sos "((((A<0 * (A<1 * A<2)) * R<1) + ((A<=1 * (A<1 * R<1)) * (R<1 * [1]^2))))")
           then have "Re (g t') \<noteq> Re z" using \<open>\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z\<close> by auto
           then show "Re (g2 t) \<noteq> Re z" unfolding g2_def subpath_def t'_def by auto
         qed
         have "2*Re (winding_number g2 z) = jumpF_pathfinish g2 z - jumpF_pathstart g2 z"
           apply (rule Re_winding[OF *])
           subgoal by (metis add.commute add.right_neutral g2_def mult_zero_right subNEq.hyps(2) 
                 subpath_def that)
           subgoal by (simp add: \<open>g2 \<equiv> subpath s 1 g\<close> subNEq.prems(3) subpath_def)
           subgoal using g2_def subNEq.hyps(1) subNEq.prems(1) valid_path_subpath by fastforce
           subgoal by (metis (no_types, hide_lams) Path_Connected.path_image_subpath_subset 
                 \<open>path g\<close> atLeastAtMost_iff atLeastLessThan_iff g2_def less_eq_real_def subNEq.hyps(1) 
                 subNEq.prems(2) subsetCE zero_le_one)
           done
         moreover have "cindex_pathE g2 z = jumpF_pathstart g2 z - jumpF_pathfinish g2 z" 
           apply (rule index_ends[OF *])
           using g2_def subNEq.hyps(1) subNEq.prems(1) valid_path_subpath by fastforce
         ultimately show ?thesis by auto
       qed
       ultimately show ?thesis by auto
     qed 
     also have "... = - cindex_pathE g z"
     proof -
       have "finite_ReZ_segments g z"
         unfolding finite_ReZ_segments_def
         apply (rule finite_Psegments.insertI_2[of s])
         subgoal using \<open>s \<in> {0..<1}\<close> by auto
         subgoal using \<open>s = 0 \<or> Re (g s) = Re z\<close> by auto
         subgoal using \<open>\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z\<close> by auto
         subgoal 
         proof -
           have "finite_Psegments (\<lambda>t. Re (g (s * t)) = Re z) 0 1"
             using \<open>finite_ReZ_segments (subpath 0 s g) z\<close>
             unfolding subpath_def finite_ReZ_segments_def by auto
           from finite_Psegments_pos_linear[of _ "1/s" 0 0 s,simplified,OF this] 
           show "finite_Psegments (\<lambda>t. Re (g t - z) = 0) 0 s" 
             using \<open>s>0\<close> unfolding comp_def by auto
         qed
         done
       then show ?thesis    
         using cindex_pathE_subpath_combine[OF _ \<open>path g\<close>,of z 0 s 1,folded g1_def g2_def,simplified]
           \<open>s\<in>{0..<1}\<close> by auto
     qed
     finally show ?thesis .
   qed
   ultimately show ?case by auto
 qed
 
 theorem winding_number_cindex_pathE:
   fixes g::"real \<Rightarrow> complex"
   assumes "finite_ReZ_segments g z" and "valid_path g" "z \<notin> path_image g" and
     loop: "pathfinish g = pathstart g"
   shows "winding_number g z = - cindex_pathE g z / 2" 
 proof (rule finite_ReZ_segment_cases[OF assms(1)])    
   fix s assume "s \<in> {0..<1}" "s = 0 \<or> Re (g s) = Re z" 
           and const:"\<forall>t\<in>{s<..<1}. Re (g t) = Re z" 
           and finite:"finite_ReZ_segments (subpath 0 s g) z" 
   have "Re (g 1) = Re z"
     apply(rule continuous_constant_on_closure[of "{s<..<1}" "\<lambda>t. Re(g t)"])
     subgoal using valid_path_imp_path[OF \<open>valid_path g\<close>,unfolded path_def] \<open>s\<in>{0..<1}\<close>
       by (auto intro!:continuous_intros continuous_Re elim:continuous_on_subset)
     subgoal using const by auto
     subgoal using \<open>s\<in>{0..<1}\<close> by auto
     done
   moreover then have "Re (g 0) = Re z" using loop unfolding path_defs by auto
   ultimately have "2 * Re (winding_number g z) = - cindex_pathE g z"
     using winding_number_cindex_pathE_aux[of g z] assms(1-3) by auto
   moreover have "winding_number g z \<in> \<int>" 
     using integer_winding_number[OF _ loop \<open>z\<notin>path_image g\<close>] valid_path_imp_path[OF \<open>valid_path g\<close>]
     by auto
   ultimately show "winding_number g z = - cindex_pathE g z / 2"  
     by (metis add.right_neutral complex_eq complex_is_Int_iff mult_zero_right 
         nonzero_mult_div_cancel_left of_real_0 zero_neq_numeral)
 next
   fix s assume "s \<in> {0..<1}" "s = 0 \<or> Re (g s) = Re z"
           and Re_neq:"\<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z" 
           and finite:"finite_ReZ_segments (subpath 0 s g) z"  
   have "path g" using \<open>valid_path g\<close> valid_path_imp_path by auto
   let ?goal = "2 * Re (winding_number g z) = - cindex_pathE g z"       
   have ?goal when "s=0"
   proof -
     have index_ends:"cindex_pathE h z = jumpF_pathstart h z - jumpF_pathfinish h z"
       when Re_neq:"\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z" and "valid_path h" for h
     proof -
       define f where "f = (\<lambda>t. Im (h t - z) / Re (h t - z))"
       define Ri where "Ri = {x. jumpF f (at_right x) \<noteq> 0 \<and> 0 \<le> x \<and> x < 1}"
       define Le where "Le = {x. jumpF f (at_left x) \<noteq> 0 \<and> 0 < x \<and> x \<le> 1}"
       have "path h" using \<open>valid_path h\<close> valid_path_imp_path by auto
       have jumpF_eq0: "jumpF f (at_left x) = 0" "jumpF f (at_right x) = 0" when "x\<in>{0<..<1}" for x
       proof -
         have "Re (h x) \<noteq> Re z"
           using  \<open>\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z\<close> that by blast
         then have "isCont f x"
           unfolding f_def using continuous_on_interior[OF \<open>path h\<close>[unfolded path_def]] that
           by (auto intro!: continuous_intros isCont_Im isCont_Re)
         then show "jumpF f (at_left x) = 0" "jumpF f (at_right x) = 0"
           unfolding continuous_at_split by (auto intro: jumpF_not_infinity)
       qed
       have "cindex_pathE h z  = cindexE 0 1 f" 
         unfolding cindex_pathE_def f_def by simp
       also have "... = sum (\<lambda>x. jumpF f (at_right x)) Ri - sum (\<lambda>x. jumpF f (at_left x)) Le"
         unfolding cindexE_def Ri_def Le_def by auto
       also have "... = jumpF f (at_right 0) -  jumpF f (at_left 1)"
       proof -
         have "sum (\<lambda>x. jumpF f (at_right x)) Ri = jumpF f (at_right 0)"
         proof (cases "jumpF f (at_right 0) = 0")
           case True
           hence False if "x \<in> Ri" for x using that
             by (cases "x = 0") (auto simp: jumpF_eq0 Ri_def)
           hence "Ri = {}" by blast
           then show ?thesis using True by auto
         next
           case False
           hence "x \<in> Ri \<longleftrightarrow> x = 0" for x using that
             by (cases "x = 0") (auto simp: jumpF_eq0 Ri_def)
           then have "Ri = {0}" by blast
           then show ?thesis by auto
         qed
         moreover have "sum (\<lambda>x. jumpF f (at_left x)) Le = jumpF f (at_left 1)"
         proof (cases "jumpF f (at_left 1) = 0")
           case True
           then have "Le = {}"
             unfolding Le_def using jumpF_eq0(1) greaterThanLessThan_iff by fastforce
           then show ?thesis using True by auto
         next
           case False
           then have "Le = {1}"
             unfolding Le_def using jumpF_eq0(1) greaterThanLessThan_iff by fastforce
           then show ?thesis by auto
         qed  
         ultimately show ?thesis by auto
       qed
       also have "... = jumpF_pathstart h z - jumpF_pathfinish h z"
         unfolding jumpF_pathstart_def jumpF_pathfinish_def f_def by simp
       finally show ?thesis .
     qed
     define fI where "fI=(\<lambda>t. Im (g t - z))"
     define fR where "fR=(\<lambda>t. Re (g t - z))"
     have fI: "(fI \<longlongrightarrow> fI 0) (at_right 0)" "(fI \<longlongrightarrow> fI 1) (at_left 1)" 
     proof -
       have "continuous (at_right 0) fI"
         apply (rule continuous_on_at_right[of _ 1])
         using \<open>path g\<close> unfolding fI_def path_def by (auto intro:continuous_intros) 
       then show "(fI \<longlongrightarrow> fI 0) (at_right 0)" by (simp add: continuous_within)
     next
       have "continuous (at_left 1) fI"
         apply (rule continuous_on_at_left[of 0])
         using \<open>path g\<close> unfolding fI_def path_def by (auto intro:continuous_intros) 
       then show "(fI \<longlongrightarrow> fI 1) (at_left 1)" by (simp add: continuous_within)
     qed  
     have fR: "(fR \<longlongrightarrow> 0) (at_right 0)" "(fR \<longlongrightarrow> 0) (at_left 1)" when "Re (g 0) = Re z" 
     proof -
       have "continuous (at_right 0) fR"
         apply (rule continuous_on_at_right[of _ 1])
         using \<open>path g\<close> unfolding fR_def path_def by (auto intro:continuous_intros) 
       then show "(fR \<longlongrightarrow> 0) (at_right 0)" using that unfolding fR_def by (simp add: continuous_within)
     next
       have "continuous (at_left 1) fR"
         apply (rule continuous_on_at_left[of 0])
         using \<open>path g\<close> unfolding fR_def path_def by (auto intro:continuous_intros) 
       then show "(fR \<longlongrightarrow> 0) (at_left 1)" 
         using that loop unfolding fR_def path_defs by (simp add: continuous_within)
     qed
     have "(\<forall>t\<in>{0<..<1}. Re (g t) > Re z) \<or> (\<forall>t\<in>{0<..<1}. Re (g t) < Re z)" 
     proof (rule ccontr)
       assume " \<not> ((\<forall>t\<in>{0<..<1}. Re z < Re (g t)) \<or> (\<forall>t\<in>{0<..<1}. Re (g t) < Re z))"
       then obtain t1 t2 where t:"t1\<in>{0<..<1}" "t2\<in>{0<..<1}" "Re (g t1)\<le>Re z" "Re (g t2)\<ge>Re z"
         unfolding path_image_def by auto 
       have False when "t1\<le>t2"
       proof -
         have "continuous_on {t1..t2} (\<lambda>t. Re (g t))" 
           using valid_path_imp_path[OF \<open>valid_path g\<close>] t unfolding path_def 
           by (metis (full_types) atLeastatMost_subset_iff continuous_on_Re continuous_on_subset 
             eucl_less_le_not_le greaterThanLessThan_iff)
         then obtain t' where t':"t'\<ge>t1" "t'\<le>t2" "Re (g t') = Re z"
           using  IVT'[of "\<lambda>t. Re (g t)" t1 _ t2] t \<open>t1\<le>t2\<close> by auto
         then have "t'\<in>{0<..<1}" using t by auto
         then have "Re (g t') \<noteq> Re z" using Re_neq \<open>s=0\<close> by auto
         then show False using \<open>Re (g t') = Re z\<close> by simp 
       qed
       moreover have False when "t1\<ge>t2"
       proof -
         have "continuous_on {t2..t1} (\<lambda>t. Re (g t))" 
           using valid_path_imp_path[OF \<open>valid_path g\<close>] t unfolding path_def 
           by (metis (full_types) atLeastatMost_subset_iff continuous_on_Re continuous_on_subset 
             eucl_less_le_not_le greaterThanLessThan_iff)
         then obtain t' where t':"t'\<le>t1" "t'\<ge>t2" "Re (g t') = Re z"
           using  IVT2'[of "\<lambda>t. Re (g t)" t1 _ t2] t \<open>t1\<ge>t2\<close> by auto
         then have "t'\<in>{0<..<1}" using t by auto
         then have "Re (g t') \<noteq> Re z" using Re_neq \<open>s=0\<close> by auto
         then show False using \<open>Re (g t') = Re z\<close> by simp 
       qed
       ultimately show False by linarith
     qed
     moreover have ?thesis when Re_pos:"\<forall>t\<in>{0<..<1}. Re (g t) > Re z"
     proof -
       have "Re (winding_number g z) = 0"
       proof -
         have "\<forall>p\<in>path_image g. Re z \<le> Re p"
         proof 
           fix p assume "p \<in> path_image g"
           then obtain t where "0\<le>t" "t\<le>1" "p = g t" unfolding path_image_def by auto
           have "Re z \<le> Re (g t)"
             apply (rule continuous_ge_on_closure[of "{0<..<1}" "\<lambda>t. Re (g t)" t "Re z",simplified])
             subgoal using valid_path_imp_path[OF \<open>valid_path g\<close>,unfolded path_def] 
               by (auto intro:continuous_intros)
             subgoal using \<open>0\<le>t\<close> \<open>t\<le>1\<close> by auto
             subgoal for x using that[rule_format,of x] by auto
             done
           then show "Re z \<le> Re p" using \<open>p = g t\<close> by auto
         qed
         from Re_winding_number_half_right[OF this \<open>valid_path g\<close> \<open>z\<notin>path_image g\<close>] loop
         show ?thesis by auto
       qed
       moreover have "cindex_pathE g z = 0"
       proof -
         have "cindex_pathE g z = jumpF_pathstart g z - jumpF_pathfinish g z"
           using index_ends[OF _ \<open>valid_path g\<close>] Re_neq \<open>s=0\<close> by auto
         moreover have "jumpF_pathstart g z = jumpF_pathfinish g z" when "Re (g 0) \<noteq> Re z"
         proof -
           have "jumpF_pathstart g z = 0" 
             using jumpF_pathstart_eq_0[OF \<open>path g\<close>] that unfolding path_defs by auto
           moreover have "jumpF_pathfinish g z=0"
             using jumpF_pathfinish_eq_0[OF \<open>path g\<close>] that loop unfolding path_defs by auto 
           ultimately show ?thesis by auto
         qed
         moreover have "jumpF_pathstart g z = jumpF_pathfinish g z" when "Re (g 0) = Re z"  
         proof -
           have [simp]:"(fR has_sgnx 1) (at_right 0)" 
             unfolding fR_def has_sgnx_def eventually_at_right
             apply (rule exI[where x=1])
             using Re_pos by auto
           have [simp]:"(fR has_sgnx 1) (at_left 1)" 
             unfolding fR_def has_sgnx_def eventually_at_left
             apply (rule exI[where x=0])
             using Re_pos by auto
           have "fI 0\<noteq>0" 
           proof (rule ccontr)
             assume "\<not> fI 0 \<noteq> 0"
             then have "g 0 =z" using \<open>Re (g 0) = Re z\<close> 
               unfolding fI_def by (simp add: complex.expand)
             then show False using \<open>z \<notin> path_image g\<close> unfolding path_image_def by auto
           qed
           moreover have ?thesis when "fI 0>0"
           proof -
             have "jumpF_pathstart g z = 1/2"
             proof -
               have "(LIM x at_right 0. fI x / fR x :> at_top)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 0"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathstart_def fI_def fR_def jumpF_def by auto
             qed
             moreover have "jumpF_pathfinish g z = 1/2"
             proof -
               have "fI 1>0" using loop that unfolding path_defs fI_def by auto
               then have "(LIM x at_left 1. fI x / fR x :> at_top)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 1"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathfinish_def fI_def fR_def jumpF_def by auto
             qed  
             ultimately show ?thesis by simp
           qed
           moreover have ?thesis when "fI 0<0" 
           proof -
             have "jumpF_pathstart g z = - 1/2"
             proof -
               have "(LIM x at_right 0. fI x / fR x :> at_bot)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 0"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathstart_def fI_def fR_def jumpF_def by auto
             qed
             moreover have "jumpF_pathfinish g z = - 1/2"
             proof -
               have "fI 1<0" using loop that unfolding path_defs fI_def by auto
               then have "(LIM x at_left 1. fI x / fR x :> at_bot)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 1"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathfinish_def fI_def fR_def jumpF_def by auto
             qed  
             ultimately show ?thesis by simp
           qed  
           ultimately show ?thesis by linarith
         qed
         ultimately show ?thesis by auto
       qed
       ultimately show ?thesis by auto
     qed
     moreover have ?thesis when Re_neg:"\<forall>t\<in>{0<..<1}. Re (g t) < Re z"
     proof -
       have "Re (winding_number g z) = 0"
       proof -
         have "\<forall>p\<in>path_image g. Re z \<ge> Re p"
         proof 
           fix p assume "p \<in> path_image g"
           then obtain t where "0\<le>t" "t\<le>1" "p = g t" unfolding path_image_def by auto
           have "Re z \<ge> Re (g t)"
             apply (rule continuous_le_on_closure[of "{0<..<1}" "\<lambda>t. Re (g t)" t "Re z",simplified])
             subgoal using valid_path_imp_path[OF \<open>valid_path g\<close>,unfolded path_def] 
               by (auto intro:continuous_intros)
             subgoal using \<open>0\<le>t\<close> \<open>t\<le>1\<close> by auto
             subgoal for x using that[rule_format,of x] by auto
             done
           then show "Re z \<ge> Re p" using \<open>p = g t\<close> by auto
         qed
         from Re_winding_number_half_left[OF this \<open>valid_path g\<close> \<open>z\<notin>path_image g\<close>] loop
         show ?thesis by auto
       qed
       moreover have "cindex_pathE g z = 0"
       proof -
         have "cindex_pathE g z = jumpF_pathstart g z - jumpF_pathfinish g z"
           using index_ends[OF _ \<open>valid_path g\<close>] Re_neq \<open>s=0\<close> by auto
         moreover have "jumpF_pathstart g z = jumpF_pathfinish g z" when "Re (g 0) \<noteq> Re z"
         proof -
           have "jumpF_pathstart g z = 0" 
             using jumpF_pathstart_eq_0[OF \<open>path g\<close>] that unfolding path_defs by auto
           moreover have "jumpF_pathfinish g z=0"
             using jumpF_pathfinish_eq_0[OF \<open>path g\<close>] that loop unfolding path_defs by auto 
           ultimately show ?thesis by auto
         qed
         moreover have "jumpF_pathstart g z = jumpF_pathfinish g z" when "Re (g 0) = Re z"  
         proof -
           have [simp]:"(fR has_sgnx - 1) (at_right 0)" 
             unfolding fR_def has_sgnx_def eventually_at_right
             apply (rule exI[where x=1])
             using Re_neg by auto
           have [simp]:"(fR has_sgnx - 1) (at_left 1)" 
             unfolding fR_def has_sgnx_def eventually_at_left
             apply (rule exI[where x=0])
             using Re_neg by auto
           have "fI 0\<noteq>0" 
           proof (rule ccontr)
             assume "\<not> fI 0 \<noteq> 0"
             then have "g 0 =z" using \<open>Re (g 0) = Re z\<close> 
               unfolding fI_def by (simp add: complex.expand)
             then show False using \<open>z \<notin> path_image g\<close> unfolding path_image_def by auto
           qed
           moreover have ?thesis when "fI 0>0"
           proof -
             have "jumpF_pathstart g z = - 1/2"
             proof -
               have "(LIM x at_right 0. fI x / fR x :> at_bot)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 0"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathstart_def fI_def fR_def jumpF_def by auto
             qed
             moreover have "jumpF_pathfinish g z = - 1/2"
             proof -
               have "fI 1>0" using loop that unfolding path_defs fI_def by auto
               then have "(LIM x at_left 1. fI x / fR x :> at_bot)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 1"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathfinish_def fI_def fR_def jumpF_def by auto
             qed  
             ultimately show ?thesis by simp
           qed
           moreover have ?thesis when "fI 0<0" 
           proof -
             have "jumpF_pathstart g z = 1/2"
             proof -
               have "(LIM x at_right 0. fI x / fR x :> at_top)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 0"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathstart_def fI_def fR_def jumpF_def by auto
             qed
             moreover have "jumpF_pathfinish g z = 1/2"
             proof -
               have "fI 1<0" using loop that unfolding path_defs fI_def by auto
               then have "(LIM x at_left 1. fI x / fR x :> at_top)"
                 apply (subst filterlim_divide_at_bot_at_top_iff[of _ "fI 1"])
                 using that fI fR[OF \<open>Re (g 0) = Re z\<close>] by simp_all
               then show ?thesis unfolding jumpF_pathfinish_def fI_def fR_def jumpF_def by auto
             qed  
             ultimately show ?thesis by simp
           qed  
           ultimately show ?thesis by linarith
         qed
         ultimately show ?thesis by auto
       qed
       ultimately show ?thesis by auto
     qed  
     ultimately show ?thesis by auto
   qed
   moreover have ?goal when "s\<noteq>0"
   proof -
     have "Re (g s) = Re z" using \<open>s = 0 \<or> Re (g s) = Re z\<close> that by auto 
     define g' where "g' = shiftpath s g"
     have "2 * Re (winding_number g' z) = - cindex_pathE g' z"
     proof (rule winding_number_cindex_pathE_aux)
       show "Re (g' 1) = Re z" "Re (g' 0) = Re z"
         using \<open>Re (g s) = Re z\<close> \<open>s\<in>{0..<1}\<close> \<open>s\<noteq>0\<close> 
         unfolding g'_def shiftpath_def by simp_all
       show "valid_path g'" 
         using valid_path_shiftpath[OF \<open>valid_path g\<close> loop,of s,folded g'_def] \<open>s\<in>{0..<1}\<close>
         by auto
       show "z \<notin> path_image g'"
         using \<open>s \<in> {0..<1}\<close> assms(3) g'_def loop path_image_shiftpath by fastforce
       show "finite_ReZ_segments g' z"
         using finite_ReZ_segments_shiftpah[OF \<open>finite_ReZ_segments g z\<close> _ \<open>path g\<close> loop] \<open>s\<in>{0..<1}\<close>
         unfolding g'_def by auto
     qed
     moreover have "winding_number g' z = winding_number g z"
       unfolding g'_def
       apply (rule winding_number_shiftpath[OF \<open>path g\<close> \<open>z \<notin> path_image g\<close> loop])
       using \<open>s\<in>{0..<1}\<close> by auto
     moreover have "cindex_pathE g' z = cindex_pathE g z"
       unfolding g'_def
       apply (rule cindex_pathE_shiftpath[OF \<open>finite_ReZ_segments g z\<close> _ \<open>path g\<close> loop])
       using \<open>s\<in>{0..<1}\<close> by auto
     ultimately show ?thesis by auto
   qed
   ultimately have ?goal by auto
   moreover have "winding_number g z \<in> \<int>" 
     using integer_winding_number[OF _ loop \<open>z\<notin>path_image g\<close>] valid_path_imp_path[OF \<open>valid_path g\<close>]
     by auto
   ultimately show "winding_number g z = - cindex_pathE g z / 2"  
     by (metis add.right_neutral complex_eq complex_is_Int_iff mult_zero_right 
         nonzero_mult_div_cancel_left of_real_0 zero_neq_numeral)
 qed
  
 text \<open>REMARK: The usual statement of Cauchy's Index theorem (i.e. Analytic Theory of Polynomials 
   (2002): Theorem 11.1.3) is about the equality between the number of polynomial roots and
   the Cauchy index, which is the joint application of @{thm winding_number_cindex_pathE} and
   @{thm argument_principle}.\<close>  
   
 end
   
   
diff --git a/thys/Winding_Number_Eval/Missing_Algebraic.thy b/thys/Winding_Number_Eval/Missing_Algebraic.thy
--- a/thys/Winding_Number_Eval/Missing_Algebraic.thy
+++ b/thys/Winding_Number_Eval/Missing_Algebraic.thy
@@ -1,371 +1,371 @@
 (*
     Author:     Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
 *)
 
 section \<open>Some useful lemmas in algebra\<close>
 
 theory Missing_Algebraic imports
   "HOL-Computational_Algebra.Polynomial_Factorial"
   "HOL-Computational_Algebra.Fundamental_Theorem_Algebra"
-  "HOL-Analysis.Analysis"
+  "HOL-Complex_Analysis.Complex_Analysis"
   Missing_Topology
   "Budan_Fourier.BF_Misc"
 begin
 
 subsection \<open>Misc\<close>
 
 lemma poly_holomorphic_on[simp]:
   "(poly p) holomorphic_on s"
   apply (rule holomorphic_onI)
   apply (unfold field_differentiable_def)
   apply (rule_tac x="poly (pderiv p) x" in exI)
   by (simp add:has_field_derivative_at_within)
 
 lemma order_zorder:
   fixes p::"complex poly" and z::complex
   assumes "p\<noteq>0"
   shows "order z p = nat (zorder (poly p) z)"
 proof -
   define n where "n=nat (zorder (poly p) z)" 
   define h where "h=zor_poly (poly p) z"
   have "\<exists>w. poly p w \<noteq> 0" using assms poly_all_0_iff_0 by auto
   then obtain r where "0 < r" "cball z r \<subseteq> UNIV" and 
       h_holo: "h holomorphic_on cball z r" and
       poly_prod:"(\<forall>w\<in>cball z r. poly p w = h w * (w - z) ^ n \<and> h w \<noteq> 0)"
     using zorder_exist_zero[of "poly p" UNIV z,folded h_def] poly_holomorphic_on
     unfolding n_def by auto
   then have "h holomorphic_on ball z r"
     and "(\<forall>w\<in>ball z r. poly p w = h w * (w - z) ^ n)" 
     and "h z\<noteq>0"
     by auto
   then have "order z p = n" using \<open>p\<noteq>0\<close>
   proof (induct n arbitrary:p h)
     case 0
     then have "poly p z=h z" using \<open>r>0\<close> by auto 
     then have "poly p z\<noteq>0" using \<open>h z\<noteq>0\<close> by auto
     then show ?case using order_root by blast
   next
     case (Suc n)
     define sn where "sn=Suc n"
     define h' where "h'\<equiv> \<lambda>w. deriv h w * (w-z)+ sn * h w"
     have "(poly p has_field_derivative poly (pderiv p) w) (at w)" for w
       using poly_DERIV[of p w] .
     moreover have "(poly p has_field_derivative (h' w)*(w-z)^n ) (at w)" when "w\<in>ball z r" for w
     proof (subst DERIV_cong_ev[of w w "poly p" "\<lambda>w.  h w * (w - z) ^ Suc n" ],simp_all)
       show "\<forall>\<^sub>F x in nhds w. poly p x = h x * ((x - z) * (x - z) ^ n)"
         unfolding eventually_nhds using Suc(3) \<open>w\<in>ball z r\<close>
         apply (intro exI[where x="ball z r"])
         by auto
       next
         have "(h has_field_derivative deriv h w) (at w)" 
           using \<open>h holomorphic_on ball z r\<close> \<open>w\<in>ball z r\<close> holomorphic_on_imp_differentiable_at 
           by (simp add: holomorphic_derivI)
         then have "((\<lambda>w. h w * ((w - z) ^ sn)) 
                       has_field_derivative h' w * (w - z) ^ (sn - 1)) (at w)"
           unfolding h'_def
           apply (auto intro!: derivative_eq_intros simp add:field_simps)
           by (auto simp add:field_simps sn_def)
         then show "((\<lambda>w. h w * ((w - z) * (w - z) ^ n)) 
                       has_field_derivative h' w * (w - z) ^ n) (at w)"
           unfolding sn_def by auto
       qed
     ultimately have "\<forall>w\<in>ball z r. poly (pderiv p) w = h' w * (w - z) ^ n"
       using DERIV_unique by blast  
     moreover have "h' holomorphic_on ball z r"
       unfolding h'_def using \<open>h holomorphic_on ball z r\<close>
       by (auto intro!: holomorphic_intros)
     moreover have "h' z\<noteq>0" unfolding h'_def sn_def using \<open>h z \<noteq> 0\<close> of_nat_neq_0 by auto
     moreover have "pderiv p \<noteq> 0"  
     proof 
       assume "pderiv p = 0"
       obtain c where "p=[:c:]" using \<open>pderiv p = 0\<close> using pderiv_iszero by blast
       then have "c=0"
         using Suc(3)[rule_format,of z] \<open>r>0\<close> by auto
       then show False using \<open>p\<noteq>0\<close> using \<open>p=[:c:]\<close> by auto
     qed
     ultimately have "order z (pderiv p) = n" by (auto elim: Suc.hyps)
     moreover have "order z p \<noteq> 0"
       using Suc(3)[rule_format,of z] \<open>r>0\<close> order_root \<open>p\<noteq>0\<close> by auto
     ultimately show ?case using order_pderiv[OF \<open>pderiv p \<noteq> 0\<close>] by auto
   qed
   then show ?thesis unfolding n_def .
 qed  
 
 lemma pcompose_pCons_0:"pcompose p [:a:] = [:poly p a:]"
   apply (induct p)
   by (auto simp add:pcompose_pCons algebra_simps)       
    
 lemma pcompose_coeff_0:
   "coeff (pcompose p q) 0 = poly p (coeff q 0)"
   apply (induct p)
   by (auto simp add:pcompose_pCons coeff_mult)  
     
 lemma poly_field_differentiable_at[simp]:
   "poly p field_differentiable (at x within s)"
 apply (unfold field_differentiable_def)
 apply (rule_tac x="poly (pderiv p) x" in exI)
 by (simp add:has_field_derivative_at_within)  
   
 lemma deriv_pderiv:
   "deriv (poly p) = poly (pderiv p)"
 apply (rule ext)  
 apply (rule DERIV_imp_deriv)  
   using poly_DERIV . 
     
 lemma lead_coeff_map_poly_nz:
   assumes "f (lead_coeff p) \<noteq>0" "f 0=0"
   shows "lead_coeff (map_poly f p) = f (lead_coeff p) "
 proof -
   have "lead_coeff (Poly (map f (coeffs p))) = f (lead_coeff p)"
     by (metis (mono_tags, lifting) antisym assms(1) assms(2) coeff_0_degree_minus_1 coeff_map_poly 
         degree_Poly degree_eq_length_coeffs le_degree length_map map_poly_def)
   then show ?thesis
     by (simp add: map_poly_def)
 qed 
 
 lemma filterlim_poly_at_infinity:
   fixes p::"'a::real_normed_field poly"
   assumes "degree p>0"
   shows "filterlim (poly p) at_infinity at_infinity"
 using assms
 proof (induct p)
   case 0
   then show ?case by auto
 next
   case (pCons a p)
   have ?case when "degree p=0"
   proof -
     obtain c where c_def:"p=[:c:]" using \<open>degree p = 0\<close> degree_eq_zeroE by blast
     then have "c\<noteq>0" using \<open>0 < degree (pCons a p)\<close> by auto
     then show ?thesis unfolding c_def 
       apply (auto intro!:tendsto_add_filterlim_at_infinity)
       apply (subst mult.commute)
       by (auto intro!:tendsto_mult_filterlim_at_infinity filterlim_ident)
   qed
   moreover have ?case when "degree p\<noteq>0"
   proof -
     have "filterlim (poly p) at_infinity at_infinity"
       using that by (auto intro:pCons)
     then show ?thesis 
       by (auto intro!:tendsto_add_filterlim_at_infinity filterlim_at_infinity_times filterlim_ident)
   qed
   ultimately show ?case by auto
 qed  
        
 lemma poly_divide_tendsto_aux:
   fixes p::"'a::real_normed_field poly"
   shows "((\<lambda>x. poly p x/x^(degree p)) \<longlongrightarrow> lead_coeff p) at_infinity"  
 proof (induct p)
   case 0
   then show ?case by (auto intro:tendsto_eq_intros)
 next
   case (pCons a p)
   have ?case when "p=0"
     using that by auto
   moreover have ?case when "p\<noteq>0"
   proof -
     define g where "g=(\<lambda>x. a/(x*x^degree p))"
     define f where "f=(\<lambda>x. poly p x/x^degree p)"
     have "\<forall>\<^sub>Fx in at_infinity. poly (pCons a p) x / x ^ degree (pCons a p) = g x + f x"
     proof (rule eventually_at_infinityI[of 1])
       fix x::'a assume "norm x\<ge>1"
       then have "x\<noteq>0" by auto
       then show "poly (pCons a p) x / x ^ degree (pCons a p) = g x + f x"
         using that unfolding g_def f_def by (auto simp add:field_simps)
     qed
     moreover have "((\<lambda>x. g x+f x) \<longlongrightarrow>  lead_coeff (pCons a p)) at_infinity"
     proof -
       have "(g \<longlongrightarrow>  0) at_infinity"
         unfolding g_def using filterlim_poly_at_infinity[of "monom 1 (Suc (degree p))"]
         apply (auto intro!:tendsto_intros tendsto_divide_0 simp add: degree_monom_eq)
         apply (subst filterlim_cong[where g="poly (monom 1 (Suc (degree p)))"])
         by (auto simp add:poly_monom)
       moreover have "(f \<longlongrightarrow>  lead_coeff (pCons a p)) at_infinity"
         using pCons \<open>p\<noteq>0\<close> unfolding f_def by auto
       ultimately show ?thesis by (auto intro:tendsto_eq_intros)
     qed
     ultimately show ?thesis by (auto dest:tendsto_cong)  
   qed
   ultimately show ?case by auto  
 qed
   
 lemma filterlim_power_at_infinity:
   assumes "n\<noteq>0"
   shows "filterlim (\<lambda>x::'a::real_normed_field. x^n) at_infinity at_infinity" 
   using filterlim_poly_at_infinity[of "monom 1 n"] assms
   apply (subst filterlim_cong[where g="poly (monom 1 n)"])
   by (auto simp add:poly_monom degree_monom_eq)
    
 lemma poly_divide_tendsto_0_at_infinity: 
   fixes p::"'a::real_normed_field poly"
   assumes "degree p > degree q" 
   shows "((\<lambda>x. poly q x / poly p x) \<longlongrightarrow> 0 ) at_infinity" 
 proof -
   define pp where "pp=(\<lambda>x. x^(degree p) / poly p x)"    
   define qq where "qq=(\<lambda>x. poly q x/x^(degree q))"
   define dd where "dd=(\<lambda>x::'a. 1/x^(degree p - degree q))"
   have "\<forall>\<^sub>Fx in at_infinity.  poly q x / poly p x = qq x * pp x * dd x"
   proof (rule eventually_at_infinityI[of 1])
     fix x::'a assume "norm x\<ge>1"
     then have "x\<noteq>0" by auto
     then show "poly q x / poly p x = qq x * pp x * dd x"
       unfolding qq_def pp_def dd_def using assms 
       by (auto simp add:field_simps divide_simps power_diff) 
   qed
   moreover have "((\<lambda>x. qq x * pp x * dd x) \<longlongrightarrow> 0) at_infinity"
   proof -
     have "(qq \<longlongrightarrow> lead_coeff q) at_infinity" 
       unfolding qq_def using poly_divide_tendsto_aux[of q] .
     moreover have "(pp \<longlongrightarrow> 1/lead_coeff p) at_infinity"
     proof -
       have "p\<noteq>0" using assms by auto
       then show ?thesis
         unfolding pp_def using poly_divide_tendsto_aux[of p] 
         apply (drule_tac tendsto_inverse)
         by (auto simp add:inverse_eq_divide)
     qed  
     moreover have "(dd \<longlongrightarrow> 0) at_infinity" 
       unfolding dd_def
       apply (rule tendsto_divide_0)
       by (auto intro!: filterlim_power_at_infinity simp add:assms)
     ultimately show ?thesis by (auto intro:tendsto_eq_intros)
   qed
   ultimately show ?thesis by (auto dest:tendsto_cong)
 qed
     
 lemma lead_coeff_list_def:
   "lead_coeff p= (if coeffs p=[] then 0 else last (coeffs p))"
 by (simp add: last_coeffs_eq_coeff_degree)
   
 lemma poly_linepath_comp: 
   fixes a::"'a::{real_normed_vector,comm_semiring_0,real_algebra_1}"
   shows "poly p o (linepath a b) = poly (p \<circ>\<^sub>p [:a, b-a:]) o of_real"
   apply rule
   by (auto simp add:poly_pcompose linepath_def scaleR_conv_of_real algebra_simps)
 
 lemma poly_eventually_not_zero:
   fixes p::"real poly"
   assumes "p\<noteq>0"
   shows "eventually (\<lambda>x. poly p x\<noteq>0) at_infinity"
 proof (rule eventually_at_infinityI[of "Max (norm ` {x. poly p x=0}) + 1"])
   fix x::real assume asm:"Max (norm ` {x. poly p x=0}) + 1 \<le> norm x" 
   have False when "poly p x=0"
   proof - 
     define S where "S=norm `{x. poly p x = 0}"
     have "norm x\<in>S" using that unfolding S_def by auto
     moreover have "finite S" using \<open>p\<noteq>0\<close> poly_roots_finite unfolding S_def by blast
     ultimately have "norm x\<le>Max S" by simp
     moreover have "Max S + 1 \<le> norm x" using asm unfolding S_def by simp
     ultimately show False by argo
   qed
   then show "poly p x \<noteq> 0" by auto
 qed
     
 subsection \<open>More about @{term degree}\<close>
    
 lemma degree_div_less:
   fixes x y::"'a::field poly"
   assumes "degree x\<noteq>0" "degree y\<noteq>0"
   shows "degree (x div y) < degree x"
 proof -
   have "x\<noteq>0" "y\<noteq>0" using assms by auto
   define q r where "q=x div y" and "r=x mod y"
   have *:"eucl_rel_poly x y (q, r)" unfolding q_def r_def 
     by (simp add: eucl_rel_poly)
   then have "r = 0 \<or> degree r < degree y" using \<open>y\<noteq>0\<close> unfolding eucl_rel_poly_iff by auto
   moreover have ?thesis when "r=0"
   proof -
     have "x = q * y" using * that unfolding eucl_rel_poly_iff by auto
     then have "q\<noteq>0" using \<open>x\<noteq>0\<close> \<open>y\<noteq>0\<close> by auto
     from degree_mult_eq[OF this \<open>y\<noteq>0\<close>] \<open>x = q * y\<close> 
     have "degree x = degree q +degree y" by auto
     then show ?thesis unfolding q_def using assms by auto
   qed
   moreover have ?thesis when "degree r<degree y"
   proof (cases "degree y>degree x")
     case True
     then have "q=0" unfolding q_def using div_poly_less by auto
     then show ?thesis unfolding q_def using assms(1) by auto
   next
     case False
     then have "degree x>degree r" using that by auto
     then have "degree x = degree (x-r)" using degree_add_eq_right[of "-r" x] by auto
     have "x-r = q*y" using * unfolding eucl_rel_poly_iff by auto
     then have "q\<noteq>0" using \<open>degree r < degree x\<close> by auto
     have "degree x = degree q +degree y" 
       using  degree_mult_eq[OF \<open>q\<noteq>0\<close> \<open>y\<noteq>0\<close>] \<open>x-r = q*y\<close> \<open>degree x = degree (x-r)\<close> by auto
     then show ?thesis unfolding q_def using assms by auto
   qed
   ultimately show ?thesis by auto
 qed  
   
 lemma map_poly_degree_eq:
   assumes "f (lead_coeff p) \<noteq>0"
   shows "degree (map_poly f p) = degree p"  
   using assms
   unfolding map_poly_def degree_eq_length_coeffs coeffs_Poly lead_coeff_list_def
   by (metis (full_types) last_conv_nth_default length_map no_trailing_unfold nth_default_coeffs_eq 
       nth_default_map_eq strip_while_idem)
   
 lemma map_poly_degree_less:
   assumes "f (lead_coeff p) =0" "degree p\<noteq>0"
   shows "degree (map_poly f p) < degree p" 
 proof -
   have "length (coeffs p) >1" 
     using \<open>degree p\<noteq>0\<close> by (simp add: degree_eq_length_coeffs)  
   then obtain xs x where xs_def:"coeffs p=xs@[x]" "length xs>0"  
     by (metis One_nat_def add.commute add_diff_cancel_left' append_Nil assms(2) 
         degree_eq_length_coeffs length_greater_0_conv list.size(3) list.size(4) not_less_zero
         rev_exhaust) 
   have "f x=0" using assms(1) by (simp add: lead_coeff_list_def xs_def(1))
   have "degree (map_poly f p) = length (strip_while ((=) 0) (map f (xs@[x]))) - 1" 
     unfolding map_poly_def degree_eq_length_coeffs coeffs_Poly
     by (subst xs_def,auto)
   also have "... = length (strip_while ((=) 0) (map f xs)) - 1"   
     using \<open>f x=0\<close> by simp
   also have "... \<le> length xs -1"
     using length_strip_while_le by (metis diff_le_mono length_map)
   also have "... < length (xs@[x]) - 1"
     using xs_def(2) by auto
   also have "... = degree p"
     unfolding degree_eq_length_coeffs xs_def by simp
   finally show ?thesis .
 qed
   
 lemma map_poly_degree_leq[simp]:
   shows "degree (map_poly f p) \<le> degree p"
   unfolding map_poly_def degree_eq_length_coeffs
   by (metis coeffs_Poly diff_le_mono length_map length_strip_while_le)  
     
 subsection \<open>roots / zeros of a univariate function\<close>
 
 definition roots_within::"('a \<Rightarrow> 'b::zero) \<Rightarrow> 'a set \<Rightarrow> 'a set" where
   "roots_within f s = {x\<in>s. f x = 0}"
   
 abbreviation roots::"('a \<Rightarrow> 'b::zero) \<Rightarrow> 'a set" where
   "roots f \<equiv> roots_within f UNIV"
 
 subsection \<open>The argument principle specialised to polynomials.\<close>
 
 lemma argument_principle_poly:
   assumes "p\<noteq>0" and valid:"valid_path g" and loop: "pathfinish g = pathstart g" 
     and no_proots:"path_image g \<subseteq> - proots p"  
   shows "contour_integral g (\<lambda>x. deriv (poly p) x / poly p x) = 2 * of_real pi * \<i> * 
             (\<Sum>x\<in>proots p. winding_number g x * of_nat (order x p))"  
 proof -
   have "contour_integral g (\<lambda>x. deriv (poly p) x / poly p x) = 2 * of_real pi * \<i> * 
           (\<Sum>x | poly p x = 0. winding_number g x * of_int (zorder (poly p) x))"
     apply (rule argument_principle[of UNIV "poly p" "{}" "\<lambda>_. 1" g,simplified,OF _ valid loop])
     using no_proots[unfolded proots_def] by (auto simp add:poly_roots_finite[OF \<open>p\<noteq>0\<close>] )
   also have "... =  2 * of_real pi * \<i> * (\<Sum>x\<in>proots p. winding_number g x * of_nat (order x p))"
   proof -
     have "nat (zorder (poly p) x) = order x p" when "x\<in>proots p" for x 
       using order_zorder[OF \<open>p\<noteq>0\<close>] that unfolding proots_def by auto
     then show ?thesis unfolding proots_def 
       apply (auto intro!:comm_monoid_add_class.sum.cong)
       by (metis assms(1) nat_eq_iff2 of_nat_nat order_root)
   qed
   finally show ?thesis . 
 qed  
   
 end
diff --git a/thys/Winding_Number_Eval/Missing_Analysis.thy b/thys/Winding_Number_Eval/Missing_Analysis.thy
--- a/thys/Winding_Number_Eval/Missing_Analysis.thy
+++ b/thys/Winding_Number_Eval/Missing_Analysis.thy
@@ -1,154 +1,153 @@
 (*
     Author:     Wenda Li <wl302@cam.ac.uk / liwenda1990@hotmail.com>
 *)
 
 section \<open>Some useful lemmas in analysis\<close>
 
-theory Missing_Analysis imports "HOL-Analysis.Analysis"
+theory Missing_Analysis
+  imports "HOL-Complex_Analysis.Complex_Analysis"
 begin  
 
 subsection \<open>More about paths\<close>
    
 lemma pathfinish_offset[simp]:
   "pathfinish (\<lambda>t. g t - z) = pathfinish g - z"
   unfolding pathfinish_def by simp 
     
 lemma pathstart_offset[simp]:
   "pathstart (\<lambda>t. g t - z) = pathstart g - z"
   unfolding pathstart_def by simp
     
 lemma pathimage_offset[simp]:
   fixes g :: "_ \<Rightarrow> 'b::topological_group_add"
   shows "p \<in> path_image (\<lambda>t. g t - z) \<longleftrightarrow> p+z \<in> path_image g " 
 unfolding path_image_def by (auto simp:algebra_simps)
   
 lemma path_offset[simp]:
  fixes g :: "_ \<Rightarrow> 'b::topological_group_add"
  shows "path (\<lambda>t. g t - z) \<longleftrightarrow> path g"
 unfolding path_def
 proof 
   assume "continuous_on {0..1} (\<lambda>t. g t - z)" 
   hence "continuous_on {0..1} (\<lambda>t. (g t - z) + z)" 
     apply (rule continuous_intros)
     by (intro continuous_intros)
   then show "continuous_on {0..1} g" by auto
 qed (auto intro:continuous_intros)   
   
 lemma not_on_circlepathI:
   assumes "cmod (z-z0) \<noteq> \<bar>r\<bar>"
   shows "z \<notin> path_image (part_circlepath z0 r st tt)"
 proof (rule ccontr)
   assume "\<not> z \<notin> path_image (part_circlepath z0 r st tt)"
   then have "z\<in>path_image (part_circlepath z0 r st tt)" by simp
   then obtain t where "t\<in>{0..1}" and *:"z = z0 + r * exp (\<i> * (linepath st tt t))"
     unfolding path_image_def image_def part_circlepath_def by blast
   define \<theta> where "\<theta> = linepath st tt t"
   then have "z-z0 = r * exp (\<i> * \<theta>)" using * by auto
   then have "cmod (z-z0) = cmod (r * exp (\<i> * \<theta>))" by auto
   also have "\<dots> = \<bar>r\<bar> * cmod (exp (\<i> * \<theta>))" by (simp add: norm_mult)
   also have "\<dots> = \<bar>r\<bar>" by auto
   finally have "cmod (z-z0) = \<bar>r\<bar>" .
   then show False using assms by auto
 qed    
 
 lemma circlepath_inj_on: 
   assumes "r>0"
   shows "inj_on (circlepath z r) {0..<1}"
 proof (rule inj_onI)
   fix x y assume asm: "x \<in> {0..<1}" "y \<in> {0..<1}" "circlepath z r x = circlepath z r y"
   define c where "c=2 * pi * \<i>"
   have "c\<noteq>0" unfolding c_def by auto 
   from asm(3) have "exp (c * x) =exp (c * y)"
     unfolding circlepath c_def using \<open>r>0\<close> by auto
   then obtain n where "c * x =c * (y + of_int n)"
     by (auto simp add:exp_eq c_def algebra_simps)
   then have "x=y+n" using \<open>c\<noteq>0\<close>
     by (meson mult_cancel_left of_real_eq_iff)
   then show "x=y" using asm(1,2) by auto
 qed
 
 subsection \<open>More lemmas related to @{term winding_number}\<close>  
   
 lemma winding_number_comp:
   assumes "open s" "f holomorphic_on s" "path_image \<gamma> \<subseteq> s"  
     "valid_path \<gamma>" "z \<notin> path_image (f \<circ> \<gamma>)" 
   shows "winding_number (f \<circ> \<gamma>) z = 1/(2*pi*\<i>)* contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))"
 proof -
   obtain spikes where "finite spikes" and \<gamma>_diff: "\<gamma> C1_differentiable_on {0..1} - spikes"
     using \<open>valid_path \<gamma>\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto  
   have "valid_path (f \<circ> \<gamma>)" 
     using valid_path_compose_holomorphic assms by blast
   moreover have "contour_integral (f \<circ> \<gamma>) (\<lambda>w. 1 / (w - z)) 
       = contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))"
     unfolding contour_integral_integral
   proof (rule integral_spike[rule_format,OF negligible_finite[OF \<open>finite spikes\<close>]])
     fix t::real assume t:"t \<in> {0..1} - spikes"
     then have "\<gamma> differentiable at t" 
       using \<gamma>_diff unfolding C1_differentiable_on_eq by auto
     moreover have "f field_differentiable at (\<gamma> t)" 
     proof -
       have "\<gamma> t \<in> s" using \<open>path_image \<gamma> \<subseteq> s\<close> t unfolding path_image_def by auto 
       thus ?thesis 
         using \<open>open s\<close> \<open>f holomorphic_on s\<close>  holomorphic_on_imp_differentiable_at by blast
     qed
     ultimately show " deriv f (\<gamma> t) / (f (\<gamma> t) - z) * vector_derivative \<gamma> (at t) =
          1 / ((f \<circ> \<gamma>) t - z) * vector_derivative (f \<circ> \<gamma>) (at t)"
       apply (subst vector_derivative_chain_at_general)
       by (simp_all add:field_simps)
   qed
   moreover note \<open>z \<notin> path_image (f \<circ> \<gamma>)\<close> 
   ultimately show ?thesis
     apply (subst winding_number_valid_path)
     by simp_all
 qed  
   
 lemma winding_number_uminus_comp:
   assumes "valid_path \<gamma>" "- z \<notin> path_image \<gamma>" 
   shows "winding_number (uminus \<circ> \<gamma>) z = winding_number \<gamma> (-z)"
 proof -
   define c where "c= 2 * pi * \<i>"
   have "winding_number (uminus \<circ> \<gamma>) z = 1/c * contour_integral \<gamma> (\<lambda>w. deriv uminus w / (-w-z)) "
   proof (rule winding_number_comp[of UNIV, folded c_def])
     show "open UNIV" "uminus holomorphic_on UNIV" "path_image \<gamma> \<subseteq> UNIV" "valid_path \<gamma>"
       using \<open>valid_path \<gamma>\<close> by (auto intro:holomorphic_intros)
     show "z \<notin> path_image (uminus \<circ> \<gamma>)" 
       unfolding path_image_compose using \<open>- z \<notin> path_image \<gamma>\<close> by auto
   qed
   also have "\<dots> = 1/c * contour_integral \<gamma> (\<lambda>w. 1 / (w- (-z)))"
-    apply (auto intro!:contour_integral_eq simp add:field_simps)
-    apply (subst deriv_linear[of "-1",simplified])
-    by (simp add: minus_divide_right)
+    by (auto intro!:contour_integral_eq simp add:field_simps minus_divide_right)
   also have "\<dots> = winding_number \<gamma> (-z)"
     using winding_number_valid_path[OF \<open>valid_path \<gamma>\<close> \<open>- z \<notin> path_image \<gamma>\<close>,folded c_def]
     by simp
   finally show ?thesis by auto
 qed  
   
 lemma winding_number_comp_linear:
   assumes "c\<noteq>0" "valid_path \<gamma>" and not_image: "(z-b)/c \<notin> path_image \<gamma>"
   shows "winding_number ((\<lambda>x. c*x+b) \<circ> \<gamma>) z = winding_number \<gamma> ((z-b)/c)" (is "?L = ?R")
 proof -
   define cc where "cc=1 / (complex_of_real (2 * pi) * \<i>)"
   define zz where "zz=(z-b)/c"
   have "?L = cc * contour_integral \<gamma> (\<lambda>w. deriv (\<lambda>x. c * x + b) w / (c * w + b - z))"
     apply (subst winding_number_comp[of UNIV,simplified])
     subgoal by (auto intro:holomorphic_intros)
     subgoal using \<open>valid_path \<gamma>\<close> .
     subgoal using not_image \<open>c\<noteq>0\<close> unfolding path_image_compose by auto
     subgoal unfolding cc_def by auto
     done
   also have "\<dots> = cc * contour_integral \<gamma> (\<lambda>w.1 / (w - zz))"
   proof -
     have "deriv (\<lambda>x. c * x + b) = (\<lambda>x. c)"
       by (auto intro:derivative_intros) 
     then show ?thesis
       unfolding zz_def cc_def using \<open>c\<noteq>0\<close>
       by (auto simp:field_simps)
   qed
   also have "\<dots> = winding_number \<gamma> zz"
     using winding_number_valid_path[OF \<open>valid_path \<gamma>\<close> not_image,folded zz_def cc_def]
     by simp
   finally show "winding_number ((\<lambda>x. c * x + b) \<circ> \<gamma>) z = winding_number \<gamma> zz" .
 qed  
  
 end
diff --git a/thys/Winding_Number_Eval/ROOT b/thys/Winding_Number_Eval/ROOT
--- a/thys/Winding_Number_Eval/ROOT
+++ b/thys/Winding_Number_Eval/ROOT
@@ -1,15 +1,16 @@
 chapter AFP
 
-session "Winding_Number_Eval" (AFP) = "HOL-Analysis" +
+session "Winding_Number_Eval" (AFP) = "HOL-Complex_Analysis" +
   options [timeout = 600]
   sessions
+    "HOL-Analysis"
     "HOL-Computational_Algebra"
     "HOL-Eisbach"
     "Sturm_Tarski"
     "Budan_Fourier"
   theories
     Winding_Number_Eval
     Winding_Number_Eval_Examples
   document_files
     "root.tex"
     "root.bib"