diff --git a/thys/Poincare_Bendixson/Analysis_Misc.thy b/thys/Poincare_Bendixson/Analysis_Misc.thy
--- a/thys/Poincare_Bendixson/Analysis_Misc.thy
+++ b/thys/Poincare_Bendixson/Analysis_Misc.thy
@@ -1,1135 +1,1135 @@
-section \<open>Additions to HOL-Analysis\<close>
-theory Analysis_Misc
-  imports 
-    Ordinary_Differential_Equations.ODE_Analysis
-begin
-
-subsection \<open>Unsorted Lemmas (TODO: sort!)\<close>
-
-lemma uminus_uminus_image: "uminus ` uminus ` S = S"
-  for S::"'r::ab_group_add set"
-  by (auto simp: image_image)
-
-lemma in_uminus_image_iff[simp]: "x \<in> uminus ` S \<longleftrightarrow> - x \<in> S"
-  for S::"'r::ab_group_add set"
-  by force
-
-lemma closed_subsegmentI:
-  "w + t *\<^sub>R (z - w) \<in> {x--y}"
-  if "w \<in> {x -- y}" "z \<in> {x -- y}" and t: "0 \<le> t" "t\<le> 1"
-proof -
-  from that obtain u v where
-    w_def: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y" and u: "0 \<le> u" "u \<le> 1"
-    and z_def: "z = (1 - v) *\<^sub>R x + v *\<^sub>R y" and v: "0 \<le> v" "v \<le> 1"
-    by (auto simp: in_segment)
-  have "w + t *\<^sub>R (z - w) =
-    (1 - (u - t * (u - v))) *\<^sub>R x + (u - t * (u - v)) *\<^sub>R y"
-    by (simp add: algebra_simps w_def z_def)
-  also have "\<dots> \<in> {x -- y}"
-    unfolding closed_segment_image_interval
-    apply (rule imageI)
-    using t u v
-    apply auto
-     apply (metis (full_types) diff_0_right diff_left_mono linear mult_left_le_one_le mult_nonneg_nonpos order.trans)
-    by (smt mult_left_le_one_le mult_nonneg_nonneg vector_space_over_itself.scale_right_diff_distrib)
-  finally show ?thesis .
-qed
-
-lemma tendsto_minus_cancel_right: "((\<lambda>x. -g x) \<longlongrightarrow> l) F \<longleftrightarrow> (g \<longlongrightarrow> -l) F"
-  \<comment> \<open>cf @{thm tendsto_minus_cancel_left}\<close>
-  for g::"_ \<Rightarrow> 'b::topological_group_add"
-  by (simp add: tendsto_minus_cancel_left)
-
-lemma tendsto_nhds_continuousI: "(f \<longlongrightarrow> l) (nhds x)" if "(f \<longlongrightarrow> l) (at x)" "f x = l"
-  \<comment> \<open>TODO: the assumption is continuity of f at x\<close>
-proof (rule topological_tendstoI)
-  fix S::"'b set" assume "open S" "l \<in> S"
-  from topological_tendstoD[OF that(1) this]
-  have "\<forall>\<^sub>F x in at x. f x \<in> S" .
-  then show "\<forall>\<^sub>F x in nhds x. f x \<in> S"
-    unfolding eventually_at_filter
-    by eventually_elim (auto simp: that \<open>l \<in> S\<close>)
-qed
-
-lemma inj_composeD:
-  assumes "inj (\<lambda>x. g (t x))"
-  shows "inj t"
-  using assms
-  by (auto simp: inj_def)
-
-lemma compact_sequentialE:
-  fixes S T::"'a::first_countable_topology set"
-  assumes "compact S"
-  assumes "infinite T"
-  assumes "T \<subseteq> S"
-  obtains t::"nat \<Rightarrow> 'a" and l::'a
-  where "\<And>n. t n \<in> T" "\<And>n. t n \<noteq> l" "t \<longlonglongrightarrow> l" "l \<in> S"
-proof -
-  from Heine_Borel_imp_Bolzano_Weierstrass[OF assms]
-  obtain l where "l \<in> S" "l islimpt T" by metis
-  then obtain t where "t n \<in> T" "t n \<noteq> l" "t \<longlonglongrightarrow> l" "l \<in> S" for n unfolding islimpt_sequential
-    by auto
-  then show ?thesis ..
-qed
-
-lemma infinite_countable_subsetE:
-  fixes S::"'a set"
-  assumes "infinite S"
-  obtains g::"nat\<Rightarrow>'a" where "inj g" "range g \<subseteq> S"
-  using assms
-  by atomize_elim (simp add: infinite_countable_subset)
-
-lemma real_quad_ge: "2 * (an * bn) \<le> an * an + bn * bn" for an bn::real
-  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [an + ~1*bn]^2))))")
-
-lemma inner_quad_ge: "2 * (a \<bullet> b) \<le> a \<bullet> a + b \<bullet> b"
-  for a b::"'a::euclidean_space"\<comment> \<open>generalize?\<close>
-proof -
-  show ?thesis
-    by (subst (1 2 3) euclidean_inner)
-      (auto simp add: sum.distrib[symmetric] sum_distrib_left intro!: sum_mono real_quad_ge)
-qed
-
-lemma inner_quad_gt: "2 * (a \<bullet> b) < a \<bullet> a + b \<bullet> b"
-  if "a \<noteq> b"
-  for a b::"'a::euclidean_space"\<comment> \<open>generalize?\<close>
-proof -
-  from that obtain i where "i \<in> Basis" "a \<bullet> i \<noteq> b \<bullet> i"
-    by (auto simp: euclidean_eq_iff[where 'a='a])
-  then have "2 * (a \<bullet> i * (b \<bullet> i)) < a \<bullet> i * (a \<bullet> i) + b \<bullet> i * (b \<bullet> i)"
-    using sum_sqs_eq[of "a\<bullet>i" "b\<bullet>i"]
-    by (auto intro!: le_neq_trans real_quad_ge)
-  then show ?thesis
-    by (subst (1 2 3) euclidean_inner)
-      (auto simp add: \<open>i \<in> Basis\<close> sum.distrib[symmetric] sum_distrib_left
-        intro!: sum_strict_mono_ex1 real_quad_ge)
-qed
-
-lemma closed_segment_line_hyperplanes:
-  "{a -- b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) \<le> x \<bullet> (b - a) \<and> x \<bullet> (b - a) \<le> b \<bullet> (b - a)}"
-  if "a \<noteq> b"
-  for a b::"'a::euclidean_space"
-proof safe
-  fix x assume x: "x \<in> {a--b}"
-  then obtain u where u: "0 \<le> u" "u \<le> 1" and x_eq: "x = a + u *\<^sub>R (b - a)"
-    by (auto simp add: in_segment algebra_simps)
-  show "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))" using x_eq by auto
-  have "2 * (a \<bullet> b) \<le> a \<bullet> a + b \<bullet> b"
-    by (rule inner_quad_ge)
-  then have "u * (2 * (a \<bullet> b) - a \<bullet> a - b \<bullet> b) \<le> 0"
-    "0 \<le> (1 - u) * (a \<bullet> a + b \<bullet> b - a \<bullet> b * 2)"
-    by (simp_all add: mult_le_0_iff u)
-  then show " a \<bullet> (b - a) \<le> x \<bullet> (b - a)" "x \<bullet> (b - a) \<le> b \<bullet> (b - a)"
-    by (auto simp: x_eq algebra_simps power2_eq_square inner_commute)
-next
-  fix u assume
-    "a \<bullet> (b - a) \<le> (a + u *\<^sub>R (b - a)) \<bullet> (b - a)"
-    "(a + u *\<^sub>R (b - a)) \<bullet> (b - a) \<le> b \<bullet> (b - a)"
-  then have "0 \<le> u * ((b - a) \<bullet> (b - a))" "0 \<le> (1 - u) * ((b - a) \<bullet> (b - a))"
-    by (auto simp: algebra_simps)
-  then have "0 \<le> u" "u \<le> 1"
-    using inner_ge_zero[of "(b - a)"] that
-    by (auto simp add: zero_le_mult_iff)
-  then show "a + u *\<^sub>R (b - a) \<in> {a--b}"
-    by (auto simp: in_segment algebra_simps)
-qed
-
-lemma open_segment_line_hyperplanes:
-  "{a <--< b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
-  if "a \<noteq> b"
-  for a b::"'a::euclidean_space"
-proof safe
-  fix x assume x: "x \<in> {a<--<b}"
-  then obtain u where u: "0 < u" "u < 1" and x_eq: "x = a + u *\<^sub>R (b - a)"
-    by (auto simp add: in_segment algebra_simps)
-  show "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))" using x_eq by auto
-  have "2 * (a \<bullet> b) < a \<bullet> a + b \<bullet> b" using that
-    by (rule inner_quad_gt)
-  then have "u * (2 * (a \<bullet> b) - a \<bullet> a - b \<bullet> b) < 0"
-    "0 < (1 - u) * (a \<bullet> a + b \<bullet> b - a \<bullet> b * 2)"
-    by (simp_all add: mult_less_0_iff u)
-  then show " a \<bullet> (b - a) < x \<bullet> (b - a)" "x \<bullet> (b - a) < b \<bullet> (b - a)"
-    by (auto simp: x_eq algebra_simps power2_eq_square inner_commute)
-next
-  fix u assume
-    "a \<bullet> (b - a) < (a + u *\<^sub>R (b - a)) \<bullet> (b - a)"
-    "(a + u *\<^sub>R (b - a)) \<bullet> (b - a) < b \<bullet> (b - a)"
-  then have "0 < u * ((b - a) \<bullet> (b - a))" "0 < (1 - u) * ((b - a) \<bullet> (b - a))"
-    by (auto simp: algebra_simps)
-  then have "0 < u" "u < 1"
-    using inner_ge_zero[of "(b - a)"] that
-    by (auto simp add: zero_less_mult_iff)
-  then show "a + u *\<^sub>R (b - a) \<in> {a<--<b}"
-    by (auto simp: in_segment algebra_simps that)
-qed
-
-lemma at_within_interior: "NO_MATCH UNIV S \<Longrightarrow> x \<in> interior S \<Longrightarrow> at x within S = at x"
-  by (auto intro: at_within_interior)
-
-lemma tendsto_at_topI:
-  "(f \<longlongrightarrow> l) at_top" if "\<And>e. 0 < e \<Longrightarrow> \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e"
-for f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  using that
-  apply (intro tendstoI)
-  unfolding eventually_at_top_linorder
-  by auto
-
-lemma tendsto_at_topE:
-  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  assumes "(f \<longlongrightarrow> l) at_top"
-  assumes "e > 0"
-  obtains x0 where "\<And>x. x \<ge> x0 \<Longrightarrow> dist (f x) l < e"
-proof -
-  from assms(1)[THEN tendstoD, OF assms(2)]
-  have "\<forall>\<^sub>F x in at_top. dist (f x) l < e" .
-  then show ?thesis
-    unfolding eventually_at_top_linorder
-    by (auto intro: that)
-qed
-lemma tendsto_at_top_iff: "(f \<longlongrightarrow> l) at_top \<longleftrightarrow> (\<forall>e>0. \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e)"
-  for f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  by (auto intro!: tendsto_at_topI elim!: tendsto_at_topE)
-
-lemma tendsto_at_top_eq_left:
-  fixes f g::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  assumes "(f \<longlongrightarrow> l) at_top"
-  assumes "\<And>x. x \<ge> x0 \<Longrightarrow> f x = g x"
-  shows "(g \<longlongrightarrow> l) at_top"
-  unfolding tendsto_at_top_iff
-  by (metis (no_types, hide_lams) assms(1) assms(2) linear order_trans tendsto_at_topE)
-
-lemma lim_divide_n: "(\<lambda>x. e / real x) \<longlonglongrightarrow> 0"
-proof -
-  have "(\<lambda>x. e * inverse (real x)) \<longlonglongrightarrow> 0"
-    by (auto intro: tendsto_eq_intros lim_inverse_n)
-  then show ?thesis by (simp add: inverse_eq_divide)
-qed
-
-definition at_top_within :: "('a::order) set \<Rightarrow> 'a filter"
-  where "at_top_within s = (INF k \<in> s. principal ({k ..} \<inter> s)) "
-
-lemma at_top_within_at_top[simp]:
-  shows "at_top_within UNIV = at_top"
-  unfolding at_top_within_def at_top_def
-  by (auto)
-
-lemma at_top_within_empty[simp]:
-  shows "at_top_within {} = top"
-  unfolding at_top_within_def
-  by (auto)
-
-definition "nhds_set X = (INF S\<in>{S. open S \<and> X \<subseteq> S}. principal S)"
-
-lemma eventually_nhds_set:
-  "(\<forall>\<^sub>F x in nhds_set X. P x) \<longleftrightarrow> (\<exists>S. open S \<and> X \<subseteq> S \<and> (\<forall>x\<in>S. P x))"
-  unfolding nhds_set_def by (subst eventually_INF_base) (auto simp: eventually_principal)
-
-term "filterlim f (nhds_set (frontier X)) F" \<comment> \<open>f tends to the boundary of X?\<close>
-
-
-text \<open>somewhat inspired by @{thm islimpt_range_imp_convergent_subsequence} and its dependencies.
-The class constraints seem somewhat arbitrary, perhaps this can be generalized in some way.
-\<close>
-lemma limpt_closed_imp_exploding_subsequence:\<comment>\<open>TODO: improve name?!\<close>
-  fixes f::"'a::{heine_borel,real_normed_vector} \<Rightarrow> 'b::{first_countable_topology, t2_space}"
-  assumes cont[THEN continuous_on_compose2, continuous_intros]: "continuous_on T f"
-  assumes closed: "closed T"
-  assumes bound: "\<And>t. t \<in> T \<Longrightarrow> f t \<noteq> l"
-  assumes limpt: "l islimpt (f ` T)"
-  obtains s where
-    "(f \<circ> s) \<longlonglongrightarrow> l"
-    "\<And>i. s i \<in> T"
-    "\<And>C. compact C \<Longrightarrow> C \<subseteq> T \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C"
-proof -
-  from countable_basis_at_decseq[of l]
-  obtain A where A: "\<And>i. open (A i)" "\<And>i. l \<in> A i"
-    and evA: "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
-    by blast
-
-  from closed_Union_compact_subsets[OF closed]
-  obtain C
-    where C: "(\<And>n. compact (C n))" "(\<And>n. C n \<subseteq> T)" "(\<And>n. C n \<subseteq> C (Suc n))" "\<Union> (range C) = T"
-      and evC: "(\<And>K. compact K \<Longrightarrow> K \<subseteq> T \<Longrightarrow> \<forall>\<^sub>F i in sequentially. K \<subseteq> C i)"
-    by (metis eventually_sequentially)
-
-  have AC: "l \<in> A i - f ` C i" "open (A i - f ` C i)" for i
-    using C bound
-    by (fastforce intro!: open_Diff A compact_imp_closed compact_continuous_image continuous_intros)+
-
-  from islimptE[OF limpt AC] have "\<exists>t\<in>T. f t \<in> A i - f ` C i \<and> f t \<noteq> l" for i by blast  
-  then obtain t where t: "\<And>i. t i \<in> T" "\<And>i. f (t i) \<in> A i - f ` C i" "\<And>i. f (t i) \<noteq> l"
-    by metis
-
-  have "(f o t) \<longlonglongrightarrow> l"
-    using t
-    by (auto intro!: topological_tendstoI dest!: evA elim!: eventually_mono)
-  moreover
-  have "\<And>i. t i \<in> T" by fact
-  moreover
-  have "\<forall>\<^sub>F i in sequentially. t i \<notin> K" if "compact K" "K \<subseteq> T" for K
-    using evC[OF that]
-    by eventually_elim (use t in auto)
-  ultimately show ?thesis ..
-qed  
-
-lemma Inf_islimpt: "bdd_below S \<Longrightarrow> Inf S \<notin> S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> Inf S islimpt S" for S::"real set"
-  by (auto simp: islimpt_in_closure intro!: closure_contains_Inf)
-
-context linorder
-begin
-
-text \<open>HOL-analysis doesn't seem to have these, maybe they were never needed.
-  Some variants are around @{thm Int_atLeastAtMost}, but with old-style naming conventions.
-  Change to the "modern" I.. convention there?\<close>
-
-lemma Int_Ico[simp]:
-  shows "{a..} \<inter> {b..} = {max a b ..}"
-  by (auto)
-
-lemma Int_Ici_Ico[simp]:
-  shows "{a..} \<inter> {b..<c} = {max a b ..<c}"
-  by auto
-
-lemma Int_Ico_Ici[simp]:
-  shows "{a..<c} \<inter> {b..} = {max a b ..<c}"
-  by auto
-
-lemma subset_Ico_iff[simp]:
-  "{a..<b} \<subseteq> {c..<b} \<longleftrightarrow> b \<le> a \<or> c \<le> a"
-  unfolding atLeastLessThan_def
-  by auto
-
-lemma Ico_subset_Ioo_iff[simp]:
-  "{a..<b} \<subseteq> {c<..<b} \<longleftrightarrow> b \<le> a \<or> c < a"
-  unfolding greaterThanLessThan_def atLeastLessThan_def
-  by auto
-
-lemma Icc_Un_Ici[simp]:
-  shows "{a..b} \<union> {b..} = {min a b..}"
-  unfolding atLeastAtMost_def atLeast_def atMost_def min_def
-  by auto
-
-end
-
-lemma at_top_within_at_top_unbounded_right:
-  fixes a::"'a::linorder"
-  shows "at_top_within {a..} = at_top"
-  unfolding at_top_within_def at_top_def
-  apply (auto intro!: INF_eq)
-  by (metis linorder_class.linear linorder_class.max.cobounded1 linorder_class.max.idem ord_class.atLeast_iff)
-
-lemma at_top_within_at_top_unbounded_rightI:
-  fixes a::"'a::linorder"
-  assumes "{a..} \<subseteq> s"
-  shows "at_top_within s = at_top"
-  unfolding at_top_within_def at_top_def
-  apply (auto intro!: INF_eq)
-   apply (meson Ici_subset_Ioi_iff Ioi_le_Ico assms dual_order.refl dual_order.trans leI)
-  by (metis assms atLeast_iff atLeast_subset_iff inf.cobounded1 linear subsetD)
-
-lemma at_top_within_at_top_bounded_right:
-  fixes a b::"'a::{dense_order,linorder_topology}"
-  assumes "a < b"
-  shows "at_top_within {a..<b} = at_left b"
-  unfolding at_top_within_def at_left_eq[OF assms(1)]
-  apply (auto intro!: INF_eq)
-   apply (smt atLeastLessThan_iff greaterThanLessThan_iff le_less lessThan_iff max.absorb1 subset_eq)
-  by (metis assms atLeastLessThan_iff dense linear max.absorb1 not_less order_trans)
-
-lemma at_top_within_at_top_bounded_right':
-  fixes a b::"'a::{dense_order,linorder_topology}"
-  assumes "a < b"
-  shows "at_top_within {..<b} = at_left b"
-  unfolding at_top_within_def at_left_eq[OF assms(1)]
-  apply (auto intro!: INF_eq)
-   apply (meson atLeast_iff greaterThanLessThan_iff le_less lessThan_iff subset_eq)
-  by (metis Ico_subset_Ioo_iff atLeastLessThan_def dense lessThan_iff)
-
-lemma eventually_at_top_within_linorder:
-  assumes sn:"s \<noteq> {}"
-  shows "eventually P (at_top_within s) \<longleftrightarrow> (\<exists>x0::'a::{linorder_topology} \<in> s. \<forall>x \<ge> x0. x\<in> s \<longrightarrow> P x)"
-  unfolding at_top_within_def
-  apply (subst eventually_INF_base)
-    apply (auto simp:eventually_principal sn)
-  by (metis atLeast_subset_iff inf.coboundedI2 inf_commute linear)
-
-lemma tendsto_at_top_withinI:
-  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  assumes "s \<noteq> {}"
-  assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>x0 \<in> s. \<forall>x \<in> {x0..} \<inter> s. dist (f x) l < e"
-  shows  "(f \<longlongrightarrow> l) (at_top_within s)"
-  apply(intro tendstoI)
-  unfolding at_top_within_def apply (subst eventually_INF_base)
-    apply (auto simp:eventually_principal assms)
-  by (metis atLeast_subset_iff inf.coboundedI2 inf_commute linear)
-
-lemma tendsto_at_top_withinE:
-  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  assumes "s \<noteq> {}"
-  assumes "(f \<longlongrightarrow> l) (at_top_within s)"
-  assumes "e > 0"
-  obtains x0 where "x0 \<in> s" "\<And>x. x \<in> {x0..} \<inter> s \<Longrightarrow> dist (f x) l < e"
-proof -
-  from assms(2)[THEN tendstoD, OF assms(3)]
-  have "\<forall>\<^sub>F x in at_top_within s. dist (f x) l < e" .
-  then show ?thesis unfolding eventually_at_top_within_linorder[OF \<open>s \<noteq> {}\<close>] 
-    by (auto intro: that)
-qed
-
-lemma tendsto_at_top_within_iff:
-  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
-  assumes "s \<noteq> {}"
-  shows "(f \<longlongrightarrow> l) (at_top_within s) \<longleftrightarrow> (\<forall>e>0. \<exists>x0 \<in> s. \<forall>x \<in> {x0..} \<inter> s. dist (f x) l < e)"
-  by (auto intro!: tendsto_at_top_withinI[OF \<open>s \<noteq> {}\<close>] elim!: tendsto_at_top_withinE[OF \<open>s \<noteq> {}\<close>])
-
-lemma filterlim_at_top_at_top_within_bounded_right:
-  fixes a b::"'a::{dense_order,linorder_topology}"
-  fixes f::"'a \<Rightarrow> real"
-  assumes "a < b"
-  shows "filterlim f at_top (at_top_within {..<b}) = (f \<longlongrightarrow> \<infinity>) (at_left b)"
-  unfolding filterlim_at_top_dense
-    at_top_within_at_top_bounded_right'[OF assms(1)]
-    eventually_at_left[OF assms(1)]
-    tendsto_PInfty
-  by auto
-
-text \<open>Extract a sequence (going to infinity) bounded away from l\<close>
-
-lemma not_tendsto_frequentlyE:
-  assumes "\<not>((f \<longlongrightarrow> l) F)"
-  obtains S where "open S" "l \<in> S" "\<exists>\<^sub>F x in F. f x \<notin> S"
-  using assms
-  by (auto simp: tendsto_def not_eventually)
-
-lemma not_tendsto_frequently_metricE:
-  assumes "\<not>((f \<longlongrightarrow> l) F)"
-  obtains e where "e > 0" "\<exists>\<^sub>F x in F. e \<le> dist (f x) l"
-  using assms
-  by (auto simp: tendsto_iff not_eventually not_less)
-
-lemma eventually_frequently_conj: "frequently P F \<Longrightarrow> eventually Q F \<Longrightarrow> frequently (\<lambda>x. P x \<and> Q x) F"
-  unfolding frequently_def
-  apply (erule contrapos_nn)
-  subgoal premises prems
-    using prems by eventually_elim auto
-  done
-
-lemma frequently_at_top:
-  "(\<exists>\<^sub>F t in at_top. P t) \<longleftrightarrow> (\<forall>t0. \<exists>t>t0. P t)"
-  for P::"'a::{linorder,no_top}\<Rightarrow>bool" 
-  by (auto simp: frequently_def eventually_at_top_dense)
-
-lemma frequently_at_topE:
-  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_"
-  assumes freq[rule_format]: "\<forall>n. \<exists>\<^sub>F a in at_top. P n a"
-  obtains s::"nat\<Rightarrow>'a"
-  where "\<And>i. P i (s i)" "strict_mono s"
-proof -
-  have "\<exists>f. \<forall>n. P n (f n) \<and> f n < f (Suc n)"
-  proof (rule dependent_nat_choice)
-    from frequently_ex[OF freq[of 0]] show "\<exists>x. P 0 x" .
-    fix x n assume "P n x"
-    from freq[unfolded frequently_at_top, rule_format, of x "Suc n"]
-    obtain y where "P (Suc n) y" "y > x" by auto
-    then show "\<exists>y. P (Suc n) y \<and> x < y"
-      by auto
-  qed
-  then obtain s where "\<And>i. P i (s i)" "strict_mono s"
-    unfolding strict_mono_Suc_iff by auto
-  then show ?thesis ..
-qed
-
-lemma frequently_at_topE':
-  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_"
-  assumes freq[rule_format]: "\<forall>n. \<exists>\<^sub>F a in at_top. P n a"
-    and g: "filterlim g at_top sequentially"
-  obtains s::"nat\<Rightarrow>'a"
-  where "\<And>i. P i (s i)" "strict_mono s" "\<And>n. g n \<le> s n"
-proof -
-  have "\<forall>n. \<exists>\<^sub>F a in at_top. P n a \<and> g n \<le> a"
-    using freq
-    by (auto intro!: eventually_frequently_conj)
-  from frequently_at_topE[OF this] obtain s where "\<And>i. P i (s i)" "strict_mono s" "\<And>n. g n \<le> s n"
-    by metis
-  then show ?thesis ..
-qed
-
-lemma frequently_at_top_at_topE:
-  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_" and g::"nat\<Rightarrow>'a"
-  assumes "\<forall>n. \<exists>\<^sub>F a in at_top. P n a" "filterlim g at_top sequentially"
-  obtains s::"nat\<Rightarrow>'a"
-  where "\<And>i. P i (s i)" "filterlim s at_top sequentially"
-proof -
-  from frequently_at_topE'[OF assms]
-  obtain s where s: "(\<And>i. P i (s i))" "strict_mono s" "(\<And>n. g n \<le> s n)" by blast
-  have s_at_top: "filterlim s at_top sequentially"
-    by (rule filterlim_at_top_mono) (use assms s in auto)
-  with s(1) show ?thesis ..
-qed
-
-(* Extract a strict monotone and sequence converging to something other than l *)
-lemma not_tendsto_convergent_seq:
-  fixes f::"real \<Rightarrow> 'a::metric_space"
-  assumes X: "compact (X::'a set)"
-  assumes im: "\<And>x. x \<ge> 0 \<Longrightarrow> f x \<in> X"
-  assumes nl: "\<not> ((f \<longlongrightarrow> (l::'a)) at_top)"
-  obtains s k where
-    "k \<in> X" "k \<noteq> l" "(f \<circ> s) \<longlonglongrightarrow> k" "strict_mono s" "\<forall>n. s n \<ge> n"
-proof -
-  from not_tendsto_frequentlyE[OF nl]
-  obtain S where "open S" "l \<in> S" "\<exists>\<^sub>F x in at_top. f x \<notin> S" .
-  have "\<forall>n. \<exists>\<^sub>F x in at_top. f x \<notin> S \<and> real n \<le> x"
-    apply (rule allI)
-    apply (rule eventually_frequently_conj)
-     apply fact
-    by (rule eventually_ge_at_top)
-  from frequently_at_topE[OF this]
-  obtain s where "\<And>i. f (s i) \<notin> S" and s: "strict_mono s" and s_ge: "(\<And>i. real i \<le> s i)" by metis
-  then have "0 \<le> s i" for i using dual_order.trans of_nat_0_le_iff by blast
-  then have "\<forall>n. (f \<circ> s) n \<in> X" using im by auto
-  from X[unfolded compact_def, THEN spec, THEN mp, OF this]
-  obtain k r where k: "k \<in> X" and r: "strict_mono r" and kLim: "(f \<circ> s \<circ> r) \<longlonglongrightarrow> k" by metis
-  have "k \<in> X - S"
-    by (rule Lim_in_closed_set[of "X - S", OF _ _ _ kLim])
-      (auto simp: im \<open>0 \<le> s _\<close>  \<open>\<And>i. f (s i) \<notin> S\<close> intro!: \<open>open S\<close> X intro: compact_imp_closed)
-
-  note k
-  moreover have "k \<noteq> l" using \<open>k \<in> X - S\<close> \<open>l \<in> S\<close> by auto
-  moreover have "(f \<circ> (s \<circ> r)) \<longlonglongrightarrow> k" using kLim by (simp add: o_assoc)
-  moreover have "strict_mono (s \<circ> r)" using s r by (rule strict_mono_o)
-  moreover have "\<forall>n. (s \<circ> r) n \<ge> n" using s_ge r
-    by (metis comp_apply dual_order.trans of_nat_le_iff seq_suble)
-  ultimately show ?thesis ..
-qed
-
-lemma harmonic_bound:
-  shows "1 / 2 ^(Suc n) < 1 / real (Suc n)"
-proof (induction n)
-  case 0
-  then show ?case by auto
-next
-  case (Suc n)
-  then show ?case
-    by (smt frac_less2 of_nat_0_less_iff of_nat_less_two_power zero_less_Suc)
-qed
-
-lemma INF_bounded_imp_convergent_seq:
-  fixes f::"real \<Rightarrow> real"
-  assumes cont: "continuous_on {a..} f"
-  assumes bound: "\<And>t. t \<ge> a \<Longrightarrow> f t > l"
-  assumes inf: "(INF t\<in>{a..}. f t) = l"
-  obtains s where
-    "(f \<circ> s) \<longlonglongrightarrow> l"
-    "\<And>i. s i \<in> {a..}"
-    "filterlim s at_top sequentially"
-proof -
-  have bound': "t \<in> {a..} \<Longrightarrow> f t \<noteq> l" for t using bound[of t] by auto
-  have limpt: "l islimpt f ` {a..}"
-  proof -
-    have "Inf (f ` {a..}) islimpt f ` {a..}"
-      by (rule Inf_islimpt) (auto simp: inf intro!: bdd_belowI2[where m=l] dest: bound)
-    then show ?thesis by (simp add: inf)
-  qed
-  from limpt_closed_imp_exploding_subsequence[OF cont closed_atLeast bound' limpt]
-  obtain s where s: "(f \<circ> s) \<longlonglongrightarrow> l"
-    "\<And>i. s i \<in> {a..}"
-    "compact C \<Longrightarrow> C \<subseteq> {a..} \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C" for C
-    by metis
-  have "\<forall>\<^sub>F i in sequentially. s i \<ge> n" for n
-    using s(3)[of "{a..n}"] s(2)
-    by (auto elim!: eventually_mono)
-  then have "filterlim s at_top sequentially"
-    unfolding filterlim_at_top
-    by auto
-  from s(1) s(2) this
-  show ?thesis ..
-qed
-
-(* Generalizes to other combinations of strict_mono and filterlim *)
-lemma filterlim_at_top_strict_mono:
-  fixes s :: "_ \<Rightarrow> 'a::linorder"
-  fixes r :: "nat \<Rightarrow> _"
-  assumes "strict_mono s"
-  assumes "strict_mono r"
-  assumes "filterlim s at_top F"
-  shows "filterlim (s \<circ> r) at_top F"
-  apply (rule filterlim_at_top_mono[OF assms(3)])
-  by (simp add: assms(1) assms(2) seq_suble strict_mono_leD)
-
-lemma LIMSEQ_lb:
-  assumes fl: "s \<longlonglongrightarrow> (l::real)"
-  assumes u: "l < u"
-  shows "\<exists>n0. \<forall>n\<ge>n0. s n < u"
-proof -
-  from fl have "\<exists>no>0. \<forall>n\<ge>no. dist (s n) l < u-l" unfolding LIMSEQ_iff_nz using u
-    by simp
-  thus ?thesis using dist_real_def by fastforce
-qed
-
-(* Used to sharpen a tendsto with additional information*)
-lemma filterlim_at_top_choose_lower:
-  assumes "filterlim s at_top sequentially"
-  assumes "(f \<circ> s) \<longlonglongrightarrow> l"
-  obtains t where
-    "filterlim t at_top sequentially"
-    "(f \<circ> t) \<longlonglongrightarrow> l"
-    "\<forall>n. t n \<ge> (b::real)"
-proof -
-  obtain k where k: "\<forall>n \<ge> k. s n \<ge> b" using assms(1)
-    unfolding filterlim_at_top eventually_sequentially by blast
-  define t where "t = (\<lambda>n. s (n+k))"
-  then have "\<forall>n. t n \<ge> b" using k by simp
-  have "filterlim t at_top sequentially" using assms(1)
-    unfolding filterlim_at_top eventually_sequentially t_def
-    by (metis (full_types) add.commute trans_le_add2)
-  from LIMSEQ_ignore_initial_segment[OF assms(2), of "k"]
-  have "(\<lambda>n. (f \<circ> s) (n + k)) \<longlonglongrightarrow> l" .
-  then have "(f \<circ> t) \<longlonglongrightarrow> l" unfolding t_def o_def by simp
-  show ?thesis
-    using \<open>(f \<circ> t) \<longlonglongrightarrow> l\<close> \<open>\<forall>n. b \<le> t n\<close> \<open>filterlim t at_top sequentially\<close> that by blast
-qed
-
-lemma frequently_at_top_realE:
-  fixes P::"nat \<Rightarrow> real \<Rightarrow> bool"
-  assumes "\<forall>n. \<exists>\<^sub>F t in at_top. P n t"
-  obtains s::"nat\<Rightarrow>real"
-  where "\<And>i. P i (s i)" "filterlim s at_top at_top"
-  by (metis assms frequently_at_top_at_topE[OF _ filterlim_real_sequentially])
-
-lemma approachable_sequenceE:
-  fixes f::"real \<Rightarrow> 'a::metric_space"
-  assumes "\<And>t e. 0 \<le> t \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>tt\<ge>t. dist (f tt) p < e"
-  obtains s where "filterlim s at_top sequentially" "(f \<circ> s) \<longlonglongrightarrow> p"
-proof -
-  have "\<forall>n. \<exists>\<^sub>F i in at_top. dist (f i) p < 1/real (Suc n)"
-    unfolding frequently_at_top
-    apply (auto )
-    subgoal for n m
-      using assms[of "max 0 (m+1)" "1/(Suc n)"]
-      by force
-    done
-  from frequently_at_top_realE[OF this]
-  obtain s where s: "\<And>i. dist (f (s i)) p < 1 / real (Suc i)" "filterlim s at_top sequentially"
-    by metis
-  note this(2)
-  moreover
-  have "(f o s) \<longlonglongrightarrow> p"
-  proof (rule tendstoI)
-    fix e::real assume "e > 0"
-    have "\<forall>\<^sub>F i in sequentially. 1 / real (Suc i) < e"
-      apply (rule order_tendstoD[OF _ \<open>0 < e\<close>])
-      apply (rule real_tendsto_divide_at_top)
-       apply (rule tendsto_intros)
-      by (rule filterlim_compose[OF filterlim_real_sequentially filterlim_Suc])
-    then show "\<forall>\<^sub>F x in sequentially. dist ((f \<circ> s) x) p < e"
-      by eventually_elim (use dual_order.strict_trans s \<open>e > 0\<close> in auto)
-  qed
-  ultimately show ?thesis ..
-qed
-
-lemma mono_inc_bdd_above_has_limit_at_topI:
-  fixes f::"real \<Rightarrow> real"
-  assumes "mono f"
-  assumes "\<And>x. f x \<le> u"
-  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
-proof -
-  define l where "l = Sup (range (\<lambda>n. f (real n)))"
-  have t:"(\<lambda>n. f (real n)) \<longlonglongrightarrow> l" unfolding l_def
-    apply (rule LIMSEQ_incseq_SUP)
-     apply (meson assms(2) bdd_aboveI2)
-    by (meson assms(1) mono_def of_nat_mono)
-  from tendsto_at_topI_sequentially_real[OF assms(1) t]
-  have "(f \<longlongrightarrow> l) at_top" .
-  thus ?thesis by blast  
-qed
-
-lemma gen_mono_inc_bdd_above_has_limit_at_topI:
-  fixes f::"real \<Rightarrow> real"
-  assumes "\<And>x y. x \<ge> b \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
-  assumes "\<And>x. x \<ge> b \<Longrightarrow> f x \<le> u"
-  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
-proof -
-  define ff where "ff = (\<lambda>x. if x \<ge> b then f x else f b)"
-  have m1:"mono ff" unfolding ff_def mono_def using assms(1) by simp
-  have m2:"\<And>x. ff x \<le> u" unfolding ff_def using assms(2) by simp
-  from mono_inc_bdd_above_has_limit_at_topI[OF m1 m2]
-  obtain l where "(ff \<longlongrightarrow> l) at_top" by blast
-  thus ?thesis
-    by (meson \<open>(ff \<longlongrightarrow> l) at_top\<close> ff_def tendsto_at_top_eq_left)
-qed
-
-lemma gen_mono_dec_bdd_below_has_limit_at_topI:
-  fixes f::"real \<Rightarrow> real"
-  assumes "\<And>x y. x \<ge> b \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
-  assumes "\<And>x. x \<ge> b \<Longrightarrow> f x \<ge> u"
-  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
-proof -
-  define ff where "ff = (\<lambda>x. if x \<ge> b then f x else f b)"
-  have m1:"mono (-ff)" unfolding ff_def mono_def using assms(1) by simp
-  have m2:"\<And>x. (-ff) x \<le> -u" unfolding ff_def using assms(2) by simp
-  from mono_inc_bdd_above_has_limit_at_topI[OF m1 m2]
-  obtain l where "(-ff \<longlongrightarrow> l) at_top" by blast
-  then have "(ff \<longlongrightarrow> -l) at_top"
-    using tendsto_at_top_eq_left tendsto_minus_cancel_left by fastforce  
-  thus ?thesis
-    by (meson \<open>(ff \<longlongrightarrow> -l) at_top\<close> ff_def tendsto_at_top_eq_left)
-qed
-
-lemma infdist_closed:
-  shows "closed ({z. infdist z S \<ge> e})"
-  by (auto intro!:closed_Collect_le simp add:continuous_on_infdist)
-
-(* TODO: this is a copy of LIMSEQ_norm_0 where the sequence
-  is bounded above in norm by a geometric series *)
-lemma LIMSEQ_norm_0_pow:
-  assumes "k > 0" "b > 1"
-  assumes  "\<And>n::nat. norm (s n) \<le> k / b^n"
-  shows "s \<longlonglongrightarrow> 0"
-proof (rule metric_LIMSEQ_I)
-  fix e
-  assume "e > (0::real)"
-  then have "k / e > 0" using assms(1) by auto
-  obtain N where N: "b^(N::nat) > k / e" using assms(2)
-    using real_arch_pow by blast
-  then have "norm (s n) < e" if "n \<ge> N" for n
-  proof -
-    have "k / b^n \<le> k / b^N"
-      by (smt assms(1) assms(2) frac_le leD power_less_imp_less_exp that zero_less_power)
-    also have " ... < e" using N
-      by (metis \<open>0 < e\<close> assms(2) less_trans mult.commute pos_divide_less_eq zero_less_one zero_less_power)
-    finally show ?thesis
-      by (meson assms less_eq_real_def not_le order_trans)
-  qed
-  then show "\<exists>no. \<forall>n\<ge>no. dist (s n) 0 < e"
-    by auto
-qed
-
-lemma filterlim_apply_filtermap:
-  assumes g: "filterlim g G F"
-  shows "filterlim (\<lambda>x. m (g x)) (filtermap m G) F"
-  by (metis filterlim_def filterlim_filtermap filtermap_mono g)
-
-lemma eventually_at_right_field_le:
-  "eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y \<le> b \<longrightarrow> P y)"
-  for x :: "'a::{linordered_field, linorder_topology}"
-  by (smt dense eventually_at_right_field le_less_trans less_le_not_le order.strict_trans1)
-
-subsection \<open>indexing euclidean space with natural numbers\<close>
-
-definition  nth_eucl :: "'a::executable_euclidean_space \<Rightarrow> nat \<Rightarrow> real" where
-  "nth_eucl x i = x \<bullet> (Basis_list ! i)"
-  \<comment> \<open>TODO: why is that and some sort of \<open>lambda_eucl\<close> nowhere available?\<close>
-definition lambda_eucl :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::executable_euclidean_space" where
-  "lambda_eucl (f::nat\<Rightarrow>real) = (\<Sum>i<DIM('a). f i *\<^sub>R Basis_list ! i)"
-
-lemma eucl_eq_iff: "x = y \<longleftrightarrow> (\<forall>i<DIM('a). nth_eucl x i = nth_eucl y i)"
-  for x y::"'a::executable_euclidean_space"
-  apply (auto simp: nth_eucl_def euclidean_eq_iff[where 'a='a])
-  by (metis eucl_of_list_list_of_eucl list_of_eucl_eq_iff)
-
-bundle eucl_notation begin
-notation nth_eucl (infixl "$\<^sub>e" 90)
-end
-bundle no_eucl_notation begin
-no_notation nth_eucl (infixl "$\<^sub>e" 90)
-end
-
-unbundle eucl_notation
-
-lemma eucl_of_list_eucl_nth:
-  "(eucl_of_list xs::'a) $\<^sub>e i = xs ! i"
-  if "length xs = DIM('a::executable_euclidean_space)"
-    "i < DIM('a)"
-  using that
-  apply (auto simp: nth_eucl_def)
-  by (metis list_of_eucl_eucl_of_list list_of_eucl_nth)
-
-lemma eucl_of_list_inner:
-  "(eucl_of_list xs::'a) \<bullet> eucl_of_list ys = (\<Sum>(x,y)\<leftarrow>zip xs ys. x * y)"
-  if "length xs = DIM('a::executable_euclidean_space)"
-    "length ys = DIM('a::executable_euclidean_space)"
-  using that
-  by (auto simp: nth_eucl_def eucl_of_list_inner_eq inner_lv_rel_def)
-
-lemma self_eq_eucl_of_list: "x = eucl_of_list (map (\<lambda>i. x $\<^sub>e i) [0..<DIM('a)])"
-  for x::"'a::executable_euclidean_space"
-  by (auto simp: eucl_eq_iff[where 'a='a] eucl_of_list_eucl_nth)
-
-lemma inner_nth_eucl: "x \<bullet> y = (\<Sum>i<DIM('a). x $\<^sub>e i * y $\<^sub>e i)"
-  for x y::"'a::executable_euclidean_space"
-  apply (subst self_eq_eucl_of_list[where x=x])
-  apply (subst self_eq_eucl_of_list[where x=y])
-  apply (subst eucl_of_list_inner)
-  by (auto simp: map2_map_map atLeast_upt interv_sum_list_conv_sum_set_nat)
-
-lemma norm_nth_eucl: "norm x = L2_set (\<lambda>i. x $\<^sub>e i) {..<DIM('a)}"
-  for x::"'a::executable_euclidean_space"
-  unfolding norm_eq_sqrt_inner inner_nth_eucl L2_set_def
-  by (auto simp: power2_eq_square)
-
-
-lemma plus_nth_eucl: "(x + y) $\<^sub>e i = x $\<^sub>e i + y $\<^sub>e i"
-  and minus_nth_eucl: "(x - y) $\<^sub>e i = x $\<^sub>e i - y $\<^sub>e i"
-  and uminus_nth_eucl: "(-x) $\<^sub>e i = - x $\<^sub>e i"
-  and scaleR_nth_eucl: "(c *\<^sub>R x) $\<^sub>e i = c *\<^sub>R (x $\<^sub>e i)"
-  by (auto simp: nth_eucl_def algebra_simps)
-
-lemma inf_nth_eucl: "inf x y $\<^sub>e i = min (x $\<^sub>e i) (y $\<^sub>e i)"
-  if "i < DIM('a)"
-  for x::"'a::executable_euclidean_space"
-  by (auto simp: nth_eucl_def algebra_simps inner_Basis_inf_left that inf_min)
-lemma sup_nth_eucl: "sup x y $\<^sub>e i = max (x $\<^sub>e i) (y $\<^sub>e i)"
-  if "i < DIM('a)"
-  for x::"'a::executable_euclidean_space"
-  by (auto simp: nth_eucl_def algebra_simps inner_Basis_sup_left that sup_max)
-
-lemma le_iff_le_nth_eucl: "x \<le> y \<longleftrightarrow> (\<forall>i<DIM('a). (x $\<^sub>e i) \<le> (y $\<^sub>e i))"
-  for x::"'a::executable_euclidean_space"
-  apply (auto simp: nth_eucl_def algebra_simps eucl_le[where 'a='a])
-  by (meson eucl_le eucl_le_Basis_list_iff)
-
-lemma eucl_less_iff_less_nth_eucl: "eucl_less x y \<longleftrightarrow> (\<forall>i<DIM('a). (x $\<^sub>e i) < (y $\<^sub>e i))"
-  for x::"'a::executable_euclidean_space"
-  apply (auto simp: nth_eucl_def algebra_simps eucl_less_def[where 'a='a])
-  by (metis Basis_zero eucl_eq_iff inner_not_same_Basis inner_zero_left length_Basis_list
-      nth_Basis_list_in_Basis nth_eucl_def)
-
-lemma continuous_on_nth_eucl[continuous_intros]:
-  "continuous_on X (\<lambda>x. f x $\<^sub>e i)"
-  if "continuous_on X f"
-  by (auto simp: nth_eucl_def intro!: continuous_intros that)
-
-
-subsection \<open>derivatives\<close>
-
-lemma eventually_at_ne[intro, simp]: "\<forall>\<^sub>F x in at x0. x \<noteq> x0"
-  by (auto simp: eventually_at_filter)
-
-lemma has_vector_derivative_withinD:
-  fixes f::"real \<Rightarrow> 'b::euclidean_space"
-  assumes "(f has_vector_derivative f') (at x0 within S)"
-  shows "((\<lambda>x. (f x - f x0) /\<^sub>R (x - x0)) \<longlongrightarrow> f') (at x0 within S)"
-  apply (rule LIM_zero_cancel)
-  apply (rule tendsto_norm_zero_cancel)
-  apply (rule Lim_transform_eventually)
-proof -
-  show "\<forall>\<^sub>F x in at x0 within S. norm ((f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0)) =
-    norm ((f x - f x0) /\<^sub>R (x - x0) - f')"
-    (is "\<forall>\<^sub>F x in _. ?th x")
-    unfolding eventually_at_filter
-  proof (safe intro!: eventuallyI)
-    fix x assume x: "x \<noteq> x0"
-    then have "norm ((f x - f x0) /\<^sub>R (x - x0) - f') = norm (sgn (x - x0) *\<^sub>R ((f x - f x0) /\<^sub>R (x - x0) - f'))"
-      by simp
-    also have "sgn (x - x0) *\<^sub>R ((f x - f x0) /\<^sub>R (x - x0) - f') = ((f x - f x0) /\<^sub>R norm (x - x0) - (x - x0) *\<^sub>R f' /\<^sub>R norm (x - x0))"
-      by (auto simp add: algebra_simps sgn_div_norm divide_simps)
-        (metis add.commute add_divide_distrib diff_add_cancel scaleR_add_left)
-    also have "\<dots> = (f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0)" by (simp add: algebra_simps)
-    finally show "?th x" ..
-  qed
-  show "((\<lambda>x. norm ((f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0))) \<longlongrightarrow> 0) (at x0 within S)"
-    by (rule tendsto_norm_zero)
-      (use assms in \<open>auto simp: has_vector_derivative_def has_derivative_at_within\<close>)
-qed
-
-text \<open>A \<open>path_connected\<close> set \<open>S\<close> entering both \<open>T\<close> and \<open>-T\<close>
-      must cross the frontier of \<open>T\<close> \<close>
-lemma path_connected_frontier:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "path_connected S"
-  assumes "S \<inter> T \<noteq> {}"
-  assumes "S \<inter> -T \<noteq> {}"
-  obtains s where "s \<in> S" "s \<in> frontier T"
-proof -
-  obtain st where st:"st \<in> S \<inter> T" using assms(2) by blast
-  obtain sn where sn:"sn \<in> S \<inter> -T" using assms(3) by blast
-  obtain g where g: "path g" "path_image g \<subseteq> S"
-    "pathstart g = st" "pathfinish g = sn"
-    using assms(1) st sn unfolding path_connected_def by blast
-  have a1:"pathstart g \<in> closure T" using st g(3) closure_Un_frontier by fastforce
-  have a2:"pathfinish g \<notin> T" using sn g(4) by auto
-  from exists_path_subpath_to_frontier[OF g(1) a1 a2]
-  obtain h where "path_image h \<subseteq> path_image g" "pathfinish h \<in> frontier T" by metis
-  thus ?thesis using g(2)
-    by (meson in_mono pathfinish_in_path_image that) 
-qed
-
-lemma path_connected_not_frontier_subset:
-  fixes S :: "'a::real_normed_vector set"
-  assumes "path_connected S"
-  assumes "S \<inter> T \<noteq> {}"
-  assumes "S \<inter> frontier T = {}"
-  shows "S \<subseteq> T"
-  using path_connected_frontier assms by auto  
-
-lemma compact_attains_bounds:
-  fixes f::"'a::topological_space \<Rightarrow> 'b::linorder_topology"
-  assumes compact: "compact S"
-  assumes ne: "S \<noteq> {}"
-  assumes cont: "continuous_on S f"
-  obtains l u where "l \<in> S" "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<in> {f l .. f u}"
-proof -
-  from compact_continuous_image[OF cont compact]
-  have compact_image: "compact (f ` S)" .
-  have ne_image: "f ` S \<noteq> {}" using ne by simp
-  from compact_attains_inf[OF compact_image ne_image]
-  obtain l where "l \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f l \<le> f x" by auto
-  moreover
-  from compact_attains_sup[OF compact_image ne_image]
-  obtain u where "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<le> f u" by auto
-  ultimately
-  have "l \<in> S" "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<in> {f l .. f u}" by auto
-  then show ?thesis ..
-qed
-
-lemma uniform_limit_const[uniform_limit_intros]:
-  "uniform_limit S (\<lambda>x y. f x) (\<lambda>_. l) F" if "(f \<longlongrightarrow> l) F"
-  apply (auto simp: uniform_limit_iff)
-  subgoal for e
-    using tendstoD[OF that(1), of e]
-    by (auto simp: eventually_mono)
-  done
-
-subsection \<open>Segments\<close>
-
-text \<open>\<open>closed_segment\<close> throws away the order that our intuition keeps\<close>
-
-definition line::"'a::real_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
-  ("{_ -- _}\<^bsub>_\<^esub>")
-  where "{a -- b}\<^bsub>u\<^esub> = a + u *\<^sub>R (b - a)"
-
-abbreviation "line_image a b U \<equiv>(\<lambda>u. {a -- b}\<^bsub>u\<^esub>) ` U"
-notation line_image ("{_ -- _}\<^bsub>`_\<^esub>")
-
-lemma in_closed_segment_iff_line: "x \<in> {a -- b} \<longleftrightarrow> (\<exists>c\<in>{0..1}. x = line a b c)"
-  by (auto simp: in_segment line_def algebra_simps)
-
-lemma in_open_segment_iff_line: "x \<in> {a <--< b} \<longleftrightarrow> (\<exists>c\<in>{0<..<1}. a \<noteq> b \<and> x = line a b c)"
-  by (auto simp: in_segment line_def algebra_simps)
-
-lemma line_convex_combination1: "(1 - u) *\<^sub>R line a b i + u *\<^sub>R b = line a b (i + u - i * u)"
-  by (auto simp: line_def algebra_simps)
-
-lemma line_convex_combination2: "(1 - u) *\<^sub>R a + u *\<^sub>R line a b i = line a b (i*u)"
-  by (auto simp: line_def algebra_simps)
-
-lemma line_convex_combination12: "(1 - u) *\<^sub>R line a b i + u *\<^sub>R line a b j = line a b (i + u * (j - i))"
-  by (auto simp: line_def algebra_simps)
-
-lemma mult_less_one_less_self: "0 < x \<Longrightarrow> i < 1 \<Longrightarrow> i * x < x" for i x::real
-  by auto
-
-lemma plus_times_le_one_lemma: "i + u - i * u \<le> 1" if "i \<le> 1" "u \<le> 1" for i u::real
-  by (simp add: diff_le_eq sum_le_prod1 that)
-
-lemma plus_times_less_one_lemma: "i + u - i * u < 1" if "i < 1" "u < 1" for i u::real
-proof -
-  have "u * (1 - i) < 1 - i"
-    using that by force
-  then show ?thesis by (simp add: algebra_simps)
-qed
-
-lemma line_eq_endpoint_iff[simp]:
-  "line a b i = b \<longleftrightarrow> (a = b \<or> i = 1)"
-  "a = line a b i \<longleftrightarrow> (a = b \<or> i = 0)"
-  by (auto simp: line_def algebra_simps)
-
-lemma line_eq_iff[simp]: "line a b x = line a b y \<longleftrightarrow> (x = y \<or> a = b)"
-  by (auto simp: line_def)
-
-lemma line_open_segment_iff:
-  "{line a b i<--<b} = line a b ` {i<..<1}"
-  if "i < 1" "a \<noteq> b"
-  using that
-  apply (auto simp: in_segment line_convex_combination1 plus_times_less_one_lemma)
-  subgoal for j
-    apply (rule exI[where x="(j - i)/(1 - i)"])
-    apply (auto simp: divide_simps algebra_simps)
-    by (metis add_diff_cancel less_numeral_extra(4) mult_2_right plus_times_less_one_lemma that(1))
-  done
-
-lemma open_segment_line_iff:
-  "{a<--<line a b i} = line a b ` {0<..<i}"
-  if "0 < i" "a \<noteq> b"
-  using that
-  apply (auto simp: in_segment line_convex_combination2 plus_times_less_one_lemma)
-  subgoal for j
-    apply (rule exI[where x="j/i"])
-    by (auto simp: )
-  done
-
-lemma line_closed_segment_iff:
-  "{line a b i--b} = line a b ` {i..1}"
-  if "i \<le> 1" "a \<noteq> b"
-  using that
-  apply (auto simp: in_segment line_convex_combination1 mult_le_cancel_right2 plus_times_le_one_lemma)
-  subgoal for j
-    apply (rule exI[where x="(j - i)/(1 - i)"])
-    apply (auto simp: divide_simps algebra_simps)
-    by (metis add_diff_cancel less_numeral_extra(4) mult_2_right plus_times_less_one_lemma that(1))
-  done
-
-lemma closed_segment_line_iff:
-  "{a--line a b i} = line a b ` {0..i}"
-  if "0 < i" "a \<noteq> b"
-  using that
-  apply (auto simp: in_segment line_convex_combination2 plus_times_less_one_lemma)
-  subgoal for j
-    apply (rule exI[where x="j/i"])
-    by (auto simp: )
-  done
-
-lemma closed_segment_line_line_iff: "{line a b i1--line a b i2} = line a b ` {i1..i2}" if "i1 \<le> i2"
-  using that
-  apply (auto simp: in_segment line_convex_combination12 intro!: imageI)
-   apply (smt mult_left_le_one_le)
-  subgoal for u
-    by (rule exI[where x="(u - i1)/(i2-i1)"]) auto
-  done
-
-lemma line_line1: "line (line a b c) b x = line a b (c + x - c * x)"
-  by (simp add: line_def algebra_simps)
-
-lemma line_line2: "line a (line a b c) x = line a b (c*x)"
-  by (simp add: line_def algebra_simps)
-
-lemma line_in_subsegment:
-  "i1 < 1 \<Longrightarrow> i2 < 1 \<Longrightarrow> a \<noteq> b \<Longrightarrow> line a b i1 \<in> {line a b i2<--<b} \<longleftrightarrow> i2 < i1"
-  by (auto simp: line_open_segment_iff intro!: imageI)
-
-lemma line_in_subsegment2:
-  "0 < i2 \<Longrightarrow> 0 < i1 \<Longrightarrow> a \<noteq> b \<Longrightarrow> line a b i1 \<in> {a<--<line a b i2} \<longleftrightarrow> i1 < i2"
-  by (auto simp: open_segment_line_iff intro!: imageI)
-
-lemma line_in_open_segment_iff[simp]:
-  "line a b i \<in> {a<--<b} \<longleftrightarrow> (a \<noteq> b \<and> 0 < i \<and> i < 1)"
-  by (auto simp: in_open_segment_iff_line)
-
-subsection \<open>Open Segments\<close>
-
-lemma open_segment_subsegment:
-  assumes "x1 \<in> {x0<--<x3}"
-    "x2 \<in> {x1<--<x3}"
-  shows "x1 \<in> {x0<--<x2}"
-  using assms
-proof -\<comment> \<open>TODO: use \<open>line\<close>\<close>
-  from assms obtain u v::real where
-    ne: "x0 \<noteq> x3" "(1 - u) *\<^sub>R x0 + u *\<^sub>R x3 \<noteq> x3"
-    and x1_def: "x1 = (1 - u) *\<^sub>R x0 + u *\<^sub>R x3"
-    and x2_def: "x2 = (1 - v) *\<^sub>R ((1 - u) *\<^sub>R x0 + u *\<^sub>R x3) + v *\<^sub>R x3"
-    and uv: \<open>0 < u\<close> \<open>0 < v\<close> \<open>u < 1\<close> \<open>v < 1\<close>
-    by (auto simp: in_segment)
-  let ?d = "(u + v - u * v)"
-  have "?d > 0" using uv
-    by (auto simp: add_nonneg_pos pos_add_strict)
-  with \<open>x0 \<noteq> x3\<close> have "0 \<noteq> ?d *\<^sub>R (x3 - x0)" by simp
-  moreover
-  define ua where "ua = u / ?d"
-  have "ua * (u * v - u - v) - - u = 0"
-    by (auto simp: ua_def algebra_simps divide_simps)
-      (metis uv add_less_same_cancel1 add_strict_mono mult.right_neutral
-        mult_less_cancel_left_pos not_real_square_gt_zero vector_space_over_itself.scale_zero_left)
-  then have "(ua * (u * v - u - v) - - u) *\<^sub>R (x3 - x0) = 0"
-    by simp
-  moreover
-  have "0 < ua" "ua < 1"
-    using \<open>0 < u\<close> \<open>0 < v\<close> \<open>u < 1\<close> \<open>v < 1\<close>
-    by (auto simp: ua_def pos_add_strict intro!: divide_pos_pos)
-  ultimately show ?thesis
-    unfolding x1_def x2_def
-    by (auto intro!: exI[where x=ua] simp: algebra_simps in_segment)
-qed
-
-
-subsection \<open>Syntax\<close>
-
-abbreviation sequentially_at_top::"(nat\<Rightarrow>real)\<Rightarrow>bool"
-  ("_ \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>") \<comment> \<open>the \<open>\<^bsub>\<^esub>\<close> is to disambiguate syntax...\<close>
-  where "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>  \<equiv> filterlim s at_top sequentially"
-
-abbreviation sequentially_at_bot::"(nat\<Rightarrow>real)\<Rightarrow>bool"
-  ("_ \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>")
-  where "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>  \<equiv> filterlim s at_bot sequentially"
-
-
-subsection \<open>Paths\<close>
-
-lemma subpath0_linepath:
-  shows "subpath 0 u (linepath t t') = linepath t (t + u * (t' - t))"
-  unfolding subpath_def linepath_def
-  apply (rule ext)
-  apply auto
-proof -
-  fix x :: real
-  have f1: "\<And>r ra rb rc. (r::real) + ra * rb - ra * rc = r - ra * (rc - rb)"
-    by (simp add: right_diff_distrib')
-  have f2: "\<And>r ra. (r::real) - r * ra = r * (1 - ra)"
-    by (simp add: right_diff_distrib')
-  have f3: "\<And>r ra rb. (r::real) - ra + rb + ra - r = rb"
-    by auto
-  have f4: "\<And>r. (r::real) + (1 - 1) = r"
-    by linarith
-  have f5: "\<And>r ra. (r::real) + ra = ra + r"
-    by force
-  have f6: "\<And>r ra. (r::real) + (1 - (r + 1) + ra) = ra"
-    by linarith
-  have "t - x * (t - (t + u * (t' - t))) = t' * (u * x) + (t - t * (u * x))"
-    by (simp add: right_diff_distrib')
-  then show "(1 - u * x) * t + u * x * t' = (1 - x) * t + x * (t + u * (t' - t))"
-    using f6 f5 f4 f3 f2 f1 by (metis (no_types) mult.commute)
-qed
-
-lemma linepath_image0_right_open_real:
-  assumes "t < (t'::real)"
-  shows "linepath t t' ` {0..<1} = {t..<t'}"
-  unfolding linepath_def
-  apply auto
-    apply (metis add.commute add_diff_cancel_left' assms diff_diff_eq2 diff_le_eq less_eq_real_def mult.commute mult.right_neutral mult_right_mono right_diff_distrib')
-   apply (smt assms comm_semiring_class.distrib mult_diff_mult semiring_normalization_rules(2) zero_le_mult_iff)
-proof -
-  fix x
-  assume "t \<le> x" "x < t'"
-  let ?u = "(x-t)/(t'-t)"
-  have "?u \<ge> 0"
-    using \<open>t \<le> x\<close> assms by auto
-  moreover have "?u < 1"
-    by (simp add: \<open>x < t'\<close> assms)
-  moreover have "x = (1-?u) * t + ?u*t'"
-  proof -
-    have f1: "\<forall>r ra. (ra::real) * - r = r * - ra"
-      by simp
-    have "t + (t' + - t) * ((x + - t) / (t' + - t)) = x"
-      using assms by force
-    then have "t' * ((x + - t) / (t' + - t)) + t * (1 + - ((x + - t) / (t' + - t))) = x"
-      using f1 by (metis (no_types) add.left_commute distrib_left mult.commute mult.right_neutral)
-    then show ?thesis
-      by (simp add: mult.commute)
-  qed
-  ultimately show "x \<in> (\<lambda>x. (1 - x) * t + x * t') ` {0..<1}"
-    using atLeastLessThan_iff by blast 
-qed
-
-lemma oriented_subsegment_scale:
-  assumes "x1 \<in> {a<--<b}"
-  assumes "x2 \<in> {x1<--<b}"
-  obtains e where "e > 0" "b-a = e *\<^sub>R (x2-x1)"
-proof -
-  from assms(1) obtain u where u : "u > 0" "u < 1" "x1 = (1 - u) *\<^sub>R a + u *\<^sub>R b"
-    unfolding in_segment by blast
-  from assms(2) obtain v where v: "v > 0" "v < 1" "x2 = (1 - v) *\<^sub>R x1 + v *\<^sub>R b"
-    unfolding in_segment by blast
-  have "x2-x1 = -v *\<^sub>R x1 + v *\<^sub>R b" using v
-    by (metis add.commute add_diff_cancel_right diff_minus_eq_add scaleR_collapse scaleR_left.minus)
-  also have "... = (-v) *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b)  + v *\<^sub>R b" using u by auto
-  also have "... = v *\<^sub>R ((1-u)*\<^sub>R b - (1-u)*\<^sub>R a )"
-    by (smt add_diff_cancel diff_diff_add diff_minus_eq_add minus_diff_eq scaleR_collapse scale_minus_left scale_right_diff_distrib)
-  finally have x2x1:"x2-x1 = (v *(1-u)) *\<^sub>R (b - a)"
-    by (metis scaleR_scaleR scale_right_diff_distrib)
-  have "v * (1-u) > 0"  using u(2) v(1) by simp
-  then have "(x2-x1)/\<^sub>R (v * (1-u)) = (b-a)" unfolding x2x1
-    by (smt field_class.field_inverse scaleR_one scaleR_scaleR) 
-  thus ?thesis
-    using \<open>0 < v * (1 - u)\<close> positive_imp_inverse_positive that by fastforce
-qed
-
-end
+section \<open>Additions to HOL-Analysis\<close>
+theory Analysis_Misc
+  imports 
+    Ordinary_Differential_Equations.ODE_Analysis
+begin
+
+subsection \<open>Unsorted Lemmas (TODO: sort!)\<close>
+
+lemma uminus_uminus_image: "uminus ` uminus ` S = S"
+  for S::"'r::ab_group_add set"
+  by (auto simp: image_image)
+
+lemma in_uminus_image_iff[simp]: "x \<in> uminus ` S \<longleftrightarrow> - x \<in> S"
+  for S::"'r::ab_group_add set"
+  by force
+
+lemma closed_subsegmentI:
+  "w + t *\<^sub>R (z - w) \<in> {x--y}"
+  if "w \<in> {x -- y}" "z \<in> {x -- y}" and t: "0 \<le> t" "t\<le> 1"
+proof -
+  from that obtain u v where
+    w_def: "w = (1 - u) *\<^sub>R x + u *\<^sub>R y" and u: "0 \<le> u" "u \<le> 1"
+    and z_def: "z = (1 - v) *\<^sub>R x + v *\<^sub>R y" and v: "0 \<le> v" "v \<le> 1"
+    by (auto simp: in_segment)
+  have "w + t *\<^sub>R (z - w) =
+    (1 - (u - t * (u - v))) *\<^sub>R x + (u - t * (u - v)) *\<^sub>R y"
+    by (simp add: algebra_simps w_def z_def)
+  also have "\<dots> \<in> {x -- y}"
+    unfolding closed_segment_image_interval
+    apply (rule imageI)
+    using t u v
+    apply auto
+     apply (metis (full_types) diff_0_right diff_left_mono linear mult_left_le_one_le mult_nonneg_nonpos order.trans)
+    by (smt mult_left_le_one_le mult_nonneg_nonneg vector_space_over_itself.scale_right_diff_distrib)
+  finally show ?thesis .
+qed
+
+lemma tendsto_minus_cancel_right: "((\<lambda>x. -g x) \<longlongrightarrow> l) F \<longleftrightarrow> (g \<longlongrightarrow> -l) F"
+  \<comment> \<open>cf @{thm tendsto_minus_cancel_left}\<close>
+  for g::"_ \<Rightarrow> 'b::topological_group_add"
+  by (simp add: tendsto_minus_cancel_left)
+
+lemma tendsto_nhds_continuousI: "(f \<longlongrightarrow> l) (nhds x)" if "(f \<longlongrightarrow> l) (at x)" "f x = l"
+  \<comment> \<open>TODO: the assumption is continuity of f at x\<close>
+proof (rule topological_tendstoI)
+  fix S::"'b set" assume "open S" "l \<in> S"
+  from topological_tendstoD[OF that(1) this]
+  have "\<forall>\<^sub>F x in at x. f x \<in> S" .
+  then show "\<forall>\<^sub>F x in nhds x. f x \<in> S"
+    unfolding eventually_at_filter
+    by eventually_elim (auto simp: that \<open>l \<in> S\<close>)
+qed
+
+lemma inj_composeD:
+  assumes "inj (\<lambda>x. g (t x))"
+  shows "inj t"
+  using assms
+  by (auto simp: inj_def)
+
+lemma compact_sequentialE:
+  fixes S T::"'a::first_countable_topology set"
+  assumes "compact S"
+  assumes "infinite T"
+  assumes "T \<subseteq> S"
+  obtains t::"nat \<Rightarrow> 'a" and l::'a
+  where "\<And>n. t n \<in> T" "\<And>n. t n \<noteq> l" "t \<longlonglongrightarrow> l" "l \<in> S"
+proof -
+  from Heine_Borel_imp_Bolzano_Weierstrass[OF assms]
+  obtain l where "l \<in> S" "l islimpt T" by metis
+  then obtain t where "t n \<in> T" "t n \<noteq> l" "t \<longlonglongrightarrow> l" "l \<in> S" for n unfolding islimpt_sequential
+    by auto
+  then show ?thesis ..
+qed
+
+lemma infinite_countable_subsetE:
+  fixes S::"'a set"
+  assumes "infinite S"
+  obtains g::"nat\<Rightarrow>'a" where "inj g" "range g \<subseteq> S"
+  using assms
+  by atomize_elim (simp add: infinite_countable_subset)
+
+lemma real_quad_ge: "2 * (an * bn) \<le> an * an + bn * bn" for an bn::real
+  by (sos "(((A<0 * R<1) + (R<1 * (R<1 * [an + ~1*bn]^2))))")
+
+lemma inner_quad_ge: "2 * (a \<bullet> b) \<le> a \<bullet> a + b \<bullet> b"
+  for a b::"'a::euclidean_space"\<comment> \<open>generalize?\<close>
+proof -
+  show ?thesis
+    by (subst (1 2 3) euclidean_inner)
+      (auto simp add: sum.distrib[symmetric] sum_distrib_left intro!: sum_mono real_quad_ge)
+qed
+
+lemma inner_quad_gt: "2 * (a \<bullet> b) < a \<bullet> a + b \<bullet> b"
+  if "a \<noteq> b"
+  for a b::"'a::euclidean_space"\<comment> \<open>generalize?\<close>
+proof -
+  from that obtain i where "i \<in> Basis" "a \<bullet> i \<noteq> b \<bullet> i"
+    by (auto simp: euclidean_eq_iff[where 'a='a])
+  then have "2 * (a \<bullet> i * (b \<bullet> i)) < a \<bullet> i * (a \<bullet> i) + b \<bullet> i * (b \<bullet> i)"
+    using sum_sqs_eq[of "a\<bullet>i" "b\<bullet>i"]
+    by (auto intro!: le_neq_trans real_quad_ge)
+  then show ?thesis
+    by (subst (1 2 3) euclidean_inner)
+      (auto simp add: \<open>i \<in> Basis\<close> sum.distrib[symmetric] sum_distrib_left
+        intro!: sum_strict_mono_ex1 real_quad_ge)
+qed
+
+lemma closed_segment_line_hyperplanes:
+  "{a -- b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) \<le> x \<bullet> (b - a) \<and> x \<bullet> (b - a) \<le> b \<bullet> (b - a)}"
+  if "a \<noteq> b"
+  for a b::"'a::euclidean_space"
+proof safe
+  fix x assume x: "x \<in> {a--b}"
+  then obtain u where u: "0 \<le> u" "u \<le> 1" and x_eq: "x = a + u *\<^sub>R (b - a)"
+    by (auto simp add: in_segment algebra_simps)
+  show "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))" using x_eq by auto
+  have "2 * (a \<bullet> b) \<le> a \<bullet> a + b \<bullet> b"
+    by (rule inner_quad_ge)
+  then have "u * (2 * (a \<bullet> b) - a \<bullet> a - b \<bullet> b) \<le> 0"
+    "0 \<le> (1 - u) * (a \<bullet> a + b \<bullet> b - a \<bullet> b * 2)"
+    by (simp_all add: mult_le_0_iff u)
+  then show " a \<bullet> (b - a) \<le> x \<bullet> (b - a)" "x \<bullet> (b - a) \<le> b \<bullet> (b - a)"
+    by (auto simp: x_eq algebra_simps power2_eq_square inner_commute)
+next
+  fix u assume
+    "a \<bullet> (b - a) \<le> (a + u *\<^sub>R (b - a)) \<bullet> (b - a)"
+    "(a + u *\<^sub>R (b - a)) \<bullet> (b - a) \<le> b \<bullet> (b - a)"
+  then have "0 \<le> u * ((b - a) \<bullet> (b - a))" "0 \<le> (1 - u) * ((b - a) \<bullet> (b - a))"
+    by (auto simp: algebra_simps)
+  then have "0 \<le> u" "u \<le> 1"
+    using inner_ge_zero[of "(b - a)"] that
+    by (auto simp add: zero_le_mult_iff)
+  then show "a + u *\<^sub>R (b - a) \<in> {a--b}"
+    by (auto simp: in_segment algebra_simps)
+qed
+
+lemma open_segment_line_hyperplanes:
+  "{a <--< b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
+  if "a \<noteq> b"
+  for a b::"'a::euclidean_space"
+proof safe
+  fix x assume x: "x \<in> {a<--<b}"
+  then obtain u where u: "0 < u" "u < 1" and x_eq: "x = a + u *\<^sub>R (b - a)"
+    by (auto simp add: in_segment algebra_simps)
+  show "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))" using x_eq by auto
+  have "2 * (a \<bullet> b) < a \<bullet> a + b \<bullet> b" using that
+    by (rule inner_quad_gt)
+  then have "u * (2 * (a \<bullet> b) - a \<bullet> a - b \<bullet> b) < 0"
+    "0 < (1 - u) * (a \<bullet> a + b \<bullet> b - a \<bullet> b * 2)"
+    by (simp_all add: mult_less_0_iff u)
+  then show " a \<bullet> (b - a) < x \<bullet> (b - a)" "x \<bullet> (b - a) < b \<bullet> (b - a)"
+    by (auto simp: x_eq algebra_simps power2_eq_square inner_commute)
+next
+  fix u assume
+    "a \<bullet> (b - a) < (a + u *\<^sub>R (b - a)) \<bullet> (b - a)"
+    "(a + u *\<^sub>R (b - a)) \<bullet> (b - a) < b \<bullet> (b - a)"
+  then have "0 < u * ((b - a) \<bullet> (b - a))" "0 < (1 - u) * ((b - a) \<bullet> (b - a))"
+    by (auto simp: algebra_simps)
+  then have "0 < u" "u < 1"
+    using inner_ge_zero[of "(b - a)"] that
+    by (auto simp add: zero_less_mult_iff)
+  then show "a + u *\<^sub>R (b - a) \<in> {a<--<b}"
+    by (auto simp: in_segment algebra_simps that)
+qed
+
+lemma at_within_interior: "NO_MATCH UNIV S \<Longrightarrow> x \<in> interior S \<Longrightarrow> at x within S = at x"
+  by (auto intro: at_within_interior)
+
+lemma tendsto_at_topI:
+  "(f \<longlongrightarrow> l) at_top" if "\<And>e. 0 < e \<Longrightarrow> \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e"
+for f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  using that
+  apply (intro tendstoI)
+  unfolding eventually_at_top_linorder
+  by auto
+
+lemma tendsto_at_topE:
+  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  assumes "(f \<longlongrightarrow> l) at_top"
+  assumes "e > 0"
+  obtains x0 where "\<And>x. x \<ge> x0 \<Longrightarrow> dist (f x) l < e"
+proof -
+  from assms(1)[THEN tendstoD, OF assms(2)]
+  have "\<forall>\<^sub>F x in at_top. dist (f x) l < e" .
+  then show ?thesis
+    unfolding eventually_at_top_linorder
+    by (auto intro: that)
+qed
+lemma tendsto_at_top_iff: "(f \<longlongrightarrow> l) at_top \<longleftrightarrow> (\<forall>e>0. \<exists>x0. \<forall>x\<ge>x0. dist (f x) l < e)"
+  for f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  by (auto intro!: tendsto_at_topI elim!: tendsto_at_topE)
+
+lemma tendsto_at_top_eq_left:
+  fixes f g::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  assumes "(f \<longlongrightarrow> l) at_top"
+  assumes "\<And>x. x \<ge> x0 \<Longrightarrow> f x = g x"
+  shows "(g \<longlongrightarrow> l) at_top"
+  unfolding tendsto_at_top_iff
+  by (metis (no_types, hide_lams) assms(1) assms(2) linear order_trans tendsto_at_topE)
+
+lemma lim_divide_n: "(\<lambda>x. e / real x) \<longlonglongrightarrow> 0"
+proof -
+  have "(\<lambda>x. e * inverse (real x)) \<longlonglongrightarrow> 0"
+    by (auto intro: tendsto_eq_intros lim_inverse_n)
+  then show ?thesis by (simp add: inverse_eq_divide)
+qed
+
+definition at_top_within :: "('a::order) set \<Rightarrow> 'a filter"
+  where "at_top_within s = (INF k \<in> s. principal ({k ..} \<inter> s)) "
+
+lemma at_top_within_at_top[simp]:
+  shows "at_top_within UNIV = at_top"
+  unfolding at_top_within_def at_top_def
+  by (auto)
+
+lemma at_top_within_empty[simp]:
+  shows "at_top_within {} = top"
+  unfolding at_top_within_def
+  by (auto)
+
+definition "nhds_set X = (INF S\<in>{S. open S \<and> X \<subseteq> S}. principal S)"
+
+lemma eventually_nhds_set:
+  "(\<forall>\<^sub>F x in nhds_set X. P x) \<longleftrightarrow> (\<exists>S. open S \<and> X \<subseteq> S \<and> (\<forall>x\<in>S. P x))"
+  unfolding nhds_set_def by (subst eventually_INF_base) (auto simp: eventually_principal)
+
+term "filterlim f (nhds_set (frontier X)) F" \<comment> \<open>f tends to the boundary of X?\<close>
+
+
+text \<open>somewhat inspired by @{thm islimpt_range_imp_convergent_subsequence} and its dependencies.
+The class constraints seem somewhat arbitrary, perhaps this can be generalized in some way.
+\<close>
+lemma limpt_closed_imp_exploding_subsequence:\<comment>\<open>TODO: improve name?!\<close>
+  fixes f::"'a::{heine_borel,real_normed_vector} \<Rightarrow> 'b::{first_countable_topology, t2_space}"
+  assumes cont[THEN continuous_on_compose2, continuous_intros]: "continuous_on T f"
+  assumes closed: "closed T"
+  assumes bound: "\<And>t. t \<in> T \<Longrightarrow> f t \<noteq> l"
+  assumes limpt: "l islimpt (f ` T)"
+  obtains s where
+    "(f \<circ> s) \<longlonglongrightarrow> l"
+    "\<And>i. s i \<in> T"
+    "\<And>C. compact C \<Longrightarrow> C \<subseteq> T \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C"
+proof -
+  from countable_basis_at_decseq[of l]
+  obtain A where A: "\<And>i. open (A i)" "\<And>i. l \<in> A i"
+    and evA: "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+    by blast
+
+  from closed_Union_compact_subsets[OF closed]
+  obtain C
+    where C: "(\<And>n. compact (C n))" "(\<And>n. C n \<subseteq> T)" "(\<And>n. C n \<subseteq> C (Suc n))" "\<Union> (range C) = T"
+      and evC: "(\<And>K. compact K \<Longrightarrow> K \<subseteq> T \<Longrightarrow> \<forall>\<^sub>F i in sequentially. K \<subseteq> C i)"
+    by (metis eventually_sequentially)
+
+  have AC: "l \<in> A i - f ` C i" "open (A i - f ` C i)" for i
+    using C bound
+    by (fastforce intro!: open_Diff A compact_imp_closed compact_continuous_image continuous_intros)+
+
+  from islimptE[OF limpt AC] have "\<exists>t\<in>T. f t \<in> A i - f ` C i \<and> f t \<noteq> l" for i by blast  
+  then obtain t where t: "\<And>i. t i \<in> T" "\<And>i. f (t i) \<in> A i - f ` C i" "\<And>i. f (t i) \<noteq> l"
+    by metis
+
+  have "(f o t) \<longlonglongrightarrow> l"
+    using t
+    by (auto intro!: topological_tendstoI dest!: evA elim!: eventually_mono)
+  moreover
+  have "\<And>i. t i \<in> T" by fact
+  moreover
+  have "\<forall>\<^sub>F i in sequentially. t i \<notin> K" if "compact K" "K \<subseteq> T" for K
+    using evC[OF that]
+    by eventually_elim (use t in auto)
+  ultimately show ?thesis ..
+qed  
+
+lemma Inf_islimpt: "bdd_below S \<Longrightarrow> Inf S \<notin> S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> Inf S islimpt S" for S::"real set"
+  by (auto simp: islimpt_in_closure intro!: closure_contains_Inf)
+
+context linorder
+begin
+
+text \<open>HOL-analysis doesn't seem to have these, maybe they were never needed.
+  Some variants are around @{thm Int_atLeastAtMost}, but with old-style naming conventions.
+  Change to the "modern" I.. convention there?\<close>
+
+lemma Int_Ico[simp]:
+  shows "{a..} \<inter> {b..} = {max a b ..}"
+  by (auto)
+
+lemma Int_Ici_Ico[simp]:
+  shows "{a..} \<inter> {b..<c} = {max a b ..<c}"
+  by auto
+
+lemma Int_Ico_Ici[simp]:
+  shows "{a..<c} \<inter> {b..} = {max a b ..<c}"
+  by auto
+
+lemma subset_Ico_iff[simp]:
+  "{a..<b} \<subseteq> {c..<b} \<longleftrightarrow> b \<le> a \<or> c \<le> a"
+  unfolding atLeastLessThan_def
+  by auto
+
+lemma Ico_subset_Ioo_iff[simp]:
+  "{a..<b} \<subseteq> {c<..<b} \<longleftrightarrow> b \<le> a \<or> c < a"
+  unfolding greaterThanLessThan_def atLeastLessThan_def
+  by auto
+
+lemma Icc_Un_Ici[simp]:
+  shows "{a..b} \<union> {b..} = {min a b..}"
+  unfolding atLeastAtMost_def atLeast_def atMost_def min_def
+  by auto
+
+end
+
+lemma at_top_within_at_top_unbounded_right:
+  fixes a::"'a::linorder"
+  shows "at_top_within {a..} = at_top"
+  unfolding at_top_within_def at_top_def
+  apply (auto intro!: INF_eq)
+  by (metis linorder_class.linear linorder_class.max.cobounded1 linorder_class.max.idem ord_class.atLeast_iff)
+
+lemma at_top_within_at_top_unbounded_rightI:
+  fixes a::"'a::linorder"
+  assumes "{a..} \<subseteq> s"
+  shows "at_top_within s = at_top"
+  unfolding at_top_within_def at_top_def
+  apply (auto intro!: INF_eq)
+   apply (meson Ici_subset_Ioi_iff Ioi_le_Ico assms dual_order.refl dual_order.trans leI)
+  by (metis assms atLeast_iff atLeast_subset_iff inf.cobounded1 linear subsetD)
+
+lemma at_top_within_at_top_bounded_right:
+  fixes a b::"'a::{dense_order,linorder_topology}"
+  assumes "a < b"
+  shows "at_top_within {a..<b} = at_left b"
+  unfolding at_top_within_def at_left_eq[OF assms(1)]
+  apply (auto intro!: INF_eq)
+   apply (smt atLeastLessThan_iff greaterThanLessThan_iff le_less lessThan_iff max.absorb1 subset_eq)
+  by (metis assms atLeastLessThan_iff dense linear max.absorb1 not_less order_trans)
+
+lemma at_top_within_at_top_bounded_right':
+  fixes a b::"'a::{dense_order,linorder_topology}"
+  assumes "a < b"
+  shows "at_top_within {..<b} = at_left b"
+  unfolding at_top_within_def at_left_eq[OF assms(1)]
+  apply (auto intro!: INF_eq)
+   apply (meson atLeast_iff greaterThanLessThan_iff le_less lessThan_iff subset_eq)
+  by (metis Ico_subset_Ioo_iff atLeastLessThan_def dense lessThan_iff)
+
+lemma eventually_at_top_within_linorder:
+  assumes sn:"s \<noteq> {}"
+  shows "eventually P (at_top_within s) \<longleftrightarrow> (\<exists>x0::'a::{linorder_topology} \<in> s. \<forall>x \<ge> x0. x\<in> s \<longrightarrow> P x)"
+  unfolding at_top_within_def
+  apply (subst eventually_INF_base)
+    apply (auto simp:eventually_principal sn)
+  by (metis atLeast_subset_iff inf.coboundedI2 inf_commute linear)
+
+lemma tendsto_at_top_withinI:
+  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  assumes "s \<noteq> {}"
+  assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>x0 \<in> s. \<forall>x \<in> {x0..} \<inter> s. dist (f x) l < e"
+  shows  "(f \<longlongrightarrow> l) (at_top_within s)"
+  apply(intro tendstoI)
+  unfolding at_top_within_def apply (subst eventually_INF_base)
+    apply (auto simp:eventually_principal assms)
+  by (metis atLeast_subset_iff inf.coboundedI2 inf_commute linear)
+
+lemma tendsto_at_top_withinE:
+  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  assumes "s \<noteq> {}"
+  assumes "(f \<longlongrightarrow> l) (at_top_within s)"
+  assumes "e > 0"
+  obtains x0 where "x0 \<in> s" "\<And>x. x \<in> {x0..} \<inter> s \<Longrightarrow> dist (f x) l < e"
+proof -
+  from assms(2)[THEN tendstoD, OF assms(3)]
+  have "\<forall>\<^sub>F x in at_top_within s. dist (f x) l < e" .
+  then show ?thesis unfolding eventually_at_top_within_linorder[OF \<open>s \<noteq> {}\<close>] 
+    by (auto intro: that)
+qed
+
+lemma tendsto_at_top_within_iff:
+  fixes f::"'a::linorder_topology \<Rightarrow> 'b::metric_space"
+  assumes "s \<noteq> {}"
+  shows "(f \<longlongrightarrow> l) (at_top_within s) \<longleftrightarrow> (\<forall>e>0. \<exists>x0 \<in> s. \<forall>x \<in> {x0..} \<inter> s. dist (f x) l < e)"
+  by (auto intro!: tendsto_at_top_withinI[OF \<open>s \<noteq> {}\<close>] elim!: tendsto_at_top_withinE[OF \<open>s \<noteq> {}\<close>])
+
+lemma filterlim_at_top_at_top_within_bounded_right:
+  fixes a b::"'a::{dense_order,linorder_topology}"
+  fixes f::"'a \<Rightarrow> real"
+  assumes "a < b"
+  shows "filterlim f at_top (at_top_within {..<b}) = (f \<longlongrightarrow> \<infinity>) (at_left b)"
+  unfolding filterlim_at_top_dense
+    at_top_within_at_top_bounded_right'[OF assms(1)]
+    eventually_at_left[OF assms(1)]
+    tendsto_PInfty
+  by auto
+
+text \<open>Extract a sequence (going to infinity) bounded away from l\<close>
+
+lemma not_tendsto_frequentlyE:
+  assumes "\<not>((f \<longlongrightarrow> l) F)"
+  obtains S where "open S" "l \<in> S" "\<exists>\<^sub>F x in F. f x \<notin> S"
+  using assms
+  by (auto simp: tendsto_def not_eventually)
+
+lemma not_tendsto_frequently_metricE:
+  assumes "\<not>((f \<longlongrightarrow> l) F)"
+  obtains e where "e > 0" "\<exists>\<^sub>F x in F. e \<le> dist (f x) l"
+  using assms
+  by (auto simp: tendsto_iff not_eventually not_less)
+
+lemma eventually_frequently_conj: "frequently P F \<Longrightarrow> eventually Q F \<Longrightarrow> frequently (\<lambda>x. P x \<and> Q x) F"
+  unfolding frequently_def
+  apply (erule contrapos_nn)
+  subgoal premises prems
+    using prems by eventually_elim auto
+  done
+
+lemma frequently_at_top:
+  "(\<exists>\<^sub>F t in at_top. P t) \<longleftrightarrow> (\<forall>t0. \<exists>t>t0. P t)"
+  for P::"'a::{linorder,no_top}\<Rightarrow>bool" 
+  by (auto simp: frequently_def eventually_at_top_dense)
+
+lemma frequently_at_topE:
+  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_"
+  assumes freq[rule_format]: "\<forall>n. \<exists>\<^sub>F a in at_top. P n a"
+  obtains s::"nat\<Rightarrow>'a"
+  where "\<And>i. P i (s i)" "strict_mono s"
+proof -
+  have "\<exists>f. \<forall>n. P n (f n) \<and> f n < f (Suc n)"
+  proof (rule dependent_nat_choice)
+    from frequently_ex[OF freq[of 0]] show "\<exists>x. P 0 x" .
+    fix x n assume "P n x"
+    from freq[unfolded frequently_at_top, rule_format, of x "Suc n"]
+    obtain y where "P (Suc n) y" "y > x" by auto
+    then show "\<exists>y. P (Suc n) y \<and> x < y"
+      by auto
+  qed
+  then obtain s where "\<And>i. P i (s i)" "strict_mono s"
+    unfolding strict_mono_Suc_iff by auto
+  then show ?thesis ..
+qed
+
+lemma frequently_at_topE':
+  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_"
+  assumes freq[rule_format]: "\<forall>n. \<exists>\<^sub>F a in at_top. P n a"
+    and g: "filterlim g at_top sequentially"
+  obtains s::"nat\<Rightarrow>'a"
+  where "\<And>i. P i (s i)" "strict_mono s" "\<And>n. g n \<le> s n"
+proof -
+  have "\<forall>n. \<exists>\<^sub>F a in at_top. P n a \<and> g n \<le> a"
+    using freq
+    by (auto intro!: eventually_frequently_conj)
+  from frequently_at_topE[OF this] obtain s where "\<And>i. P i (s i)" "strict_mono s" "\<And>n. g n \<le> s n"
+    by metis
+  then show ?thesis ..
+qed
+
+lemma frequently_at_top_at_topE:
+  fixes P::"nat \<Rightarrow> 'a::{linorder,no_top}\<Rightarrow>_" and g::"nat\<Rightarrow>'a"
+  assumes "\<forall>n. \<exists>\<^sub>F a in at_top. P n a" "filterlim g at_top sequentially"
+  obtains s::"nat\<Rightarrow>'a"
+  where "\<And>i. P i (s i)" "filterlim s at_top sequentially"
+proof -
+  from frequently_at_topE'[OF assms]
+  obtain s where s: "(\<And>i. P i (s i))" "strict_mono s" "(\<And>n. g n \<le> s n)" by blast
+  have s_at_top: "filterlim s at_top sequentially"
+    by (rule filterlim_at_top_mono) (use assms s in auto)
+  with s(1) show ?thesis ..
+qed
+
+(* Extract a strict monotone and sequence converging to something other than l *)
+lemma not_tendsto_convergent_seq:
+  fixes f::"real \<Rightarrow> 'a::metric_space"
+  assumes X: "compact (X::'a set)"
+  assumes im: "\<And>x. x \<ge> 0 \<Longrightarrow> f x \<in> X"
+  assumes nl: "\<not> ((f \<longlongrightarrow> (l::'a)) at_top)"
+  obtains s k where
+    "k \<in> X" "k \<noteq> l" "(f \<circ> s) \<longlonglongrightarrow> k" "strict_mono s" "\<forall>n. s n \<ge> n"
+proof -
+  from not_tendsto_frequentlyE[OF nl]
+  obtain S where "open S" "l \<in> S" "\<exists>\<^sub>F x in at_top. f x \<notin> S" .
+  have "\<forall>n. \<exists>\<^sub>F x in at_top. f x \<notin> S \<and> real n \<le> x"
+    apply (rule allI)
+    apply (rule eventually_frequently_conj)
+     apply fact
+    by (rule eventually_ge_at_top)
+  from frequently_at_topE[OF this]
+  obtain s where "\<And>i. f (s i) \<notin> S" and s: "strict_mono s" and s_ge: "(\<And>i. real i \<le> s i)" by metis
+  then have "0 \<le> s i" for i using dual_order.trans of_nat_0_le_iff by blast
+  then have "\<forall>n. (f \<circ> s) n \<in> X" using im by auto
+  from X[unfolded compact_def, THEN spec, THEN mp, OF this]
+  obtain k r where k: "k \<in> X" and r: "strict_mono r" and kLim: "(f \<circ> s \<circ> r) \<longlonglongrightarrow> k" by metis
+  have "k \<in> X - S"
+    by (rule Lim_in_closed_set[of "X - S", OF _ _ _ kLim])
+      (auto simp: im \<open>0 \<le> s _\<close>  \<open>\<And>i. f (s i) \<notin> S\<close> intro!: \<open>open S\<close> X intro: compact_imp_closed)
+
+  note k
+  moreover have "k \<noteq> l" using \<open>k \<in> X - S\<close> \<open>l \<in> S\<close> by auto
+  moreover have "(f \<circ> (s \<circ> r)) \<longlonglongrightarrow> k" using kLim by (simp add: o_assoc)
+  moreover have "strict_mono (s \<circ> r)" using s r by (rule strict_mono_o)
+  moreover have "\<forall>n. (s \<circ> r) n \<ge> n" using s_ge r
+    by (metis comp_apply dual_order.trans of_nat_le_iff seq_suble)
+  ultimately show ?thesis ..
+qed
+
+lemma harmonic_bound:
+  shows "1 / 2 ^(Suc n) < 1 / real (Suc n)"
+proof (induction n)
+  case 0
+  then show ?case by auto
+next
+  case (Suc n)
+  then show ?case
+    by (smt frac_less2 of_nat_0_less_iff of_nat_less_two_power zero_less_Suc)
+qed
+
+lemma INF_bounded_imp_convergent_seq:
+  fixes f::"real \<Rightarrow> real"
+  assumes cont: "continuous_on {a..} f"
+  assumes bound: "\<And>t. t \<ge> a \<Longrightarrow> f t > l"
+  assumes inf: "(INF t\<in>{a..}. f t) = l"
+  obtains s where
+    "(f \<circ> s) \<longlonglongrightarrow> l"
+    "\<And>i. s i \<in> {a..}"
+    "filterlim s at_top sequentially"
+proof -
+  have bound': "t \<in> {a..} \<Longrightarrow> f t \<noteq> l" for t using bound[of t] by auto
+  have limpt: "l islimpt f ` {a..}"
+  proof -
+    have "Inf (f ` {a..}) islimpt f ` {a..}"
+      by (rule Inf_islimpt) (auto simp: inf intro!: bdd_belowI2[where m=l] dest: bound)
+    then show ?thesis by (simp add: inf)
+  qed
+  from limpt_closed_imp_exploding_subsequence[OF cont closed_atLeast bound' limpt]
+  obtain s where s: "(f \<circ> s) \<longlonglongrightarrow> l"
+    "\<And>i. s i \<in> {a..}"
+    "compact C \<Longrightarrow> C \<subseteq> {a..} \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C" for C
+    by metis
+  have "\<forall>\<^sub>F i in sequentially. s i \<ge> n" for n
+    using s(3)[of "{a..n}"] s(2)
+    by (auto elim!: eventually_mono)
+  then have "filterlim s at_top sequentially"
+    unfolding filterlim_at_top
+    by auto
+  from s(1) s(2) this
+  show ?thesis ..
+qed
+
+(* Generalizes to other combinations of strict_mono and filterlim *)
+lemma filterlim_at_top_strict_mono:
+  fixes s :: "_ \<Rightarrow> 'a::linorder"
+  fixes r :: "nat \<Rightarrow> _"
+  assumes "strict_mono s"
+  assumes "strict_mono r"
+  assumes "filterlim s at_top F"
+  shows "filterlim (s \<circ> r) at_top F"
+  apply (rule filterlim_at_top_mono[OF assms(3)])
+  by (simp add: assms(1) assms(2) seq_suble strict_mono_leD)
+
+lemma LIMSEQ_lb:
+  assumes fl: "s \<longlonglongrightarrow> (l::real)"
+  assumes u: "l < u"
+  shows "\<exists>n0. \<forall>n\<ge>n0. s n < u"
+proof -
+  from fl have "\<exists>no>0. \<forall>n\<ge>no. dist (s n) l < u-l" unfolding LIMSEQ_iff_nz using u
+    by simp
+  thus ?thesis using dist_real_def by fastforce
+qed
+
+(* Used to sharpen a tendsto with additional information*)
+lemma filterlim_at_top_choose_lower:
+  assumes "filterlim s at_top sequentially"
+  assumes "(f \<circ> s) \<longlonglongrightarrow> l"
+  obtains t where
+    "filterlim t at_top sequentially"
+    "(f \<circ> t) \<longlonglongrightarrow> l"
+    "\<forall>n. t n \<ge> (b::real)"
+proof -
+  obtain k where k: "\<forall>n \<ge> k. s n \<ge> b" using assms(1)
+    unfolding filterlim_at_top eventually_sequentially by blast
+  define t where "t = (\<lambda>n. s (n+k))"
+  then have "\<forall>n. t n \<ge> b" using k by simp
+  have "filterlim t at_top sequentially" using assms(1)
+    unfolding filterlim_at_top eventually_sequentially t_def
+    by (metis (full_types) add.commute trans_le_add2)
+  from LIMSEQ_ignore_initial_segment[OF assms(2), of "k"]
+  have "(\<lambda>n. (f \<circ> s) (n + k)) \<longlonglongrightarrow> l" .
+  then have "(f \<circ> t) \<longlonglongrightarrow> l" unfolding t_def o_def by simp
+  show ?thesis
+    using \<open>(f \<circ> t) \<longlonglongrightarrow> l\<close> \<open>\<forall>n. b \<le> t n\<close> \<open>filterlim t at_top sequentially\<close> that by blast
+qed
+
+lemma frequently_at_top_realE:
+  fixes P::"nat \<Rightarrow> real \<Rightarrow> bool"
+  assumes "\<forall>n. \<exists>\<^sub>F t in at_top. P n t"
+  obtains s::"nat\<Rightarrow>real"
+  where "\<And>i. P i (s i)" "filterlim s at_top at_top"
+  by (metis assms frequently_at_top_at_topE[OF _ filterlim_real_sequentially])
+
+lemma approachable_sequenceE:
+  fixes f::"real \<Rightarrow> 'a::metric_space"
+  assumes "\<And>t e. 0 \<le> t \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>tt\<ge>t. dist (f tt) p < e"
+  obtains s where "filterlim s at_top sequentially" "(f \<circ> s) \<longlonglongrightarrow> p"
+proof -
+  have "\<forall>n. \<exists>\<^sub>F i in at_top. dist (f i) p < 1/real (Suc n)"
+    unfolding frequently_at_top
+    apply (auto )
+    subgoal for n m
+      using assms[of "max 0 (m+1)" "1/(Suc n)"]
+      by force
+    done
+  from frequently_at_top_realE[OF this]
+  obtain s where s: "\<And>i. dist (f (s i)) p < 1 / real (Suc i)" "filterlim s at_top sequentially"
+    by metis
+  note this(2)
+  moreover
+  have "(f o s) \<longlonglongrightarrow> p"
+  proof (rule tendstoI)
+    fix e::real assume "e > 0"
+    have "\<forall>\<^sub>F i in sequentially. 1 / real (Suc i) < e"
+      apply (rule order_tendstoD[OF _ \<open>0 < e\<close>])
+      apply (rule real_tendsto_divide_at_top)
+       apply (rule tendsto_intros)
+      by (rule filterlim_compose[OF filterlim_real_sequentially filterlim_Suc])
+    then show "\<forall>\<^sub>F x in sequentially. dist ((f \<circ> s) x) p < e"
+      by eventually_elim (use dual_order.strict_trans s \<open>e > 0\<close> in auto)
+  qed
+  ultimately show ?thesis ..
+qed
+
+lemma mono_inc_bdd_above_has_limit_at_topI:
+  fixes f::"real \<Rightarrow> real"
+  assumes "mono f"
+  assumes "\<And>x. f x \<le> u"
+  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
+proof -
+  define l where "l = Sup (range (\<lambda>n. f (real n)))"
+  have t:"(\<lambda>n. f (real n)) \<longlonglongrightarrow> l" unfolding l_def
+    apply (rule LIMSEQ_incseq_SUP)
+     apply (meson assms(2) bdd_aboveI2)
+    by (meson assms(1) mono_def of_nat_mono)
+  from tendsto_at_topI_sequentially_real[OF assms(1) t]
+  have "(f \<longlongrightarrow> l) at_top" .
+  thus ?thesis by blast  
+qed
+
+lemma gen_mono_inc_bdd_above_has_limit_at_topI:
+  fixes f::"real \<Rightarrow> real"
+  assumes "\<And>x y. x \<ge> b \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
+  assumes "\<And>x. x \<ge> b \<Longrightarrow> f x \<le> u"
+  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
+proof -
+  define ff where "ff = (\<lambda>x. if x \<ge> b then f x else f b)"
+  have m1:"mono ff" unfolding ff_def mono_def using assms(1) by simp
+  have m2:"\<And>x. ff x \<le> u" unfolding ff_def using assms(2) by simp
+  from mono_inc_bdd_above_has_limit_at_topI[OF m1 m2]
+  obtain l where "(ff \<longlongrightarrow> l) at_top" by blast
+  thus ?thesis
+    by (meson \<open>(ff \<longlongrightarrow> l) at_top\<close> ff_def tendsto_at_top_eq_left)
+qed
+
+lemma gen_mono_dec_bdd_below_has_limit_at_topI:
+  fixes f::"real \<Rightarrow> real"
+  assumes "\<And>x y. x \<ge> b \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<ge> f y"
+  assumes "\<And>x. x \<ge> b \<Longrightarrow> f x \<ge> u"
+  shows "\<exists>l. (f \<longlongrightarrow> l) at_top"
+proof -
+  define ff where "ff = (\<lambda>x. if x \<ge> b then f x else f b)"
+  have m1:"mono (-ff)" unfolding ff_def mono_def using assms(1) by simp
+  have m2:"\<And>x. (-ff) x \<le> -u" unfolding ff_def using assms(2) by simp
+  from mono_inc_bdd_above_has_limit_at_topI[OF m1 m2]
+  obtain l where "(-ff \<longlongrightarrow> l) at_top" by blast
+  then have "(ff \<longlongrightarrow> -l) at_top"
+    using tendsto_at_top_eq_left tendsto_minus_cancel_left by fastforce  
+  thus ?thesis
+    by (meson \<open>(ff \<longlongrightarrow> -l) at_top\<close> ff_def tendsto_at_top_eq_left)
+qed
+
+lemma infdist_closed:
+  shows "closed ({z. infdist z S \<ge> e})"
+  by (auto intro!:closed_Collect_le simp add:continuous_on_infdist)
+
+(* TODO: this is a copy of LIMSEQ_norm_0 where the sequence
+  is bounded above in norm by a geometric series *)
+lemma LIMSEQ_norm_0_pow:
+  assumes "k > 0" "b > 1"
+  assumes  "\<And>n::nat. norm (s n) \<le> k / b^n"
+  shows "s \<longlonglongrightarrow> 0"
+proof (rule metric_LIMSEQ_I)
+  fix e
+  assume "e > (0::real)"
+  then have "k / e > 0" using assms(1) by auto
+  obtain N where N: "b^(N::nat) > k / e" using assms(2)
+    using real_arch_pow by blast
+  then have "norm (s n) < e" if "n \<ge> N" for n
+  proof -
+    have "k / b^n \<le> k / b^N"
+      by (smt assms(1) assms(2) frac_le leD power_less_imp_less_exp that zero_less_power)
+    also have " ... < e" using N
+      by (metis \<open>0 < e\<close> assms(2) less_trans mult.commute pos_divide_less_eq zero_less_one zero_less_power)
+    finally show ?thesis
+      by (meson assms less_eq_real_def not_le order_trans)
+  qed
+  then show "\<exists>no. \<forall>n\<ge>no. dist (s n) 0 < e"
+    by auto
+qed
+
+lemma filterlim_apply_filtermap:
+  assumes g: "filterlim g G F"
+  shows "filterlim (\<lambda>x. m (g x)) (filtermap m G) F"
+  by (metis filterlim_def filterlim_filtermap filtermap_mono g)
+
+lemma eventually_at_right_field_le:
+  "eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y \<le> b \<longrightarrow> P y)"
+  for x :: "'a::{linordered_field, linorder_topology}"
+  by (smt dense eventually_at_right_field le_less_trans less_le_not_le order.strict_trans1)
+
+subsection \<open>indexing euclidean space with natural numbers\<close>
+
+definition  nth_eucl :: "'a::executable_euclidean_space \<Rightarrow> nat \<Rightarrow> real" where
+  "nth_eucl x i = x \<bullet> (Basis_list ! i)"
+  \<comment> \<open>TODO: why is that and some sort of \<open>lambda_eucl\<close> nowhere available?\<close>
+definition lambda_eucl :: "(nat \<Rightarrow> real) \<Rightarrow> 'a::executable_euclidean_space" where
+  "lambda_eucl (f::nat\<Rightarrow>real) = (\<Sum>i<DIM('a). f i *\<^sub>R Basis_list ! i)"
+
+lemma eucl_eq_iff: "x = y \<longleftrightarrow> (\<forall>i<DIM('a). nth_eucl x i = nth_eucl y i)"
+  for x y::"'a::executable_euclidean_space"
+  apply (auto simp: nth_eucl_def euclidean_eq_iff[where 'a='a])
+  by (metis eucl_of_list_list_of_eucl list_of_eucl_eq_iff)
+
+bundle eucl_notation begin
+notation nth_eucl (infixl "$\<^sub>e" 90)
+end
+bundle no_eucl_notation begin
+no_notation nth_eucl (infixl "$\<^sub>e" 90)
+end
+
+unbundle eucl_notation
+
+lemma eucl_of_list_eucl_nth:
+  "(eucl_of_list xs::'a) $\<^sub>e i = xs ! i"
+  if "length xs = DIM('a::executable_euclidean_space)"
+    "i < DIM('a)"
+  using that
+  apply (auto simp: nth_eucl_def)
+  by (metis list_of_eucl_eucl_of_list list_of_eucl_nth)
+
+lemma eucl_of_list_inner:
+  "(eucl_of_list xs::'a) \<bullet> eucl_of_list ys = (\<Sum>(x,y)\<leftarrow>zip xs ys. x * y)"
+  if "length xs = DIM('a::executable_euclidean_space)"
+    "length ys = DIM('a::executable_euclidean_space)"
+  using that
+  by (auto simp: nth_eucl_def eucl_of_list_inner_eq inner_lv_rel_def)
+
+lemma self_eq_eucl_of_list: "x = eucl_of_list (map (\<lambda>i. x $\<^sub>e i) [0..<DIM('a)])"
+  for x::"'a::executable_euclidean_space"
+  by (auto simp: eucl_eq_iff[where 'a='a] eucl_of_list_eucl_nth)
+
+lemma inner_nth_eucl: "x \<bullet> y = (\<Sum>i<DIM('a). x $\<^sub>e i * y $\<^sub>e i)"
+  for x y::"'a::executable_euclidean_space"
+  apply (subst self_eq_eucl_of_list[where x=x])
+  apply (subst self_eq_eucl_of_list[where x=y])
+  apply (subst eucl_of_list_inner)
+  by (auto simp: map2_map_map atLeast_upt interv_sum_list_conv_sum_set_nat)
+
+lemma norm_nth_eucl: "norm x = L2_set (\<lambda>i. x $\<^sub>e i) {..<DIM('a)}"
+  for x::"'a::executable_euclidean_space"
+  unfolding norm_eq_sqrt_inner inner_nth_eucl L2_set_def
+  by (auto simp: power2_eq_square)
+
+
+lemma plus_nth_eucl: "(x + y) $\<^sub>e i = x $\<^sub>e i + y $\<^sub>e i"
+  and minus_nth_eucl: "(x - y) $\<^sub>e i = x $\<^sub>e i - y $\<^sub>e i"
+  and uminus_nth_eucl: "(-x) $\<^sub>e i = - x $\<^sub>e i"
+  and scaleR_nth_eucl: "(c *\<^sub>R x) $\<^sub>e i = c *\<^sub>R (x $\<^sub>e i)"
+  by (auto simp: nth_eucl_def algebra_simps)
+
+lemma inf_nth_eucl: "inf x y $\<^sub>e i = min (x $\<^sub>e i) (y $\<^sub>e i)"
+  if "i < DIM('a)"
+  for x::"'a::executable_euclidean_space"
+  by (auto simp: nth_eucl_def algebra_simps inner_Basis_inf_left that inf_min)
+lemma sup_nth_eucl: "sup x y $\<^sub>e i = max (x $\<^sub>e i) (y $\<^sub>e i)"
+  if "i < DIM('a)"
+  for x::"'a::executable_euclidean_space"
+  by (auto simp: nth_eucl_def algebra_simps inner_Basis_sup_left that sup_max)
+
+lemma le_iff_le_nth_eucl: "x \<le> y \<longleftrightarrow> (\<forall>i<DIM('a). (x $\<^sub>e i) \<le> (y $\<^sub>e i))"
+  for x::"'a::executable_euclidean_space"
+  apply (auto simp: nth_eucl_def algebra_simps eucl_le[where 'a='a])
+  by (meson eucl_le eucl_le_Basis_list_iff)
+
+lemma eucl_less_iff_less_nth_eucl: "eucl_less x y \<longleftrightarrow> (\<forall>i<DIM('a). (x $\<^sub>e i) < (y $\<^sub>e i))"
+  for x::"'a::executable_euclidean_space"
+  apply (auto simp: nth_eucl_def algebra_simps eucl_less_def[where 'a='a])
+  by (metis Basis_zero eucl_eq_iff inner_not_same_Basis inner_zero_left length_Basis_list
+      nth_Basis_list_in_Basis nth_eucl_def)
+
+lemma continuous_on_nth_eucl[continuous_intros]:
+  "continuous_on X (\<lambda>x. f x $\<^sub>e i)"
+  if "continuous_on X f"
+  by (auto simp: nth_eucl_def intro!: continuous_intros that)
+
+
+subsection \<open>derivatives\<close>
+
+lemma eventually_at_ne[intro, simp]: "\<forall>\<^sub>F x in at x0. x \<noteq> x0"
+  by (auto simp: eventually_at_filter)
+
+lemma has_vector_derivative_withinD:
+  fixes f::"real \<Rightarrow> 'b::euclidean_space"
+  assumes "(f has_vector_derivative f') (at x0 within S)"
+  shows "((\<lambda>x. (f x - f x0) /\<^sub>R (x - x0)) \<longlongrightarrow> f') (at x0 within S)"
+  apply (rule LIM_zero_cancel)
+  apply (rule tendsto_norm_zero_cancel)
+  apply (rule Lim_transform_eventually)
+proof -
+  show "\<forall>\<^sub>F x in at x0 within S. norm ((f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0)) =
+    norm ((f x - f x0) /\<^sub>R (x - x0) - f')"
+    (is "\<forall>\<^sub>F x in _. ?th x")
+    unfolding eventually_at_filter
+  proof (safe intro!: eventuallyI)
+    fix x assume x: "x \<noteq> x0"
+    then have "norm ((f x - f x0) /\<^sub>R (x - x0) - f') = norm (sgn (x - x0) *\<^sub>R ((f x - f x0) /\<^sub>R (x - x0) - f'))"
+      by simp
+    also have "sgn (x - x0) *\<^sub>R ((f x - f x0) /\<^sub>R (x - x0) - f') = ((f x - f x0) /\<^sub>R norm (x - x0) - (x - x0) *\<^sub>R f' /\<^sub>R norm (x - x0))"
+      by (auto simp add: algebra_simps sgn_div_norm divide_simps)
+        (metis add.commute add_divide_distrib diff_add_cancel scaleR_add_left)
+    also have "\<dots> = (f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0)" by (simp add: algebra_simps)
+    finally show "?th x" ..
+  qed
+  show "((\<lambda>x. norm ((f x - f x0 - (x - x0) *\<^sub>R f') /\<^sub>R norm (x - x0))) \<longlongrightarrow> 0) (at x0 within S)"
+    by (rule tendsto_norm_zero)
+      (use assms in \<open>auto simp: has_vector_derivative_def has_derivative_at_within\<close>)
+qed
+
+text \<open>A \<open>path_connected\<close> set \<open>S\<close> entering both \<open>T\<close> and \<open>-T\<close>
+      must cross the frontier of \<open>T\<close> \<close>
+lemma path_connected_frontier:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "path_connected S"
+  assumes "S \<inter> T \<noteq> {}"
+  assumes "S \<inter> -T \<noteq> {}"
+  obtains s where "s \<in> S" "s \<in> frontier T"
+proof -
+  obtain st where st:"st \<in> S \<inter> T" using assms(2) by blast
+  obtain sn where sn:"sn \<in> S \<inter> -T" using assms(3) by blast
+  obtain g where g: "path g" "path_image g \<subseteq> S"
+    "pathstart g = st" "pathfinish g = sn"
+    using assms(1) st sn unfolding path_connected_def by blast
+  have a1:"pathstart g \<in> closure T" using st g(3) closure_Un_frontier by fastforce
+  have a2:"pathfinish g \<notin> T" using sn g(4) by auto
+  from exists_path_subpath_to_frontier[OF g(1) a1 a2]
+  obtain h where "path_image h \<subseteq> path_image g" "pathfinish h \<in> frontier T" by metis
+  thus ?thesis using g(2)
+    by (meson in_mono pathfinish_in_path_image that) 
+qed
+
+lemma path_connected_not_frontier_subset:
+  fixes S :: "'a::real_normed_vector set"
+  assumes "path_connected S"
+  assumes "S \<inter> T \<noteq> {}"
+  assumes "S \<inter> frontier T = {}"
+  shows "S \<subseteq> T"
+  using path_connected_frontier assms by auto  
+
+lemma compact_attains_bounds:
+  fixes f::"'a::topological_space \<Rightarrow> 'b::linorder_topology"
+  assumes compact: "compact S"
+  assumes ne: "S \<noteq> {}"
+  assumes cont: "continuous_on S f"
+  obtains l u where "l \<in> S" "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<in> {f l .. f u}"
+proof -
+  from compact_continuous_image[OF cont compact]
+  have compact_image: "compact (f ` S)" .
+  have ne_image: "f ` S \<noteq> {}" using ne by simp
+  from compact_attains_inf[OF compact_image ne_image]
+  obtain l where "l \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f l \<le> f x" by auto
+  moreover
+  from compact_attains_sup[OF compact_image ne_image]
+  obtain u where "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<le> f u" by auto
+  ultimately
+  have "l \<in> S" "u \<in> S" "\<And>x. x \<in> S \<Longrightarrow> f x \<in> {f l .. f u}" by auto
+  then show ?thesis ..
+qed
+
+lemma uniform_limit_const[uniform_limit_intros]:
+  "uniform_limit S (\<lambda>x y. f x) (\<lambda>_. l) F" if "(f \<longlongrightarrow> l) F"
+  apply (auto simp: uniform_limit_iff)
+  subgoal for e
+    using tendstoD[OF that(1), of e]
+    by (auto simp: eventually_mono)
+  done
+
+subsection \<open>Segments\<close>
+
+text \<open>\<open>closed_segment\<close> throws away the order that our intuition keeps\<close>
+
+definition line::"'a::real_vector \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"
+  ("{_ -- _}\<^bsub>_\<^esub>")
+  where "{a -- b}\<^bsub>u\<^esub> = a + u *\<^sub>R (b - a)"
+
+abbreviation "line_image a b U \<equiv>(\<lambda>u. {a -- b}\<^bsub>u\<^esub>) ` U"
+notation line_image ("{_ -- _}\<^bsub>`_\<^esub>")
+
+lemma in_closed_segment_iff_line: "x \<in> {a -- b} \<longleftrightarrow> (\<exists>c\<in>{0..1}. x = line a b c)"
+  by (auto simp: in_segment line_def algebra_simps)
+
+lemma in_open_segment_iff_line: "x \<in> {a <--< b} \<longleftrightarrow> (\<exists>c\<in>{0<..<1}. a \<noteq> b \<and> x = line a b c)"
+  by (auto simp: in_segment line_def algebra_simps)
+
+lemma line_convex_combination1: "(1 - u) *\<^sub>R line a b i + u *\<^sub>R b = line a b (i + u - i * u)"
+  by (auto simp: line_def algebra_simps)
+
+lemma line_convex_combination2: "(1 - u) *\<^sub>R a + u *\<^sub>R line a b i = line a b (i*u)"
+  by (auto simp: line_def algebra_simps)
+
+lemma line_convex_combination12: "(1 - u) *\<^sub>R line a b i + u *\<^sub>R line a b j = line a b (i + u * (j - i))"
+  by (auto simp: line_def algebra_simps)
+
+lemma mult_less_one_less_self: "0 < x \<Longrightarrow> i < 1 \<Longrightarrow> i * x < x" for i x::real
+  by auto
+
+lemma plus_times_le_one_lemma: "i + u - i * u \<le> 1" if "i \<le> 1" "u \<le> 1" for i u::real
+  by (simp add: diff_le_eq sum_le_prod1 that)
+
+lemma plus_times_less_one_lemma: "i + u - i * u < 1" if "i < 1" "u < 1" for i u::real
+proof -
+  have "u * (1 - i) < 1 - i"
+    using that by force
+  then show ?thesis by (simp add: algebra_simps)
+qed
+
+lemma line_eq_endpoint_iff[simp]:
+  "line a b i = b \<longleftrightarrow> (a = b \<or> i = 1)"
+  "a = line a b i \<longleftrightarrow> (a = b \<or> i = 0)"
+  by (auto simp: line_def algebra_simps)
+
+lemma line_eq_iff[simp]: "line a b x = line a b y \<longleftrightarrow> (x = y \<or> a = b)"
+  by (auto simp: line_def)
+
+lemma line_open_segment_iff:
+  "{line a b i<--<b} = line a b ` {i<..<1}"
+  if "i < 1" "a \<noteq> b"
+  using that
+  apply (auto simp: in_segment line_convex_combination1 plus_times_less_one_lemma)
+  subgoal for j
+    apply (rule exI[where x="(j - i)/(1 - i)"])
+    apply (auto simp: divide_simps algebra_simps)
+    by (metis add_diff_cancel less_numeral_extra(4) mult_2_right plus_times_less_one_lemma that(1))
+  done
+
+lemma open_segment_line_iff:
+  "{a<--<line a b i} = line a b ` {0<..<i}"
+  if "0 < i" "a \<noteq> b"
+  using that
+  apply (auto simp: in_segment line_convex_combination2 plus_times_less_one_lemma)
+  subgoal for j
+    apply (rule exI[where x="j/i"])
+    by (auto simp: )
+  done
+
+lemma line_closed_segment_iff:
+  "{line a b i--b} = line a b ` {i..1}"
+  if "i \<le> 1" "a \<noteq> b"
+  using that
+  apply (auto simp: in_segment line_convex_combination1 mult_le_cancel_right2 plus_times_le_one_lemma)
+  subgoal for j
+    apply (rule exI[where x="(j - i)/(1 - i)"])
+    apply (auto simp: divide_simps algebra_simps)
+    by (metis add_diff_cancel less_numeral_extra(4) mult_2_right plus_times_less_one_lemma that(1))
+  done
+
+lemma closed_segment_line_iff:
+  "{a--line a b i} = line a b ` {0..i}"
+  if "0 < i" "a \<noteq> b"
+  using that
+  apply (auto simp: in_segment line_convex_combination2 plus_times_less_one_lemma)
+  subgoal for j
+    apply (rule exI[where x="j/i"])
+    by (auto simp: )
+  done
+
+lemma closed_segment_line_line_iff: "{line a b i1--line a b i2} = line a b ` {i1..i2}" if "i1 \<le> i2"
+  using that
+  apply (auto simp: in_segment line_convex_combination12 intro!: imageI)
+   apply (smt mult_left_le_one_le)
+  subgoal for u
+    by (rule exI[where x="(u - i1)/(i2-i1)"]) auto
+  done
+
+lemma line_line1: "line (line a b c) b x = line a b (c + x - c * x)"
+  by (simp add: line_def algebra_simps)
+
+lemma line_line2: "line a (line a b c) x = line a b (c*x)"
+  by (simp add: line_def algebra_simps)
+
+lemma line_in_subsegment:
+  "i1 < 1 \<Longrightarrow> i2 < 1 \<Longrightarrow> a \<noteq> b \<Longrightarrow> line a b i1 \<in> {line a b i2<--<b} \<longleftrightarrow> i2 < i1"
+  by (auto simp: line_open_segment_iff intro!: imageI)
+
+lemma line_in_subsegment2:
+  "0 < i2 \<Longrightarrow> 0 < i1 \<Longrightarrow> a \<noteq> b \<Longrightarrow> line a b i1 \<in> {a<--<line a b i2} \<longleftrightarrow> i1 < i2"
+  by (auto simp: open_segment_line_iff intro!: imageI)
+
+lemma line_in_open_segment_iff[simp]:
+  "line a b i \<in> {a<--<b} \<longleftrightarrow> (a \<noteq> b \<and> 0 < i \<and> i < 1)"
+  by (auto simp: in_open_segment_iff_line)
+
+subsection \<open>Open Segments\<close>
+
+lemma open_segment_subsegment:
+  assumes "x1 \<in> {x0<--<x3}"
+    "x2 \<in> {x1<--<x3}"
+  shows "x1 \<in> {x0<--<x2}"
+  using assms
+proof -\<comment> \<open>TODO: use \<open>line\<close>\<close>
+  from assms obtain u v::real where
+    ne: "x0 \<noteq> x3" "(1 - u) *\<^sub>R x0 + u *\<^sub>R x3 \<noteq> x3"
+    and x1_def: "x1 = (1 - u) *\<^sub>R x0 + u *\<^sub>R x3"
+    and x2_def: "x2 = (1 - v) *\<^sub>R ((1 - u) *\<^sub>R x0 + u *\<^sub>R x3) + v *\<^sub>R x3"
+    and uv: \<open>0 < u\<close> \<open>0 < v\<close> \<open>u < 1\<close> \<open>v < 1\<close>
+    by (auto simp: in_segment)
+  let ?d = "(u + v - u * v)"
+  have "?d > 0" using uv
+    by (auto simp: add_nonneg_pos pos_add_strict)
+  with \<open>x0 \<noteq> x3\<close> have "0 \<noteq> ?d *\<^sub>R (x3 - x0)" by simp
+  moreover
+  define ua where "ua = u / ?d"
+  have "ua * (u * v - u - v) - - u = 0"
+    by (auto simp: ua_def algebra_simps divide_simps)
+      (metis uv add_less_same_cancel1 add_strict_mono mult.right_neutral
+        mult_less_cancel_left_pos not_real_square_gt_zero vector_space_over_itself.scale_zero_left)
+  then have "(ua * (u * v - u - v) - - u) *\<^sub>R (x3 - x0) = 0"
+    by simp
+  moreover
+  have "0 < ua" "ua < 1"
+    using \<open>0 < u\<close> \<open>0 < v\<close> \<open>u < 1\<close> \<open>v < 1\<close>
+    by (auto simp: ua_def pos_add_strict intro!: divide_pos_pos)
+  ultimately show ?thesis
+    unfolding x1_def x2_def
+    by (auto intro!: exI[where x=ua] simp: algebra_simps in_segment)
+qed
+
+
+subsection \<open>Syntax\<close>
+
+abbreviation sequentially_at_top::"(nat\<Rightarrow>real)\<Rightarrow>bool"
+  ("_ \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>") \<comment> \<open>the \<open>\<^bsub>\<^esub>\<close> is to disambiguate syntax...\<close>
+  where "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>  \<equiv> filterlim s at_top sequentially"
+
+abbreviation sequentially_at_bot::"(nat\<Rightarrow>real)\<Rightarrow>bool"
+  ("_ \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>")
+  where "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>  \<equiv> filterlim s at_bot sequentially"
+
+
+subsection \<open>Paths\<close>
+
+lemma subpath0_linepath:
+  shows "subpath 0 u (linepath t t') = linepath t (t + u * (t' - t))"
+  unfolding subpath_def linepath_def
+  apply (rule ext)
+  apply auto
+proof -
+  fix x :: real
+  have f1: "\<And>r ra rb rc. (r::real) + ra * rb - ra * rc = r - ra * (rc - rb)"
+    by (simp add: right_diff_distrib')
+  have f2: "\<And>r ra. (r::real) - r * ra = r * (1 - ra)"
+    by (simp add: right_diff_distrib')
+  have f3: "\<And>r ra rb. (r::real) - ra + rb + ra - r = rb"
+    by auto
+  have f4: "\<And>r. (r::real) + (1 - 1) = r"
+    by linarith
+  have f5: "\<And>r ra. (r::real) + ra = ra + r"
+    by force
+  have f6: "\<And>r ra. (r::real) + (1 - (r + 1) + ra) = ra"
+    by linarith
+  have "t - x * (t - (t + u * (t' - t))) = t' * (u * x) + (t - t * (u * x))"
+    by (simp add: right_diff_distrib')
+  then show "(1 - u * x) * t + u * x * t' = (1 - x) * t + x * (t + u * (t' - t))"
+    using f6 f5 f4 f3 f2 f1 by (metis (no_types) mult.commute)
+qed
+
+lemma linepath_image0_right_open_real:
+  assumes "t < (t'::real)"
+  shows "linepath t t' ` {0..<1} = {t..<t'}"
+  unfolding linepath_def
+  apply auto
+    apply (metis add.commute add_diff_cancel_left' assms diff_diff_eq2 diff_le_eq less_eq_real_def mult.commute mult.right_neutral mult_right_mono right_diff_distrib')
+   apply (smt assms comm_semiring_class.distrib mult_diff_mult semiring_normalization_rules(2) zero_le_mult_iff)
+proof -
+  fix x
+  assume "t \<le> x" "x < t'"
+  let ?u = "(x-t)/(t'-t)"
+  have "?u \<ge> 0"
+    using \<open>t \<le> x\<close> assms by auto
+  moreover have "?u < 1"
+    by (simp add: \<open>x < t'\<close> assms)
+  moreover have "x = (1-?u) * t + ?u*t'"
+  proof -
+    have f1: "\<forall>r ra. (ra::real) * - r = r * - ra"
+      by simp
+    have "t + (t' + - t) * ((x + - t) / (t' + - t)) = x"
+      using assms by force
+    then have "t' * ((x + - t) / (t' + - t)) + t * (1 + - ((x + - t) / (t' + - t))) = x"
+      using f1 by (metis (no_types) add.left_commute distrib_left mult.commute mult.right_neutral)
+    then show ?thesis
+      by (simp add: mult.commute)
+  qed
+  ultimately show "x \<in> (\<lambda>x. (1 - x) * t + x * t') ` {0..<1}"
+    using atLeastLessThan_iff by blast 
+qed
+
+lemma oriented_subsegment_scale:
+  assumes "x1 \<in> {a<--<b}"
+  assumes "x2 \<in> {x1<--<b}"
+  obtains e where "e > 0" "b-a = e *\<^sub>R (x2-x1)"
+proof -
+  from assms(1) obtain u where u : "u > 0" "u < 1" "x1 = (1 - u) *\<^sub>R a + u *\<^sub>R b"
+    unfolding in_segment by blast
+  from assms(2) obtain v where v: "v > 0" "v < 1" "x2 = (1 - v) *\<^sub>R x1 + v *\<^sub>R b"
+    unfolding in_segment by blast
+  have "x2-x1 = -v *\<^sub>R x1 + v *\<^sub>R b" using v
+    by (metis add.commute add_diff_cancel_right diff_minus_eq_add scaleR_collapse scaleR_left.minus)
+  also have "... = (-v) *\<^sub>R ((1 - u) *\<^sub>R a + u *\<^sub>R b)  + v *\<^sub>R b" using u by auto
+  also have "... = v *\<^sub>R ((1-u)*\<^sub>R b - (1-u)*\<^sub>R a )"
+    by (smt add_diff_cancel diff_diff_add diff_minus_eq_add minus_diff_eq scaleR_collapse scale_minus_left scale_right_diff_distrib)
+  finally have x2x1:"x2-x1 = (v *(1-u)) *\<^sub>R (b - a)"
+    by (metis scaleR_scaleR scale_right_diff_distrib)
+  have "v * (1-u) > 0"  using u(2) v(1) by simp
+  then have "(x2-x1)/\<^sub>R (v * (1-u)) = (b-a)" unfolding x2x1
+    by (smt field_class.field_inverse scaleR_one scaleR_scaleR) 
+  thus ?thesis
+    using \<open>0 < v * (1 - u)\<close> positive_imp_inverse_positive that by fastforce
+qed
+
+end
diff --git a/thys/Poincare_Bendixson/Examples.thy b/thys/Poincare_Bendixson/Examples.thy
--- a/thys/Poincare_Bendixson/Examples.thy
+++ b/thys/Poincare_Bendixson/Examples.thy
@@ -1,462 +1,462 @@
-section \<open>Examples\<close>
-theory Examples
-  imports Poincare_Bendixson
-    "HOL-ODE-Numerics.ODE_Numerics"
-    Affine_Arithmetic_Misc
-begin
-
-subsection \<open>Simple\<close>
-
-context
-begin
-
-text \<open>coordinate functions\<close>
-definition "cx x y = -y + x * (1 - x^2 - y^2)"
-definition "cy x y = x + y * (1 - x^2 - y^2)"
-
-lemmas c_defs = cx_def cy_def
-
-text \<open>partial derivatives\<close>
-definition C11::"real\<Rightarrow>real\<Rightarrow>real" where "C11 x y = 1 - 3 * x^2 - y^2"
-definition C12::"real\<Rightarrow>real\<Rightarrow>real" where "C12 x y = -1 - 2 * x * y"
-definition C21::"real\<Rightarrow>real\<Rightarrow>real" where "C21 x y = 1 - 2 * x * y"
-definition C22::"real\<Rightarrow>real\<Rightarrow>real" where "C22 x y = 1 - x^2 - 3 * y^2"
-
-lemmas C_partials = C11_def C12_def C21_def C22_def
-
-text \<open>Jacobian as linear map\<close>
-definition C :: "real \<Rightarrow> real \<Rightarrow> (real \<times> real) \<Rightarrow>\<^sub>L (real \<times> real)" where
-  "C x y = blinfun_of_matrix
-    ((\<lambda>_ _. 0)
-      ((1,0) := (\<lambda>_. 0)((1, 0):=C11 x y, (0, 1):=C12 x y),
-       (0, 1):= (\<lambda>_. 0)((1, 0):=C21 x y, (0, 1):=C22 x y)))"
-
-lemma C_simp[simp]: "blinfun_apply (C x y) (dx, dy) =
-  (dx * C11 x y + dy * C12 x y,
-   dx * C21 x y + dy * C22 x y)"
-  by (auto simp: C_def blinfun_of_matrix_apply Basis_prod_def)
-
-lemma C_continuous[continuous_intros]:
-  "continuous_on S (\<lambda>x. local.C (f x) (g x))"
-  if "continuous_on S f" "continuous_on S g"
-  unfolding C_def
-  by (auto intro!: continuous_on_blinfun_of_matrix continuous_intros that
-      simp: Basis_prod_def C_partials)
-
-interpretation c: c1_on_open_R2 "\<lambda>(x::real, y::real). (cx x y, cy x y)::real*real"
-  "\<lambda>(x, y). C x y" UNIV
-  by unfold_locales
-    (auto intro!: derivative_eq_intros ext continuous_intros simp: split_beta algebra_simps
-      c_defs C_partials power2_eq_square)
-
-definition "trapC = cball (0::real,0::real) 2 - ball (0::real,0::real) (1/2)"
-
-lemma trapC_eq:
-  shows "trapC = {p. (fst p)^2 + (snd p)^2 - 4 \<le> 0} \<inter> {p. 1/4 - ((fst p)^2 + (snd p)^2) \<le> 0}"
-  unfolding trapC_def
-  apply (auto simp add: dist_Pair_Pair)
-  using real_sqrt_le_iff apply fastforce
-    apply (smt four_x_squared one_le_power real_sqrt_ge_0_iff real_sqrt_pow2)
-  using real_sqrt_le_mono apply fastforce
-proof -
-  fix a :: real and b :: real
-  assume a1: "sqrt (a\<^sup>2 + b\<^sup>2) * 2 < 1"
-  assume a2: "1 \<le> a\<^sup>2 * 4 + b\<^sup>2 * 4"
-  have "\<forall>r. 1 \<le> sqrt r \<or> \<not> 1 \<le> r"
-    by simp
-  then show False
-    using a2 a1 by (metis (no_types) Groups.mult_ac(2) distrib_left linorder_not_le real_sqrt_four real_sqrt_mult)
-qed
-
-lemma x_in_trapC:
-  shows "(2,0) \<in> trapC"
-  unfolding trapC_def
-  by (auto simp add: dist_Pair_Pair)
-
-lemma compact_trapC:
-  shows "compact trapC"
-  unfolding trapC_def
-  using compact_cball compact_diff by blast
-
-lemma nonempty_trapC:
-  shows "trapC \<noteq> {}"
-  using x_in_trapC by auto
-
-lemma origin_fixpoint:
-  assumes "(\<lambda>(x, y). (cx x y, cy x y)) (a,b) = 0"
-  shows "a = (0::real)" "b = (0::real)"
-  using assms unfolding cx_def cy_def zero_prod_def apply auto
-  apply (sos "((((A<0 * R<1) + (([28859/65536*a + 5089/8192*b + ~1/2] * A=0) + (([~5089/8192*a + 17219/65536*b + ~1/2] * A=1) + (R<1 * ((R<11853/65536 * [~16384/11853*a^2 + ~11585/11853*b^2 + 302/1317*a*b + a + 1940/3951*b]^2) + ((R<73630271/776798208 * [a^2 + 64177444/73630271*b^2 + 44531712/73630271*a*b + ~131061126/73630271*b]^2) + ((R<70211653911/4825433440256 * [~77895776116/70211653911*b^2 + 5825642465/10030236273*a*b + b]^2) + ((R<48375415273/657341564387328 * [~36776393918/48375415273*b^2 + a*b]^2) + (R<18852430195/11096159253659648 * [b^2]^2)))))))))) & (((A<0 * (A<0 * R<1)) + (([b] * A=0) + (([~1*a] * A=1) + (R<1 * (R<1 * [b]^2)))))))")
-proof -
-  assume a1: "a * (1 - a\<^sup>2 - b\<^sup>2) = b"
-  assume a2: "a + b * (1 - a\<^sup>2 - b\<^sup>2) = 0"
-  have f3: "\<forall>r ra. - (ra::real) * r = ra * - r"
-    by simp
-  have "- b * (1 - a\<^sup>2 - b\<^sup>2) = a"
-    using a2 by simp
-  then have "\<exists>r ra. b * b - ra * (r * (ra * - r)) = 0"
-    using f3 a1 by (metis (no_types) c.vec_simps(15) right_minus_eq)
-  then have "\<exists>r. b * b - r * - r = 0"
-    using f3 by (metis (no_types) c.vec_simps(14))
-  then show "b = 0"
-    by simp
-qed
-
-lemma origin_not_trapC:
-  shows "0 \<notin> trapC"
-  unfolding trapC_def zero_prod_def 
-  by auto
-
-lemma regular_trapC:
-  shows "0 \<notin> (\<lambda>(x, y). (cx x y, cy x y)) ` trapC"
-  using origin_fixpoint origin_not_trapC
-  by (smt UNIV_I UNIV_I UNIV_def case_prodE2 imageE c.flow_initial_time_if
-      c.rev.flow_initial_time_if mem_Collect_eq zero_prod_def)
-
-lemma positively_invariant_outer:
-  shows "c.positively_invariant  {p. (\<lambda>p. (fst p)\<^sup>2 + (snd p)\<^sup>2 - 4) p \<le> 0}"
-  apply (rule c.positively_invariant_le[of "\<lambda>p.-2*((fst p)^2+(snd p)^2)" _  "\<lambda>x p. 2 * fst x * fst p + 2 * snd x * snd p" ])
-    apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding cx_def cy_def
-  by (sos "(((A<0 * R<1) + (R<1 * ((R<6 * [a]^2) + (R<6 * [b]^2)))))")
-
-
-lemma positively_invariant_inner:
-  shows "c.positively_invariant  {p. (\<lambda>p. 1/4 - ((fst p)\<^sup>2 + (snd p)\<^sup>2)) p \<le> 0}"
-  apply (rule c.positively_invariant_le[of "\<lambda>p.-2*((fst p)^2+(snd p)^2)" _ "\<lambda>x p. - 2 * fst x * fst p - 2 * snd x * snd p"])
-    apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding cx_def cy_def
-  by (sos "(((A<0 * R<1) + (R<1 * ((R<3/2 * [a]^2) + (R<3/2 * [b]^2)))))")
-
-lemma positively_invariant_trapC:
-  shows "c.positively_invariant trapC"
-  unfolding trapC_eq
-  apply (rule c.positively_invariant_conj)
-  using positively_invariant_outer
-   apply (metis (no_types, lifting) Collect_cong case_prodE case_prodI2 case_prod_conv)
-  using positively_invariant_inner
-  by (metis (no_types, lifting) Collect_cong case_prodE case_prodI2 case_prod_conv)
-
-theorem c_has_periodic_orbit:
-  obtains y where "c.periodic_orbit y" "c.flow0 y ` UNIV \<subseteq> trapC"
-proof -
-  from c.poincare_bendixson_applied[OF compact_trapC _ nonempty_trapC positively_invariant_trapC regular_trapC]
-  show ?thesis using that by blast
-qed
-
-text \<open>Real-Arithmetic\<close>
-schematic_goal c_fas:
-  "[-(-(X!1) + (X!0) * (1 - (X!0)^2 - (X!1)^2)), -((X!0) + (X!1) * (1 - (X!0)^2 - (X!1)^2))] = interpret_floatariths ?fas X"
-  by (reify_floatariths)
-
-concrete_definition c_fas uses c_fas
-
-interpretation crev: ode_interpretation true_form UNIV c_fas
-  "-(\<lambda>(x, y). (cx x y, cy x y)::real*real)"
-  "d::2" for d
-  by unfold_locales (auto simp: c_fas_def less_Suc_eq_0_disj nth_Basis_list_prod Basis_list_real_def
-      cx_def cy_def eval_nat_numeral
-      mk_ode_ops_def eucl_of_list_prod power2_eq_square intro!: isFDERIV_I)
-
-lemma crev: "t \<in> {1/8 .. 1/8} \<longrightarrow> (x, y) \<in> {(2, 0) .. (2, 0)} \<longrightarrow>
-   t \<in> c.rev.existence_ivl0 (x, y) \<and> c.rev.flow0 (x, y) t \<in> {(5.15, -0.651)..(5.18, -0.647)}"
-  by (tactic \<open>ode_bnds_tac @{thms c_fas_def} 30 20 7 12 [(0, 1, "0x000000")] (* "crev.out" *) "" @{context} 1\<close>)
-
-theorem c_has_limit_cycle:
-  obtains y where "c.limit_cycle y" "range (c.flow0 y) \<subseteq> trapC"
-proof -
-  define E where "E = {(5.15, -0.651)..(5.18, -0.647)::real*real}"
-  from crev have "c.rev.flow0 (2, 0) (1/8) \<in> E"
-    by (auto simp: E_def)
-  moreover
-  have "E \<inter> trapC = {}"
-  proof -
-    have "norm x > 2" if "x \<in> E" for x
-      using that
-      apply (auto simp: norm_prod_def less_eq_prod_def E_def)
-      by (smt power2_less_eq_zero_iff real_less_rsqrt zero_compare_simps(9))
-    moreover have "norm x \<le> 2" if "x \<in> trapC" for x
-      using that
-      by (auto simp: trapC_def dist_prod_def norm_prod_def)
-    ultimately show ?thesis by force
-  qed
-  ultimately have "c.rev.flow0 (2, 0) (1 / 8) \<notin> trapC" by blast
-  from c.poincare_bendixson_limit_cycle[OF compact_trapC subset_UNIV x_in_trapC positively_invariant_trapC regular_trapC this] that
-  show ?thesis by blast
-qed
-
-end
-
-
-subsection \<open>Glycolysis\<close>
-
-text \<open>Strogatz, Example 7.3.2\<close>
-
-context
-begin
-
-text \<open>coordinate functions\<close>
-definition "gx x y = -x + 0.08 * y + x\<^sup>2 * y"
-definition "gy x y = 0.6 - 0.08 * y - x\<^sup>2 * y"
-
-lemmas g_defs = gx_def gy_def
-
-text \<open>partial derivatives\<close>
-definition A11::"real\<Rightarrow>real\<Rightarrow>real" where "A11 x y = -1 + 2 * x * y"
-definition A12::"real\<Rightarrow>real\<Rightarrow>real" where "A12 x y = (0.08 + x\<^sup>2)"
-definition A21::"real\<Rightarrow>real\<Rightarrow>real" where "A21 x y = -2*x*y"
-definition A22::"real\<Rightarrow>real\<Rightarrow>real" where "A22 x y = -(0.08 + x\<^sup>2)"
-
-lemmas A_partials = A11_def A12_def A21_def A22_def
-
-text \<open>Jacobian as linear map\<close>
-definition A :: "real \<Rightarrow> real \<Rightarrow> (real \<times> real) \<Rightarrow>\<^sub>L (real \<times> real)" where
-  "A x y = blinfun_of_matrix
-    ((\<lambda>_ _. 0)
-      ((1,0) := (\<lambda>_. 0)((1, 0):=A11 x y, (0, 1):=A12 x y),
-       (0, 1):= (\<lambda>_. 0)((1, 0):=A21 x y, (0, 1):=A22 x y)))"
-
-lemma A_simp[simp]: "blinfun_apply (A x y) (dx, dy) =
-  (dx * A11 x y + dy * A12 x y,
-   dx * A21 x y + dy * A22 x y)"
-  by (auto simp: A_def blinfun_of_matrix_apply Basis_prod_def)
-
-lemma A_continuous[continuous_intros]:
-  "continuous_on S (\<lambda>x. local.A (f x) (g x))"
-  if "continuous_on S f" "continuous_on S g"
-  unfolding A_def
-  by (auto intro!: continuous_on_blinfun_of_matrix continuous_intros that
-      simp: Basis_prod_def A_partials)
-
-interpretation g: c1_on_open_R2 "\<lambda>(x::real, y::real). (gx x y, gy x y)::real*real"
-  "\<lambda>(x, y). A x y" UNIV
-  by unfold_locales
-    (auto intro!: derivative_eq_intros ext continuous_intros simp: split_beta algebra_simps
-      g_defs A_partials)
-
-(*
-  The outer invariant is the convex set formed by the axes, y \<le> 7.51, and x+y\<le>8.12
-*)
-definition "(pos_quad::(real \<times> real) set) =  {p . - snd p \<le> 0} \<inter> {p . - fst p \<le> 0}"
-
-definition "(trapG1::(real \<times> real) set) = pos_quad \<inter> ({p. (snd p) - 751/100 \<le> 0} \<inter> {p. (fst p) + (snd p) - 812/100 \<le> 0})"
-
-lemma positively_invariant_y:
-  shows "g.positively_invariant {p . - snd p \<le> 0}"
-  apply (rule g.positively_invariant_le[of "\<lambda>p. -(0.08 + (fst p)^2)" _ "\<lambda>x p. - snd p"])
-    apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding gy_def
-  by (sos "()")
-
-lemma positively_invariant_pos_quad:
-  shows "g.positively_invariant pos_quad"
-  unfolding pos_quad_def
-  apply (rule g.positively_invariant_le_domain[OF positively_invariant_y, of "\<lambda>p. fst p * snd p -1"])
-    apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding gx_def
-  by (sos "(((A<0 * R<1) + (((A<0 * R<1) * (R<11/14 * [1]^2)) + ((A<=0 * R<1) * (R<1/7 * [1]^2)))))")
-
-lemma positively_invariant_y_upper:
-  shows "g.positively_invariant {p. (snd p) - 751/100 \<le> 0}"
-  apply (rule g.positively_invariant_barrier)
-    apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding gy_def
-  by (sos "((R<1 + ((R<1 * (R<18775/2 * [a]^2)) + ((A<=0 * R<1) * (R<1250 * [1]^2)))))")
-
-lemma arith2:
-  shows "(y::real) \<le> 751/100 \<and> x + (y::real) = 812/100 \<Longrightarrow> 3/5 - (x::real) < 0"
-  by linarith
-
-lemma positively_invariant_trapG1:
-  shows "g.positively_invariant trapG1"
-  unfolding trapG1_def
-  apply (rule g.positively_invariant_conj[OF positively_invariant_pos_quad])
-  apply (rule g.positively_invariant_barrier_domain[OF positively_invariant_y_upper])
-  apply (auto intro!: continuous_intros derivative_eq_intros)
-  unfolding gx_def gy_def by auto
-
-(* Polynomial in invariant *)
-definition "p1 (x::real) (y::real) = -(21/34) - (69*x)/38 + (19*x^2)/15 - (9*x^3)/28 - (6*x^4)/43 + ( 14*y)/29 + (31*x*y)/21 + (182*x^2*y)/47 - (35*x^3*y)/16 - ( 3*y^2)/17 - (2*x*y^2)/9 - (31*x^2*y^2)/20 +y^3/102 + (x*y^3)/59"
-
-definition "p1d x xa = 38 * (fst xa * fst x) / 15 - 69 * fst xa / 38 -
-          27 * (fst xa * (fst x)\<^sup>2) / 28 -
-          24 * (fst xa * fst x ^ 3) / 43 +
-          14 * snd xa / 29 +
-          (651 * (fst x * snd xa) +
-           651 * (fst xa * snd x)) /
-          441 +
-          (8554 * ((fst x)\<^sup>2 * snd xa) +
-           17108 * (fst xa * (fst x * snd x))) /
-          2209 -
-          (560 * (fst x ^ 3 * snd xa) +
-           1680 * (fst xa * ((fst x)\<^sup>2 * snd x))) /
-          256 -
-          6 * (snd xa * snd x) / 17 -
-          (36 * (fst x * (snd xa * snd x)) +
-           18 * (fst xa * (snd x)\<^sup>2)) /
-          81 -
-          (1240 * ((fst x)\<^sup>2 * (snd xa * snd x)) +
-           1240 * (fst xa * (fst x * (snd x)\<^sup>2))) /
-          400 +
-          snd xa * (snd x)\<^sup>2 / 34 +
-          (177 * (fst x * (snd xa * (snd x)\<^sup>2)) +
-           fst xa * snd x ^ 3 * 59) /
-          3481"
-
-lemma p1_has_derivative:
-  shows "((\<lambda>x. p1 (fst x) (snd x)) has_derivative p1d x) (at x)"
-  unfolding p1_def p1d_def
-  by (auto intro!: continuous_intros derivative_eq_intros)
-
-(* p1 excludes equilibria for free *)
-lemma p1_not_equil:
-  shows " p1 x y \<le> 0 \<Longrightarrow> gx x y \<noteq> 0 \<or> gy x y \<noteq> 0"
-  unfolding gx_def gy_def p1_def
-  by (sos "()")
-
-definition "trapG = trapG1 \<inter> {p. p1 (fst p) (snd p) \<le> 0}"
-
-text \<open>Real-Arithmetic\<close>
-definition "g_arith a b = (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b - p1d (a, b) (gx a b, gy a b)"
-
-schematic_goal g_arith_fas:
-  "[g_arith (X!0) (X!1)] = interpret_floatariths ?fas X"
-  unfolding g_arith_def p1_def p1d_def gx_def gy_def fst_conv snd_conv
-  by (reify_floatariths)
-
-concrete_definition g_arith_fas uses g_arith_fas
-
-lemma list_interval2: "list_interval [a, b] [c, d] = {[x, y] | x y. x \<in> {a .. c} \<and> y \<in> {b .. d}}"
-  apply (auto simp: list_interval_def)
-  subgoal for x
-    apply (cases x)
-    apply auto
-    subgoal for y zs
-      apply (cases zs)
-      by auto
-    done
-  done
-
-lemma g_arith_nonneg: "g_arith a b \<ge> 0"
-  if a: "0 \<le> a" "a \<le> 8.24" and b: "0 \<le> b" "b \<le> 7.51"
-proof -
-  have "prove_nonneg [(0, 1, ''0x000000'')] 1000000 30 (slp_of_fas [hd g_arith_fas]) [aforms_of_ivls [0, 0]
-    [float_divr 30 824 100, float_divr 30 751 100]]"
-    by eval\<comment> \<open>slow: 60s\<close>
-  from prove_nonneg[OF this]
-  have "0 \<le> interpret_floatarith (hd g_arith_fas) [a, b]"
-    apply (auto simp: g_arith_fas)
-    apply (subst (asm) Joints_aforms_of_ivls)
-     apply (auto )
-      apply (smt divide_nonneg_nonneg float_divr float_numeral rel_simps(27))
-     apply (smt divide_nonneg_nonneg float_divr float_numeral rel_simps(27))
-    apply (subst (asm) list_interval2)
-    apply auto
-    apply (drule spec[where x="[a, b]"])
-    using a b
-    apply auto
-    subgoal by (rule order_trans[OF _ float_divr]) simp
-    subgoal by (rule order_trans[OF _ float_divr]) simp
-    done
-  also have "\<dots> = g_arith a b"
-    by (auto simp: g_arith_fas_def g_arith_def p1_def p1d_def gx_def gy_def)
-  finally show ?thesis .
-qed
-
-lemma trap_arithmetic:
-  "p1d (a, b) (gx a b, gy a b) \<le> (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b" if "(a, b) \<in> trapG1"
-proof -
-  from that
-  have b: "0 \<le> b" "b \<le> 7.51"
-    and a: "0 \<le> a" "a \<le> 8.24"
-    by (auto simp: trapG1_def pos_quad_def)
-  from g_arith_nonneg[OF a b] show ?thesis
-    by (simp add: g_arith_def)
-qed
-
-lemma positively_invariant_trapG:
-  shows "g.positively_invariant trapG"
-  unfolding trapG_def
-  apply (rule g.positively_invariant_le_domain[OF positively_invariant_trapG1 _ p1_has_derivative,
-        of "\<lambda>p. -1.08 - (fst p)^2 + 2 * fst p * snd p"])
-  subgoal by (auto intro!: continuous_intros derivative_eq_intros simp add: pos_quad_def)
-  apply auto
-  by (rule trap_arithmetic)
-
-lemma regular_trapG:
-  shows "0 \<notin> (\<lambda>(x, y). (gx x y, gy x y)) ` trapG"
-  unfolding trapG_def apply auto using p1_not_equil
-  by force
-
-lemma arith:
-  "\<And>a b::real. 0 \<le> b \<Longrightarrow>
-           0 \<le> a \<Longrightarrow>
-           b * 100 \<le> 751 \<Longrightarrow>
-           a * 25 + b * 25 \<le> 203 \<Longrightarrow> norm a + norm b \<le> 20"
-  by auto
-
-lemma trapG1_subset:
-  shows "trapG1 \<subseteq> cball (0::real \<times> real) 20"
-  unfolding trapG1_def pos_quad_def
-  apply auto
-  using arith norm_Pair_le
-  by smt
-
-lemma compact_subset_closed:
-  assumes "compact S" "closed T"
-  assumes "T \<subseteq> S"
-  shows "compact T"
-  using compact_Int_closed[OF assms(1-2)] assms(3)
-  by (simp add: inf_absorb2)
-
-lemma compact_trapG1:
-  shows "compact trapG1"
-  apply (auto intro!: compact_subset_closed[OF _ _ trapG1_subset])
-  unfolding trapG1_def pos_quad_def
-  by (auto intro!: closed_Collect_le continuous_intros)
-
-lemma compact_trapG:
-  shows "compact trapG"
-  unfolding trapG_def
-  by (auto intro!: compact_Int_closed compact_trapG1 closed_Collect_le continuous_intros simp add: p1_def)
-
-lemma x_in_trapG:
-  shows "(1,0) \<in> trapG"
-  unfolding trapG_def trapG1_def pos_quad_def p1_def
-  by (auto simp add: dist_Pair_Pair)
-
-schematic_goal g_fas:
-  "[- (- (X!0) + 8 / 100 * (X!1) + (X!0)^2 * (X!1)),-( 6 / 10 - 8 / 100 * (X!1) - (X!0)^2 * (X!1))] = interpret_floatariths ?fas X"
-  by (reify_floatariths)
-
-concrete_definition g_fas uses g_fas
-
-interpretation grev: ode_interpretation true_form UNIV g_fas
-  "-(\<lambda>(x, y). (gx x y, gy x y)::real*real)"
-  "d::2" for d
-  by unfold_locales (auto simp: g_fas_def less_Suc_eq_0_disj nth_Basis_list_prod Basis_list_real_def
-      gx_def gy_def eval_nat_numeral
-      mk_ode_ops_def eucl_of_list_prod power2_eq_square intro!: isFDERIV_I)
-
-lemma grev: "t \<in> {1/8 .. 1/8} \<longrightarrow> (x, y) \<in> {(1, 0) .. (1, 0)} \<longrightarrow>
-   t \<in> g.rev.existence_ivl0 (x, y) \<and> g.rev.flow0 (x, y) t \<in>
-    {(1.1, -0.09) .. (1.2, -0.08)}"
-  by (tactic \<open>ode_bnds_tac @{thms g_fas_def} 30 20 7 12 [(0, 1, "0x000000")] (* "grev.out" *) "" @{context} 1\<close>)
-
-theorem g_has_limit_cycle:
-  obtains y where "g.limit_cycle y" "range (g.flow0 y) \<subseteq> trapG"
-proof -
-  define E::"(real*real) set" where "E = {(1.1, -0.09) .. (1.2, -0.08)}"
-  from grev have "g.rev.flow0 (1, 0) (1/8) \<in> E"
-    by (auto simp: E_def)
-  moreover
-  have "E \<inter> trapG = {}"
-    by (auto simp: trapG_def E_def trapG1_def pos_quad_def)
-  ultimately have "g.rev.flow0 (1, 0) (1 / 8) \<notin> trapG" by blast
-  from g.poincare_bendixson_limit_cycle[OF compact_trapG subset_UNIV x_in_trapG positively_invariant_trapG regular_trapG this] that
-  show ?thesis by blast
-qed
-
-end
-
+section \<open>Examples\<close>
+theory Examples
+  imports Poincare_Bendixson
+    "HOL-ODE-Numerics.ODE_Numerics"
+    Affine_Arithmetic_Misc
+begin
+
+subsection \<open>Simple\<close>
+
+context
+begin
+
+text \<open>coordinate functions\<close>
+definition "cx x y = -y + x * (1 - x^2 - y^2)"
+definition "cy x y = x + y * (1 - x^2 - y^2)"
+
+lemmas c_defs = cx_def cy_def
+
+text \<open>partial derivatives\<close>
+definition C11::"real\<Rightarrow>real\<Rightarrow>real" where "C11 x y = 1 - 3 * x^2 - y^2"
+definition C12::"real\<Rightarrow>real\<Rightarrow>real" where "C12 x y = -1 - 2 * x * y"
+definition C21::"real\<Rightarrow>real\<Rightarrow>real" where "C21 x y = 1 - 2 * x * y"
+definition C22::"real\<Rightarrow>real\<Rightarrow>real" where "C22 x y = 1 - x^2 - 3 * y^2"
+
+lemmas C_partials = C11_def C12_def C21_def C22_def
+
+text \<open>Jacobian as linear map\<close>
+definition C :: "real \<Rightarrow> real \<Rightarrow> (real \<times> real) \<Rightarrow>\<^sub>L (real \<times> real)" where
+  "C x y = blinfun_of_matrix
+    ((\<lambda>_ _. 0)
+      ((1,0) := (\<lambda>_. 0)((1, 0):=C11 x y, (0, 1):=C12 x y),
+       (0, 1):= (\<lambda>_. 0)((1, 0):=C21 x y, (0, 1):=C22 x y)))"
+
+lemma C_simp[simp]: "blinfun_apply (C x y) (dx, dy) =
+  (dx * C11 x y + dy * C12 x y,
+   dx * C21 x y + dy * C22 x y)"
+  by (auto simp: C_def blinfun_of_matrix_apply Basis_prod_def)
+
+lemma C_continuous[continuous_intros]:
+  "continuous_on S (\<lambda>x. local.C (f x) (g x))"
+  if "continuous_on S f" "continuous_on S g"
+  unfolding C_def
+  by (auto intro!: continuous_on_blinfun_of_matrix continuous_intros that
+      simp: Basis_prod_def C_partials)
+
+interpretation c: c1_on_open_R2 "\<lambda>(x::real, y::real). (cx x y, cy x y)::real*real"
+  "\<lambda>(x, y). C x y" UNIV
+  by unfold_locales
+    (auto intro!: derivative_eq_intros ext continuous_intros simp: split_beta algebra_simps
+      c_defs C_partials power2_eq_square)
+
+definition "trapC = cball (0::real,0::real) 2 - ball (0::real,0::real) (1/2)"
+
+lemma trapC_eq:
+  shows "trapC = {p. (fst p)^2 + (snd p)^2 - 4 \<le> 0} \<inter> {p. 1/4 - ((fst p)^2 + (snd p)^2) \<le> 0}"
+  unfolding trapC_def
+  apply (auto simp add: dist_Pair_Pair)
+  using real_sqrt_le_iff apply fastforce
+    apply (smt four_x_squared one_le_power real_sqrt_ge_0_iff real_sqrt_pow2)
+  using real_sqrt_le_mono apply fastforce
+proof -
+  fix a :: real and b :: real
+  assume a1: "sqrt (a\<^sup>2 + b\<^sup>2) * 2 < 1"
+  assume a2: "1 \<le> a\<^sup>2 * 4 + b\<^sup>2 * 4"
+  have "\<forall>r. 1 \<le> sqrt r \<or> \<not> 1 \<le> r"
+    by simp
+  then show False
+    using a2 a1 by (metis (no_types) Groups.mult_ac(2) distrib_left linorder_not_le real_sqrt_four real_sqrt_mult)
+qed
+
+lemma x_in_trapC:
+  shows "(2,0) \<in> trapC"
+  unfolding trapC_def
+  by (auto simp add: dist_Pair_Pair)
+
+lemma compact_trapC:
+  shows "compact trapC"
+  unfolding trapC_def
+  using compact_cball compact_diff by blast
+
+lemma nonempty_trapC:
+  shows "trapC \<noteq> {}"
+  using x_in_trapC by auto
+
+lemma origin_fixpoint:
+  assumes "(\<lambda>(x, y). (cx x y, cy x y)) (a,b) = 0"
+  shows "a = (0::real)" "b = (0::real)"
+  using assms unfolding cx_def cy_def zero_prod_def apply auto
+  apply (sos "((((A<0 * R<1) + (([28859/65536*a + 5089/8192*b + ~1/2] * A=0) + (([~5089/8192*a + 17219/65536*b + ~1/2] * A=1) + (R<1 * ((R<11853/65536 * [~16384/11853*a^2 + ~11585/11853*b^2 + 302/1317*a*b + a + 1940/3951*b]^2) + ((R<73630271/776798208 * [a^2 + 64177444/73630271*b^2 + 44531712/73630271*a*b + ~131061126/73630271*b]^2) + ((R<70211653911/4825433440256 * [~77895776116/70211653911*b^2 + 5825642465/10030236273*a*b + b]^2) + ((R<48375415273/657341564387328 * [~36776393918/48375415273*b^2 + a*b]^2) + (R<18852430195/11096159253659648 * [b^2]^2)))))))))) & (((A<0 * (A<0 * R<1)) + (([b] * A=0) + (([~1*a] * A=1) + (R<1 * (R<1 * [b]^2)))))))")
+proof -
+  assume a1: "a * (1 - a\<^sup>2 - b\<^sup>2) = b"
+  assume a2: "a + b * (1 - a\<^sup>2 - b\<^sup>2) = 0"
+  have f3: "\<forall>r ra. - (ra::real) * r = ra * - r"
+    by simp
+  have "- b * (1 - a\<^sup>2 - b\<^sup>2) = a"
+    using a2 by simp
+  then have "\<exists>r ra. b * b - ra * (r * (ra * - r)) = 0"
+    using f3 a1 by (metis (no_types) c.vec_simps(15) right_minus_eq)
+  then have "\<exists>r. b * b - r * - r = 0"
+    using f3 by (metis (no_types) c.vec_simps(14))
+  then show "b = 0"
+    by simp
+qed
+
+lemma origin_not_trapC:
+  shows "0 \<notin> trapC"
+  unfolding trapC_def zero_prod_def 
+  by auto
+
+lemma regular_trapC:
+  shows "0 \<notin> (\<lambda>(x, y). (cx x y, cy x y)) ` trapC"
+  using origin_fixpoint origin_not_trapC
+  by (smt UNIV_I UNIV_I UNIV_def case_prodE2 imageE c.flow_initial_time_if
+      c.rev.flow_initial_time_if mem_Collect_eq zero_prod_def)
+
+lemma positively_invariant_outer:
+  shows "c.positively_invariant  {p. (\<lambda>p. (fst p)\<^sup>2 + (snd p)\<^sup>2 - 4) p \<le> 0}"
+  apply (rule c.positively_invariant_le[of "\<lambda>p.-2*((fst p)^2+(snd p)^2)" _  "\<lambda>x p. 2 * fst x * fst p + 2 * snd x * snd p" ])
+    apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding cx_def cy_def
+  by (sos "(((A<0 * R<1) + (R<1 * ((R<6 * [a]^2) + (R<6 * [b]^2)))))")
+
+
+lemma positively_invariant_inner:
+  shows "c.positively_invariant  {p. (\<lambda>p. 1/4 - ((fst p)\<^sup>2 + (snd p)\<^sup>2)) p \<le> 0}"
+  apply (rule c.positively_invariant_le[of "\<lambda>p.-2*((fst p)^2+(snd p)^2)" _ "\<lambda>x p. - 2 * fst x * fst p - 2 * snd x * snd p"])
+    apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding cx_def cy_def
+  by (sos "(((A<0 * R<1) + (R<1 * ((R<3/2 * [a]^2) + (R<3/2 * [b]^2)))))")
+
+lemma positively_invariant_trapC:
+  shows "c.positively_invariant trapC"
+  unfolding trapC_eq
+  apply (rule c.positively_invariant_conj)
+  using positively_invariant_outer
+   apply (metis (no_types, lifting) Collect_cong case_prodE case_prodI2 case_prod_conv)
+  using positively_invariant_inner
+  by (metis (no_types, lifting) Collect_cong case_prodE case_prodI2 case_prod_conv)
+
+theorem c_has_periodic_orbit:
+  obtains y where "c.periodic_orbit y" "c.flow0 y ` UNIV \<subseteq> trapC"
+proof -
+  from c.poincare_bendixson_applied[OF compact_trapC _ nonempty_trapC positively_invariant_trapC regular_trapC]
+  show ?thesis using that by blast
+qed
+
+text \<open>Real-Arithmetic\<close>
+schematic_goal c_fas:
+  "[-(-(X!1) + (X!0) * (1 - (X!0)^2 - (X!1)^2)), -((X!0) + (X!1) * (1 - (X!0)^2 - (X!1)^2))] = interpret_floatariths ?fas X"
+  by (reify_floatariths)
+
+concrete_definition c_fas uses c_fas
+
+interpretation crev: ode_interpretation true_form UNIV c_fas
+  "-(\<lambda>(x, y). (cx x y, cy x y)::real*real)"
+  "d::2" for d
+  by unfold_locales (auto simp: c_fas_def less_Suc_eq_0_disj nth_Basis_list_prod Basis_list_real_def
+      cx_def cy_def eval_nat_numeral
+      mk_ode_ops_def eucl_of_list_prod power2_eq_square intro!: isFDERIV_I)
+
+lemma crev: "t \<in> {1/8 .. 1/8} \<longrightarrow> (x, y) \<in> {(2, 0) .. (2, 0)} \<longrightarrow>
+   t \<in> c.rev.existence_ivl0 (x, y) \<and> c.rev.flow0 (x, y) t \<in> {(5.15, -0.651)..(5.18, -0.647)}"
+  by (tactic \<open>ode_bnds_tac @{thms c_fas_def} 30 20 7 12 [(0, 1, "0x000000")] (* "crev.out" *) "" @{context} 1\<close>)
+
+theorem c_has_limit_cycle:
+  obtains y where "c.limit_cycle y" "range (c.flow0 y) \<subseteq> trapC"
+proof -
+  define E where "E = {(5.15, -0.651)..(5.18, -0.647)::real*real}"
+  from crev have "c.rev.flow0 (2, 0) (1/8) \<in> E"
+    by (auto simp: E_def)
+  moreover
+  have "E \<inter> trapC = {}"
+  proof -
+    have "norm x > 2" if "x \<in> E" for x
+      using that
+      apply (auto simp: norm_prod_def less_eq_prod_def E_def)
+      by (smt power2_less_eq_zero_iff real_less_rsqrt zero_compare_simps(9))
+    moreover have "norm x \<le> 2" if "x \<in> trapC" for x
+      using that
+      by (auto simp: trapC_def dist_prod_def norm_prod_def)
+    ultimately show ?thesis by force
+  qed
+  ultimately have "c.rev.flow0 (2, 0) (1 / 8) \<notin> trapC" by blast
+  from c.poincare_bendixson_limit_cycle[OF compact_trapC subset_UNIV x_in_trapC positively_invariant_trapC regular_trapC this] that
+  show ?thesis by blast
+qed
+
+end
+
+
+subsection \<open>Glycolysis\<close>
+
+text \<open>Strogatz, Example 7.3.2\<close>
+
+context
+begin
+
+text \<open>coordinate functions\<close>
+definition "gx x y = -x + 0.08 * y + x\<^sup>2 * y"
+definition "gy x y = 0.6 - 0.08 * y - x\<^sup>2 * y"
+
+lemmas g_defs = gx_def gy_def
+
+text \<open>partial derivatives\<close>
+definition A11::"real\<Rightarrow>real\<Rightarrow>real" where "A11 x y = -1 + 2 * x * y"
+definition A12::"real\<Rightarrow>real\<Rightarrow>real" where "A12 x y = (0.08 + x\<^sup>2)"
+definition A21::"real\<Rightarrow>real\<Rightarrow>real" where "A21 x y = -2*x*y"
+definition A22::"real\<Rightarrow>real\<Rightarrow>real" where "A22 x y = -(0.08 + x\<^sup>2)"
+
+lemmas A_partials = A11_def A12_def A21_def A22_def
+
+text \<open>Jacobian as linear map\<close>
+definition A :: "real \<Rightarrow> real \<Rightarrow> (real \<times> real) \<Rightarrow>\<^sub>L (real \<times> real)" where
+  "A x y = blinfun_of_matrix
+    ((\<lambda>_ _. 0)
+      ((1,0) := (\<lambda>_. 0)((1, 0):=A11 x y, (0, 1):=A12 x y),
+       (0, 1):= (\<lambda>_. 0)((1, 0):=A21 x y, (0, 1):=A22 x y)))"
+
+lemma A_simp[simp]: "blinfun_apply (A x y) (dx, dy) =
+  (dx * A11 x y + dy * A12 x y,
+   dx * A21 x y + dy * A22 x y)"
+  by (auto simp: A_def blinfun_of_matrix_apply Basis_prod_def)
+
+lemma A_continuous[continuous_intros]:
+  "continuous_on S (\<lambda>x. local.A (f x) (g x))"
+  if "continuous_on S f" "continuous_on S g"
+  unfolding A_def
+  by (auto intro!: continuous_on_blinfun_of_matrix continuous_intros that
+      simp: Basis_prod_def A_partials)
+
+interpretation g: c1_on_open_R2 "\<lambda>(x::real, y::real). (gx x y, gy x y)::real*real"
+  "\<lambda>(x, y). A x y" UNIV
+  by unfold_locales
+    (auto intro!: derivative_eq_intros ext continuous_intros simp: split_beta algebra_simps
+      g_defs A_partials)
+
+(*
+  The outer invariant is the convex set formed by the axes, y \<le> 7.51, and x+y\<le>8.12
+*)
+definition "(pos_quad::(real \<times> real) set) =  {p . - snd p \<le> 0} \<inter> {p . - fst p \<le> 0}"
+
+definition "(trapG1::(real \<times> real) set) = pos_quad \<inter> ({p. (snd p) - 751/100 \<le> 0} \<inter> {p. (fst p) + (snd p) - 812/100 \<le> 0})"
+
+lemma positively_invariant_y:
+  shows "g.positively_invariant {p . - snd p \<le> 0}"
+  apply (rule g.positively_invariant_le[of "\<lambda>p. -(0.08 + (fst p)^2)" _ "\<lambda>x p. - snd p"])
+    apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding gy_def
+  by (sos "()")
+
+lemma positively_invariant_pos_quad:
+  shows "g.positively_invariant pos_quad"
+  unfolding pos_quad_def
+  apply (rule g.positively_invariant_le_domain[OF positively_invariant_y, of "\<lambda>p. fst p * snd p -1"])
+    apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding gx_def
+  by (sos "(((A<0 * R<1) + (((A<0 * R<1) * (R<11/14 * [1]^2)) + ((A<=0 * R<1) * (R<1/7 * [1]^2)))))")
+
+lemma positively_invariant_y_upper:
+  shows "g.positively_invariant {p. (snd p) - 751/100 \<le> 0}"
+  apply (rule g.positively_invariant_barrier)
+    apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding gy_def
+  by (sos "((R<1 + ((R<1 * (R<18775/2 * [a]^2)) + ((A<=0 * R<1) * (R<1250 * [1]^2)))))")
+
+lemma arith2:
+  shows "(y::real) \<le> 751/100 \<and> x + (y::real) = 812/100 \<Longrightarrow> 3/5 - (x::real) < 0"
+  by linarith
+
+lemma positively_invariant_trapG1:
+  shows "g.positively_invariant trapG1"
+  unfolding trapG1_def
+  apply (rule g.positively_invariant_conj[OF positively_invariant_pos_quad])
+  apply (rule g.positively_invariant_barrier_domain[OF positively_invariant_y_upper])
+  apply (auto intro!: continuous_intros derivative_eq_intros)
+  unfolding gx_def gy_def by auto
+
+(* Polynomial in invariant *)
+definition "p1 (x::real) (y::real) = -(21/34) - (69*x)/38 + (19*x^2)/15 - (9*x^3)/28 - (6*x^4)/43 + ( 14*y)/29 + (31*x*y)/21 + (182*x^2*y)/47 - (35*x^3*y)/16 - ( 3*y^2)/17 - (2*x*y^2)/9 - (31*x^2*y^2)/20 +y^3/102 + (x*y^3)/59"
+
+definition "p1d x xa = 38 * (fst xa * fst x) / 15 - 69 * fst xa / 38 -
+          27 * (fst xa * (fst x)\<^sup>2) / 28 -
+          24 * (fst xa * fst x ^ 3) / 43 +
+          14 * snd xa / 29 +
+          (651 * (fst x * snd xa) +
+           651 * (fst xa * snd x)) /
+          441 +
+          (8554 * ((fst x)\<^sup>2 * snd xa) +
+           17108 * (fst xa * (fst x * snd x))) /
+          2209 -
+          (560 * (fst x ^ 3 * snd xa) +
+           1680 * (fst xa * ((fst x)\<^sup>2 * snd x))) /
+          256 -
+          6 * (snd xa * snd x) / 17 -
+          (36 * (fst x * (snd xa * snd x)) +
+           18 * (fst xa * (snd x)\<^sup>2)) /
+          81 -
+          (1240 * ((fst x)\<^sup>2 * (snd xa * snd x)) +
+           1240 * (fst xa * (fst x * (snd x)\<^sup>2))) /
+          400 +
+          snd xa * (snd x)\<^sup>2 / 34 +
+          (177 * (fst x * (snd xa * (snd x)\<^sup>2)) +
+           fst xa * snd x ^ 3 * 59) /
+          3481"
+
+lemma p1_has_derivative:
+  shows "((\<lambda>x. p1 (fst x) (snd x)) has_derivative p1d x) (at x)"
+  unfolding p1_def p1d_def
+  by (auto intro!: continuous_intros derivative_eq_intros)
+
+(* p1 excludes equilibria for free *)
+lemma p1_not_equil:
+  shows " p1 x y \<le> 0 \<Longrightarrow> gx x y \<noteq> 0 \<or> gy x y \<noteq> 0"
+  unfolding gx_def gy_def p1_def
+  by (sos "()")
+
+definition "trapG = trapG1 \<inter> {p. p1 (fst p) (snd p) \<le> 0}"
+
+text \<open>Real-Arithmetic\<close>
+definition "g_arith a b = (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b - p1d (a, b) (gx a b, gy a b)"
+
+schematic_goal g_arith_fas:
+  "[g_arith (X!0) (X!1)] = interpret_floatariths ?fas X"
+  unfolding g_arith_def p1_def p1d_def gx_def gy_def fst_conv snd_conv
+  by (reify_floatariths)
+
+concrete_definition g_arith_fas uses g_arith_fas
+
+lemma list_interval2: "list_interval [a, b] [c, d] = {[x, y] | x y. x \<in> {a .. c} \<and> y \<in> {b .. d}}"
+  apply (auto simp: list_interval_def)
+  subgoal for x
+    apply (cases x)
+    apply auto
+    subgoal for y zs
+      apply (cases zs)
+      by auto
+    done
+  done
+
+lemma g_arith_nonneg: "g_arith a b \<ge> 0"
+  if a: "0 \<le> a" "a \<le> 8.24" and b: "0 \<le> b" "b \<le> 7.51"
+proof -
+  have "prove_nonneg [(0, 1, ''0x000000'')] 1000000 30 (slp_of_fas [hd g_arith_fas]) [aforms_of_ivls [0, 0]
+    [float_divr 30 824 100, float_divr 30 751 100]]"
+    by eval\<comment> \<open>slow: 60s\<close>
+  from prove_nonneg[OF this]
+  have "0 \<le> interpret_floatarith (hd g_arith_fas) [a, b]"
+    apply (auto simp: g_arith_fas)
+    apply (subst (asm) Joints_aforms_of_ivls)
+     apply (auto )
+      apply (smt divide_nonneg_nonneg float_divr float_numeral rel_simps(27))
+     apply (smt divide_nonneg_nonneg float_divr float_numeral rel_simps(27))
+    apply (subst (asm) list_interval2)
+    apply auto
+    apply (drule spec[where x="[a, b]"])
+    using a b
+    apply auto
+    subgoal by (rule order_trans[OF _ float_divr]) simp
+    subgoal by (rule order_trans[OF _ float_divr]) simp
+    done
+  also have "\<dots> = g_arith a b"
+    by (auto simp: g_arith_fas_def g_arith_def p1_def p1d_def gx_def gy_def)
+  finally show ?thesis .
+qed
+
+lemma trap_arithmetic:
+  "p1d (a, b) (gx a b, gy a b) \<le> (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b" if "(a, b) \<in> trapG1"
+proof -
+  from that
+  have b: "0 \<le> b" "b \<le> 7.51"
+    and a: "0 \<le> a" "a \<le> 8.24"
+    by (auto simp: trapG1_def pos_quad_def)
+  from g_arith_nonneg[OF a b] show ?thesis
+    by (simp add: g_arith_def)
+qed
+
+lemma positively_invariant_trapG:
+  shows "g.positively_invariant trapG"
+  unfolding trapG_def
+  apply (rule g.positively_invariant_le_domain[OF positively_invariant_trapG1 _ p1_has_derivative,
+        of "\<lambda>p. -1.08 - (fst p)^2 + 2 * fst p * snd p"])
+  subgoal by (auto intro!: continuous_intros derivative_eq_intros simp add: pos_quad_def)
+  apply auto
+  by (rule trap_arithmetic)
+
+lemma regular_trapG:
+  shows "0 \<notin> (\<lambda>(x, y). (gx x y, gy x y)) ` trapG"
+  unfolding trapG_def apply auto using p1_not_equil
+  by force
+
+lemma arith:
+  "\<And>a b::real. 0 \<le> b \<Longrightarrow>
+           0 \<le> a \<Longrightarrow>
+           b * 100 \<le> 751 \<Longrightarrow>
+           a * 25 + b * 25 \<le> 203 \<Longrightarrow> norm a + norm b \<le> 20"
+  by auto
+
+lemma trapG1_subset:
+  shows "trapG1 \<subseteq> cball (0::real \<times> real) 20"
+  unfolding trapG1_def pos_quad_def
+  apply auto
+  using arith norm_Pair_le
+  by smt
+
+lemma compact_subset_closed:
+  assumes "compact S" "closed T"
+  assumes "T \<subseteq> S"
+  shows "compact T"
+  using compact_Int_closed[OF assms(1-2)] assms(3)
+  by (simp add: inf_absorb2)
+
+lemma compact_trapG1:
+  shows "compact trapG1"
+  apply (auto intro!: compact_subset_closed[OF _ _ trapG1_subset])
+  unfolding trapG1_def pos_quad_def
+  by (auto intro!: closed_Collect_le continuous_intros)
+
+lemma compact_trapG:
+  shows "compact trapG"
+  unfolding trapG_def
+  by (auto intro!: compact_Int_closed compact_trapG1 closed_Collect_le continuous_intros simp add: p1_def)
+
+lemma x_in_trapG:
+  shows "(1,0) \<in> trapG"
+  unfolding trapG_def trapG1_def pos_quad_def p1_def
+  by (auto simp add: dist_Pair_Pair)
+
+schematic_goal g_fas:
+  "[- (- (X!0) + 8 / 100 * (X!1) + (X!0)^2 * (X!1)),-( 6 / 10 - 8 / 100 * (X!1) - (X!0)^2 * (X!1))] = interpret_floatariths ?fas X"
+  by (reify_floatariths)
+
+concrete_definition g_fas uses g_fas
+
+interpretation grev: ode_interpretation true_form UNIV g_fas
+  "-(\<lambda>(x, y). (gx x y, gy x y)::real*real)"
+  "d::2" for d
+  by unfold_locales (auto simp: g_fas_def less_Suc_eq_0_disj nth_Basis_list_prod Basis_list_real_def
+      gx_def gy_def eval_nat_numeral
+      mk_ode_ops_def eucl_of_list_prod power2_eq_square intro!: isFDERIV_I)
+
+lemma grev: "t \<in> {1/8 .. 1/8} \<longrightarrow> (x, y) \<in> {(1, 0) .. (1, 0)} \<longrightarrow>
+   t \<in> g.rev.existence_ivl0 (x, y) \<and> g.rev.flow0 (x, y) t \<in>
+    {(1.1, -0.09) .. (1.2, -0.08)}"
+  by (tactic \<open>ode_bnds_tac @{thms g_fas_def} 30 20 7 12 [(0, 1, "0x000000")] (* "grev.out" *) "" @{context} 1\<close>)
+
+theorem g_has_limit_cycle:
+  obtains y where "g.limit_cycle y" "range (g.flow0 y) \<subseteq> trapG"
+proof -
+  define E::"(real*real) set" where "E = {(1.1, -0.09) .. (1.2, -0.08)}"
+  from grev have "g.rev.flow0 (1, 0) (1/8) \<in> E"
+    by (auto simp: E_def)
+  moreover
+  have "E \<inter> trapG = {}"
+    by (auto simp: trapG_def E_def trapG1_def pos_quad_def)
+  ultimately have "g.rev.flow0 (1, 0) (1 / 8) \<notin> trapG" by blast
+  from g.poincare_bendixson_limit_cycle[OF compact_trapG subset_UNIV x_in_trapG positively_invariant_trapG regular_trapG this] that
+  show ?thesis by blast
+qed
+
+end
+
 end
\ No newline at end of file
diff --git a/thys/Poincare_Bendixson/Invariance.thy b/thys/Poincare_Bendixson/Invariance.thy
--- a/thys/Poincare_Bendixson/Invariance.thy
+++ b/thys/Poincare_Bendixson/Invariance.thy
@@ -1,299 +1,299 @@
-section \<open>Invariance\<close>
-theory Invariance
-  imports ODE_Misc
-begin
-
-context auto_ll_on_open begin
-
-definition "invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped x M)"
-
-definition "positively_invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped_forward x M)"
-
-definition "negatively_invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped_backward x M)"
-
-lemma positively_invariant_iff:
-  "positively_invariant M \<longleftrightarrow>
-  (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {0..})) \<subseteq> M"
-  unfolding positively_invariant_def trapped_forward_def
-  by auto
-
-lemma negatively_invariant_iff:
-  "negatively_invariant M \<longleftrightarrow>
-  (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {..0})) \<subseteq> M"
-  unfolding negatively_invariant_def trapped_backward_def
-  by auto
-
-lemma invariant_iff_pos_and_neg_invariant:
-  "invariant M \<longleftrightarrow> positively_invariant M \<and> negatively_invariant M"
-  unfolding invariant_def trapped_def positively_invariant_def negatively_invariant_def
-  by blast
-
-lemma invariant_iff:
-  "invariant M \<longleftrightarrow> (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x)) \<subseteq>  M"
-  unfolding invariant_iff_pos_and_neg_invariant positively_invariant_iff negatively_invariant_iff
-  by (metis (mono_tags) SUP_le_iff invariant_def invariant_iff_pos_and_neg_invariant negatively_invariant_iff positively_invariant_iff trapped_iff_on_existence_ivl0)
-
-lemma positively_invariant_restrict_dom: "positively_invariant M = positively_invariant (M \<inter> X)"
-  unfolding positively_invariant_def trapped_forward_def
-  by (auto intro!: flow_in_domain dest: mem_existence_ivl_iv_defined)
-
-lemma negatively_invariant_restrict_dom: "negatively_invariant M = negatively_invariant (M \<inter> X)"
-  unfolding negatively_invariant_def trapped_backward_def
-  by (auto intro!: flow_in_domain dest: mem_existence_ivl_iv_defined)
-
-lemma invariant_restrict_dom: "invariant M = invariant (M \<inter> X)"
-  using invariant_iff_pos_and_neg_invariant
-    negatively_invariant_restrict_dom
-    positively_invariant_restrict_dom by auto
-    (*
-lemma positively_invariant_imp_subset:
-  "M \<subseteq> X" if "positively_invariant M"
-  using positively_invariant_iff that by blast
-
-lemma negatively_invariant_imp_subset:
-  "M \<subseteq> X" if "negatively_invariant M"
-  using negatively_invariant_iff that by blast
-
-lemma invariant_imp_subset:
-  "M \<subseteq> X" if "invariant M"
-  using invariant_iff that by blast
-*)
-
-end context auto_ll_on_open begin
-
-lemma positively_invariant_eq_rev: "positively_invariant M = rev.negatively_invariant M"
-  unfolding positively_invariant_def rev.negatively_invariant_def
-  by (simp add: rev.trapped_backward_iff_rev_trapped_forward)
-
-lemma negatively_invariant_eq_rev: "negatively_invariant M = rev.positively_invariant M"
-  unfolding negatively_invariant_def rev.positively_invariant_def
-  by (simp add: trapped_backward_iff_rev_trapped_forward)
-
-lemma invariant_eq_rev: "invariant M = rev.invariant M"
-  unfolding invariant_iff_pos_and_neg_invariant rev.invariant_iff_pos_and_neg_invariant
-    positively_invariant_eq_rev negatively_invariant_eq_rev by auto
-
-lemma negatively_invariant_complI: "negatively_invariant (X-M)" if "positively_invariant M"
-  unfolding negatively_invariant_def trapped_backward_def
-proof clarsimp
-  fix x t
-  assume x: "x \<in> X" "x \<notin> M" "t \<in> existence_ivl0 x" "t \<le> 0"
-  have a1:"flow0 x t \<in> X" using x
-    using flow_in_domain by blast
-  have a2:"flow0 x t \<notin> M"
-  proof (rule ccontr)
-    assume "\<not> flow0 x t \<notin> M"
-    then have "trapped_forward (flow0 x t) M"
-      using positively_invariant_def that by auto
-    moreover have "flow0 (flow0 x t) (-t) = x"
-      using \<open>t \<in> existence_ivl0 x\<close> flows_reverse by auto
-    moreover have "-t \<in> existence_ivl0 (flow0 x t) \<inter> {0..}"
-      using existence_ivl_reverse x(3) x(4) by auto
-    ultimately have "x \<in> M" unfolding trapped_forward_def
-      by (metis image_subset_iff)
-    thus False using x(2) by auto
-  qed
-  show "flow0 x t \<in> X \<and> flow0 x t \<notin> M" using a1 a2 by auto
-qed
-
-end context auto_ll_on_open begin
-
-lemma negatively_invariant_complD: "positively_invariant M" if "negatively_invariant (X-M)"
-proof -
-  have "rev.positively_invariant (X-M)" using that
-    by (simp add: negatively_invariant_eq_rev)
-  then have "rev.negatively_invariant (X-(X-M))"
-    by (simp add: rev.negatively_invariant_complI)
-  then have "positively_invariant (X-(X-M))"
-    using rev.negatively_invariant_eq_rev by auto
-  thus ?thesis using Diff_Diff_Int
-    by (metis inf_commute positively_invariant_restrict_dom) 
-qed
-
-lemma pos_invariant_iff_compl_neg_invariant: "positively_invariant M \<longleftrightarrow> negatively_invariant (X - M)"
-  by (safe intro!: negatively_invariant_complI dest!: negatively_invariant_complD)
-
-lemma neg_invariant_iff_compl_pos_invariant:
-  shows "negatively_invariant M \<longleftrightarrow> positively_invariant (X - M)"
-  by (simp add: auto_ll_on_open.pos_invariant_iff_compl_neg_invariant negatively_invariant_eq_rev positively_invariant_eq_rev rev.auto_ll_on_open_axioms)
-
-lemma invariant_iff_compl_invariant:
-  shows "invariant M \<longleftrightarrow> invariant (X - M)"
-  using invariant_iff_pos_and_neg_invariant neg_invariant_iff_compl_pos_invariant pos_invariant_iff_compl_neg_invariant by blast
-
-lemma invariant_iff_pos_invariant_and_compl_pos_invariant:
-  shows "invariant M \<longleftrightarrow> positively_invariant M \<and> positively_invariant (X-M)"
-  by (simp add: invariant_iff_pos_and_neg_invariant neg_invariant_iff_compl_pos_invariant)
-
-end
-
-subsection \<open>Tools for proving invariance\<close>
-
-context auto_ll_on_open begin
-
-lemma positively_invariant_left_inter:
-  assumes "positively_invariant C"
-  assumes "\<forall>x \<in> C \<inter> D. trapped_forward x D"
-  shows "positively_invariant (C \<inter> D)"
-  using assms positively_invariant_def trapped_forward_def by auto
-
-lemma trapped_forward_le:
-  fixes V :: "'a \<Rightarrow> real"
-  assumes "V x \<le> 0"
-  assumes contg: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"
-  assumes "\<And>x. (V has_derivative V' x) (at x)"
-  assumes "\<And>s. s \<in> existence_ivl0 x \<inter> {0..} \<Longrightarrow> V' (flow0 x s) (f (flow0 x s)) \<le> g (flow0 x s) * V (flow0 x s)"
-  shows "trapped_forward x {x. V x \<le> 0}"
-  unfolding trapped_forward_def
-proof clarsimp
-  fix t
-  assume t: "t \<in> existence_ivl0 x" "0 \<le> t"
-  then have ex:"{0..t} \<subseteq> existence_ivl0 x"
-    by (simp add: local.ivl_subset_existence_ivl)
-  have contV: "continuous_on UNIV V"
-    using assms(3) has_derivative_continuous_on by blast
-  have 1: "continuous_on {0..t} (g \<circ> flow0 x)"
-    apply (rule continuous_on_compose)
-    using continuous_on_subset ex local.flow_continuous_on apply blast
-    by (meson Int_subset_iff atLeastAtMost_iff atLeast_iff contg continuous_on_subset ex image_mono subsetI)
-  have 2: "(\<And>s. s \<in> {0..t} \<Longrightarrow>
-         (V \<circ> flow0 x has_real_derivative (V' (flow0 x s) \<circ> f \<circ> flow0 x) s) (at s))"
-    apply (auto simp add:o_def has_field_derivative_def)
-  proof -                              
-    fix s
-    assume "0 \<le> s" "s \<le> t"
-    then have "s \<in> existence_ivl0 x" using ex by auto
-    from flow_has_derivative[OF this] have
-      "(flow0 x has_derivative (\<lambda>i. i *\<^sub>R f (flow0 x s))) (at s)" .
-    from has_derivative_compose[OF this assms(3)]
-    have "((\<lambda>t. V (flow0 x t)) has_derivative (\<lambda>t. V' (flow0 x s)  (t *\<^sub>R f (flow0 x s)))) (at s)" .
-    moreover have "linear (V' (flow0 x s))"  using assms(3) has_derivative_linear by blast
-    ultimately 
-    have "((\<lambda>t. V (flow0 x t)) has_derivative (\<lambda>t. t *\<^sub>R V' (flow0 x s) (f (flow0 x s)))) (at s)" 
-      unfolding linear_cmul[OF \<open>linear (V' (flow0 x s))\<close>] by blast
-    thus "((\<lambda>t. V (flow0 x t)) has_derivative (*) (V' (flow0 x s) (f (flow0 x s)))) (at s)"
-      by (auto intro!: derivative_eq_intros simp add: mult_commute_abs)
-  qed
-  have 3: "(\<And>s. s \<in> {0..t} \<Longrightarrow>
-         (V' (flow0 x s)  \<circ> f \<circ> flow0 x) s \<le> (g \<circ> flow0 x) s *\<^sub>R (V \<circ> flow0 x) s)"
-    using ex by (auto intro!:assms(4))
-  from comparison_principle_le_linear[OF 1 2 _ 3] assms(1)
-  have "\<forall>s \<in> {0..t}. (V \<circ> flow0 x) s \<le> 0"
-    using local.mem_existence_ivl_iv_defined(2) t(1) by auto
-  thus " V (flow0 x t) \<le> 0"
-    by (simp add: t(2))
-qed
-
-lemma positively_invariant_le_domain:
-  fixes V :: "'a \<Rightarrow> real"
-  assumes "positively_invariant D"
-  assumes contg: "continuous_on D g"
-  assumes "\<And>x. (V has_derivative V' x) (at x)" (* TODO: domain can be added here too ? *)
-  assumes "\<And>s. s \<in> D \<Longrightarrow> V' s (f s) \<le> g s * V s"
-  shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
-  apply (auto intro!:positively_invariant_left_inter[OF assms(1)])
-proof -
-  fix x
-  assume "x \<in> D" "V x \<le> 0"
-  have "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"
-    by (meson \<open>x \<in> D\<close> assms(1) contg continuous_on_subset positively_invariant_def trapped_forward_def)
-  from trapped_forward_le[OF \<open>V x \<le> 0\<close> this assms(3)]
-  show "trapped_forward x {x. V x \<le> 0}" using assms(4)
-    using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
-qed
-
-lemma positively_invariant_barrier_domain:
-  fixes V :: "'a \<Rightarrow> real"
-  assumes "positively_invariant D"
-  assumes "\<And>x. (V has_derivative V' x) (at x)"
-  assumes "continuous_on D (\<lambda>x. V' x (f x))"
-  assumes "\<And>s. s \<in> D \<Longrightarrow> V s = 0 \<Longrightarrow> V' s (f s) < 0"
-  shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
-  apply (auto intro!:positively_invariant_left_inter[OF assms(1)])
-proof -
-  fix x
-  assume "x \<in> D" "V x \<le> 0"
-  have contV: "continuous_on UNIV V" using assms(2) has_derivative_continuous_on by blast
-  then have *: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) V"
-    using continuous_on_subset by blast
-  have sub: "flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> D"
-    using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
-  then have contV': "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) (\<lambda>x. V' x (f x))"
-    by (metis assms(3) continuous_on_subset)
-  have nz: "\<And>i t. t \<in> existence_ivl0 x \<Longrightarrow>
-       0 \<le> t \<Longrightarrow>  max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0"
-  proof -
-    fix i t
-    assume "t \<in> existence_ivl0 x" "0 \<le> t"
-    then have "flow0 x t \<in> D"
-      using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
-    then have "V (flow0 x t) = 0 \<Longrightarrow> - V' (flow0 x t) (f (flow0 x t)) > 0" using assms(4) by simp
-    then have "(V (flow0 x t))^2 > 0 \<or> - V' (flow0 x t) (f (flow0 x t)) > 0" by simp
-    thus "max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0" unfolding less_max_iff_disj
-      by auto
-  qed
-  have *: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) (\<lambda>x. V' x (f x) * V x / max (- V' x (f x)) ((V x)^2))"
-    apply (auto intro!:continuous_intros continuous_on_max simp add: * contV')
-    using nz by fastforce
-  have "(\<And>t. t \<in> existence_ivl0 x \<inter> {0..} \<Longrightarrow>
-        V' (flow0 x t) (f (flow0 x t)) \<le>
-        (V' (flow0 x t) (f (flow0 x t)) * V (flow0 x t)
-        / max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)) * V (flow0 x t))"
-  proof clarsimp
-    fix t
-    assume "t \<in> existence_ivl0 x" "0 \<le> t"
-    then have p: "max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0" using nz by auto
-    have " V' (flow0 x t) (f (flow0 x t)) * max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)
-      \<le>  V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t))\<^sup>2"
-      by (smt mult_minus_left mult_minus_right power2_eq_square real_mult_le_cancel_iff2)
-    then have "V' (flow0 x t) (f (flow0 x t))
-      \<le>  V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t))\<^sup>2
-      / max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)"
-      using p pos_le_divide_eq by blast
-    thus " V' (flow0 x t) (f (flow0 x t))
-         \<le> V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t)) * V (flow0 x t) /
-           max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)"
-      by (simp add: power2_eq_square)
-  qed
-  from trapped_forward_le[OF \<open>V x \<le> 0\<close> * assms(2) this]
-  show "trapped_forward x {x. V x \<le> 0}" by auto
-qed
-
-lemma positively_invariant_UNIV:
-  shows "positively_invariant UNIV"
-  using positively_invariant_iff by blast
-
-lemma positively_invariant_conj:
-  assumes "positively_invariant C"
-  assumes "positively_invariant D"
-  shows "positively_invariant (C \<inter> D)"
-  using assms positively_invariant_def
-  using positively_invariant_left_inter by auto
-
-lemma positively_invariant_le:
-  fixes V :: "'a \<Rightarrow> real"
-  assumes contg: "continuous_on UNIV g"
-  assumes "\<And>x. (V has_derivative V' x) (at x)"
-  assumes "\<And>s. V' s (f s) \<le> g s * V s"
-  shows "positively_invariant {x. V x \<le> 0}"
-proof -
-  from positively_invariant_le_domain[OF positively_invariant_UNIV assms]  
-  have "positively_invariant (UNIV \<inter> {x. V x \<le> 0})" .
-  thus ?thesis by auto
-qed
-
-lemma positively_invariant_barrier:
-  fixes V :: "'a \<Rightarrow> real"
-  assumes "\<And>x. (V has_derivative V' x) (at x)"
-  assumes "continuous_on UNIV (\<lambda>x. V' x (f x))"
-  assumes "\<And>s. V s = 0 \<Longrightarrow> V' s (f s) < 0"
-  shows "positively_invariant {x. V x \<le> 0}"
-proof -
-  from positively_invariant_barrier_domain[OF positively_invariant_UNIV assms]  
-  have "positively_invariant (UNIV \<inter> {x. V x \<le> 0})" .
-  thus ?thesis by auto
-qed
-
-end
-
+section \<open>Invariance\<close>
+theory Invariance
+  imports ODE_Misc
+begin
+
+context auto_ll_on_open begin
+
+definition "invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped x M)"
+
+definition "positively_invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped_forward x M)"
+
+definition "negatively_invariant M \<longleftrightarrow> (\<forall>x\<in>M. trapped_backward x M)"
+
+lemma positively_invariant_iff:
+  "positively_invariant M \<longleftrightarrow>
+  (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {0..})) \<subseteq> M"
+  unfolding positively_invariant_def trapped_forward_def
+  by auto
+
+lemma negatively_invariant_iff:
+  "negatively_invariant M \<longleftrightarrow>
+  (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {..0})) \<subseteq> M"
+  unfolding negatively_invariant_def trapped_backward_def
+  by auto
+
+lemma invariant_iff_pos_and_neg_invariant:
+  "invariant M \<longleftrightarrow> positively_invariant M \<and> negatively_invariant M"
+  unfolding invariant_def trapped_def positively_invariant_def negatively_invariant_def
+  by blast
+
+lemma invariant_iff:
+  "invariant M \<longleftrightarrow> (\<Union>x\<in>M. flow0 x ` (existence_ivl0 x)) \<subseteq>  M"
+  unfolding invariant_iff_pos_and_neg_invariant positively_invariant_iff negatively_invariant_iff
+  by (metis (mono_tags) SUP_le_iff invariant_def invariant_iff_pos_and_neg_invariant negatively_invariant_iff positively_invariant_iff trapped_iff_on_existence_ivl0)
+
+lemma positively_invariant_restrict_dom: "positively_invariant M = positively_invariant (M \<inter> X)"
+  unfolding positively_invariant_def trapped_forward_def
+  by (auto intro!: flow_in_domain dest: mem_existence_ivl_iv_defined)
+
+lemma negatively_invariant_restrict_dom: "negatively_invariant M = negatively_invariant (M \<inter> X)"
+  unfolding negatively_invariant_def trapped_backward_def
+  by (auto intro!: flow_in_domain dest: mem_existence_ivl_iv_defined)
+
+lemma invariant_restrict_dom: "invariant M = invariant (M \<inter> X)"
+  using invariant_iff_pos_and_neg_invariant
+    negatively_invariant_restrict_dom
+    positively_invariant_restrict_dom by auto
+    (*
+lemma positively_invariant_imp_subset:
+  "M \<subseteq> X" if "positively_invariant M"
+  using positively_invariant_iff that by blast
+
+lemma negatively_invariant_imp_subset:
+  "M \<subseteq> X" if "negatively_invariant M"
+  using negatively_invariant_iff that by blast
+
+lemma invariant_imp_subset:
+  "M \<subseteq> X" if "invariant M"
+  using invariant_iff that by blast
+*)
+
+end context auto_ll_on_open begin
+
+lemma positively_invariant_eq_rev: "positively_invariant M = rev.negatively_invariant M"
+  unfolding positively_invariant_def rev.negatively_invariant_def
+  by (simp add: rev.trapped_backward_iff_rev_trapped_forward)
+
+lemma negatively_invariant_eq_rev: "negatively_invariant M = rev.positively_invariant M"
+  unfolding negatively_invariant_def rev.positively_invariant_def
+  by (simp add: trapped_backward_iff_rev_trapped_forward)
+
+lemma invariant_eq_rev: "invariant M = rev.invariant M"
+  unfolding invariant_iff_pos_and_neg_invariant rev.invariant_iff_pos_and_neg_invariant
+    positively_invariant_eq_rev negatively_invariant_eq_rev by auto
+
+lemma negatively_invariant_complI: "negatively_invariant (X-M)" if "positively_invariant M"
+  unfolding negatively_invariant_def trapped_backward_def
+proof clarsimp
+  fix x t
+  assume x: "x \<in> X" "x \<notin> M" "t \<in> existence_ivl0 x" "t \<le> 0"
+  have a1:"flow0 x t \<in> X" using x
+    using flow_in_domain by blast
+  have a2:"flow0 x t \<notin> M"
+  proof (rule ccontr)
+    assume "\<not> flow0 x t \<notin> M"
+    then have "trapped_forward (flow0 x t) M"
+      using positively_invariant_def that by auto
+    moreover have "flow0 (flow0 x t) (-t) = x"
+      using \<open>t \<in> existence_ivl0 x\<close> flows_reverse by auto
+    moreover have "-t \<in> existence_ivl0 (flow0 x t) \<inter> {0..}"
+      using existence_ivl_reverse x(3) x(4) by auto
+    ultimately have "x \<in> M" unfolding trapped_forward_def
+      by (metis image_subset_iff)
+    thus False using x(2) by auto
+  qed
+  show "flow0 x t \<in> X \<and> flow0 x t \<notin> M" using a1 a2 by auto
+qed
+
+end context auto_ll_on_open begin
+
+lemma negatively_invariant_complD: "positively_invariant M" if "negatively_invariant (X-M)"
+proof -
+  have "rev.positively_invariant (X-M)" using that
+    by (simp add: negatively_invariant_eq_rev)
+  then have "rev.negatively_invariant (X-(X-M))"
+    by (simp add: rev.negatively_invariant_complI)
+  then have "positively_invariant (X-(X-M))"
+    using rev.negatively_invariant_eq_rev by auto
+  thus ?thesis using Diff_Diff_Int
+    by (metis inf_commute positively_invariant_restrict_dom) 
+qed
+
+lemma pos_invariant_iff_compl_neg_invariant: "positively_invariant M \<longleftrightarrow> negatively_invariant (X - M)"
+  by (safe intro!: negatively_invariant_complI dest!: negatively_invariant_complD)
+
+lemma neg_invariant_iff_compl_pos_invariant:
+  shows "negatively_invariant M \<longleftrightarrow> positively_invariant (X - M)"
+  by (simp add: auto_ll_on_open.pos_invariant_iff_compl_neg_invariant negatively_invariant_eq_rev positively_invariant_eq_rev rev.auto_ll_on_open_axioms)
+
+lemma invariant_iff_compl_invariant:
+  shows "invariant M \<longleftrightarrow> invariant (X - M)"
+  using invariant_iff_pos_and_neg_invariant neg_invariant_iff_compl_pos_invariant pos_invariant_iff_compl_neg_invariant by blast
+
+lemma invariant_iff_pos_invariant_and_compl_pos_invariant:
+  shows "invariant M \<longleftrightarrow> positively_invariant M \<and> positively_invariant (X-M)"
+  by (simp add: invariant_iff_pos_and_neg_invariant neg_invariant_iff_compl_pos_invariant)
+
+end
+
+subsection \<open>Tools for proving invariance\<close>
+
+context auto_ll_on_open begin
+
+lemma positively_invariant_left_inter:
+  assumes "positively_invariant C"
+  assumes "\<forall>x \<in> C \<inter> D. trapped_forward x D"
+  shows "positively_invariant (C \<inter> D)"
+  using assms positively_invariant_def trapped_forward_def by auto
+
+lemma trapped_forward_le:
+  fixes V :: "'a \<Rightarrow> real"
+  assumes "V x \<le> 0"
+  assumes contg: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"
+  assumes "\<And>x. (V has_derivative V' x) (at x)"
+  assumes "\<And>s. s \<in> existence_ivl0 x \<inter> {0..} \<Longrightarrow> V' (flow0 x s) (f (flow0 x s)) \<le> g (flow0 x s) * V (flow0 x s)"
+  shows "trapped_forward x {x. V x \<le> 0}"
+  unfolding trapped_forward_def
+proof clarsimp
+  fix t
+  assume t: "t \<in> existence_ivl0 x" "0 \<le> t"
+  then have ex:"{0..t} \<subseteq> existence_ivl0 x"
+    by (simp add: local.ivl_subset_existence_ivl)
+  have contV: "continuous_on UNIV V"
+    using assms(3) has_derivative_continuous_on by blast
+  have 1: "continuous_on {0..t} (g \<circ> flow0 x)"
+    apply (rule continuous_on_compose)
+    using continuous_on_subset ex local.flow_continuous_on apply blast
+    by (meson Int_subset_iff atLeastAtMost_iff atLeast_iff contg continuous_on_subset ex image_mono subsetI)
+  have 2: "(\<And>s. s \<in> {0..t} \<Longrightarrow>
+         (V \<circ> flow0 x has_real_derivative (V' (flow0 x s) \<circ> f \<circ> flow0 x) s) (at s))"
+    apply (auto simp add:o_def has_field_derivative_def)
+  proof -                              
+    fix s
+    assume "0 \<le> s" "s \<le> t"
+    then have "s \<in> existence_ivl0 x" using ex by auto
+    from flow_has_derivative[OF this] have
+      "(flow0 x has_derivative (\<lambda>i. i *\<^sub>R f (flow0 x s))) (at s)" .
+    from has_derivative_compose[OF this assms(3)]
+    have "((\<lambda>t. V (flow0 x t)) has_derivative (\<lambda>t. V' (flow0 x s)  (t *\<^sub>R f (flow0 x s)))) (at s)" .
+    moreover have "linear (V' (flow0 x s))"  using assms(3) has_derivative_linear by blast
+    ultimately 
+    have "((\<lambda>t. V (flow0 x t)) has_derivative (\<lambda>t. t *\<^sub>R V' (flow0 x s) (f (flow0 x s)))) (at s)" 
+      unfolding linear_cmul[OF \<open>linear (V' (flow0 x s))\<close>] by blast
+    thus "((\<lambda>t. V (flow0 x t)) has_derivative (*) (V' (flow0 x s) (f (flow0 x s)))) (at s)"
+      by (auto intro!: derivative_eq_intros simp add: mult_commute_abs)
+  qed
+  have 3: "(\<And>s. s \<in> {0..t} \<Longrightarrow>
+         (V' (flow0 x s)  \<circ> f \<circ> flow0 x) s \<le> (g \<circ> flow0 x) s *\<^sub>R (V \<circ> flow0 x) s)"
+    using ex by (auto intro!:assms(4))
+  from comparison_principle_le_linear[OF 1 2 _ 3] assms(1)
+  have "\<forall>s \<in> {0..t}. (V \<circ> flow0 x) s \<le> 0"
+    using local.mem_existence_ivl_iv_defined(2) t(1) by auto
+  thus " V (flow0 x t) \<le> 0"
+    by (simp add: t(2))
+qed
+
+lemma positively_invariant_le_domain:
+  fixes V :: "'a \<Rightarrow> real"
+  assumes "positively_invariant D"
+  assumes contg: "continuous_on D g"
+  assumes "\<And>x. (V has_derivative V' x) (at x)" (* TODO: domain can be added here too ? *)
+  assumes "\<And>s. s \<in> D \<Longrightarrow> V' s (f s) \<le> g s * V s"
+  shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
+  apply (auto intro!:positively_invariant_left_inter[OF assms(1)])
+proof -
+  fix x
+  assume "x \<in> D" "V x \<le> 0"
+  have "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) g"
+    by (meson \<open>x \<in> D\<close> assms(1) contg continuous_on_subset positively_invariant_def trapped_forward_def)
+  from trapped_forward_le[OF \<open>V x \<le> 0\<close> this assms(3)]
+  show "trapped_forward x {x. V x \<le> 0}" using assms(4)
+    using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
+qed
+
+lemma positively_invariant_barrier_domain:
+  fixes V :: "'a \<Rightarrow> real"
+  assumes "positively_invariant D"
+  assumes "\<And>x. (V has_derivative V' x) (at x)"
+  assumes "continuous_on D (\<lambda>x. V' x (f x))"
+  assumes "\<And>s. s \<in> D \<Longrightarrow> V s = 0 \<Longrightarrow> V' s (f s) < 0"
+  shows "positively_invariant (D \<inter> {x. V x \<le> 0})"
+  apply (auto intro!:positively_invariant_left_inter[OF assms(1)])
+proof -
+  fix x
+  assume "x \<in> D" "V x \<le> 0"
+  have contV: "continuous_on UNIV V" using assms(2) has_derivative_continuous_on by blast
+  then have *: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) V"
+    using continuous_on_subset by blast
+  have sub: "flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> D"
+    using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
+  then have contV': "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) (\<lambda>x. V' x (f x))"
+    by (metis assms(3) continuous_on_subset)
+  have nz: "\<And>i t. t \<in> existence_ivl0 x \<Longrightarrow>
+       0 \<le> t \<Longrightarrow>  max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0"
+  proof -
+    fix i t
+    assume "t \<in> existence_ivl0 x" "0 \<le> t"
+    then have "flow0 x t \<in> D"
+      using \<open>x \<in> D\<close> assms(1) positively_invariant_def trapped_forward_def by auto
+    then have "V (flow0 x t) = 0 \<Longrightarrow> - V' (flow0 x t) (f (flow0 x t)) > 0" using assms(4) by simp
+    then have "(V (flow0 x t))^2 > 0 \<or> - V' (flow0 x t) (f (flow0 x t)) > 0" by simp
+    thus "max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0" unfolding less_max_iff_disj
+      by auto
+  qed
+  have *: "continuous_on (flow0 x ` (existence_ivl0 x \<inter> {0..})) (\<lambda>x. V' x (f x) * V x / max (- V' x (f x)) ((V x)^2))"
+    apply (auto intro!:continuous_intros continuous_on_max simp add: * contV')
+    using nz by fastforce
+  have "(\<And>t. t \<in> existence_ivl0 x \<inter> {0..} \<Longrightarrow>
+        V' (flow0 x t) (f (flow0 x t)) \<le>
+        (V' (flow0 x t) (f (flow0 x t)) * V (flow0 x t)
+        / max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)) * V (flow0 x t))"
+  proof clarsimp
+    fix t
+    assume "t \<in> existence_ivl0 x" "0 \<le> t"
+    then have p: "max (-V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2) > 0" using nz by auto
+    have " V' (flow0 x t) (f (flow0 x t)) * max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)
+      \<le>  V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t))\<^sup>2"
+      by (smt mult_minus_left mult_minus_right power2_eq_square real_mult_le_cancel_iff2)
+    then have "V' (flow0 x t) (f (flow0 x t))
+      \<le>  V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t))\<^sup>2
+      / max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)"
+      using p pos_le_divide_eq by blast
+    thus " V' (flow0 x t) (f (flow0 x t))
+         \<le> V' (flow0 x t) (f (flow0 x t)) * (V (flow0 x t)) * V (flow0 x t) /
+           max (- V' (flow0 x t) (f (flow0 x t))) ((V (flow0 x t))\<^sup>2)"
+      by (simp add: power2_eq_square)
+  qed
+  from trapped_forward_le[OF \<open>V x \<le> 0\<close> * assms(2) this]
+  show "trapped_forward x {x. V x \<le> 0}" by auto
+qed
+
+lemma positively_invariant_UNIV:
+  shows "positively_invariant UNIV"
+  using positively_invariant_iff by blast
+
+lemma positively_invariant_conj:
+  assumes "positively_invariant C"
+  assumes "positively_invariant D"
+  shows "positively_invariant (C \<inter> D)"
+  using assms positively_invariant_def
+  using positively_invariant_left_inter by auto
+
+lemma positively_invariant_le:
+  fixes V :: "'a \<Rightarrow> real"
+  assumes contg: "continuous_on UNIV g"
+  assumes "\<And>x. (V has_derivative V' x) (at x)"
+  assumes "\<And>s. V' s (f s) \<le> g s * V s"
+  shows "positively_invariant {x. V x \<le> 0}"
+proof -
+  from positively_invariant_le_domain[OF positively_invariant_UNIV assms]  
+  have "positively_invariant (UNIV \<inter> {x. V x \<le> 0})" .
+  thus ?thesis by auto
+qed
+
+lemma positively_invariant_barrier:
+  fixes V :: "'a \<Rightarrow> real"
+  assumes "\<And>x. (V has_derivative V' x) (at x)"
+  assumes "continuous_on UNIV (\<lambda>x. V' x (f x))"
+  assumes "\<And>s. V s = 0 \<Longrightarrow> V' s (f s) < 0"
+  shows "positively_invariant {x. V x \<le> 0}"
+proof -
+  from positively_invariant_barrier_domain[OF positively_invariant_UNIV assms]  
+  have "positively_invariant (UNIV \<inter> {x. V x \<le> 0})" .
+  thus ?thesis by auto
+qed
+
+end
+
 end
\ No newline at end of file
diff --git a/thys/Poincare_Bendixson/Limit_Set.thy b/thys/Poincare_Bendixson/Limit_Set.thy
--- a/thys/Poincare_Bendixson/Limit_Set.thy
+++ b/thys/Poincare_Bendixson/Limit_Set.thy
@@ -1,473 +1,473 @@
-section \<open>Limit Sets\<close>
-theory Limit_Set
-  imports Invariance
-begin
-
-context auto_ll_on_open begin
-text \<open>Positive limit point, assuming \<open>{0..} \<subseteq> existence_ivl0 x\<close>\<close>
-
-(* TODO: Perhaps a more intrinsic definition would be better here *)
-definition "\<omega>_limit_point x p \<longleftrightarrow>
-  {0..} \<subseteq> existence_ivl0 x \<and>
-  (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p)"
-
-text \<open> Also called the \<open>\<omega>\<close>-limit set of x \<close>
-definition "\<omega>_limit_set x = {p. \<omega>_limit_point x p}"
-
-definition "\<alpha>_limit_point x p \<longleftrightarrow>
-  {..0} \<subseteq> existence_ivl0 x \<and>
-  (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p)"
-
-text \<open> Also called the \<open>\<alpha>\<close>-limit set of x \<close>
-definition "\<alpha>_limit_set x =
-  {p. \<alpha>_limit_point x p}"
-
-end context auto_ll_on_open begin
-
-lemma \<alpha>_limit_point_eq_rev: "\<alpha>_limit_point x p = rev.\<omega>_limit_point x p"
-  unfolding \<alpha>_limit_point_def rev.\<omega>_limit_point_def
-  apply (auto simp: rev_eq_flow[abs_def] o_def filterlim_uminus_at_bot rev_existence_ivl_eq0 subset_iff
-      intro: exI[where x="uminus o s" for s])
-  using neg_0_le_iff_le by fastforce
-
-lemma \<alpha>_limit_set_eq_rev: "\<alpha>_limit_set x = rev.\<omega>_limit_set x"
-  unfolding \<alpha>_limit_set_def rev.\<omega>_limit_set_def \<alpha>_limit_point_eq_rev ..
-
-lemma \<omega>_limit_pointE:
-  assumes "\<omega>_limit_point x p"
-  obtains s where
-    "filterlim s at_top sequentially"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>n. b \<le> s n"
-  using assms filterlim_at_top_choose_lower \<omega>_limit_point_def by blast
-
-lemma \<omega>_limit_set_eq:
-  assumes "{0..} \<subseteq> existence_ivl0 x"
-  shows "\<omega>_limit_set x = (INF \<tau> \<in> {0..}. closure (flow0 x ` {\<tau>..}))"
-  unfolding \<omega>_limit_set_def
-proof safe
-  fix p t
-  assume pt: "0 \<le> (t::real)" "\<omega>_limit_point x p"
-  from \<omega>_limit_pointE[OF pt(2)]
-  obtain s where
-    "filterlim s at_top sequentially"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>n. t \<le> s n" by blast
-  thus "p \<in> closure (flow0 x ` {t..})" unfolding closure_sequential
-    by (metis atLeast_iff comp_apply imageI)
-next
-  fix p
-  assume "p \<in> (\<Inter>\<tau>\<in>{0..}. closure (flow0 x ` {\<tau>..}))"
-  then have "\<And>t. t \<ge>0 \<Longrightarrow> p \<in> closure (flow0 x ` {t..})" by blast
-  then have "\<And>t e. t \<ge>0 \<Longrightarrow> e > 0 \<Longrightarrow> (\<exists>tt \<ge> t. dist (flow0 x tt) p < e)"
-    unfolding closure_approachable
-    by fastforce
-  from approachable_sequenceE[OF this]
-  obtain s where "filterlim s at_top sequentially" "(flow0 x \<circ> s) \<longlonglongrightarrow> p" by auto
-  thus "\<omega>_limit_point x p" unfolding \<omega>_limit_point_def using assms by auto
-qed
-
-lemma \<omega>_limit_set_empty:
-  assumes "\<not> ({0..} \<subseteq> existence_ivl0 x)"
-  shows "\<omega>_limit_set x = {}"
-  unfolding \<omega>_limit_set_def \<omega>_limit_point_def
-  by (simp add: assms)
-
-lemma \<omega>_limit_set_closed: "closed (\<omega>_limit_set x)"
-  using \<omega>_limit_set_eq
-  by (metis \<omega>_limit_set_empty closed_INT closed_closure closed_empty)
-
-(* TODO: Walter gives a more direct proof *)
-lemma \<omega>_limit_set_positively_invariant:
-  shows "positively_invariant (\<omega>_limit_set x)"
-  unfolding positively_invariant_def trapped_forward_def
-proof safe
-  fix dummy p t
-  assume xa: "p \<in> \<omega>_limit_set x"
-    "t \<in> existence_ivl0 p"
-    "0 \<le> t"
-  have "p \<in> X" using mem_existence_ivl_iv_defined(2) xa(2) by blast
-  have exist: "{0..} \<subseteq> existence_ivl0 x" using xa(1)
-    unfolding \<omega>_limit_set_def \<omega>_limit_point_def by auto
-  from xa(1)
-  obtain s where s:
-    "filterlim s at_top sequentially"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>n. 0 \<le> s n"
-    unfolding \<omega>_limit_set_def by (auto elim!:\<omega>_limit_pointE)
-  define r where "r = (\<lambda>n. t + s n)"
-  have rlim: "filterlim r at_top sequentially" unfolding r_def
-    by (auto intro: filterlim_tendsto_add_at_top[OF _ s(1)])
-  define dom where "dom = image (flow0 x) {0..} \<union> {p} "
-  have domin: "\<forall>n. (flow0 x \<circ> s) n \<in> dom" "p \<in> dom" unfolding dom_def o_def
-    using exist by(auto simp add: s(3))
-  have xt: "\<And>x. x \<in> dom \<Longrightarrow> t \<in> existence_ivl0 x" unfolding dom_def using xa(2)
-    apply auto
-    apply (rule existence_ivl_trans')
-    using exist xa(3) apply auto[1]
-    using exist  by auto
-  have cont: "continuous_on dom (\<lambda>x. flow0 x t)"
-    apply (rule flow_continuous_on_compose)
-       apply auto
-    using \<open>p \<in> X\<close>  exist local.dom_def local.flow_in_domain apply auto[1]
-    using xt .
-  then have f1: "((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) \<longlonglongrightarrow> flow0 p t" using domin s(2)
-    unfolding continuous_on_sequentially
-    by blast
-  have ff:"\<And>n. (flow0 x \<circ> r) n = ((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) n"
-    unfolding o_def r_def
-  proof -
-    fix n
-    have s:"s n \<in> existence_ivl0 x"
-      using s(3) exist by auto
-    then have t:"t \<in> existence_ivl0 (flow0 x (s n))"
-      using domin(1) xt by auto
-    from flow_trans[OF s t]
-    show "flow0 x (t + s n) = flow0 (flow0 x (s n)) t"
-      by (simp add: add.commute)
-  qed
-  have f2: "(flow0 x \<circ> r) \<longlonglongrightarrow> flow0 p t" using f1 unfolding ff .
-  show "flow0 p t \<in> \<omega>_limit_set x" using exist f2 rlim
-    unfolding \<omega>_limit_set_def \<omega>_limit_point_def
-    using flow_in_domain r_def s(3) xa(2) xa(3) by auto
-qed
-
-lemma \<omega>_limit_set_invariant:
-  shows "invariant (\<omega>_limit_set x)"
-  unfolding invariant_iff_pos_invariant_and_compl_pos_invariant
-proof safe
-  show "positively_invariant (\<omega>_limit_set x)"
-    using \<omega>_limit_set_positively_invariant .
-next
-  show "positively_invariant (X - \<omega>_limit_set x)"
-    unfolding positively_invariant_def trapped_forward_def
-    apply safe
-    using local.flow_in_domain apply blast
-  proof -
-    fix dummy p t 
-    assume xa: "p \<in> X" "p \<notin> \<omega>_limit_set x"
-      "t \<in> existence_ivl0 p" "0 \<le> t"
-      and f: "flow0 p t \<in> \<omega>_limit_set x"
-    have exist: "{0..} \<subseteq> existence_ivl0 x" using f
-      unfolding \<omega>_limit_set_def \<omega>_limit_point_def by auto
-    from f
-    obtain s where s:
-      "filterlim s at_top sequentially"
-      "(flow0 x \<circ> s) \<longlonglongrightarrow> flow0 p t"
-      "\<forall>n. t \<le> s n"
-      unfolding \<omega>_limit_set_def by (auto elim!:\<omega>_limit_pointE)
-    define r where "r = (\<lambda>n. (-t) + s n)"
-    have "(\<lambda>x. -t) \<longlonglongrightarrow> -t" by simp
-    from filterlim_tendsto_add_at_top[OF this s(1)]
-    have rlim: "filterlim r at_top sequentially" unfolding r_def by simp
-    define dom where "dom = image (flow0 x) {t..} \<union> {flow0 p t} "
-    have domin: "\<forall>n. (flow0 x \<circ> s) n \<in> dom" "flow0 p t \<in> dom" unfolding dom_def o_def
-      using exist by(auto simp add: s(3))
-    have xt: "\<And>x. x \<in> dom \<Longrightarrow> -t \<in> existence_ivl0 x" unfolding dom_def using xa(2)
-      apply auto
-      using local.existence_ivl_reverse xa(3) apply auto[1]
-      by (metis exist atLeast_iff diff_conv_add_uminus diff_ge_0_iff_ge linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add local.existence_ivl_trans' order_trans subset_iff xa(4))
-    have cont: "continuous_on dom (\<lambda>x. flow0 x (-t))"
-      apply (rule flow_continuous_on_compose)
-         apply auto
-      using local.mem_existence_ivl_iv_defined(2) xt apply blast
-      by (simp add: xt)
-    then have f1: "((\<lambda>x. flow0 x (-t)) \<circ> (flow0 x \<circ> s)) \<longlonglongrightarrow> flow0 (flow0 p t) (-t)" using domin s(2)
-      unfolding continuous_on_sequentially
-      by blast
-    have ff:"\<And>n. (flow0 x \<circ> r) n = ((\<lambda>x. flow0 x (-t)) \<circ> (flow0 x \<circ> s)) n"
-      unfolding o_def r_def
-    proof -
-      fix n
-      have s:"s n \<in> existence_ivl0 x"
-        using s(3) exist \<open>0\<le> t\<close> by (meson atLeast_iff order_trans subset_eq)
-      then have t:"-t \<in> existence_ivl0 (flow0 x (s n))"
-        using domin(1) xt by auto
-      from flow_trans[OF s t]
-      show "flow0 x (-t + s n) = flow0 (flow0 x (s n)) (-t)"
-        by (simp add: add.commute)
-    qed
-    have "(flow0 x \<circ> r) \<longlonglongrightarrow> flow0 (flow0 p t) (-t)" using f1 unfolding ff .
-    then have f2: "(flow0 x \<circ> r) \<longlonglongrightarrow> p" using flows_reverse xa(3) by auto
-    then have "p \<in> \<omega>_limit_set x" unfolding \<omega>_limit_set_def \<omega>_limit_point_def
-      using rlim exist by auto
-    thus False using xa(2) by auto
-  qed
-qed
-
-end context auto_ll_on_open begin
-
-lemma \<alpha>_limit_set_eq:
-  assumes "{..0} \<subseteq> existence_ivl0 x"
-  shows "\<alpha>_limit_set x = (INF \<tau> \<in> {..0}. closure (flow0 x ` {..\<tau>}))"
-  using rev.\<omega>_limit_set_eq[of x, OF assms[folded infinite_rev_existence_ivl0_rewrites]]
-  unfolding \<alpha>_limit_set_eq_rev rev_flow_image_eq image_uminus_atLeast 
-  by (smt INT_extend_simps(10) Sup.SUP_cong image_uminus_atMost)
-
-lemma \<alpha>_limit_set_closed:
-  shows "closed (\<alpha>_limit_set x)"
-  unfolding \<alpha>_limit_set_eq_rev by (rule rev.\<omega>_limit_set_closed)
-
-lemma \<alpha>_limit_set_positively_invariant:
-  shows "negatively_invariant (\<alpha>_limit_set x)"
-  unfolding negatively_invariant_eq_rev \<alpha>_limit_set_eq_rev
-  by (simp add: rev.\<omega>_limit_set_positively_invariant)
-
-lemma \<alpha>_limit_set_invariant:
-  shows "invariant (\<alpha>_limit_set x)"
-  unfolding invariant_eq_rev \<alpha>_limit_set_eq_rev
-  by (simp add: rev.\<omega>_limit_set_invariant)
-
-text \<open> Fundamental properties of the positive limit set \<close>
-
-context
-  fixes x K
-  assumes K: "compact K" "K \<subseteq> X"
-  assumes x: "x \<in> X" "trapped_forward x K"
-begin
-
-text \<open>Bunch of facts for what's to come\<close>
-private lemma props:
-  shows "{0..} \<subseteq> existence_ivl0 x" "seq_compact K"
-   apply (rule trapped_sol_right)
-  using x K by (auto simp add: compact_imp_seq_compact)
-
-private lemma flowimg:
-  shows "flow0 x ` (existence_ivl0 x \<inter> {0..}) = flow0 x ` {0..}"
-  using props(1) by auto
-
-lemma \<omega>_limit_set_in_compact_subset:
-  shows "\<omega>_limit_set x \<subseteq> K"
-  unfolding \<omega>_limit_set_def
-proof safe
-  fix p s
-  assume "\<omega>_limit_point x p"
-  from \<omega>_limit_pointE[OF this]
-  obtain s where s:
-    "filterlim s at_top sequentially"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>n. 0 \<le> s n" by blast
-  then have fin: "\<forall>n. (flow0 x \<circ> s) n \<in> K" using s(3) x K props(1)
-    unfolding trapped_forward_def
-    by (simp add: subset_eq)
-  from seq_compactE[OF props(2) fin]
-  show "p \<in> K" using s(2)
-    by (metis LIMSEQ_subseq_LIMSEQ LIMSEQ_unique)
-qed
-
-lemma \<omega>_limit_set_in_compact_compact:
-  shows "compact (\<omega>_limit_set x)"
-proof -
-  from \<omega>_limit_set_in_compact_subset 
-  have "bounded (\<omega>_limit_set x)"
-    using bounded_subset compact_imp_bounded
-    using K(1) by auto
-  thus ?thesis using \<omega>_limit_set_closed
-    by (simp add: compact_eq_bounded_closed)
-qed
-
-lemma \<omega>_limit_set_in_compact_nonempty:
-  shows "\<omega>_limit_set x \<noteq> {}"
-proof -
-  have fin: "\<forall>n. (flow0 x \<circ> real) n \<in> K" using x K props(1)
-    by (simp add: flowimg image_subset_iff trapped_forward_def)
-  from seq_compactE[OF props(2) this]
-  obtain r l where "l \<in> K" "strict_mono r" "(flow0 x \<circ> real \<circ> r) \<longlonglongrightarrow> l" by blast
-  then have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1)
-    by (smt comp_def filterlim_sequentially_iff_filterlim_real filterlim_subseq tendsto_at_top_eq_left)
-  thus ?thesis unfolding \<omega>_limit_set_def by auto
-qed
-
-lemma \<omega>_limit_set_in_compact_existence:
-  shows "\<And>y. y \<in> \<omega>_limit_set x \<Longrightarrow> existence_ivl0 y = UNIV"
-proof -
-  fix y
-  assume y: "y \<in> \<omega>_limit_set x"
-  then have "y \<in> X" using \<omega>_limit_set_in_compact_subset K by blast 
-  from \<omega>_limit_set_invariant
-  have "\<And>t. t \<in> existence_ivl0 y \<Longrightarrow> flow0 y t \<in> \<omega>_limit_set x"
-    unfolding invariant_def trapped_iff_on_existence_ivl0 using y by blast 
-  then have t: "\<And>t. t \<in> existence_ivl0 y \<Longrightarrow> flow0 y t \<in> K"
-    using \<omega>_limit_set_in_compact_subset by blast
-  thus "existence_ivl0 y = UNIV" 
-    by (meson \<open>y \<in> X\<close> existence_ivl_zero existence_ivl_initial_time_iff existence_ivl_subset mem_compact_implies_subset_existence_interval subset_antisym K)
-qed
-
-lemma \<omega>_limit_set_in_compact_tendsto:
-  shows "((\<lambda>t. infdist (flow0 x t) (\<omega>_limit_set x)) \<longlongrightarrow> 0) at_top"
-proof (rule ccontr)
-  assume "\<not> ((\<lambda>t. infdist (flow0 x t) (\<omega>_limit_set x)) \<longlongrightarrow> 0) at_top"
-  from not_tendsto_frequentlyE[OF this]
-  obtain S where S: "open S" "0 \<in> S"
-    "\<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<notin> S" .
-  then obtain e where "e > 0" "ball 0 e \<subseteq> S" using openE by blast
-  then have "\<And>x. x \<ge>0 \<Longrightarrow> x \<notin> S \<Longrightarrow> x \<ge> e" by force
-  then have "\<forall>xa. infdist (flow0 x xa) (\<omega>_limit_set x) \<notin> S \<longrightarrow>
-        infdist (flow0 x xa) (\<omega>_limit_set x) \<ge> e" using infdist_nonneg by blast  
-  from frequently_mono[OF this S(3)]
-  have "\<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<ge> e" by blast
-  then have "\<forall>n. \<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<ge> e \<and> real n \<le> t"
-    by (auto intro!: eventually_frequently_conj)
-  from frequently_at_topE[OF this]
-  obtain s where s: "\<And>i. e \<le> infdist (flow0 x (s i)) (\<omega>_limit_set x)"
-    "\<And>i. real i \<le> s i" "strict_mono s" by force
-  then have sf: "filterlim s at_top sequentially"
-    using filterlim_at_top_mono filterlim_real_sequentially not_eventuallyD by blast 
-  have fin: "\<forall>n. (flow0 x \<circ> s) n \<in> K"  using x K props(1) s unfolding flowimg trapped_forward_def
-    by (metis atLeast_iff comp_apply image_subset_iff of_nat_0_le_iff order_trans)
-  from seq_compactE[OF props(2) this]
-  obtain r l where r:"strict_mono r" and l: "l \<in> K" "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l" by blast
-  moreover from filterlim_at_top_strict_mono[OF s(3) r(1) sf]
-  have "filterlim (s \<circ> r) at_top sequentially" .
-  moreover have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1) calculation
-    by (metis comp_assoc)
-  ultimately have "infdist l (\<omega>_limit_set x) = 0" by (simp add: \<omega>_limit_set_def)
-  then have c1:"((\<lambda>y. infdist y (\<omega>_limit_set x)) \<circ> (flow0 x \<circ> s \<circ> r)) \<longlonglongrightarrow> 0"
-    by (auto intro!: tendsto_compose_at[OF l(2)] tendsto_eq_intros)
-  have c2: "\<And>i. e \<le> infdist (flow0 x ((s \<circ> r) i)) (\<omega>_limit_set x)" using s(1) by simp
-  show False using c1 c2 \<open>e > 0\<close> unfolding o_def
-    using Lim_bounded2
-    by (metis (no_types, lifting) ball_eq_empty centre_in_ball empty_iff)
-qed
-
-lemma \<omega>_limit_set_in_compact_connected:
-  shows "connected (\<omega>_limit_set x)"
-  unfolding connected_closed_set[OF \<omega>_limit_set_closed]
-proof clarsimp
-  fix Apre Bpre
-  assume pre: "closed Apre" "Apre \<union> Bpre = \<omega>_limit_set x" "closed Bpre"
-    "Apre \<noteq> {}" "Bpre \<noteq> {}" "Apre \<inter> Bpre = {}"
-    (* TODO: this move is very strange *)
-  then obtain A B where "Apre \<subseteq> A" "Bpre \<subseteq> B" "open A" "open B" and disj:"A \<inter> B = {}"
-    by (meson t4_space)
-  then have "\<omega>_limit_set x \<subseteq> A \<union> B"
-    "\<omega>_limit_set x \<inter> A \<noteq> {}" "\<omega>_limit_set x \<inter> B \<noteq> {}" using pre by auto
-  then obtain p q where
-    p: "\<omega>_limit_point x p" "p \<in> A"
-    and q: "\<omega>_limit_point x q" "q \<in> B"
-    using \<omega>_limit_set_def by auto
-  from \<omega>_limit_pointE[OF p(1)]
-  obtain ps where ps: "filterlim ps at_top sequentially"
-    "(flow0 x \<circ> ps) \<longlonglongrightarrow> p" "\<forall>n. 0 \<le> ps n" by blast
-  from \<omega>_limit_pointE[OF q(1)]
-  obtain qs where qs: "filterlim qs at_top sequentially"
-    "(flow0 x \<circ> qs) \<longlonglongrightarrow> q" "\<forall>n. 0 \<le> qs n" by blast
-  have "\<forall>n. \<exists>\<^sub>F t in at_top. flow0 x t \<notin> A \<and> flow0 x t \<notin> B" unfolding frequently_at_top
-  proof safe
-    fix dummy mpre
-    obtain m where "m \<ge> (0::real)" "m > mpre"
-      by (meson approximation_preproc_push_neg(1) gt_ex le_cases order_trans) 
-    from ps obtain a where a:"a > m" "(flow0 x a) \<in> A"
-      using \<open>open A\<close> p unfolding tendsto_def filterlim_at_top eventually_sequentially
-      by (metis approximation_preproc_push_neg(1) comp_apply gt_ex le_cases order_trans)
-    from qs obtain b where b:"b > a" "(flow0 x b) \<in> B"
-      using \<open>open B\<close> q unfolding tendsto_def filterlim_at_top eventually_sequentially
-      by (metis approximation_preproc_push_neg(1) comp_apply gt_ex le_cases order_trans)
-    have "continuous_on {a..b} (flow0 x)"
-      by (metis Icc_subset_Ici_iff \<open>0 \<le> m\<close> \<open>m < a\<close> approximation_preproc_push_neg(2) atMost_iff atMost_subset_iff continuous_on_subset le_cases local.flow_continuous_on props(1) subset_eq)
-    from connected_continuous_image[OF this connected_Icc]
-    have c:"connected (flow0 x ` {a..b})" .
-    have "\<exists>t\<in> {a..b}. flow0 x t \<notin> A \<and> flow0 x t \<notin> B"
-    proof (rule ccontr)
-      assume "\<not> (\<exists>t\<in>{a..b}. flow0 x t \<notin> A \<and> flow0 x t \<notin> B)"
-      then have "flow0 x ` {a..b} \<subseteq> A \<union> B" by blast
-      from topological_space_class.connectedD[OF c \<open>open A\<close> \<open>open B\<close> _ this]
-      show False using a b disj by force
-    qed
-    thus "\<exists>n>mpre. flow0 x n \<notin> A \<and> flow0 x n \<notin> B"
-      by (smt \<open>mpre < m\<close> a(1) atLeastAtMost_iff)
-  qed
-  from frequently_at_topE'[OF this filterlim_real_sequentially]
-  obtain s where s: "\<forall>i. flow0 x (s i) \<notin> A \<and> flow0 x (s i) \<notin> B"
-    "strict_mono s" "\<And>n. real n \<le> s n" by blast
-  then have "\<forall>n. (flow0 x \<circ> s) n \<in> K"
-    by (smt atLeast_iff comp_apply flowimg image_subset_iff of_nat_0_le_iff trapped_forward_def x(2))
-  from seq_compactE[OF props(2) this]
-  obtain r l where r: "l \<in> K" "strict_mono r" "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l" by blast
-  have "filterlim s at_top sequentially"
-    using s filterlim_at_top_mono filterlim_real_sequentially not_eventuallyD by blast 
-  from filterlim_at_top_strict_mono[OF s(2) r(2) this]
-  have "filterlim (s \<circ> r) at_top sequentially" .
-  then have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1) r
-    by (metis comp_assoc)
-  moreover have "l \<notin> A" using s(1) r(3) \<open>open A\<close> unfolding tendsto_def by auto
-  moreover have "l \<notin> B" using s(1) r(3) \<open>open B\<close> unfolding tendsto_def by auto
-  ultimately show False using \<open>\<omega>_limit_set x \<subseteq> A \<union> B\<close> unfolding \<omega>_limit_set_def
-    by auto
-qed
-
-lemma \<omega>_limit_set_in_compact_\<omega>_limit_set_contained:
-  shows "\<forall>y \<in> \<omega>_limit_set x. \<omega>_limit_set y \<subseteq> \<omega>_limit_set x"
-proof safe
-  fix y z
-  assume "y \<in> \<omega>_limit_set x" "z \<in> \<omega>_limit_set y"
-  then have "\<omega>_limit_point y z" unfolding \<omega>_limit_set_def by auto
-  from \<omega>_limit_pointE[OF this]
-  obtain s where s: "(flow0 y \<circ> s) \<longlonglongrightarrow> z" .
-  have "\<forall>n. (flow0 y \<circ> s) n \<in> \<omega>_limit_set x"
-    using \<open>y \<in> \<omega>_limit_set x\<close> invariant_def
-      \<omega>_limit_set_in_compact_existence \<omega>_limit_set_invariant trapped_iff_on_existence_ivl0
-    by force
-  thus "z \<in> \<omega>_limit_set x" using closed_sequential_limits s \<omega>_limit_set_closed
-    by blast
-qed
-
-lemma \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained:
-  assumes zpx: "z \<in> \<omega>_limit_set x"
-  shows "\<alpha>_limit_set z \<subseteq> \<omega>_limit_set x"
-proof
-  fix w assume "w \<in> \<alpha>_limit_set z"
-  then obtain s where s: "(flow0 z \<circ> s) \<longlonglongrightarrow> w"
-    unfolding \<alpha>_limit_set_def \<alpha>_limit_point_def
-    by auto
-  from \<omega>_limit_set_invariant have "invariant (\<omega>_limit_set x)" .
-  then have "\<forall>n. (flow0 z \<circ> s) n \<in> \<omega>_limit_set x"
-    using \<omega>_limit_set_in_compact_existence[OF zpx] zpx
-    using invariant_def trapped_iff_on_existence_ivl0 by fastforce
-  from closed_sequentially[OF \<omega>_limit_set_closed this s]
-  show "w \<in> \<omega>_limit_set x" .
-qed
-
-end
-
-text \<open> Fundamental properties of the negative limit set \<close>
-
-end context auto_ll_on_open begin
-
-context
-  fixes x K
-  assumes x: "x \<in> X" "trapped_backward x K"
-  assumes K: "compact K" "K \<subseteq> X"
-begin
-
-private lemma xrev: "x \<in> X" "rev.trapped_forward x K"
-  using trapped_backward_iff_rev_trapped_forward x(2) 
-  by (auto simp: rev_existence_ivl_eq0 rev_eq_flow x(1))
-
-lemma \<alpha>_limit_set_in_compact_subset: "\<alpha>_limit_set x \<subseteq> K"
-  and \<alpha>_limit_set_in_compact_compact: "compact (\<alpha>_limit_set x)"
-  and \<alpha>_limit_set_in_compact_nonempty: "\<alpha>_limit_set x \<noteq> {}"
-  and \<alpha>_limit_set_in_compact_connected: "connected (\<alpha>_limit_set x)"
-  and \<alpha>_limit_set_in_compact_\<alpha>_limit_set_contained:
-  "\<forall>y \<in> \<alpha>_limit_set x. \<alpha>_limit_set y \<subseteq> \<alpha>_limit_set x"
-  and \<alpha>_limit_set_in_compact_tendsto: "((\<lambda>t. infdist (flow0 x t) (\<alpha>_limit_set x)) \<longlongrightarrow> 0) at_bot"
-  using rev.\<omega>_limit_set_in_compact_subset[OF K xrev]
-  using rev.\<omega>_limit_set_in_compact_compact[OF K xrev]
-  using rev.\<omega>_limit_set_in_compact_nonempty[OF K xrev]
-  using rev.\<omega>_limit_set_in_compact_connected[OF K xrev]
-  using rev.\<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF K xrev]
-  using rev.\<omega>_limit_set_in_compact_tendsto[OF K xrev]
-  unfolding invariant_eq_rev \<alpha>_limit_set_eq_rev existence_ivl_eq_rev flow_eq_rev0 filterlim_at_bot_mirror
-    minus_minus
-  .
-
-lemma \<alpha>_limit_set_in_compact_existence:
-  shows "\<And>y. y \<in> \<alpha>_limit_set x \<Longrightarrow> existence_ivl0 y = UNIV"
-  using rev.\<omega>_limit_set_in_compact_existence[OF K xrev]
-  unfolding \<alpha>_limit_set_eq_rev existence_ivl_eq_rev0
-  by auto
-
-end
-end
-
-end
+section \<open>Limit Sets\<close>
+theory Limit_Set
+  imports Invariance
+begin
+
+context auto_ll_on_open begin
+text \<open>Positive limit point, assuming \<open>{0..} \<subseteq> existence_ivl0 x\<close>\<close>
+
+(* TODO: Perhaps a more intrinsic definition would be better here *)
+definition "\<omega>_limit_point x p \<longleftrightarrow>
+  {0..} \<subseteq> existence_ivl0 x \<and>
+  (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p)"
+
+text \<open> Also called the \<open>\<omega>\<close>-limit set of x \<close>
+definition "\<omega>_limit_set x = {p. \<omega>_limit_point x p}"
+
+definition "\<alpha>_limit_point x p \<longleftrightarrow>
+  {..0} \<subseteq> existence_ivl0 x \<and>
+  (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> p)"
+
+text \<open> Also called the \<open>\<alpha>\<close>-limit set of x \<close>
+definition "\<alpha>_limit_set x =
+  {p. \<alpha>_limit_point x p}"
+
+end context auto_ll_on_open begin
+
+lemma \<alpha>_limit_point_eq_rev: "\<alpha>_limit_point x p = rev.\<omega>_limit_point x p"
+  unfolding \<alpha>_limit_point_def rev.\<omega>_limit_point_def
+  apply (auto simp: rev_eq_flow[abs_def] o_def filterlim_uminus_at_bot rev_existence_ivl_eq0 subset_iff
+      intro: exI[where x="uminus o s" for s])
+  using neg_0_le_iff_le by fastforce
+
+lemma \<alpha>_limit_set_eq_rev: "\<alpha>_limit_set x = rev.\<omega>_limit_set x"
+  unfolding \<alpha>_limit_set_def rev.\<omega>_limit_set_def \<alpha>_limit_point_eq_rev ..
+
+lemma \<omega>_limit_pointE:
+  assumes "\<omega>_limit_point x p"
+  obtains s where
+    "filterlim s at_top sequentially"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>n. b \<le> s n"
+  using assms filterlim_at_top_choose_lower \<omega>_limit_point_def by blast
+
+lemma \<omega>_limit_set_eq:
+  assumes "{0..} \<subseteq> existence_ivl0 x"
+  shows "\<omega>_limit_set x = (INF \<tau> \<in> {0..}. closure (flow0 x ` {\<tau>..}))"
+  unfolding \<omega>_limit_set_def
+proof safe
+  fix p t
+  assume pt: "0 \<le> (t::real)" "\<omega>_limit_point x p"
+  from \<omega>_limit_pointE[OF pt(2)]
+  obtain s where
+    "filterlim s at_top sequentially"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>n. t \<le> s n" by blast
+  thus "p \<in> closure (flow0 x ` {t..})" unfolding closure_sequential
+    by (metis atLeast_iff comp_apply imageI)
+next
+  fix p
+  assume "p \<in> (\<Inter>\<tau>\<in>{0..}. closure (flow0 x ` {\<tau>..}))"
+  then have "\<And>t. t \<ge>0 \<Longrightarrow> p \<in> closure (flow0 x ` {t..})" by blast
+  then have "\<And>t e. t \<ge>0 \<Longrightarrow> e > 0 \<Longrightarrow> (\<exists>tt \<ge> t. dist (flow0 x tt) p < e)"
+    unfolding closure_approachable
+    by fastforce
+  from approachable_sequenceE[OF this]
+  obtain s where "filterlim s at_top sequentially" "(flow0 x \<circ> s) \<longlonglongrightarrow> p" by auto
+  thus "\<omega>_limit_point x p" unfolding \<omega>_limit_point_def using assms by auto
+qed
+
+lemma \<omega>_limit_set_empty:
+  assumes "\<not> ({0..} \<subseteq> existence_ivl0 x)"
+  shows "\<omega>_limit_set x = {}"
+  unfolding \<omega>_limit_set_def \<omega>_limit_point_def
+  by (simp add: assms)
+
+lemma \<omega>_limit_set_closed: "closed (\<omega>_limit_set x)"
+  using \<omega>_limit_set_eq
+  by (metis \<omega>_limit_set_empty closed_INT closed_closure closed_empty)
+
+(* TODO: Walter gives a more direct proof *)
+lemma \<omega>_limit_set_positively_invariant:
+  shows "positively_invariant (\<omega>_limit_set x)"
+  unfolding positively_invariant_def trapped_forward_def
+proof safe
+  fix dummy p t
+  assume xa: "p \<in> \<omega>_limit_set x"
+    "t \<in> existence_ivl0 p"
+    "0 \<le> t"
+  have "p \<in> X" using mem_existence_ivl_iv_defined(2) xa(2) by blast
+  have exist: "{0..} \<subseteq> existence_ivl0 x" using xa(1)
+    unfolding \<omega>_limit_set_def \<omega>_limit_point_def by auto
+  from xa(1)
+  obtain s where s:
+    "filterlim s at_top sequentially"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>n. 0 \<le> s n"
+    unfolding \<omega>_limit_set_def by (auto elim!:\<omega>_limit_pointE)
+  define r where "r = (\<lambda>n. t + s n)"
+  have rlim: "filterlim r at_top sequentially" unfolding r_def
+    by (auto intro: filterlim_tendsto_add_at_top[OF _ s(1)])
+  define dom where "dom = image (flow0 x) {0..} \<union> {p} "
+  have domin: "\<forall>n. (flow0 x \<circ> s) n \<in> dom" "p \<in> dom" unfolding dom_def o_def
+    using exist by(auto simp add: s(3))
+  have xt: "\<And>x. x \<in> dom \<Longrightarrow> t \<in> existence_ivl0 x" unfolding dom_def using xa(2)
+    apply auto
+    apply (rule existence_ivl_trans')
+    using exist xa(3) apply auto[1]
+    using exist  by auto
+  have cont: "continuous_on dom (\<lambda>x. flow0 x t)"
+    apply (rule flow_continuous_on_compose)
+       apply auto
+    using \<open>p \<in> X\<close>  exist local.dom_def local.flow_in_domain apply auto[1]
+    using xt .
+  then have f1: "((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) \<longlonglongrightarrow> flow0 p t" using domin s(2)
+    unfolding continuous_on_sequentially
+    by blast
+  have ff:"\<And>n. (flow0 x \<circ> r) n = ((\<lambda>x. flow0 x t) \<circ> (flow0 x \<circ> s)) n"
+    unfolding o_def r_def
+  proof -
+    fix n
+    have s:"s n \<in> existence_ivl0 x"
+      using s(3) exist by auto
+    then have t:"t \<in> existence_ivl0 (flow0 x (s n))"
+      using domin(1) xt by auto
+    from flow_trans[OF s t]
+    show "flow0 x (t + s n) = flow0 (flow0 x (s n)) t"
+      by (simp add: add.commute)
+  qed
+  have f2: "(flow0 x \<circ> r) \<longlonglongrightarrow> flow0 p t" using f1 unfolding ff .
+  show "flow0 p t \<in> \<omega>_limit_set x" using exist f2 rlim
+    unfolding \<omega>_limit_set_def \<omega>_limit_point_def
+    using flow_in_domain r_def s(3) xa(2) xa(3) by auto
+qed
+
+lemma \<omega>_limit_set_invariant:
+  shows "invariant (\<omega>_limit_set x)"
+  unfolding invariant_iff_pos_invariant_and_compl_pos_invariant
+proof safe
+  show "positively_invariant (\<omega>_limit_set x)"
+    using \<omega>_limit_set_positively_invariant .
+next
+  show "positively_invariant (X - \<omega>_limit_set x)"
+    unfolding positively_invariant_def trapped_forward_def
+    apply safe
+    using local.flow_in_domain apply blast
+  proof -
+    fix dummy p t 
+    assume xa: "p \<in> X" "p \<notin> \<omega>_limit_set x"
+      "t \<in> existence_ivl0 p" "0 \<le> t"
+      and f: "flow0 p t \<in> \<omega>_limit_set x"
+    have exist: "{0..} \<subseteq> existence_ivl0 x" using f
+      unfolding \<omega>_limit_set_def \<omega>_limit_point_def by auto
+    from f
+    obtain s where s:
+      "filterlim s at_top sequentially"
+      "(flow0 x \<circ> s) \<longlonglongrightarrow> flow0 p t"
+      "\<forall>n. t \<le> s n"
+      unfolding \<omega>_limit_set_def by (auto elim!:\<omega>_limit_pointE)
+    define r where "r = (\<lambda>n. (-t) + s n)"
+    have "(\<lambda>x. -t) \<longlonglongrightarrow> -t" by simp
+    from filterlim_tendsto_add_at_top[OF this s(1)]
+    have rlim: "filterlim r at_top sequentially" unfolding r_def by simp
+    define dom where "dom = image (flow0 x) {t..} \<union> {flow0 p t} "
+    have domin: "\<forall>n. (flow0 x \<circ> s) n \<in> dom" "flow0 p t \<in> dom" unfolding dom_def o_def
+      using exist by(auto simp add: s(3))
+    have xt: "\<And>x. x \<in> dom \<Longrightarrow> -t \<in> existence_ivl0 x" unfolding dom_def using xa(2)
+      apply auto
+      using local.existence_ivl_reverse xa(3) apply auto[1]
+      by (metis exist atLeast_iff diff_conv_add_uminus diff_ge_0_iff_ge linordered_ab_group_add_class.zero_le_double_add_iff_zero_le_single_add local.existence_ivl_trans' order_trans subset_iff xa(4))
+    have cont: "continuous_on dom (\<lambda>x. flow0 x (-t))"
+      apply (rule flow_continuous_on_compose)
+         apply auto
+      using local.mem_existence_ivl_iv_defined(2) xt apply blast
+      by (simp add: xt)
+    then have f1: "((\<lambda>x. flow0 x (-t)) \<circ> (flow0 x \<circ> s)) \<longlonglongrightarrow> flow0 (flow0 p t) (-t)" using domin s(2)
+      unfolding continuous_on_sequentially
+      by blast
+    have ff:"\<And>n. (flow0 x \<circ> r) n = ((\<lambda>x. flow0 x (-t)) \<circ> (flow0 x \<circ> s)) n"
+      unfolding o_def r_def
+    proof -
+      fix n
+      have s:"s n \<in> existence_ivl0 x"
+        using s(3) exist \<open>0\<le> t\<close> by (meson atLeast_iff order_trans subset_eq)
+      then have t:"-t \<in> existence_ivl0 (flow0 x (s n))"
+        using domin(1) xt by auto
+      from flow_trans[OF s t]
+      show "flow0 x (-t + s n) = flow0 (flow0 x (s n)) (-t)"
+        by (simp add: add.commute)
+    qed
+    have "(flow0 x \<circ> r) \<longlonglongrightarrow> flow0 (flow0 p t) (-t)" using f1 unfolding ff .
+    then have f2: "(flow0 x \<circ> r) \<longlonglongrightarrow> p" using flows_reverse xa(3) by auto
+    then have "p \<in> \<omega>_limit_set x" unfolding \<omega>_limit_set_def \<omega>_limit_point_def
+      using rlim exist by auto
+    thus False using xa(2) by auto
+  qed
+qed
+
+end context auto_ll_on_open begin
+
+lemma \<alpha>_limit_set_eq:
+  assumes "{..0} \<subseteq> existence_ivl0 x"
+  shows "\<alpha>_limit_set x = (INF \<tau> \<in> {..0}. closure (flow0 x ` {..\<tau>}))"
+  using rev.\<omega>_limit_set_eq[of x, OF assms[folded infinite_rev_existence_ivl0_rewrites]]
+  unfolding \<alpha>_limit_set_eq_rev rev_flow_image_eq image_uminus_atLeast 
+  by (smt INT_extend_simps(10) Sup.SUP_cong image_uminus_atMost)
+
+lemma \<alpha>_limit_set_closed:
+  shows "closed (\<alpha>_limit_set x)"
+  unfolding \<alpha>_limit_set_eq_rev by (rule rev.\<omega>_limit_set_closed)
+
+lemma \<alpha>_limit_set_positively_invariant:
+  shows "negatively_invariant (\<alpha>_limit_set x)"
+  unfolding negatively_invariant_eq_rev \<alpha>_limit_set_eq_rev
+  by (simp add: rev.\<omega>_limit_set_positively_invariant)
+
+lemma \<alpha>_limit_set_invariant:
+  shows "invariant (\<alpha>_limit_set x)"
+  unfolding invariant_eq_rev \<alpha>_limit_set_eq_rev
+  by (simp add: rev.\<omega>_limit_set_invariant)
+
+text \<open> Fundamental properties of the positive limit set \<close>
+
+context
+  fixes x K
+  assumes K: "compact K" "K \<subseteq> X"
+  assumes x: "x \<in> X" "trapped_forward x K"
+begin
+
+text \<open>Bunch of facts for what's to come\<close>
+private lemma props:
+  shows "{0..} \<subseteq> existence_ivl0 x" "seq_compact K"
+   apply (rule trapped_sol_right)
+  using x K by (auto simp add: compact_imp_seq_compact)
+
+private lemma flowimg:
+  shows "flow0 x ` (existence_ivl0 x \<inter> {0..}) = flow0 x ` {0..}"
+  using props(1) by auto
+
+lemma \<omega>_limit_set_in_compact_subset:
+  shows "\<omega>_limit_set x \<subseteq> K"
+  unfolding \<omega>_limit_set_def
+proof safe
+  fix p s
+  assume "\<omega>_limit_point x p"
+  from \<omega>_limit_pointE[OF this]
+  obtain s where s:
+    "filterlim s at_top sequentially"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>n. 0 \<le> s n" by blast
+  then have fin: "\<forall>n. (flow0 x \<circ> s) n \<in> K" using s(3) x K props(1)
+    unfolding trapped_forward_def
+    by (simp add: subset_eq)
+  from seq_compactE[OF props(2) fin]
+  show "p \<in> K" using s(2)
+    by (metis LIMSEQ_subseq_LIMSEQ LIMSEQ_unique)
+qed
+
+lemma \<omega>_limit_set_in_compact_compact:
+  shows "compact (\<omega>_limit_set x)"
+proof -
+  from \<omega>_limit_set_in_compact_subset 
+  have "bounded (\<omega>_limit_set x)"
+    using bounded_subset compact_imp_bounded
+    using K(1) by auto
+  thus ?thesis using \<omega>_limit_set_closed
+    by (simp add: compact_eq_bounded_closed)
+qed
+
+lemma \<omega>_limit_set_in_compact_nonempty:
+  shows "\<omega>_limit_set x \<noteq> {}"
+proof -
+  have fin: "\<forall>n. (flow0 x \<circ> real) n \<in> K" using x K props(1)
+    by (simp add: flowimg image_subset_iff trapped_forward_def)
+  from seq_compactE[OF props(2) this]
+  obtain r l where "l \<in> K" "strict_mono r" "(flow0 x \<circ> real \<circ> r) \<longlonglongrightarrow> l" by blast
+  then have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1)
+    by (smt comp_def filterlim_sequentially_iff_filterlim_real filterlim_subseq tendsto_at_top_eq_left)
+  thus ?thesis unfolding \<omega>_limit_set_def by auto
+qed
+
+lemma \<omega>_limit_set_in_compact_existence:
+  shows "\<And>y. y \<in> \<omega>_limit_set x \<Longrightarrow> existence_ivl0 y = UNIV"
+proof -
+  fix y
+  assume y: "y \<in> \<omega>_limit_set x"
+  then have "y \<in> X" using \<omega>_limit_set_in_compact_subset K by blast 
+  from \<omega>_limit_set_invariant
+  have "\<And>t. t \<in> existence_ivl0 y \<Longrightarrow> flow0 y t \<in> \<omega>_limit_set x"
+    unfolding invariant_def trapped_iff_on_existence_ivl0 using y by blast 
+  then have t: "\<And>t. t \<in> existence_ivl0 y \<Longrightarrow> flow0 y t \<in> K"
+    using \<omega>_limit_set_in_compact_subset by blast
+  thus "existence_ivl0 y = UNIV" 
+    by (meson \<open>y \<in> X\<close> existence_ivl_zero existence_ivl_initial_time_iff existence_ivl_subset mem_compact_implies_subset_existence_interval subset_antisym K)
+qed
+
+lemma \<omega>_limit_set_in_compact_tendsto:
+  shows "((\<lambda>t. infdist (flow0 x t) (\<omega>_limit_set x)) \<longlongrightarrow> 0) at_top"
+proof (rule ccontr)
+  assume "\<not> ((\<lambda>t. infdist (flow0 x t) (\<omega>_limit_set x)) \<longlongrightarrow> 0) at_top"
+  from not_tendsto_frequentlyE[OF this]
+  obtain S where S: "open S" "0 \<in> S"
+    "\<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<notin> S" .
+  then obtain e where "e > 0" "ball 0 e \<subseteq> S" using openE by blast
+  then have "\<And>x. x \<ge>0 \<Longrightarrow> x \<notin> S \<Longrightarrow> x \<ge> e" by force
+  then have "\<forall>xa. infdist (flow0 x xa) (\<omega>_limit_set x) \<notin> S \<longrightarrow>
+        infdist (flow0 x xa) (\<omega>_limit_set x) \<ge> e" using infdist_nonneg by blast  
+  from frequently_mono[OF this S(3)]
+  have "\<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<ge> e" by blast
+  then have "\<forall>n. \<exists>\<^sub>F t in at_top. infdist (flow0 x t) (\<omega>_limit_set x) \<ge> e \<and> real n \<le> t"
+    by (auto intro!: eventually_frequently_conj)
+  from frequently_at_topE[OF this]
+  obtain s where s: "\<And>i. e \<le> infdist (flow0 x (s i)) (\<omega>_limit_set x)"
+    "\<And>i. real i \<le> s i" "strict_mono s" by force
+  then have sf: "filterlim s at_top sequentially"
+    using filterlim_at_top_mono filterlim_real_sequentially not_eventuallyD by blast 
+  have fin: "\<forall>n. (flow0 x \<circ> s) n \<in> K"  using x K props(1) s unfolding flowimg trapped_forward_def
+    by (metis atLeast_iff comp_apply image_subset_iff of_nat_0_le_iff order_trans)
+  from seq_compactE[OF props(2) this]
+  obtain r l where r:"strict_mono r" and l: "l \<in> K" "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l" by blast
+  moreover from filterlim_at_top_strict_mono[OF s(3) r(1) sf]
+  have "filterlim (s \<circ> r) at_top sequentially" .
+  moreover have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1) calculation
+    by (metis comp_assoc)
+  ultimately have "infdist l (\<omega>_limit_set x) = 0" by (simp add: \<omega>_limit_set_def)
+  then have c1:"((\<lambda>y. infdist y (\<omega>_limit_set x)) \<circ> (flow0 x \<circ> s \<circ> r)) \<longlonglongrightarrow> 0"
+    by (auto intro!: tendsto_compose_at[OF l(2)] tendsto_eq_intros)
+  have c2: "\<And>i. e \<le> infdist (flow0 x ((s \<circ> r) i)) (\<omega>_limit_set x)" using s(1) by simp
+  show False using c1 c2 \<open>e > 0\<close> unfolding o_def
+    using Lim_bounded2
+    by (metis (no_types, lifting) ball_eq_empty centre_in_ball empty_iff)
+qed
+
+lemma \<omega>_limit_set_in_compact_connected:
+  shows "connected (\<omega>_limit_set x)"
+  unfolding connected_closed_set[OF \<omega>_limit_set_closed]
+proof clarsimp
+  fix Apre Bpre
+  assume pre: "closed Apre" "Apre \<union> Bpre = \<omega>_limit_set x" "closed Bpre"
+    "Apre \<noteq> {}" "Bpre \<noteq> {}" "Apre \<inter> Bpre = {}"
+    (* TODO: this move is very strange *)
+  then obtain A B where "Apre \<subseteq> A" "Bpre \<subseteq> B" "open A" "open B" and disj:"A \<inter> B = {}"
+    by (meson t4_space)
+  then have "\<omega>_limit_set x \<subseteq> A \<union> B"
+    "\<omega>_limit_set x \<inter> A \<noteq> {}" "\<omega>_limit_set x \<inter> B \<noteq> {}" using pre by auto
+  then obtain p q where
+    p: "\<omega>_limit_point x p" "p \<in> A"
+    and q: "\<omega>_limit_point x q" "q \<in> B"
+    using \<omega>_limit_set_def by auto
+  from \<omega>_limit_pointE[OF p(1)]
+  obtain ps where ps: "filterlim ps at_top sequentially"
+    "(flow0 x \<circ> ps) \<longlonglongrightarrow> p" "\<forall>n. 0 \<le> ps n" by blast
+  from \<omega>_limit_pointE[OF q(1)]
+  obtain qs where qs: "filterlim qs at_top sequentially"
+    "(flow0 x \<circ> qs) \<longlonglongrightarrow> q" "\<forall>n. 0 \<le> qs n" by blast
+  have "\<forall>n. \<exists>\<^sub>F t in at_top. flow0 x t \<notin> A \<and> flow0 x t \<notin> B" unfolding frequently_at_top
+  proof safe
+    fix dummy mpre
+    obtain m where "m \<ge> (0::real)" "m > mpre"
+      by (meson approximation_preproc_push_neg(1) gt_ex le_cases order_trans) 
+    from ps obtain a where a:"a > m" "(flow0 x a) \<in> A"
+      using \<open>open A\<close> p unfolding tendsto_def filterlim_at_top eventually_sequentially
+      by (metis approximation_preproc_push_neg(1) comp_apply gt_ex le_cases order_trans)
+    from qs obtain b where b:"b > a" "(flow0 x b) \<in> B"
+      using \<open>open B\<close> q unfolding tendsto_def filterlim_at_top eventually_sequentially
+      by (metis approximation_preproc_push_neg(1) comp_apply gt_ex le_cases order_trans)
+    have "continuous_on {a..b} (flow0 x)"
+      by (metis Icc_subset_Ici_iff \<open>0 \<le> m\<close> \<open>m < a\<close> approximation_preproc_push_neg(2) atMost_iff atMost_subset_iff continuous_on_subset le_cases local.flow_continuous_on props(1) subset_eq)
+    from connected_continuous_image[OF this connected_Icc]
+    have c:"connected (flow0 x ` {a..b})" .
+    have "\<exists>t\<in> {a..b}. flow0 x t \<notin> A \<and> flow0 x t \<notin> B"
+    proof (rule ccontr)
+      assume "\<not> (\<exists>t\<in>{a..b}. flow0 x t \<notin> A \<and> flow0 x t \<notin> B)"
+      then have "flow0 x ` {a..b} \<subseteq> A \<union> B" by blast
+      from topological_space_class.connectedD[OF c \<open>open A\<close> \<open>open B\<close> _ this]
+      show False using a b disj by force
+    qed
+    thus "\<exists>n>mpre. flow0 x n \<notin> A \<and> flow0 x n \<notin> B"
+      by (smt \<open>mpre < m\<close> a(1) atLeastAtMost_iff)
+  qed
+  from frequently_at_topE'[OF this filterlim_real_sequentially]
+  obtain s where s: "\<forall>i. flow0 x (s i) \<notin> A \<and> flow0 x (s i) \<notin> B"
+    "strict_mono s" "\<And>n. real n \<le> s n" by blast
+  then have "\<forall>n. (flow0 x \<circ> s) n \<in> K"
+    by (smt atLeast_iff comp_apply flowimg image_subset_iff of_nat_0_le_iff trapped_forward_def x(2))
+  from seq_compactE[OF props(2) this]
+  obtain r l where r: "l \<in> K" "strict_mono r" "(flow0 x \<circ> s \<circ> r) \<longlonglongrightarrow> l" by blast
+  have "filterlim s at_top sequentially"
+    using s filterlim_at_top_mono filterlim_real_sequentially not_eventuallyD by blast 
+  from filterlim_at_top_strict_mono[OF s(2) r(2) this]
+  have "filterlim (s \<circ> r) at_top sequentially" .
+  then have "\<omega>_limit_point x l" unfolding \<omega>_limit_point_def using props(1) r
+    by (metis comp_assoc)
+  moreover have "l \<notin> A" using s(1) r(3) \<open>open A\<close> unfolding tendsto_def by auto
+  moreover have "l \<notin> B" using s(1) r(3) \<open>open B\<close> unfolding tendsto_def by auto
+  ultimately show False using \<open>\<omega>_limit_set x \<subseteq> A \<union> B\<close> unfolding \<omega>_limit_set_def
+    by auto
+qed
+
+lemma \<omega>_limit_set_in_compact_\<omega>_limit_set_contained:
+  shows "\<forall>y \<in> \<omega>_limit_set x. \<omega>_limit_set y \<subseteq> \<omega>_limit_set x"
+proof safe
+  fix y z
+  assume "y \<in> \<omega>_limit_set x" "z \<in> \<omega>_limit_set y"
+  then have "\<omega>_limit_point y z" unfolding \<omega>_limit_set_def by auto
+  from \<omega>_limit_pointE[OF this]
+  obtain s where s: "(flow0 y \<circ> s) \<longlonglongrightarrow> z" .
+  have "\<forall>n. (flow0 y \<circ> s) n \<in> \<omega>_limit_set x"
+    using \<open>y \<in> \<omega>_limit_set x\<close> invariant_def
+      \<omega>_limit_set_in_compact_existence \<omega>_limit_set_invariant trapped_iff_on_existence_ivl0
+    by force
+  thus "z \<in> \<omega>_limit_set x" using closed_sequential_limits s \<omega>_limit_set_closed
+    by blast
+qed
+
+lemma \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained:
+  assumes zpx: "z \<in> \<omega>_limit_set x"
+  shows "\<alpha>_limit_set z \<subseteq> \<omega>_limit_set x"
+proof
+  fix w assume "w \<in> \<alpha>_limit_set z"
+  then obtain s where s: "(flow0 z \<circ> s) \<longlonglongrightarrow> w"
+    unfolding \<alpha>_limit_set_def \<alpha>_limit_point_def
+    by auto
+  from \<omega>_limit_set_invariant have "invariant (\<omega>_limit_set x)" .
+  then have "\<forall>n. (flow0 z \<circ> s) n \<in> \<omega>_limit_set x"
+    using \<omega>_limit_set_in_compact_existence[OF zpx] zpx
+    using invariant_def trapped_iff_on_existence_ivl0 by fastforce
+  from closed_sequentially[OF \<omega>_limit_set_closed this s]
+  show "w \<in> \<omega>_limit_set x" .
+qed
+
+end
+
+text \<open> Fundamental properties of the negative limit set \<close>
+
+end context auto_ll_on_open begin
+
+context
+  fixes x K
+  assumes x: "x \<in> X" "trapped_backward x K"
+  assumes K: "compact K" "K \<subseteq> X"
+begin
+
+private lemma xrev: "x \<in> X" "rev.trapped_forward x K"
+  using trapped_backward_iff_rev_trapped_forward x(2) 
+  by (auto simp: rev_existence_ivl_eq0 rev_eq_flow x(1))
+
+lemma \<alpha>_limit_set_in_compact_subset: "\<alpha>_limit_set x \<subseteq> K"
+  and \<alpha>_limit_set_in_compact_compact: "compact (\<alpha>_limit_set x)"
+  and \<alpha>_limit_set_in_compact_nonempty: "\<alpha>_limit_set x \<noteq> {}"
+  and \<alpha>_limit_set_in_compact_connected: "connected (\<alpha>_limit_set x)"
+  and \<alpha>_limit_set_in_compact_\<alpha>_limit_set_contained:
+  "\<forall>y \<in> \<alpha>_limit_set x. \<alpha>_limit_set y \<subseteq> \<alpha>_limit_set x"
+  and \<alpha>_limit_set_in_compact_tendsto: "((\<lambda>t. infdist (flow0 x t) (\<alpha>_limit_set x)) \<longlongrightarrow> 0) at_bot"
+  using rev.\<omega>_limit_set_in_compact_subset[OF K xrev]
+  using rev.\<omega>_limit_set_in_compact_compact[OF K xrev]
+  using rev.\<omega>_limit_set_in_compact_nonempty[OF K xrev]
+  using rev.\<omega>_limit_set_in_compact_connected[OF K xrev]
+  using rev.\<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF K xrev]
+  using rev.\<omega>_limit_set_in_compact_tendsto[OF K xrev]
+  unfolding invariant_eq_rev \<alpha>_limit_set_eq_rev existence_ivl_eq_rev flow_eq_rev0 filterlim_at_bot_mirror
+    minus_minus
+  .
+
+lemma \<alpha>_limit_set_in_compact_existence:
+  shows "\<And>y. y \<in> \<alpha>_limit_set x \<Longrightarrow> existence_ivl0 y = UNIV"
+  using rev.\<omega>_limit_set_in_compact_existence[OF K xrev]
+  unfolding \<alpha>_limit_set_eq_rev existence_ivl_eq_rev0
+  by auto
+
+end
+end
+
+end
diff --git a/thys/Poincare_Bendixson/ODE_Misc.thy b/thys/Poincare_Bendixson/ODE_Misc.thy
--- a/thys/Poincare_Bendixson/ODE_Misc.thy
+++ b/thys/Poincare_Bendixson/ODE_Misc.thy
@@ -1,1111 +1,1111 @@
-section \<open>Additions to the ODE Library\<close>
-theory ODE_Misc
-  imports
-    Ordinary_Differential_Equations.ODE_Analysis
-    Analysis_Misc
-begin
-
-lemma local_lipschitz_compact_bicomposeE:
-  assumes ll: "local_lipschitz T X f"
-  assumes cf: "\<And>x. x \<in> X \<Longrightarrow> continuous_on I (\<lambda>t. f t x)"
-  assumes cI: "compact I"
-  assumes "I \<subseteq> T"
-  assumes cv: "continuous_on I v"
-  assumes cw: "continuous_on I w"
-  assumes v: "v ` I \<subseteq> X"
-  assumes w: "w ` I \<subseteq> X"
-  obtains L where "L > 0" "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L * dist (v x) (w x)"
-proof -
-  from v w have "v ` I \<union> w ` I \<subseteq> X" by auto
-  with ll \<open>I \<subseteq> T\<close> have llI:"local_lipschitz I (v ` I \<union> w ` I) f"
-    by (rule local_lipschitz_subset)
-  have cvwI: "compact (v ` I \<union> w ` I)"
-    by (auto intro!: compact_continuous_image cv cw cI)    
-
-  from local_lipschitz_compact_implies_lipschitz[OF llI cvwI \<open>compact I\<close> cf]
-  obtain L where L: "\<And>t. t \<in> I \<Longrightarrow> L-lipschitz_on (v ` I \<union> w ` I) (f t)"
-    using v w
-    by blast
-  define L' where "L' = max L 1"
-  with L have "L' > 0" "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L' * dist (v x) (w x)"
-     apply (auto simp: lipschitz_on_def L'_def)
-    by (smt Un_iff image_eqI mult_right_mono zero_le_dist)
-  then show ?thesis ..
-qed
-
-subsection \<open>Comparison Principle\<close>
-
-lemma comparison_principle_le:
-  fixes f::"real \<Rightarrow> real \<Rightarrow> real"
-    and \<phi> \<psi>::"real \<Rightarrow> real"
-  assumes ll: "local_lipschitz X Y f"
-  assumes cf: "\<And>x. x \<in> Y \<Longrightarrow> continuous_on {a..b} (\<lambda>t. f t x)"
-  assumes abX: "{a .. b} \<subseteq> X"
-  assumes \<phi>': "\<And>x. x \<in> {a .. b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' x) (at x)"
-  assumes \<psi>': "\<And>x. x \<in> {a .. b} \<Longrightarrow> (\<psi> has_real_derivative \<psi>' x) (at x)"
-  assumes \<phi>_in: "\<phi> ` {a..b} \<subseteq> Y"
-  assumes \<psi>_in: "\<psi> ` {a..b} \<subseteq> Y"
-  assumes init: "\<phi> a \<le> \<psi> a"
-  assumes defect: "\<And>x. x \<in> {a .. b} \<Longrightarrow> \<phi>' x - f x (\<phi> x) \<le> \<psi>' x - f x (\<psi> x)"
-  shows "\<forall>x \<in> {a .. b}. \<phi> x \<le> \<psi> x" (is "?th1")
-    (*
-    "(\<forall>x \<in> {a .. b}. \<phi> x < \<psi> x) \<or> (\<exists>c\<in>{a..b}. (\<forall>x\<in>{a..c}. \<phi> x \<le> \<psi> x) \<and> (\<forall>x\<in>{c<..b}. \<phi> x < \<psi> x))"
-    (is "?th2")
-*)
-  unfolding atomize_conj
-  apply (cases "a \<le> b")
-   defer subgoal by simp
-proof -
-  assume "a \<le> b"
-  note \<phi>_cont = has_real_derivative_imp_continuous_on[OF \<phi>']
-  note \<psi>_cont = has_real_derivative_imp_continuous_on[OF \<psi>']
-  from local_lipschitz_compact_bicomposeE[OF ll cf compact_Icc abX \<phi>_cont \<psi>_cont \<phi>_in \<psi>_in]
-  obtain L where L: "L > 0" "\<And>x. x \<in> {a..b} \<Longrightarrow> dist (f x (\<phi> x)) (f x (\<psi> x)) \<le> L * dist (\<phi> x) (\<psi> x)" by blast
-  define w where "w x = \<psi> x - \<phi> x" for x
-
-  have w'[derivative_intros]: "\<And>x. x \<in> {a .. b} \<Longrightarrow> (w has_real_derivative \<psi>' x - \<phi>' x) (at x)"
-    using \<phi>' \<psi>'
-    by (auto simp: has_vderiv_on_def w_def[abs_def] intro!: derivative_eq_intros)
-  note w_cont[continuous_intros] = has_real_derivative_imp_continuous_on[OF w', THEN continuous_on_compose2]
-  have "w d \<ge> 0" if "d \<in> {a .. b}" for d
-  proof (rule ccontr, unfold not_le)
-    assume "w d < 0"
-    let ?N = "(w -` {..0} \<inter> {a .. d})"
-    from \<open>w d < 0\<close> that have "d \<in> ?N" by auto
-    then have "?N \<noteq> {}" by auto
-    have "closed ?N"
-      unfolding compact_eq_bounded_closed
-      using that
-      by (intro conjI closed_vimage_Int) (auto intro!: continuous_intros)
-
-    let ?N' = "{a0 \<in> {a .. d}. w ` {a0 .. d} \<subseteq> {..0}}"
-    from \<open>w d < 0\<close> that have "d \<in> ?N'" by simp
-    then have "?N' \<noteq> {}" by auto
-    have "compact ?N'"
-      unfolding compact_eq_bounded_closed
-    proof
-      have "?N' \<subseteq> {a .. d}" using that by auto
-      then show "bounded ?N'"
-        by (rule bounded_subset[rotated]) simp
-      have "w u \<le> 0" if "(\<forall>n. x n \<in> ?N')" "x \<longlonglongrightarrow> l" "l \<le> u" "u \<le> d" for x l u
-      proof cases
-        assume "l = u"
-        have "\<forall>n. x n \<in> ?N" using that(1) by force
-        from closed_sequentially[OF \<open>closed ?N\<close> this \<open>x \<longlonglongrightarrow> l\<close>]
-        show ?thesis by (auto simp: \<open>l = u\<close>)
-      next
-        assume "l \<noteq> u" with that have "l < u" by auto
-        from order_tendstoD(2)[OF \<open>x \<longlonglongrightarrow> l\<close> \<open>l < u\<close>] obtain n where "x n < u"
-          by (auto dest: eventually_happens)
-        with that show ?thesis using \<open>l < u\<close>
-          by (auto dest!: spec[where x=n] simp: image_subset_iff)
-      qed
-      then show "closed ?N'"
-        unfolding closed_sequential_limits
-        by (auto simp: Lim_bounded Lim_bounded2)
-    qed
-
-    from compact_attains_inf[OF \<open>compact ?N'\<close> \<open>?N' \<noteq> {}\<close>]
-    obtain a0 where a0: "a \<le> a0" "a0 \<le> d" "w ` {a0..d} \<subseteq> {..0}"
-      and a0_least: "\<And>x. a \<le> x \<Longrightarrow> x \<le> d \<Longrightarrow> w ` {x..d} \<subseteq> {..0} \<Longrightarrow> a0 \<le> x"
-      by auto
-    have a0d: "{a0 .. d} \<subseteq> {a .. b}" using that a0
-      by auto
-    have L_w_bound: "L * w x \<le> \<psi>' x - \<phi>' x" if "x \<in> {a0 .. d}" for x
-    proof -
-      from set_mp[OF a0d that] have "x \<in> {a .. b}" .
-      from defect[OF this]
-      have "\<phi>' x - \<psi>' x \<le> dist (f x (\<phi> x)) (f x (\<psi> x))"
-        by (simp add: dist_real_def)
-      also have "\<dots> \<le> L * dist (\<phi> x) (\<psi> x)"
-        using \<open>x \<in> {a .. b}\<close>
-        by (rule L)
-      also have "\<dots> \<le> -L * w x"
-        using \<open>0 < L\<close> a0 that
-        by (force simp add: dist_real_def abs_real_def w_def algebra_split_simps )
-      finally show ?thesis
-        by simp
-    qed
-    have mono: "mono_on (\<lambda>x. w x * exp(-L*x)) {a0..d}"
-      apply (rule mono_onI)
-      apply (rule DERIV_nonneg_imp_nondecreasing, assumption)
-      using a0d
-      by (auto intro!: exI[where x="(\<psi>' x - \<phi>' x) * exp (- (L * x)) - exp (- (L * x)) * L * w x" for x]
-          derivative_eq_intros L_w_bound simp: )
-    then have "w a0 * exp (-L * a0) \<le> w d * exp (-L * d)"
-      by (rule mono_onD) (use that a0 in auto)
-    also have "\<dots> < 0" using \<open>w d < 0\<close> by (simp add: algebra_split_simps)
-    finally have "w a0 * exp (- L * a0) < 0" .
-    then have "w a0 < 0" by (simp add: algebra_split_simps)
-    have "a0 \<le> a"
-    proof (rule ccontr, unfold not_le)
-      assume "a < a0"
-      have "continuous_on {a.. a0} w"
-        by (rule continuous_intros, assumption) (use a0 a0d in auto)
-      from continuous_on_Icc_at_leftD[OF this \<open>a < a0\<close>]
-      have "(w \<longlongrightarrow> w a0) (at_left a0)" .
-      from order_tendstoD(2)[OF this \<open>w a0 < 0\<close>] have "\<forall>\<^sub>F x in at_left a0. w x < 0" .
-      moreover have "\<forall>\<^sub>F x in at_left a0. a < x"
-        by (rule order_tendstoD) (auto intro!: \<open>a < a0\<close>)
-      ultimately have "\<forall>\<^sub>F x in at_left a0. a < x \<and> w x < 0" by eventually_elim auto
-      then obtain a1' where "a1'<a0" and a1_neg: "\<And>y. y > a1' \<Longrightarrow> y < a0 \<Longrightarrow> a < y \<and> w y < 0"
-        unfolding eventually_at_left_field by auto
-      define a1 where "a1 = (a1' + a0)/2"
-      have "a1 < a0" using \<open>a1' < a0\<close> by (auto simp: a1_def)
-      have "a \<le> a1"
-        using \<open>a < a0\<close> a1_neg by (force simp: a1_def)
-      moreover have "a1 \<le> d"
-        using \<open>a1' < a0\<close> a0(2) by (auto simp: a1_def)
-      moreover have "w ` {a1..a0} \<subseteq> {..0}"
-        using \<open>w a0 < 0\<close> a1_neg a0(3)
-        by (auto simp: a1_def) smt
-      moreover have "w ` {a0..d} \<subseteq> {..0}" using a0 by auto
-      ultimately
-      have "a0 \<le> a1"
-        apply (intro a0_least) apply assumption apply assumption
-        by (smt atLeastAtMost_iff image_subset_iff)
-      with \<open>a1<a0\<close> show False by simp
-    qed
-    then have "a0 = a" using \<open>a \<le> a0\<close> by simp
-    with \<open>w a0 < 0\<close> have "w a < 0" by simp
-    with init show False
-      by (auto simp: w_def)
-  qed
-  then show ?thesis
-    by (auto simp: w_def)
-qed
-
-lemma local_lipschitz_mult:
-  shows "local_lipschitz  (UNIV::real set) (UNIV::real set) (*)"
-  apply (auto intro!: c1_implies_local_lipschitz[where f'="\<lambda>p. blinfun_mult_left (fst p)"])
-   apply (simp add: has_derivative_mult_right mult_commute_abs)
-  by (auto intro!: continuous_intros)
-
-lemma comparison_principle_le_linear:
-  fixes \<phi> :: "real \<Rightarrow> real"
-  assumes "continuous_on {a..b} g"
-  assumes "(\<And>t. t \<in> {a..b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' t) (at t))"
-  assumes "\<phi> a \<le> 0"
-  assumes "(\<And>t. t \<in> {a..b} \<Longrightarrow> \<phi>' t \<le> g t *\<^sub>R \<phi> t)"
-  shows "\<forall>t\<in>{a..b}. \<phi> t \<le> 0"
-proof -
-  have *: "\<And>x. continuous_on {a..b} (\<lambda>t. g t * x)"
-    using assms(1) continuous_on_mult_right by blast
-  then have "local_lipschitz (g`{a..b}) UNIV (*)"
-    using local_lipschitz_subset[OF local_lipschitz_mult] by blast 
-  from local_lipschitz_compose1[OF this assms(1)]
-  have "local_lipschitz {a..b} UNIV (\<lambda>t. (*) (g t))" .
-  from comparison_principle_le[OF this _ _ assms(2) _ _ _ assms(3), of b "\<lambda>t.0"] * assms(4)
-  show ?thesis by auto
-qed
-
-subsection \<open>Locally Lipschitz ODEs\<close>
-
-context ll_on_open_it begin
-
-lemma flow_lipschitzE:
-  assumes "{a .. b} \<subseteq> existence_ivl t0 x"
-  obtains L where "L-lipschitz_on {a .. b} (flow t0 x)"
-proof -
-  have f': "(flow t0 x has_derivative (\<lambda>i. i *\<^sub>R f t (flow t0 x t))) (at t within {a .. b})" if "t \<in> {a .. b}" for t
-    using flow_has_derivative[of t x] assms that
-    by (auto simp: has_derivative_at_withinI)
-
-  have "compact ((\<lambda>t. f t (flow t0 x t)) ` {a .. b})"
-    using assms
-    apply (auto intro!: compact_continuous_image continuous_intros)
-    using local.existence_ivl_empty2 apply fastforce
-     apply (meson atLeastAtMost_iff general.existence_ivl_subset in_mono)
-    by (simp add: general.flow_in_domain subset_iff)
-  then obtain C where "t \<in> {a .. b} \<Longrightarrow> norm (f t (flow t0 x t)) \<le> C" for t
-    by (fastforce dest!: compact_imp_bounded simp: bounded_iff intro: that)
-  then have "t \<in> {a..b} \<Longrightarrow> onorm (\<lambda>i. i *\<^sub>R f t (flow t0 x t)) \<le> max 0 C" for t
-    apply (subst onorm_scaleR_left) 
-     apply (auto simp: onorm_id max_def)
-    by (metis diff_0_right diff_mono diff_self norm_ge_zero)
-  from bounded_derivative_imp_lipschitz[OF f' _ this]
-  have "(max 0 C)-lipschitz_on {a..b} (flow t0 x)"
-    by auto
-  then show ?thesis ..
-qed
-
-lemma flow_undefined0: "t \<notin> existence_ivl t0 x \<Longrightarrow> flow t0 x t = 0"
-  unfolding flow_def by auto
-
-lemma csols_undefined: "x \<notin> X \<Longrightarrow> csols t0 x = {}"
-  apply (auto simp: csols_def)
-  using general.existence_ivl_empty2 general.existence_ivl_maximal_segment
-  apply blast
-  done
-
-lemmas existence_ivl_undefined = existence_ivl_empty2
-
-end
-
-subsection \<open>Reverse flow as Sublocale\<close>
-
-lemma range_preflect_0[simp]: "range (preflect 0) = UNIV"
-  by (auto simp: preflect_def)
-lemma range_uminus[simp]: "range uminus = (UNIV::'a::ab_group_add set)"
-  by auto
-
-context auto_ll_on_open begin
-
-sublocale rev: auto_ll_on_open "-f" rewrites "-(-f) = f"
-   apply unfold_locales
-  using auto_local_lipschitz auto_open_domain
-  unfolding fun_Compl_def local_lipschitz_minus
-  by auto
-
-lemma existence_ivl_eq_rev0: "existence_ivl0 y = uminus ` rev.existence_ivl0 y" for y
-  by (auto simp: existence_ivl_eq_rev rev.existence_ivl0_def preflect_def)
-
-lemma rev_existence_ivl_eq0: "rev.existence_ivl0 y = uminus ` existence_ivl0 y" for y
-  using uminus_uminus_image[of "rev.existence_ivl0 y"]
-  by (simp add: existence_ivl_eq_rev0)
-
-lemma flow_eq_rev0: "flow0 y t = rev.flow0 y (-t)" for y t
-  apply (cases "t \<in> existence_ivl0 y")
-  subgoal
-    apply (subst flow_eq_rev(2), assumption)
-    apply (subst rev.flow0_def)
-    by (simp add: preflect_def)
-  subgoal
-    apply (frule flow_undefined0)
-    by (auto simp: existence_ivl_eq_rev0 rev.flow_undefined0)
-  done
-
-lemma rev_eq_flow: "rev.flow0 y t = flow0 y (-t)" for y t
-  apply (subst flow_eq_rev0)
-  using uminus_uminus_image[of "rev.existence_ivl0 y"]
-  apply -
-  apply (subst (asm) existence_ivl_eq_rev0[symmetric])
-  by auto
-
-lemma rev_flow_image_eq: "rev.flow0 x ` S = flow0 x ` (uminus ` S)"
-  unfolding rev_eq_flow[abs_def]
-  by force
-
-lemma flow_image_eq_rev: "flow0 x ` S = rev.flow0 x ` (uminus ` S)"
-  unfolding rev_eq_flow[abs_def]
-  by force
-
-end
-
-context c1_on_open begin
-
-sublocale rev: c1_on_open "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
-  by (rule c1_on_open_rev) auto
-
-end
-
-context c1_on_open_euclidean begin
-
-sublocale rev: c1_on_open_euclidean "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
-  by unfold_locales auto
-
-end
-
-
-subsection \<open>Autonomous LL ODE : Existence Interval and trapping on the interval\<close>
-
-lemma bdd_above_is_intervalI: "bdd_above I"
-  if "is_interval I" "a \<le> b" "a \<in> I" "b \<notin> I" for I::"real set"
-  by (meson bdd_above_def is_interval_1 le_cases that) 
-
-lemma bdd_below_is_intervalI: "bdd_below I"
-  if "is_interval I" "a \<le> b" "a \<notin> I" "b \<in> I" for I::"real set"
-  by (meson bdd_below_def is_interval_1 le_cases that) 
-
-context auto_ll_on_open begin
-
-lemma open_existence_ivl0:
-  assumes x : "x \<in> X"
-  shows "\<exists>a b. a < 0 \<and> 0 < b \<and> {a..b} \<subseteq> existence_ivl0 x"
-proof -
-  have a1:"0 \<in> existence_ivl0 x"
-    by (simp add: x)
-  have a2: "open (existence_ivl0 x)"
-    by (simp add: x)
-  from a1 a2 obtain d where "d > 0" "ball 0 d \<subseteq> existence_ivl0 x"
-    using openE by blast
-  have "{-d/2..d/2} \<subseteq> ball 0 d"
-    using \<open>0 < d\<close> dist_norm mem_ball by auto
-  thus ?thesis
-    by (smt \<open>0 < d\<close> \<open>ball 0 d \<subseteq> existence_ivl0 x\<close> divide_minus_left half_gt_zero order_trans)
-qed
-
-lemma open_existence_ivl':
-  assumes x : "x \<in> X"
-  obtains a where "a > 0"  "{-a..a} \<subseteq> existence_ivl0 x"
-proof -
-  from open_existence_ivl0[OF assms(1)]
-  obtain a b where ab: "a < 0" "0 < b" "{a..b} \<subseteq> existence_ivl0 x" by auto
-  then have "min (-a) b > 0" by linarith
-  have "{-min (-a) b .. min(-a) b} \<subseteq> {a..b}" by auto
-  thus ?thesis using ab(3) that[OF \<open>min (-a) b > 0\<close>] by blast
-qed
-
-lemma open_existence_ivl_on_compact:
-  assumes C: "C \<subseteq> X" and "compact C" "C \<noteq> {}"
-  obtains a where "a > 0" "\<And>x. x \<in> C \<Longrightarrow> {-a..a} \<subseteq> existence_ivl0 x"
-proof -
-  from existence_ivl_cballs
-  have "\<forall>x\<in>C. \<exists>e>0. \<exists>t>0. \<forall>y\<in>cball x e. cball 0 t\<subseteq>existence_ivl0 y"
-    by (metis (full_types) C Int_absorb1 Int_iff UNIV_I)
-  then
-  obtain d' t' where *:
-    "\<forall>x\<in>C. 0 < d' x \<and> t' x > 0 \<and> (\<forall>y\<in>cball x (d' x). cball 0 (t' x) \<subseteq> existence_ivl0 y)"
-    by metis
-  with compactE_image[OF \<open>compact C\<close>, of C "\<lambda>x. ball x (d' x)"]
-  obtain C' where "C' \<subseteq> C" and [simp]: "finite C'" and C_subset: "C \<subseteq> (\<Union>c\<in>C'. ball c (d' c))"
-    by force
-  from C_subset \<open>C \<noteq> {}\<close> have [simp]: "C' \<noteq> {}" by auto
-  define d where "d = Min (d' ` C')"
-  define t where "t = Min (t' ` C')"
-  have "t > 0" using * \<open>C' \<subseteq> C\<close>
-    by (auto simp: t_def)
-  moreover have "{-t .. t} \<subseteq> existence_ivl0 x" if "x \<in> C" for x
-  proof -
-    from C_subset that \<open>C' \<subseteq> C\<close>
-    obtain c where c: "c \<in> C'" "x \<in> ball c (d' c)" "c \<in> C" by force
-    then have "{-t .. t} \<subseteq> cball 0 (t' c)"
-      by (auto simp: abs_real_def t_def minus_le_iff)
-    also
-    from c have "cball 0 (t' c) \<subseteq> existence_ivl0 x"
-      using *[rule_format, OF \<open>c \<in> C\<close>] by auto
-    finally show ?thesis .
-  qed
-  ultimately show ?thesis ..
-qed
-
-definition "trapped_forward x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> K)"
-  \<comment> \<open>TODO: use this for backwards trapped, invariant, and all assumptions\<close>
-
-definition "trapped_backward x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x \<inter> {..0}) \<subseteq> K)"
-
-definition "trapped x K \<longleftrightarrow> trapped_forward x K \<and> trapped_backward x K"
-
-lemma trapped_iff_on_existence_ivl0:
-  "trapped x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x) \<subseteq> K)"
-  unfolding trapped_def trapped_forward_def trapped_backward_def
-  apply (auto)
-  by (metis IntI atLeast_iff atMost_iff image_subset_iff less_eq_real_def linorder_not_less)
-end
-
-context auto_ll_on_open begin
-
-lemma infinite_rev_existence_ivl0_rewrites:
-  "{0..} \<subseteq> rev.existence_ivl0 x \<longleftrightarrow> {..0} \<subseteq> existence_ivl0 x"
-  "{..0} \<subseteq> rev.existence_ivl0 x \<longleftrightarrow> {0..} \<subseteq> existence_ivl0 x"
-   apply (auto simp add: rev.rev_existence_ivl_eq0 subset_iff)
-  using neg_le_0_iff_le apply fastforce
-  using neg_0_le_iff_le by fastforce
-
-lemma trapped_backward_iff_rev_trapped_forward:
-  "trapped_backward x K  \<longleftrightarrow> rev.trapped_forward x K"
-  unfolding trapped_backward_def rev.trapped_forward_def
-  by (auto simp add: rev_flow_image_eq existence_ivl_eq_rev0 image_subset_iff)
-
-text \<open>If solution is trapped in a compact set at some time
-  on its existence interval then it is trapped forever\<close>
-lemma trapped_sol_right:
-  \<comment> \<open>TODO: when building on afp-devel (??? outdated):
-    \<^url>\<open>https://bitbucket.org/isa-afp/afp-devel/commits/0c3edf9248d5389197f248c723b625c419e4d3eb\<close>\<close>
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  shows "{0..} \<subseteq> existence_ivl0 x"
-proof (rule ccontr)
-  assume "\<not> {0..} \<subseteq> existence_ivl0 x"
-  from this obtain t where "0 \<le> t" "t \<notin> existence_ivl0 x" by blast
-  then have bdd: "bdd_above (existence_ivl0 x)" 
-    by (auto intro!: bdd_above_is_intervalI \<open>x \<in> X\<close>)
-  from flow_leaves_compact_ivl_right [OF UNIV_I \<open>x \<in> X\<close> bdd UNIV_I assms(1-2)]
-  show False by (metis assms(4) trapped_forward_def IntI atLeast_iff image_subset_iff)
-qed
-
-lemma trapped_sol_right_gen:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "t \<in> existence_ivl0 x" "trapped_forward (flow0 x t) K"
-  shows "{t..} \<subseteq> existence_ivl0 x"
-proof -
-  have "x \<in> X"
-    using assms(3) local.existence_ivl_empty_iff by fastforce 
-  have xtk: "flow0 x t \<in> X"
-    by (simp add: assms(3) local.flow_in_domain)
-  from trapped_sol_right[OF assms(1-2) xtk assms(4)] have "{0..} \<subseteq> existence_ivl0 (flow0 x t)" .
-  thus "{t..} \<subseteq> existence_ivl0 x"
-    using existence_ivl_trans[OF assms(3)]
-    by (metis add.commute atLeast_iff diff_add_cancel le_add_same_cancel1 subset_iff)
-qed
-
-lemma trapped_sol_left:
-  \<comment> \<open>TODO: when building on afp-devel:
-    \<^url>\<open>https://bitbucket.org/isa-afp/afp-devel/commits/0c3edf9248d5389197f248c723b625c419e4d3eb\<close>\<close>
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_backward x K"
-  shows "{..0} \<subseteq> existence_ivl0 x"
-proof (rule ccontr)
-  assume "\<not> {..0} \<subseteq> existence_ivl0 x"
-  from this obtain t where "t \<le> 0" "t \<notin> existence_ivl0 x" by blast
-  then have bdd: "bdd_below (existence_ivl0 x)" 
-    by (auto intro!: bdd_below_is_intervalI \<open>x \<in> X\<close>)
-  from flow_leaves_compact_ivl_left [OF UNIV_I \<open>x \<in> X\<close> bdd UNIV_I assms(1-2)]
-  show False
-    by (metis IntI assms(4) atMost_iff auto_ll_on_open.trapped_backward_def auto_ll_on_open_axioms image_subset_iff)
-qed
-
-lemma trapped_sol_left_gen:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "t \<in> existence_ivl0 x" "trapped_backward (flow0 x t) K"
-  shows "{..t} \<subseteq> existence_ivl0 x"
-proof -
-  have "x \<in> X"
-    using assms(3) local.existence_ivl_empty_iff by fastforce 
-  have xtk: "flow0 x t \<in> X"
-    by (simp add: assms(3) local.flow_in_domain)
-  from trapped_sol_left[OF assms(1-2) xtk assms(4)] have "{..0} \<subseteq> existence_ivl0 (flow0 x t)" .
-  thus "{..t} \<subseteq> existence_ivl0 x"
-    using existence_ivl_trans[OF assms(3)]
-    by (metis add.commute add_le_same_cancel1 atMost_iff diff_add_cancel subset_eq)
-qed
-
-lemma trapped_sol:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped x K"
-  shows "existence_ivl0 x = UNIV"
-  by (metis (mono_tags, lifting) assms existence_ivl_zero image_subset_iff interval local.existence_ivl_initial_time_iff local.existence_ivl_subset local.subset_mem_compact_implies_subset_existence_interval order_refl subset_antisym trapped_iff_on_existence_ivl0)
-
-(* Follows from rectification *)
-lemma regular_locally_noteq:\<comment> \<open>TODO: should be true in \<open>ll_on_open_it\<close>\<close>
-  assumes "x \<in> X" "f x \<noteq> 0"
-  shows "eventually (\<lambda>t. flow0 x t \<noteq> x) (at 0)"
-proof -
-  have nf:"norm (f x) > 0" by (simp add: assms(2)) 
-      (* By continuity of solutions and f probably *)
-  obtain a where
-    a: "a>0"
-    "{-a--a} \<subseteq> existence_ivl0 x"
-    "0 \<in> {-a--a}"
-    "\<And>t. t \<in> {-a--a} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2"
-  proof -
-    from open_existence_ivl'[OF assms(1)]
-    obtain a1 where a1: "a1 > 0" "{-a1..a1} \<subseteq> existence_ivl0 x" .
-    have "continuous (at 0) (\<lambda>t. norm(f (flow0 x t) - f (flow0 x 0) ))"
-      apply (auto intro!: continuous_intros)
-      by (simp add: assms(1) local.f_flow_continuous)
-    then obtain a2 where "a2>0"
-      "\<forall>t. norm t < a2 \<longrightarrow>
-             norm (f (flow0 x t) - f (flow0 x 0)) < norm(f x)/2"
-      unfolding continuous_at_real_range
-      by (metis abs_norm_cancel cancel_comm_monoid_add_class.diff_cancel diff_zero half_gt_zero nf norm_zero)
-    then have 
-      t: "\<And>t. t \<in> {-a2<--<a2} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2"
-      by (smt open_segment_bound(2) open_segment_bound1 real_norm_def)
-    define a where "a = min a1 (a2/2)"
-    have t1:"a > 0" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close> by auto
-    then have t3:"0 \<in>{-a--a}"
-      using closed_segment_eq_real_ivl by auto
-    have "{-a--a} \<subseteq> {-a1..a1}" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close>
-      using ODE_Auxiliarities.closed_segment_eq_real_ivl by auto
-    then have t2:"{-a--a} \<subseteq> existence_ivl0 x" using a1 by auto
-    have "{-a--a} \<subseteq> {-a2<--<a2}" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close>
-      by (smt Diff_iff closed_segment_eq_real_ivl atLeastAtMost_iff empty_iff half_gt_zero insert_iff pos_half_less segment(1) subset_eq)
-    then have t4:"\<And>t. t \<in> {-a--a} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2" using t by auto
-    show ?thesis using t1 t2 t3 t4 that by auto
-  qed
-  have "\<And>t. t \<in> {-a--a} \<Longrightarrow> (flow0 x has_vector_derivative f (flow0 x t)) (at t within {-a--a})"
-    apply (rule has_vector_derivative_at_within)
-    using a(2) by (auto intro!:flow_has_vector_derivative)
-  from vector_differentiable_bound_linearization[OF this _ a(4)]
-  have nb:"\<And>c d. {c--d} \<subseteq> {-a--a} \<Longrightarrow>
-   norm (flow0 x d - flow0 x c - (d - c) *\<^sub>R f (flow0 x 0))  \<le> norm (d - c) * (norm (f x) / 2)"
-    using a(3) by blast
-  have "\<And>t. dist t 0 < a \<Longrightarrow> t \<noteq> 0 \<Longrightarrow> flow0 x t \<noteq> x"
-  proof (rule ccontr)
-    fix t
-    assume "dist t 0 < a" "t \<noteq> 0" "\<not> flow0 x t \<noteq> x"
-    then have tx:"flow0 x t = x" by auto
-    have "t \<in> {-a--a}"
-      using closed_segment_eq_real_ivl \<open>dist t 0 < a\<close> by auto
-    have "t > 0 \<or> t < 0" using \<open>t \<noteq> 0\<close> by linarith
-    moreover {
-      assume "t > 0"
-      then have "{0--t} \<subseteq> {-a--a}"
-        by (simp add: \<open>t \<in> {-a--a}\<close> a(3) subset_closed_segment) 
-      from nb[OF this] have
-        "norm (flow0 x t - x - t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)"
-        by (simp add: assms(1))
-      then have "norm (t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)" using tx by auto
-      then have False using nf
-        using \<open>0 < t\<close> by auto 
-    }
-    moreover {
-      assume "t < 0"
-      then have "{t--0} \<subseteq> {-a--a}"
-        by (simp add: \<open>t \<in> {-a--a}\<close> a(3) subset_closed_segment) 
-      from nb[OF this] have
-        "norm (x - flow0 x t + t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)"
-        by (simp add: assms(1))
-      then have "norm (t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)" using tx by auto
-      then have False using nf
-        using \<open>t < 0\<close> by auto 
-    }
-    ultimately show False by blast
-  qed
-  thus ?thesis unfolding eventually_at
-    using a(1) by blast
-qed
-
-lemma compact_max_time_flow_in_closed:
-  assumes "closed M" and t_ex: "t \<in> existence_ivl0 x"
-  shows "compact {s \<in> {0..t}. flow0 x ` {0..s} \<subseteq> M}" (is "compact ?C")
-  unfolding compact_eq_bounded_closed
-proof
-  have "bounded {0 .. t}" by auto
-  then show "bounded ?C"
-    by (rule bounded_subset) auto
-  show "closed ?C"
-    unfolding closed_def
-  proof (rule topological_space_class.openI, clarsimp)
-    \<comment> \<open>TODO: there must be a more abstract argument for this, e.g., with
-      @{thm continuous_on_closed_vimageI} and then reasoning about the connected component around 0?\<close>
-    fix s
-    assume notM: "s \<le> t \<longrightarrow> 0 \<le> s \<longrightarrow> \<not> flow0 x ` {0..s} \<subseteq> M"
-    consider "0 \<le> s" "s \<le> t" "flow0 x s \<notin> M" | "0 \<le> s" "s \<le> t" "flow0 x s \<in> M" | "s < 0" | "s > t"
-      by arith
-    then show "\<exists>T. open T \<and> s \<in> T \<and> T \<subseteq> - {s. 0 \<le> s \<and> s \<le> t \<and> flow0 x ` {0..s} \<subseteq> M}"
-    proof cases
-      assume s: "0 \<le> s" "s \<le> t" and sM: "flow0 x s \<notin> M"
-      have "isCont (flow0 x) s"
-        using s ivl_subset_existence_ivl[OF t_ex]
-        by (auto intro!: flow_continuous)
-      from this[unfolded continuous_at_open, rule_format, of "-M"] sM \<open>closed M\<close>
-      obtain S where "open S" "s \<in> S" "(\<forall>x'\<in>S. flow0 x x' \<in> - M)"
-        by auto
-      then show ?thesis
-        by (force intro!: exI[where x=S])
-    next
-      assume s: "0 \<le> s" "s \<le> t" and sM: "flow0 x s \<in> M"
-      from this notM obtain s0 where s0: "0 \<le> s0" "s0 < s" "flow0 x s0 \<notin> M"
-        by force
-      from order_tendstoD(1)[OF tendsto_ident_at \<open>s0 < s\<close>, of UNIV, unfolded eventually_at_topological]
-      obtain S where "open S" "s \<in> S" "\<And>x. x \<in> S \<Longrightarrow> x \<noteq> s \<Longrightarrow> s0 < x"
-        by auto
-      then show ?thesis using s0
-        by (auto simp: intro!: exI[where x=S]) (smt atLeastAtMost_iff image_subset_iff)
-    qed (force intro: exI[where x="{t<..}"] exI[where x="{..<0}"])+
-  qed
-qed
-
-lemma flow_in_closed_max_timeE:
-  assumes "closed M" "t \<in> existence_ivl0 x" "0 \<le> t" "x \<in> M"
-  obtains T where "0 \<le> T" "T \<le> t" "flow0 x ` {0..T} \<subseteq> M"
-    "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> T"
-proof -
-  let ?C = "{s \<in> {0..t}. flow0 x ` {0..s} \<subseteq> M}"
-  have "?C \<noteq> {}"
-    using assms
-    using local.mem_existence_ivl_iv_defined
-    by (auto intro!: exI[where x=0])
-  from compact_max_time_flow_in_closed[OF assms(1,2)]
-  have "compact ?C" .
-  from compact_attains_sup[OF this \<open>?C \<noteq> {}\<close>]
-  obtain s where s: "0 \<le> s" "s \<le> t" "flow0 x ` {0..s} \<subseteq> M"
-    and s_max: "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> s"
-    by auto
-  then show ?thesis ..
-qed
-
-lemma flow_leaves_closed_at_frontierE:
-  assumes "closed M" and t_ex: "t \<in> existence_ivl0 x" and "0 \<le> t" "x \<in> M" "flow0 x t \<notin> M"
-  obtains s where "0 \<le> s" "s < t" "flow0 x ` {0..s} \<subseteq> M"
-    "flow0 x s \<in> frontier M"
-    "\<exists>\<^sub>F s' in at_right s. flow0 x s' \<notin> M"
-proof -
-  from flow_in_closed_max_timeE[OF assms(1-4)] assms(5)
-  obtain s where s: "0 \<le> s" "s < t" "flow0 x ` {0..s} \<subseteq> M"
-    and s_max: "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> s"
-    by (smt atLeastAtMost_iff image_subset_iff)
-  note s
-  moreover
-  have "flow0 x s \<notin> interior M"
-  proof
-    assume interior: "flow0 x s \<in> interior M"
-    have "s \<in> existence_ivl0 x" using ivl_subset_existence_ivl[OF \<open>t \<in> _\<close>] s by auto
-    from flow_continuous[OF this, THEN isContD, THEN topological_tendstoD, OF open_interior interior]
-    have "\<forall>\<^sub>F s' in at s. flow0 x s' \<in> interior M" by auto
-    then have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> interior M"
-      by (auto simp: eventually_at_split)
-    moreover have "\<forall>\<^sub>F s' in at_right s. s' < t"
-      using tendsto_ident_at \<open>s < t\<close>
-      by (rule order_tendstoD)
-    ultimately have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> M \<and> s' < t"
-      by eventually_elim (use interior_subset[of M] in auto)
-    then obtain s' where s': "s < s'" "s' < t" "\<And>y. y > s \<Longrightarrow> y \<le> s' \<Longrightarrow> flow0 x y \<in> M"
-      by (auto simp: eventually_at_right_field_le)
-    have s'_ivl: "flow0 x ` {0..s'} \<subseteq> M"
-    proof safe
-      fix s'' assume "s'' \<in> {0 .. s'}"
-      then show "flow0 x s'' \<in> M"
-        using s interior_subset[of M] s'
-        by (cases "s'' \<le> s") auto
-    qed
-    with s_max[of s'] \<open>s' < t\<close> \<open>0 \<le> s\<close> \<open>s < s'\<close> show False by auto
-  qed
-  then have "flow0 x s \<in> frontier M"
-    using s closure_subset[of M]
-    by (force simp: frontier_def)
-  moreover
-  have "compact (flow0 x -` M \<inter> {s..t})" (is "compact ?C")
-    unfolding compact_eq_bounded_closed
-  proof
-    have "bounded {s .. t}" by simp
-    then show "bounded ?C"
-      by (rule bounded_subset) auto
-    show "closed ?C"
-      using \<open>closed M\<close> assms mem_existence_ivl_iv_defined(2)[OF t_ex] ivl_subset_existence_ivl[OF t_ex] \<open>0 \<le> s\<close>
-      by (intro closed_vimage_Int) (auto intro!: continuous_intros)
-  qed
-  have "\<exists>\<^sub>F s' in at_right s. flow0 x s' \<notin> M"
-    apply (rule ccontr)
-    unfolding not_frequently
-  proof -
-    assume "\<forall>\<^sub>F s' in at_right s. \<not> flow0 x s' \<notin> M"
-    moreover have "\<forall>\<^sub>F s' in at_right s. s' < t"
-      using tendsto_ident_at \<open>s < t\<close>
-      by (rule order_tendstoD)
-    ultimately have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> M \<and> s' < t" by eventually_elim auto
-    then obtain s' where s': "s < s'"
-      "\<And>y. y > s \<Longrightarrow> y < s' \<Longrightarrow> flow0 x y \<in> M"
-      "\<And>y. y > s \<Longrightarrow> y < s' \<Longrightarrow> y < t"
-      by (auto simp: eventually_at_right_field)
-    define s'' where "s'' = (s + s') / 2"
-    have "0 \<le> s''" "s'' \<le> t"  "s < s''" "s'' < s'"
-      using s s'
-      by (auto simp del: divide_le_eq_numeral1 le_divide_eq_numeral1 simp: s''_def) fastforce
-    then have "flow0 x ` {0..s''} \<subseteq> M"
-      using s s'
-      apply (auto simp: )
-      subgoal for u
-        by (cases "u\<le>s") auto
-      done
-    from s_max[OF \<open>0 \<le> s''\<close> \<open>s''\<le> t\<close> this] \<open>s'' > s\<close>
-    show False by simp
-  qed
-  ultimately show ?thesis ..
-qed
-
-
-subsection \<open>Connectedness\<close>
-
-lemma fcontX:
-  shows "continuous_on X f"
-  using auto_local_lipschitz local_lipschitz_continuous_on by blast
-
-lemma fcontx:
-  assumes "x \<in> X"
-  shows "continuous (at x) f"
-proof -
-  have "open X" by simp
-  from continuous_on_eq_continuous_at[OF this]
-  show ?thesis using fcontX assms(1) by blast
-qed
-
-lemma continuous_at_imp_cball:
-  assumes "continuous (at x) g"
-  assumes "g x > (0::real)"
-  obtains r where "r > 0" "\<forall>y \<in> cball x r. g y > 0"
-proof -
-  from assms(1)
-  obtain d where "d>0" "g ` (ball x d) \<subseteq> ball (g x) ((g x)/2)"
-    by (meson assms(2) continuous_at_ball half_gt_zero)
-  then have "\<forall>y \<in> cball x (d/2). g y > 0"
-    by (smt assms(2) dist_norm image_subset_iff mem_ball mem_cball pos_half_less real_norm_def)
-  thus ?thesis
-    using \<open>0 < d\<close> that half_gt_zero by blast
-qed
-
-text \<open> \<open>flow0\<close> is \<open>path_connected\<close> \<close>
-lemma flow0_path_connected_time:
-  assumes "ts \<subseteq> existence_ivl0 x" "path_connected ts"
-  shows "path_connected (flow0 x ` ts)"
-proof -
-  have "continuous_on ts (flow0 x)"
-    by (meson assms continuous_on_sequentially flow_continuous_on subsetD)
-  from path_connected_continuous_image[OF this assms(2)] 
-  show ?thesis .
-qed
-
-lemma flow0_path_connected:
-  assumes "path_connected D"
-    "path_connected ts"
-    "\<And>x. x \<in> D \<Longrightarrow> ts \<subseteq> existence_ivl0 x"
-  shows "path_connected ( (\<lambda>(x, y). flow0 x y) ` (D \<times> ts))"
-proof -
-  have "D \<times> ts \<subseteq> Sigma X existence_ivl0"
-    using assms(3) subset_iff by fastforce
-  then have a1:"continuous_on (D \<times> ts) (\<lambda>(x, y). flow0 x y)"
-    using flow_continuous_on_state_space continuous_on_subset by blast 
-  have a2 : "path_connected (D \<times> ts)" using path_connected_Times assms by auto
-  from path_connected_continuous_image[OF a1 a2]
-  show ?thesis .
-qed
-
-end
-
-subsection \<open>Return Time and Implicit Function Theorem\<close>
-
-context c1_on_open_euclidean begin
-
-lemma flow_implicit_function:
-  \<comment> \<open>TODO: generalization of @{thm returns_to_implicit_function}!\<close>
-  fixes s::"'a::euclidean_space \<Rightarrow> real" and S::"'a set"
-  assumes t: "t \<in> existence_ivl0 x" and x: "x \<in> X" and st: "s (flow0 x t) = 0"
-  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
-  assumes DsC: "isCont Ds (flow0 x t)"
-  assumes nz: "Ds (flow0 x t) (f (flow0 x t)) \<noteq> 0"
-  obtains u e
-  where "s (flow0 x (u x)) = 0"
-    "u x = t"
-    "(\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0)"
-    "continuous_on (cball x e) u"
-    "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"
-    "0 < e" "(u has_derivative (- blinfun_scaleR_left
-                   (inverse (blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)))) o\<^sub>L
-                      (Ds (flow0 x t) o\<^sub>L flowderiv x t) o\<^sub>L embed1_blinfun)) (at x)"
-proof -
-  note [derivative_intros] = has_derivative_compose[OF _ Ds]
-  have cont_s: "continuous_on UNIV s" by (rule has_derivative_continuous_on[OF Ds])
-  note cls[simp, intro] = closed_levelset[OF cont_s]
-  then have xt1: "(x, t) \<in> Sigma X existence_ivl0"
-    by (auto simp: t x)
-  have D: "(\<And>x. x \<in> Sigma X existence_ivl0 \<Longrightarrow>
-      ((\<lambda>(x, t). s (flow0 x t)) has_derivative
-       blinfun_apply (Ds (flow0 (fst x) (snd x)) o\<^sub>L (flowderiv (fst x) (snd x))))
-       (at x))"
-    by (auto intro!: derivative_eq_intros)
-  have C: "isCont (\<lambda>x. Ds (flow0 (fst x) (snd x)) o\<^sub>L flowderiv (fst x) (snd x))
-   (x, t)"
-    using flowderiv_continuous_on[unfolded continuous_on_eq_continuous_within,
-        rule_format, OF xt1]
-    using at_within_open[OF xt1 open_state_space]
-    by (auto intro!: continuous_intros tendsto_eq_intros x t
-        isCont_tendsto_compose[OF DsC, unfolded poincare_map_def]
-        simp: split_beta' isCont_def)
-  have Z: "(case (x, t) of (x, t) \<Rightarrow> s (flow0 x t)) = 0"
-    by (auto simp: st)
-  have I1: "blinfun_scaleR_left (inverse (Ds (flow0 x t)(f (flow0 x t)))) o\<^sub>L 
-    ((Ds (flow0 (fst (x, t))
-            (snd (x, t))) o\<^sub>L
-       flowderiv (fst (x, t))
-        (snd (x, t))) o\<^sub>L
-      embed2_blinfun)
-     = 1\<^sub>L"
-    using nz
-    by (auto intro!: blinfun_eqI
-        simp: flowderiv_def blinfun.bilinear_simps inverse_eq_divide poincare_map_def)
-  have I2: "((Ds (flow0 (fst (x, t))
-            (snd (x, t))) o\<^sub>L
-       flowderiv (fst (x, t))
-        (snd (x, t))) o\<^sub>L
-      embed2_blinfun) o\<^sub>L blinfun_scaleR_left (inverse (Ds (flow0 x t)(f (flow0 x t))))
-     = 1\<^sub>L"
-    using nz
-    by (auto intro!: blinfun_eqI
-        simp: flowderiv_def blinfun.bilinear_simps inverse_eq_divide poincare_map_def)
-  show ?thesis
-    apply (rule implicit_function_theorem[where f="\<lambda>(x, t). s (flow0 x t)"
-          and S="Sigma X existence_ivl0", OF D xt1 open_state_space order_refl C Z I1 I2])
-     apply blast
-    unfolding split_beta' fst_conv snd_conv poincare_map_def[symmetric]
-    ..
-qed
-
-lemma flow_implicit_function_at:
-  fixes s::"'a::euclidean_space \<Rightarrow> real" and S::"'a set"
-  assumes x: "x \<in> X" and st: "s x = 0"
-  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
-  assumes DsC: "isCont Ds x"
-  assumes nz: "Ds x (f x) \<noteq> 0"
-  assumes pos: "e > 0"
-  obtains u d
-  where
-    "0 < d"
-    "u x = 0"
-    "\<And>y. y \<in> cball x d \<Longrightarrow> s (flow0 y (u y)) = 0"
-    "\<And>y. y \<in> cball x d \<Longrightarrow> \<bar>u y\<bar> < e"
-    "\<And>y. y \<in> cball x d \<Longrightarrow> u y \<in> existence_ivl0 y"
-    "continuous_on (cball x d) u"
-    "(u has_derivative -Ds x /\<^sub>R (Ds x) (f x)) (at x)"
-proof -
-  have x0: "flow0 x 0 = x" by (simp add: x)
-  from flow_implicit_function[OF existence_ivl_zero[OF x] x, unfolded x0, of s, OF st Ds DsC nz]
-  obtain u d0 where
-    s0: "s (flow0 x (u x)) = 0"
-    and u0: "u x = 0"
-    and u: "\<And>y. y \<in> cball x d0 \<Longrightarrow> s (flow0 y (u y)) = 0"
-    and uc: "continuous_on (cball x d0) u"
-    and uex: "(\<lambda>t. (t, u t)) ` cball x d0 \<subseteq> Sigma X existence_ivl0"
-    and d0: "0 < d0"
-    and u': "(u has_derivative
-     blinfun_apply
-      (- blinfun_scaleR_left (inverse (blinfun_apply (Ds x) (f x))) o\<^sub>L (Ds x o\<^sub>L flowderiv x 0) o\<^sub>L embed1_blinfun))
-     (at x)"
-    by blast
-  have "at x within cball x d0 = at x" by (rule at_within_interior) (auto simp: \<open>0 < d0\<close>)
-  then have "(u \<longlongrightarrow> 0) (at x)"
-    using uc d0 by (auto simp: continuous_on_def u0 dest!: bspec[where x=x])
-  from tendstoD[OF this \<open>0 < e\<close>] pos u0
-  obtain d1 where d1: "0 < d1" "\<And>xa. dist xa x \<le> d1 \<Longrightarrow> \<bar>u xa\<bar> < e"
-    unfolding eventually_at_le
-    by force
-  define d where "d = min d0 d1"
-  have "0 < d" by (auto simp: d_def d0 d1)
-  moreover note u0
-  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> s (flow0 y (u y)) = 0" by (auto intro!: u simp: d_def)
-  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> \<bar>u y\<bar> < e" using d1 by (auto simp: d_def dist_commute)
-  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> u y \<in> existence_ivl0 y"
-    using uex by (force simp: d_def)
-  moreover have "continuous_on (cball x d) u"
-    using uc by (rule continuous_on_subset) (auto simp: d_def)
-  moreover
-  have "(u has_derivative -Ds x /\<^sub>R (Ds x) (f x)) (at x)"
-    using u'
-    by (rule has_derivative_subst) (auto intro!: ext simp: x x0 flowderiv_def blinfun.bilinear_simps)
-  ultimately show ?thesis ..
-qed
-
-lemma returns_to_implicit_function_gen:
-  \<comment> \<open>TODO: generalizes proof of @{thm returns_to_implicit_function}!\<close>
-  fixes s::"'a::euclidean_space \<Rightarrow> real"
-  assumes rt: "returns_to {x \<in> S. s x = 0} x" (is "returns_to ?P x")
-  assumes cS: "closed S"
-  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
-    "isCont Ds (poincare_map ?P x)"
-    "Ds (poincare_map ?P x) (f (poincare_map ?P x)) \<noteq> 0"
-  obtains u e
-  where "s (flow0 x (u x)) = 0"
-    "u x = return_time ?P x"
-    "(\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0)"
-    "continuous_on (cball x e) u"
-    "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"
-    "0 < e" "(u has_derivative (- blinfun_scaleR_left
-                   (inverse (blinfun_apply (Ds (poincare_map ?P x)) (f (poincare_map ?P x)))) o\<^sub>L
-                      (Ds (poincare_map ?P x) o\<^sub>L flowderiv x (return_time ?P x)) o\<^sub>L embed1_blinfun)) (at x)"
-proof -
-  note [derivative_intros] = has_derivative_compose[OF _ Ds(1)]
-  have cont_s: "continuous_on UNIV s" by (rule has_derivative_continuous_on[OF Ds(1)])
-  note cls[simp, intro] = closed_levelset[OF cont_s]
-  let ?t1 = "return_time ?P x"
-  have cls[simp, intro]: "closed {x \<in> S. s x = 0}"
-    by (rule closed_levelset_within) (auto intro!: cS continuous_on_subset[OF cont_s])
-
-  have *: "poincare_map ?P x = flow0 x (return_time {x \<in> S. s x = 0} x)"
-    by (simp add: poincare_map_def)
-  have "return_time {x \<in> S. s x = 0} x \<in> existence_ivl0 x"
-    "x \<in> X"
-    "s (poincare_map ?P x) = 0"
-    using poincare_map_returns rt
-    by (auto intro!: return_time_exivl rt)
-  note E = flow_implicit_function[of "return_time ?P x" x s Ds, OF this[unfolded *] Ds[unfolded *],
-      folded *]
-  show ?thesis
-    by (rule E) rule
-qed
-
-text \<open>c.f. Perko Section 3.7 Lemma 2 part 1.\<close>
-
-lemma flow_transversal_surface_finite_intersections:
-  fixes s::"'a \<Rightarrow> 'b::real_normed_vector"
-    and Ds::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
-  assumes "closed S"
-  assumes "\<And>x. (s has_derivative (Ds x)) (at x)"
-  assumes "\<And>x. x \<in> S \<Longrightarrow> s x = 0 \<Longrightarrow> Ds x (f x) \<noteq> 0"
-  assumes "a \<le> b" "{a .. b} \<subseteq> existence_ivl0 x"
-  shows "finite {t\<in>{a..b}. flow0 x t \<in> {x \<in> S. s x = 0}}"
-    \<comment> \<open>TODO: define notion of (compact/closed)-(continuous/differentiable/C1)-surface?\<close>
-proof cases
-  note Ds = \<open>\<And>x. (s has_derivative (Ds x)) (at x)\<close>
-  note transversal = \<open>\<And>x. x \<in> S \<Longrightarrow> s x = 0 \<Longrightarrow> Ds x (f x) \<noteq> 0\<close>
-  assume "a < b"
-  show ?thesis
-  proof (rule ccontr)
-    let ?S = "{x \<in> S. s x = 0}"
-    let ?T = "{t\<in>{a..b}. flow0 x t \<in> {x \<in> S. s x = 0}}"
-    define \<phi> where "\<phi> = flow0 x"
-    have [THEN continuous_on_compose2, continuous_intros]: "continuous_on S s"
-      by (auto simp: intro!: has_derivative_continuous_on Ds intro: has_derivative_at_withinI)
-    assume "infinite ?T"
-    from compact_sequentialE[OF compact_Icc[of a b] this]
-    obtain t tl where t: "t n \<in> ?T" "flow0 x (t n) \<in> ?S" "t n \<in> {a .. b}" "t n \<noteq> tl"
-      and tl: "t \<longlonglongrightarrow> tl" "tl \<in> {a..b}"
-    for n
-      by force
-    have tl_ex: "tl \<in> existence_ivl0 x" using \<open>{a .. b} \<subseteq> existence_ivl0 x\<close> \<open>tl \<in> {a .. b}\<close> by auto
-    have "closed ?S"
-      by (auto intro!: closed_levelset_within \<open>closed S\<close> continuous_intros)
-    moreover
-    have "\<forall>n. flow0 x (t n) \<in> ?S"
-      using t by auto
-    moreover
-    have flow_t: "(\<lambda>n. flow0 x (t n)) \<longlonglongrightarrow> flow0 x tl"
-      by (auto intro!: tendsto_eq_intros tl_ex tl)
-    ultimately have "flow0 x tl \<in> ?S"
-      by (rule closed_sequentially)
-
-    let ?qt = "\<lambda>t. (flow0 x t - flow0 x tl) /\<^sub>R (t - tl)"
-    from flow_has_vector_derivative[OF tl_ex, THEN has_vector_derivative_withinD]
-    have qt_tendsto: "?qt \<midarrow>tl\<rightarrow> f (flow0 x tl)" .
-    let ?q = "\<lambda>n. ?qt (t n)"
-    have "filterlim t (at tl) sequentially"
-      using tl(1)
-      by (rule filterlim_atI) (simp add: t)
-    with qt_tendsto have "?q \<longlonglongrightarrow> f (flow0 x tl)"
-      by (rule filterlim_compose)
-    then have "((\<lambda>n. Ds (flow0 x tl) (?q n))) \<longlonglongrightarrow> Ds (flow0 x tl) (f (flow0 x tl))"
-      by (auto intro!: tendsto_intros)
-    moreover
-
-    from flow_lipschitzE[OF \<open>{a .. b} \<subseteq> existence_ivl0 x\<close>] obtain L' where L': "L'-lipschitz_on {a..b} (flow0 x)" .
-    define L where "L = L' + 1"
-    from lipschitz_on_le[OF L', of L] lipschitz_on_nonneg[OF L']
-    have L: "L-lipschitz_on {a .. b} (flow0 x)" and "L > 0"
-      by (auto simp: L_def)
-    from flow_lipschitzE[OF \<open>{a .. b} \<subseteq> existence_ivl0 x\<close>] obtain L' where "L'-lipschitz_on {a..b} (flow0 x)" .
-        \<comment> \<open>TODO: is this reasoning (below) with this Lipschitz constant really necessary?\<close>
-    have s[simp]: "s (flow0 x (t n)) = 0""s (flow0 x tl) = 0"
-      for n
-      using t \<open>flow0 x tl \<in> ?S\<close>
-      by auto
-
-    from Ds(1)[of "flow0 x tl", unfolded has_derivative_within]
-    have "(\<lambda>y. (1 / norm (y - flow0 x tl)) *\<^sub>R (s y - (s (flow0 x tl) + blinfun_apply (Ds (flow0 x tl)) (y - flow0 x tl)))) \<midarrow>flow0 x tl\<rightarrow> 0"
-      by auto
-    then have "((\<lambda>y. (1 / norm (y - flow0 x tl)) *\<^sub>R (s y - (s (flow0 x tl) + blinfun_apply (Ds (flow0 x tl)) (y - flow0 x tl)))) \<longlongrightarrow> 0)
-      (nhds (flow0 x tl))"
-      by (rule tendsto_nhds_continuousI) simp
-
-    from filterlim_compose[OF this flow_t]
-    have "(\<lambda>xa. (blinfun_apply (Ds (flow0 x tl)) (flow0 x (t xa) - flow0 x tl)) /\<^sub>R norm (flow0 x (t xa) - flow0 x tl))
-      \<longlonglongrightarrow> 0"
-      using t
-      by (auto simp: inverse_eq_divide tendsto_minus_cancel_right)
-    from tendsto_mult[OF tendsto_const[of "L"] tendsto_norm[OF this, simplified, simplified divide_inverse_commute[symmetric]]]\<comment> \<open>TODO: uuugly\<close>
-    have Ds0: "(\<lambda>xa. norm (blinfun_apply (Ds (flow0 x tl)) (flow0 x (t xa) - flow0 x tl)) / (norm (flow0 x (t xa) - flow0 x tl)/(L))) \<longlonglongrightarrow> 0"
-      by (auto simp: ac_simps)
-
-    from _ Ds0 have "((\<lambda>n. Ds (flow0 x tl) (?q n)) \<longlonglongrightarrow> 0)"
-      apply (rule Lim_null_comparison)
-      apply (rule eventuallyI)
-      unfolding norm_scaleR blinfun.scaleR_right abs_inverse divide_inverse_commute[symmetric]
-      subgoal for n
-        apply (cases "flow0 x (t n) = flow0 x tl")
-        subgoal by (simp add: blinfun.bilinear_simps)
-        subgoal
-          apply (rule divide_left_mono)
-          using lipschitz_onD[OF L, of "t n" tl] \<open>0 < L\<close> t(3) tl(2)
-          by (auto simp: algebra_split_simps zero_less_divide_iff dist_norm pos_divide_le_eq
-              intro!: add_pos_nonneg)
-        done
-      done
-    ultimately have "Ds (flow0 x tl) (f (flow0 x tl)) = 0"
-      by (rule LIMSEQ_unique)
-    moreover have "Ds (flow0 x tl) (f (flow0 x tl)) \<noteq> 0"
-      by (rule transversal) (use \<open>flow0 x tl \<in> ?S\<close> in auto)
-    ultimately show False by auto
-  qed
-qed (use assms in auto)
-
-lemma uniform_limit_flow0_state:\<comment> \<open>TODO: is that something more general?\<close>
-  assumes "compact C"
-  assumes "C \<subseteq> X"
-  shows "uniform_limit C (\<lambda>s x. flow0 x s) (\<lambda>x. flow0 x 0) (at 0)"
-proof (cases "C = {}")
-  case True then show ?thesis by auto
-next
-  case False show ?thesis
-  proof (rule uniform_limitI)
-    fix e::real assume "0 < e"
-    {
-      fix x assume "x \<in> C"
-      with assms have "x \<in> X" by auto
-      from existence_ivl_cballs[OF UNIV_I \<open>x \<in> X\<close>]
-      obtain t L u where "\<And>y. y \<in> cball x u \<Longrightarrow> cball 0 t \<subseteq> existence_ivl0 y"
-        "\<And>s y. y \<in> cball x u \<Longrightarrow> s \<in> cball 0 t \<Longrightarrow> flow0 y s \<in> cball y u"
-        "L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)"
-        "\<And>y. y \<in> cball x u \<Longrightarrow> cball y u \<subseteq> X"
-        "0 < t" "0 < u"
-        by metis
-      then have "\<exists>L. \<exists>u>0. \<exists>t>0. L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)" by blast
-    } then have "\<forall>x\<in>C. \<exists>L. \<exists>u>0. \<exists>t>0. L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)" ..
-    then obtain L d' u' where
-      L: "\<And>x. x \<in> C \<Longrightarrow> (L x)-lipschitz_on (cball 0 (d' x)\<times>cball x (u' x)) (\<lambda>(t, x). flow0 x t)"
-      and d': "\<And>x. x \<in> C \<Longrightarrow> d' x > 0"
-      and u': "\<And>x. x \<in> C \<Longrightarrow> u' x > 0"
-      by metis
-    have "C \<subseteq> (\<Union>c\<in>C. ball c (u' c))" using u' by auto
-    from compactE_image[OF \<open>compact C\<close> _ this]
-    obtain C' where "C' \<subseteq> C" and [simp]: "finite C'" and C'_cover: "C \<subseteq> (\<Union>c\<in>C'. ball c (u' c))"
-      by auto
-    from C'_cover obtain c' where c': "x \<in> C \<Longrightarrow> x \<in> ball (c' x) (u' (c' x))" "x \<in> C \<Longrightarrow> c' x \<in> C'" for x
-      by (auto simp: subset_iff) metis
-    have "\<forall>\<^sub>F s in at 0. \<forall>x\<in>ball c (u' c). dist (flow0 x s) (flow0 x 0) < e" if "c \<in> C'" for c
-    proof -
-      have cC: "c \<in> C"
-        using c' \<open>c \<in> C'\<close> d' \<open>C' \<subseteq> C\<close>
-        by auto
-      have *: "dist (flow0 x s) (flow0 x 0) \<le> L c * \<bar>s\<bar>"
-        if "x\<in>ball c (u' c)"
-          "s \<in> cball 0 (d' c)"
-        for x s
-      proof -
-        from L[OF cC, THEN lipschitz_onD, of "(0, x)" "(s, x)"] d'[OF cC] that
-        show ?thesis
-          by (auto simp: dist_prod_def dist_commute)
-      qed
-      have "\<forall>\<^sub>F s in at 0. abs s < d' c"
-        by (rule order_tendstoD tendsto_intros)+ (use d' cC in auto)
-      moreover have "\<forall>\<^sub>F s in at 0. L c * \<bar>s\<bar> < e"
-        by (rule order_tendstoD tendsto_intros)+ (use \<open>0 < e\<close> in auto)
-      ultimately show ?thesis
-        apply eventually_elim
-        apply (use * in auto)
-        by smt
-    qed
-    then have "\<forall>\<^sub>F s in at 0. \<forall>c\<in>C'. \<forall>x\<in>ball c (u' c). dist (flow0 x s) (flow0 x 0) < e"
-      by (simp add: eventually_ball_finite_distrib)
-    then show "\<forall>\<^sub>F s in at 0. \<forall>x\<in>C. dist (flow0 x s) (flow0 x 0) < e"
-      apply eventually_elim
-      apply (auto simp: )
-      subgoal for s x
-        apply (drule bspec[where x="c' x"])
-         apply (simp add: c'(2))
-        apply (drule bspec) prefer 2 apply assumption
-        apply auto
-        using c'(1) by auto
-      done
-  qed
-qed
-
-end
-
-subsection \<open>Fixpoints\<close>
-
-context auto_ll_on_open begin
-
-lemma fixpoint_sol:
-  assumes "x \<in> X" "f x = 0"
-  shows "existence_ivl0 x = UNIV" "flow0 x t = x"
-proof -
-  have sol: "((\<lambda>t::real. x) solves_ode (\<lambda>_. f)) UNIV X"
-    apply (rule solves_odeI)
-    by(auto simp add: assms intro!: derivative_intros)
-  from maximal_existence_flow[OF sol] have
-    "UNIV \<subseteq> existence_ivl0 x" "flow0 x t = x" by auto
-  thus "existence_ivl0 x = UNIV" "flow0 x t = x" by auto
-qed
-
-end
-
+section \<open>Additions to the ODE Library\<close>
+theory ODE_Misc
+  imports
+    Ordinary_Differential_Equations.ODE_Analysis
+    Analysis_Misc
+begin
+
+lemma local_lipschitz_compact_bicomposeE:
+  assumes ll: "local_lipschitz T X f"
+  assumes cf: "\<And>x. x \<in> X \<Longrightarrow> continuous_on I (\<lambda>t. f t x)"
+  assumes cI: "compact I"
+  assumes "I \<subseteq> T"
+  assumes cv: "continuous_on I v"
+  assumes cw: "continuous_on I w"
+  assumes v: "v ` I \<subseteq> X"
+  assumes w: "w ` I \<subseteq> X"
+  obtains L where "L > 0" "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L * dist (v x) (w x)"
+proof -
+  from v w have "v ` I \<union> w ` I \<subseteq> X" by auto
+  with ll \<open>I \<subseteq> T\<close> have llI:"local_lipschitz I (v ` I \<union> w ` I) f"
+    by (rule local_lipschitz_subset)
+  have cvwI: "compact (v ` I \<union> w ` I)"
+    by (auto intro!: compact_continuous_image cv cw cI)    
+
+  from local_lipschitz_compact_implies_lipschitz[OF llI cvwI \<open>compact I\<close> cf]
+  obtain L where L: "\<And>t. t \<in> I \<Longrightarrow> L-lipschitz_on (v ` I \<union> w ` I) (f t)"
+    using v w
+    by blast
+  define L' where "L' = max L 1"
+  with L have "L' > 0" "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L' * dist (v x) (w x)"
+     apply (auto simp: lipschitz_on_def L'_def)
+    by (smt Un_iff image_eqI mult_right_mono zero_le_dist)
+  then show ?thesis ..
+qed
+
+subsection \<open>Comparison Principle\<close>
+
+lemma comparison_principle_le:
+  fixes f::"real \<Rightarrow> real \<Rightarrow> real"
+    and \<phi> \<psi>::"real \<Rightarrow> real"
+  assumes ll: "local_lipschitz X Y f"
+  assumes cf: "\<And>x. x \<in> Y \<Longrightarrow> continuous_on {a..b} (\<lambda>t. f t x)"
+  assumes abX: "{a .. b} \<subseteq> X"
+  assumes \<phi>': "\<And>x. x \<in> {a .. b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' x) (at x)"
+  assumes \<psi>': "\<And>x. x \<in> {a .. b} \<Longrightarrow> (\<psi> has_real_derivative \<psi>' x) (at x)"
+  assumes \<phi>_in: "\<phi> ` {a..b} \<subseteq> Y"
+  assumes \<psi>_in: "\<psi> ` {a..b} \<subseteq> Y"
+  assumes init: "\<phi> a \<le> \<psi> a"
+  assumes defect: "\<And>x. x \<in> {a .. b} \<Longrightarrow> \<phi>' x - f x (\<phi> x) \<le> \<psi>' x - f x (\<psi> x)"
+  shows "\<forall>x \<in> {a .. b}. \<phi> x \<le> \<psi> x" (is "?th1")
+    (*
+    "(\<forall>x \<in> {a .. b}. \<phi> x < \<psi> x) \<or> (\<exists>c\<in>{a..b}. (\<forall>x\<in>{a..c}. \<phi> x \<le> \<psi> x) \<and> (\<forall>x\<in>{c<..b}. \<phi> x < \<psi> x))"
+    (is "?th2")
+*)
+  unfolding atomize_conj
+  apply (cases "a \<le> b")
+   defer subgoal by simp
+proof -
+  assume "a \<le> b"
+  note \<phi>_cont = has_real_derivative_imp_continuous_on[OF \<phi>']
+  note \<psi>_cont = has_real_derivative_imp_continuous_on[OF \<psi>']
+  from local_lipschitz_compact_bicomposeE[OF ll cf compact_Icc abX \<phi>_cont \<psi>_cont \<phi>_in \<psi>_in]
+  obtain L where L: "L > 0" "\<And>x. x \<in> {a..b} \<Longrightarrow> dist (f x (\<phi> x)) (f x (\<psi> x)) \<le> L * dist (\<phi> x) (\<psi> x)" by blast
+  define w where "w x = \<psi> x - \<phi> x" for x
+
+  have w'[derivative_intros]: "\<And>x. x \<in> {a .. b} \<Longrightarrow> (w has_real_derivative \<psi>' x - \<phi>' x) (at x)"
+    using \<phi>' \<psi>'
+    by (auto simp: has_vderiv_on_def w_def[abs_def] intro!: derivative_eq_intros)
+  note w_cont[continuous_intros] = has_real_derivative_imp_continuous_on[OF w', THEN continuous_on_compose2]
+  have "w d \<ge> 0" if "d \<in> {a .. b}" for d
+  proof (rule ccontr, unfold not_le)
+    assume "w d < 0"
+    let ?N = "(w -` {..0} \<inter> {a .. d})"
+    from \<open>w d < 0\<close> that have "d \<in> ?N" by auto
+    then have "?N \<noteq> {}" by auto
+    have "closed ?N"
+      unfolding compact_eq_bounded_closed
+      using that
+      by (intro conjI closed_vimage_Int) (auto intro!: continuous_intros)
+
+    let ?N' = "{a0 \<in> {a .. d}. w ` {a0 .. d} \<subseteq> {..0}}"
+    from \<open>w d < 0\<close> that have "d \<in> ?N'" by simp
+    then have "?N' \<noteq> {}" by auto
+    have "compact ?N'"
+      unfolding compact_eq_bounded_closed
+    proof
+      have "?N' \<subseteq> {a .. d}" using that by auto
+      then show "bounded ?N'"
+        by (rule bounded_subset[rotated]) simp
+      have "w u \<le> 0" if "(\<forall>n. x n \<in> ?N')" "x \<longlonglongrightarrow> l" "l \<le> u" "u \<le> d" for x l u
+      proof cases
+        assume "l = u"
+        have "\<forall>n. x n \<in> ?N" using that(1) by force
+        from closed_sequentially[OF \<open>closed ?N\<close> this \<open>x \<longlonglongrightarrow> l\<close>]
+        show ?thesis by (auto simp: \<open>l = u\<close>)
+      next
+        assume "l \<noteq> u" with that have "l < u" by auto
+        from order_tendstoD(2)[OF \<open>x \<longlonglongrightarrow> l\<close> \<open>l < u\<close>] obtain n where "x n < u"
+          by (auto dest: eventually_happens)
+        with that show ?thesis using \<open>l < u\<close>
+          by (auto dest!: spec[where x=n] simp: image_subset_iff)
+      qed
+      then show "closed ?N'"
+        unfolding closed_sequential_limits
+        by (auto simp: Lim_bounded Lim_bounded2)
+    qed
+
+    from compact_attains_inf[OF \<open>compact ?N'\<close> \<open>?N' \<noteq> {}\<close>]
+    obtain a0 where a0: "a \<le> a0" "a0 \<le> d" "w ` {a0..d} \<subseteq> {..0}"
+      and a0_least: "\<And>x. a \<le> x \<Longrightarrow> x \<le> d \<Longrightarrow> w ` {x..d} \<subseteq> {..0} \<Longrightarrow> a0 \<le> x"
+      by auto
+    have a0d: "{a0 .. d} \<subseteq> {a .. b}" using that a0
+      by auto
+    have L_w_bound: "L * w x \<le> \<psi>' x - \<phi>' x" if "x \<in> {a0 .. d}" for x
+    proof -
+      from set_mp[OF a0d that] have "x \<in> {a .. b}" .
+      from defect[OF this]
+      have "\<phi>' x - \<psi>' x \<le> dist (f x (\<phi> x)) (f x (\<psi> x))"
+        by (simp add: dist_real_def)
+      also have "\<dots> \<le> L * dist (\<phi> x) (\<psi> x)"
+        using \<open>x \<in> {a .. b}\<close>
+        by (rule L)
+      also have "\<dots> \<le> -L * w x"
+        using \<open>0 < L\<close> a0 that
+        by (force simp add: dist_real_def abs_real_def w_def algebra_split_simps )
+      finally show ?thesis
+        by simp
+    qed
+    have mono: "mono_on (\<lambda>x. w x * exp(-L*x)) {a0..d}"
+      apply (rule mono_onI)
+      apply (rule DERIV_nonneg_imp_nondecreasing, assumption)
+      using a0d
+      by (auto intro!: exI[where x="(\<psi>' x - \<phi>' x) * exp (- (L * x)) - exp (- (L * x)) * L * w x" for x]
+          derivative_eq_intros L_w_bound simp: )
+    then have "w a0 * exp (-L * a0) \<le> w d * exp (-L * d)"
+      by (rule mono_onD) (use that a0 in auto)
+    also have "\<dots> < 0" using \<open>w d < 0\<close> by (simp add: algebra_split_simps)
+    finally have "w a0 * exp (- L * a0) < 0" .
+    then have "w a0 < 0" by (simp add: algebra_split_simps)
+    have "a0 \<le> a"
+    proof (rule ccontr, unfold not_le)
+      assume "a < a0"
+      have "continuous_on {a.. a0} w"
+        by (rule continuous_intros, assumption) (use a0 a0d in auto)
+      from continuous_on_Icc_at_leftD[OF this \<open>a < a0\<close>]
+      have "(w \<longlongrightarrow> w a0) (at_left a0)" .
+      from order_tendstoD(2)[OF this \<open>w a0 < 0\<close>] have "\<forall>\<^sub>F x in at_left a0. w x < 0" .
+      moreover have "\<forall>\<^sub>F x in at_left a0. a < x"
+        by (rule order_tendstoD) (auto intro!: \<open>a < a0\<close>)
+      ultimately have "\<forall>\<^sub>F x in at_left a0. a < x \<and> w x < 0" by eventually_elim auto
+      then obtain a1' where "a1'<a0" and a1_neg: "\<And>y. y > a1' \<Longrightarrow> y < a0 \<Longrightarrow> a < y \<and> w y < 0"
+        unfolding eventually_at_left_field by auto
+      define a1 where "a1 = (a1' + a0)/2"
+      have "a1 < a0" using \<open>a1' < a0\<close> by (auto simp: a1_def)
+      have "a \<le> a1"
+        using \<open>a < a0\<close> a1_neg by (force simp: a1_def)
+      moreover have "a1 \<le> d"
+        using \<open>a1' < a0\<close> a0(2) by (auto simp: a1_def)
+      moreover have "w ` {a1..a0} \<subseteq> {..0}"
+        using \<open>w a0 < 0\<close> a1_neg a0(3)
+        by (auto simp: a1_def) smt
+      moreover have "w ` {a0..d} \<subseteq> {..0}" using a0 by auto
+      ultimately
+      have "a0 \<le> a1"
+        apply (intro a0_least) apply assumption apply assumption
+        by (smt atLeastAtMost_iff image_subset_iff)
+      with \<open>a1<a0\<close> show False by simp
+    qed
+    then have "a0 = a" using \<open>a \<le> a0\<close> by simp
+    with \<open>w a0 < 0\<close> have "w a < 0" by simp
+    with init show False
+      by (auto simp: w_def)
+  qed
+  then show ?thesis
+    by (auto simp: w_def)
+qed
+
+lemma local_lipschitz_mult:
+  shows "local_lipschitz  (UNIV::real set) (UNIV::real set) (*)"
+  apply (auto intro!: c1_implies_local_lipschitz[where f'="\<lambda>p. blinfun_mult_left (fst p)"])
+   apply (simp add: has_derivative_mult_right mult_commute_abs)
+  by (auto intro!: continuous_intros)
+
+lemma comparison_principle_le_linear:
+  fixes \<phi> :: "real \<Rightarrow> real"
+  assumes "continuous_on {a..b} g"
+  assumes "(\<And>t. t \<in> {a..b} \<Longrightarrow> (\<phi> has_real_derivative \<phi>' t) (at t))"
+  assumes "\<phi> a \<le> 0"
+  assumes "(\<And>t. t \<in> {a..b} \<Longrightarrow> \<phi>' t \<le> g t *\<^sub>R \<phi> t)"
+  shows "\<forall>t\<in>{a..b}. \<phi> t \<le> 0"
+proof -
+  have *: "\<And>x. continuous_on {a..b} (\<lambda>t. g t * x)"
+    using assms(1) continuous_on_mult_right by blast
+  then have "local_lipschitz (g`{a..b}) UNIV (*)"
+    using local_lipschitz_subset[OF local_lipschitz_mult] by blast 
+  from local_lipschitz_compose1[OF this assms(1)]
+  have "local_lipschitz {a..b} UNIV (\<lambda>t. (*) (g t))" .
+  from comparison_principle_le[OF this _ _ assms(2) _ _ _ assms(3), of b "\<lambda>t.0"] * assms(4)
+  show ?thesis by auto
+qed
+
+subsection \<open>Locally Lipschitz ODEs\<close>
+
+context ll_on_open_it begin
+
+lemma flow_lipschitzE:
+  assumes "{a .. b} \<subseteq> existence_ivl t0 x"
+  obtains L where "L-lipschitz_on {a .. b} (flow t0 x)"
+proof -
+  have f': "(flow t0 x has_derivative (\<lambda>i. i *\<^sub>R f t (flow t0 x t))) (at t within {a .. b})" if "t \<in> {a .. b}" for t
+    using flow_has_derivative[of t x] assms that
+    by (auto simp: has_derivative_at_withinI)
+
+  have "compact ((\<lambda>t. f t (flow t0 x t)) ` {a .. b})"
+    using assms
+    apply (auto intro!: compact_continuous_image continuous_intros)
+    using local.existence_ivl_empty2 apply fastforce
+     apply (meson atLeastAtMost_iff general.existence_ivl_subset in_mono)
+    by (simp add: general.flow_in_domain subset_iff)
+  then obtain C where "t \<in> {a .. b} \<Longrightarrow> norm (f t (flow t0 x t)) \<le> C" for t
+    by (fastforce dest!: compact_imp_bounded simp: bounded_iff intro: that)
+  then have "t \<in> {a..b} \<Longrightarrow> onorm (\<lambda>i. i *\<^sub>R f t (flow t0 x t)) \<le> max 0 C" for t
+    apply (subst onorm_scaleR_left) 
+     apply (auto simp: onorm_id max_def)
+    by (metis diff_0_right diff_mono diff_self norm_ge_zero)
+  from bounded_derivative_imp_lipschitz[OF f' _ this]
+  have "(max 0 C)-lipschitz_on {a..b} (flow t0 x)"
+    by auto
+  then show ?thesis ..
+qed
+
+lemma flow_undefined0: "t \<notin> existence_ivl t0 x \<Longrightarrow> flow t0 x t = 0"
+  unfolding flow_def by auto
+
+lemma csols_undefined: "x \<notin> X \<Longrightarrow> csols t0 x = {}"
+  apply (auto simp: csols_def)
+  using general.existence_ivl_empty2 general.existence_ivl_maximal_segment
+  apply blast
+  done
+
+lemmas existence_ivl_undefined = existence_ivl_empty2
+
+end
+
+subsection \<open>Reverse flow as Sublocale\<close>
+
+lemma range_preflect_0[simp]: "range (preflect 0) = UNIV"
+  by (auto simp: preflect_def)
+lemma range_uminus[simp]: "range uminus = (UNIV::'a::ab_group_add set)"
+  by auto
+
+context auto_ll_on_open begin
+
+sublocale rev: auto_ll_on_open "-f" rewrites "-(-f) = f"
+   apply unfold_locales
+  using auto_local_lipschitz auto_open_domain
+  unfolding fun_Compl_def local_lipschitz_minus
+  by auto
+
+lemma existence_ivl_eq_rev0: "existence_ivl0 y = uminus ` rev.existence_ivl0 y" for y
+  by (auto simp: existence_ivl_eq_rev rev.existence_ivl0_def preflect_def)
+
+lemma rev_existence_ivl_eq0: "rev.existence_ivl0 y = uminus ` existence_ivl0 y" for y
+  using uminus_uminus_image[of "rev.existence_ivl0 y"]
+  by (simp add: existence_ivl_eq_rev0)
+
+lemma flow_eq_rev0: "flow0 y t = rev.flow0 y (-t)" for y t
+  apply (cases "t \<in> existence_ivl0 y")
+  subgoal
+    apply (subst flow_eq_rev(2), assumption)
+    apply (subst rev.flow0_def)
+    by (simp add: preflect_def)
+  subgoal
+    apply (frule flow_undefined0)
+    by (auto simp: existence_ivl_eq_rev0 rev.flow_undefined0)
+  done
+
+lemma rev_eq_flow: "rev.flow0 y t = flow0 y (-t)" for y t
+  apply (subst flow_eq_rev0)
+  using uminus_uminus_image[of "rev.existence_ivl0 y"]
+  apply -
+  apply (subst (asm) existence_ivl_eq_rev0[symmetric])
+  by auto
+
+lemma rev_flow_image_eq: "rev.flow0 x ` S = flow0 x ` (uminus ` S)"
+  unfolding rev_eq_flow[abs_def]
+  by force
+
+lemma flow_image_eq_rev: "flow0 x ` S = rev.flow0 x ` (uminus ` S)"
+  unfolding rev_eq_flow[abs_def]
+  by force
+
+end
+
+context c1_on_open begin
+
+sublocale rev: c1_on_open "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
+  by (rule c1_on_open_rev) auto
+
+end
+
+context c1_on_open_euclidean begin
+
+sublocale rev: c1_on_open_euclidean "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
+  by unfold_locales auto
+
+end
+
+
+subsection \<open>Autonomous LL ODE : Existence Interval and trapping on the interval\<close>
+
+lemma bdd_above_is_intervalI: "bdd_above I"
+  if "is_interval I" "a \<le> b" "a \<in> I" "b \<notin> I" for I::"real set"
+  by (meson bdd_above_def is_interval_1 le_cases that) 
+
+lemma bdd_below_is_intervalI: "bdd_below I"
+  if "is_interval I" "a \<le> b" "a \<notin> I" "b \<in> I" for I::"real set"
+  by (meson bdd_below_def is_interval_1 le_cases that) 
+
+context auto_ll_on_open begin
+
+lemma open_existence_ivl0:
+  assumes x : "x \<in> X"
+  shows "\<exists>a b. a < 0 \<and> 0 < b \<and> {a..b} \<subseteq> existence_ivl0 x"
+proof -
+  have a1:"0 \<in> existence_ivl0 x"
+    by (simp add: x)
+  have a2: "open (existence_ivl0 x)"
+    by (simp add: x)
+  from a1 a2 obtain d where "d > 0" "ball 0 d \<subseteq> existence_ivl0 x"
+    using openE by blast
+  have "{-d/2..d/2} \<subseteq> ball 0 d"
+    using \<open>0 < d\<close> dist_norm mem_ball by auto
+  thus ?thesis
+    by (smt \<open>0 < d\<close> \<open>ball 0 d \<subseteq> existence_ivl0 x\<close> divide_minus_left half_gt_zero order_trans)
+qed
+
+lemma open_existence_ivl':
+  assumes x : "x \<in> X"
+  obtains a where "a > 0"  "{-a..a} \<subseteq> existence_ivl0 x"
+proof -
+  from open_existence_ivl0[OF assms(1)]
+  obtain a b where ab: "a < 0" "0 < b" "{a..b} \<subseteq> existence_ivl0 x" by auto
+  then have "min (-a) b > 0" by linarith
+  have "{-min (-a) b .. min(-a) b} \<subseteq> {a..b}" by auto
+  thus ?thesis using ab(3) that[OF \<open>min (-a) b > 0\<close>] by blast
+qed
+
+lemma open_existence_ivl_on_compact:
+  assumes C: "C \<subseteq> X" and "compact C" "C \<noteq> {}"
+  obtains a where "a > 0" "\<And>x. x \<in> C \<Longrightarrow> {-a..a} \<subseteq> existence_ivl0 x"
+proof -
+  from existence_ivl_cballs
+  have "\<forall>x\<in>C. \<exists>e>0. \<exists>t>0. \<forall>y\<in>cball x e. cball 0 t\<subseteq>existence_ivl0 y"
+    by (metis (full_types) C Int_absorb1 Int_iff UNIV_I)
+  then
+  obtain d' t' where *:
+    "\<forall>x\<in>C. 0 < d' x \<and> t' x > 0 \<and> (\<forall>y\<in>cball x (d' x). cball 0 (t' x) \<subseteq> existence_ivl0 y)"
+    by metis
+  with compactE_image[OF \<open>compact C\<close>, of C "\<lambda>x. ball x (d' x)"]
+  obtain C' where "C' \<subseteq> C" and [simp]: "finite C'" and C_subset: "C \<subseteq> (\<Union>c\<in>C'. ball c (d' c))"
+    by force
+  from C_subset \<open>C \<noteq> {}\<close> have [simp]: "C' \<noteq> {}" by auto
+  define d where "d = Min (d' ` C')"
+  define t where "t = Min (t' ` C')"
+  have "t > 0" using * \<open>C' \<subseteq> C\<close>
+    by (auto simp: t_def)
+  moreover have "{-t .. t} \<subseteq> existence_ivl0 x" if "x \<in> C" for x
+  proof -
+    from C_subset that \<open>C' \<subseteq> C\<close>
+    obtain c where c: "c \<in> C'" "x \<in> ball c (d' c)" "c \<in> C" by force
+    then have "{-t .. t} \<subseteq> cball 0 (t' c)"
+      by (auto simp: abs_real_def t_def minus_le_iff)
+    also
+    from c have "cball 0 (t' c) \<subseteq> existence_ivl0 x"
+      using *[rule_format, OF \<open>c \<in> C\<close>] by auto
+    finally show ?thesis .
+  qed
+  ultimately show ?thesis ..
+qed
+
+definition "trapped_forward x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> K)"
+  \<comment> \<open>TODO: use this for backwards trapped, invariant, and all assumptions\<close>
+
+definition "trapped_backward x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x \<inter> {..0}) \<subseteq> K)"
+
+definition "trapped x K \<longleftrightarrow> trapped_forward x K \<and> trapped_backward x K"
+
+lemma trapped_iff_on_existence_ivl0:
+  "trapped x K \<longleftrightarrow> (flow0 x ` (existence_ivl0 x) \<subseteq> K)"
+  unfolding trapped_def trapped_forward_def trapped_backward_def
+  apply (auto)
+  by (metis IntI atLeast_iff atMost_iff image_subset_iff less_eq_real_def linorder_not_less)
+end
+
+context auto_ll_on_open begin
+
+lemma infinite_rev_existence_ivl0_rewrites:
+  "{0..} \<subseteq> rev.existence_ivl0 x \<longleftrightarrow> {..0} \<subseteq> existence_ivl0 x"
+  "{..0} \<subseteq> rev.existence_ivl0 x \<longleftrightarrow> {0..} \<subseteq> existence_ivl0 x"
+   apply (auto simp add: rev.rev_existence_ivl_eq0 subset_iff)
+  using neg_le_0_iff_le apply fastforce
+  using neg_0_le_iff_le by fastforce
+
+lemma trapped_backward_iff_rev_trapped_forward:
+  "trapped_backward x K  \<longleftrightarrow> rev.trapped_forward x K"
+  unfolding trapped_backward_def rev.trapped_forward_def
+  by (auto simp add: rev_flow_image_eq existence_ivl_eq_rev0 image_subset_iff)
+
+text \<open>If solution is trapped in a compact set at some time
+  on its existence interval then it is trapped forever\<close>
+lemma trapped_sol_right:
+  \<comment> \<open>TODO: when building on afp-devel (??? outdated):
+    \<^url>\<open>https://bitbucket.org/isa-afp/afp-devel/commits/0c3edf9248d5389197f248c723b625c419e4d3eb\<close>\<close>
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  shows "{0..} \<subseteq> existence_ivl0 x"
+proof (rule ccontr)
+  assume "\<not> {0..} \<subseteq> existence_ivl0 x"
+  from this obtain t where "0 \<le> t" "t \<notin> existence_ivl0 x" by blast
+  then have bdd: "bdd_above (existence_ivl0 x)" 
+    by (auto intro!: bdd_above_is_intervalI \<open>x \<in> X\<close>)
+  from flow_leaves_compact_ivl_right [OF UNIV_I \<open>x \<in> X\<close> bdd UNIV_I assms(1-2)]
+  show False by (metis assms(4) trapped_forward_def IntI atLeast_iff image_subset_iff)
+qed
+
+lemma trapped_sol_right_gen:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "t \<in> existence_ivl0 x" "trapped_forward (flow0 x t) K"
+  shows "{t..} \<subseteq> existence_ivl0 x"
+proof -
+  have "x \<in> X"
+    using assms(3) local.existence_ivl_empty_iff by fastforce 
+  have xtk: "flow0 x t \<in> X"
+    by (simp add: assms(3) local.flow_in_domain)
+  from trapped_sol_right[OF assms(1-2) xtk assms(4)] have "{0..} \<subseteq> existence_ivl0 (flow0 x t)" .
+  thus "{t..} \<subseteq> existence_ivl0 x"
+    using existence_ivl_trans[OF assms(3)]
+    by (metis add.commute atLeast_iff diff_add_cancel le_add_same_cancel1 subset_iff)
+qed
+
+lemma trapped_sol_left:
+  \<comment> \<open>TODO: when building on afp-devel:
+    \<^url>\<open>https://bitbucket.org/isa-afp/afp-devel/commits/0c3edf9248d5389197f248c723b625c419e4d3eb\<close>\<close>
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_backward x K"
+  shows "{..0} \<subseteq> existence_ivl0 x"
+proof (rule ccontr)
+  assume "\<not> {..0} \<subseteq> existence_ivl0 x"
+  from this obtain t where "t \<le> 0" "t \<notin> existence_ivl0 x" by blast
+  then have bdd: "bdd_below (existence_ivl0 x)" 
+    by (auto intro!: bdd_below_is_intervalI \<open>x \<in> X\<close>)
+  from flow_leaves_compact_ivl_left [OF UNIV_I \<open>x \<in> X\<close> bdd UNIV_I assms(1-2)]
+  show False
+    by (metis IntI assms(4) atMost_iff auto_ll_on_open.trapped_backward_def auto_ll_on_open_axioms image_subset_iff)
+qed
+
+lemma trapped_sol_left_gen:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "t \<in> existence_ivl0 x" "trapped_backward (flow0 x t) K"
+  shows "{..t} \<subseteq> existence_ivl0 x"
+proof -
+  have "x \<in> X"
+    using assms(3) local.existence_ivl_empty_iff by fastforce 
+  have xtk: "flow0 x t \<in> X"
+    by (simp add: assms(3) local.flow_in_domain)
+  from trapped_sol_left[OF assms(1-2) xtk assms(4)] have "{..0} \<subseteq> existence_ivl0 (flow0 x t)" .
+  thus "{..t} \<subseteq> existence_ivl0 x"
+    using existence_ivl_trans[OF assms(3)]
+    by (metis add.commute add_le_same_cancel1 atMost_iff diff_add_cancel subset_eq)
+qed
+
+lemma trapped_sol:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped x K"
+  shows "existence_ivl0 x = UNIV"
+  by (metis (mono_tags, lifting) assms existence_ivl_zero image_subset_iff interval local.existence_ivl_initial_time_iff local.existence_ivl_subset local.subset_mem_compact_implies_subset_existence_interval order_refl subset_antisym trapped_iff_on_existence_ivl0)
+
+(* Follows from rectification *)
+lemma regular_locally_noteq:\<comment> \<open>TODO: should be true in \<open>ll_on_open_it\<close>\<close>
+  assumes "x \<in> X" "f x \<noteq> 0"
+  shows "eventually (\<lambda>t. flow0 x t \<noteq> x) (at 0)"
+proof -
+  have nf:"norm (f x) > 0" by (simp add: assms(2)) 
+      (* By continuity of solutions and f probably *)
+  obtain a where
+    a: "a>0"
+    "{-a--a} \<subseteq> existence_ivl0 x"
+    "0 \<in> {-a--a}"
+    "\<And>t. t \<in> {-a--a} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2"
+  proof -
+    from open_existence_ivl'[OF assms(1)]
+    obtain a1 where a1: "a1 > 0" "{-a1..a1} \<subseteq> existence_ivl0 x" .
+    have "continuous (at 0) (\<lambda>t. norm(f (flow0 x t) - f (flow0 x 0) ))"
+      apply (auto intro!: continuous_intros)
+      by (simp add: assms(1) local.f_flow_continuous)
+    then obtain a2 where "a2>0"
+      "\<forall>t. norm t < a2 \<longrightarrow>
+             norm (f (flow0 x t) - f (flow0 x 0)) < norm(f x)/2"
+      unfolding continuous_at_real_range
+      by (metis abs_norm_cancel cancel_comm_monoid_add_class.diff_cancel diff_zero half_gt_zero nf norm_zero)
+    then have 
+      t: "\<And>t. t \<in> {-a2<--<a2} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2"
+      by (smt open_segment_bound(2) open_segment_bound1 real_norm_def)
+    define a where "a = min a1 (a2/2)"
+    have t1:"a > 0" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close> by auto
+    then have t3:"0 \<in>{-a--a}"
+      using closed_segment_eq_real_ivl by auto
+    have "{-a--a} \<subseteq> {-a1..a1}" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close>
+      using ODE_Auxiliarities.closed_segment_eq_real_ivl by auto
+    then have t2:"{-a--a} \<subseteq> existence_ivl0 x" using a1 by auto
+    have "{-a--a} \<subseteq> {-a2<--<a2}" unfolding a_def using \<open>a1 > 0\<close> \<open>a2 > 0\<close>
+      by (smt Diff_iff closed_segment_eq_real_ivl atLeastAtMost_iff empty_iff half_gt_zero insert_iff pos_half_less segment(1) subset_eq)
+    then have t4:"\<And>t. t \<in> {-a--a} \<Longrightarrow> norm(f (flow0 x t) - f (flow0 x 0)) \<le> norm(f x)/2" using t by auto
+    show ?thesis using t1 t2 t3 t4 that by auto
+  qed
+  have "\<And>t. t \<in> {-a--a} \<Longrightarrow> (flow0 x has_vector_derivative f (flow0 x t)) (at t within {-a--a})"
+    apply (rule has_vector_derivative_at_within)
+    using a(2) by (auto intro!:flow_has_vector_derivative)
+  from vector_differentiable_bound_linearization[OF this _ a(4)]
+  have nb:"\<And>c d. {c--d} \<subseteq> {-a--a} \<Longrightarrow>
+   norm (flow0 x d - flow0 x c - (d - c) *\<^sub>R f (flow0 x 0))  \<le> norm (d - c) * (norm (f x) / 2)"
+    using a(3) by blast
+  have "\<And>t. dist t 0 < a \<Longrightarrow> t \<noteq> 0 \<Longrightarrow> flow0 x t \<noteq> x"
+  proof (rule ccontr)
+    fix t
+    assume "dist t 0 < a" "t \<noteq> 0" "\<not> flow0 x t \<noteq> x"
+    then have tx:"flow0 x t = x" by auto
+    have "t \<in> {-a--a}"
+      using closed_segment_eq_real_ivl \<open>dist t 0 < a\<close> by auto
+    have "t > 0 \<or> t < 0" using \<open>t \<noteq> 0\<close> by linarith
+    moreover {
+      assume "t > 0"
+      then have "{0--t} \<subseteq> {-a--a}"
+        by (simp add: \<open>t \<in> {-a--a}\<close> a(3) subset_closed_segment) 
+      from nb[OF this] have
+        "norm (flow0 x t - x - t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)"
+        by (simp add: assms(1))
+      then have "norm (t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)" using tx by auto
+      then have False using nf
+        using \<open>0 < t\<close> by auto 
+    }
+    moreover {
+      assume "t < 0"
+      then have "{t--0} \<subseteq> {-a--a}"
+        by (simp add: \<open>t \<in> {-a--a}\<close> a(3) subset_closed_segment) 
+      from nb[OF this] have
+        "norm (x - flow0 x t + t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)"
+        by (simp add: assms(1))
+      then have "norm (t *\<^sub>R f x) \<le> norm t * (norm (f x) / 2)" using tx by auto
+      then have False using nf
+        using \<open>t < 0\<close> by auto 
+    }
+    ultimately show False by blast
+  qed
+  thus ?thesis unfolding eventually_at
+    using a(1) by blast
+qed
+
+lemma compact_max_time_flow_in_closed:
+  assumes "closed M" and t_ex: "t \<in> existence_ivl0 x"
+  shows "compact {s \<in> {0..t}. flow0 x ` {0..s} \<subseteq> M}" (is "compact ?C")
+  unfolding compact_eq_bounded_closed
+proof
+  have "bounded {0 .. t}" by auto
+  then show "bounded ?C"
+    by (rule bounded_subset) auto
+  show "closed ?C"
+    unfolding closed_def
+  proof (rule topological_space_class.openI, clarsimp)
+    \<comment> \<open>TODO: there must be a more abstract argument for this, e.g., with
+      @{thm continuous_on_closed_vimageI} and then reasoning about the connected component around 0?\<close>
+    fix s
+    assume notM: "s \<le> t \<longrightarrow> 0 \<le> s \<longrightarrow> \<not> flow0 x ` {0..s} \<subseteq> M"
+    consider "0 \<le> s" "s \<le> t" "flow0 x s \<notin> M" | "0 \<le> s" "s \<le> t" "flow0 x s \<in> M" | "s < 0" | "s > t"
+      by arith
+    then show "\<exists>T. open T \<and> s \<in> T \<and> T \<subseteq> - {s. 0 \<le> s \<and> s \<le> t \<and> flow0 x ` {0..s} \<subseteq> M}"
+    proof cases
+      assume s: "0 \<le> s" "s \<le> t" and sM: "flow0 x s \<notin> M"
+      have "isCont (flow0 x) s"
+        using s ivl_subset_existence_ivl[OF t_ex]
+        by (auto intro!: flow_continuous)
+      from this[unfolded continuous_at_open, rule_format, of "-M"] sM \<open>closed M\<close>
+      obtain S where "open S" "s \<in> S" "(\<forall>x'\<in>S. flow0 x x' \<in> - M)"
+        by auto
+      then show ?thesis
+        by (force intro!: exI[where x=S])
+    next
+      assume s: "0 \<le> s" "s \<le> t" and sM: "flow0 x s \<in> M"
+      from this notM obtain s0 where s0: "0 \<le> s0" "s0 < s" "flow0 x s0 \<notin> M"
+        by force
+      from order_tendstoD(1)[OF tendsto_ident_at \<open>s0 < s\<close>, of UNIV, unfolded eventually_at_topological]
+      obtain S where "open S" "s \<in> S" "\<And>x. x \<in> S \<Longrightarrow> x \<noteq> s \<Longrightarrow> s0 < x"
+        by auto
+      then show ?thesis using s0
+        by (auto simp: intro!: exI[where x=S]) (smt atLeastAtMost_iff image_subset_iff)
+    qed (force intro: exI[where x="{t<..}"] exI[where x="{..<0}"])+
+  qed
+qed
+
+lemma flow_in_closed_max_timeE:
+  assumes "closed M" "t \<in> existence_ivl0 x" "0 \<le> t" "x \<in> M"
+  obtains T where "0 \<le> T" "T \<le> t" "flow0 x ` {0..T} \<subseteq> M"
+    "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> T"
+proof -
+  let ?C = "{s \<in> {0..t}. flow0 x ` {0..s} \<subseteq> M}"
+  have "?C \<noteq> {}"
+    using assms
+    using local.mem_existence_ivl_iv_defined
+    by (auto intro!: exI[where x=0])
+  from compact_max_time_flow_in_closed[OF assms(1,2)]
+  have "compact ?C" .
+  from compact_attains_sup[OF this \<open>?C \<noteq> {}\<close>]
+  obtain s where s: "0 \<le> s" "s \<le> t" "flow0 x ` {0..s} \<subseteq> M"
+    and s_max: "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> s"
+    by auto
+  then show ?thesis ..
+qed
+
+lemma flow_leaves_closed_at_frontierE:
+  assumes "closed M" and t_ex: "t \<in> existence_ivl0 x" and "0 \<le> t" "x \<in> M" "flow0 x t \<notin> M"
+  obtains s where "0 \<le> s" "s < t" "flow0 x ` {0..s} \<subseteq> M"
+    "flow0 x s \<in> frontier M"
+    "\<exists>\<^sub>F s' in at_right s. flow0 x s' \<notin> M"
+proof -
+  from flow_in_closed_max_timeE[OF assms(1-4)] assms(5)
+  obtain s where s: "0 \<le> s" "s < t" "flow0 x ` {0..s} \<subseteq> M"
+    and s_max: "\<And>s'. 0 \<le> s' \<Longrightarrow> s' \<le> t \<Longrightarrow> flow0 x ` {0..s'} \<subseteq> M \<Longrightarrow> s' \<le> s"
+    by (smt atLeastAtMost_iff image_subset_iff)
+  note s
+  moreover
+  have "flow0 x s \<notin> interior M"
+  proof
+    assume interior: "flow0 x s \<in> interior M"
+    have "s \<in> existence_ivl0 x" using ivl_subset_existence_ivl[OF \<open>t \<in> _\<close>] s by auto
+    from flow_continuous[OF this, THEN isContD, THEN topological_tendstoD, OF open_interior interior]
+    have "\<forall>\<^sub>F s' in at s. flow0 x s' \<in> interior M" by auto
+    then have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> interior M"
+      by (auto simp: eventually_at_split)
+    moreover have "\<forall>\<^sub>F s' in at_right s. s' < t"
+      using tendsto_ident_at \<open>s < t\<close>
+      by (rule order_tendstoD)
+    ultimately have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> M \<and> s' < t"
+      by eventually_elim (use interior_subset[of M] in auto)
+    then obtain s' where s': "s < s'" "s' < t" "\<And>y. y > s \<Longrightarrow> y \<le> s' \<Longrightarrow> flow0 x y \<in> M"
+      by (auto simp: eventually_at_right_field_le)
+    have s'_ivl: "flow0 x ` {0..s'} \<subseteq> M"
+    proof safe
+      fix s'' assume "s'' \<in> {0 .. s'}"
+      then show "flow0 x s'' \<in> M"
+        using s interior_subset[of M] s'
+        by (cases "s'' \<le> s") auto
+    qed
+    with s_max[of s'] \<open>s' < t\<close> \<open>0 \<le> s\<close> \<open>s < s'\<close> show False by auto
+  qed
+  then have "flow0 x s \<in> frontier M"
+    using s closure_subset[of M]
+    by (force simp: frontier_def)
+  moreover
+  have "compact (flow0 x -` M \<inter> {s..t})" (is "compact ?C")
+    unfolding compact_eq_bounded_closed
+  proof
+    have "bounded {s .. t}" by simp
+    then show "bounded ?C"
+      by (rule bounded_subset) auto
+    show "closed ?C"
+      using \<open>closed M\<close> assms mem_existence_ivl_iv_defined(2)[OF t_ex] ivl_subset_existence_ivl[OF t_ex] \<open>0 \<le> s\<close>
+      by (intro closed_vimage_Int) (auto intro!: continuous_intros)
+  qed
+  have "\<exists>\<^sub>F s' in at_right s. flow0 x s' \<notin> M"
+    apply (rule ccontr)
+    unfolding not_frequently
+  proof -
+    assume "\<forall>\<^sub>F s' in at_right s. \<not> flow0 x s' \<notin> M"
+    moreover have "\<forall>\<^sub>F s' in at_right s. s' < t"
+      using tendsto_ident_at \<open>s < t\<close>
+      by (rule order_tendstoD)
+    ultimately have "\<forall>\<^sub>F s' in at_right s. flow0 x s' \<in> M \<and> s' < t" by eventually_elim auto
+    then obtain s' where s': "s < s'"
+      "\<And>y. y > s \<Longrightarrow> y < s' \<Longrightarrow> flow0 x y \<in> M"
+      "\<And>y. y > s \<Longrightarrow> y < s' \<Longrightarrow> y < t"
+      by (auto simp: eventually_at_right_field)
+    define s'' where "s'' = (s + s') / 2"
+    have "0 \<le> s''" "s'' \<le> t"  "s < s''" "s'' < s'"
+      using s s'
+      by (auto simp del: divide_le_eq_numeral1 le_divide_eq_numeral1 simp: s''_def) fastforce
+    then have "flow0 x ` {0..s''} \<subseteq> M"
+      using s s'
+      apply (auto simp: )
+      subgoal for u
+        by (cases "u\<le>s") auto
+      done
+    from s_max[OF \<open>0 \<le> s''\<close> \<open>s''\<le> t\<close> this] \<open>s'' > s\<close>
+    show False by simp
+  qed
+  ultimately show ?thesis ..
+qed
+
+
+subsection \<open>Connectedness\<close>
+
+lemma fcontX:
+  shows "continuous_on X f"
+  using auto_local_lipschitz local_lipschitz_continuous_on by blast
+
+lemma fcontx:
+  assumes "x \<in> X"
+  shows "continuous (at x) f"
+proof -
+  have "open X" by simp
+  from continuous_on_eq_continuous_at[OF this]
+  show ?thesis using fcontX assms(1) by blast
+qed
+
+lemma continuous_at_imp_cball:
+  assumes "continuous (at x) g"
+  assumes "g x > (0::real)"
+  obtains r where "r > 0" "\<forall>y \<in> cball x r. g y > 0"
+proof -
+  from assms(1)
+  obtain d where "d>0" "g ` (ball x d) \<subseteq> ball (g x) ((g x)/2)"
+    by (meson assms(2) continuous_at_ball half_gt_zero)
+  then have "\<forall>y \<in> cball x (d/2). g y > 0"
+    by (smt assms(2) dist_norm image_subset_iff mem_ball mem_cball pos_half_less real_norm_def)
+  thus ?thesis
+    using \<open>0 < d\<close> that half_gt_zero by blast
+qed
+
+text \<open> \<open>flow0\<close> is \<open>path_connected\<close> \<close>
+lemma flow0_path_connected_time:
+  assumes "ts \<subseteq> existence_ivl0 x" "path_connected ts"
+  shows "path_connected (flow0 x ` ts)"
+proof -
+  have "continuous_on ts (flow0 x)"
+    by (meson assms continuous_on_sequentially flow_continuous_on subsetD)
+  from path_connected_continuous_image[OF this assms(2)] 
+  show ?thesis .
+qed
+
+lemma flow0_path_connected:
+  assumes "path_connected D"
+    "path_connected ts"
+    "\<And>x. x \<in> D \<Longrightarrow> ts \<subseteq> existence_ivl0 x"
+  shows "path_connected ( (\<lambda>(x, y). flow0 x y) ` (D \<times> ts))"
+proof -
+  have "D \<times> ts \<subseteq> Sigma X existence_ivl0"
+    using assms(3) subset_iff by fastforce
+  then have a1:"continuous_on (D \<times> ts) (\<lambda>(x, y). flow0 x y)"
+    using flow_continuous_on_state_space continuous_on_subset by blast 
+  have a2 : "path_connected (D \<times> ts)" using path_connected_Times assms by auto
+  from path_connected_continuous_image[OF a1 a2]
+  show ?thesis .
+qed
+
+end
+
+subsection \<open>Return Time and Implicit Function Theorem\<close>
+
+context c1_on_open_euclidean begin
+
+lemma flow_implicit_function:
+  \<comment> \<open>TODO: generalization of @{thm returns_to_implicit_function}!\<close>
+  fixes s::"'a::euclidean_space \<Rightarrow> real" and S::"'a set"
+  assumes t: "t \<in> existence_ivl0 x" and x: "x \<in> X" and st: "s (flow0 x t) = 0"
+  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
+  assumes DsC: "isCont Ds (flow0 x t)"
+  assumes nz: "Ds (flow0 x t) (f (flow0 x t)) \<noteq> 0"
+  obtains u e
+  where "s (flow0 x (u x)) = 0"
+    "u x = t"
+    "(\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0)"
+    "continuous_on (cball x e) u"
+    "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"
+    "0 < e" "(u has_derivative (- blinfun_scaleR_left
+                   (inverse (blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)))) o\<^sub>L
+                      (Ds (flow0 x t) o\<^sub>L flowderiv x t) o\<^sub>L embed1_blinfun)) (at x)"
+proof -
+  note [derivative_intros] = has_derivative_compose[OF _ Ds]
+  have cont_s: "continuous_on UNIV s" by (rule has_derivative_continuous_on[OF Ds])
+  note cls[simp, intro] = closed_levelset[OF cont_s]
+  then have xt1: "(x, t) \<in> Sigma X existence_ivl0"
+    by (auto simp: t x)
+  have D: "(\<And>x. x \<in> Sigma X existence_ivl0 \<Longrightarrow>
+      ((\<lambda>(x, t). s (flow0 x t)) has_derivative
+       blinfun_apply (Ds (flow0 (fst x) (snd x)) o\<^sub>L (flowderiv (fst x) (snd x))))
+       (at x))"
+    by (auto intro!: derivative_eq_intros)
+  have C: "isCont (\<lambda>x. Ds (flow0 (fst x) (snd x)) o\<^sub>L flowderiv (fst x) (snd x))
+   (x, t)"
+    using flowderiv_continuous_on[unfolded continuous_on_eq_continuous_within,
+        rule_format, OF xt1]
+    using at_within_open[OF xt1 open_state_space]
+    by (auto intro!: continuous_intros tendsto_eq_intros x t
+        isCont_tendsto_compose[OF DsC, unfolded poincare_map_def]
+        simp: split_beta' isCont_def)
+  have Z: "(case (x, t) of (x, t) \<Rightarrow> s (flow0 x t)) = 0"
+    by (auto simp: st)
+  have I1: "blinfun_scaleR_left (inverse (Ds (flow0 x t)(f (flow0 x t)))) o\<^sub>L 
+    ((Ds (flow0 (fst (x, t))
+            (snd (x, t))) o\<^sub>L
+       flowderiv (fst (x, t))
+        (snd (x, t))) o\<^sub>L
+      embed2_blinfun)
+     = 1\<^sub>L"
+    using nz
+    by (auto intro!: blinfun_eqI
+        simp: flowderiv_def blinfun.bilinear_simps inverse_eq_divide poincare_map_def)
+  have I2: "((Ds (flow0 (fst (x, t))
+            (snd (x, t))) o\<^sub>L
+       flowderiv (fst (x, t))
+        (snd (x, t))) o\<^sub>L
+      embed2_blinfun) o\<^sub>L blinfun_scaleR_left (inverse (Ds (flow0 x t)(f (flow0 x t))))
+     = 1\<^sub>L"
+    using nz
+    by (auto intro!: blinfun_eqI
+        simp: flowderiv_def blinfun.bilinear_simps inverse_eq_divide poincare_map_def)
+  show ?thesis
+    apply (rule implicit_function_theorem[where f="\<lambda>(x, t). s (flow0 x t)"
+          and S="Sigma X existence_ivl0", OF D xt1 open_state_space order_refl C Z I1 I2])
+     apply blast
+    unfolding split_beta' fst_conv snd_conv poincare_map_def[symmetric]
+    ..
+qed
+
+lemma flow_implicit_function_at:
+  fixes s::"'a::euclidean_space \<Rightarrow> real" and S::"'a set"
+  assumes x: "x \<in> X" and st: "s x = 0"
+  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
+  assumes DsC: "isCont Ds x"
+  assumes nz: "Ds x (f x) \<noteq> 0"
+  assumes pos: "e > 0"
+  obtains u d
+  where
+    "0 < d"
+    "u x = 0"
+    "\<And>y. y \<in> cball x d \<Longrightarrow> s (flow0 y (u y)) = 0"
+    "\<And>y. y \<in> cball x d \<Longrightarrow> \<bar>u y\<bar> < e"
+    "\<And>y. y \<in> cball x d \<Longrightarrow> u y \<in> existence_ivl0 y"
+    "continuous_on (cball x d) u"
+    "(u has_derivative -Ds x /\<^sub>R (Ds x) (f x)) (at x)"
+proof -
+  have x0: "flow0 x 0 = x" by (simp add: x)
+  from flow_implicit_function[OF existence_ivl_zero[OF x] x, unfolded x0, of s, OF st Ds DsC nz]
+  obtain u d0 where
+    s0: "s (flow0 x (u x)) = 0"
+    and u0: "u x = 0"
+    and u: "\<And>y. y \<in> cball x d0 \<Longrightarrow> s (flow0 y (u y)) = 0"
+    and uc: "continuous_on (cball x d0) u"
+    and uex: "(\<lambda>t. (t, u t)) ` cball x d0 \<subseteq> Sigma X existence_ivl0"
+    and d0: "0 < d0"
+    and u': "(u has_derivative
+     blinfun_apply
+      (- blinfun_scaleR_left (inverse (blinfun_apply (Ds x) (f x))) o\<^sub>L (Ds x o\<^sub>L flowderiv x 0) o\<^sub>L embed1_blinfun))
+     (at x)"
+    by blast
+  have "at x within cball x d0 = at x" by (rule at_within_interior) (auto simp: \<open>0 < d0\<close>)
+  then have "(u \<longlongrightarrow> 0) (at x)"
+    using uc d0 by (auto simp: continuous_on_def u0 dest!: bspec[where x=x])
+  from tendstoD[OF this \<open>0 < e\<close>] pos u0
+  obtain d1 where d1: "0 < d1" "\<And>xa. dist xa x \<le> d1 \<Longrightarrow> \<bar>u xa\<bar> < e"
+    unfolding eventually_at_le
+    by force
+  define d where "d = min d0 d1"
+  have "0 < d" by (auto simp: d_def d0 d1)
+  moreover note u0
+  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> s (flow0 y (u y)) = 0" by (auto intro!: u simp: d_def)
+  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> \<bar>u y\<bar> < e" using d1 by (auto simp: d_def dist_commute)
+  moreover have "\<And>y. y \<in> cball x d \<Longrightarrow> u y \<in> existence_ivl0 y"
+    using uex by (force simp: d_def)
+  moreover have "continuous_on (cball x d) u"
+    using uc by (rule continuous_on_subset) (auto simp: d_def)
+  moreover
+  have "(u has_derivative -Ds x /\<^sub>R (Ds x) (f x)) (at x)"
+    using u'
+    by (rule has_derivative_subst) (auto intro!: ext simp: x x0 flowderiv_def blinfun.bilinear_simps)
+  ultimately show ?thesis ..
+qed
+
+lemma returns_to_implicit_function_gen:
+  \<comment> \<open>TODO: generalizes proof of @{thm returns_to_implicit_function}!\<close>
+  fixes s::"'a::euclidean_space \<Rightarrow> real"
+  assumes rt: "returns_to {x \<in> S. s x = 0} x" (is "returns_to ?P x")
+  assumes cS: "closed S"
+  assumes Ds: "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"
+    "isCont Ds (poincare_map ?P x)"
+    "Ds (poincare_map ?P x) (f (poincare_map ?P x)) \<noteq> 0"
+  obtains u e
+  where "s (flow0 x (u x)) = 0"
+    "u x = return_time ?P x"
+    "(\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0)"
+    "continuous_on (cball x e) u"
+    "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"
+    "0 < e" "(u has_derivative (- blinfun_scaleR_left
+                   (inverse (blinfun_apply (Ds (poincare_map ?P x)) (f (poincare_map ?P x)))) o\<^sub>L
+                      (Ds (poincare_map ?P x) o\<^sub>L flowderiv x (return_time ?P x)) o\<^sub>L embed1_blinfun)) (at x)"
+proof -
+  note [derivative_intros] = has_derivative_compose[OF _ Ds(1)]
+  have cont_s: "continuous_on UNIV s" by (rule has_derivative_continuous_on[OF Ds(1)])
+  note cls[simp, intro] = closed_levelset[OF cont_s]
+  let ?t1 = "return_time ?P x"
+  have cls[simp, intro]: "closed {x \<in> S. s x = 0}"
+    by (rule closed_levelset_within) (auto intro!: cS continuous_on_subset[OF cont_s])
+
+  have *: "poincare_map ?P x = flow0 x (return_time {x \<in> S. s x = 0} x)"
+    by (simp add: poincare_map_def)
+  have "return_time {x \<in> S. s x = 0} x \<in> existence_ivl0 x"
+    "x \<in> X"
+    "s (poincare_map ?P x) = 0"
+    using poincare_map_returns rt
+    by (auto intro!: return_time_exivl rt)
+  note E = flow_implicit_function[of "return_time ?P x" x s Ds, OF this[unfolded *] Ds[unfolded *],
+      folded *]
+  show ?thesis
+    by (rule E) rule
+qed
+
+text \<open>c.f. Perko Section 3.7 Lemma 2 part 1.\<close>
+
+lemma flow_transversal_surface_finite_intersections:
+  fixes s::"'a \<Rightarrow> 'b::real_normed_vector"
+    and Ds::"'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b"
+  assumes "closed S"
+  assumes "\<And>x. (s has_derivative (Ds x)) (at x)"
+  assumes "\<And>x. x \<in> S \<Longrightarrow> s x = 0 \<Longrightarrow> Ds x (f x) \<noteq> 0"
+  assumes "a \<le> b" "{a .. b} \<subseteq> existence_ivl0 x"
+  shows "finite {t\<in>{a..b}. flow0 x t \<in> {x \<in> S. s x = 0}}"
+    \<comment> \<open>TODO: define notion of (compact/closed)-(continuous/differentiable/C1)-surface?\<close>
+proof cases
+  note Ds = \<open>\<And>x. (s has_derivative (Ds x)) (at x)\<close>
+  note transversal = \<open>\<And>x. x \<in> S \<Longrightarrow> s x = 0 \<Longrightarrow> Ds x (f x) \<noteq> 0\<close>
+  assume "a < b"
+  show ?thesis
+  proof (rule ccontr)
+    let ?S = "{x \<in> S. s x = 0}"
+    let ?T = "{t\<in>{a..b}. flow0 x t \<in> {x \<in> S. s x = 0}}"
+    define \<phi> where "\<phi> = flow0 x"
+    have [THEN continuous_on_compose2, continuous_intros]: "continuous_on S s"
+      by (auto simp: intro!: has_derivative_continuous_on Ds intro: has_derivative_at_withinI)
+    assume "infinite ?T"
+    from compact_sequentialE[OF compact_Icc[of a b] this]
+    obtain t tl where t: "t n \<in> ?T" "flow0 x (t n) \<in> ?S" "t n \<in> {a .. b}" "t n \<noteq> tl"
+      and tl: "t \<longlonglongrightarrow> tl" "tl \<in> {a..b}"
+    for n
+      by force
+    have tl_ex: "tl \<in> existence_ivl0 x" using \<open>{a .. b} \<subseteq> existence_ivl0 x\<close> \<open>tl \<in> {a .. b}\<close> by auto
+    have "closed ?S"
+      by (auto intro!: closed_levelset_within \<open>closed S\<close> continuous_intros)
+    moreover
+    have "\<forall>n. flow0 x (t n) \<in> ?S"
+      using t by auto
+    moreover
+    have flow_t: "(\<lambda>n. flow0 x (t n)) \<longlonglongrightarrow> flow0 x tl"
+      by (auto intro!: tendsto_eq_intros tl_ex tl)
+    ultimately have "flow0 x tl \<in> ?S"
+      by (rule closed_sequentially)
+
+    let ?qt = "\<lambda>t. (flow0 x t - flow0 x tl) /\<^sub>R (t - tl)"
+    from flow_has_vector_derivative[OF tl_ex, THEN has_vector_derivative_withinD]
+    have qt_tendsto: "?qt \<midarrow>tl\<rightarrow> f (flow0 x tl)" .
+    let ?q = "\<lambda>n. ?qt (t n)"
+    have "filterlim t (at tl) sequentially"
+      using tl(1)
+      by (rule filterlim_atI) (simp add: t)
+    with qt_tendsto have "?q \<longlonglongrightarrow> f (flow0 x tl)"
+      by (rule filterlim_compose)
+    then have "((\<lambda>n. Ds (flow0 x tl) (?q n))) \<longlonglongrightarrow> Ds (flow0 x tl) (f (flow0 x tl))"
+      by (auto intro!: tendsto_intros)
+    moreover
+
+    from flow_lipschitzE[OF \<open>{a .. b} \<subseteq> existence_ivl0 x\<close>] obtain L' where L': "L'-lipschitz_on {a..b} (flow0 x)" .
+    define L where "L = L' + 1"
+    from lipschitz_on_le[OF L', of L] lipschitz_on_nonneg[OF L']
+    have L: "L-lipschitz_on {a .. b} (flow0 x)" and "L > 0"
+      by (auto simp: L_def)
+    from flow_lipschitzE[OF \<open>{a .. b} \<subseteq> existence_ivl0 x\<close>] obtain L' where "L'-lipschitz_on {a..b} (flow0 x)" .
+        \<comment> \<open>TODO: is this reasoning (below) with this Lipschitz constant really necessary?\<close>
+    have s[simp]: "s (flow0 x (t n)) = 0""s (flow0 x tl) = 0"
+      for n
+      using t \<open>flow0 x tl \<in> ?S\<close>
+      by auto
+
+    from Ds(1)[of "flow0 x tl", unfolded has_derivative_within]
+    have "(\<lambda>y. (1 / norm (y - flow0 x tl)) *\<^sub>R (s y - (s (flow0 x tl) + blinfun_apply (Ds (flow0 x tl)) (y - flow0 x tl)))) \<midarrow>flow0 x tl\<rightarrow> 0"
+      by auto
+    then have "((\<lambda>y. (1 / norm (y - flow0 x tl)) *\<^sub>R (s y - (s (flow0 x tl) + blinfun_apply (Ds (flow0 x tl)) (y - flow0 x tl)))) \<longlongrightarrow> 0)
+      (nhds (flow0 x tl))"
+      by (rule tendsto_nhds_continuousI) simp
+
+    from filterlim_compose[OF this flow_t]
+    have "(\<lambda>xa. (blinfun_apply (Ds (flow0 x tl)) (flow0 x (t xa) - flow0 x tl)) /\<^sub>R norm (flow0 x (t xa) - flow0 x tl))
+      \<longlonglongrightarrow> 0"
+      using t
+      by (auto simp: inverse_eq_divide tendsto_minus_cancel_right)
+    from tendsto_mult[OF tendsto_const[of "L"] tendsto_norm[OF this, simplified, simplified divide_inverse_commute[symmetric]]]\<comment> \<open>TODO: uuugly\<close>
+    have Ds0: "(\<lambda>xa. norm (blinfun_apply (Ds (flow0 x tl)) (flow0 x (t xa) - flow0 x tl)) / (norm (flow0 x (t xa) - flow0 x tl)/(L))) \<longlonglongrightarrow> 0"
+      by (auto simp: ac_simps)
+
+    from _ Ds0 have "((\<lambda>n. Ds (flow0 x tl) (?q n)) \<longlonglongrightarrow> 0)"
+      apply (rule Lim_null_comparison)
+      apply (rule eventuallyI)
+      unfolding norm_scaleR blinfun.scaleR_right abs_inverse divide_inverse_commute[symmetric]
+      subgoal for n
+        apply (cases "flow0 x (t n) = flow0 x tl")
+        subgoal by (simp add: blinfun.bilinear_simps)
+        subgoal
+          apply (rule divide_left_mono)
+          using lipschitz_onD[OF L, of "t n" tl] \<open>0 < L\<close> t(3) tl(2)
+          by (auto simp: algebra_split_simps zero_less_divide_iff dist_norm pos_divide_le_eq
+              intro!: add_pos_nonneg)
+        done
+      done
+    ultimately have "Ds (flow0 x tl) (f (flow0 x tl)) = 0"
+      by (rule LIMSEQ_unique)
+    moreover have "Ds (flow0 x tl) (f (flow0 x tl)) \<noteq> 0"
+      by (rule transversal) (use \<open>flow0 x tl \<in> ?S\<close> in auto)
+    ultimately show False by auto
+  qed
+qed (use assms in auto)
+
+lemma uniform_limit_flow0_state:\<comment> \<open>TODO: is that something more general?\<close>
+  assumes "compact C"
+  assumes "C \<subseteq> X"
+  shows "uniform_limit C (\<lambda>s x. flow0 x s) (\<lambda>x. flow0 x 0) (at 0)"
+proof (cases "C = {}")
+  case True then show ?thesis by auto
+next
+  case False show ?thesis
+  proof (rule uniform_limitI)
+    fix e::real assume "0 < e"
+    {
+      fix x assume "x \<in> C"
+      with assms have "x \<in> X" by auto
+      from existence_ivl_cballs[OF UNIV_I \<open>x \<in> X\<close>]
+      obtain t L u where "\<And>y. y \<in> cball x u \<Longrightarrow> cball 0 t \<subseteq> existence_ivl0 y"
+        "\<And>s y. y \<in> cball x u \<Longrightarrow> s \<in> cball 0 t \<Longrightarrow> flow0 y s \<in> cball y u"
+        "L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)"
+        "\<And>y. y \<in> cball x u \<Longrightarrow> cball y u \<subseteq> X"
+        "0 < t" "0 < u"
+        by metis
+      then have "\<exists>L. \<exists>u>0. \<exists>t>0. L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)" by blast
+    } then have "\<forall>x\<in>C. \<exists>L. \<exists>u>0. \<exists>t>0. L-lipschitz_on (cball 0 t\<times>cball x u) (\<lambda>(t, x). flow0 x t)" ..
+    then obtain L d' u' where
+      L: "\<And>x. x \<in> C \<Longrightarrow> (L x)-lipschitz_on (cball 0 (d' x)\<times>cball x (u' x)) (\<lambda>(t, x). flow0 x t)"
+      and d': "\<And>x. x \<in> C \<Longrightarrow> d' x > 0"
+      and u': "\<And>x. x \<in> C \<Longrightarrow> u' x > 0"
+      by metis
+    have "C \<subseteq> (\<Union>c\<in>C. ball c (u' c))" using u' by auto
+    from compactE_image[OF \<open>compact C\<close> _ this]
+    obtain C' where "C' \<subseteq> C" and [simp]: "finite C'" and C'_cover: "C \<subseteq> (\<Union>c\<in>C'. ball c (u' c))"
+      by auto
+    from C'_cover obtain c' where c': "x \<in> C \<Longrightarrow> x \<in> ball (c' x) (u' (c' x))" "x \<in> C \<Longrightarrow> c' x \<in> C'" for x
+      by (auto simp: subset_iff) metis
+    have "\<forall>\<^sub>F s in at 0. \<forall>x\<in>ball c (u' c). dist (flow0 x s) (flow0 x 0) < e" if "c \<in> C'" for c
+    proof -
+      have cC: "c \<in> C"
+        using c' \<open>c \<in> C'\<close> d' \<open>C' \<subseteq> C\<close>
+        by auto
+      have *: "dist (flow0 x s) (flow0 x 0) \<le> L c * \<bar>s\<bar>"
+        if "x\<in>ball c (u' c)"
+          "s \<in> cball 0 (d' c)"
+        for x s
+      proof -
+        from L[OF cC, THEN lipschitz_onD, of "(0, x)" "(s, x)"] d'[OF cC] that
+        show ?thesis
+          by (auto simp: dist_prod_def dist_commute)
+      qed
+      have "\<forall>\<^sub>F s in at 0. abs s < d' c"
+        by (rule order_tendstoD tendsto_intros)+ (use d' cC in auto)
+      moreover have "\<forall>\<^sub>F s in at 0. L c * \<bar>s\<bar> < e"
+        by (rule order_tendstoD tendsto_intros)+ (use \<open>0 < e\<close> in auto)
+      ultimately show ?thesis
+        apply eventually_elim
+        apply (use * in auto)
+        by smt
+    qed
+    then have "\<forall>\<^sub>F s in at 0. \<forall>c\<in>C'. \<forall>x\<in>ball c (u' c). dist (flow0 x s) (flow0 x 0) < e"
+      by (simp add: eventually_ball_finite_distrib)
+    then show "\<forall>\<^sub>F s in at 0. \<forall>x\<in>C. dist (flow0 x s) (flow0 x 0) < e"
+      apply eventually_elim
+      apply (auto simp: )
+      subgoal for s x
+        apply (drule bspec[where x="c' x"])
+         apply (simp add: c'(2))
+        apply (drule bspec) prefer 2 apply assumption
+        apply auto
+        using c'(1) by auto
+      done
+  qed
+qed
+
+end
+
+subsection \<open>Fixpoints\<close>
+
+context auto_ll_on_open begin
+
+lemma fixpoint_sol:
+  assumes "x \<in> X" "f x = 0"
+  shows "existence_ivl0 x = UNIV" "flow0 x t = x"
+proof -
+  have sol: "((\<lambda>t::real. x) solves_ode (\<lambda>_. f)) UNIV X"
+    apply (rule solves_odeI)
+    by(auto simp add: assms intro!: derivative_intros)
+  from maximal_existence_flow[OF sol] have
+    "UNIV \<subseteq> existence_ivl0 x" "flow0 x t = x" by auto
+  thus "existence_ivl0 x = UNIV" "flow0 x t = x" by auto
+qed
+
+end
+
 end
\ No newline at end of file
diff --git a/thys/Poincare_Bendixson/Periodic_Orbit.thy b/thys/Poincare_Bendixson/Periodic_Orbit.thy
--- a/thys/Poincare_Bendixson/Periodic_Orbit.thy
+++ b/thys/Poincare_Bendixson/Periodic_Orbit.thy
@@ -1,541 +1,541 @@
-section \<open>Periodic Orbits\<close>
-theory Periodic_Orbit
-  imports 
-    Ordinary_Differential_Equations.ODE_Analysis
-    Analysis_Misc
-    ODE_Misc
-    Limit_Set
-begin
-
-text \<open> Definition of closed and periodic orbits and their associated properties \<close>
-
-context auto_ll_on_open
-begin
-
-text \<open>
-   TODO: not sure if the "closed orbit" terminology is standard
-   Closed orbits have some non-zero recurrence time T where the flow returns to the initial state
-   The period of a closed orbit is the infimum of all positive recurrence times
-   Periodic orbits are the subset of closed orbits where the period is non-zero
-  \<close>
-
-definition "closed_orbit x \<longleftrightarrow>
-  (\<exists>T \<in> existence_ivl0 x. T \<noteq> 0 \<and> flow0 x T = x)"
-
-definition "period x =
-  Inf {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
-
-definition "periodic_orbit x \<longleftrightarrow>
-  closed_orbit x \<and> period x > 0"
-
-lemma recurrence_time_flip_sign:
-  assumes "T \<in> existence_ivl0 x" "flow0 x T = x"
-  shows "-T \<in> existence_ivl0 x" "flow0 x (-T) = x"
-  using assms existence_ivl_reverse apply fastforce
-  using assms flows_reverse by fastforce  
-
-lemma closed_orbit_recurrence_times_nonempty:
-  assumes "closed_orbit x"
-  shows " {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x} \<noteq> {}"
-  apply auto
-  using assms(1) unfolding closed_orbit_def
-  by (smt recurrence_time_flip_sign)
-
-lemma closed_orbit_recurrence_times_bdd_below:
-  shows "bdd_below {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
-  unfolding bdd_below_def
-  by (auto) (meson le_cases not_le)
-
-lemma closed_orbit_period_nonneg:
-  assumes "closed_orbit x"
-  shows "period x \<ge> 0"
-  unfolding period_def
-  using assms(1) unfolding closed_orbit_def apply (auto intro!:cInf_greatest)
-  by (smt recurrence_time_flip_sign)
-
-lemma closed_orbit_in_domain:
-  assumes "closed_orbit x"
-  shows "x \<in> X"
-  using assms unfolding closed_orbit_def
-  using local.mem_existence_ivl_iv_defined(2) by blast
-
-lemma closed_orbit_global_existence:
-  assumes "closed_orbit x"
-  shows "existence_ivl0 x = UNIV"
-proof -
-  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
-    unfolding closed_orbit_def by blast
-  then obtain T where T: "T > 0" "T \<in> existence_ivl0 x" "flow0 x T = x"
-    by (smt recurrence_time_flip_sign)
-  have apos: "real n * T \<in> existence_ivl0 x \<and> flow0 x (real n * T) = x" for n
-  proof (induction n)
-    case 0
-    then show ?case using closed_orbit_in_domain assms by auto
-  next
-    case (Suc n)
-    fix n
-    assume ih:"real n * T \<in> existence_ivl0 x \<and> flow0 x (real n * T) = x"
-    then have "T \<in> existence_ivl0 (flow0 x (real n * T))" using T by metis
-    then have l:"real n * T + T \<in> existence_ivl0 x" using ih
-      using existence_ivl_trans by blast 
-    have "flow0 (flow0 x (real n * T)) T = x" using ih T by metis
-    then have r: "flow0 x (real n * T + T) = x"
-      by (simp add: T(2) ih local.flow_trans)
-    show "real (Suc n) * T \<in> existence_ivl0 x \<and> flow0 x (real (Suc n) * T) = x"
-      using l r
-      by (simp add: add.commute distrib_left mult.commute)
-  qed
-  then have aneg: "-real n * T \<in> existence_ivl0 x \<and> flow0 x (-real n * T) = x" for n
-    by (simp add: recurrence_time_flip_sign)
-  have "\<forall>t. t \<in> existence_ivl0 x"
-  proof safe
-    fix t
-    have "t \<ge> 0 \<or> t \<le> (0::real)" by linarith
-    moreover {
-      assume "t \<ge> 0"
-      obtain k where "real k * T > t"
-        using T(1) ex_less_of_nat_mult by blast
-      then have "t \<in> existence_ivl0 x" using apos
-        by (meson \<open>0 \<le> t\<close> atLeastAtMost_iff less_eq_real_def local.ivl_subset_existence_ivl subset_eq) 
-    }
-    moreover {
-      assume "t \<le> 0"
-      obtain k where "- real k * T < t"
-        by (metis T(1) add.inverse_inverse ex_less_of_nat_mult mult.commute mult_minus_right neg_less_iff_less)
-      then have "t \<in> existence_ivl0 x" using aneg
-        by (smt apos atLeastAtMost_iff calculation(2) local.existence_ivl_trans' local.ivl_subset_existence_ivl mult_minus_left subset_eq)
-    }
-    ultimately show "t \<in> existence_ivl0 x" by blast
-  qed
-  thus ?thesis by auto
-qed
-
-lemma recurrence_time_multiples:
-  fixes n::nat
-  assumes "T \<in> existence_ivl0 x" "T \<noteq> 0" "flow0 x T = x"
-  shows "\<And>t. flow0 x (t+T*n) = flow0 x t"
-proof (induction n)
-  case 0
-  then show ?case by auto
-next
-  case (Suc n)
-  fix n t
-  assume ih : "(\<And>t. flow0 x (t + T * real n) = flow0 x t)"
-  have "closed_orbit x" using assms unfolding closed_orbit_def by auto
-  from closed_orbit_global_existence[OF this] have ex:"existence_ivl0 x = UNIV" .
-  have "flow0 x (t + T * real (Suc n)) = flow0 x (t+T*real n + T)"
-    by (simp add: Groups.add_ac(3) add.commute distrib_left)
-  also have "... = flow0 (flow0 x (t+ T*real n)) T" using ex
-    by (simp add: local.existence_ivl_trans' local.flow_trans)
-  also have "... = flow0 (flow0 x t) T" using ih by auto
-  also have "... = flow0 (flow0 x T) t" using ex
-    by (metis UNIV_I add.commute local.existence_ivl_trans' local.flow_trans)
-  finally show "flow0 x (t + T * real (Suc n)) = flow0 x t" using assms(3) by simp
-qed
-
-lemma nasty_arithmetic1:
-  fixes t T::real
-  assumes "T > 0" "t \<ge> 0"
-  obtains q r where "t = (q::nat) * T + r" "0 \<le> r" "r < T"
-proof -
-  define q where "q = floor (t / T)"
-  have q:"q \<ge> 0" using assms unfolding q_def by auto
-  from floor_divide_lower[OF assms(1), of t]
-  have ql: "q * T \<le> t" unfolding q_def .
-  from floor_divide_upper[OF assms(1), of t]
-  have qu: "t < (q + 1)* T" unfolding q_def by auto
-  define r where "r = t - q * T"
-  have rl:"0 \<le> r" using ql unfolding r_def by auto
-  have ru:"r < T" using qu unfolding r_def  by (simp add: distrib_right)
-  show ?thesis using q r_def rl ru
-    by (metis le_add_diff_inverse of_int_of_nat_eq plus_int_code(2) ql that zle_iff_zadd)
-qed
-
-lemma nasty_arithmetic2:
-  fixes t T::real
-  assumes "T > 0" "t \<le> 0"
-  obtains q r where "t = (q::nat) * (-T) + r" "0 \<le> r" "r < T"
-proof -
-  have "-t \<ge> 0" using assms(2) by linarith
-  from nasty_arithmetic1[OF assms(1) this]
-  obtain q r where qr: "-t = (q::nat) * T + r" "0 \<le> r" "r < T" by blast
-  then have "t = q * (-T) - r"  by auto
-  then have "t = (q+(1::nat)) * (-T) + (T-r)" by (simp add: distrib_right)
-  thus ?thesis using qr(2-3)
-    by (smt \<open>t = real q * - T - r\<close> that) 
-qed
-
-lemma recurrence_time_restricts_compact_flow:
-  assumes "T \<in> existence_ivl0 x" "T > 0" "flow0 x T = x"
-  shows "flow0 x ` UNIV = flow0 x ` {0..T}"
-  apply auto
-proof -
-  fix t
-  have "t \<ge> 0 \<or> t \<le> (0::real)" by linarith
-  moreover {
-    assume "t \<ge> 0"
-    from nasty_arithmetic1[OF assms(2) this]
-    obtain q r where qr:"t = (q::nat) * T + r" "0 \<le> r" "r < T" by blast
-    have "T \<noteq> 0" using assms(2) by auto
-    from recurrence_time_multiples[OF assms(1) this assms(3),of "r" "q"]
-    have "flow0 x t = flow0 x r"
-      by (simp add: qr(1) add.commute mult.commute)
-    then have "flow0 x t \<in> flow0 x ` {0..<T}" using qr by auto
-  }
-  moreover {
-    assume "t \<le> 0"
-    from nasty_arithmetic2[OF assms(2) this]
-    obtain q r where qr:"t = (q::nat) * (-T) + r" "0 \<le> r" "r < T" by blast
-    have "-T \<in> existence_ivl0 x" "-T \<noteq> 0" "flow0 x (-T) = x" using recurrence_time_flip_sign assms by auto
-    from recurrence_time_multiples[OF this, of r q]
-    have "flow0 x t = flow0 x r"
-      by (simp add: mult.commute qr(1))
-    then have "flow0 x t \<in> flow0 x ` {0..<T}" using qr by auto
-  }
-  ultimately show "flow0 x t \<in> flow0 x ` {0..T}"
-    by auto
-qed
-
-lemma closed_orbitI:
-  assumes "t \<noteq> t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
-  assumes "flow0 y t = flow0 y t'"
-  shows "closed_orbit y"
-  unfolding closed_orbit_def
-  by (smt assms local.existence_ivl_reverse local.existence_ivl_trans local.flow_trans local.flows_reverse)
-
-(* TODO: can be considerably generalized *)
-lemma flow0_image_UNIV:
-  assumes "existence_ivl0 x = UNIV"
-  shows "flow0 (flow0 x t) ` S = flow0 x ` (\<lambda>s. s + t) ` S"
-  apply auto
-   apply (metis UNIV_I add.commute assms image_eqI local.existence_ivl_trans' local.flow_trans)
-  by (metis UNIV_I add.commute assms imageI local.existence_ivl_trans' local.flow_trans)
-
-lemma recurrence_time_restricts_compact_flow':
-  assumes "t < t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
-  assumes "flow0 y t = flow0 y t'"
-  shows "flow0 y ` UNIV = flow0 y ` {t..t'}"
-proof -
-  have "closed_orbit y"
-    using assms(1-4) closed_orbitI inf.strict_order_iff by blast
-  from closed_orbit_global_existence[OF this]
-  have yex: "existence_ivl0 y = UNIV" .
-  have a1:"t'- t \<in> existence_ivl0 (flow0 y t)"
-    by (simp add: assms(2-3) local.existence_ivl_trans')
-  have a2:"t' -t > 0" using assms(1) by auto
-  have a3:"flow0 (flow0 y t) (t' - t) = flow0 y t"
-    using a1 assms(2) assms(4) local.flow_trans by fastforce 
-  from recurrence_time_restricts_compact_flow[OF a1 a2 a3]
-  have eq:"flow0 (flow0 y t) ` UNIV = flow0 (flow0 y t) ` {0.. t'-t}" .
-  from flow0_image_UNIV[OF yex, of _ "UNIV"]
-  have eql:"flow0 (flow0 y t) ` UNIV = flow0 y ` UNIV"
-    by (metis (no_types) add.commute surj_def surj_plus)
-  from flow0_image_UNIV[OF yex, of _ "{0..t'-t}"]
-  have eqr:"flow0 (flow0 y t) ` {0.. t'-t} = flow0 y ` {t..t'}" by auto
-  show ?thesis using eq eql eqr by auto
-qed
-
-lemma closed_orbitE':
-  assumes "closed_orbit x"
-  obtains T where "T > 0" "\<And>t (n::nat). flow0 x (t+T*n) = flow0 x t"
-proof -
-  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
-    unfolding closed_orbit_def by blast
-  then obtain T where T: "T > 0" "T \<in> existence_ivl0 x" "flow0 x T = x"
-    by (smt recurrence_time_flip_sign)
-  thus ?thesis using  recurrence_time_multiples T that by blast 
-qed
-
-lemma closed_orbitE:
-  assumes "closed_orbit x"
-  obtains T where "T > 0" "\<And>t. flow0 x (t+T) = flow0 x t"
-  using closed_orbitE'
-  by (metis assms mult.commute reals_Archimedean3)
-
-lemma closed_orbit_flow_compact:
-  assumes "closed_orbit x"
-  shows "compact(flow0 x ` UNIV)"
-proof -
-  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
-    unfolding closed_orbit_def by blast
-  then obtain T where T: "T \<in> existence_ivl0 x" "T > 0" "flow0 x T = x"
-    by (smt recurrence_time_flip_sign)
-  from recurrence_time_restricts_compact_flow[OF this]
-  have feq: "flow0 x ` UNIV = flow0 x ` {0..T}" .
-  have "continuous_on {0..T} (flow0 x)"
-    by (meson T(1) continuous_on_sequentially in_mono local.flow_continuous_on local.ivl_subset_existence_ivl) 
-  from compact_continuous_image[OF this]
-  have "compact (flow0 x ` {0..T})" by auto
-  thus ?thesis using feq by auto
-qed
-
-lemma fixed_point_imp_closed_orbit_period_zero:
-  assumes "x \<in> X"
-  assumes "f x = 0"
-  shows "closed_orbit x" "period x = 0"
-proof -
-  from fixpoint_sol[OF assms] have fp:"existence_ivl0 x = UNIV" "\<And>t. flow0 x t = x" by auto
-  then have co:"closed_orbit x" unfolding closed_orbit_def by blast
-  have a: "\<forall>y>0. \<exists>a\<in>{T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}. a < y"
-    apply auto
-    using fp
-    by (simp add: dense) 
-  from cInf_le_iff[OF closed_orbit_recurrence_times_nonempty[OF co]
-      closed_orbit_recurrence_times_bdd_below , of 0]
-  have "period x \<le> 0" unfolding period_def using a by auto
-  from closed_orbit_period_nonneg[OF co] have "period x \<ge> 0" .
-  then have "period x = 0" using \<open>period x \<le> 0\<close> by linarith
-  thus "closed_orbit x" "period x = 0" using co by auto
-qed
-
-lemma closed_orbit_period_zero_fixed_point:
-  assumes "closed_orbit x" "period x = 0"
-  shows "f x = 0"
-proof (rule ccontr)
-  assume "f x \<noteq> 0"
-  from regular_locally_noteq[OF closed_orbit_in_domain[OF assms(1)] this]
-  have "\<forall>\<^sub>F t in at 0. flow0 x t \<noteq> x " .
-  then obtain r where "r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 x t \<noteq> x" unfolding eventually_at
-    by auto
-  then have "period x \<ge> r" unfolding period_def
-    apply (auto intro!:cInf_greatest)
-     apply (meson assms(1) closed_orbit_def linorder_neqE_linordered_idom neg_0_less_iff_less recurrence_time_flip_sign)
-    using not_le by force
-  thus False using assms(2) \<open>r >0\<close> by linarith
-qed
-
-lemma closed_orbit_subset_\<omega>_limit_set:
-  assumes "closed_orbit x"
-  shows "flow0 x ` UNIV \<subseteq> \<omega>_limit_set x"
-  unfolding \<omega>_limit_set_def \<omega>_limit_point_def
-proof clarsimp
-  fix t
-  from closed_orbitE'[OF assms]
-  obtain T where T: "0 < T" "\<And>t n. flow0 x (t + T* real n) = flow0 x t" by blast
-  define s where "s = (\<lambda>n::nat. t + T * real n)"
-  have exist: "{0..} \<subseteq> existence_ivl0 x"
-    by (simp add: assms closed_orbit_global_existence)
-  have l:"filterlim s at_top sequentially" unfolding s_def
-    using T(1)
-    by (auto intro!:filterlim_tendsto_add_at_top filterlim_tendsto_pos_mult_at_top
-        simp add: filterlim_real_sequentially)
-  have "flow0 x \<circ> s = (\<lambda>n. flow0 x t)" unfolding o_def s_def using T(2) by simp
-  then have r:"(flow0 x \<circ> s) \<longlonglongrightarrow> flow0 x t" by auto
-  show "{0..} \<subseteq> existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> flow0 x t)"
-    using exist l r by blast
-qed
-
-lemma closed_orbit_\<omega>_limit_set:
-  assumes "closed_orbit x"
-  shows "flow0 x ` UNIV = \<omega>_limit_set x"
-proof -
-  have "\<omega>_limit_set x \<subseteq> flow0 x ` UNIV"
-    using closed_orbit_global_existence[OF assms]
-    by (intro \<omega>_limit_set_in_compact_subset)
-      (auto intro!: flow_in_domain
-        simp add: assms closed_orbit_in_domain image_subset_iff trapped_forward_def
-        closed_orbit_flow_compact)
-  thus ?thesis using closed_orbit_subset_\<omega>_limit_set[OF assms] by auto
-qed
-
-lemma flow0_inj_on:
-  assumes "t \<le> t'"
-  assumes "{t..t'} \<subseteq> existence_ivl0 x"
-  assumes "\<And>s. t < s \<Longrightarrow> s \<le> t' \<Longrightarrow> flow0 x s \<noteq> flow0 x t"
-  shows "inj_on (flow0 x) {t..t'}"
-  apply (rule inj_onI)
-proof (rule ccontr)
-  fix u v
-  assume uv: "u \<in> {t..t'}" "v \<in> {t..t'}" "flow0 x u = flow0 x v" "u \<noteq> v"
-  have "u < v \<or> v < u" using uv(4) by linarith
-  moreover {
-    assume "u < v"
-    from recurrence_time_restricts_compact_flow'[OF this _ _ uv(3)]
-    have "flow0 x ` UNIV = flow0 x ` {u..v}"  using uv(1-2) assms(2) by blast
-    then have "flow0 x t \<in> flow0 x ` {u..v}" by auto
-    moreover have "u = t \<or> flow0 x t \<notin> flow0 x ` {u..v}" using assms(3)
-      by (smt atLeastAtMost_iff image_iff uv(1) uv(2))
-    ultimately have False using uv assms(3)
-      by force
-  }
-  moreover {
-    assume "v < u"
-    from recurrence_time_restricts_compact_flow'[OF this _ _ ]
-    have "flow0 x ` UNIV = flow0 x ` {v..u}"
-      by (metis assms(2) subset_iff uv(1) uv(2) uv(3))
-    then have "flow0 x t \<in> flow0 x ` {v..u}" by auto
-    moreover have "v = t \<or> flow0 x t \<notin> flow0 x ` {v..u}" using assms(3)
-      by (smt atLeastAtMost_iff image_iff uv(1) uv(2))
-    ultimately have False using uv assms(3) by force
-  }
-  ultimately show False by blast
-qed
-
-(* TODO: Probably has a simpler proof *)
-lemma finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes "finite (\<omega>_limit_set x)"
-  obtains y where "\<omega>_limit_set x = {y}" "f y = 0"
-proof -
-  from connected_finite_iff_sing[OF \<omega>_limit_set_in_compact_connected]
-  obtain y where y: "\<omega>_limit_set x = {y}"
-    using \<omega>_limit_set_in_compact_nonempty assms by auto
-  have "f y = 0"
-  proof (rule ccontr)
-    assume fy:"f y \<noteq> 0"
-    from \<omega>_limit_set_in_compact_existence[OF assms(1-4)]
-    have yex: "existence_ivl0 y = UNIV"
-      by (simp add: y)
-    then have "y \<in> X"
-      by (simp add: local.mem_existence_ivl_iv_defined(2))
-    from regular_locally_noteq[OF this fy]
-    have "\<forall>\<^sub>F t in at 0. flow0 y t \<noteq> y" .
-    then obtain r where r: "r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 y t \<noteq> flow0 y 0"
-      unfolding eventually_at using \<open>y \<in> X\<close>
-      by auto
-    then have "\<And>s. 0 < s \<Longrightarrow> s \<le> r/2 \<Longrightarrow> flow0 y s \<noteq> flow0 y 0" by simp
-    from flow0_inj_on[OF _ _ this, of "r/2"]
-    obtain "inj_on(flow0 y) {0..r/2}" using r yex by simp
-    then have "infinite (flow0 y`{0..r/2})" by (simp add: finite_image_iff r(1))
-    moreover from \<omega>_limit_set_invariant[of x]
-    have "flow0 y `{0..r/2} \<subseteq> \<omega>_limit_set x" using y yex
-      unfolding invariant_def trapped_iff_on_existence_ivl0 by auto
-    ultimately show False using y
-      by (metis assms(5) finite.emptyI subset_singleton_iff)
-  qed
-  thus ?thesis using that y by auto
-qed
-
-lemma closed_orbit_periodic:
-  assumes "closed_orbit x" "f x \<noteq> 0"
-  shows "periodic_orbit x"
-  unfolding periodic_orbit_def
-  using assms(1) apply auto
-proof (rule ccontr)
-  assume "closed_orbit x"
-  from closed_orbit_period_nonneg[OF assms(1)] have nneg: "period x \<ge> 0" .
-  assume "\<not> 0 < period x"
-  then have "period x = 0" using nneg by linarith
-  from closed_orbit_period_zero_fixed_point[OF assms(1) this]
-  have "f x = 0" . 
-  thus False using assms(2) by linarith
-qed
-
-lemma periodic_orbitI:
-  assumes "t \<noteq> t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
-  assumes "flow0 y t = flow0 y t'"
-  assumes "f y \<noteq> 0"
-  shows "periodic_orbit y"
-proof -
-  have y:"y \<in> X"
-    using assms(3) local.mem_existence_ivl_iv_defined(2) by blast
-  from closed_orbitI[OF assms(1-4)] have "closed_orbit y" .
-  from closed_orbit_periodic[OF this assms(5)]
-  show ?thesis .
-qed
-
-lemma periodic_orbit_recurrence_times_closed:
-  assumes "periodic_orbit x"
-  shows "closed {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
-proof -
-  have a1:"x \<in> X"
-    using assms closed_orbit_in_domain periodic_orbit_def by auto 
-  have a2:"f x \<noteq> 0"
-    using assms closed_orbit_in_domain fixed_point_imp_closed_orbit_period_zero(2) periodic_orbit_def by auto
-  from regular_locally_noteq[OF a1 a2] have
-    "\<forall>\<^sub>F t in at 0. flow0 x t \<noteq> x" .
-  then obtain r where r:"r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 x t \<noteq> x" unfolding eventually_at
-    by auto
-  show ?thesis
-  proof (auto intro!:discrete_imp_closed[OF r(1)])
-    fix t1 t2
-    assume t12: "t1 > 0" "flow0 x t1 = x" "t2 > 0" "flow0 x t2 = x" "dist t2 t1 < r"
-    then have fx: "flow0 x (t1-t2) = x"
-      by (smt a1 assms closed_orbit_global_existence existence_ivl_zero general.existence_ivl_initial_time_iff local.flow_trans periodic_orbit_def)
-    have "dist (t1-t2) 0 < r" using t12(5)
-      by (simp add: dist_norm) 
-    thus "t2 = t1" using r fx
-      by smt
-  qed
-qed
-
-lemma periodic_orbit_period:
-  assumes "periodic_orbit x"
-  shows "period x > 0" "flow0 x (period x) = x"
-proof -
-  from periodic_orbit_recurrence_times_closed[OF assms(1)]
-  have cl: "closed {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}" .
-  have "closed_orbit x" using assms(1) unfolding periodic_orbit_def by auto
-  from closed_contains_Inf[OF closed_orbit_recurrence_times_nonempty[OF this]
-      closed_orbit_recurrence_times_bdd_below cl]
-  have "period x \<in>  {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}" unfolding period_def .
-  thus "period x > 0" "flow0 x (period x) = x" by auto
-qed
-
-lemma closed_orbit_flow0:
-  assumes "closed_orbit x"
-  shows "closed_orbit (flow0 x t)"
-proof -
-  from closed_orbit_global_existence[OF assms]
-  have "existence_ivl0 x = UNIV" .
-  from closed_orbitE[OF assms]
-  obtain T where "T > 0" "flow0 x (t+T) = flow0 x t"
-    by metis
-  thus ?thesis unfolding closed_orbit_def
-    by (metis UNIV_I \<open>existence_ivl0 x = UNIV\<close> less_irrefl local.existence_ivl_trans' local.flow_trans)
-qed
-
-lemma periodic_orbit_imp_flow0_regular:
-  assumes "periodic_orbit x"
-  shows "f (flow0 x t) \<noteq> 0"
-  by (metis UNIV_I assms closed_orbit_flow0 closed_orbit_global_existence closed_orbit_in_domain fixed_point_imp_closed_orbit_period_zero(2) fixpoint_sol(2) less_irrefl local.flows_reverse periodic_orbit_def)
-
-lemma fixed_point_imp_\<omega>_limit_set:
-  assumes "x \<in> X" "f x = 0"
-  shows "\<omega>_limit_set x = {x}"
-proof -
-  have "closed_orbit x"
-    by (metis assms fixed_point_imp_closed_orbit_period_zero(1))
-  from closed_orbit_\<omega>_limit_set[OF this]
-  have "flow0 x ` UNIV = \<omega>_limit_set x" .
-  thus ?thesis
-    by (metis assms(1) assms(2) fixpoint_sol(2) image_empty image_insert image_subset_iff insertI1 rangeI subset_antisym)
-qed
-
-end
-
-context auto_ll_on_open begin
-
-lemma closed_orbit_eq_rev: "closed_orbit x = rev.closed_orbit x"
-  unfolding closed_orbit_def rev.closed_orbit_def rev_eq_flow rev_existence_ivl_eq0
-  by auto
-
-lemma closed_orbit_\<alpha>_limit_set:
-  assumes "closed_orbit x"
-  shows "flow0 x ` UNIV = \<alpha>_limit_set x"
-  using rev.closed_orbit_\<omega>_limit_set assms
-  unfolding closed_orbit_eq_rev \<alpha>_limit_set_eq_rev flow_image_eq_rev range_uminus .
-
-lemma fixed_point_imp_\<alpha>_limit_set:
-  assumes "x \<in> X" "f x = 0"
-  shows "\<alpha>_limit_set x = {x}"
-  using rev.fixed_point_imp_\<omega>_limit_set assms
-  unfolding \<alpha>_limit_set_eq_rev
-  by auto
-
-lemma finite_\<alpha>_limit_set_in_compact_imp_unique_fixed_point:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X"  "trapped_backward x K"
-  assumes "finite (\<alpha>_limit_set x)"
-  obtains y where "\<alpha>_limit_set x = {y}" "f y = 0"
-proof -
-  from rev.finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF
-      assms(1-5)[unfolded trapped_backward_iff_rev_trapped_forward \<alpha>_limit_set_eq_rev]]
-  show ?thesis using that
-    unfolding \<alpha>_limit_set_eq_rev
-    by auto
-qed
-end
-
+section \<open>Periodic Orbits\<close>
+theory Periodic_Orbit
+  imports 
+    Ordinary_Differential_Equations.ODE_Analysis
+    Analysis_Misc
+    ODE_Misc
+    Limit_Set
+begin
+
+text \<open> Definition of closed and periodic orbits and their associated properties \<close>
+
+context auto_ll_on_open
+begin
+
+text \<open>
+   TODO: not sure if the "closed orbit" terminology is standard
+   Closed orbits have some non-zero recurrence time T where the flow returns to the initial state
+   The period of a closed orbit is the infimum of all positive recurrence times
+   Periodic orbits are the subset of closed orbits where the period is non-zero
+  \<close>
+
+definition "closed_orbit x \<longleftrightarrow>
+  (\<exists>T \<in> existence_ivl0 x. T \<noteq> 0 \<and> flow0 x T = x)"
+
+definition "period x =
+  Inf {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
+
+definition "periodic_orbit x \<longleftrightarrow>
+  closed_orbit x \<and> period x > 0"
+
+lemma recurrence_time_flip_sign:
+  assumes "T \<in> existence_ivl0 x" "flow0 x T = x"
+  shows "-T \<in> existence_ivl0 x" "flow0 x (-T) = x"
+  using assms existence_ivl_reverse apply fastforce
+  using assms flows_reverse by fastforce  
+
+lemma closed_orbit_recurrence_times_nonempty:
+  assumes "closed_orbit x"
+  shows " {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x} \<noteq> {}"
+  apply auto
+  using assms(1) unfolding closed_orbit_def
+  by (smt recurrence_time_flip_sign)
+
+lemma closed_orbit_recurrence_times_bdd_below:
+  shows "bdd_below {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
+  unfolding bdd_below_def
+  by (auto) (meson le_cases not_le)
+
+lemma closed_orbit_period_nonneg:
+  assumes "closed_orbit x"
+  shows "period x \<ge> 0"
+  unfolding period_def
+  using assms(1) unfolding closed_orbit_def apply (auto intro!:cInf_greatest)
+  by (smt recurrence_time_flip_sign)
+
+lemma closed_orbit_in_domain:
+  assumes "closed_orbit x"
+  shows "x \<in> X"
+  using assms unfolding closed_orbit_def
+  using local.mem_existence_ivl_iv_defined(2) by blast
+
+lemma closed_orbit_global_existence:
+  assumes "closed_orbit x"
+  shows "existence_ivl0 x = UNIV"
+proof -
+  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
+    unfolding closed_orbit_def by blast
+  then obtain T where T: "T > 0" "T \<in> existence_ivl0 x" "flow0 x T = x"
+    by (smt recurrence_time_flip_sign)
+  have apos: "real n * T \<in> existence_ivl0 x \<and> flow0 x (real n * T) = x" for n
+  proof (induction n)
+    case 0
+    then show ?case using closed_orbit_in_domain assms by auto
+  next
+    case (Suc n)
+    fix n
+    assume ih:"real n * T \<in> existence_ivl0 x \<and> flow0 x (real n * T) = x"
+    then have "T \<in> existence_ivl0 (flow0 x (real n * T))" using T by metis
+    then have l:"real n * T + T \<in> existence_ivl0 x" using ih
+      using existence_ivl_trans by blast 
+    have "flow0 (flow0 x (real n * T)) T = x" using ih T by metis
+    then have r: "flow0 x (real n * T + T) = x"
+      by (simp add: T(2) ih local.flow_trans)
+    show "real (Suc n) * T \<in> existence_ivl0 x \<and> flow0 x (real (Suc n) * T) = x"
+      using l r
+      by (simp add: add.commute distrib_left mult.commute)
+  qed
+  then have aneg: "-real n * T \<in> existence_ivl0 x \<and> flow0 x (-real n * T) = x" for n
+    by (simp add: recurrence_time_flip_sign)
+  have "\<forall>t. t \<in> existence_ivl0 x"
+  proof safe
+    fix t
+    have "t \<ge> 0 \<or> t \<le> (0::real)" by linarith
+    moreover {
+      assume "t \<ge> 0"
+      obtain k where "real k * T > t"
+        using T(1) ex_less_of_nat_mult by blast
+      then have "t \<in> existence_ivl0 x" using apos
+        by (meson \<open>0 \<le> t\<close> atLeastAtMost_iff less_eq_real_def local.ivl_subset_existence_ivl subset_eq) 
+    }
+    moreover {
+      assume "t \<le> 0"
+      obtain k where "- real k * T < t"
+        by (metis T(1) add.inverse_inverse ex_less_of_nat_mult mult.commute mult_minus_right neg_less_iff_less)
+      then have "t \<in> existence_ivl0 x" using aneg
+        by (smt apos atLeastAtMost_iff calculation(2) local.existence_ivl_trans' local.ivl_subset_existence_ivl mult_minus_left subset_eq)
+    }
+    ultimately show "t \<in> existence_ivl0 x" by blast
+  qed
+  thus ?thesis by auto
+qed
+
+lemma recurrence_time_multiples:
+  fixes n::nat
+  assumes "T \<in> existence_ivl0 x" "T \<noteq> 0" "flow0 x T = x"
+  shows "\<And>t. flow0 x (t+T*n) = flow0 x t"
+proof (induction n)
+  case 0
+  then show ?case by auto
+next
+  case (Suc n)
+  fix n t
+  assume ih : "(\<And>t. flow0 x (t + T * real n) = flow0 x t)"
+  have "closed_orbit x" using assms unfolding closed_orbit_def by auto
+  from closed_orbit_global_existence[OF this] have ex:"existence_ivl0 x = UNIV" .
+  have "flow0 x (t + T * real (Suc n)) = flow0 x (t+T*real n + T)"
+    by (simp add: Groups.add_ac(3) add.commute distrib_left)
+  also have "... = flow0 (flow0 x (t+ T*real n)) T" using ex
+    by (simp add: local.existence_ivl_trans' local.flow_trans)
+  also have "... = flow0 (flow0 x t) T" using ih by auto
+  also have "... = flow0 (flow0 x T) t" using ex
+    by (metis UNIV_I add.commute local.existence_ivl_trans' local.flow_trans)
+  finally show "flow0 x (t + T * real (Suc n)) = flow0 x t" using assms(3) by simp
+qed
+
+lemma nasty_arithmetic1:
+  fixes t T::real
+  assumes "T > 0" "t \<ge> 0"
+  obtains q r where "t = (q::nat) * T + r" "0 \<le> r" "r < T"
+proof -
+  define q where "q = floor (t / T)"
+  have q:"q \<ge> 0" using assms unfolding q_def by auto
+  from floor_divide_lower[OF assms(1), of t]
+  have ql: "q * T \<le> t" unfolding q_def .
+  from floor_divide_upper[OF assms(1), of t]
+  have qu: "t < (q + 1)* T" unfolding q_def by auto
+  define r where "r = t - q * T"
+  have rl:"0 \<le> r" using ql unfolding r_def by auto
+  have ru:"r < T" using qu unfolding r_def  by (simp add: distrib_right)
+  show ?thesis using q r_def rl ru
+    by (metis le_add_diff_inverse of_int_of_nat_eq plus_int_code(2) ql that zle_iff_zadd)
+qed
+
+lemma nasty_arithmetic2:
+  fixes t T::real
+  assumes "T > 0" "t \<le> 0"
+  obtains q r where "t = (q::nat) * (-T) + r" "0 \<le> r" "r < T"
+proof -
+  have "-t \<ge> 0" using assms(2) by linarith
+  from nasty_arithmetic1[OF assms(1) this]
+  obtain q r where qr: "-t = (q::nat) * T + r" "0 \<le> r" "r < T" by blast
+  then have "t = q * (-T) - r"  by auto
+  then have "t = (q+(1::nat)) * (-T) + (T-r)" by (simp add: distrib_right)
+  thus ?thesis using qr(2-3)
+    by (smt \<open>t = real q * - T - r\<close> that) 
+qed
+
+lemma recurrence_time_restricts_compact_flow:
+  assumes "T \<in> existence_ivl0 x" "T > 0" "flow0 x T = x"
+  shows "flow0 x ` UNIV = flow0 x ` {0..T}"
+  apply auto
+proof -
+  fix t
+  have "t \<ge> 0 \<or> t \<le> (0::real)" by linarith
+  moreover {
+    assume "t \<ge> 0"
+    from nasty_arithmetic1[OF assms(2) this]
+    obtain q r where qr:"t = (q::nat) * T + r" "0 \<le> r" "r < T" by blast
+    have "T \<noteq> 0" using assms(2) by auto
+    from recurrence_time_multiples[OF assms(1) this assms(3),of "r" "q"]
+    have "flow0 x t = flow0 x r"
+      by (simp add: qr(1) add.commute mult.commute)
+    then have "flow0 x t \<in> flow0 x ` {0..<T}" using qr by auto
+  }
+  moreover {
+    assume "t \<le> 0"
+    from nasty_arithmetic2[OF assms(2) this]
+    obtain q r where qr:"t = (q::nat) * (-T) + r" "0 \<le> r" "r < T" by blast
+    have "-T \<in> existence_ivl0 x" "-T \<noteq> 0" "flow0 x (-T) = x" using recurrence_time_flip_sign assms by auto
+    from recurrence_time_multiples[OF this, of r q]
+    have "flow0 x t = flow0 x r"
+      by (simp add: mult.commute qr(1))
+    then have "flow0 x t \<in> flow0 x ` {0..<T}" using qr by auto
+  }
+  ultimately show "flow0 x t \<in> flow0 x ` {0..T}"
+    by auto
+qed
+
+lemma closed_orbitI:
+  assumes "t \<noteq> t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
+  assumes "flow0 y t = flow0 y t'"
+  shows "closed_orbit y"
+  unfolding closed_orbit_def
+  by (smt assms local.existence_ivl_reverse local.existence_ivl_trans local.flow_trans local.flows_reverse)
+
+(* TODO: can be considerably generalized *)
+lemma flow0_image_UNIV:
+  assumes "existence_ivl0 x = UNIV"
+  shows "flow0 (flow0 x t) ` S = flow0 x ` (\<lambda>s. s + t) ` S"
+  apply auto
+   apply (metis UNIV_I add.commute assms image_eqI local.existence_ivl_trans' local.flow_trans)
+  by (metis UNIV_I add.commute assms imageI local.existence_ivl_trans' local.flow_trans)
+
+lemma recurrence_time_restricts_compact_flow':
+  assumes "t < t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
+  assumes "flow0 y t = flow0 y t'"
+  shows "flow0 y ` UNIV = flow0 y ` {t..t'}"
+proof -
+  have "closed_orbit y"
+    using assms(1-4) closed_orbitI inf.strict_order_iff by blast
+  from closed_orbit_global_existence[OF this]
+  have yex: "existence_ivl0 y = UNIV" .
+  have a1:"t'- t \<in> existence_ivl0 (flow0 y t)"
+    by (simp add: assms(2-3) local.existence_ivl_trans')
+  have a2:"t' -t > 0" using assms(1) by auto
+  have a3:"flow0 (flow0 y t) (t' - t) = flow0 y t"
+    using a1 assms(2) assms(4) local.flow_trans by fastforce 
+  from recurrence_time_restricts_compact_flow[OF a1 a2 a3]
+  have eq:"flow0 (flow0 y t) ` UNIV = flow0 (flow0 y t) ` {0.. t'-t}" .
+  from flow0_image_UNIV[OF yex, of _ "UNIV"]
+  have eql:"flow0 (flow0 y t) ` UNIV = flow0 y ` UNIV"
+    by (metis (no_types) add.commute surj_def surj_plus)
+  from flow0_image_UNIV[OF yex, of _ "{0..t'-t}"]
+  have eqr:"flow0 (flow0 y t) ` {0.. t'-t} = flow0 y ` {t..t'}" by auto
+  show ?thesis using eq eql eqr by auto
+qed
+
+lemma closed_orbitE':
+  assumes "closed_orbit x"
+  obtains T where "T > 0" "\<And>t (n::nat). flow0 x (t+T*n) = flow0 x t"
+proof -
+  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
+    unfolding closed_orbit_def by blast
+  then obtain T where T: "T > 0" "T \<in> existence_ivl0 x" "flow0 x T = x"
+    by (smt recurrence_time_flip_sign)
+  thus ?thesis using  recurrence_time_multiples T that by blast 
+qed
+
+lemma closed_orbitE:
+  assumes "closed_orbit x"
+  obtains T where "T > 0" "\<And>t. flow0 x (t+T) = flow0 x t"
+  using closed_orbitE'
+  by (metis assms mult.commute reals_Archimedean3)
+
+lemma closed_orbit_flow_compact:
+  assumes "closed_orbit x"
+  shows "compact(flow0 x ` UNIV)"
+proof -
+  obtain Tp where "Tp \<noteq> 0" "Tp \<in> existence_ivl0 x" "flow0 x Tp = x" using assms
+    unfolding closed_orbit_def by blast
+  then obtain T where T: "T \<in> existence_ivl0 x" "T > 0" "flow0 x T = x"
+    by (smt recurrence_time_flip_sign)
+  from recurrence_time_restricts_compact_flow[OF this]
+  have feq: "flow0 x ` UNIV = flow0 x ` {0..T}" .
+  have "continuous_on {0..T} (flow0 x)"
+    by (meson T(1) continuous_on_sequentially in_mono local.flow_continuous_on local.ivl_subset_existence_ivl) 
+  from compact_continuous_image[OF this]
+  have "compact (flow0 x ` {0..T})" by auto
+  thus ?thesis using feq by auto
+qed
+
+lemma fixed_point_imp_closed_orbit_period_zero:
+  assumes "x \<in> X"
+  assumes "f x = 0"
+  shows "closed_orbit x" "period x = 0"
+proof -
+  from fixpoint_sol[OF assms] have fp:"existence_ivl0 x = UNIV" "\<And>t. flow0 x t = x" by auto
+  then have co:"closed_orbit x" unfolding closed_orbit_def by blast
+  have a: "\<forall>y>0. \<exists>a\<in>{T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}. a < y"
+    apply auto
+    using fp
+    by (simp add: dense) 
+  from cInf_le_iff[OF closed_orbit_recurrence_times_nonempty[OF co]
+      closed_orbit_recurrence_times_bdd_below , of 0]
+  have "period x \<le> 0" unfolding period_def using a by auto
+  from closed_orbit_period_nonneg[OF co] have "period x \<ge> 0" .
+  then have "period x = 0" using \<open>period x \<le> 0\<close> by linarith
+  thus "closed_orbit x" "period x = 0" using co by auto
+qed
+
+lemma closed_orbit_period_zero_fixed_point:
+  assumes "closed_orbit x" "period x = 0"
+  shows "f x = 0"
+proof (rule ccontr)
+  assume "f x \<noteq> 0"
+  from regular_locally_noteq[OF closed_orbit_in_domain[OF assms(1)] this]
+  have "\<forall>\<^sub>F t in at 0. flow0 x t \<noteq> x " .
+  then obtain r where "r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 x t \<noteq> x" unfolding eventually_at
+    by auto
+  then have "period x \<ge> r" unfolding period_def
+    apply (auto intro!:cInf_greatest)
+     apply (meson assms(1) closed_orbit_def linorder_neqE_linordered_idom neg_0_less_iff_less recurrence_time_flip_sign)
+    using not_le by force
+  thus False using assms(2) \<open>r >0\<close> by linarith
+qed
+
+lemma closed_orbit_subset_\<omega>_limit_set:
+  assumes "closed_orbit x"
+  shows "flow0 x ` UNIV \<subseteq> \<omega>_limit_set x"
+  unfolding \<omega>_limit_set_def \<omega>_limit_point_def
+proof clarsimp
+  fix t
+  from closed_orbitE'[OF assms]
+  obtain T where T: "0 < T" "\<And>t n. flow0 x (t + T* real n) = flow0 x t" by blast
+  define s where "s = (\<lambda>n::nat. t + T * real n)"
+  have exist: "{0..} \<subseteq> existence_ivl0 x"
+    by (simp add: assms closed_orbit_global_existence)
+  have l:"filterlim s at_top sequentially" unfolding s_def
+    using T(1)
+    by (auto intro!:filterlim_tendsto_add_at_top filterlim_tendsto_pos_mult_at_top
+        simp add: filterlim_real_sequentially)
+  have "flow0 x \<circ> s = (\<lambda>n. flow0 x t)" unfolding o_def s_def using T(2) by simp
+  then have r:"(flow0 x \<circ> s) \<longlonglongrightarrow> flow0 x t" by auto
+  show "{0..} \<subseteq> existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity> \<and> (flow0 x \<circ> s) \<longlonglongrightarrow> flow0 x t)"
+    using exist l r by blast
+qed
+
+lemma closed_orbit_\<omega>_limit_set:
+  assumes "closed_orbit x"
+  shows "flow0 x ` UNIV = \<omega>_limit_set x"
+proof -
+  have "\<omega>_limit_set x \<subseteq> flow0 x ` UNIV"
+    using closed_orbit_global_existence[OF assms]
+    by (intro \<omega>_limit_set_in_compact_subset)
+      (auto intro!: flow_in_domain
+        simp add: assms closed_orbit_in_domain image_subset_iff trapped_forward_def
+        closed_orbit_flow_compact)
+  thus ?thesis using closed_orbit_subset_\<omega>_limit_set[OF assms] by auto
+qed
+
+lemma flow0_inj_on:
+  assumes "t \<le> t'"
+  assumes "{t..t'} \<subseteq> existence_ivl0 x"
+  assumes "\<And>s. t < s \<Longrightarrow> s \<le> t' \<Longrightarrow> flow0 x s \<noteq> flow0 x t"
+  shows "inj_on (flow0 x) {t..t'}"
+  apply (rule inj_onI)
+proof (rule ccontr)
+  fix u v
+  assume uv: "u \<in> {t..t'}" "v \<in> {t..t'}" "flow0 x u = flow0 x v" "u \<noteq> v"
+  have "u < v \<or> v < u" using uv(4) by linarith
+  moreover {
+    assume "u < v"
+    from recurrence_time_restricts_compact_flow'[OF this _ _ uv(3)]
+    have "flow0 x ` UNIV = flow0 x ` {u..v}"  using uv(1-2) assms(2) by blast
+    then have "flow0 x t \<in> flow0 x ` {u..v}" by auto
+    moreover have "u = t \<or> flow0 x t \<notin> flow0 x ` {u..v}" using assms(3)
+      by (smt atLeastAtMost_iff image_iff uv(1) uv(2))
+    ultimately have False using uv assms(3)
+      by force
+  }
+  moreover {
+    assume "v < u"
+    from recurrence_time_restricts_compact_flow'[OF this _ _ ]
+    have "flow0 x ` UNIV = flow0 x ` {v..u}"
+      by (metis assms(2) subset_iff uv(1) uv(2) uv(3))
+    then have "flow0 x t \<in> flow0 x ` {v..u}" by auto
+    moreover have "v = t \<or> flow0 x t \<notin> flow0 x ` {v..u}" using assms(3)
+      by (smt atLeastAtMost_iff image_iff uv(1) uv(2))
+    ultimately have False using uv assms(3) by force
+  }
+  ultimately show False by blast
+qed
+
+(* TODO: Probably has a simpler proof *)
+lemma finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes "finite (\<omega>_limit_set x)"
+  obtains y where "\<omega>_limit_set x = {y}" "f y = 0"
+proof -
+  from connected_finite_iff_sing[OF \<omega>_limit_set_in_compact_connected]
+  obtain y where y: "\<omega>_limit_set x = {y}"
+    using \<omega>_limit_set_in_compact_nonempty assms by auto
+  have "f y = 0"
+  proof (rule ccontr)
+    assume fy:"f y \<noteq> 0"
+    from \<omega>_limit_set_in_compact_existence[OF assms(1-4)]
+    have yex: "existence_ivl0 y = UNIV"
+      by (simp add: y)
+    then have "y \<in> X"
+      by (simp add: local.mem_existence_ivl_iv_defined(2))
+    from regular_locally_noteq[OF this fy]
+    have "\<forall>\<^sub>F t in at 0. flow0 y t \<noteq> y" .
+    then obtain r where r: "r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 y t \<noteq> flow0 y 0"
+      unfolding eventually_at using \<open>y \<in> X\<close>
+      by auto
+    then have "\<And>s. 0 < s \<Longrightarrow> s \<le> r/2 \<Longrightarrow> flow0 y s \<noteq> flow0 y 0" by simp
+    from flow0_inj_on[OF _ _ this, of "r/2"]
+    obtain "inj_on(flow0 y) {0..r/2}" using r yex by simp
+    then have "infinite (flow0 y`{0..r/2})" by (simp add: finite_image_iff r(1))
+    moreover from \<omega>_limit_set_invariant[of x]
+    have "flow0 y `{0..r/2} \<subseteq> \<omega>_limit_set x" using y yex
+      unfolding invariant_def trapped_iff_on_existence_ivl0 by auto
+    ultimately show False using y
+      by (metis assms(5) finite.emptyI subset_singleton_iff)
+  qed
+  thus ?thesis using that y by auto
+qed
+
+lemma closed_orbit_periodic:
+  assumes "closed_orbit x" "f x \<noteq> 0"
+  shows "periodic_orbit x"
+  unfolding periodic_orbit_def
+  using assms(1) apply auto
+proof (rule ccontr)
+  assume "closed_orbit x"
+  from closed_orbit_period_nonneg[OF assms(1)] have nneg: "period x \<ge> 0" .
+  assume "\<not> 0 < period x"
+  then have "period x = 0" using nneg by linarith
+  from closed_orbit_period_zero_fixed_point[OF assms(1) this]
+  have "f x = 0" . 
+  thus False using assms(2) by linarith
+qed
+
+lemma periodic_orbitI:
+  assumes "t \<noteq> t'" "t \<in> existence_ivl0 y" "t' \<in> existence_ivl0 y"
+  assumes "flow0 y t = flow0 y t'"
+  assumes "f y \<noteq> 0"
+  shows "periodic_orbit y"
+proof -
+  have y:"y \<in> X"
+    using assms(3) local.mem_existence_ivl_iv_defined(2) by blast
+  from closed_orbitI[OF assms(1-4)] have "closed_orbit y" .
+  from closed_orbit_periodic[OF this assms(5)]
+  show ?thesis .
+qed
+
+lemma periodic_orbit_recurrence_times_closed:
+  assumes "periodic_orbit x"
+  shows "closed {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}"
+proof -
+  have a1:"x \<in> X"
+    using assms closed_orbit_in_domain periodic_orbit_def by auto 
+  have a2:"f x \<noteq> 0"
+    using assms closed_orbit_in_domain fixed_point_imp_closed_orbit_period_zero(2) periodic_orbit_def by auto
+  from regular_locally_noteq[OF a1 a2] have
+    "\<forall>\<^sub>F t in at 0. flow0 x t \<noteq> x" .
+  then obtain r where r:"r>0" "\<forall>t. t \<noteq> 0 \<and> dist t 0 < r \<longrightarrow> flow0 x t \<noteq> x" unfolding eventually_at
+    by auto
+  show ?thesis
+  proof (auto intro!:discrete_imp_closed[OF r(1)])
+    fix t1 t2
+    assume t12: "t1 > 0" "flow0 x t1 = x" "t2 > 0" "flow0 x t2 = x" "dist t2 t1 < r"
+    then have fx: "flow0 x (t1-t2) = x"
+      by (smt a1 assms closed_orbit_global_existence existence_ivl_zero general.existence_ivl_initial_time_iff local.flow_trans periodic_orbit_def)
+    have "dist (t1-t2) 0 < r" using t12(5)
+      by (simp add: dist_norm) 
+    thus "t2 = t1" using r fx
+      by smt
+  qed
+qed
+
+lemma periodic_orbit_period:
+  assumes "periodic_orbit x"
+  shows "period x > 0" "flow0 x (period x) = x"
+proof -
+  from periodic_orbit_recurrence_times_closed[OF assms(1)]
+  have cl: "closed {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}" .
+  have "closed_orbit x" using assms(1) unfolding periodic_orbit_def by auto
+  from closed_contains_Inf[OF closed_orbit_recurrence_times_nonempty[OF this]
+      closed_orbit_recurrence_times_bdd_below cl]
+  have "period x \<in>  {T \<in> existence_ivl0 x. T > 0 \<and> flow0 x T = x}" unfolding period_def .
+  thus "period x > 0" "flow0 x (period x) = x" by auto
+qed
+
+lemma closed_orbit_flow0:
+  assumes "closed_orbit x"
+  shows "closed_orbit (flow0 x t)"
+proof -
+  from closed_orbit_global_existence[OF assms]
+  have "existence_ivl0 x = UNIV" .
+  from closed_orbitE[OF assms]
+  obtain T where "T > 0" "flow0 x (t+T) = flow0 x t"
+    by metis
+  thus ?thesis unfolding closed_orbit_def
+    by (metis UNIV_I \<open>existence_ivl0 x = UNIV\<close> less_irrefl local.existence_ivl_trans' local.flow_trans)
+qed
+
+lemma periodic_orbit_imp_flow0_regular:
+  assumes "periodic_orbit x"
+  shows "f (flow0 x t) \<noteq> 0"
+  by (metis UNIV_I assms closed_orbit_flow0 closed_orbit_global_existence closed_orbit_in_domain fixed_point_imp_closed_orbit_period_zero(2) fixpoint_sol(2) less_irrefl local.flows_reverse periodic_orbit_def)
+
+lemma fixed_point_imp_\<omega>_limit_set:
+  assumes "x \<in> X" "f x = 0"
+  shows "\<omega>_limit_set x = {x}"
+proof -
+  have "closed_orbit x"
+    by (metis assms fixed_point_imp_closed_orbit_period_zero(1))
+  from closed_orbit_\<omega>_limit_set[OF this]
+  have "flow0 x ` UNIV = \<omega>_limit_set x" .
+  thus ?thesis
+    by (metis assms(1) assms(2) fixpoint_sol(2) image_empty image_insert image_subset_iff insertI1 rangeI subset_antisym)
+qed
+
+end
+
+context auto_ll_on_open begin
+
+lemma closed_orbit_eq_rev: "closed_orbit x = rev.closed_orbit x"
+  unfolding closed_orbit_def rev.closed_orbit_def rev_eq_flow rev_existence_ivl_eq0
+  by auto
+
+lemma closed_orbit_\<alpha>_limit_set:
+  assumes "closed_orbit x"
+  shows "flow0 x ` UNIV = \<alpha>_limit_set x"
+  using rev.closed_orbit_\<omega>_limit_set assms
+  unfolding closed_orbit_eq_rev \<alpha>_limit_set_eq_rev flow_image_eq_rev range_uminus .
+
+lemma fixed_point_imp_\<alpha>_limit_set:
+  assumes "x \<in> X" "f x = 0"
+  shows "\<alpha>_limit_set x = {x}"
+  using rev.fixed_point_imp_\<omega>_limit_set assms
+  unfolding \<alpha>_limit_set_eq_rev
+  by auto
+
+lemma finite_\<alpha>_limit_set_in_compact_imp_unique_fixed_point:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X"  "trapped_backward x K"
+  assumes "finite (\<alpha>_limit_set x)"
+  obtains y where "\<alpha>_limit_set x = {y}" "f y = 0"
+proof -
+  from rev.finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF
+      assms(1-5)[unfolded trapped_backward_iff_rev_trapped_forward \<alpha>_limit_set_eq_rev]]
+  show ?thesis using that
+    unfolding \<alpha>_limit_set_eq_rev
+    by auto
+qed
+end
+
 end
\ No newline at end of file
diff --git a/thys/Poincare_Bendixson/Poincare_Bendixson.thy b/thys/Poincare_Bendixson/Poincare_Bendixson.thy
--- a/thys/Poincare_Bendixson/Poincare_Bendixson.thy
+++ b/thys/Poincare_Bendixson/Poincare_Bendixson.thy
@@ -1,2821 +1,2821 @@
-section \<open>Poincare Bendixson Theory\<close>
-theory Poincare_Bendixson
-  imports 
-    Ordinary_Differential_Equations.ODE_Analysis
-    Analysis_Misc ODE_Misc Periodic_Orbit
-begin
-
-
-subsection \<open>Flow to Path\<close>
-
-context auto_ll_on_open begin
-
-(* The path along the flow starting at time t to time t' *)
-definition "flow_to_path x t t' = flow0 x \<circ> linepath t t'"
-
-lemma pathstart_flow_to_path[simp]:
-  shows "pathstart (flow_to_path x t t') = flow0 x t"
-  unfolding flow_to_path_def
-  by (auto simp add: pathstart_compose)
-
-lemma pathfinish_flow_to_path[simp]:
-  shows "pathfinish (flow_to_path x t t') = flow0 x t'"
-  unfolding flow_to_path_def
-  by (auto simp add: pathfinish_compose)
-
-lemma flow_to_path_unfold:
-  shows "flow_to_path x t t' s = flow0 x ((1 - s) * t + s * t')"
-  unfolding flow_to_path_def o_def linepath_def by auto
-
-lemma subpath0_flow_to_path:
-  shows "(subpath 0 u (flow_to_path x t t')) = flow_to_path x t (t + u*(t'-t))"
-  unfolding flow_to_path_def subpath_image subpath0_linepath
-  by auto
-
-lemma path_image_flow_to_path[simp]:
-  assumes "t \<le> t'"
-  shows "path_image (flow_to_path x t t') = flow0 x ` {t..t'}"
-  unfolding flow_to_path_def path_image_compose path_image_linepath
-  using assms real_Icc_closed_segment by auto
-
-lemma flow_to_path_image0_right_open[simp]:
-  assumes "t < t'"
-  shows "flow_to_path x t t' ` {0..<1} = flow0 x `{t..<t'}"
-  unfolding flow_to_path_def image_comp[symmetric] linepath_image0_right_open_real[OF assms]
-  by auto
-
-lemma flow_to_path_path:
-  assumes "t \<le> t'"
-  assumes "{t..t'} \<subseteq> existence_ivl0 x"
-  shows "path (flow_to_path x t t')"
-proof -
-  have "x \<in> X"
-    using assms(1) assms(2) subset_empty by fastforce
-  have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow>  (1 - xa) * t + xa * t' \<le> t'"
-    by (simp add: assms(1) convex_bound_le)
-  moreover have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow> t \<le> (1 - xa) * t + xa * t'" using assms(1)
-    by (metis add.commute add_diff_cancel_left' diff_diff_eq2 diff_le_eq mult.commute mult.right_neutral mult_right_mono right_diff_distrib')
-  ultimately have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow> (1 - xa) * t + xa * t' \<in> existence_ivl0 x"
-    using assms(2) by auto
-  thus ?thesis unfolding path_def flow_to_path_def linepath_def
-    by (auto intro!:continuous_intros simp add :\<open>x \<in> X\<close>)
-qed
-
-lemma flow_to_path_arc:
-  assumes "t \<le> t'"
-  assumes "{t..t'} \<subseteq> existence_ivl0 x"
-  assumes "\<forall>s \<in> {t<..<t'}. flow0 x s \<noteq> flow0 x t"
-  assumes "flow0 x t \<noteq> flow0 x t'"
-  shows "arc (flow_to_path x t t')"
-  unfolding arc_def
-proof safe
-  from flow_to_path_path[OF assms(1-2)]
-  show "path (flow_to_path x t t')" .
-next
-  show "inj_on (flow_to_path x t t') {0..1}"
-    unfolding flow_to_path_def
-    apply (rule comp_inj_on)
-     apply (metis assms(4) inj_on_linepath)
-    using assms path_image_linepath[of t t'] apply (auto intro!:flow0_inj_on)
-    using flow0_inj_on greaterThanLessThan_iff linepath_image_01 real_Icc_closed_segment by fastforce
-qed
-
-end
-
-locale c1_on_open_R2 = c1_on_open_euclidean f f' X for f::"'a::executable_euclidean_space \<Rightarrow> _" and f' and X +
-  assumes dim2: "DIM('a) = 2"
-begin
-
-
-subsection \<open>2D Line segments\<close>
-
-text \<open>Line segments are specified by two endpoints
-      The closed line segment from x to y is given by the set {x--y}
-      and {x<--<y} for the open segment\<close>
-
-text \<open> Rotates a vector clockwise 90 degrees \<close>
-definition "rot (v::'a) = (eucl_of_list [nth_eucl v 1, -nth_eucl v 0]::'a)"
-
-lemma exhaust2_nat: "(\<forall>i<(2::nat). P i) \<longleftrightarrow> P 0 \<and> P 1"
-  using less_2_cases by auto
-lemma sum2_nat: "(\<Sum>i<(2::nat). P i) = P 0 + P 1"
-  by (simp add: eval_nat_numeral)
-
-lemmas vec_simps =
-  eucl_eq_iff[where 'a='a] dim2 eucl_of_list_eucl_nth exhaust2_nat
-  plus_nth_eucl
-  minus_nth_eucl
-  uminus_nth_eucl
-  scaleR_nth_eucl
-  inner_nth_eucl
-  sum2_nat
-  algebra_simps
-
-lemma minus_expand:
-  shows "(x::'a)-y = (eucl_of_list [x$\<^sub>e0 - y$\<^sub>e0, x$\<^sub>e1 - y$\<^sub>e1])"
-  by (simp add:vec_simps)
-
-lemma dot_ortho[simp]: "x \<bullet> rot x = 0"
-  unfolding rot_def minus_expand
-  by (simp add: vec_simps)
-
-lemma nrm_dot:
-  shows "((x::'a)-y) \<bullet> (rot (x-y)) = 0"
-  unfolding rot_def minus_expand
-  by (simp add: vec_simps)
-
-lemma nrm_reverse: "a \<bullet> (rot (x-y)) = - a \<bullet> (rot (y-x))" for x y::'a
-  unfolding rot_def
-  by (simp add:vec_simps)
-
-lemma norm_rot: "norm (rot v) = norm v" for v::'a
-  unfolding rot_def
-  by (simp add:vec_simps norm_nth_eucl L2_set_def)
-
-lemma rot_rot[simp]:
-  shows "rot (rot v) = -v"
-  unfolding rot_def
-  by (simp add:vec_simps)
-
-lemma rot_scaleR[simp]:
-  shows "rot ( u *\<^sub>R v) =  u *\<^sub>R (rot v)"
-  unfolding rot_def
-  by (simp add:vec_simps)
-
-lemma rot_0[simp]: "rot 0 = 0"
-  using rot_scaleR[of 0] by simp
-
-lemma rot_eq_0_iff[simp]: "rot x = 0 \<longleftrightarrow> x = 0"
-  apply (auto simp: rot_def)
-   apply (metis One_nat_def norm_eq_zero norm_rot norm_zero rot_def)
-  using rot_0 rot_def by auto
-
-lemma in_segment_inner_rot:
-  "(x - a) \<bullet> rot (b - a) = 0"
-  if "x \<in> {a--b}"
-proof -
-  from that obtain u where x: "x = a + u *\<^sub>R (b - a)" "0 \<le> u" "u \<le> 1"
-    by (auto simp: in_segment algebra_simps)
-  show ?thesis
-    unfolding x
-    by (simp add: inner_add_left nrm_dot)
-qed
-
-lemma inner_rot_in_segment:
-  "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))"
-  if "(x - a) \<bullet> rot (b - a) = 0" "a \<noteq> b"
-proof -
-  from that have
-    x0: "b $\<^sub>e 0 = a $\<^sub>e 0 \<Longrightarrow> x $\<^sub>e 0 =
-      (a $\<^sub>e 0 * b $\<^sub>e Suc 0 - b $\<^sub>e 0 * a $\<^sub>e Suc 0 + (b $\<^sub>e 0 - a $\<^sub>e 0) * x $\<^sub>e Suc 0) /
-      (b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0)"
-    and x1: "b $\<^sub>e 0 \<noteq> a $\<^sub>e 0 \<Longrightarrow> x $\<^sub>e Suc 0 =
-      ((b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0) * x $\<^sub>e 0 - a $\<^sub>e 0 * b $\<^sub>e Suc 0 + b $\<^sub>e 0 * a $\<^sub>e Suc 0) / (b $\<^sub>e 0 - a $\<^sub>e 0)"
-    by (auto simp: rot_def vec_simps divide_simps)
-  define u where "u = (if b $\<^sub>e 0 - a $\<^sub>e 0 \<noteq> 0
-    then ((x $\<^sub>e 0 - a $\<^sub>e 0) / (b $\<^sub>e 0 -  a $\<^sub>e 0))
-    else ((x $\<^sub>e 1 - a $\<^sub>e 1) / (b $\<^sub>e 1 -  a $\<^sub>e 1)))
-    "
-  show ?thesis
-    apply (cases "b $\<^sub>e 0 - a $\<^sub>e 0 = 0")
-    subgoal
-      using that(2)
-      apply (auto intro!: image_eqI[where x="((x $\<^sub>e 1 - a $\<^sub>e 1) / (b $\<^sub>e 1 -  a $\<^sub>e 1))"]
-          simp: vec_simps x0 divide_simps algebra_simps)
-       apply (metis ab_semigroup_mult_class.mult_ac(1) mult.commute sum_sqs_eq)
-      by (metis mult.commute mult.left_commute sum_sqs_eq)
-    subgoal
-      apply (auto intro!: image_eqI[where x="((x $\<^sub>e 0 - a $\<^sub>e 0) / (b $\<^sub>e 0 -  a $\<^sub>e 0))"]
-          simp: vec_simps x1 divide_simps algebra_simps)
-       apply (metis ab_semigroup_mult_class.mult_ac(1) mult.commute sum_sqs_eq)
-      by (metis mult.commute mult.left_commute sum_sqs_eq)
-    done
-qed
-
-lemma in_open_segment_iff_rot:
-  "x \<in> {a<--<b} \<longleftrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) <..< b \<bullet> (b - a)}"
-  if "a \<noteq> b"
-  unfolding open_segment_line_hyperplanes[OF that]
-  by (auto simp: nrm_dot intro!: inner_rot_in_segment)
-
-lemma in_open_segment_rotD:
-  "x \<in> {a<--<b} \<Longrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) <..< b \<bullet> (b - a)}"
-  by (subst in_open_segment_iff_rot[symmetric]) auto
-
-lemma in_closed_segment_iff_rot:
-  "x \<in> {a--b} \<longleftrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) .. b \<bullet> (b - a)}"
-  if "a \<noteq> b"
-  unfolding closed_segment_line_hyperplanes[OF that] using that
-  by (auto simp: nrm_dot intro!: inner_rot_in_segment)
-
-lemma in_segment_inner_rot2:
-  "(x - y) \<bullet> rot (a - b) = 0"
-  if "x \<in> {a--b}" "y \<in> {a--b}"
-proof -
-  from that obtain u where x: "x = a + u *\<^sub>R (b - a)" "0 \<le> u" "u \<le> 1"
-    by (auto simp: in_segment algebra_simps)
-  from that obtain v where y: "y = a + v *\<^sub>R (b - a)" "0 \<le> v" "v \<le> 1"
-    by (auto simp: in_segment algebra_simps)
-  show ?thesis
-    unfolding x y
-    apply (auto simp: inner_add_left )
-    by (smt add_diff_cancel_left' in_segment_inner_rot inner_diff_left minus_diff_eq nrm_reverse that(1) that(2) x(1) y(1))
-qed
-
-lemma closed_segment_surface:
-  "a \<noteq> b \<Longrightarrow> {a--b} = { x \<in> {x. x \<bullet> (b - a) \<in> {a\<bullet>(b - a) .. b \<bullet> (b - a)}}. (x - a) \<bullet> rot (b - a) = 0}"
-  by (auto simp: in_closed_segment_iff_rot)
-
-lemma rot_diff_commute: "rot (b - a) = -rot(a - b)"
-  apply (auto simp: rot_def algebra_simps)
-  by (metis One_nat_def minus_diff_eq rot_def rot_rot)
-
-
-subsection \<open>Bijection Real-Complex for Jordan Curve Theorem\<close>
-
-definition "complex_of (x::'a) = x$\<^sub>e0 + \<i> * x$\<^sub>e1"
-
-definition "real_of (x::complex) = (eucl_of_list [Re x, Im x]::'a)"
-
-lemma complex_of_linear:
-  shows "linear complex_of"
-  unfolding complex_of_def
-  apply (auto intro!:linearI simp add: distrib_left plus_nth_eucl)
-  by (simp add: of_real_def scaleR_add_right scaleR_nth_eucl)
-
-lemma complex_of_bounded_linear:
-  shows "bounded_linear complex_of"
-  unfolding complex_of_def
-  apply (auto intro!:bounded_linearI' simp add: distrib_left plus_nth_eucl)
-  by (simp add: of_real_def scaleR_add_right scaleR_nth_eucl)
-
-lemma real_of_linear:
-  shows "linear real_of"
-  unfolding real_of_def
-  by (auto intro!:linearI simp add: vec_simps)
-
-lemma real_of_bounded_linear:
-  shows "bounded_linear real_of"
-  unfolding real_of_def
-  by (auto intro!:bounded_linearI' simp add: vec_simps)
-
-lemma complex_of_real_of:
-  "(complex_of \<circ> real_of) = id"
-  unfolding complex_of_def real_of_def
-  using complex_eq by (auto simp add:vec_simps)
-
-lemma real_of_complex_of:
-  "(real_of \<circ> complex_of) = id"
-  unfolding complex_of_def real_of_def
-  using complex_eq by (auto simp add:vec_simps)
-
-lemma complex_of_bij:
-  shows "bij (complex_of)"
-  using o_bij[OF real_of_complex_of complex_of_real_of] .
-
-lemma real_of_bij:
-  shows "bij (real_of)"
-  using o_bij[OF complex_of_real_of real_of_complex_of] .
-
-lemma real_of_inj:
-  shows "inj (real_of)"
-  using real_of_bij
-  using bij_betw_imp_inj_on by auto
-
-lemma Jordan_curve_R2:
-  fixes c :: "real \<Rightarrow> 'a"
-  assumes "simple_path c" "pathfinish c = pathstart c"
-  obtains inside outside where
-    "inside \<noteq> {}" "open inside" "connected inside"
-    "outside \<noteq> {}" "open outside" "connected outside"
-    "bounded inside" "\<not> bounded outside"
-    "inside \<inter> outside = {}"
-    "inside \<union> outside = - path_image c"
-    "frontier inside = path_image c"
-    "frontier outside = path_image c"
-proof -
-  from simple_path_linear_image_eq[OF complex_of_linear]
-  have a1:"simple_path (complex_of \<circ> c)" using assms(1) complex_of_bij
-    using bij_betw_imp_inj_on by blast
-  have a2:"pathfinish (complex_of \<circ> c) = pathstart (complex_of \<circ> c)"
-    using assms(2) by (simp add:pathstart_compose pathfinish_compose)
-
-  from Jordan_curve[OF a1 a2]
-  obtain inside outside where io:
-    "inside \<noteq> {}" "open inside" "connected inside"
-    "outside \<noteq> {}" "open outside" "connected outside"
-    "bounded inside" "\<not> bounded outside" "inside \<inter> outside = {}"
-    "inside \<union> outside = - path_image (complex_of \<circ> c)"
-    "frontier inside = path_image (complex_of \<circ> c)"
-    "frontier outside = path_image (complex_of \<circ> c)" by blast
-  let ?rin = "real_of ` inside"
-  let ?rout = "real_of ` outside"
-  have i: "inside = complex_of ` ?rin" using complex_of_real_of unfolding image_comp
-    by auto
-  have o: "outside = complex_of ` ?rout" using complex_of_real_of unfolding image_comp
-    by auto
-  have c: "path_image(complex_of \<circ> c) = complex_of ` (path_image c)"
-    by (simp add: path_image_compose)
-  have "?rin \<noteq> {}" using io by auto
-  moreover from open_bijective_linear_image_eq[OF real_of_linear real_of_bij]
-  have "open ?rin" using io by auto
-  moreover from connected_linear_image[OF real_of_linear]
-  have "connected ?rin" using io by auto
-  moreover have "?rout \<noteq> {}" using io by auto
-  moreover from open_bijective_linear_image_eq[OF real_of_linear real_of_bij]
-  have "open ?rout" using io by auto
-  moreover from connected_linear_image[OF real_of_linear]
-  have "connected ?rout" using io by auto
-  moreover from bounded_linear_image[OF io(7) real_of_bounded_linear]
-  have "bounded ?rin" .
-  moreover from bounded_linear_image[OF _ complex_of_bounded_linear]
-  have "\<not> bounded ?rout" using io(8) o
-    by force
-  from image_Int[OF real_of_inj]
-  have "?rin \<inter> ?rout = {}" using io(9) by auto
-  moreover from bij_image_Compl_eq[OF complex_of_bij]
-  have "?rin \<union> ?rout = - path_image c" using io(10) unfolding c
-    by (metis id_apply image_Un image_comp image_cong image_ident real_of_complex_of)
-  moreover from closure_injective_linear_image[OF real_of_linear real_of_inj]
-  have "frontier ?rin = path_image c" using io(11)
-    unfolding frontier_closures c
-    by (metis \<open>\<And>B A. real_of ` (A \<inter> B) = real_of ` A \<inter> real_of ` B\<close> bij_image_Compl_eq c calculation(9) compl_sup double_compl io(10) real_of_bij)
-  moreover from closure_injective_linear_image[OF real_of_linear real_of_inj]
-  have "frontier ?rout = path_image c" using io(12)
-    unfolding frontier_closures c
-    by (metis \<open>\<And>B A. real_of ` (A \<inter> B) = real_of ` A \<inter> real_of ` B\<close> bij_image_Compl_eq c calculation(10) frontier_closures io(11) real_of_bij)
-  ultimately show ?thesis
-    by (meson \<open>\<not> bounded (real_of ` outside)\<close> that)
-qed
-
-(* copied *)
-corollary Jordan_inside_outside_R2:
-  fixes c :: "real \<Rightarrow> 'a"
-  assumes "simple_path c" "pathfinish c = pathstart c"
-  shows "inside(path_image c) \<noteq> {} \<and>
-          open(inside(path_image c)) \<and>
-          connected(inside(path_image c)) \<and>
-          outside(path_image c) \<noteq> {} \<and>
-          open(outside(path_image c)) \<and>
-          connected(outside(path_image c)) \<and>
-          bounded(inside(path_image c)) \<and>
-          \<not> bounded(outside(path_image c)) \<and>
-          inside(path_image c) \<inter> outside(path_image c) = {} \<and>
-          inside(path_image c) \<union> outside(path_image c) =
-          - path_image c \<and>
-          frontier(inside(path_image c)) = path_image c \<and>
-          frontier(outside(path_image c)) = path_image c"
-proof -
-  obtain inner outer
-    where *: "inner \<noteq> {}" "open inner" "connected inner"
-      "outer \<noteq> {}" "open outer" "connected outer"
-      "bounded inner" "\<not> bounded outer" "inner \<inter> outer = {}"
-      "inner \<union> outer = - path_image c"
-      "frontier inner = path_image c"
-      "frontier outer = path_image c"
-    using Jordan_curve_R2 [OF assms] by blast
-  then have inner: "inside(path_image c) = inner"
-    by (metis dual_order.antisym inside_subset interior_eq interior_inside_frontier)
-  have outer: "outside(path_image c) = outer"
-    using \<open>inner \<union> outer = - path_image c\<close> \<open>inside (path_image c) = inner\<close>
-      outside_inside \<open>inner \<inter> outer = {}\<close> by auto
-  show ?thesis
-    using * by (auto simp: inner outer)
-qed
-
-lemma jordan_points_inside_outside:
-  fixes p :: "real \<Rightarrow> 'a"
-  assumes "0 < e"
-  assumes jordan: "simple_path p" "pathfinish p = pathstart p"
-  assumes x: "x \<in> path_image p"
-  obtains y z where "y \<in> inside (path_image p)" "y \<in> ball x e" 
-    "z \<in> outside (path_image p)" "z \<in> ball x e"
-proof -
-  from Jordan_inside_outside_R2[OF jordan]
-  have xi: "x \<in> frontier(inside (path_image p))" and
-    xo: "x \<in> frontier(outside (path_image p))"
-    using x by auto
-  obtain y where y:"y \<in> inside (path_image p)" "y \<in> ball x e" using \<open>0 < e\<close> xi
-    unfolding frontier_straddle
-    by auto
-  obtain z where z:"z \<in> outside (path_image p)" "z \<in> ball x e" using \<open>0 < e\<close> xo
-    unfolding frontier_straddle
-    by auto
-  show ?thesis using y z that by blast
-qed  
-
-lemma eventually_at_open_segment:
-  assumes "x \<in> {a<--<b}"
-  shows "\<forall>\<^sub>F y in at x. (y-a) \<bullet> rot(a-b) = 0 \<longrightarrow> y \<in> {a <--< b}"
-proof -
-  from assms have "a \<noteq> b" by auto
-  from assms have x: "(x - a) \<bullet> rot (b - a) = 0" "x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
-    unfolding in_open_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
-    by auto
-  then have "\<forall>\<^sub>F y in at x. y \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
-    by (intro topological_tendstoD) (auto intro!: tendsto_intros)
-  then show ?thesis
-    by eventually_elim (auto simp: in_open_segment_iff_rot[OF \<open>a \<noteq> b\<close>] nrm_reverse[of _ a b] algebra_simps dist_commute)
-qed
-
-lemma linepath_ball:
-  assumes "x \<in> {a<--<b}"
-  obtains e where "e > 0" "ball x e \<inter> {y. (y-a) \<bullet> rot(a-b) = 0} \<subseteq> {a <--< b}"
-proof -
-  from eventually_at_open_segment[OF assms] assms
-  obtain e where "0 < e" "ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0} \<subseteq> {a<--<b}"
-    by (force simp: eventually_at in_open_segment_iff_rot dist_commute)
-  then show ?thesis ..
-qed
-
-lemma linepath_ball_inside_outside:
-  fixes p :: "real \<Rightarrow> 'a"
-  assumes jordan: "simple_path (p +++ linepath a b)" "pathfinish p = a" "pathstart p = b"
-  assumes x: "x \<in> {a<--<b}"
-  obtains e where "e > 0" "ball x e \<inter> path_image p = {}"
-    "ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0} \<subseteq> inside (path_image (p +++ linepath a b)) \<and>
-     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0} \<subseteq> outside (path_image (p +++ linepath a b))
-     \<or> 
-     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0} \<subseteq> inside (path_image (p +++ linepath a b)) \<and>
-     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0} \<subseteq> outside (path_image (p +++ linepath a b))"
-proof -
-  let ?lp = "p +++ linepath a b"
-  have "a \<noteq> b" using x by auto
-  have pp:"path p" using jordan path_join pathfinish_linepath simple_path_imp_path
-    by fastforce
-  have "path_image p \<inter> path_image (linepath a b) \<subseteq> {a,b}"
-    using jordan simple_path_join_loop_eq
-    by (metis (no_types, lifting) inf_sup_aci(1) insert_commute path_join_path_ends path_linepath simple_path_imp_path simple_path_joinE)
-  then have "x \<notin> path_image p" using x unfolding path_image_linepath
-    by (metis DiffE Int_iff le_iff_inf open_segment_def)
-  then have "\<forall>\<^sub>F y in at x. y \<notin> path_image p"
-    by (intro eventually_not_in_closed) (auto simp: closed_path_image \<open>path p\<close>)
-  moreover 
-  have "\<forall>\<^sub>F y in at x. (y - a) \<bullet> rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b}"
-    by (rule eventually_at_open_segment[OF x])
-  ultimately have "\<forall>\<^sub>F y in at x. y \<notin> path_image p \<and> ((y - a) \<bullet> rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b})"
-    by eventually_elim auto
-  then obtain e where e: "e > 0" "ball x e \<inter> path_image p = {}"
-    "ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0} \<subseteq> {a<--<b}"
-    using \<open>x \<notin> path_image p\<close> x in_open_segment_rotD[OF x]
-    apply (auto simp: eventually_at subset_iff dist_commute dest!: )
-    by (metis Int_iff all_not_in_conv dist_commute mem_ball) 
-  have a1: "pathfinish ?lp = pathstart ?lp"
-    by (auto simp add: jordan)
-  have "x \<in> path_image ?lp"
-    using jordan(1) open_closed_segment path_image_join path_join_path_ends simple_path_imp_path x by fastforce
-  from jordan_points_inside_outside[OF e(1) jordan(1) a1 this]
-  obtain y z where y: "y \<in> inside (path_image ?lp)" "y \<in> ball x e" 
-    and z: "z \<in> outside (path_image ?lp)" "z \<in> ball x e" by blast
-  have jordancurve:
-    "inside (path_image ?lp) \<inter> outside(path_image ?lp) = {}"
-    "frontier (inside (path_image ?lp)) = path_image ?lp"
-    "frontier (outside (path_image ?lp)) = path_image ?lp"
-    using Jordan_inside_outside_R2[OF jordan(1) a1] by auto
-  define b1 where "b1 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0}"
-  define b2 where "b2 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0}"
-  define b3 where "b3 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) = 0}"
-  have "path_connected b1" unfolding b1_def
-    apply (auto intro!: convex_imp_path_connected convex_Int simp add:inner_diff_left)
-    using convex_halfspace_gt[of "a \<bullet> rot (a - b)" "rot(a-b)"] inner_commute
-    by (metis (no_types, lifting) Collect_cong)
-  have "path_connected b2" unfolding b2_def
-    apply (auto intro!: convex_imp_path_connected convex_Int simp add:inner_diff_left)
-    using convex_halfspace_lt[of "rot(a-b)" "a \<bullet> rot (a - b)"] inner_commute
-    by (metis (no_types, lifting) Collect_cong)
-  have "b1 \<inter> path_image(linepath a b) = {}" unfolding path_image_linepath b1_def
-    using closed_segment_surface[OF \<open>a \<noteq> b\<close>] in_segment_inner_rot2 by auto 
-  then have b1i:"b1 \<inter> path_image ?lp = {}"
-    by (metis IntD2 b1_def disjoint_iff_not_equal e(2) inf_sup_aci(1) not_in_path_image_join)
-  have "b2 \<inter> path_image(linepath a b) = {}" unfolding path_image_linepath b2_def
-    using closed_segment_surface[OF \<open>a \<noteq> b\<close>] in_segment_inner_rot2 by auto 
-  then have b2i:"b2 \<inter> path_image ?lp = {}"
-    by (metis IntD2 b2_def disjoint_iff_not_equal e(2) inf_sup_aci(1) not_in_path_image_join)
-  have bsplit: "ball x e = b1 \<union> b2 \<union> b3"
-    unfolding b1_def b2_def b3_def
-    by auto
-  have "z \<notin> b3"
-  proof clarsimp
-    assume "z \<in> b3"
-    then have "z \<in> {a<--<b}" unfolding b3_def using e by blast
-    then have "z \<in> path_image(linepath a b)" by (auto simp add: open_segment_def)
-    then have "z \<in> path_image ?lp"
-      by (simp add: jordan(2) path_image_join)
-    thus False using z
-      using inside_Un_outside by fastforce
-  qed
-  then have z12: "z \<in> b1 \<or> z \<in> b2" using z bsplit by blast
-  have "y \<notin> b3"
-  proof clarsimp
-    assume "y \<in> b3"
-    then have "y \<in> {a<--<b}" unfolding b3_def using e by auto
-    then have "y \<in> path_image(linepath a b)" by (auto simp add: open_segment_def)
-    then have "y \<in> path_image ?lp"
-      by (simp add: jordan(2) path_image_join)
-    thus False using y
-      using inside_Un_outside by fastforce
-  qed
-  then have "y \<in> b1 \<or> y \<in> b2" using y bsplit by blast
-  moreover {
-    assume "y \<in> b1"
-    then have "b1 \<inter> inside (path_image ?lp) \<noteq> {}" using y by blast
-    from path_connected_not_frontier_subset[OF \<open>path_connected b1\<close> this]
-    have 1:"b1 \<subseteq> inside (path_image ?lp)" unfolding jordancurve using b1i
-      by blast
-    then have "z \<in> b2" using jordancurve(1) z(1) z12 by blast
-    then have "b2 \<inter> outside (path_image ?lp) \<noteq> {}" using z by blast
-    from path_connected_not_frontier_subset[OF \<open>path_connected b2\<close> this]
-    have 2:"b2 \<subseteq> outside (path_image ?lp)" unfolding jordancurve using b2i
-      by blast
-    note conjI[OF 1 2]
-  }
-  moreover {
-    assume "y \<in> b2"
-    then have "b2 \<inter> inside (path_image ?lp) \<noteq> {}" using y by blast
-    from path_connected_not_frontier_subset[OF \<open>path_connected b2\<close> this]
-    have 1:"b2 \<subseteq> inside (path_image ?lp)" unfolding jordancurve using b2i
-      by blast
-    then have "z \<in> b1" using jordancurve(1) z(1) z12 by blast
-    then have "b1 \<inter> outside (path_image ?lp) \<noteq> {}" using z by blast
-    from path_connected_not_frontier_subset[OF \<open>path_connected b1\<close> this]
-    have 2:"b1 \<subseteq> outside (path_image ?lp)" unfolding jordancurve using b1i
-      by blast
-    note conjI[OF 1 2]
-  } 
-  ultimately show ?thesis unfolding b1_def b2_def using that[OF e(1-2)] by auto
-qed
-
-subsection \<open>Transversal Segments\<close>\<comment> \<open>TODO: Transversal surface in Euclidean space?!\<close>
-
-definition "transversal_segment a b \<longleftrightarrow>
-  a \<noteq> b \<and> {a--b} \<subseteq> X \<and>
-  (\<forall>z \<in> {a--b}. f z \<bullet> rot (a-b) \<noteq> 0)"
-
-lemma transversal_segment_reverse:
-  assumes "transversal_segment x y"
-  shows "transversal_segment y x"
-  unfolding transversal_segment_def
-  by (metis (no_types, hide_lams) add.left_neutral add_uminus_conv_diff assms closed_segment_commute inner_diff_left inner_zero_left nrm_reverse transversal_segment_def) 
-
-lemma transversal_segment_commute: "transversal_segment x y \<longleftrightarrow> transversal_segment y x"
-  using transversal_segment_reverse by blast
-
-lemma transversal_segment_neg:
-  assumes "transversal_segment x y"
-  assumes w: "w \<in> {x -- y}" and "f w \<bullet> rot (x-y) < 0"
-  shows "\<forall>z \<in> {x--y}. f(z) \<bullet> rot (x-y) < 0"
-proof (rule ccontr)
-  assume " \<not> (\<forall>z\<in>{x--y}. f z \<bullet> rot (x-y) < 0)"
-  then obtain z where z: "z \<in> {x--y}" "f z \<bullet> rot (x-y) \<ge> 0" by auto
-  define ff where "ff = (\<lambda>s. f (w + s *\<^sub>R (z - w)) \<bullet> rot (x-y))"
-  have f0:"ff 0 \<le> 0" unfolding ff_def using assms(3)
-    by simp 
-  have fu:"ff 1 \<ge> 0"
-    by (auto simp: ff_def z)
-  from assms(2) obtain u where u: "0 \<le> u" "u \<le> 1" "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
-    unfolding in_segment by blast
-  have "{x--y} \<subseteq> X" using assms(1) unfolding transversal_segment_def by blast
-  then have "continuous_on {0..1} ff" unfolding ff_def 
-    using assms(2)
-    by (auto intro!:continuous_intros closed_subsegmentI z elim!: set_mp)
-  from IVT'[of ff, OF f0 fu zero_le_one this]
-  obtain s where s: "s \<ge> 0" "s \<le> 1" "ff s = 0" by blast
-  have "w + s *\<^sub>R (z - w) \<in> {x -- y}"
-    by (auto intro!: closed_subsegmentI z s w)
-  with \<open>ff s = 0\<close> show False
-    using s assms(1) unfolding transversal_segment_def ff_def by blast
-qed
-
-lemmas transversal_segment_sign_less = transversal_segment_neg[OF _ ends_in_segment(1)]
-
-lemma transversal_segment_pos:
-  assumes "transversal_segment x y"
-  assumes w: "w \<in> {x -- y}" "f w \<bullet> rot (x-y) > 0"
-  shows "\<forall>z \<in> {x--y}. f(z) \<bullet> rot (x-y) > 0"
-  using transversal_segment_neg[OF transversal_segment_reverse[OF assms(1)], of w] w
-  by (auto simp: rot_diff_commute[of x y] closed_segment_commute)
-
-lemma transversal_segment_posD:
-  assumes "transversal_segment x y"
-    and pos: "z \<in> {x -- y}" "f z \<bullet> rot (x - y) > 0"
-  shows "x \<noteq> y" "{x--y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x-y) > 0"
-  using assms(1) transversal_segment_pos[OF assms]
-  by (auto simp: transversal_segment_def)
-
-lemma transversal_segment_negD:
-  assumes "transversal_segment x y"
-    and pos: "z \<in> {x -- y}" "f z \<bullet> rot (x - y) < 0"
-  shows "x \<noteq> y" "{x--y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x-y) < 0"
-  using assms(1) transversal_segment_neg[OF assms]
-  by (auto simp: transversal_segment_def)
-
-lemma transversal_segmentE:
-  assumes "transversal_segment x y"
-  obtains "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x - y) > 0"
-  |  "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (y - x) > 0"
-proof (cases "f x \<bullet> rot (x - y) < 0")
-  case True
-  from transversal_segment_negD[OF assms ends_in_segment(1) True]
-  have "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (y - x) > 0"
-    by (auto simp: rot_diff_commute[of x y])
-  then show ?thesis ..
-next
-  case False
-  then have "f x \<bullet> rot (x - y) > 0" using assms
-    by (auto simp: transversal_segment_def algebra_split_simps not_less order.order_iff_strict)
-  from transversal_segment_posD[OF assms ends_in_segment(1) this]
-  show ?thesis ..
-qed
-
-lemma dist_add_vec:
-  shows "dist (x + s *\<^sub>R v) x = abs s * norm v"
-  by (simp add: dist_cancel_add1)
-
-lemma transversal_segment_exists:
-  assumes "x \<in> X" "f x \<noteq> 0"
-  obtains a b where "x \<in> {a<--<b}"
-    "transversal_segment a b"
-proof -
-  (* Line through x perpendicular to f x *)
-  define l where "l = (\<lambda>s::real. x + (s/norm(f x)) *\<^sub>R rot (f x))"
-  have "norm (f x) > 0" using assms(2) using zero_less_norm_iff by blast 
-  then have distl: "\<forall>s. dist (l s) x = abs s" unfolding l_def dist_add_vec
-    by (auto simp add: norm_rot)
-  obtain d where d:"d > 0" "cball x d \<subseteq> X"
-    by (meson UNIV_I assms(1) local.local_unique_solution)
-  then have lb: "l`{-d..d} \<subseteq> cball x d" using distl by (simp add: abs_le_iff dist_commute image_subset_iff)
-  from fcontx[OF assms(1)] have "continuous (at x) f" .
-  then have c:"continuous (at 0) ((\<lambda>y. (f y \<bullet> f x)) \<circ> l)" unfolding l_def
-    by (auto intro!:continuous_intros simp add: assms(2))
-  have "((\<lambda>y. f y \<bullet> f x) \<circ> l) 0 > 0" using assms(2) unfolding l_def o_def by auto
-  from continuous_at_imp_cball[OF c this]
-  obtain r where r:"r > 0" " \<forall>z\<in>cball 0 r. 0 < ((\<lambda>y. f y \<bullet> f x) \<circ> l) z" by blast
-  then have rc:"\<forall>z \<in> l`{-r..r}. 0 < f z \<bullet> f x" using real_norm_def by auto 
-  define dr where "dr = min r d"
-  have t1:"l (-dr) \<noteq> l dr" unfolding l_def dr_def
-    by (smt \<open>0 < d\<close> \<open>0 < norm (f x)\<close> \<open>0 < r\<close> add_left_imp_eq divide_cancel_right norm_rot norm_zero scale_cancel_right)
-  have "x = midpoint (l (-dr)) (l dr)" unfolding midpoint_def l_def by auto
-  then have xin:"x \<in> {l (-dr)<--<(l dr)}" using t1 by auto
-      (* TODO: actually this should be equality, but l is affine ...
-     also the existing stuff about -- is a little too specific *)
-  have lsub:"{l (-dr)--l dr} \<subseteq> l`{-dr..dr}"
-  proof safe
-    fix z
-    assume "z \<in> {l (- dr)--l dr}"
-    then obtain u where u: "0 \<le> u" "u \<le> 1" "z = (1 - u) *\<^sub>R (l (-dr)) + u *\<^sub>R (l dr)"
-      unfolding in_segment by blast
-    then have "z = x - (1-u) *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x) + u *\<^sub>R  (dr/norm(f x)) *\<^sub>R rot (f x) "
-      unfolding l_def
-      by (simp add: l_def scaleR_add_right scale_right_diff_distrib u(3))
-    also have "... = x - (1 - 2 * u) *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x)"
-      by (auto simp add: algebra_simps divide_simps simp flip: scaleR_add_left)
-    also have "... =  x + (((2 * u - 1) * dr)/norm(f x)) *\<^sub>R rot (f x)"
-      by (smt add_uminus_conv_diff scaleR_scaleR scale_minus_left times_divide_eq_right)
-    finally have zeq: "z = l ((2*u-1)*dr)" unfolding l_def .
-    have ub: " 2* u - 1 \<le> 1 \<and> -1 \<le>  2* u - 1 " using u by linarith
-    thus "z \<in> l ` {- dr..dr}" using zeq
-      by (smt atLeastAtMost_iff d(1) dr_def image_eqI mult.commute mult_left_le mult_minus_left r(1)) 
-  qed
-  have t2: "{l (- dr)--l dr} \<subseteq> X" using lsub
-    by (smt atLeastAtMost_iff d(2) dist_commute distl dr_def image_subset_iff mem_cball order_trans)
-  have "l (- dr) - l dr = -2 *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x)" unfolding l_def
-    by (simp add: algebra_simps flip: scaleR_add_left)
-  then have req: "rot (l (- dr) - l dr) = (2 * dr/norm(f x)) *\<^sub>R f x"
-    by auto (metis add.inverse_inverse rot_rot rot_scaleR)
-  have "l`{-dr..dr} \<subseteq> l ` {-r ..r}"
-    by (simp add: dr_def image_mono)
-  then have "{l (- dr)--l dr} \<subseteq> l ` {-r .. r}" using lsub by auto
-  then have "\<forall>z \<in> {l (- dr)--l dr}. 0 < f z \<bullet> f x" using rc by blast
-  moreover have "(dr / norm (f x)) > 0"
-    using \<open>0 < norm (f x)\<close> d(1) dr_def r(1) by auto 
-  ultimately have t3: "\<forall>z \<in> {l (- dr)--l dr}. f z \<bullet> rot (l (- dr)- l dr) > 0" unfolding req
-    by (smt divide_divide_eq_right inner_scaleR_right mult_2 norm_not_less_zero scaleR_2 times_divide_eq_left times_divide_eq_right zero_less_divide_iff)
-  have "transversal_segment (l (-dr)) (l dr)" using t1 t2 t3 unfolding transversal_segment_def by auto
-  thus ?thesis using xin
-    using that by auto 
-qed
-
-text \<open>Perko Section 3.7 Lemma 2 part 1.\<close> 
-lemma flow_transversal_segment_finite_intersections:
-  assumes "transversal_segment a b"
-  assumes "t \<le> t'" "{t .. t'} \<subseteq> existence_ivl0 x"
-  shows "finite {s\<in>{t..t'}. flow0 x s \<in> {a--b}}"
-proof -
-  from assms have "a \<noteq> b" by (simp add: transversal_segment_def)
-  show ?thesis
-    unfolding closed_segment_surface[OF \<open>a \<noteq> b\<close>]
-    apply (rule flow_transversal_surface_finite_intersections[where Ds="\<lambda>_. blinfun_inner_left (rot (b - a))"])
-    by
-      (use assms in \<open>auto intro!: closed_Collect_conj closed_halfspace_component_ge closed_halfspace_component_le
-        derivative_eq_intros
-        simp: transversal_segment_def nrm_reverse[where x=a] in_closed_segment_iff_rot\<close>)
-qed
-
-lemma transversal_bound_posE:
-  assumes transversal: "transversal_segment a b"
-  assumes direction: "z \<in> {a -- b}" "f z \<bullet> (rot (a - b)) > 0"
-  obtains d B where "d > 0" "0 < B"
-    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (a - b)) \<ge> B"
-proof -
-  let ?a = "(\<lambda>y. (f y) \<bullet> (rot (a - b)))"
-  from transversal_segment_posD[OF transversal direction]
-  have seg: "a \<noteq> b" "{a--b} \<subseteq> X" "z \<in> {a--b} \<Longrightarrow> 0 < f z \<bullet> rot (a - b)" for z
-    by auto
-  {
-    fix x
-    assume "x \<in> {a--b}"
-    then have "x \<in> X" "f x \<noteq> 0" "a \<noteq> b" using transversal by (auto simp: transversal_segment_def)
-    then have "?a \<midarrow>x\<rightarrow> ?a x"
-      by (auto intro!: tendsto_eq_intros)
-    moreover have "?a x > 0"
-      using seg \<open>x \<in> {a -- b}\<close> \<open>f x \<noteq> 0\<close>
-      by (auto simp: simp del: divide_const_simps
-          intro!: divide_pos_pos mult_pos_pos)
-    ultimately have "\<forall>\<^sub>F x in at x. ?a x > 0"
-      by (rule order_tendstoD)
-    moreover have "\<forall>\<^sub>F x in at x. x \<in> X"
-      by (rule topological_tendstoD[OF tendsto_ident_at open_dom \<open>x \<in> X\<close>])
-    moreover have "\<forall>\<^sub>F x in at x. f x \<noteq> 0"
-      by (rule tendsto_imp_eventually_ne tendsto_intros \<open>x \<in> X\<close> \<open>f x \<noteq> 0\<close>)+
-    ultimately have "\<forall>\<^sub>F x in at x. ?a x>0 \<and> x \<in> X \<and> f x \<noteq> 0" by eventually_elim auto
-    then obtain d where d: "0 < d" "\<And>y. y \<in> cball x d \<Longrightarrow> ?a y > 0 \<and> y \<in> X \<and> f y \<noteq> 0"
-      using \<open>?a x > 0\<close> \<open>x \<in> X\<close>
-      by (force simp: eventually_at_le dist_commute)
-
-    have "continuous_on (cball x d) ?a"
-      using d \<open>a \<noteq> b\<close>
-      by (auto intro!: continuous_intros)
-    from compact_continuous_image[OF this compact_cball]
-    have "compact (?a ` cball x d)" .
-    from compact_attains_inf[OF this] obtain s where "s \<in> cball x d" "\<forall>x\<in>cball x d. ?a x \<ge> ?a s"
-      using \<open>d > 0\<close>
-      by auto
-    then have "\<exists>d>0. \<exists>b>0. \<forall>x \<in> cball x d. ?a x \<ge> b"
-      using d
-      by (force simp: intro: exI[where x="?a s"])
-  } then obtain dx Bx where dB:
-    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> y\<in>cball x (dx x) \<Longrightarrow> ?a y \<ge> Bx x"
-    "\<And>x. x \<in> {a -- b} \<Longrightarrow> Bx x > 0"
-    "\<And>x. x \<in> {a -- b} \<Longrightarrow> dx x > 0"
-    by metis
-  define d' where "d' = (\<lambda>x. dx x / 2)"
-  have d':
-    "\<And>x. x \<in> {a -- b} \<Longrightarrow> \<forall>y\<in>cball x (d' x). ?a y \<ge> Bx x"
-    "\<And>x. x \<in> {a -- b} \<Longrightarrow> d' x > 0"
-    using dB(1,3) by (force simp: d'_def)+
-  have d'B: "\<And>x. x \<in> {a -- b} \<Longrightarrow> \<forall>y\<in>cball x (d' x). ?a y \<ge> Bx x"
-    using d' by auto
-  have "{a--b} \<subseteq> \<Union>((\<lambda>x. ball x (d' x)) ` {a -- b})"
-    using d'(2) by auto
-  from compactE_image[OF compact_segment _ this]
-  obtain X where X: "X \<subseteq> {a--b}"
-    and [simp]: "finite X"
-    and cover: "{a--b} \<subseteq> (\<Union>x\<in>X. ball x (d' x))"
-    by auto
-  have [simp]: "X \<noteq> {}" using X cover by auto
-  define d where "d = Min (d' ` X)"
-  define B where "B = Min (Bx ` X)"
-  have "d > 0"
-    using X d'
-    by (auto simp: d_def d'_def)
-  moreover have "B > 0"
-    using X dB
-    by (auto simp: B_def simp del: divide_const_simps)
-  moreover have "B \<le> ?a y" if "x \<in> {a -- b}" "dist x y \<le> d" for x y
-  proof -
-    from \<open>x \<in> {a -- b}\<close> obtain xc where xc: "xc \<in> X" "x \<in> ball xc (d' xc)"
-      using cover by auto
-    have "?a y \<ge> Bx xc"
-    proof (rule dB)
-      show "xc \<in> {a -- b}" using xc \<open>X \<subseteq> _\<close> by auto
-      have "dist xc y \<le> dist xc x + dist x y" by norm
-      also have "dist xc x \<le> d' xc" using xc by auto
-      also note \<open>dist x y \<le> d\<close>
-      also have "d \<le> d' xc"
-        using xc
-        by (auto simp: d_def)
-      also have "d' xc + d' xc = dx xc" by (simp add: d'_def)
-      finally show "y \<in> cball xc (dx xc)" by simp
-    qed
-    also have "B \<le> Bx xc"
-      using xc
-      unfolding B_def
-      by (auto simp: B_def)
-    finally (xtrans) show ?thesis .
-  qed
-  ultimately show ?thesis ..
-qed
-
-lemma transversal_bound_negE:
-  assumes transversal: "transversal_segment a b"
-  assumes direction: "z \<in> {a -- b}" "f z \<bullet> (rot (a - b)) < 0"
-  obtains d B where "d > 0" "0 < B"
-    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (b - a)) \<ge> B"
-proof -
-  from direction have "z \<in> {b -- a}" "f z \<bullet> (rot (b - a)) > 0"
-    by (auto simp: closed_segment_commute rot_diff_commute[of b a])
-  from transversal_bound_posE[OF transversal_segment_reverse[OF transversal] this]
-  obtain d B where "d > 0" "0 < B"
-    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (b - a)) \<ge> B"
-    by (auto simp: closed_segment_commute)
-  then show ?thesis ..
-qed
-
-lemma leaves_transversal_segmentE:
-  assumes transversal: "transversal_segment a b"
-  obtains T n where "T > 0" "n = a - b \<or> n = b - a"
-    "\<And>x. x \<in> {a -- b} \<Longrightarrow> {-T..T} \<subseteq> existence_ivl0 x"
-    "\<And>x s. x \<in> {a -- b} \<Longrightarrow> 0 < s \<Longrightarrow> s \<le> T \<Longrightarrow>
-    (flow0 x s - x) \<bullet> rot n > 0"
-    "\<And>x s. x \<in> {a -- b} \<Longrightarrow> -T \<le> s \<Longrightarrow> s < 0 \<Longrightarrow>
-    (flow0 x s - x) \<bullet> rot n < 0"
-proof -
-  from transversal_segmentE[OF assms(1)] obtain n
-    where n: "n = (a - b) \<or> n = (b - a)"
-      and seg: "a \<noteq> b" "{a -- b} \<subseteq> X" "\<And>z. z \<in> {a--b} \<Longrightarrow> f z \<bullet> rot n > 0"
-    by metis
-  from open_existence_ivl_on_compact[OF \<open>{a -- b} \<subseteq> X\<close>]
-  obtain t where "0 < t" and t: "x \<in> {a--b} \<Longrightarrow> {- t..t} \<subseteq> existence_ivl0 x" for x
-    by auto
-  from n obtain d B where B: "0 < d" "0 < B" "(\<And>x y. x \<in> {a--b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> B \<le> f y \<bullet> rot n)"
-  proof
-    assume n_def: "n = a - b"
-    with seg have pos:  "0 < f a \<bullet> rot (a - b)"
-      by auto
-    from transversal_bound_posE[OF transversal ends_in_segment(1) pos, folded n_def]
-    show ?thesis using that by blast
-  next
-    assume n_def: "n = b - a"
-    with seg have pos:  "0 > f a \<bullet> rot (a - b)"
-      by (auto simp: rot_diff_commute[of a b])
-    from transversal_bound_negE[OF transversal ends_in_segment(1) this, folded n_def]
-    show ?thesis using that by blast
-  qed
-  define S where "S = \<Union>((\<lambda>x. ball x d) ` {a -- b})"
-  have S: "x \<in> S \<Longrightarrow> B \<le> f x \<bullet> rot n" for x
-    by (auto simp: S_def intro!: B)
-  have "open S" by (auto simp: S_def)
-  have "{a -- b} \<subseteq> S"
-    by (auto simp: S_def \<open>0 < d\<close>)
-  have "\<forall>\<^sub>F (t, x) in at (0, x). flow0 x t \<in> S" if "x \<in> {a -- b}" for x
-    unfolding split_beta'
-    apply (rule topological_tendstoD tendsto_intros)+
-    using set_mp[OF \<open>{a -- b} \<subseteq> X\<close> that] \<open>0 < d\<close> that \<open>open S\<close> \<open>{a -- b} \<subseteq> S\<close>
-    by (force simp: )+
-  then obtain d' where d':
-    "\<And>x. x \<in> {a--b} \<Longrightarrow> d' x > 0"
-    "\<And>x y s. x \<in> {a--b} \<Longrightarrow> (s = 0 \<longrightarrow> y \<noteq> x) \<Longrightarrow> dist (s, y) (0, x) < d' x \<Longrightarrow> flow0 y s \<in> S"
-    by (auto simp: eventually_at) metis
-  define d2 where "d2 x = d' x / 2" for x
-  have d2: "\<And>x. x \<in> {a--b} \<Longrightarrow> d2 x > 0" using d' by (auto simp: d2_def)
-  have C: "{a--b} \<subseteq> \<Union>((\<lambda>x. ball x (d2 x)) ` {a -- b})"
-    using d2 by auto
-  from compactE_image[OF compact_segment _ C]
-  obtain C' where "C' \<subseteq> {a--b}" and [simp]: "finite C'"
-    and C'_cover: "{a--b} \<subseteq> (\<Union>c\<in>C'. ball c (d2 c))" by auto
-
-  define T where "T = Min (insert t (d2 ` C'))"
-
-  have "T > 0"
-    using \<open>0 < t\<close> d2 \<open>C' \<subseteq> _\<close> 
-    by (auto simp: T_def)
-  moreover
-  note n
-  moreover
-  have T_ex: "{-T..T} \<subseteq> existence_ivl0 x" if "x \<in> {a--b}" for x
-    by (rule order_trans[OF _ t[OF that]]) (auto simp: T_def)
-  moreover
-  have B_le: "B \<le> f (flow0 x \<xi>) \<bullet> rot n"
-    if "x \<in> {a -- b}"
-      and c': "c' \<in> C'" "x \<in> ball c' (d2 c')"
-      and "\<xi> \<noteq> 0" and \<xi>_le: "\<bar>\<xi>\<bar> < d2 c'"
-    for x c' \<xi>
-  proof -
-    have "c' \<in> {a -- b}" using c' \<open>C' \<subseteq> _\<close> by auto
-    moreover have "\<xi> = 0 \<longrightarrow> x \<noteq> c'" using \<open>\<xi> \<noteq> 0\<close> by simp
-    moreover have "dist (\<xi>, x) (0, c') < d' c'"
-    proof -
-      have "dist (\<xi>, x) (0, c') \<le> dist (\<xi>, x) (\<xi>, c') + dist (\<xi>, c') (0, c')"
-        by norm
-      also have "dist (\<xi>, x) (\<xi>, c') < d2 c'"
-        using c'
-        by (simp add: dist_prod_def dist_commute)
-      also
-      have "T \<le> d2 c'" using c'
-        by (auto simp: T_def)
-      then have "dist (\<xi>, c') (0, c') < d2 c'"
-        using \<xi>_le
-        by (simp add: dist_prod_def)
-      also have "d2 c' + d2 c' = d' c'" by (simp add: d2_def)
-      finally show ?thesis by simp
-    qed
-    ultimately have "flow0 x \<xi> \<in> S"
-      by (rule d')
-    then show ?thesis
-      by (rule S)
-  qed
-  let ?g = "(\<lambda>x t. (flow0 x t - x) \<bullet> rot n)"
-  have cont: "continuous_on {-T .. T} (?g x)"
-    if "x \<in> {a--b}" for x
-    using T_ex that
-    by (force intro!: continuous_intros)
-  have deriv: "-T \<le> s' \<Longrightarrow> s' \<le> T \<Longrightarrow> ((?g x) has_derivative
-      (\<lambda>t. t * (f (flow0 x s') \<bullet> rot n))) (at s')"
-    if "x \<in> {a--b}" for x s'
-    using T_ex that
-    by (force intro!: derivative_eq_intros simp: flowderiv_def blinfun.bilinear_simps)
-
-  have "(flow0 x s - x) \<bullet> rot n > 0" if "x \<in> {a -- b}" "0 < s" "s \<le> T" for x s
-  proof (rule ccontr, unfold not_less)
-    have [simp]: "x \<in> X" using that \<open>{a -- b} \<subseteq> X\<close> by auto
-    assume H: "(flow0 x s - x) \<bullet> rot n \<le> 0"
-    have cont: "continuous_on {0 .. s} (?g x)"
-      using cont by (rule continuous_on_subset) (use that in auto)
-    from mvt[OF \<open>0 < s\<close> cont deriv] that
-    obtain \<xi> where \<xi>: "0 < \<xi>" "\<xi> < s" "(flow0 x s - x) \<bullet> rot n = s * (f (flow0 x \<xi>) \<bullet> rot n)"
-      by (auto intro: continuous_on_subset)
-    note \<open>0 < B\<close>
-    also
-    from C'_cover that obtain c' where c': "c' \<in> C'" "x \<in> ball c' (d2 c')" by auto
-    have "B \<le> f (flow0 x \<xi>) \<bullet> rot n"
-    proof (rule B_le[OF that(1) c'])
-      show "\<xi> \<noteq> 0" using \<open>0 < \<xi>\<close> by simp
-      have "T \<le> d2 c'" using c'
-        by (auto simp: T_def)
-      then show "\<bar>\<xi>\<bar> < d2 c'"
-        using \<open>0 < \<xi>\<close> \<open>\<xi> < s\<close> \<open>s \<le> T\<close>
-        by (simp add: dist_prod_def)
-    qed
-    also from \<xi> H have "\<dots> \<le> 0"
-      by (auto simp add: algebra_split_simps not_less split: if_splits)
-    finally show False by simp
-  qed
-  moreover
-  have "(flow0 x s - x) \<bullet> rot n < 0" if "x \<in> {a -- b}" "-T \<le> s" "s < 0" for x s
-  proof (rule ccontr, unfold not_less)
-    have [simp]: "x \<in> X" using that \<open>{a -- b} \<subseteq> X\<close> by auto
-    assume H: "(flow0 x s - x) \<bullet> rot n \<ge> 0"
-    have cont: "continuous_on {s .. 0} (?g x)"
-      using cont by (rule continuous_on_subset) (use that in auto)
-    from mvt[OF \<open>s < 0\<close> cont deriv] that
-    obtain \<xi> where \<xi>: "s < \<xi>" "\<xi> < 0" "(flow0 x s - x) \<bullet> rot n = s * (f (flow0 x \<xi>) \<bullet> rot n)"
-      by auto
-    note \<open>0 < B\<close>
-    also
-    from C'_cover that obtain c' where c': "c' \<in> C'" "x \<in> ball c' (d2 c')" by auto
-    have "B \<le> (f (flow0 x \<xi>) \<bullet> rot n)"
-    proof (rule B_le[OF that(1) c'])
-      show "\<xi> \<noteq> 0" using \<open>0 > \<xi>\<close> by simp
-      have "T \<le> d2 c'" using c'
-        by (auto simp: T_def)
-      then show "\<bar>\<xi>\<bar> < d2 c'"
-        using \<open>0 > \<xi>\<close> \<open>\<xi> > s\<close> \<open>s \<ge> - T\<close>
-        by (simp add: dist_prod_def)
-    qed
-    also from \<xi> H have "\<dots> \<le> 0"
-      by (simp add: algebra_split_simps)
-    finally show False by simp
-  qed
-  ultimately show ?thesis ..
-qed
-
-
-lemma inner_rot_pos_move_base: "(x - a) \<bullet> rot (a - b) > 0"
-  if "(x - y) \<bullet> rot (a - b) > 0" "y \<in> {a -- b}"
-  by (smt in_segment_inner_rot inner_diff_left inner_minus_right minus_diff_eq rot_rot that)
-
-lemma inner_rot_neg_move_base: "(x - a) \<bullet> rot (a - b) < 0"
-  if "(x - y) \<bullet> rot (a - b) < 0" "y \<in> {a -- b}"
-  by (smt in_segment_inner_rot inner_diff_left inner_minus_right minus_diff_eq rot_rot that)
-
-lemma inner_pos_move_base: "(x - a) \<bullet> n > 0"
-  if "(a - b) \<bullet> n = 0" "(x - y) \<bullet> n > 0" "y \<in> {a -- b}"
-proof -
-  from that(3) obtain u where y_def: "y = (1 - u) *\<^sub>R a + u *\<^sub>R b" and u: "0 \<le> u" "u \<le> 1"
-    by (auto simp: in_segment)
-  have "(x - a) \<bullet> n = (x - y) \<bullet> n - u * ((a - b) \<bullet> n)"
-    by (simp add: algebra_simps y_def)
-  also have "\<dots> = (x - y) \<bullet> n"
-    by (simp add: that)
-  also note \<open>\<dots> > 0\<close>
-  finally show ?thesis .
-qed
-
-lemma inner_neg_move_base: "(x - a) \<bullet> n < 0"
-  if "(a - b) \<bullet> n = 0" "(x - y) \<bullet> n < 0" "y \<in> {a -- b}"
-proof -
-  from that(3) obtain u where y_def: "y = (1 - u) *\<^sub>R a + u *\<^sub>R b" and u: "0 \<le> u" "u \<le> 1"
-    by (auto simp: in_segment)
-  have "(x - a) \<bullet> n = (x - y) \<bullet> n - u * ((a - b) \<bullet> n)"
-    by (simp add: algebra_simps y_def)
-  also have "\<dots> = (x - y) \<bullet> n"
-    by (simp add: that)
-  also note \<open>\<dots> < 0\<close>
-  finally show ?thesis .
-qed
-
-lemma rot_same_dir:
-  assumes "x1 \<in> {a<--<b}"
-  assumes "x2 \<in> {x1<--<b}"
-  shows "(y \<bullet> rot (a-b) > 0) = (y \<bullet> rot(x1-x2) > 0)"  "(y \<bullet> rot (a-b) < 0) = (y \<bullet> rot(x1-x2) < 0)"
-  using oriented_subsegment_scale[OF assms]
-   apply (smt inner_scaleR_right nrm_reverse rot_scaleR zero_less_mult_iff)
-  by (smt \<open>\<And>thesis. (\<And>e. \<lbrakk>0 < e; b - a = e *\<^sub>R (x2 - x1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> inner_minus_right inner_scaleR_right rot_diff_commute rot_scaleR zero_less_mult_iff)
-
-
-subsection \<open>Monotone Step Lemma\<close>
-
-lemma flow0_transversal_segment_monotone_step:
-  assumes "transversal_segment a b"
-  assumes "t1 \<le> t2" "{t1..t2} \<subseteq> existence_ivl0 x"
-  assumes x1: "flow0 x t1 \<in> {a<--<b}"
-  assumes x2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
-  assumes "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
-  assumes "t > t2" "t \<in> existence_ivl0 x"
-  shows "flow0 x t \<notin> {a<--<flow0 x t2}"
-proof -
-  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
-  note t1t2 = \<open>\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}\<close>
-    (* Basic properties of the segment *)
-  have x1neqx2: "flow0 x t1 \<noteq> flow0 x t2"
-    using open_segment_def x2 by force 
-  then have t1neqt2: "t1 \<noteq> t2" by auto
-
-  have [simp]: "a \<noteq> b" and \<open>{a -- b} \<subseteq> X\<close> using \<open>transversal_segment a b\<close>
-    by (auto simp: transversal_segment_def)
-
-  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
-    by (auto simp: in_open_segment_iff_line)
-  from x2 obtain i2 where i2: "flow0 x t2 = line a b i2" "0 < i1" "i1 < i2"
-    by (auto simp: i1 line_open_segment_iff)
-
-
-  have "{a <--< flow0 x t1} \<subseteq> {a<--<b}"
-    by (simp add: open_closed_segment subset_open_segment x1) 
-  have t12sub: "{flow0 x t1--flow0 x t2} \<subseteq> {a<--<b}"
-    by (metis ends_in_segment(2) open_closed_segment subset_co_segment subset_eq subset_open_segment x1 x2)
-  have subr: "{flow0 x t1<--<flow0 x t2} \<subseteq> {flow0 x t1 <--<b}"
-    by (simp add: open_closed_segment subset_open_segment x2)
-  have "flow0 x t1 \<in> {a <--<flow0 x t2}" using x1 x2
-    by (rule open_segment_subsegment)
-  then have subl: "{flow0 x t1<--<flow0 x t2} \<subseteq> {a <--< flow0 x t2}" using x1 x2
-    by (simp add: open_closed_segment subset_open_segment x2)
-  then have subl2: "{flow0 x t1--<flow0 x t2} \<subseteq> {a <--< flow0 x t2}" using x1 x2
-    by (smt DiffE DiffI \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> half_open_segment_def insert_iff open_segment_def subset_eq)
-
-  have sub1b: "{flow0 x t1--b} \<subseteq> {a--b}"
-    by (simp add: open_closed_segment subset_closed_segment x1)
-  have suba2: "{a--flow0 x t2} \<subseteq> {a -- b}"
-    using open_closed_segment subset_closed_segment t12sub by blast
-  then have suba2o: "{a<--<flow0 x t2} \<subseteq> {a -- b}"
-    using open_closed_segment subset_closed_segment t12sub by blast
-  have x2_notmem: "flow0 x t2 \<notin> {a--flow0 x t1}"
-    using i1 i2
-    by (auto simp: closed_segment_line_iff)
-  have suba12: "{a--flow0 x t1} \<subseteq> {a--flow0 x t2}"
-    by (simp add: \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> open_closed_segment subset_closed_segment)
-  then have suba12_open: "{a<--<flow0 x t1} \<subseteq> {a<--<flow0 x t2}"
-    using x2_notmem
-    by (auto simp: open_segment_def)
-  have "flow0 x t2 \<in> {a--b}"
-    using suba2 by auto
-
-  have intereq: "\<And>t. t1 \<le> t \<Longrightarrow> t \<le> t2 \<Longrightarrow> flow0 x t \<in> {a<--<b} \<Longrightarrow>  t = t1 \<or> t = t2"
-  proof (rule ccontr)
-    fix t
-    assume t: "t1 \<le> t" "t \<le> t2" "flow0 x t \<in> {a<--<b}" "\<not>(t= t1 \<or> t = t2)"
-    then have "t \<in> {t1<..<t2}" by auto
-    then have "flow0 x t \<notin> {a<--<b}" using t1t2 by blast
-    thus False using t by auto
-  qed
-  then have intereqt12: "\<And>t. t1 \<le> t \<Longrightarrow> t \<le> t2 \<Longrightarrow> flow0 x t \<in> {flow0 x t1--flow0 x t2} \<Longrightarrow>  t = t1 \<or> t = t2"
-    using t12sub by blast
-
-(* The Jordan curve *)
-  define J1 where "J1 = flow_to_path x t1 t2"
-  define J2 where "J2 = linepath (flow0 x t2) (flow0 x t1)" 
-  define J where "J = J1 +++ J2"
-    (* Proof that J is a Jordan curve *)
-  have "pathfinish J = pathstart J" unfolding J_def J1_def J2_def
-    by (auto simp add: pathstart_compose pathfinish_compose)
-  have piJ: "path_image J = path_image J1 \<union> path_image J2"
-    unfolding J_def J1_def J2_def
-    apply (rule path_image_join)
-    by auto
-  have "flow0 x t1 \<in> flow0 x ` {t1..t2} \<and> flow0 x t2 \<in> flow0 x ` {t1..t2}"
-    using atLeastAtMost_iff \<open>t1 \<le> t2\<close> by blast 
-  then have piD: "path_image J = path_image J1 \<union> {flow0 x t1 <--<flow0 x t2}"
-    unfolding piJ J1_def J2_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
-      path_image_linepath open_segment_def
-    by (smt Diff_idemp Diff_insert2 Un_Diff_cancel closed_segment_commute mk_disjoint_insert)
-  have "\<forall>s\<in>{t1<..<t2}. flow0 x s \<noteq> flow0 x t1"
-    using x1 t1t2 by fastforce
-  from flow_to_path_arc[OF \<open>t1 \<le> t2\<close> exist this x1neqx2]
-  have "arc J1" using J1_def assms flow_to_path_arc by auto
-  then have "simple_path J" unfolding J_def
-    using \<open>arc J1\<close> J1_def J2_def assms x1neqx2 t1neqt2 apply (auto intro!:simple_path_join_loop)
-    using intereqt12 closed_segment_commute by blast
-
-  from Jordan_inside_outside_R2[OF this \<open>pathfinish J = pathstart J\<close>]
-  obtain inner outer where inner_def: "inner = inside (path_image J)"
-    and outer_def: "outer = outside (path_image J)"
-    and io:
-    "inner \<noteq> {}" "open inner" "connected inner"
-    "outer \<noteq> {}" "open outer" "connected outer"
-    "bounded inner" "\<not> bounded outer" "inner \<inter> outer = {}"
-    "inner \<union> outer = - path_image J"
-    "frontier inner = path_image J"
-    "frontier outer = path_image J" by metis
-  from io have io2: "outer \<inter> inner = {}" "outer \<union> inner = - path_image J" by auto
-
-  have swap_side: "\<And>y t. y \<in> side2 \<Longrightarrow>
-    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
-    flow0 y t \<in> closure side1 \<Longrightarrow>
-    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> side2) \<and>
-        flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
-    if "side1 \<inter> side2 = {}"
-      "open side2"
-      "frontier side1 = path_image J"
-      "frontier side2 = path_image J"
-      "side1 \<union> side2 = - path_image J"
-    for side1 side2
-  proof -
-    fix y t
-    assume yt: "y \<in> side2" "0 \<le> t" "t \<in> existence_ivl0 y"
-      "flow0 y t \<in> closure side1"
-    define fp where "fp = flow_to_path y 0 t"
-    have ex:"{0..t} \<subseteq> existence_ivl0 y"
-      using ivl_subset_existence_ivl yt(3) by blast
-    then have y0:"flow0 y 0 = y"
-      using mem_existence_ivl_iv_defined(2) yt(3) by auto
-    then have tpos: "t > 0" using yt(2)  \<open>side1 \<inter> side2 = {}\<close>
-      using yt(1) yt(4)
-      by (metis closure_iff_nhds_not_empty less_eq_real_def order_refl that(2)) 
-    from flow_to_path_path[OF yt(2) ex]
-    have a1: "path fp" unfolding fp_def .
-    have "y \<in> closure side2" using yt(1)
-      by (simp add: assms closure_def)
-    then have a2: "pathstart fp \<in> closure side2" unfolding fp_def using y0 by auto
-    have a3:"pathfinish fp \<notin> side2" using yt(4) \<open>side1 \<inter> side2 = {}\<close>
-      unfolding fp_def apply auto
-      using closure_iff_nhds_not_empty that(2) by blast
-    from subpath_to_frontier_strong[OF a1 a3]
-    obtain u where u:"0 \<le> u" "u \<le> 1"
-      "fp u \<notin> interior side2"
-      "u = 0 \<or>
-      (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow>
-        subpath 0 u fp x \<in> interior side2) \<and> fp u \<in> closure side2" by blast
-    have p1:"path_image (subpath 0 u fp) =  flow0 y ` {0 ..  u*t}"
-      unfolding fp_def subpath0_flow_to_path using path_image_flow_to_path
-      by (simp add: u(1) yt(2))
-    have p2:"fp u = flow0 y (u*t)" unfolding fp_def flow_to_path_unfold by simp
-    have inout:"interior side2 = side2" using \<open>open side2\<close>
-      by (simp add: interior_eq)
-    then have iemp: "side2 \<inter> path_image J = {}"
-      using \<open>frontier side2 = path_image J\<close>
-      by (metis frontier_disjoint_eq inf_sup_aci(1) interior_eq)
-    have "u \<noteq> 0" using inout u(3) y0 p2 yt(1) by force
-    then have c1:"u * t > 0" using tpos u y0  \<open>side1 \<inter> side2 = {}\<close>
-      using frontier_disjoint_eq io(5) yt(1) zero_less_mult_iff by fastforce
-    have uim:"fp u \<in> path_image J" using u \<open>u \<noteq> 0\<close>
-      using \<open>frontier side2 = path_image J\<close>
-      by (metis ComplI IntI closure_subset frontier_closures inout subsetD) 
-    have c2:"u * t \<le> t"  using u(1-2) tpos by auto
-    have"(flow_to_path y 0 (u * t) ` {0..<1} \<subseteq> side2)"
-      using \<open>u \<noteq> 0\<close> u inout unfolding fp_def subpath0_flow_to_path by auto
-    then have c3:"\<forall>s \<in>{0..<u*t}. flow0 y s \<in> side2" by auto
-    have c4: "flow0 y (u*t) \<in> path_image J"
-      using uim path_image_join_subset
-      by (simp add: p2)
-    have "flow0 y (u*t) \<notin> path_image J1 \<or> flow0 y (u*t) = flow0 x t1"
-    proof clarsimp
-      assume "flow0 y (u*t) \<in> path_image J1"
-      then obtain s where s: "t1 \<le> s" "s \<le> t2" "flow0 x s = flow0 y (u*t)"
-        using J1_def \<open>t1 \<le> t2\<close> by auto
-      have "s = t1"
-      proof (rule ccontr)
-        assume "s \<noteq> t1"
-        then have st1:"s > t1" using s(1) by linarith
-        define sc where "sc = min (s-t1) (u*t)"
-        have scd: "s-sc \<in> {t1..t2}" unfolding sc_def
-          using c1 s(1) s(2) by auto
-        then have *:"flow0 x (s-sc) \<in> path_image J1" unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
-          by blast
-        have "flow0 x (s-sc) = flow0 (flow0 x s) (-sc)"
-          by (smt exist atLeastAtMost_iff existence_ivl_trans' flow_trans s(1) s(2) scd subsetD)
-        then have **:"flow0 (flow0 y (u*t)) (-sc)  \<in> path_image J1"
-          using s(3) * by auto
-        have b:"u*t - sc \<in> {0..<u*t}" unfolding sc_def by (simp add: st1 c1 s(1))
-        then have "u*t - sc \<in> existence_ivl0 y"
-          using c2 ex by auto 
-        then have "flow0 y (u*t - sc) \<in> path_image J1" using **
-          by (smt atLeastAtMost_iff diff_existence_ivl_trans ex flow_trans mult_left_le_one_le mult_nonneg_nonneg subset_eq u(1) u(2) yt(2))
-        thus False using b c3 iemp piJ by blast
-      qed
-      thus "flow0 y (u * t) = flow0 x t1" using s by simp
-    qed
-    thus "\<exists>T>0. T \<le> t \<and> (\<forall>s\<in>{0..<T}. flow0 y s \<in> side2) \<and>
-          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
-      using c1 c2 c3 c4 unfolding piD
-      by (metis DiffE UnE ends_in_segment(1) half_open_segment_closed_segmentI insertCI open_segment_def x1neqx2)
-  qed
-  have outside_in: "\<And>y t. y \<in> outer \<Longrightarrow>
-    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
-    flow0 y t \<in> closure inner \<Longrightarrow>
-    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> outer) \<and>
-          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
-    by (rule swap_side; (rule io | assumption))
-  have inside_out: "\<And>y t. y \<in> inner \<Longrightarrow>
-    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
-    flow0 y t \<in> closure outer \<Longrightarrow>
-    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> inner) \<and>
-          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
-    by (rule swap_side; (rule io2 io | assumption))
-
-  from leaves_transversal_segmentE[OF assms(1)]
-  obtain d n where d: "d > (0::real)"
-    and n: "n = a - b \<or> n = b - a"
-    and d_ex: "\<And>x. x \<in> {a -- b} \<Longrightarrow> {-d..d} \<subseteq> existence_ivl0 x"
-    and d_above: "\<And>x s. x \<in> {a -- b} \<Longrightarrow> 0 < s \<Longrightarrow> s \<le> d \<Longrightarrow> (flow0 x s - x) \<bullet> rot n > 0"
-    and d_below: "\<And>x s. x \<in> {a -- b} \<Longrightarrow> -d \<le> s \<Longrightarrow> s < 0 \<Longrightarrow> (flow0 x s - x) \<bullet> rot n < 0"
-    by blast
-
-  have ortho: "(a - b) \<bullet> rot n = 0"
-    using n by (auto simp: algebra_simps)
-
-(* These "rectangles" are either fully inside or fully outside
-           |-----------------------|
-           |           r1          | (flow d)
-    a --- (t1) --- rp --- (t2) --- b
-    |          r2           | (flow -d)
-    |-----------------------|
-   *)
-  define r1 where "r1 = (\<lambda>(x, y). flow0 x y)`({flow0 x t1<--<b} \<times> {0<..<d}) "
-  have r1a1: "path_connected {flow0 x t1 <--<b}" by simp
-  have r1a2: "path_connected {0<..<d}" by simp
-  have "{flow0 x t1<--<b} \<subseteq> {a--b}"
-    by (simp add: open_closed_segment subset_oc_segment x1)
-  then have r1a3: "y \<in> {flow0 x t1<--<b} \<Longrightarrow> {0<..<d} \<subseteq> existence_ivl0 y" for y
-    using d_ex[of y]
-    by force
-  from flow0_path_connected[OF r1a1 r1a2 r1a3]
-  have pcr1:"path_connected r1" unfolding r1_def by auto
-  have pir1J1: "r1 \<inter> path_image J1 = {}"
-    unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
-  proof (rule ccontr)
-    assume "r1 \<inter> flow0 x ` {t1..t2} \<noteq> {}"
-    then obtain xx tt ss where
-      eq: "flow0 xx tt = flow0 x ss"
-      and xx: "xx \<in> {flow0 x t1<--<b}"
-      and ss: "t1 \<le> ss" "ss \<le> t2"
-      and tt: "0 < tt" "tt < d"
-      unfolding r1_def
-      by force
-    have "xx \<in> {a -- b}"
-      using sub1b
-      apply (rule set_mp)
-      using xx by (simp add: open_closed_segment)
-    then have [simp]: "xx \<in> X" using \<open>transversal_segment a b\<close> by (auto simp: transversal_segment_def)
-    from ss have ss_ex: "ss \<in> existence_ivl0 x" using exist
-      by auto
-    from d_ex[OF \<open>xx \<in> {a -- b}\<close>] tt
-    have tt_ex: "tt \<in> existence_ivl0 xx" by auto
-    then have neg_tt_ex: "- tt \<in> existence_ivl0 (flow0 xx tt)"
-      by (rule existence_ivl_reverse[simplified])
-    from eq have "flow0 (flow0 xx tt) (-tt) = flow0 (flow0 x ss) (-tt)"
-      by simp
-    then have "xx = flow0 x (ss - tt)"
-      apply (subst (asm) flow_trans[symmetric])
-        apply (rule tt_ex)
-       apply (rule neg_tt_ex)
-      apply (subst (asm) flow_trans[symmetric])
-        apply (rule ss_ex)
-       apply (subst eq[symmetric])
-       apply (rule neg_tt_ex)
-      by simp
-    moreover
-    define e where "e = ss - t1"
-    consider "e > tt" | "e \<le> tt" by arith
-    then show False
-    proof cases
-      case 1
-      have "flow0 (flow0 x ss) (-tt) \<notin> {a<--<b}"
-        apply (subst flow_trans[symmetric])
-          apply fact
-        subgoal using neg_tt_ex eq by simp
-        apply (rule t1t2)
-        using 1 ss tt
-        unfolding e_def
-        by auto
-      moreover have "flow0 (flow0 x ss) (-tt) \<in> {a<--<b}"
-        unfolding eq[symmetric] using tt_ex xx
-        apply (subst flow_trans[symmetric])
-          apply (auto simp add: neg_tt_ex)
-        by (metis (no_types, hide_lams) sub1b subset_eq subset_open_segment)
-      ultimately show ?thesis by simp
-    next
-      case 2
-      have les: "0 \<le> tt - e" "tt - e \<le> d"
-        using tt ss 2 e_def
-        by auto
-      have xxtte: "flow0 xx (tt - e) = flow0 x t1"
-        apply (simp add: e_def)
-        by (smt \<open>0 \<le> tt - e\<close> \<open>{- d..d} \<subseteq> existence_ivl0 xx\<close> atLeastAtMost_iff e_def eq
-            local.existence_ivl_reverse local.existence_ivl_trans local.flow_trans ss(1) ss_ex subset_iff tt(2))
-      show False
-      proof (cases "tt = e")
-        case True
-        with xxtte have "xx = flow0 x t1"
-          by (simp add: )
-        with xx show ?thesis
-          apply auto
-          by (auto simp: open_segment_def)
-      next
-        case False
-        with les have "0 < tt - e" by (simp)
-        from d_above[OF \<open>xx \<in> {a -- b}\<close> this \<open>tt - e \<le> d\<close>]
-        have "flow0 xx (tt - e) \<notin> {a -- b}"
-          apply (simp add: in_closed_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
-              not_le )
-          by (smt \<open>xx \<in> {a--b}\<close> inner_minus_right inner_rot_neg_move_base inner_rot_pos_move_base n rot_diff_commute)
-        with xxtte show ?thesis
-          using \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> suba2o by auto
-      qed
-    qed
-  qed
-    (* for sufficiently small d, the flow does not return to the line *)
-  moreover
-  have pir1J2: "r1 \<inter> path_image J2 = {}"
-  proof -
-    have "r1 \<subseteq> {x. (x - a) \<bullet> rot n > 0}"
-      unfolding r1_def
-    proof safe
-      fix aa ba
-      assume "aa \<in> {flow0 x t1<--<b}" "ba \<in> {0<..<d}"
-      with sub1b show "0 < (flow0 aa ba - a) \<bullet> rot n"
-        using segment_open_subset_closed[of "flow0 x t1" b]
-        by (intro inner_pos_move_base[OF ortho d_above]) auto
-    qed
-    also have "\<dots> \<inter> {a -- b} = {}"
-      using in_segment_inner_rot in_segment_inner_rot2 n by auto
-    finally show ?thesis
-      unfolding J2_def path_image_linepath
-      using t12sub open_closed_segment
-      by (force simp: closed_segment_commute)
-  qed
-  ultimately have pir1:"r1 \<inter> (path_image J) = {}" unfolding J_def
-    by (metis disjoint_iff_not_equal not_in_path_image_join)
-
-  define r2 where "r2 =(\<lambda>(x, y). flow0 x y)`({a <--< flow0 x t2} \<times> {-d<..<0})"
-  have r2a1:"path_connected {a <--< flow0 x t2}" by simp
-  have r2a2:"path_connected {-d<..<0}" by simp
-  have "{a <--< flow0 x t2} \<subseteq> {a -- b}"
-    by (meson ends_in_segment(1) open_closed_segment subset_co_segment subset_oc_segment t12sub)
-  then have r2a3: "y \<in> {a <--< flow0 x t2} \<Longrightarrow> {-d<..<0} \<subseteq> existence_ivl0 y" for y
-    using d_ex[of y]
-    by force
-  from flow0_path_connected[OF r2a1 r2a2 r2a3]
-  have pcr2:"path_connected r2" unfolding r2_def by auto
-  have pir2J2: "r2 \<inter> path_image J1 = {}"
-    unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
-  proof (rule ccontr)
-    assume "r2 \<inter> flow0 x ` {t1..t2} \<noteq> {}"
-    then obtain xx tt ss where
-      eq: "flow0 xx tt = flow0 x ss"
-      and xx: "xx \<in> {a<--<flow0 x t2}"
-      and ss: "t1 \<le> ss" "ss \<le> t2"
-      and tt: "-d < tt" "tt < 0"
-      unfolding r2_def
-      by force
-    have "xx \<in> {a -- b}"
-      using suba2
-      apply (rule set_mp)
-      using xx by (simp add: open_closed_segment)
-    then have [simp]: "xx \<in> X" using \<open>transversal_segment a b\<close> by (auto simp: transversal_segment_def)
-    from ss have ss_ex: "ss \<in> existence_ivl0 x" using exist
-      by auto
-    from d_ex[OF \<open>xx \<in> {a -- b}\<close>] tt
-    have tt_ex: "tt \<in> existence_ivl0 xx" by auto
-    then have neg_tt_ex: "- tt \<in> existence_ivl0 (flow0 xx tt)"
-      by (rule existence_ivl_reverse[simplified])
-    from eq have "flow0 (flow0 xx tt) (-tt) = flow0 (flow0 x ss) (-tt)"
-      by simp
-    then have "xx = flow0 x (ss - tt)"
-      apply (subst (asm) flow_trans[symmetric])
-        apply (rule tt_ex)
-       apply (rule neg_tt_ex)
-      apply (subst (asm) flow_trans[symmetric])
-        apply (rule ss_ex)
-       apply (subst eq[symmetric])
-       apply (rule neg_tt_ex)
-      by simp
-    moreover
-    define e where "e = t2 - ss"
-    consider "e > - tt" | "e \<le> -tt" by arith
-    then show False
-    proof cases
-      case 1
-      have "flow0 (flow0 x ss) (-tt) \<notin> {a<--<b}"
-        apply (subst flow_trans[symmetric])
-          apply fact
-        subgoal using neg_tt_ex eq by simp
-        apply (rule t1t2)
-        using 1 ss tt
-        unfolding e_def
-        by auto
-      moreover have "flow0 (flow0 x ss) (-tt) \<in> {a<--<b}"
-        unfolding eq[symmetric] using tt_ex xx
-        apply (subst flow_trans[symmetric])
-          apply (auto simp add: neg_tt_ex)
-        by (metis (no_types, hide_lams) suba2 subset_eq subset_open_segment)
-      ultimately show ?thesis by simp
-    next
-      case 2
-      have les: "tt + e \<le> 0" "-d \<le> tt + e"
-        using tt ss 2 e_def
-        by auto
-      have xxtte: "flow0 xx (tt + e) = flow0 x t2"
-        apply (simp add: e_def)
-        by (smt atLeastAtMost_iff calculation eq exist local.existence_ivl_trans' local.flow_trans neg_tt_ex ss_ex subset_iff \<open>t1 \<le> t2\<close>)
-      show False
-      proof (cases "tt=-e")
-        case True
-        with xxtte have "xx = flow0 x t2"
-          by (simp add: )
-        with xx show ?thesis
-          apply auto
-          by (auto simp: open_segment_def)
-      next
-        case False
-        with les have "tt+e < 0" by simp
-        from d_below[OF \<open>xx \<in> {a -- b}\<close>  \<open>-d \<le> tt + e\<close> this]
-        have "flow0 xx (tt + e) \<notin> {a -- b}"
-          apply (simp add: in_closed_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
-              not_le )
-          by (smt \<open>xx \<in> {a--b}\<close> inner_minus_right inner_rot_neg_move_base inner_rot_pos_move_base n rot_diff_commute)
-        with xxtte show ?thesis
-          using \<open>flow0 x t2 \<in> {a--b}\<close> by simp
-      qed
-    qed
-  qed
-  moreover
-  have pir2J2: "r2 \<inter> path_image J2 = {}"
-  proof -
-    have "r2 \<subseteq> {x. (x - a) \<bullet> rot n < 0}"
-      unfolding r2_def
-    proof safe
-      fix aa ba
-      assume "aa \<in> {a<--<flow0 x t2}" "ba \<in> {-d<..<0}"
-      with suba2 show "0 > (flow0 aa ba - a) \<bullet> rot n"
-        using segment_open_subset_closed[of a "flow0 x t2"]
-        by (intro inner_neg_move_base[OF ortho d_below]) auto
-    qed
-    also have "\<dots> \<inter> {a -- b} = {}"
-      using in_segment_inner_rot in_segment_inner_rot2 n by auto
-    finally show ?thesis
-      unfolding J2_def path_image_linepath
-      using t12sub open_closed_segment
-      by (force simp: closed_segment_commute)
-  qed
-  ultimately have pir2:"r2 \<inter> (path_image J) = {}"
-    unfolding J_def
-    by (metis disjoint_iff_not_equal not_in_path_image_join)
-
-  define rp where "rp = midpoint (flow0 x t1) (flow0 x t2)"
-  have rpi: "rp \<in> {flow0 x t1<--<flow0 x t2}" unfolding rp_def
-    by (simp add: x1neqx2)
-  have "rp \<in> {a -- b}"
-    using rpi suba2o subl by blast
-  then have [simp]: "rp \<in> X"
-    using \<open>{a--b} \<subseteq> X\<close> by blast
-
-(* The fundamental case distinction *)
-  have *: "pathfinish J1 = flow0 x t2"
-    "pathstart J1 = flow0 x t1"
-    "rp \<in> {flow0 x t2<--<flow0 x t1}"
-    using rpi
-    by (auto simp: open_segment_commute J1_def)
-  have "{y. 0 < (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1)} = {y. 0 < (y - rp) \<bullet> rot (flow0 x t2 - flow0 x t1)}"
-    by (smt Collect_cong in_open_segment_rotD inner_diff_left nrm_dot rpi)
-  also have "... =  {y. 0 > (y - rp) \<bullet> rot (flow0 x t1 - flow0 x t2)}"
-    by (smt Collect_cong inner_minus_left nrm_reverse)
-  also have " ... = {y. 0 > (y - rp) \<bullet> rot (a - b) }"
-    by (metis rot_same_dir(2) x1 x2)
-  finally have side1: "{y. 0 < (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1)} = {y. 0 > (y - rp) \<bullet> rot (a - b) }"
-    (is "_ = ?lower1") .
-  have "{y. (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0} = {y. (y - rp) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0}"
-    by (smt Collect_cong in_open_segment_rotD inner_diff_left nrm_dot rpi)
-  also have "... =  {y. (y - rp) \<bullet> rot (flow0 x t1 - flow0 x t2) > 0}"
-    by (smt Collect_cong inner_minus_left nrm_reverse)
-  also have " ... = {y. 0 < (y - rp) \<bullet> rot (a - b) }"
-    by (metis rot_same_dir(1) x1 x2)
-  finally have side2: "{y. (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0} = {y. 0 < (y - rp) \<bullet> rot (a - b) }"
-    (is "_ = ?upper1") .
-  from linepath_ball_inside_outside[OF \<open>simple_path J\<close>[unfolded J_def J2_def] *,
-      folded J2_def J_def, unfolded side1 side2]
-  obtain e where e0: "0 < e"
-    "ball rp e \<inter> path_image J1 = {}"
-    "ball rp e \<inter> ?lower1 \<subseteq> inner \<and>
-        ball rp e \<inter> ?upper1 \<subseteq> outer \<or>
-        ball rp e \<inter> ?upper1 \<subseteq> inner \<and>
-        ball rp e \<inter> ?lower1 \<subseteq> outer"
-    by (auto simp: inner_def outer_def)
-
-  let ?lower = "{y. 0 > (y - rp) \<bullet> rot n }"
-  let ?upper = "{y. 0 < (y - rp) \<bullet> rot n }"
-  have "?lower1 = {y. 0 < (y - rp) \<bullet> rot n } \<and> ?upper1 = {y. 0 > (y - rp) \<bullet> rot n } \<or>
-      ?lower1 = {y. 0 > (y - rp) \<bullet> rot n } \<and> ?upper1 =  {y. 0 < (y - rp) \<bullet> rot n }"
-    using n rot_diff_commute[of a b]
-    by auto
-  from this e0 have e: "0 < e"
-    "ball rp e \<inter> path_image J1 = {}"
-    "ball rp e \<inter> ?lower \<subseteq> inner \<and>
-        ball rp e \<inter> ?upper \<subseteq> outer \<or>
-        ball rp e \<inter> ?upper \<subseteq> inner \<and>
-        ball rp e \<inter> ?lower \<subseteq> outer"
-    by auto
-
-  have "\<forall>\<^sub>F t in at_right 0. t < d"
-    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
-  then have evr: "\<forall>\<^sub>F t in at_right 0. 0 < (flow0 rp t - rp) \<bullet> rot n"
-    unfolding eventually_at_filter
-    by eventually_elim (auto intro!: \<open>rp \<in> {a--b}\<close> d_above)
-  have "\<forall>\<^sub>F t in at_left 0. t > -d"
-    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
-  then have evl: "\<forall>\<^sub>F t in at_left 0. 0 > (flow0 rp t - rp) \<bullet> rot n"
-    unfolding eventually_at_filter
-    by eventually_elim (auto intro!: \<open>rp \<in> {a--b}\<close> d_below)
-  have "\<forall>\<^sub>F t in at 0. flow0 rp t \<in> ball rp e"
-    unfolding mem_ball
-    apply (subst dist_commute)
-    apply (rule tendstoD)
-    by (auto intro!: tendsto_eq_intros \<open>0 < e\<close>)
-  then have evl2: "(\<forall>\<^sub>F t in at_left 0. flow0 rp t \<in> ball rp e)"
-    and evr2: "(\<forall>\<^sub>F t in at_right 0. flow0 rp t \<in> ball rp e)"
-    unfolding eventually_at_split by auto
-  have evl3: "(\<forall>\<^sub>F t in at_left 0. t > -d)"
-    and evr3: "(\<forall>\<^sub>F t in at_right 0. t < d)"
-    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
-  have evl4: "(\<forall>\<^sub>F t in at_left 0. t < 0)"
-    and evr4: "(\<forall>\<^sub>F t in at_right 0. t > 0)"
-    by (auto simp: eventually_at_filter)
-  from evl evl2 evl3 evl4
-  have "\<forall>\<^sub>F t in at_left 0. (flow0 rp t - rp) \<bullet> rot n < 0 \<and> flow0 rp t \<in> ball rp e \<and> t > -d \<and> t < 0"
-    by eventually_elim auto
-  from eventually_happens[OF this]
-  obtain dl where dl: "(flow0 rp dl - rp) \<bullet> rot n < 0" "flow0 rp dl \<in> ball rp e" "- d < dl" "dl < 0"
-    by auto
-  from evr evr2 evr3 evr4
-  have "\<forall>\<^sub>F t in at_right 0. (flow0 rp t - rp) \<bullet> rot n > 0 \<and> flow0 rp t \<in> ball rp e \<and> t < d \<and> t > 0"
-    by eventually_elim auto
-  from eventually_happens[OF this]
-  obtain dr where dr: "(flow0 rp dr - rp) \<bullet> rot n > 0" "flow0 rp dr \<in> ball rp e" "d > dr" "dr > 0"
-    by auto
-
-  have "rp \<in> {flow0 x t1<--<b}" using rpi subr by auto
-  then have rpr1:"flow0 rp (dr) \<in> r1" unfolding r1_def using \<open>d > dr\<close> \<open>dr > 0\<close>
-    by auto
-  have "rp \<in> {a<--<flow0 x t2}" using rpi subl by auto
-  then have rpr2:"flow0 rp (dl) \<in> r2" unfolding r2_def using \<open>-d < dl\<close> \<open>dl < 0\<close>
-    by auto
-
-  from e(3) dr dl
-  have "flow0 rp (dr) \<in> outer \<and> flow0 rp (dl) \<in> inner \<or> flow0 rp (dr) \<in> inner \<and> flow0 rp (dl) \<in> outer"
-    by auto
-  moreover {
-    assume "flow0 rp dr \<in> outer" "flow0 rp dl \<in> inner"
-    then have
-      r1o: "r1 \<inter> outer \<noteq> {}" and
-      r2i: "r2 \<inter> inner \<noteq> {}" using rpr1 rpr2 by auto
-    from path_connected_not_frontier_subset[OF pcr1 r1o]
-    have "r1 \<subseteq> outer" using pir1 by (simp add: io(12))
-    from path_connected_not_frontier_subset[OF pcr2 r2i]
-    have "r2 \<subseteq> inner" using pir2 by (simp add: io(11))
-    have "(\<lambda>(x, y). flow0 x y)`({flow0 x t2} \<times> {0<..<d}) \<subseteq> r1" unfolding r1_def
-      by (auto intro!:image_mono simp add: x2)
-    then have *:"\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 (flow0 x t2) t \<in> outer"
-      by (smt \<open>r1 \<subseteq> outer\<close> greaterThanLessThan_iff mem_Sigma_iff pair_imageI r1_def subset_eq x2)
-
-    then have t2o: "\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 x (t2 + t) \<in> outer"
-      using r1a3[OF x2] exist flow_trans
-      by (metis (no_types, hide_lams) closed_segment_commute ends_in_segment(1) local.existence_ivl_trans' local.flow_undefined0 real_Icc_closed_segment subset_eq \<open>t1 \<le> t2\<close>)
-
-(* Construct a sequence of times converging to these points in r2 \<subseteq> inner *)
-    have inner: "{a <--< flow0 x t2} \<subseteq> closure inner"
-    proof (rule subsetI)
-      fix y
-      assume y: "y \<in> {a <--< flow0 x t2}"
-      have [simp]: "y \<in> X"
-        using y suba12_open suba2o \<open>{a -- b} \<subseteq> X\<close>
-        by auto
-      have "(\<forall>n. flow0 y (- d / real (Suc (Suc n))) \<in> inner)"
-        using y
-        using suba12_open \<open>0 < d\<close> suba2o \<open>{a -- b} \<subseteq> X\<close>
-        by (auto intro!: set_mp[OF \<open>r2 \<subseteq> inner\<close>] image_eqI[where x="(y, -d/Suc (Suc n))" for n]
-            simp: r2_def divide_simps) 
-      moreover
-      have d_over_0: "(\<lambda>s. - d / real (Suc (Suc s))) \<longlonglongrightarrow> 0"
-        by (rule real_tendsto_divide_at_top)
-          (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
-      have "(\<lambda>n. flow0 y (- d / real (Suc (Suc n)))) \<longlonglongrightarrow> y"
-        apply (rule tendsto_eq_intros)
-           apply (rule tendsto_intros)
-          apply (rule d_over_0)
-        by auto
-      ultimately show "y \<in> closure inner"
-        unfolding closure_sequential
-        by (intro exI[where x="\<lambda>n. flow0 y (-d/Suc (Suc n))"]) (rule conjI)
-    qed
-    then have "{a <--< flow0 x t1} \<subseteq> closure inner"
-      using suba12_open by blast
-    then have "{flow0 x t1 -- flow0 x t2} \<subseteq> closure inner"
-      by (metis (no_types, lifting) closure_closure closure_mono closure_open_segment dual_order.trans inner subl x1neqx2)
-    have outer:"\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<in> outer"
-    proof (rule ccontr)
-      fix t
-      assume t: "t > t2" "t \<in> existence_ivl0 x" "flow0 x t \<notin> outer"
-      have "0 \<le> t- (t2+d)" using t2o t  by smt 
-      then have a2:"0 \<le> t - (t2+dr)" using d \<open>0 < dr\<close> \<open>dr < d\<close> by linarith
-      have t2d2_ex: "t2 + dr \<in> existence_ivl0 x"
-        using \<open>t1 \<le> t2\<close> exist d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a--b}\<close> \<open>0 < d\<close> \<open>0 < dr\<close> \<open>dr < d\<close> 
-        by (intro existence_ivl_trans) auto
-      then have a3: "t - (t2 + dr) \<in> existence_ivl0 (flow0 x (t2 + dr))"
-        using t(2)
-        by (intro diff_existence_ivl_trans) auto
-      then have "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) = flow0 x t"
-        by (subst flow_trans[symmetric]) (auto simp: t2d2_ex)
-      moreover have "flow0 x t \<in> closure inner" using t(3) io
-        by (metis ComplI Un_iff closure_Un_frontier)
-      ultimately have a4: "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) \<in> closure inner" by auto
-      have a1: "flow0 x (t2+dr) \<in> outer"
-        by (simp add: d t2o \<open>0 < dr\<close> \<open>dr < d\<close>)
-      from outside_in[OF a1 a2 a3 a4]
-      obtain T where T: "T > 0" "T \<le> t - (t2 + dr)"
-        "(\<forall>s\<in>{0..<T}. flow0 (flow0 x (t2 + dr)) s \<in> outer)"
-        "flow0 (flow0 x (t2 + dr)) T \<in> {flow0 x t1 --< flow0 x t2}" by blast
-      define y where "y = flow0 (flow0 x (t2 + dr)) T"
-      have "y \<in> {a <--< flow0 x t2}" unfolding y_def using T(4)
-        using subl2 by blast 
-      then have "(\<lambda>(x, y). flow0 x y)`({y} \<times> {-d<..<0}) \<subseteq> r2" unfolding r2_def
-        by (auto intro!:image_mono)
-      then have *:"\<And>t. -d < t \<Longrightarrow> t < 0 \<Longrightarrow> flow0 y t \<in> r2"
-        by (simp add: pair_imageI subsetD)
-      have "max (-T/2) dl < 0" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
-      moreover have "-d < max (-T/2) dl" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
-      ultimately have inner: "flow0 y (max (-T/2) dl) \<in> inner" using * \<open>r2 \<subseteq> inner\<close> by blast 
-      have "0\<le>(T+(max (-T/2) dl))" using T(1) by linarith
-      moreover have "(T+(max (-T/2) dl)) < T" using T(1) d \<open>0 > dl\<close> \<open>dl > -d\<close> by linarith
-      ultimately have outer: " flow0 (flow0 x (t2 + dr)) (T+(max (-T/2) dl)) \<in> outer"
-        using T by auto
-      have T_ex: "T \<in> existence_ivl0 (flow0 x (t2 + dr))"
-        apply (subst flow_trans)
-        using exist \<open>t1 \<le> t2\<close>
-        using d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a -- b}\<close> \<open>d > 0\<close> T \<open>0 < dr\<close> \<open>dr < d\<close>
-          apply (auto simp: )
-        apply (rule set_rev_mp[where A="{0 .. t - (t2 + dr)}"], force)
-        apply (rule ivl_subset_existence_ivl)
-        apply (rule existence_ivl_trans')
-         apply (rule existence_ivl_trans')
-        by (auto simp: t)
-      have T_ex2: "dr + T \<in> existence_ivl0 (flow0 x t2)"
-        by (smt T_ex ends_in_segment(2) exist local.existence_ivl_trans local.existence_ivl_trans' real_Icc_closed_segment subset_eq t2d2_ex \<open>t1 \<le> t2\<close>)
-      thus False using T \<open>t1 \<le> t2\<close> exist
-        by (smt T_ex diff_existence_ivl_trans disjoint_iff_not_equal inner io(9) local.flow_trans local.flow_undefined0 outer y_def)
-    qed
-    have "closure inner \<inter> outer = {}"
-      by (simp add: inf_sup_aci(1) io(5) io(9) open_Int_closure_eq_empty) 
-    then have "flow0 x t \<notin> {a<--<flow0 x t2}"
-      using \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close> inner outer by blast
-  }
-  moreover {
-    assume "flow0 rp dr \<in> inner" "flow0 rp dl \<in> outer"
-    then have
-      r1i: "r1 \<inter> inner \<noteq> {}" and
-      r2o: "r2 \<inter> outer \<noteq> {}" using rpr1 rpr2 by auto
-    from path_connected_not_frontier_subset[OF pcr1 r1i]
-    have "r1 \<subseteq> inner" using pir1 by (simp add: io(11))
-    from path_connected_not_frontier_subset[OF pcr2 r2o]
-    have "r2 \<subseteq> outer" using pir2 by (simp add: io(12))
-
-    have "(\<lambda>(x, y). flow0 x y)`({flow0 x t2} \<times> {0<..<d}) \<subseteq> r1" unfolding r1_def
-      by (auto intro!:image_mono simp add: x2)
-    then have
-      *:"\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 (flow0 x t2) t \<in> inner"
-      by (smt \<open>r1 \<subseteq> inner\<close> greaterThanLessThan_iff mem_Sigma_iff pair_imageI r1_def subset_eq x2)
-
-    then have t2o: "\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 x (t2 + t) \<in> inner"
-      using r1a3[OF x2] exist flow_trans
-      by (metis (no_types, hide_lams) closed_segment_commute ends_in_segment(1) local.existence_ivl_trans' local.flow_undefined0 real_Icc_closed_segment subset_eq \<open>t1 \<le> t2\<close>)
-
-(* Construct a sequence of times converging to these points in r2 \<subseteq> outer *)
-    have outer: "{a <--< flow0 x t2} \<subseteq> closure outer"
-    proof (rule subsetI)
-      fix y
-      assume y: "y \<in> {a <--< flow0 x t2}"
-      have [simp]: "y \<in> X"
-        using y suba12_open suba2o \<open>{a -- b} \<subseteq> X\<close>
-        by auto
-      have "(\<forall>n. flow0 y (- d / real (Suc (Suc n))) \<in> outer)"
-        using y
-        using suba12_open \<open>0 < d\<close> suba2o \<open>{a -- b} \<subseteq> X\<close>
-        by (auto intro!: set_mp[OF \<open>r2 \<subseteq> outer\<close>] image_eqI[where x="(y, -d/Suc (Suc n))" for n]
-            simp: r2_def divide_simps) 
-      moreover
-      have d_over_0: "(\<lambda>s. - d / real (Suc (Suc s))) \<longlonglongrightarrow> 0"
-        by (rule real_tendsto_divide_at_top)
-          (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
-      have "(\<lambda>n. flow0 y (- d / real (Suc (Suc n)))) \<longlonglongrightarrow> y"
-        apply (rule tendsto_eq_intros)
-           apply (rule tendsto_intros)
-          apply (rule d_over_0)
-        by auto
-      ultimately show "y \<in> closure outer"
-        unfolding closure_sequential
-        by (intro exI[where x="\<lambda>n. flow0 y (-d/Suc (Suc n))"]) (rule conjI)
-    qed
-    then have "{a <--< flow0 x t1} \<subseteq> closure outer"
-      using suba12_open by blast
-    then have "{flow0 x t1 -- flow0 x t2} \<subseteq> closure outer"
-      by (metis (no_types, lifting) closure_closure closure_mono closure_open_segment dual_order.trans outer subl x1neqx2)
-
-    have inner:"\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<in> inner"
-    proof (rule ccontr)
-      fix t
-      assume t: "t > t2" "t \<in> existence_ivl0 x" "flow0 x t \<notin> inner"
-      have "0 \<le> t- (t2+d)" using t2o t by smt 
-      then have a2:"0 \<le> t - (t2+dr)" using d \<open>0 < dr\<close> \<open>dr < d\<close> by linarith
-      have t2d2_ex: "t2 + dr \<in> existence_ivl0 x"
-        using \<open>t1 \<le> t2\<close> exist d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a--b}\<close> \<open>0 < d\<close> \<open>0 < dr\<close> \<open>dr < d\<close> 
-        by (intro existence_ivl_trans) auto
-      then have a3: "t - (t2 + dr) \<in> existence_ivl0 (flow0 x (t2 + dr))"
-        using t(2)
-        by (intro diff_existence_ivl_trans) auto
-      then have "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) = flow0 x t"
-        by (subst flow_trans[symmetric]) (auto simp: t2d2_ex)
-      moreover have "flow0 x t \<in> closure outer" using t(3) io
-        by (metis ComplI Un_iff closure_Un_frontier)
-      ultimately have a4: "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) \<in> closure outer" by auto
-      have a1: "flow0 x (t2+dr) \<in> inner"
-        by (simp add: d t2o \<open>0 < dr\<close> \<open>dr < d\<close>)
-      from inside_out[OF a1 a2 a3 a4]
-      obtain T where T: "T > 0" "T \<le> t - (t2 + dr)"
-        "(\<forall>s\<in>{0..<T}. flow0 (flow0 x (t2 + dr)) s \<in> inner)"
-        "flow0 (flow0 x (t2 + dr)) T \<in> {flow0 x t1 --< flow0 x t2}" by blast
-      define y where "y = flow0 (flow0 x (t2 + dr)) T"
-      have "y \<in> {a <--< flow0 x t2}" unfolding y_def using T(4)
-        using subl2 by blast 
-      then have "(\<lambda>(x, y). flow0 x y)`({y} \<times> {-d<..<0}) \<subseteq> r2" unfolding r2_def
-        by (auto intro!:image_mono)
-      then have *:"\<And>t. -d < t \<Longrightarrow> t < 0 \<Longrightarrow> flow0 y t \<in> r2"
-        by (simp add: pair_imageI subsetD)
-      have "max (-T/2) dl < 0" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
-      moreover have "-d < max (-T/2) dl" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
-      ultimately have outer: "flow0 y (max (-T/2) dl) \<in> outer" using * \<open>r2 \<subseteq> outer\<close> by blast 
-      have "0\<le>(T+(max (-T/2) dl))" using T(1) by linarith
-      moreover have "(T+(max (-T/2) dl)) < T" using T(1) d \<open>0 > dl\<close> \<open>dl > -d\<close> by linarith
-      ultimately have inner: " flow0 (flow0 x (t2 + dr)) (T+(max (-T/2) dl)) \<in> inner"
-        using T by auto
-      have T_ex: "T \<in> existence_ivl0 (flow0 x (t2 + dr))"
-        apply (subst flow_trans)
-        using exist \<open>t1 \<le> t2\<close>
-        using d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a -- b}\<close> \<open>d > 0\<close> T \<open>0 < dr\<close> \<open>dr < d\<close>
-          apply (auto simp: )
-        apply (rule set_rev_mp[where A="{0 .. t - (t2 + dr)}"], force)
-        apply (rule ivl_subset_existence_ivl)
-        apply (rule existence_ivl_trans')
-         apply (rule existence_ivl_trans')
-        by (auto simp: t)
-      have T_ex2: "dr + T \<in> existence_ivl0 (flow0 x t2)"
-        by (smt T_ex ends_in_segment(2) exist local.existence_ivl_trans local.existence_ivl_trans' real_Icc_closed_segment subset_eq t2d2_ex \<open>t1 \<le> t2\<close>)
-      thus False using T \<open>t1 \<le> t2\<close> exist
-        by (smt T_ex diff_existence_ivl_trans disjoint_iff_not_equal inner io(9) local.flow_trans local.flow_undefined0 outer y_def)
-    qed
-    have "closure outer \<inter> inner = {}"
-      by (metis inf_sup_aci(1) io(2) io2(1) open_Int_closure_eq_empty)
-    then have "flow0 x t \<notin> {a<--<flow0 x t2}"
-      using \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close> inner outer by blast
-  }
-  ultimately show
-    "flow0 x t \<notin> {a<--<flow0 x t2}" by auto
-qed
-
-lemma open_segment_trichotomy:
-  fixes x y a b::'a
-  assumes x:"x \<in> {a<--<b}"
-  assumes y:"y \<in> {a<--<b}"
-  shows "x = y \<or> y \<in> {x<--<b} \<or> y \<in> {a<--<x}"
-proof -
-  from Un_open_segment[OF y]
-  have "{a<--<y} \<union> {y} \<union> {y<--<b} = {a<--<b}" .
-  then have "x \<in> {a<--<y} \<or> x = y \<or> x \<in> {y <--<b}" using x by blast
-  moreover {
-    assume "x \<in> {a<--<y}"
-    then have "y \<in> {x<--<b}" using open_segment_subsegment
-      using open_segment_commute y by blast
-  }
-  moreover {
-    assume "x \<in> {y<--<b}"
-    from open_segment_subsegment[OF y this]
-    have "y \<in> {a<--<x}" .
-  }
-  ultimately show ?thesis by blast
-qed
-
-sublocale rev: c1_on_open_R2 "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
-  by unfold_locales (auto simp: dim2)
-
-lemma rev_transversal_segment: "rev.transversal_segment a b = transversal_segment a b"
-  by (auto simp: transversal_segment_def rev.transversal_segment_def)
-
-lemma flow0_transversal_segment_monotone_step_reverse:
-  assumes "transversal_segment a b"
-  assumes "t1 \<le> t2"
-  assumes "{t1..t2} \<subseteq> existence_ivl0 x"
-  assumes x1: "flow0 x t1 \<in> {a<--<b}"
-  assumes x2: "flow0 x t2 \<in> {a<--<flow0 x t1}"
-  assumes "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
-  assumes "t < t1" "t \<in> existence_ivl0 x"
-  shows "flow0 x t \<notin> {a<--<flow0 x t1}"
-proof -
-  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
-  note t1t2 = \<open>\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}\<close>
-  from \<open>transversal_segment a b\<close> have [simp]: "a \<noteq> b" by (simp add: transversal_segment_def)
-  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
-    by (auto simp: in_open_segment_iff_line)
-  from x2 obtain i2 where i2: "flow0 x t2 = line a b i2" "0 < i2" "i2 < i1"
-    by (auto simp: i1 open_segment_line_iff)
-
-  have t2_exist[simp]: "t2 \<in> existence_ivl0 x"
-    using \<open>t1 \<le> t2\<close> exist by auto
-  have t2_mem: "flow0 x t2 \<in> {a<--<b}"
-    and x1_mem: "flow0 x t1 \<in> {flow0 x t2<--<b}"
-    using i1 i2
-    by (auto simp: line_in_subsegment line_line1)
-
-  have transversal': "rev.transversal_segment a b"
-    using \<open>transversal_segment a b\<close> unfolding rev_transversal_segment .
-  have time': "0 \<le> t2 - t1" using \<open>t1 \<le> t2\<close> by simp
-  have [simp, intro]: "flow0 x t2 \<in> X"
-    using exist \<open>t1 \<le> t2\<close>
-    by auto
-  have exivl': "{0..t2 - t1} \<subseteq> rev.existence_ivl0 (flow0 x t2)"
-    using exist \<open>t1 \<le> t2\<close>
-    by (force simp add: rev_existence_ivl_eq0 intro!: existence_ivl_trans')
-  have step': "rev.flow0 (flow0 x t2) (t2-t) \<notin>  {a<--<rev.flow0 (flow0 x t2) (t2 - t1)}"
-    apply (rule rev.flow0_transversal_segment_monotone_step[OF transversal' time' exivl'])
-    using exist \<open>t1 \<le> t2\<close> x1 x2 t2_mem x1_mem t1t2 \<open>t < t1\<close> \<open>t \<in> existence_ivl0 x\<close>
-        apply (auto simp: rev_existence_ivl_eq0 rev_eq_flow existence_ivl_trans' flow_trans[symmetric])
-    by (subst (asm) flow_trans[symmetric]) (auto intro!: existence_ivl_trans')
-  then show ?thesis
-    unfolding rev_eq_flow
-    using \<open>t1 \<le> t2\<close> exist \<open>t < t1\<close> \<open>t \<in> existence_ivl0 x\<close>
-    by (auto simp: flow_trans[symmetric] existence_ivl_trans')
-qed
-
-lemma flow0_transversal_segment_monotone_step_reverse2:
-  assumes transversal: "transversal_segment a b"
-  assumes time: "t1 \<le> t2"
-  assumes exist: "{t1..t2} \<subseteq> existence_ivl0 x"
-  assumes t1: "flow0 x t1 \<in> {a<--<b}"
-  assumes t2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
-  assumes t1t2: "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
-  assumes t: "t < t1" "t \<in> existence_ivl0 x"
-  shows "flow0 x t \<notin> {flow0 x t1<--<b}"
-  using flow0_transversal_segment_monotone_step_reverse[of b a, OF _ time exist, of t]
-    assms
-  by (auto simp: open_segment_commute transversal_segment_commute)
-
-lemma flow0_transversal_segment_monotone_step2:
-  assumes transversal: "transversal_segment a b"
-  assumes time: "t1 \<le> t2"
-  assumes exist: "{t1..t2} \<subseteq> existence_ivl0 x"
-  assumes t1: "flow0 x t1 \<in> {a<--<b}"
-  assumes t2: "flow0 x t2 \<in> {a<--<flow0 x t1}"
-  assumes t1t2: "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
-  shows "\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<notin> {flow0 x t2<--<b}"
-  using flow0_transversal_segment_monotone_step[of b a, OF _ time exist]
-    assms
-  by (auto simp: transversal_segment_commute open_segment_commute)
-
-lemma flow0_transversal_segment_monotone:
-  assumes "transversal_segment a b"
-  assumes "t1 \<le> t2"
-  assumes "{t1..t2} \<subseteq> existence_ivl0 x"
-  assumes x1: "flow0 x t1 \<in> {a<--<b}"
-  assumes x2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
-  assumes "t > t2" "t \<in> existence_ivl0 x"
-  shows "flow0 x t \<notin> {a<--<flow0 x t2}"
-proof -
-  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
-  note t = \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close>
-  have x1neqx2: "flow0 x t1 \<noteq> flow0 x t2"
-    using open_segment_def x2 by force 
-  then have t1neqt2: "t1 \<noteq> t2" by auto
-  with \<open>t1 \<le> t2\<close> have "t1 < t2" by simp
-
-  from \<open>transversal_segment a b\<close> have [simp]: "a \<noteq> b" by (simp add: transversal_segment_def)
-  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
-    by (auto simp: in_open_segment_iff_line)
-  from x2 i1 obtain i2 where i2: "flow0 x t2 = line a b i2" "i1 < i2" "i2 < 1"
-    by (auto simp: line_open_segment_iff)
-  have t2_in: "flow0 x t2 \<in> {a<--<b}"
-    using i1 i2
-    by simp
-
-  let ?T = "{s \<in> {t1..t2}. flow0 x s \<in> {a--b}}"
-  let ?T' = "{s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}}"
-  from flow_transversal_segment_finite_intersections[OF \<open>transversal_segment a b\<close> \<open>t1 \<le> t2\<close> exist]
-  have "finite ?T" .
-  then have "finite ?T'" by (rule finite_subset[rotated]) (auto simp: open_closed_segment)
-  have "?T' \<noteq> {}"
-    by (auto intro!: exI[where x=t1] \<open>t1 < t2\<close> x1)
-  note tm_defined = \<open>finite ?T'\<close> \<open>?T' \<noteq> {}\<close>
-  define tm where "tm = Max ?T'"
-  have "tm \<in> ?T'"
-    unfolding tm_def
-    using tm_defined by (rule Max_in)
-  have tm_in: "flow0 x tm \<in> {a<--<b}"
-    using \<open>tm \<in> ?T'\<close>
-    by auto
-  have tm: "t1 \<le> tm" "tm < t2" "tm \<le> t2"
-    using \<open>tm \<in> ?T'\<close> by auto
-  have tm_Max: "t \<le> tm" if "t \<in> ?T'" for t
-    unfolding tm_def
-    using tm_defined(1) that
-    by (rule Max_ge)
-
-  have tm_exclude: "flow0 x t \<notin> {a<--<b}" if "t \<in> {tm<..<t2}" for t
-    using \<open>tm \<in> ?T'\<close> tm_Max that
-    by auto (meson approximation_preproc_push_neg(2) dual_order.strict_trans2 le_cases)
-  have "{tm..t2} \<subseteq> existence_ivl0 x"
-    using exist tm by auto
-
-  from open_segment_trichotomy[OF tm_in t2_in]
-  consider
-    "flow0 x t2 \<in> {flow0 x tm<--<b}" |
-    "flow0 x t2 \<in> {a<--<flow0 x tm}" |
-    "flow0 x tm = flow0 x t2"
-    by blast
-  then show "flow0 x t \<notin> {a<--<flow0 x t2}"
-  proof cases
-    case 1
-    from flow0_transversal_segment_monotone_step[OF \<open>transversal_segment a b\<close> \<open>tm \<le> t2\<close>
-        \<open>{tm..t2} \<subseteq> existence_ivl0 x\<close> tm_in 1 tm_exclude t]
-    show ?thesis .
-  next
-    case 2
-    have "t1 \<noteq> tm"
-      using 2 x2 i1 i2
-      by (auto simp: line_in_subsegment line_in_subsegment2)
-    then have "t1 < tm" using \<open>t1 \<le> tm\<close> by simp
-    from flow0_transversal_segment_monotone_step_reverse[OF \<open>transversal_segment a b\<close> \<open>tm \<le> t2\<close>
-        \<open>{tm..t2} \<subseteq> existence_ivl0 x\<close> tm_in 2 tm_exclude \<open>t1 < tm\<close>] exist \<open>t1 \<le> t2\<close>
-    have "flow0 x t1 \<notin> {a<--<flow0 x tm}" by auto
-    then have False
-      using x1 x2 2 i1 i2
-      apply (auto simp: line_in_subsegment line_in_subsegment2)
-      by (smt greaterThanLessThan_iff in_open_segment_iff_line line_in_subsegment2 tm_in)
-    then show ?thesis by simp
-  next
-    case 3
-    have "t1 \<noteq> tm"
-      using 3 x2
-      by (auto simp: open_segment_def)
-    then have "t1 < tm" using \<open>t1 \<le> tm\<close> by simp
-    have "range (flow0 x) = flow0 x ` {tm..t2}"
-      apply (rule recurrence_time_restricts_compact_flow'[OF \<open>tm < t2\<close> _ _ 3])
-      using exist \<open>t1 \<le> t2\<close> \<open>t1 < tm\<close> \<open>tm < t2\<close>
-      by auto
-    also have "\<dots> = flow0 x ` (insert t2 {tm<..<t2})"
-      using \<open>tm \<le> t2\<close> 3
-      apply (auto simp: )
-      by (smt greaterThanLessThan_iff image_eqI)
-    finally have "flow0 x t1 \<in> flow0 x ` (insert t2 {tm<..<t2})"
-      by auto
-    then have "flow0 x t1 \<in> flow0 x ` {tm<..<t2}" using x1neqx2
-      by auto
-    moreover have "\<dots> \<inter> {a<--<b} = {}"
-      using tm_exclude
-      by auto
-    ultimately have False using x1 by auto
-    then show ?thesis by blast
-  qed
-qed
-
-subsection \<open>Straightening\<close>
-
-text \<open> This lemma uses the implicit function theorem \<close>
-lemma cross_time_continuous:
-  assumes "transversal_segment a b"
-  assumes "x \<in> {a<--<b}"
-  assumes "e > 0"
-  obtains d t where "d > 0" "continuous_on (ball x d) t"
-    "\<And>y. y \<in> ball x d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}"
-    "\<And>y. y \<in> ball x d \<Longrightarrow> \<bar>t y\<bar> < e"
-    "continuous_on (ball x d) t"
-    "t x = 0"
-proof -
-  have "x \<in> X" using assms segment_open_subset_closed[of a b]
-    by (auto simp: transversal_segment_def)
-  have "a \<noteq> b" using assms by auto
-  define s where "s x = (x - a) \<bullet> rot (b - a)" for x
-  have "s x = 0"
-    unfolding s_def
-    by (subst in_segment_inner_rot) (auto intro!: assms open_closed_segment)
-  have Ds: "(s has_derivative blinfun_inner_left (rot (b - a))) (at x)"
-    (is "(_ has_derivative blinfun_apply (?Ds x)) _")
-    for x
-    unfolding s_def
-    by (auto intro!: derivative_eq_intros)
-  have Dsc: "isCont ?Ds x" by (auto intro!: continuous_intros)
-  have nz: "?Ds x (f x) \<noteq> 0"
-    using assms  apply auto
-    unfolding transversal_segment_def
-    by (smt inner_minus_left nrm_reverse open_closed_segment)
-
-  from flow_implicit_function_at[OF \<open>x \<in> X\<close>, of s, OF \<open>s x = 0\<close> Ds Dsc nz \<open>e > 0\<close>]
-  obtain t d1 where "0 < d1"
-    and t0: "t x = 0"
-    and d1: "(\<And>y. y \<in> cball x d1 \<Longrightarrow> s (flow0 y (t y)) = 0)"
-    "(\<And>y. y \<in> cball x d1 \<Longrightarrow> \<bar>t y\<bar> < e)"
-    "(\<And>y. y \<in> cball x d1 \<Longrightarrow> t y \<in> existence_ivl0 y)"
-    and tc: "continuous_on (cball x d1) t"
-    and t': "(t has_derivative
-        (- blinfun_inner_left (rot (b - a)) /\<^sub>R (blinfun_inner_left (rot (b - a))) (f x)))
-      (at x)"
-    by metis
-  from tc
-  have "t \<midarrow>x\<rightarrow> 0"
-    using \<open>0 < d1\<close>
-    by (auto simp: continuous_on_def at_within_interior t0 dest!: bspec[where x=x])
-  then have ftc: "((\<lambda>y. flow0 y (t y)) \<longlongrightarrow> x) (at x)"
-    by (auto intro!: tendsto_eq_intros simp: \<open>x \<in> X\<close>)
-
-
-
-  define e2 where "e2 = min (dist a x) (dist b x)"
-  have "e2 > 0"
-    using assms
-    by (auto simp: e2_def open_segment_def)
-
-  from tendstoD[OF ftc this] have "\<forall>\<^sub>F y in at x. dist (flow0 y (t y)) x < e2" .
-  moreover
-  let ?S = "{x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
-  have "open ?S" "x \<in> ?S"
-    using \<open>x \<in> {a<--<b}\<close>
-    by (auto simp add: open_segment_line_hyperplanes \<open>a \<noteq> b\<close>
-        intro!: open_Collect_conj open_halfspace_component_gt open_halfspace_component_lt)
-  from topological_tendstoD[OF ftc this] have "\<forall>\<^sub>F y in at x. flow0 y (t y) \<in> ?S" .
-  ultimately
-  have "\<forall>\<^sub>F y in at x. flow0 y (t y) \<in> ball x e2 \<inter> ?S" by eventually_elim (auto simp: dist_commute)
-  then obtain d2 where "0 < d2" and "\<And>y. x \<noteq> y \<Longrightarrow> dist y x < d2 \<Longrightarrow> flow0 y (t y) \<in> ball x e2 \<inter> ?S"
-    by (force simp: eventually_at)
-  then have d2: "dist y x < d2 \<Longrightarrow> flow0 y (t y) \<in> ball x e2 \<inter> ?S" for y
-    using \<open>0 < e2\<close> \<open>x \<in> X\<close> t0 \<open>x \<in> ?S\<close>
-    by (cases "y = x") auto
-
-  define d where "d = min d1 d2"
-  have "d > 0" using \<open>0 < d1\<close> \<open>0 < d2\<close> by (simp add: d_def)
-  moreover have "continuous_on (ball x d) t"
-    by (auto intro!:continuous_on_subset[OF tc] simp add: d_def)
-  moreover
-  have "ball x e2 \<inter> ?S \<inter> {x. s x = 0} \<subseteq> {a<--<b}"
-    by (auto simp add: in_open_segment_iff_rot \<open>a \<noteq> b\<close>) (auto simp: s_def e2_def in_segment)
-  then have "\<And>y. y \<in> ball x d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}"
-    apply (rule set_mp)
-    using d1 d2 \<open>0 < d2\<close>
-    by (auto simp: d_def e2_def dist_commute)
-  moreover have "\<And>y. y \<in> ball x d \<Longrightarrow> \<bar>t y\<bar> < e"
-    using d1 by (auto simp: d_def)
-  moreover have "continuous_on (ball x d) t"
-    using tc by (rule continuous_on_subset) (auto simp: d_def)
-  moreover have "t x = 0" by (simp add: t0)
-  ultimately show ?thesis ..
-qed
-
-lemma \<omega>_limit_crossings:
-  assumes "transversal_segment a b"
-  assumes pos_ex: "{0..} \<subseteq> existence_ivl0 x"
-  assumes "\<omega>_limit_point x p"
-  assumes "p \<in> {a<--<b}"
-  obtains s where 
-    "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
-proof -
-  from assms have "p \<in> X" by (auto simp: transversal_segment_def open_closed_segment)
-  from assms(3)
-  obtain t where
-    "t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" "(flow0 x \<circ> t) \<longlonglongrightarrow> p"
-    by (auto simp: \<omega>_limit_point_def)
-  note t = \<open>t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>\<close> \<open>(flow0 x \<circ> t) \<longlonglongrightarrow> p\<close>
-  note [tendsto_intros] = t(2)
-  from cross_time_continuous[OF assms(1,4) zero_less_one\<comment> \<open>TODO ??\<close>]
-  obtain \<tau> \<delta>
-    where "0 < \<delta>" "continuous_on (ball p \<delta>) \<tau>" 
-      "\<tau> p = 0" "(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> \<bar>\<tau> y\<bar> < 1)"
-      "(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> flow0 y (\<tau> y) \<in> {a<--<b})"
-    by metis
-  note \<tau> =
-    \<open>(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> flow0 y (\<tau> y) \<in> {a<--<b})\<close>
-    \<open>(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> \<bar>\<tau> y\<bar> < 1)\<close>
-    \<open>continuous_on (ball p \<delta>) \<tau>\<close> \<open>\<tau> p = 0\<close>
-  define s where "s n = t n + \<tau> (flow0 x (t n))" for n
-  have ev_in_ball: "\<forall>\<^sub>F n in at_top. flow0 x (t n) \<in> ball p \<delta>"
-    apply (simp add: )
-    apply (subst dist_commute)
-    apply (rule tendstoD)
-     apply (rule t[unfolded o_def])
-    apply (rule \<open>0 < \<delta>\<close>)
-    done
-  have "filterlim s at_top sequentially"
-  proof (rule filterlim_at_top_mono)
-    show "filterlim (\<lambda>n. -1 + t n) at_top sequentially"
-      by (rule filterlim_tendsto_add_at_top) (auto intro!: filterlim_tendsto_add_at_top t)
-    from ev_in_ball show "\<forall>\<^sub>F x in sequentially. -1 + t x \<le> s x"
-      apply eventually_elim
-      using \<tau>
-      by (force simp : s_def)
-  qed
-  moreover
-  have \<tau>_cont: "\<tau> \<midarrow>p\<rightarrow> \<tau> p"
-    using \<tau>(3) \<open>0 < \<delta>\<close>
-    by (auto simp: continuous_on_def at_within_ball dest!: bspec[where x=p])
-  note [tendsto_intros] = tendsto_compose_at[OF _ this, simplified]
-  have ev1: "\<forall>\<^sub>F n in sequentially. t n > 1"
-    using filterlim_at_top_dense t(1) by auto
-  then have ev_eq: "\<forall>\<^sub>F n in sequentially. flow0 ((flow0 x o t) n) ((\<tau> o (flow0 x o t)) n) = (flow0 x o s) n"
-    using ev_in_ball
-    apply (eventually_elim)
-    apply (drule \<tau>(2))
-    unfolding o_def
-    apply (subst flow_trans[symmetric])
-    using pos_ex
-      apply (auto simp: s_def)
-    apply (rule existence_ivl_trans')
-    by auto
-  then
-  have "\<forall>\<^sub>F n in sequentially.
-  (flow0 x o s) n = flow0 ((flow0 x o t) n) ((\<tau> o (flow0 x o t)) n)"
-    by (simp add: eventually_mono)
-  from \<open>(flow0 x \<circ> t) \<longlonglongrightarrow> p\<close> and \<open>\<tau> \<midarrow>p\<rightarrow> \<tau> p\<close>
-  have
-    "(\<lambda>n. flow0 ((flow0 x \<circ> t) n) ((\<tau> \<circ> (flow0 x \<circ> t)) n))
-  \<longlonglongrightarrow>
-  flow0 p (\<tau> p)"
-    using \<open>\<tau> p = 0\<close> \<tau>_cont \<open>p \<in> X\<close>
-    by (intro tendsto_eq_intros) auto
-  then have "(flow0 x o s) \<longlonglongrightarrow> flow0 p (\<tau> p)"
-    using ev_eq by (rule Lim_transform_eventually)
-  then have "(flow0 x o s) \<longlonglongrightarrow> p"
-    using \<open>p \<in> X\<close> \<open>\<tau> p = 0\<close>
-    by simp
-  moreover
-  {
-    have "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b}"
-      using ev_eq ev_in_ball
-      apply eventually_elim
-      apply (drule sym)
-      apply simp
-      apply (rule \<tau>) by simp
-    moreover have "\<forall>\<^sub>F n in sequentially. s n \<in> existence_ivl0 x"
-      using ev_in_ball ev1
-      apply (eventually_elim)
-      apply (drule \<tau>(2))
-      using pos_ex
-      by (auto simp: s_def)
-    ultimately have "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
-      by eventually_elim auto
-  }
-  ultimately show ?thesis ..
-qed
-
-(* Obvious but frequently used step *)
-lemma filterlim_at_top_tendstoE:
-  assumes "e > 0"
-  assumes "filterlim s at_top sequentially"
-  assumes "(flow0 x \<circ> s) \<longlonglongrightarrow> u"
-  assumes "\<forall>\<^sub>F n in sequentially. P (s n)"
-  obtains m where "m > b" "P m" "dist (flow0 x m) u < e"
-proof -
-  from assms(2) have "\<forall>\<^sub>F n in sequentially. b < s n"
-    by (simp add: filterlim_at_top_dense)
-  moreover have "\<forall>\<^sub>F n in sequentially. norm ((flow0 x \<circ> s) n - u) < e"
-    using assms(3)[THEN tendstoD, OF assms(1)] by (simp add: dist_norm)
-  moreover note assms(4)
-  ultimately have "\<forall>\<^sub>F n in sequentially. b < s n \<and> norm ((flow0 x \<circ> s) n - u) < e \<and> P (s n)"
-    by eventually_elim auto
-  then obtain m where "m > b" "P m" "dist (flow0 x m) u < e"
-    by (auto simp add: eventually_sequentially dist_norm)
-  then show ?thesis ..
-qed
-
-lemma open_segment_separate_left:
-  fixes u v x a b::'a
-  assumes u:"u \<in> {a <--< b}"
-  assumes v:"v \<in> {u <--< b}"
-  assumes x: "dist x u < dist u v" "x \<in> {a <--< b}"
-  shows "x \<in> {a <--< v}"
-proof -
-  have "v \<noteq> x"
-    by (smt dist_commute x(1)) 
-  moreover have "x \<notin> {v<--<b}"
-    by (smt dist_commute dist_in_open_segment open_segment_subsegment v x(1))
-  moreover have "v \<in> {a<--<b}" using v
-    by (metis ends_in_segment(1) segment_open_subset_closed subset_eq subset_segment(4) u)
-  ultimately show ?thesis using open_segment_trichotomy[OF _ x(2)]
-    by blast
-qed
-
-lemma open_segment_separate_right:
-  fixes u v x a b::'a
-  assumes u:"u \<in> {a <--< b}"
-  assumes v:"v \<in> {a <--< u}"
-  assumes x: "dist x u < dist u v" "x \<in> {a <--< b}"
-  shows "x \<in> {v <--< b}"
-proof -
-  have "v \<noteq> x"
-    by (smt dist_commute x(1))
-  moreover have "x \<notin> {a<--<v}"
-    by (smt dist_commute dist_in_open_segment open_segment_commute open_segment_subsegment v x(1))
-  moreover have "v \<in> {a<--<b}" using v
-    by (metis ends_in_segment(1) segment_open_subset_closed subset_eq subset_segment(4) u)
-  ultimately show ?thesis using open_segment_trichotomy[OF _ x(2)]
-    by blast
-qed
-
-lemma no_two_\<omega>_limit_points:
-  assumes transversal: "transversal_segment a b"
-  assumes ex_pos: "{0..} \<subseteq> existence_ivl0 x"
-  assumes u: "\<omega>_limit_point x u" "u \<in> {a<--<b}"
-  assumes v: "\<omega>_limit_point x v" "v \<in> {a<--<b}"
-  assumes uv: "v \<in> {u<--<b}"
-  shows False
-proof -
-  have unotv: "u \<noteq> v" using uv
-    using dist_in_open_segment by blast 
-  define duv where "duv = dist u v / 2"
-  have duv: "duv > 0" unfolding duv_def using unotv by simp
-  from \<omega>_limit_crossings[OF transversal ex_pos u]
-  obtain su where su: "filterlim su at_top sequentially"
-    "(flow0 x \<circ> su) \<longlonglongrightarrow> u"
-    "\<forall>\<^sub>F n in sequentially. flow0 x (su n) \<in> {a<--<b} \<and> su n \<in> existence_ivl0 x" by blast
-  from \<omega>_limit_crossings[OF transversal ex_pos v]
-  obtain sv where sv: "filterlim sv at_top sequentially"
-    "(flow0 x \<circ> sv) \<longlonglongrightarrow> v"
-    "\<forall>\<^sub>F n in sequentially. flow0 x (sv n) \<in> {a<--<b} \<and> sv n \<in> existence_ivl0 x" by blast
-  from filterlim_at_top_tendstoE[OF duv su]
-  obtain su1 where su1:"su1 > 0" "flow0 x su1 \<in> {a<--<b}"
-    "su1 \<in> existence_ivl0 x" "dist (flow0 x su1) u < duv" by auto
-  from filterlim_at_top_tendstoE[OF duv sv, of su1]
-  obtain su2 where su2:"su2 > su1" "flow0 x su2 \<in> {a<--<b}"
-    "su2 \<in> existence_ivl0 x" "dist (flow0 x su2) v < duv" by auto
-  from filterlim_at_top_tendstoE[OF duv su, of su2]
-  obtain su3 where su3:"su3 > su2" "flow0 x su3 \<in> {a<--<b}"
-    "su3 \<in> existence_ivl0 x" "dist (flow0 x su3) u < duv" by auto
-  have *: "su1 \<le> su2" "{su1..su2} \<subseteq> existence_ivl0 x" using su1 su2
-     apply linarith
-    by (metis atLeastatMost_empty_iff empty_iff mvar.closed_segment_subset_domain real_Icc_closed_segment su1(3) su2(3) subset_eq)
-
-(* by construction *)
-  have d1: "dist (flow0 x su1) v \<ge> (dist u v)/2" using su1(4) duv unfolding duv_def
-    by (smt dist_triangle_half_r)
-  have "dist (flow0 x su1) u < dist u v" using su1(4) duv unfolding duv_def by linarith
-  from open_segment_separate_left[OF u(2) uv this su1(2)]
-  have su1l:"flow0 x su1 \<in> {a<--<v}" .
-  have "dist (flow0 x su2) v < dist v (flow0 x su1)" using d1
-    by (smt dist_commute duv_def su2(4))
-  from open_segment_separate_right[OF v(2) su1l this su2(2)]
-  have su2l:"flow0 x su2 \<in> {flow0 x su1<--<b}" .
-  then have su2ll:"flow0 x su2 \<in> {u<--<b}"
-    by (smt dist_commute dist_pos_lt duv_def open_segment_subsegment pos_half_less open_segment_separate_right su2(2) su2(4) u(2) uv v(2) unotv)
-
-  have "dist (flow0 x su2) u \<ge> (dist u v)/2" using su2(4) duv unfolding duv_def
-    by (smt dist_triangle_half_r)
-  then have "dist (flow0 x su3) u < dist u (flow0 x su2)"
-    by (smt dist_commute duv_def su3(4)) 
-  from open_segment_separate_left[OF u(2) su2ll this su3(2)]
-  have su3l:"flow0 x su3 \<in> {a<--<flow0 x su2}" .
-
-  from flow0_transversal_segment_monotone[OF transversal * su1(2) su2l su3(1) su3(3)]
-  have "flow0 x su3 \<notin> {a <--<flow0 x su2}" .
-  thus False using su3l by auto
-qed
-
-
-subsection \<open>Unique Intersection\<close>
-
-text \<open>Perko Section 3.7 Remark 2\<close>
-lemma unique_transversal_segment_intersection:
-  assumes "transversal_segment a b"
-  assumes "{0..} \<subseteq> existence_ivl0 x"
-  assumes "u \<in> \<omega>_limit_set x \<inter> {a<--<b}"
-  shows "\<omega>_limit_set x \<inter> {a<--<b} = {u}"
-proof (rule ccontr)
-  assume "\<omega>_limit_set x \<inter> {a<--<b} \<noteq> {u}"
-  then
-  obtain v where uv: "u \<noteq> v"
-    and v: "\<omega>_limit_point x v" "v \<in> {a<--<b}" using assms unfolding \<omega>_limit_set_def
-    by fastforce
-  have u:"\<omega>_limit_point x u" "u \<in> {a<--<b}" using assms unfolding \<omega>_limit_set_def
-    by auto
-  show False using no_two_\<omega>_limit_points[OF \<open>transversal_segment a b\<close>]
-    by (smt dist_commute dist_in_open_segment open_segment_trichotomy u uv v assms)
-qed
-
-text \<open>Adapted from Perko Section 3.7 Lemma 4 (+ Chicone )\<close>
-lemma periodic_imp_\<omega>_limit_set:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes "periodic_orbit y"
-    "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x"
-  shows "flow0 y `UNIV = \<omega>_limit_set x"
-proof (rule ccontr)
-  note y = \<open>periodic_orbit y\<close> \<open>flow0 y ` UNIV \<subseteq> \<omega>_limit_set x\<close>
-  from trapped_sol_right[OF assms(1-4)] 
-  have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
-  assume "flow0 y `UNIV \<noteq> \<omega>_limit_set x"
-  obtain p where p: "p \<in> \<omega>_limit_set x" "p \<notin> flow0 y ` UNIV"
-    using y(2) apply auto
-    using \<open>range (flow0 y) \<noteq> \<omega>_limit_set x\<close> by blast
-  from \<omega>_limit_set_in_compact_connected[OF assms(1-4)] have
-    wcon: "connected (\<omega>_limit_set x)" .
-  from \<omega>_limit_set_invariant have
-    "invariant (\<omega>_limit_set x)" .
-  from \<omega>_limit_set_in_compact_compact[OF assms(1-4)] have
-    "compact (\<omega>_limit_set x)" .
-  then have sc: "seq_compact (\<omega>_limit_set x)"
-    using compact_imp_seq_compact by blast
-  have y1:"closed (flow0 y ` UNIV)"
-    using closed_orbit_\<omega>_limit_set periodic_orbit_def \<omega>_limit_set_closed y(1) by auto
-  have y2: "flow0 y ` UNIV \<noteq> {}" by simp 
-  let ?py = "infdist p (range (flow0 y))"
-  have "0 < ?py"
-    using y1 y2 p(2)
-    by (rule infdist_pos_not_in_closed)
-  have "\<forall>n::nat. \<exists>z. z \<in> \<omega>_limit_set x - flow0 y ` UNIV \<and>
-    infdist z (flow0 y ` UNIV) < ?py/2^n"
-  proof (rule ccontr)
-    assume " \<not> (\<forall>n. \<exists>z. z \<in> \<omega>_limit_set x - range (flow0 y) \<and>
-                infdist z (range (flow0 y))
-                < infdist p (range (flow0 y)) / 2 ^ n) "
-    then obtain n where n: "(\<forall>z \<in> \<omega>_limit_set x - range (flow0 y).
-      infdist z (range (flow0 y)) \<ge> ?py / 2 ^ n)"
-      using not_less by blast
-    define A where "A = flow0 y ` UNIV"
-    define B where "B = {z. infdist z (range (flow0 y)) \<ge> ?py / 2 ^ n}"
-    have Ac:"closed A" unfolding A_def using y1 by auto
-    have Bc:"closed B" unfolding B_def using infdist_closed by auto
-    have "A \<inter> B = {}"
-    proof (rule ccontr)
-      assume "A \<inter> B \<noteq> {}"
-      then obtain q where q: "q \<in> A" "q \<in> B" by blast
-      have qz:"infdist q (range(flow0 y)) = 0" using q(1) unfolding A_def
-        by simp
-      note \<open>0 < ?py\<close>
-      moreover have "2 ^ n > (0::real)" by auto
-      ultimately have "infdist p (range (flow0 y)) / 2 ^ n > (0::real)"
-        by simp
-      then have qnz: "infdist q(range (flow0 y)) > 0" using q(2) unfolding B_def
-        by auto
-      show False using qz qnz by auto
-    qed
-    then have a1:"A \<inter> B \<inter> \<omega>_limit_set x = {}" by auto
-    have "\<omega>_limit_set x - range(flow0 y) \<subseteq> B" using n B_def by blast
-    then have a2:"\<omega>_limit_set x \<subseteq> A \<union> B" using A_def by auto
-    from connected_closedD[OF wcon a1 a2 Ac Bc]
-    have "A \<inter> \<omega>_limit_set x = {} \<or> B \<inter> \<omega>_limit_set x = {}" .
-    moreover {
-      assume "A \<inter> \<omega>_limit_set x = {}"
-      then have False unfolding A_def using y(2) by blast
-    }
-    moreover {
-      assume "B \<inter> \<omega>_limit_set x = {}"
-      then have False unfolding B_def using p
-        using A_def B_def a2 by blast
-    }
-    ultimately show False by blast
-  qed
-  then obtain s where s: "\<forall>n::nat. (s::nat \<Rightarrow> _) n \<in> \<omega>_limit_set x - flow0 y ` UNIV \<and>
-                      infdist (s n) (flow0 y ` UNIV) < ?py/2^n"
-    by metis
-  then have "\<forall>n. s n \<in> \<omega>_limit_set x" by blast
-  from seq_compactE[OF sc this]
-  obtain l r where lr: "l \<in> \<omega>_limit_set x" "strict_mono r" "(s \<circ> r) \<longlonglongrightarrow> l" by blast
-  have "\<And>n. infdist (s n) (range (flow0 y)) \<le> ?py / 2 ^ n" using s
-    using less_eq_real_def by blast
-  then have "\<And>n. norm(infdist (s n) (range (flow0 y))) \<le> ?py / 2 ^ n"
-    by (auto simp add: infdist_nonneg)
-  from LIMSEQ_norm_0_pow[OF \<open>0 < ?py\<close> _ this]
-  have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> s) \<longlonglongrightarrow> 0"
-    by (auto simp add:o_def)
-  from LIMSEQ_subseq_LIMSEQ[OF this lr(2)]
-  have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> (s \<circ> r)) \<longlonglongrightarrow> 0" by (simp add: o_assoc)
-  moreover have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> (s \<circ> r)) \<longlonglongrightarrow> infdist l (flow0 y ` UNIV)"
-    by (auto intro!: tendsto_eq_intros tendsto_compose_at[OF lr(3)])
-  ultimately have "infdist l (flow0 y `UNIV) = 0" using LIMSEQ_unique by auto
-  then have lu: "l \<in> flow0 y ` UNIV" using in_closed_iff_infdist_zero[OF y1 y2] by auto
-  then have l1:"l \<in> X"
-    using closed_orbit_global_existence periodic_orbit_def y(1) by auto
-      (* TODO: factor out as periodic_orbitE *)
-  have l2:"f l \<noteq> 0"
-    by (smt \<open>l \<in> X\<close> \<open>l \<in> range (flow0 y)\<close> closed_orbit_global_existence fixed_point_imp_closed_orbit_period_zero(2) fixpoint_sol(2) image_iff local.flows_reverse periodic_orbit_def y(1))
-  from transversal_segment_exists[OF l1 l2]
-  obtain a b where ab: "transversal_segment a b" "l \<in> {a<--<b}" by blast
-  then have "l \<in> \<omega>_limit_set x \<inter> {a<--<b}" using lr by auto
-  from unique_transversal_segment_intersection[OF ab(1) ex_pos this]
-  have luniq: "\<omega>_limit_set x \<inter> {a<--<b} = {l} " .
-  from cross_time_continuous[OF ab, of 1]
-  obtain d t where dt: "0 < d"
-    "(\<And>y. y \<in> ball l d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b})"
-    "(\<And>y. y \<in> ball l d \<Longrightarrow> \<bar>t y\<bar> < 1)"
-    "continuous_on (ball l d) t" "t l = 0"
-    by auto
-  obtain n where "(s \<circ> r) n \<in> ball l d" using lr(3) dt(1) unfolding LIMSEQ_iff_nz
-    by (metis dist_commute mem_ball order_refl)
-  then have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> {a<--<b}" using dt by auto
-  moreover have sr: "(s \<circ> r) n \<in> \<omega>_limit_set x" "(s \<circ> r) n \<notin> flow0 y ` UNIV"
-    using s by auto
-  moreover have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> \<omega>_limit_set x"
-    using \<open>invariant (\<omega>_limit_set x)\<close> calculation unfolding invariant_def trapped_def
-    by (smt \<omega>_limit_set_in_compact_subset \<open>invariant (\<omega>_limit_set x)\<close> assms(1-4) invariant_def order_trans range_eqI subsetD trapped_iff_on_existence_ivl0 trapped_sol)
-  ultimately have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> \<omega>_limit_set x \<inter> {a<--<b}" by auto
-  from unique_transversal_segment_intersection[OF ab(1) ex_pos this]
-  have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) = l" using luniq by auto
-  then have "((s \<circ> r) n) = flow0 l (-(t ((s \<circ> r) n ))) "
-    by (smt UNIV_I \<open>(s \<circ> r) n \<in> \<omega>_limit_set x\<close> flows_reverse \<omega>_limit_set_in_compact_existence assms(1-4)) 
-  thus False using sr(2) lu
-      \<open>flow0 ((s \<circ> r) n) (t ((s \<circ> r) n)) = l\<close> \<open>flow0 ((s \<circ> r) n) (t ((s \<circ> r) n)) \<in> \<omega>_limit_set x\<close>
-      closed_orbit_global_existence image_iff local.flow_trans periodic_orbit_def \<omega>_limit_set_in_compact_existence range_eqI assms y(1)
-    by smt
-qed
-
-end context c1_on_open_R2 begin
-
-lemma \<alpha>_limit_crossings:
-  assumes "transversal_segment a b"
-  assumes pos_ex: "{..0} \<subseteq> existence_ivl0 x"
-  assumes "\<alpha>_limit_point x p"
-  assumes "p \<in> {a<--<b}"
-  obtains s where
-    "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>"
-    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>\<^sub>F n in sequentially.
-    flow0 x (s n) \<in> {a<--<b} \<and>
-    s n \<in> existence_ivl0 x"
-proof -
-  from pos_ex have "{0..} \<subseteq> uminus ` existence_ivl0 x" by force
-  from rev.\<omega>_limit_crossings[unfolded rev_transversal_segment rev_existence_ivl_eq0 rev_eq_flow
-      \<alpha>_limit_point_eq_rev[symmetric], OF assms(1) this assms(3,4)]
-  obtain s where "filterlim s at_top sequentially" "((\<lambda>t. flow0 x (- t)) \<circ> s) \<longlonglongrightarrow> p"
-    "\<forall>\<^sub>F n in sequentially. flow0 x (- s n) \<in> {a<--<b} \<and> s n \<in> uminus ` existence_ivl0 x" .
-  then have "filterlim (-s) at_bot sequentially"
-    "(flow0 x \<circ> (-s)) \<longlonglongrightarrow> p"
-    "\<forall>\<^sub>F n in sequentially. flow0 x ((-s) n) \<in> {a<--<b} \<and> (-s) n \<in> existence_ivl0 x"
-    by (auto simp: fun_Compl_def o_def filterlim_uminus_at_top)
-  then show ?thesis ..
-qed
-
-text \<open>If a positive limit point has a regular point in its positive limit set then it is periodic\<close>
-lemma \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes y: "y \<in> \<omega>_limit_set x" "f y \<noteq> 0"
-  assumes z: "z \<in> \<omega>_limit_set y \<union> \<alpha>_limit_set y" "f z \<noteq> 0"
-  shows "periodic_orbit y \<and> flow0 y ` UNIV = \<omega>_limit_set x"
-proof -
-  from trapped_sol_right[OF assms(1-4)] have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
-  from \<omega>_limit_set_in_compact_existence[OF assms(1-4) y(1)]
-  have yex: "existence_ivl0 y = UNIV" .
-  from \<omega>_limit_set_invariant
-  have "invariant (\<omega>_limit_set x)" .
-  then have yinv: "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x" using yex unfolding invariant_def
-    using trapped_iff_on_existence_ivl0 y(1) by blast 
-
-  have zy: "\<omega>_limit_point y z \<or> \<alpha>_limit_point y z"
-    using z unfolding \<omega>_limit_set_def \<alpha>_limit_set_def by auto
-
-  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
-    \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained[OF assms(1-4)]
-  have zx:"z \<in> \<omega>_limit_set x" using zy y
-    using z(1) by blast
-  then have "z \<in> X"
-    by (metis UNIV_I local.existence_ivl_initial_time_iff \<omega>_limit_set_in_compact_existence assms(1-4))
-  from transversal_segment_exists[OF this z(2)]
-  obtain a b where ab: "transversal_segment a b" "z \<in> {a<--<b}" by blast
-
-  from zy
-  obtain t1 t2 where t1: "flow0 y t1 \<in> {a<--<b}" and t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
-  proof
-    assume zy: "\<omega>_limit_point y z"
-    from \<omega>_limit_crossings[OF ab(1) _ zy ab(2), unfolded yex]
-    obtain s where s: "filterlim s at_top sequentially"
-      "(flow0 y \<circ> s) \<longlonglongrightarrow> z"
-      "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b}"
-      by auto
-    from eventually_happens[OF this(3)] obtain t1 where t1: "flow0 y t1 \<in> {a<--<b}" by auto
-    have "\<forall>\<^sub>F n in sequentially. s n > t1"
-      using filterlim_at_top_dense s(1) by auto
-    with s(3) have "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b} \<and> s n > t1"
-      by eventually_elim simp
-    from eventually_happens[OF this] obtain t2 where t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
-      by auto
-    from t1 this show ?thesis ..
-  next
-    assume zy: "\<alpha>_limit_point y z"
-    from \<alpha>_limit_crossings[OF ab(1) _ zy ab(2), unfolded yex]
-    obtain s where s: "filterlim s at_bot sequentially"
-      "(flow0 y \<circ> s) \<longlonglongrightarrow> z"
-      "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b}"
-      by auto
-    from eventually_happens[OF this(3)] obtain t1 where t1: "flow0 y t1 \<in> {a<--<b}" by auto
-    have "\<forall>\<^sub>F n in sequentially. s n < t1"
-      using filterlim_at_bot_dense s(1) by auto
-    with s(3) have "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b} \<and> s n < t1"
-      by eventually_elim simp
-    from eventually_happens[OF this] obtain t2 where t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
-      by auto
-    from t1 this show ?thesis ..
-  qed
-  have "flow0 y t1 \<in> \<omega>_limit_set x \<inter> {a<--<b}" using t1 UNIV_I yinv by auto
-  moreover have "flow0 y t2 \<in> \<omega>_limit_set x \<inter> {a<--<b}" using t2 UNIV_I yinv by auto
-  ultimately have feq:"flow0 y t1 = flow0 y t2"
-    using unique_transversal_segment_intersection[OF \<open>transversal_segment a b\<close> ex_pos]
-    by blast
-  have "t1 \<noteq> t2" "t1 \<in> existence_ivl0 y" "t2 \<in> existence_ivl0 y" using \<open>t1 \<noteq>  t2\<close>
-      apply blast
-     apply (simp add: yex)
-    by (simp add: yex)
-  from periodic_orbitI[OF this feq y(2)]
-  have 1: "periodic_orbit y" .
-  from periodic_imp_\<omega>_limit_set[OF assms(1-4) this yinv]
-  have 2: "flow0 y` UNIV = \<omega>_limit_set x" .
-  show ?thesis using 1 2 by auto
-qed
-
-subsection \<open>Poincare Bendixson Theorems\<close>
-text \<open>Perko Section 3.7 Theorem 1\<close>
-theorem poincare_bendixson:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes "0 \<notin> f ` (\<omega>_limit_set x)"
-  obtains y where "periodic_orbit y"
-    "flow0 y ` UNIV = \<omega>_limit_set x"
-proof -
-  note f = \<open>0 \<notin> f ` (\<omega>_limit_set x)\<close>
-  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-4)]
-  obtain y where y: "y \<in> \<omega>_limit_set x" by fastforce
-  from \<omega>_limit_set_in_compact_existence[OF  assms(1-4) y]
-  have yex: "existence_ivl0 y = UNIV" .
-  from \<omega>_limit_set_invariant
-  have "invariant (\<omega>_limit_set x)" .
-  then have yinv: "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x" using yex unfolding invariant_def
-    using trapped_iff_on_existence_ivl0 y by blast
-  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
-  have "\<omega>_limit_set x \<subseteq> K" .
-  then have "flow0 y ` UNIV \<subseteq> K" using yinv by auto
-  then have yk:"trapped_forward y K"
-    by (simp add: image_subsetI range_subsetD trapped_forward_def)
-  have "y \<in> X"
-    by (simp add: local.mem_existence_ivl_iv_defined(2) yex)
-
-  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-2) this _]
-  obtain z where z: "z \<in> \<omega>_limit_set y" using yk by blast
-  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
-  have zx:"z \<in> \<omega>_limit_set x" using \<open>z \<in> \<omega>_limit_set y\<close> y by auto
-
-  have yreg : "f y \<noteq> 0" using f y
-    by (metis rev_image_eqI)
-  have zreg : "f z \<noteq> 0" using f zx
-    by (metis rev_image_eqI) 
-  from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) y yreg _ zreg] z
-  show ?thesis using that by blast
-qed
-
-lemma fixed_point_in_\<omega>_limit_set_imp_\<omega>_limit_set_singleton_fixed_point:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes fp: "yfp \<in> \<omega>_limit_set x" "f yfp = 0"
-  assumes zpx: "z \<in> \<omega>_limit_set x"
-  assumes finite_fp: "finite {y \<in> K. f y = 0}" (is "finite ?S")
-  shows "(\<exists>p1 \<in> \<omega>_limit_set x. f p1 = 0 \<and> \<omega>_limit_set z = {p1}) \<and>
-    (\<exists>p2 \<in> \<omega>_limit_set x. f p2 = 0 \<and> \<alpha>_limit_set z = {p2})"
-proof -
-  let ?weq = "{y \<in> \<omega>_limit_set x. f y = 0}"
-  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
-  have wxK: "\<omega>_limit_set x \<subseteq> K" .
-  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
-  have zx: "\<omega>_limit_set z \<subseteq> \<omega>_limit_set x" using zpx by auto
-  have zX: "z \<in> X" using subset_trans[OF wxK assms(2)]
-    by (metis subset_iff zpx)
-  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
-  have "?weq \<subseteq> ?S"
-    by (smt Collect_mono_iff Int_iff inf.absorb_iff1)
-  then have "finite ?weq" using \<open>finite ?S\<close>
-    by (blast intro: rev_finite_subset) 
-
-  consider "f z = 0" | "f z \<noteq> 0" by auto
-  then show ?thesis
-  proof cases
-    assume "f z = 0"
-    from fixed_point_imp_\<omega>_limit_set[OF zX this]
-      fixed_point_imp_\<alpha>_limit_set[OF zX this]
-    show ?thesis
-      by (metis (mono_tags) \<open>f z = 0\<close> zpx)
-  next
-    assume "f z \<noteq> 0"
-    have zweq: "\<omega>_limit_set z \<subseteq> ?weq"
-      apply clarsimp
-    proof (rule ccontr)
-      fix k assume k: "k \<in> \<omega>_limit_set z" "\<not> (k \<in> \<omega>_limit_set x \<and> f k = 0)"
-      then have "f k \<noteq> 0" using zx k by auto
-      from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) zpx \<open>f z \<noteq> 0\<close> _ this] k(1)
-      have "periodic_orbit z" "range(flow0 z) = \<omega>_limit_set x" by auto
-      then have "0 \<notin> f ` (\<omega>_limit_set x)"
-        by (metis image_iff periodic_orbit_imp_flow0_regular)
-      thus False using fp
-        by (metis (mono_tags, lifting) empty_Collect_eq image_eqI)
-    qed
-    have zweq0: "\<alpha>_limit_set z \<subseteq> ?weq"
-      apply clarsimp
-    proof (rule ccontr)
-      fix k assume k: "k \<in> \<alpha>_limit_set z" "\<not> (k \<in> \<omega>_limit_set x \<and> f k = 0)"
-      then have "f k \<noteq> 0" using zx k
-          \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained[OF assms(1-4), of z] zpx
-        by auto
-      from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) zpx \<open>f z \<noteq> 0\<close> _ this] k(1)
-      have "periodic_orbit z" "range(flow0 z) = \<omega>_limit_set x" by auto
-      then have "0 \<notin> f ` (\<omega>_limit_set x)"
-        by (metis image_iff periodic_orbit_imp_flow0_regular)
-      thus False using fp
-        by (metis (mono_tags, lifting) empty_Collect_eq image_eqI)
-    qed
-    from \<omega>_limit_set_in_compact_existence[OF assms(1-4) zpx]
-    have zex: "existence_ivl0 z = UNIV" .
-    from \<omega>_limit_set_invariant
-    have "invariant (\<omega>_limit_set x)" .
-    then have zinv: "flow0 z ` UNIV \<subseteq> \<omega>_limit_set x" using zex unfolding invariant_def
-      using trapped_iff_on_existence_ivl0 zpx by blast
-    then have "flow0 z ` UNIV \<subseteq> K" using wxK by auto
-    then have a2: "trapped_forward z K" "trapped_backward z K"
-      using trapped_def trapped_iff_on_existence_ivl0 apply fastforce
-      using \<open>range (flow0 z) \<subseteq> K\<close> trapped_def trapped_iff_on_existence_ivl0 by blast
-    have a3: "finite (\<omega>_limit_set z)"
-      by (metis \<open>finite ?weq\<close> finite_subset zweq)
-    from finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-2) zX a2(1) a3]
-    obtain p1 where p1: "\<omega>_limit_set z = {p1}" "f p1 = 0" by blast
-    then have "p1 \<in> ?weq" using zweq by blast
-    moreover
-    have "finite (\<alpha>_limit_set z)"
-      by (metis \<open>finite ?weq\<close> finite_subset zweq0)
-    from finite_\<alpha>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-2) zX a2(2) this]
-    obtain p2 where p2: "\<alpha>_limit_set z = {p2}" "f p2 = 0" by blast
-    then have "p2 \<in> ?weq" using zweq0 by blast
-    ultimately show ?thesis
-      by (simp add: p1 p2)
-  qed
-qed
-
-end context c1_on_open_R2 begin
-
-text \<open>Perko Section 3.7 Theorem 2\<close>
-theorem poincare_bendixson_general:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> X" "trapped_forward x K"
-  assumes "S = {y \<in> K. f y = 0}" "finite S"
-  shows
-    "(\<exists>y \<in> S. \<omega>_limit_set x = {y}) \<or>
-  (\<exists>y. periodic_orbit y \<and>
-    flow0 y ` UNIV = \<omega>_limit_set x) \<or>
-  (\<exists>P R. \<omega>_limit_set x = P \<union> R \<and>
-    P \<subseteq> S \<and> 0 \<notin> f ` R \<and> R \<noteq> {} \<and>
-    (\<forall>z \<in> R.
-      (\<exists>p1 \<in> P. \<omega>_limit_set z = {p1}) \<and>
-      (\<exists>p2 \<in> P. \<alpha>_limit_set z = {p2})))"
-proof -
-  note S = \<open>S = {y \<in> K. f y = 0}\<close>
-  let ?wreg = "{y \<in> \<omega>_limit_set x. f y \<noteq> 0}"
-  let ?weq = "{y \<in> \<omega>_limit_set x. f y = 0}"
-  have wreqweq: "?wreg \<union> ?weq = \<omega>_limit_set x"
-    by (smt Collect_cong Collect_disj_eq mem_Collect_eq \<omega>_limit_set_def)
-
-  from trapped_sol_right[OF assms(1-4)] have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
-  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
-  have wxK: "\<omega>_limit_set x \<subseteq> K" .
-  then have "?weq \<subseteq> S" using S
-    by (smt Collect_mono_iff Int_iff inf.absorb_iff1)
-  then have "finite ?weq" using \<open>finite S\<close>
-    by (metis rev_finite_subset) 
-  from \<omega>_limit_set_invariant
-  have xinv: "invariant (\<omega>_limit_set x)" .
-
-  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-4)] wreqweq
-  consider "?wreg = {}" |
-    "?weq = {}" |
-    "?weq \<noteq> {}" "?wreg \<noteq> {}" by auto
-  then show ?thesis
-  proof cases
-    (* If w has no regular points then it is equal to a single unique fixed point *)
-    assume "?wreg = {}"
-    then have "finite (\<omega>_limit_set x)"
-      by (metis (mono_tags, lifting) \<open>{y \<in> \<omega>_limit_set x. f y = 0} \<subseteq> S\<close> \<open>finite S\<close> rev_finite_subset sup_bot.left_neutral wreqweq)
-    from finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-4) this]
-    obtain y where y: "\<omega>_limit_set x = {y}" "f y = 0" by blast
-    then have "y \<in> S"
-      by (metis Un_empty_left \<open>?weq \<subseteq> S\<close> \<open>?wreg = {}\<close> insert_subset wreqweq)
-    then show ?thesis using y by auto
-  next
-    (* If w has no fixed points, then the Poincare Bendixson theorem applies *)
-    assume "?weq = {}"
-    then have " 0 \<notin> f ` \<omega>_limit_set x"
-      by (smt empty_Collect_eq imageE)
-    from poincare_bendixson[OF assms(1-4) this]
-    have "(\<exists>y. periodic_orbit y \<and> flow0 y ` UNIV = \<omega>_limit_set x)"
-      by metis
-    then show ?thesis by blast
-  next
-    (* Otherwise, all points in the limit set converge to a finite subset of the fixed points *)
-    assume "?weq \<noteq> {}" "?wreg \<noteq> {}"
-    then obtain yfp where yfp: "yfp \<in> \<omega>_limit_set x" "f yfp = 0" by auto
-    have "0 \<notin> f ` ?wreg" by auto
-    have "(\<exists>p1\<in>\<omega>_limit_set x. f p1 = 0 \<and> \<omega>_limit_set z = {p1}) \<and>
-      (\<exists>p2\<in>\<omega>_limit_set x. f p2 = 0 \<and> \<alpha>_limit_set z = {p2})"
-      if zpx: "z \<in> \<omega>_limit_set x" for z
-      using fixed_point_in_\<omega>_limit_set_imp_\<omega>_limit_set_singleton_fixed_point[
-          OF assms(1-4) yfp zpx \<open>finite S\<close>[unfolded S]] by auto
-    then have "\<omega>_limit_set x = ?weq \<union> ?wreg \<and>
-        ?weq \<subseteq> S \<and> 0 \<notin> f ` ?wreg \<and> ?wreg \<noteq> {} \<and>
-        (\<forall>z \<in> ?wreg.
-         (\<exists>p1 \<in> ?weq. \<omega>_limit_set z = {p1}) \<and>
-         (\<exists>p2 \<in> ?weq. \<alpha>_limit_set z = {p2}))"
-      using wreqweq \<open>?weq \<subseteq> S\<close> \<open>?wreg \<noteq> {}\<close> \<open>0 \<notin> f ` ?wreg\<close>
-      by blast
-    then show ?thesis by blast
-  qed
-qed
-
-corollary poincare_bendixson_applied:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "K \<noteq> {}" "positively_invariant K"
-  assumes "0 \<notin> f ` K"
-  obtains y where "periodic_orbit y" "flow0 y ` UNIV \<subseteq> K"
-proof -
-  from assms(1-4) obtain x where "x \<in> K" "x \<in> X" by auto
-  have *: "trapped_forward x K"
-    using assms(4) \<open>x \<in> K\<close>
-    by (auto simp: positively_invariant_def)
-  have subs: "\<omega>_limit_set x \<subseteq> K"
-    by (rule \<omega>_limit_set_in_compact_subset[OF assms(1-2) \<open>x \<in> X\<close> *])
-  with assms(5) have "0 \<notin> f ` \<omega>_limit_set x" by auto
-  from poincare_bendixson[OF assms(1-2) \<open>x \<in> X\<close> * this]
-  obtain y where "periodic_orbit y" "range (flow0 y) = \<omega>_limit_set x"
-    by force
-  then have "periodic_orbit y" "flow0 y ` UNIV \<subseteq> K" using subs by auto
-  then show ?thesis ..
-qed
-
-(*
-  Limit cycles are periodic orbits that is the \<omega> (or \<alpha>)-limit set of some point not in the cycle.
-  As with periodic_orbit, limit_cycles are defined for a representative point y
-*)
-definition "limit_cycle y \<longleftrightarrow>
-  periodic_orbit y \<and>
-  (\<exists>x. x \<notin> flow0 y ` UNIV \<and>
-  (flow0 y ` UNIV = \<omega>_limit_set x \<or> flow0 y ` UNIV = \<alpha>_limit_set x))"
-
-corollary poincare_bendixson_limit_cycle:
-  assumes "compact K" "K \<subseteq> X"
-  assumes "x \<in> K" "positively_invariant K"
-  assumes "0 \<notin> f ` K"
-  assumes "rev.flow0 x t \<notin> K"
-  obtains y where "limit_cycle y" "flow0 y ` UNIV \<subseteq> K"
-proof -
-  have "x \<in> X" using assms(2-3) by blast
-  have *: "trapped_forward x K"
-    using assms(3-4)
-    by (auto simp: positively_invariant_def)
-  have subs: "\<omega>_limit_set x \<subseteq> K"
-    by (rule \<omega>_limit_set_in_compact_subset[OF assms(1-2) \<open>x \<in> X\<close> *])
-  with assms(5) have "0 \<notin> f ` \<omega>_limit_set x" by auto
-  from poincare_bendixson[OF assms(1-2) \<open>x \<in> X\<close> * this]
-  obtain y where y: "periodic_orbit y" "range (flow0 y) = \<omega>_limit_set x"
-    by force
-  then have c2: "flow0 y ` UNIV \<subseteq> K" using subs by auto
-  have exy: "existence_ivl0 y = UNIV"
-    using closed_orbit_global_existence periodic_orbit_def y(1) by blast
-  have "x \<notin> flow0 y ` UNIV"
-  proof clarsimp
-    fix tt
-    assume "x = flow0 y tt"
-    then have "rev.flow0 (flow0 y tt) t \<notin> K" using assms(6) by auto
-    moreover have "rev.flow0 (flow0 y tt) t \<in> flow0 y ` UNIV" using exy unfolding rev_eq_flow
-      using UNIV_I \<open>x = flow0 y tt\<close> closed_orbit_\<omega>_limit_set closed_orbit_flow0 periodic_orbit_def y by auto
-    ultimately show False using c2 by blast
-  qed
-  then have "limit_cycle y" "flow0 y ` UNIV \<subseteq> K" using y c2 unfolding limit_cycle_def by auto
-  then show ?thesis ..
-qed
-
-end
-
-end
+section \<open>Poincare Bendixson Theory\<close>
+theory Poincare_Bendixson
+  imports 
+    Ordinary_Differential_Equations.ODE_Analysis
+    Analysis_Misc ODE_Misc Periodic_Orbit
+begin
+
+
+subsection \<open>Flow to Path\<close>
+
+context auto_ll_on_open begin
+
+(* The path along the flow starting at time t to time t' *)
+definition "flow_to_path x t t' = flow0 x \<circ> linepath t t'"
+
+lemma pathstart_flow_to_path[simp]:
+  shows "pathstart (flow_to_path x t t') = flow0 x t"
+  unfolding flow_to_path_def
+  by (auto simp add: pathstart_compose)
+
+lemma pathfinish_flow_to_path[simp]:
+  shows "pathfinish (flow_to_path x t t') = flow0 x t'"
+  unfolding flow_to_path_def
+  by (auto simp add: pathfinish_compose)
+
+lemma flow_to_path_unfold:
+  shows "flow_to_path x t t' s = flow0 x ((1 - s) * t + s * t')"
+  unfolding flow_to_path_def o_def linepath_def by auto
+
+lemma subpath0_flow_to_path:
+  shows "(subpath 0 u (flow_to_path x t t')) = flow_to_path x t (t + u*(t'-t))"
+  unfolding flow_to_path_def subpath_image subpath0_linepath
+  by auto
+
+lemma path_image_flow_to_path[simp]:
+  assumes "t \<le> t'"
+  shows "path_image (flow_to_path x t t') = flow0 x ` {t..t'}"
+  unfolding flow_to_path_def path_image_compose path_image_linepath
+  using assms real_Icc_closed_segment by auto
+
+lemma flow_to_path_image0_right_open[simp]:
+  assumes "t < t'"
+  shows "flow_to_path x t t' ` {0..<1} = flow0 x `{t..<t'}"
+  unfolding flow_to_path_def image_comp[symmetric] linepath_image0_right_open_real[OF assms]
+  by auto
+
+lemma flow_to_path_path:
+  assumes "t \<le> t'"
+  assumes "{t..t'} \<subseteq> existence_ivl0 x"
+  shows "path (flow_to_path x t t')"
+proof -
+  have "x \<in> X"
+    using assms(1) assms(2) subset_empty by fastforce
+  have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow>  (1 - xa) * t + xa * t' \<le> t'"
+    by (simp add: assms(1) convex_bound_le)
+  moreover have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow> t \<le> (1 - xa) * t + xa * t'" using assms(1)
+    by (metis add.commute add_diff_cancel_left' diff_diff_eq2 diff_le_eq mult.commute mult.right_neutral mult_right_mono right_diff_distrib')
+  ultimately have "\<And>xa. 0 \<le> xa \<Longrightarrow> xa \<le> 1 \<Longrightarrow> (1 - xa) * t + xa * t' \<in> existence_ivl0 x"
+    using assms(2) by auto
+  thus ?thesis unfolding path_def flow_to_path_def linepath_def
+    by (auto intro!:continuous_intros simp add :\<open>x \<in> X\<close>)
+qed
+
+lemma flow_to_path_arc:
+  assumes "t \<le> t'"
+  assumes "{t..t'} \<subseteq> existence_ivl0 x"
+  assumes "\<forall>s \<in> {t<..<t'}. flow0 x s \<noteq> flow0 x t"
+  assumes "flow0 x t \<noteq> flow0 x t'"
+  shows "arc (flow_to_path x t t')"
+  unfolding arc_def
+proof safe
+  from flow_to_path_path[OF assms(1-2)]
+  show "path (flow_to_path x t t')" .
+next
+  show "inj_on (flow_to_path x t t') {0..1}"
+    unfolding flow_to_path_def
+    apply (rule comp_inj_on)
+     apply (metis assms(4) inj_on_linepath)
+    using assms path_image_linepath[of t t'] apply (auto intro!:flow0_inj_on)
+    using flow0_inj_on greaterThanLessThan_iff linepath_image_01 real_Icc_closed_segment by fastforce
+qed
+
+end
+
+locale c1_on_open_R2 = c1_on_open_euclidean f f' X for f::"'a::executable_euclidean_space \<Rightarrow> _" and f' and X +
+  assumes dim2: "DIM('a) = 2"
+begin
+
+
+subsection \<open>2D Line segments\<close>
+
+text \<open>Line segments are specified by two endpoints
+      The closed line segment from x to y is given by the set {x--y}
+      and {x<--<y} for the open segment\<close>
+
+text \<open> Rotates a vector clockwise 90 degrees \<close>
+definition "rot (v::'a) = (eucl_of_list [nth_eucl v 1, -nth_eucl v 0]::'a)"
+
+lemma exhaust2_nat: "(\<forall>i<(2::nat). P i) \<longleftrightarrow> P 0 \<and> P 1"
+  using less_2_cases by auto
+lemma sum2_nat: "(\<Sum>i<(2::nat). P i) = P 0 + P 1"
+  by (simp add: eval_nat_numeral)
+
+lemmas vec_simps =
+  eucl_eq_iff[where 'a='a] dim2 eucl_of_list_eucl_nth exhaust2_nat
+  plus_nth_eucl
+  minus_nth_eucl
+  uminus_nth_eucl
+  scaleR_nth_eucl
+  inner_nth_eucl
+  sum2_nat
+  algebra_simps
+
+lemma minus_expand:
+  shows "(x::'a)-y = (eucl_of_list [x$\<^sub>e0 - y$\<^sub>e0, x$\<^sub>e1 - y$\<^sub>e1])"
+  by (simp add:vec_simps)
+
+lemma dot_ortho[simp]: "x \<bullet> rot x = 0"
+  unfolding rot_def minus_expand
+  by (simp add: vec_simps)
+
+lemma nrm_dot:
+  shows "((x::'a)-y) \<bullet> (rot (x-y)) = 0"
+  unfolding rot_def minus_expand
+  by (simp add: vec_simps)
+
+lemma nrm_reverse: "a \<bullet> (rot (x-y)) = - a \<bullet> (rot (y-x))" for x y::'a
+  unfolding rot_def
+  by (simp add:vec_simps)
+
+lemma norm_rot: "norm (rot v) = norm v" for v::'a
+  unfolding rot_def
+  by (simp add:vec_simps norm_nth_eucl L2_set_def)
+
+lemma rot_rot[simp]:
+  shows "rot (rot v) = -v"
+  unfolding rot_def
+  by (simp add:vec_simps)
+
+lemma rot_scaleR[simp]:
+  shows "rot ( u *\<^sub>R v) =  u *\<^sub>R (rot v)"
+  unfolding rot_def
+  by (simp add:vec_simps)
+
+lemma rot_0[simp]: "rot 0 = 0"
+  using rot_scaleR[of 0] by simp
+
+lemma rot_eq_0_iff[simp]: "rot x = 0 \<longleftrightarrow> x = 0"
+  apply (auto simp: rot_def)
+   apply (metis One_nat_def norm_eq_zero norm_rot norm_zero rot_def)
+  using rot_0 rot_def by auto
+
+lemma in_segment_inner_rot:
+  "(x - a) \<bullet> rot (b - a) = 0"
+  if "x \<in> {a--b}"
+proof -
+  from that obtain u where x: "x = a + u *\<^sub>R (b - a)" "0 \<le> u" "u \<le> 1"
+    by (auto simp: in_segment algebra_simps)
+  show ?thesis
+    unfolding x
+    by (simp add: inner_add_left nrm_dot)
+qed
+
+lemma inner_rot_in_segment:
+  "x \<in> range (\<lambda>u. a + u *\<^sub>R (b - a))"
+  if "(x - a) \<bullet> rot (b - a) = 0" "a \<noteq> b"
+proof -
+  from that have
+    x0: "b $\<^sub>e 0 = a $\<^sub>e 0 \<Longrightarrow> x $\<^sub>e 0 =
+      (a $\<^sub>e 0 * b $\<^sub>e Suc 0 - b $\<^sub>e 0 * a $\<^sub>e Suc 0 + (b $\<^sub>e 0 - a $\<^sub>e 0) * x $\<^sub>e Suc 0) /
+      (b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0)"
+    and x1: "b $\<^sub>e 0 \<noteq> a $\<^sub>e 0 \<Longrightarrow> x $\<^sub>e Suc 0 =
+      ((b $\<^sub>e Suc 0 - a $\<^sub>e Suc 0) * x $\<^sub>e 0 - a $\<^sub>e 0 * b $\<^sub>e Suc 0 + b $\<^sub>e 0 * a $\<^sub>e Suc 0) / (b $\<^sub>e 0 - a $\<^sub>e 0)"
+    by (auto simp: rot_def vec_simps divide_simps)
+  define u where "u = (if b $\<^sub>e 0 - a $\<^sub>e 0 \<noteq> 0
+    then ((x $\<^sub>e 0 - a $\<^sub>e 0) / (b $\<^sub>e 0 -  a $\<^sub>e 0))
+    else ((x $\<^sub>e 1 - a $\<^sub>e 1) / (b $\<^sub>e 1 -  a $\<^sub>e 1)))
+    "
+  show ?thesis
+    apply (cases "b $\<^sub>e 0 - a $\<^sub>e 0 = 0")
+    subgoal
+      using that(2)
+      apply (auto intro!: image_eqI[where x="((x $\<^sub>e 1 - a $\<^sub>e 1) / (b $\<^sub>e 1 -  a $\<^sub>e 1))"]
+          simp: vec_simps x0 divide_simps algebra_simps)
+       apply (metis ab_semigroup_mult_class.mult_ac(1) mult.commute sum_sqs_eq)
+      by (metis mult.commute mult.left_commute sum_sqs_eq)
+    subgoal
+      apply (auto intro!: image_eqI[where x="((x $\<^sub>e 0 - a $\<^sub>e 0) / (b $\<^sub>e 0 -  a $\<^sub>e 0))"]
+          simp: vec_simps x1 divide_simps algebra_simps)
+       apply (metis ab_semigroup_mult_class.mult_ac(1) mult.commute sum_sqs_eq)
+      by (metis mult.commute mult.left_commute sum_sqs_eq)
+    done
+qed
+
+lemma in_open_segment_iff_rot:
+  "x \<in> {a<--<b} \<longleftrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) <..< b \<bullet> (b - a)}"
+  if "a \<noteq> b"
+  unfolding open_segment_line_hyperplanes[OF that]
+  by (auto simp: nrm_dot intro!: inner_rot_in_segment)
+
+lemma in_open_segment_rotD:
+  "x \<in> {a<--<b} \<Longrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) <..< b \<bullet> (b - a)}"
+  by (subst in_open_segment_iff_rot[symmetric]) auto
+
+lemma in_closed_segment_iff_rot:
+  "x \<in> {a--b} \<longleftrightarrow> (x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a\<bullet>(b - a) .. b \<bullet> (b - a)}"
+  if "a \<noteq> b"
+  unfolding closed_segment_line_hyperplanes[OF that] using that
+  by (auto simp: nrm_dot intro!: inner_rot_in_segment)
+
+lemma in_segment_inner_rot2:
+  "(x - y) \<bullet> rot (a - b) = 0"
+  if "x \<in> {a--b}" "y \<in> {a--b}"
+proof -
+  from that obtain u where x: "x = a + u *\<^sub>R (b - a)" "0 \<le> u" "u \<le> 1"
+    by (auto simp: in_segment algebra_simps)
+  from that obtain v where y: "y = a + v *\<^sub>R (b - a)" "0 \<le> v" "v \<le> 1"
+    by (auto simp: in_segment algebra_simps)
+  show ?thesis
+    unfolding x y
+    apply (auto simp: inner_add_left )
+    by (smt add_diff_cancel_left' in_segment_inner_rot inner_diff_left minus_diff_eq nrm_reverse that(1) that(2) x(1) y(1))
+qed
+
+lemma closed_segment_surface:
+  "a \<noteq> b \<Longrightarrow> {a--b} = { x \<in> {x. x \<bullet> (b - a) \<in> {a\<bullet>(b - a) .. b \<bullet> (b - a)}}. (x - a) \<bullet> rot (b - a) = 0}"
+  by (auto simp: in_closed_segment_iff_rot)
+
+lemma rot_diff_commute: "rot (b - a) = -rot(a - b)"
+  apply (auto simp: rot_def algebra_simps)
+  by (metis One_nat_def minus_diff_eq rot_def rot_rot)
+
+
+subsection \<open>Bijection Real-Complex for Jordan Curve Theorem\<close>
+
+definition "complex_of (x::'a) = x$\<^sub>e0 + \<i> * x$\<^sub>e1"
+
+definition "real_of (x::complex) = (eucl_of_list [Re x, Im x]::'a)"
+
+lemma complex_of_linear:
+  shows "linear complex_of"
+  unfolding complex_of_def
+  apply (auto intro!:linearI simp add: distrib_left plus_nth_eucl)
+  by (simp add: of_real_def scaleR_add_right scaleR_nth_eucl)
+
+lemma complex_of_bounded_linear:
+  shows "bounded_linear complex_of"
+  unfolding complex_of_def
+  apply (auto intro!:bounded_linearI' simp add: distrib_left plus_nth_eucl)
+  by (simp add: of_real_def scaleR_add_right scaleR_nth_eucl)
+
+lemma real_of_linear:
+  shows "linear real_of"
+  unfolding real_of_def
+  by (auto intro!:linearI simp add: vec_simps)
+
+lemma real_of_bounded_linear:
+  shows "bounded_linear real_of"
+  unfolding real_of_def
+  by (auto intro!:bounded_linearI' simp add: vec_simps)
+
+lemma complex_of_real_of:
+  "(complex_of \<circ> real_of) = id"
+  unfolding complex_of_def real_of_def
+  using complex_eq by (auto simp add:vec_simps)
+
+lemma real_of_complex_of:
+  "(real_of \<circ> complex_of) = id"
+  unfolding complex_of_def real_of_def
+  using complex_eq by (auto simp add:vec_simps)
+
+lemma complex_of_bij:
+  shows "bij (complex_of)"
+  using o_bij[OF real_of_complex_of complex_of_real_of] .
+
+lemma real_of_bij:
+  shows "bij (real_of)"
+  using o_bij[OF complex_of_real_of real_of_complex_of] .
+
+lemma real_of_inj:
+  shows "inj (real_of)"
+  using real_of_bij
+  using bij_betw_imp_inj_on by auto
+
+lemma Jordan_curve_R2:
+  fixes c :: "real \<Rightarrow> 'a"
+  assumes "simple_path c" "pathfinish c = pathstart c"
+  obtains inside outside where
+    "inside \<noteq> {}" "open inside" "connected inside"
+    "outside \<noteq> {}" "open outside" "connected outside"
+    "bounded inside" "\<not> bounded outside"
+    "inside \<inter> outside = {}"
+    "inside \<union> outside = - path_image c"
+    "frontier inside = path_image c"
+    "frontier outside = path_image c"
+proof -
+  from simple_path_linear_image_eq[OF complex_of_linear]
+  have a1:"simple_path (complex_of \<circ> c)" using assms(1) complex_of_bij
+    using bij_betw_imp_inj_on by blast
+  have a2:"pathfinish (complex_of \<circ> c) = pathstart (complex_of \<circ> c)"
+    using assms(2) by (simp add:pathstart_compose pathfinish_compose)
+
+  from Jordan_curve[OF a1 a2]
+  obtain inside outside where io:
+    "inside \<noteq> {}" "open inside" "connected inside"
+    "outside \<noteq> {}" "open outside" "connected outside"
+    "bounded inside" "\<not> bounded outside" "inside \<inter> outside = {}"
+    "inside \<union> outside = - path_image (complex_of \<circ> c)"
+    "frontier inside = path_image (complex_of \<circ> c)"
+    "frontier outside = path_image (complex_of \<circ> c)" by blast
+  let ?rin = "real_of ` inside"
+  let ?rout = "real_of ` outside"
+  have i: "inside = complex_of ` ?rin" using complex_of_real_of unfolding image_comp
+    by auto
+  have o: "outside = complex_of ` ?rout" using complex_of_real_of unfolding image_comp
+    by auto
+  have c: "path_image(complex_of \<circ> c) = complex_of ` (path_image c)"
+    by (simp add: path_image_compose)
+  have "?rin \<noteq> {}" using io by auto
+  moreover from open_bijective_linear_image_eq[OF real_of_linear real_of_bij]
+  have "open ?rin" using io by auto
+  moreover from connected_linear_image[OF real_of_linear]
+  have "connected ?rin" using io by auto
+  moreover have "?rout \<noteq> {}" using io by auto
+  moreover from open_bijective_linear_image_eq[OF real_of_linear real_of_bij]
+  have "open ?rout" using io by auto
+  moreover from connected_linear_image[OF real_of_linear]
+  have "connected ?rout" using io by auto
+  moreover from bounded_linear_image[OF io(7) real_of_bounded_linear]
+  have "bounded ?rin" .
+  moreover from bounded_linear_image[OF _ complex_of_bounded_linear]
+  have "\<not> bounded ?rout" using io(8) o
+    by force
+  from image_Int[OF real_of_inj]
+  have "?rin \<inter> ?rout = {}" using io(9) by auto
+  moreover from bij_image_Compl_eq[OF complex_of_bij]
+  have "?rin \<union> ?rout = - path_image c" using io(10) unfolding c
+    by (metis id_apply image_Un image_comp image_cong image_ident real_of_complex_of)
+  moreover from closure_injective_linear_image[OF real_of_linear real_of_inj]
+  have "frontier ?rin = path_image c" using io(11)
+    unfolding frontier_closures c
+    by (metis \<open>\<And>B A. real_of ` (A \<inter> B) = real_of ` A \<inter> real_of ` B\<close> bij_image_Compl_eq c calculation(9) compl_sup double_compl io(10) real_of_bij)
+  moreover from closure_injective_linear_image[OF real_of_linear real_of_inj]
+  have "frontier ?rout = path_image c" using io(12)
+    unfolding frontier_closures c
+    by (metis \<open>\<And>B A. real_of ` (A \<inter> B) = real_of ` A \<inter> real_of ` B\<close> bij_image_Compl_eq c calculation(10) frontier_closures io(11) real_of_bij)
+  ultimately show ?thesis
+    by (meson \<open>\<not> bounded (real_of ` outside)\<close> that)
+qed
+
+(* copied *)
+corollary Jordan_inside_outside_R2:
+  fixes c :: "real \<Rightarrow> 'a"
+  assumes "simple_path c" "pathfinish c = pathstart c"
+  shows "inside(path_image c) \<noteq> {} \<and>
+          open(inside(path_image c)) \<and>
+          connected(inside(path_image c)) \<and>
+          outside(path_image c) \<noteq> {} \<and>
+          open(outside(path_image c)) \<and>
+          connected(outside(path_image c)) \<and>
+          bounded(inside(path_image c)) \<and>
+          \<not> bounded(outside(path_image c)) \<and>
+          inside(path_image c) \<inter> outside(path_image c) = {} \<and>
+          inside(path_image c) \<union> outside(path_image c) =
+          - path_image c \<and>
+          frontier(inside(path_image c)) = path_image c \<and>
+          frontier(outside(path_image c)) = path_image c"
+proof -
+  obtain inner outer
+    where *: "inner \<noteq> {}" "open inner" "connected inner"
+      "outer \<noteq> {}" "open outer" "connected outer"
+      "bounded inner" "\<not> bounded outer" "inner \<inter> outer = {}"
+      "inner \<union> outer = - path_image c"
+      "frontier inner = path_image c"
+      "frontier outer = path_image c"
+    using Jordan_curve_R2 [OF assms] by blast
+  then have inner: "inside(path_image c) = inner"
+    by (metis dual_order.antisym inside_subset interior_eq interior_inside_frontier)
+  have outer: "outside(path_image c) = outer"
+    using \<open>inner \<union> outer = - path_image c\<close> \<open>inside (path_image c) = inner\<close>
+      outside_inside \<open>inner \<inter> outer = {}\<close> by auto
+  show ?thesis
+    using * by (auto simp: inner outer)
+qed
+
+lemma jordan_points_inside_outside:
+  fixes p :: "real \<Rightarrow> 'a"
+  assumes "0 < e"
+  assumes jordan: "simple_path p" "pathfinish p = pathstart p"
+  assumes x: "x \<in> path_image p"
+  obtains y z where "y \<in> inside (path_image p)" "y \<in> ball x e" 
+    "z \<in> outside (path_image p)" "z \<in> ball x e"
+proof -
+  from Jordan_inside_outside_R2[OF jordan]
+  have xi: "x \<in> frontier(inside (path_image p))" and
+    xo: "x \<in> frontier(outside (path_image p))"
+    using x by auto
+  obtain y where y:"y \<in> inside (path_image p)" "y \<in> ball x e" using \<open>0 < e\<close> xi
+    unfolding frontier_straddle
+    by auto
+  obtain z where z:"z \<in> outside (path_image p)" "z \<in> ball x e" using \<open>0 < e\<close> xo
+    unfolding frontier_straddle
+    by auto
+  show ?thesis using y z that by blast
+qed  
+
+lemma eventually_at_open_segment:
+  assumes "x \<in> {a<--<b}"
+  shows "\<forall>\<^sub>F y in at x. (y-a) \<bullet> rot(a-b) = 0 \<longrightarrow> y \<in> {a <--< b}"
+proof -
+  from assms have "a \<noteq> b" by auto
+  from assms have x: "(x - a) \<bullet> rot (b - a) = 0" "x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
+    unfolding in_open_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
+    by auto
+  then have "\<forall>\<^sub>F y in at x. y \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
+    by (intro topological_tendstoD) (auto intro!: tendsto_intros)
+  then show ?thesis
+    by eventually_elim (auto simp: in_open_segment_iff_rot[OF \<open>a \<noteq> b\<close>] nrm_reverse[of _ a b] algebra_simps dist_commute)
+qed
+
+lemma linepath_ball:
+  assumes "x \<in> {a<--<b}"
+  obtains e where "e > 0" "ball x e \<inter> {y. (y-a) \<bullet> rot(a-b) = 0} \<subseteq> {a <--< b}"
+proof -
+  from eventually_at_open_segment[OF assms] assms
+  obtain e where "0 < e" "ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0} \<subseteq> {a<--<b}"
+    by (force simp: eventually_at in_open_segment_iff_rot dist_commute)
+  then show ?thesis ..
+qed
+
+lemma linepath_ball_inside_outside:
+  fixes p :: "real \<Rightarrow> 'a"
+  assumes jordan: "simple_path (p +++ linepath a b)" "pathfinish p = a" "pathstart p = b"
+  assumes x: "x \<in> {a<--<b}"
+  obtains e where "e > 0" "ball x e \<inter> path_image p = {}"
+    "ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0} \<subseteq> inside (path_image (p +++ linepath a b)) \<and>
+     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0} \<subseteq> outside (path_image (p +++ linepath a b))
+     \<or> 
+     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0} \<subseteq> inside (path_image (p +++ linepath a b)) \<and>
+     ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0} \<subseteq> outside (path_image (p +++ linepath a b))"
+proof -
+  let ?lp = "p +++ linepath a b"
+  have "a \<noteq> b" using x by auto
+  have pp:"path p" using jordan path_join pathfinish_linepath simple_path_imp_path
+    by fastforce
+  have "path_image p \<inter> path_image (linepath a b) \<subseteq> {a,b}"
+    using jordan simple_path_join_loop_eq
+    by (metis (no_types, lifting) inf_sup_aci(1) insert_commute path_join_path_ends path_linepath simple_path_imp_path simple_path_joinE)
+  then have "x \<notin> path_image p" using x unfolding path_image_linepath
+    by (metis DiffE Int_iff le_iff_inf open_segment_def)
+  then have "\<forall>\<^sub>F y in at x. y \<notin> path_image p"
+    by (intro eventually_not_in_closed) (auto simp: closed_path_image \<open>path p\<close>)
+  moreover 
+  have "\<forall>\<^sub>F y in at x. (y - a) \<bullet> rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b}"
+    by (rule eventually_at_open_segment[OF x])
+  ultimately have "\<forall>\<^sub>F y in at x. y \<notin> path_image p \<and> ((y - a) \<bullet> rot (a - b) = 0 \<longrightarrow> y \<in> {a<--<b})"
+    by eventually_elim auto
+  then obtain e where e: "e > 0" "ball x e \<inter> path_image p = {}"
+    "ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0} \<subseteq> {a<--<b}"
+    using \<open>x \<notin> path_image p\<close> x in_open_segment_rotD[OF x]
+    apply (auto simp: eventually_at subset_iff dist_commute dest!: )
+    by (metis Int_iff all_not_in_conv dist_commute mem_ball) 
+  have a1: "pathfinish ?lp = pathstart ?lp"
+    by (auto simp add: jordan)
+  have "x \<in> path_image ?lp"
+    using jordan(1) open_closed_segment path_image_join path_join_path_ends simple_path_imp_path x by fastforce
+  from jordan_points_inside_outside[OF e(1) jordan(1) a1 this]
+  obtain y z where y: "y \<in> inside (path_image ?lp)" "y \<in> ball x e" 
+    and z: "z \<in> outside (path_image ?lp)" "z \<in> ball x e" by blast
+  have jordancurve:
+    "inside (path_image ?lp) \<inter> outside(path_image ?lp) = {}"
+    "frontier (inside (path_image ?lp)) = path_image ?lp"
+    "frontier (outside (path_image ?lp)) = path_image ?lp"
+    using Jordan_inside_outside_R2[OF jordan(1) a1] by auto
+  define b1 where "b1 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) > 0}"
+  define b2 where "b2 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) < 0}"
+  define b3 where "b3 = ball x e \<inter> {y. (y-a) \<bullet> rot (a-b) = 0}"
+  have "path_connected b1" unfolding b1_def
+    apply (auto intro!: convex_imp_path_connected convex_Int simp add:inner_diff_left)
+    using convex_halfspace_gt[of "a \<bullet> rot (a - b)" "rot(a-b)"] inner_commute
+    by (metis (no_types, lifting) Collect_cong)
+  have "path_connected b2" unfolding b2_def
+    apply (auto intro!: convex_imp_path_connected convex_Int simp add:inner_diff_left)
+    using convex_halfspace_lt[of "rot(a-b)" "a \<bullet> rot (a - b)"] inner_commute
+    by (metis (no_types, lifting) Collect_cong)
+  have "b1 \<inter> path_image(linepath a b) = {}" unfolding path_image_linepath b1_def
+    using closed_segment_surface[OF \<open>a \<noteq> b\<close>] in_segment_inner_rot2 by auto 
+  then have b1i:"b1 \<inter> path_image ?lp = {}"
+    by (metis IntD2 b1_def disjoint_iff_not_equal e(2) inf_sup_aci(1) not_in_path_image_join)
+  have "b2 \<inter> path_image(linepath a b) = {}" unfolding path_image_linepath b2_def
+    using closed_segment_surface[OF \<open>a \<noteq> b\<close>] in_segment_inner_rot2 by auto 
+  then have b2i:"b2 \<inter> path_image ?lp = {}"
+    by (metis IntD2 b2_def disjoint_iff_not_equal e(2) inf_sup_aci(1) not_in_path_image_join)
+  have bsplit: "ball x e = b1 \<union> b2 \<union> b3"
+    unfolding b1_def b2_def b3_def
+    by auto
+  have "z \<notin> b3"
+  proof clarsimp
+    assume "z \<in> b3"
+    then have "z \<in> {a<--<b}" unfolding b3_def using e by blast
+    then have "z \<in> path_image(linepath a b)" by (auto simp add: open_segment_def)
+    then have "z \<in> path_image ?lp"
+      by (simp add: jordan(2) path_image_join)
+    thus False using z
+      using inside_Un_outside by fastforce
+  qed
+  then have z12: "z \<in> b1 \<or> z \<in> b2" using z bsplit by blast
+  have "y \<notin> b3"
+  proof clarsimp
+    assume "y \<in> b3"
+    then have "y \<in> {a<--<b}" unfolding b3_def using e by auto
+    then have "y \<in> path_image(linepath a b)" by (auto simp add: open_segment_def)
+    then have "y \<in> path_image ?lp"
+      by (simp add: jordan(2) path_image_join)
+    thus False using y
+      using inside_Un_outside by fastforce
+  qed
+  then have "y \<in> b1 \<or> y \<in> b2" using y bsplit by blast
+  moreover {
+    assume "y \<in> b1"
+    then have "b1 \<inter> inside (path_image ?lp) \<noteq> {}" using y by blast
+    from path_connected_not_frontier_subset[OF \<open>path_connected b1\<close> this]
+    have 1:"b1 \<subseteq> inside (path_image ?lp)" unfolding jordancurve using b1i
+      by blast
+    then have "z \<in> b2" using jordancurve(1) z(1) z12 by blast
+    then have "b2 \<inter> outside (path_image ?lp) \<noteq> {}" using z by blast
+    from path_connected_not_frontier_subset[OF \<open>path_connected b2\<close> this]
+    have 2:"b2 \<subseteq> outside (path_image ?lp)" unfolding jordancurve using b2i
+      by blast
+    note conjI[OF 1 2]
+  }
+  moreover {
+    assume "y \<in> b2"
+    then have "b2 \<inter> inside (path_image ?lp) \<noteq> {}" using y by blast
+    from path_connected_not_frontier_subset[OF \<open>path_connected b2\<close> this]
+    have 1:"b2 \<subseteq> inside (path_image ?lp)" unfolding jordancurve using b2i
+      by blast
+    then have "z \<in> b1" using jordancurve(1) z(1) z12 by blast
+    then have "b1 \<inter> outside (path_image ?lp) \<noteq> {}" using z by blast
+    from path_connected_not_frontier_subset[OF \<open>path_connected b1\<close> this]
+    have 2:"b1 \<subseteq> outside (path_image ?lp)" unfolding jordancurve using b1i
+      by blast
+    note conjI[OF 1 2]
+  } 
+  ultimately show ?thesis unfolding b1_def b2_def using that[OF e(1-2)] by auto
+qed
+
+subsection \<open>Transversal Segments\<close>\<comment> \<open>TODO: Transversal surface in Euclidean space?!\<close>
+
+definition "transversal_segment a b \<longleftrightarrow>
+  a \<noteq> b \<and> {a--b} \<subseteq> X \<and>
+  (\<forall>z \<in> {a--b}. f z \<bullet> rot (a-b) \<noteq> 0)"
+
+lemma transversal_segment_reverse:
+  assumes "transversal_segment x y"
+  shows "transversal_segment y x"
+  unfolding transversal_segment_def
+  by (metis (no_types, hide_lams) add.left_neutral add_uminus_conv_diff assms closed_segment_commute inner_diff_left inner_zero_left nrm_reverse transversal_segment_def) 
+
+lemma transversal_segment_commute: "transversal_segment x y \<longleftrightarrow> transversal_segment y x"
+  using transversal_segment_reverse by blast
+
+lemma transversal_segment_neg:
+  assumes "transversal_segment x y"
+  assumes w: "w \<in> {x -- y}" and "f w \<bullet> rot (x-y) < 0"
+  shows "\<forall>z \<in> {x--y}. f(z) \<bullet> rot (x-y) < 0"
+proof (rule ccontr)
+  assume " \<not> (\<forall>z\<in>{x--y}. f z \<bullet> rot (x-y) < 0)"
+  then obtain z where z: "z \<in> {x--y}" "f z \<bullet> rot (x-y) \<ge> 0" by auto
+  define ff where "ff = (\<lambda>s. f (w + s *\<^sub>R (z - w)) \<bullet> rot (x-y))"
+  have f0:"ff 0 \<le> 0" unfolding ff_def using assms(3)
+    by simp 
+  have fu:"ff 1 \<ge> 0"
+    by (auto simp: ff_def z)
+  from assms(2) obtain u where u: "0 \<le> u" "u \<le> 1" "w = (1 - u) *\<^sub>R x + u *\<^sub>R y"
+    unfolding in_segment by blast
+  have "{x--y} \<subseteq> X" using assms(1) unfolding transversal_segment_def by blast
+  then have "continuous_on {0..1} ff" unfolding ff_def 
+    using assms(2)
+    by (auto intro!:continuous_intros closed_subsegmentI z elim!: set_mp)
+  from IVT'[of ff, OF f0 fu zero_le_one this]
+  obtain s where s: "s \<ge> 0" "s \<le> 1" "ff s = 0" by blast
+  have "w + s *\<^sub>R (z - w) \<in> {x -- y}"
+    by (auto intro!: closed_subsegmentI z s w)
+  with \<open>ff s = 0\<close> show False
+    using s assms(1) unfolding transversal_segment_def ff_def by blast
+qed
+
+lemmas transversal_segment_sign_less = transversal_segment_neg[OF _ ends_in_segment(1)]
+
+lemma transversal_segment_pos:
+  assumes "transversal_segment x y"
+  assumes w: "w \<in> {x -- y}" "f w \<bullet> rot (x-y) > 0"
+  shows "\<forall>z \<in> {x--y}. f(z) \<bullet> rot (x-y) > 0"
+  using transversal_segment_neg[OF transversal_segment_reverse[OF assms(1)], of w] w
+  by (auto simp: rot_diff_commute[of x y] closed_segment_commute)
+
+lemma transversal_segment_posD:
+  assumes "transversal_segment x y"
+    and pos: "z \<in> {x -- y}" "f z \<bullet> rot (x - y) > 0"
+  shows "x \<noteq> y" "{x--y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x-y) > 0"
+  using assms(1) transversal_segment_pos[OF assms]
+  by (auto simp: transversal_segment_def)
+
+lemma transversal_segment_negD:
+  assumes "transversal_segment x y"
+    and pos: "z \<in> {x -- y}" "f z \<bullet> rot (x - y) < 0"
+  shows "x \<noteq> y" "{x--y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x-y) < 0"
+  using assms(1) transversal_segment_neg[OF assms]
+  by (auto simp: transversal_segment_def)
+
+lemma transversal_segmentE:
+  assumes "transversal_segment x y"
+  obtains "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (x - y) > 0"
+  |  "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (y - x) > 0"
+proof (cases "f x \<bullet> rot (x - y) < 0")
+  case True
+  from transversal_segment_negD[OF assms ends_in_segment(1) True]
+  have "x \<noteq> y" "{x -- y} \<subseteq> X" "\<And>z. z \<in> {x--y} \<Longrightarrow> f z \<bullet> rot (y - x) > 0"
+    by (auto simp: rot_diff_commute[of x y])
+  then show ?thesis ..
+next
+  case False
+  then have "f x \<bullet> rot (x - y) > 0" using assms
+    by (auto simp: transversal_segment_def algebra_split_simps not_less order.order_iff_strict)
+  from transversal_segment_posD[OF assms ends_in_segment(1) this]
+  show ?thesis ..
+qed
+
+lemma dist_add_vec:
+  shows "dist (x + s *\<^sub>R v) x = abs s * norm v"
+  by (simp add: dist_cancel_add1)
+
+lemma transversal_segment_exists:
+  assumes "x \<in> X" "f x \<noteq> 0"
+  obtains a b where "x \<in> {a<--<b}"
+    "transversal_segment a b"
+proof -
+  (* Line through x perpendicular to f x *)
+  define l where "l = (\<lambda>s::real. x + (s/norm(f x)) *\<^sub>R rot (f x))"
+  have "norm (f x) > 0" using assms(2) using zero_less_norm_iff by blast 
+  then have distl: "\<forall>s. dist (l s) x = abs s" unfolding l_def dist_add_vec
+    by (auto simp add: norm_rot)
+  obtain d where d:"d > 0" "cball x d \<subseteq> X"
+    by (meson UNIV_I assms(1) local.local_unique_solution)
+  then have lb: "l`{-d..d} \<subseteq> cball x d" using distl by (simp add: abs_le_iff dist_commute image_subset_iff)
+  from fcontx[OF assms(1)] have "continuous (at x) f" .
+  then have c:"continuous (at 0) ((\<lambda>y. (f y \<bullet> f x)) \<circ> l)" unfolding l_def
+    by (auto intro!:continuous_intros simp add: assms(2))
+  have "((\<lambda>y. f y \<bullet> f x) \<circ> l) 0 > 0" using assms(2) unfolding l_def o_def by auto
+  from continuous_at_imp_cball[OF c this]
+  obtain r where r:"r > 0" " \<forall>z\<in>cball 0 r. 0 < ((\<lambda>y. f y \<bullet> f x) \<circ> l) z" by blast
+  then have rc:"\<forall>z \<in> l`{-r..r}. 0 < f z \<bullet> f x" using real_norm_def by auto 
+  define dr where "dr = min r d"
+  have t1:"l (-dr) \<noteq> l dr" unfolding l_def dr_def
+    by (smt \<open>0 < d\<close> \<open>0 < norm (f x)\<close> \<open>0 < r\<close> add_left_imp_eq divide_cancel_right norm_rot norm_zero scale_cancel_right)
+  have "x = midpoint (l (-dr)) (l dr)" unfolding midpoint_def l_def by auto
+  then have xin:"x \<in> {l (-dr)<--<(l dr)}" using t1 by auto
+      (* TODO: actually this should be equality, but l is affine ...
+     also the existing stuff about -- is a little too specific *)
+  have lsub:"{l (-dr)--l dr} \<subseteq> l`{-dr..dr}"
+  proof safe
+    fix z
+    assume "z \<in> {l (- dr)--l dr}"
+    then obtain u where u: "0 \<le> u" "u \<le> 1" "z = (1 - u) *\<^sub>R (l (-dr)) + u *\<^sub>R (l dr)"
+      unfolding in_segment by blast
+    then have "z = x - (1-u) *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x) + u *\<^sub>R  (dr/norm(f x)) *\<^sub>R rot (f x) "
+      unfolding l_def
+      by (simp add: l_def scaleR_add_right scale_right_diff_distrib u(3))
+    also have "... = x - (1 - 2 * u) *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x)"
+      by (auto simp add: algebra_simps divide_simps simp flip: scaleR_add_left)
+    also have "... =  x + (((2 * u - 1) * dr)/norm(f x)) *\<^sub>R rot (f x)"
+      by (smt add_uminus_conv_diff scaleR_scaleR scale_minus_left times_divide_eq_right)
+    finally have zeq: "z = l ((2*u-1)*dr)" unfolding l_def .
+    have ub: " 2* u - 1 \<le> 1 \<and> -1 \<le>  2* u - 1 " using u by linarith
+    thus "z \<in> l ` {- dr..dr}" using zeq
+      by (smt atLeastAtMost_iff d(1) dr_def image_eqI mult.commute mult_left_le mult_minus_left r(1)) 
+  qed
+  have t2: "{l (- dr)--l dr} \<subseteq> X" using lsub
+    by (smt atLeastAtMost_iff d(2) dist_commute distl dr_def image_subset_iff mem_cball order_trans)
+  have "l (- dr) - l dr = -2 *\<^sub>R (dr/norm(f x)) *\<^sub>R rot (f x)" unfolding l_def
+    by (simp add: algebra_simps flip: scaleR_add_left)
+  then have req: "rot (l (- dr) - l dr) = (2 * dr/norm(f x)) *\<^sub>R f x"
+    by auto (metis add.inverse_inverse rot_rot rot_scaleR)
+  have "l`{-dr..dr} \<subseteq> l ` {-r ..r}"
+    by (simp add: dr_def image_mono)
+  then have "{l (- dr)--l dr} \<subseteq> l ` {-r .. r}" using lsub by auto
+  then have "\<forall>z \<in> {l (- dr)--l dr}. 0 < f z \<bullet> f x" using rc by blast
+  moreover have "(dr / norm (f x)) > 0"
+    using \<open>0 < norm (f x)\<close> d(1) dr_def r(1) by auto 
+  ultimately have t3: "\<forall>z \<in> {l (- dr)--l dr}. f z \<bullet> rot (l (- dr)- l dr) > 0" unfolding req
+    by (smt divide_divide_eq_right inner_scaleR_right mult_2 norm_not_less_zero scaleR_2 times_divide_eq_left times_divide_eq_right zero_less_divide_iff)
+  have "transversal_segment (l (-dr)) (l dr)" using t1 t2 t3 unfolding transversal_segment_def by auto
+  thus ?thesis using xin
+    using that by auto 
+qed
+
+text \<open>Perko Section 3.7 Lemma 2 part 1.\<close> 
+lemma flow_transversal_segment_finite_intersections:
+  assumes "transversal_segment a b"
+  assumes "t \<le> t'" "{t .. t'} \<subseteq> existence_ivl0 x"
+  shows "finite {s\<in>{t..t'}. flow0 x s \<in> {a--b}}"
+proof -
+  from assms have "a \<noteq> b" by (simp add: transversal_segment_def)
+  show ?thesis
+    unfolding closed_segment_surface[OF \<open>a \<noteq> b\<close>]
+    apply (rule flow_transversal_surface_finite_intersections[where Ds="\<lambda>_. blinfun_inner_left (rot (b - a))"])
+    by
+      (use assms in \<open>auto intro!: closed_Collect_conj closed_halfspace_component_ge closed_halfspace_component_le
+        derivative_eq_intros
+        simp: transversal_segment_def nrm_reverse[where x=a] in_closed_segment_iff_rot\<close>)
+qed
+
+lemma transversal_bound_posE:
+  assumes transversal: "transversal_segment a b"
+  assumes direction: "z \<in> {a -- b}" "f z \<bullet> (rot (a - b)) > 0"
+  obtains d B where "d > 0" "0 < B"
+    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (a - b)) \<ge> B"
+proof -
+  let ?a = "(\<lambda>y. (f y) \<bullet> (rot (a - b)))"
+  from transversal_segment_posD[OF transversal direction]
+  have seg: "a \<noteq> b" "{a--b} \<subseteq> X" "z \<in> {a--b} \<Longrightarrow> 0 < f z \<bullet> rot (a - b)" for z
+    by auto
+  {
+    fix x
+    assume "x \<in> {a--b}"
+    then have "x \<in> X" "f x \<noteq> 0" "a \<noteq> b" using transversal by (auto simp: transversal_segment_def)
+    then have "?a \<midarrow>x\<rightarrow> ?a x"
+      by (auto intro!: tendsto_eq_intros)
+    moreover have "?a x > 0"
+      using seg \<open>x \<in> {a -- b}\<close> \<open>f x \<noteq> 0\<close>
+      by (auto simp: simp del: divide_const_simps
+          intro!: divide_pos_pos mult_pos_pos)
+    ultimately have "\<forall>\<^sub>F x in at x. ?a x > 0"
+      by (rule order_tendstoD)
+    moreover have "\<forall>\<^sub>F x in at x. x \<in> X"
+      by (rule topological_tendstoD[OF tendsto_ident_at open_dom \<open>x \<in> X\<close>])
+    moreover have "\<forall>\<^sub>F x in at x. f x \<noteq> 0"
+      by (rule tendsto_imp_eventually_ne tendsto_intros \<open>x \<in> X\<close> \<open>f x \<noteq> 0\<close>)+
+    ultimately have "\<forall>\<^sub>F x in at x. ?a x>0 \<and> x \<in> X \<and> f x \<noteq> 0" by eventually_elim auto
+    then obtain d where d: "0 < d" "\<And>y. y \<in> cball x d \<Longrightarrow> ?a y > 0 \<and> y \<in> X \<and> f y \<noteq> 0"
+      using \<open>?a x > 0\<close> \<open>x \<in> X\<close>
+      by (force simp: eventually_at_le dist_commute)
+
+    have "continuous_on (cball x d) ?a"
+      using d \<open>a \<noteq> b\<close>
+      by (auto intro!: continuous_intros)
+    from compact_continuous_image[OF this compact_cball]
+    have "compact (?a ` cball x d)" .
+    from compact_attains_inf[OF this] obtain s where "s \<in> cball x d" "\<forall>x\<in>cball x d. ?a x \<ge> ?a s"
+      using \<open>d > 0\<close>
+      by auto
+    then have "\<exists>d>0. \<exists>b>0. \<forall>x \<in> cball x d. ?a x \<ge> b"
+      using d
+      by (force simp: intro: exI[where x="?a s"])
+  } then obtain dx Bx where dB:
+    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> y\<in>cball x (dx x) \<Longrightarrow> ?a y \<ge> Bx x"
+    "\<And>x. x \<in> {a -- b} \<Longrightarrow> Bx x > 0"
+    "\<And>x. x \<in> {a -- b} \<Longrightarrow> dx x > 0"
+    by metis
+  define d' where "d' = (\<lambda>x. dx x / 2)"
+  have d':
+    "\<And>x. x \<in> {a -- b} \<Longrightarrow> \<forall>y\<in>cball x (d' x). ?a y \<ge> Bx x"
+    "\<And>x. x \<in> {a -- b} \<Longrightarrow> d' x > 0"
+    using dB(1,3) by (force simp: d'_def)+
+  have d'B: "\<And>x. x \<in> {a -- b} \<Longrightarrow> \<forall>y\<in>cball x (d' x). ?a y \<ge> Bx x"
+    using d' by auto
+  have "{a--b} \<subseteq> \<Union>((\<lambda>x. ball x (d' x)) ` {a -- b})"
+    using d'(2) by auto
+  from compactE_image[OF compact_segment _ this]
+  obtain X where X: "X \<subseteq> {a--b}"
+    and [simp]: "finite X"
+    and cover: "{a--b} \<subseteq> (\<Union>x\<in>X. ball x (d' x))"
+    by auto
+  have [simp]: "X \<noteq> {}" using X cover by auto
+  define d where "d = Min (d' ` X)"
+  define B where "B = Min (Bx ` X)"
+  have "d > 0"
+    using X d'
+    by (auto simp: d_def d'_def)
+  moreover have "B > 0"
+    using X dB
+    by (auto simp: B_def simp del: divide_const_simps)
+  moreover have "B \<le> ?a y" if "x \<in> {a -- b}" "dist x y \<le> d" for x y
+  proof -
+    from \<open>x \<in> {a -- b}\<close> obtain xc where xc: "xc \<in> X" "x \<in> ball xc (d' xc)"
+      using cover by auto
+    have "?a y \<ge> Bx xc"
+    proof (rule dB)
+      show "xc \<in> {a -- b}" using xc \<open>X \<subseteq> _\<close> by auto
+      have "dist xc y \<le> dist xc x + dist x y" by norm
+      also have "dist xc x \<le> d' xc" using xc by auto
+      also note \<open>dist x y \<le> d\<close>
+      also have "d \<le> d' xc"
+        using xc
+        by (auto simp: d_def)
+      also have "d' xc + d' xc = dx xc" by (simp add: d'_def)
+      finally show "y \<in> cball xc (dx xc)" by simp
+    qed
+    also have "B \<le> Bx xc"
+      using xc
+      unfolding B_def
+      by (auto simp: B_def)
+    finally (xtrans) show ?thesis .
+  qed
+  ultimately show ?thesis ..
+qed
+
+lemma transversal_bound_negE:
+  assumes transversal: "transversal_segment a b"
+  assumes direction: "z \<in> {a -- b}" "f z \<bullet> (rot (a - b)) < 0"
+  obtains d B where "d > 0" "0 < B"
+    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (b - a)) \<ge> B"
+proof -
+  from direction have "z \<in> {b -- a}" "f z \<bullet> (rot (b - a)) > 0"
+    by (auto simp: closed_segment_commute rot_diff_commute[of b a])
+  from transversal_bound_posE[OF transversal_segment_reverse[OF transversal] this]
+  obtain d B where "d > 0" "0 < B"
+    "\<And>x y. x \<in> {a -- b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> f y \<bullet> (rot (b - a)) \<ge> B"
+    by (auto simp: closed_segment_commute)
+  then show ?thesis ..
+qed
+
+lemma leaves_transversal_segmentE:
+  assumes transversal: "transversal_segment a b"
+  obtains T n where "T > 0" "n = a - b \<or> n = b - a"
+    "\<And>x. x \<in> {a -- b} \<Longrightarrow> {-T..T} \<subseteq> existence_ivl0 x"
+    "\<And>x s. x \<in> {a -- b} \<Longrightarrow> 0 < s \<Longrightarrow> s \<le> T \<Longrightarrow>
+    (flow0 x s - x) \<bullet> rot n > 0"
+    "\<And>x s. x \<in> {a -- b} \<Longrightarrow> -T \<le> s \<Longrightarrow> s < 0 \<Longrightarrow>
+    (flow0 x s - x) \<bullet> rot n < 0"
+proof -
+  from transversal_segmentE[OF assms(1)] obtain n
+    where n: "n = (a - b) \<or> n = (b - a)"
+      and seg: "a \<noteq> b" "{a -- b} \<subseteq> X" "\<And>z. z \<in> {a--b} \<Longrightarrow> f z \<bullet> rot n > 0"
+    by metis
+  from open_existence_ivl_on_compact[OF \<open>{a -- b} \<subseteq> X\<close>]
+  obtain t where "0 < t" and t: "x \<in> {a--b} \<Longrightarrow> {- t..t} \<subseteq> existence_ivl0 x" for x
+    by auto
+  from n obtain d B where B: "0 < d" "0 < B" "(\<And>x y. x \<in> {a--b} \<Longrightarrow> dist x y \<le> d \<Longrightarrow> B \<le> f y \<bullet> rot n)"
+  proof
+    assume n_def: "n = a - b"
+    with seg have pos:  "0 < f a \<bullet> rot (a - b)"
+      by auto
+    from transversal_bound_posE[OF transversal ends_in_segment(1) pos, folded n_def]
+    show ?thesis using that by blast
+  next
+    assume n_def: "n = b - a"
+    with seg have pos:  "0 > f a \<bullet> rot (a - b)"
+      by (auto simp: rot_diff_commute[of a b])
+    from transversal_bound_negE[OF transversal ends_in_segment(1) this, folded n_def]
+    show ?thesis using that by blast
+  qed
+  define S where "S = \<Union>((\<lambda>x. ball x d) ` {a -- b})"
+  have S: "x \<in> S \<Longrightarrow> B \<le> f x \<bullet> rot n" for x
+    by (auto simp: S_def intro!: B)
+  have "open S" by (auto simp: S_def)
+  have "{a -- b} \<subseteq> S"
+    by (auto simp: S_def \<open>0 < d\<close>)
+  have "\<forall>\<^sub>F (t, x) in at (0, x). flow0 x t \<in> S" if "x \<in> {a -- b}" for x
+    unfolding split_beta'
+    apply (rule topological_tendstoD tendsto_intros)+
+    using set_mp[OF \<open>{a -- b} \<subseteq> X\<close> that] \<open>0 < d\<close> that \<open>open S\<close> \<open>{a -- b} \<subseteq> S\<close>
+    by (force simp: )+
+  then obtain d' where d':
+    "\<And>x. x \<in> {a--b} \<Longrightarrow> d' x > 0"
+    "\<And>x y s. x \<in> {a--b} \<Longrightarrow> (s = 0 \<longrightarrow> y \<noteq> x) \<Longrightarrow> dist (s, y) (0, x) < d' x \<Longrightarrow> flow0 y s \<in> S"
+    by (auto simp: eventually_at) metis
+  define d2 where "d2 x = d' x / 2" for x
+  have d2: "\<And>x. x \<in> {a--b} \<Longrightarrow> d2 x > 0" using d' by (auto simp: d2_def)
+  have C: "{a--b} \<subseteq> \<Union>((\<lambda>x. ball x (d2 x)) ` {a -- b})"
+    using d2 by auto
+  from compactE_image[OF compact_segment _ C]
+  obtain C' where "C' \<subseteq> {a--b}" and [simp]: "finite C'"
+    and C'_cover: "{a--b} \<subseteq> (\<Union>c\<in>C'. ball c (d2 c))" by auto
+
+  define T where "T = Min (insert t (d2 ` C'))"
+
+  have "T > 0"
+    using \<open>0 < t\<close> d2 \<open>C' \<subseteq> _\<close> 
+    by (auto simp: T_def)
+  moreover
+  note n
+  moreover
+  have T_ex: "{-T..T} \<subseteq> existence_ivl0 x" if "x \<in> {a--b}" for x
+    by (rule order_trans[OF _ t[OF that]]) (auto simp: T_def)
+  moreover
+  have B_le: "B \<le> f (flow0 x \<xi>) \<bullet> rot n"
+    if "x \<in> {a -- b}"
+      and c': "c' \<in> C'" "x \<in> ball c' (d2 c')"
+      and "\<xi> \<noteq> 0" and \<xi>_le: "\<bar>\<xi>\<bar> < d2 c'"
+    for x c' \<xi>
+  proof -
+    have "c' \<in> {a -- b}" using c' \<open>C' \<subseteq> _\<close> by auto
+    moreover have "\<xi> = 0 \<longrightarrow> x \<noteq> c'" using \<open>\<xi> \<noteq> 0\<close> by simp
+    moreover have "dist (\<xi>, x) (0, c') < d' c'"
+    proof -
+      have "dist (\<xi>, x) (0, c') \<le> dist (\<xi>, x) (\<xi>, c') + dist (\<xi>, c') (0, c')"
+        by norm
+      also have "dist (\<xi>, x) (\<xi>, c') < d2 c'"
+        using c'
+        by (simp add: dist_prod_def dist_commute)
+      also
+      have "T \<le> d2 c'" using c'
+        by (auto simp: T_def)
+      then have "dist (\<xi>, c') (0, c') < d2 c'"
+        using \<xi>_le
+        by (simp add: dist_prod_def)
+      also have "d2 c' + d2 c' = d' c'" by (simp add: d2_def)
+      finally show ?thesis by simp
+    qed
+    ultimately have "flow0 x \<xi> \<in> S"
+      by (rule d')
+    then show ?thesis
+      by (rule S)
+  qed
+  let ?g = "(\<lambda>x t. (flow0 x t - x) \<bullet> rot n)"
+  have cont: "continuous_on {-T .. T} (?g x)"
+    if "x \<in> {a--b}" for x
+    using T_ex that
+    by (force intro!: continuous_intros)
+  have deriv: "-T \<le> s' \<Longrightarrow> s' \<le> T \<Longrightarrow> ((?g x) has_derivative
+      (\<lambda>t. t * (f (flow0 x s') \<bullet> rot n))) (at s')"
+    if "x \<in> {a--b}" for x s'
+    using T_ex that
+    by (force intro!: derivative_eq_intros simp: flowderiv_def blinfun.bilinear_simps)
+
+  have "(flow0 x s - x) \<bullet> rot n > 0" if "x \<in> {a -- b}" "0 < s" "s \<le> T" for x s
+  proof (rule ccontr, unfold not_less)
+    have [simp]: "x \<in> X" using that \<open>{a -- b} \<subseteq> X\<close> by auto
+    assume H: "(flow0 x s - x) \<bullet> rot n \<le> 0"
+    have cont: "continuous_on {0 .. s} (?g x)"
+      using cont by (rule continuous_on_subset) (use that in auto)
+    from mvt[OF \<open>0 < s\<close> cont deriv] that
+    obtain \<xi> where \<xi>: "0 < \<xi>" "\<xi> < s" "(flow0 x s - x) \<bullet> rot n = s * (f (flow0 x \<xi>) \<bullet> rot n)"
+      by (auto intro: continuous_on_subset)
+    note \<open>0 < B\<close>
+    also
+    from C'_cover that obtain c' where c': "c' \<in> C'" "x \<in> ball c' (d2 c')" by auto
+    have "B \<le> f (flow0 x \<xi>) \<bullet> rot n"
+    proof (rule B_le[OF that(1) c'])
+      show "\<xi> \<noteq> 0" using \<open>0 < \<xi>\<close> by simp
+      have "T \<le> d2 c'" using c'
+        by (auto simp: T_def)
+      then show "\<bar>\<xi>\<bar> < d2 c'"
+        using \<open>0 < \<xi>\<close> \<open>\<xi> < s\<close> \<open>s \<le> T\<close>
+        by (simp add: dist_prod_def)
+    qed
+    also from \<xi> H have "\<dots> \<le> 0"
+      by (auto simp add: algebra_split_simps not_less split: if_splits)
+    finally show False by simp
+  qed
+  moreover
+  have "(flow0 x s - x) \<bullet> rot n < 0" if "x \<in> {a -- b}" "-T \<le> s" "s < 0" for x s
+  proof (rule ccontr, unfold not_less)
+    have [simp]: "x \<in> X" using that \<open>{a -- b} \<subseteq> X\<close> by auto
+    assume H: "(flow0 x s - x) \<bullet> rot n \<ge> 0"
+    have cont: "continuous_on {s .. 0} (?g x)"
+      using cont by (rule continuous_on_subset) (use that in auto)
+    from mvt[OF \<open>s < 0\<close> cont deriv] that
+    obtain \<xi> where \<xi>: "s < \<xi>" "\<xi> < 0" "(flow0 x s - x) \<bullet> rot n = s * (f (flow0 x \<xi>) \<bullet> rot n)"
+      by auto
+    note \<open>0 < B\<close>
+    also
+    from C'_cover that obtain c' where c': "c' \<in> C'" "x \<in> ball c' (d2 c')" by auto
+    have "B \<le> (f (flow0 x \<xi>) \<bullet> rot n)"
+    proof (rule B_le[OF that(1) c'])
+      show "\<xi> \<noteq> 0" using \<open>0 > \<xi>\<close> by simp
+      have "T \<le> d2 c'" using c'
+        by (auto simp: T_def)
+      then show "\<bar>\<xi>\<bar> < d2 c'"
+        using \<open>0 > \<xi>\<close> \<open>\<xi> > s\<close> \<open>s \<ge> - T\<close>
+        by (simp add: dist_prod_def)
+    qed
+    also from \<xi> H have "\<dots> \<le> 0"
+      by (simp add: algebra_split_simps)
+    finally show False by simp
+  qed
+  ultimately show ?thesis ..
+qed
+
+
+lemma inner_rot_pos_move_base: "(x - a) \<bullet> rot (a - b) > 0"
+  if "(x - y) \<bullet> rot (a - b) > 0" "y \<in> {a -- b}"
+  by (smt in_segment_inner_rot inner_diff_left inner_minus_right minus_diff_eq rot_rot that)
+
+lemma inner_rot_neg_move_base: "(x - a) \<bullet> rot (a - b) < 0"
+  if "(x - y) \<bullet> rot (a - b) < 0" "y \<in> {a -- b}"
+  by (smt in_segment_inner_rot inner_diff_left inner_minus_right minus_diff_eq rot_rot that)
+
+lemma inner_pos_move_base: "(x - a) \<bullet> n > 0"
+  if "(a - b) \<bullet> n = 0" "(x - y) \<bullet> n > 0" "y \<in> {a -- b}"
+proof -
+  from that(3) obtain u where y_def: "y = (1 - u) *\<^sub>R a + u *\<^sub>R b" and u: "0 \<le> u" "u \<le> 1"
+    by (auto simp: in_segment)
+  have "(x - a) \<bullet> n = (x - y) \<bullet> n - u * ((a - b) \<bullet> n)"
+    by (simp add: algebra_simps y_def)
+  also have "\<dots> = (x - y) \<bullet> n"
+    by (simp add: that)
+  also note \<open>\<dots> > 0\<close>
+  finally show ?thesis .
+qed
+
+lemma inner_neg_move_base: "(x - a) \<bullet> n < 0"
+  if "(a - b) \<bullet> n = 0" "(x - y) \<bullet> n < 0" "y \<in> {a -- b}"
+proof -
+  from that(3) obtain u where y_def: "y = (1 - u) *\<^sub>R a + u *\<^sub>R b" and u: "0 \<le> u" "u \<le> 1"
+    by (auto simp: in_segment)
+  have "(x - a) \<bullet> n = (x - y) \<bullet> n - u * ((a - b) \<bullet> n)"
+    by (simp add: algebra_simps y_def)
+  also have "\<dots> = (x - y) \<bullet> n"
+    by (simp add: that)
+  also note \<open>\<dots> < 0\<close>
+  finally show ?thesis .
+qed
+
+lemma rot_same_dir:
+  assumes "x1 \<in> {a<--<b}"
+  assumes "x2 \<in> {x1<--<b}"
+  shows "(y \<bullet> rot (a-b) > 0) = (y \<bullet> rot(x1-x2) > 0)"  "(y \<bullet> rot (a-b) < 0) = (y \<bullet> rot(x1-x2) < 0)"
+  using oriented_subsegment_scale[OF assms]
+   apply (smt inner_scaleR_right nrm_reverse rot_scaleR zero_less_mult_iff)
+  by (smt \<open>\<And>thesis. (\<And>e. \<lbrakk>0 < e; b - a = e *\<^sub>R (x2 - x1)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> inner_minus_right inner_scaleR_right rot_diff_commute rot_scaleR zero_less_mult_iff)
+
+
+subsection \<open>Monotone Step Lemma\<close>
+
+lemma flow0_transversal_segment_monotone_step:
+  assumes "transversal_segment a b"
+  assumes "t1 \<le> t2" "{t1..t2} \<subseteq> existence_ivl0 x"
+  assumes x1: "flow0 x t1 \<in> {a<--<b}"
+  assumes x2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
+  assumes "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
+  assumes "t > t2" "t \<in> existence_ivl0 x"
+  shows "flow0 x t \<notin> {a<--<flow0 x t2}"
+proof -
+  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
+  note t1t2 = \<open>\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}\<close>
+    (* Basic properties of the segment *)
+  have x1neqx2: "flow0 x t1 \<noteq> flow0 x t2"
+    using open_segment_def x2 by force 
+  then have t1neqt2: "t1 \<noteq> t2" by auto
+
+  have [simp]: "a \<noteq> b" and \<open>{a -- b} \<subseteq> X\<close> using \<open>transversal_segment a b\<close>
+    by (auto simp: transversal_segment_def)
+
+  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
+    by (auto simp: in_open_segment_iff_line)
+  from x2 obtain i2 where i2: "flow0 x t2 = line a b i2" "0 < i1" "i1 < i2"
+    by (auto simp: i1 line_open_segment_iff)
+
+
+  have "{a <--< flow0 x t1} \<subseteq> {a<--<b}"
+    by (simp add: open_closed_segment subset_open_segment x1) 
+  have t12sub: "{flow0 x t1--flow0 x t2} \<subseteq> {a<--<b}"
+    by (metis ends_in_segment(2) open_closed_segment subset_co_segment subset_eq subset_open_segment x1 x2)
+  have subr: "{flow0 x t1<--<flow0 x t2} \<subseteq> {flow0 x t1 <--<b}"
+    by (simp add: open_closed_segment subset_open_segment x2)
+  have "flow0 x t1 \<in> {a <--<flow0 x t2}" using x1 x2
+    by (rule open_segment_subsegment)
+  then have subl: "{flow0 x t1<--<flow0 x t2} \<subseteq> {a <--< flow0 x t2}" using x1 x2
+    by (simp add: open_closed_segment subset_open_segment x2)
+  then have subl2: "{flow0 x t1--<flow0 x t2} \<subseteq> {a <--< flow0 x t2}" using x1 x2
+    by (smt DiffE DiffI \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> half_open_segment_def insert_iff open_segment_def subset_eq)
+
+  have sub1b: "{flow0 x t1--b} \<subseteq> {a--b}"
+    by (simp add: open_closed_segment subset_closed_segment x1)
+  have suba2: "{a--flow0 x t2} \<subseteq> {a -- b}"
+    using open_closed_segment subset_closed_segment t12sub by blast
+  then have suba2o: "{a<--<flow0 x t2} \<subseteq> {a -- b}"
+    using open_closed_segment subset_closed_segment t12sub by blast
+  have x2_notmem: "flow0 x t2 \<notin> {a--flow0 x t1}"
+    using i1 i2
+    by (auto simp: closed_segment_line_iff)
+  have suba12: "{a--flow0 x t1} \<subseteq> {a--flow0 x t2}"
+    by (simp add: \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> open_closed_segment subset_closed_segment)
+  then have suba12_open: "{a<--<flow0 x t1} \<subseteq> {a<--<flow0 x t2}"
+    using x2_notmem
+    by (auto simp: open_segment_def)
+  have "flow0 x t2 \<in> {a--b}"
+    using suba2 by auto
+
+  have intereq: "\<And>t. t1 \<le> t \<Longrightarrow> t \<le> t2 \<Longrightarrow> flow0 x t \<in> {a<--<b} \<Longrightarrow>  t = t1 \<or> t = t2"
+  proof (rule ccontr)
+    fix t
+    assume t: "t1 \<le> t" "t \<le> t2" "flow0 x t \<in> {a<--<b}" "\<not>(t= t1 \<or> t = t2)"
+    then have "t \<in> {t1<..<t2}" by auto
+    then have "flow0 x t \<notin> {a<--<b}" using t1t2 by blast
+    thus False using t by auto
+  qed
+  then have intereqt12: "\<And>t. t1 \<le> t \<Longrightarrow> t \<le> t2 \<Longrightarrow> flow0 x t \<in> {flow0 x t1--flow0 x t2} \<Longrightarrow>  t = t1 \<or> t = t2"
+    using t12sub by blast
+
+(* The Jordan curve *)
+  define J1 where "J1 = flow_to_path x t1 t2"
+  define J2 where "J2 = linepath (flow0 x t2) (flow0 x t1)" 
+  define J where "J = J1 +++ J2"
+    (* Proof that J is a Jordan curve *)
+  have "pathfinish J = pathstart J" unfolding J_def J1_def J2_def
+    by (auto simp add: pathstart_compose pathfinish_compose)
+  have piJ: "path_image J = path_image J1 \<union> path_image J2"
+    unfolding J_def J1_def J2_def
+    apply (rule path_image_join)
+    by auto
+  have "flow0 x t1 \<in> flow0 x ` {t1..t2} \<and> flow0 x t2 \<in> flow0 x ` {t1..t2}"
+    using atLeastAtMost_iff \<open>t1 \<le> t2\<close> by blast 
+  then have piD: "path_image J = path_image J1 \<union> {flow0 x t1 <--<flow0 x t2}"
+    unfolding piJ J1_def J2_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
+      path_image_linepath open_segment_def
+    by (smt Diff_idemp Diff_insert2 Un_Diff_cancel closed_segment_commute mk_disjoint_insert)
+  have "\<forall>s\<in>{t1<..<t2}. flow0 x s \<noteq> flow0 x t1"
+    using x1 t1t2 by fastforce
+  from flow_to_path_arc[OF \<open>t1 \<le> t2\<close> exist this x1neqx2]
+  have "arc J1" using J1_def assms flow_to_path_arc by auto
+  then have "simple_path J" unfolding J_def
+    using \<open>arc J1\<close> J1_def J2_def assms x1neqx2 t1neqt2 apply (auto intro!:simple_path_join_loop)
+    using intereqt12 closed_segment_commute by blast
+
+  from Jordan_inside_outside_R2[OF this \<open>pathfinish J = pathstart J\<close>]
+  obtain inner outer where inner_def: "inner = inside (path_image J)"
+    and outer_def: "outer = outside (path_image J)"
+    and io:
+    "inner \<noteq> {}" "open inner" "connected inner"
+    "outer \<noteq> {}" "open outer" "connected outer"
+    "bounded inner" "\<not> bounded outer" "inner \<inter> outer = {}"
+    "inner \<union> outer = - path_image J"
+    "frontier inner = path_image J"
+    "frontier outer = path_image J" by metis
+  from io have io2: "outer \<inter> inner = {}" "outer \<union> inner = - path_image J" by auto
+
+  have swap_side: "\<And>y t. y \<in> side2 \<Longrightarrow>
+    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
+    flow0 y t \<in> closure side1 \<Longrightarrow>
+    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> side2) \<and>
+        flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
+    if "side1 \<inter> side2 = {}"
+      "open side2"
+      "frontier side1 = path_image J"
+      "frontier side2 = path_image J"
+      "side1 \<union> side2 = - path_image J"
+    for side1 side2
+  proof -
+    fix y t
+    assume yt: "y \<in> side2" "0 \<le> t" "t \<in> existence_ivl0 y"
+      "flow0 y t \<in> closure side1"
+    define fp where "fp = flow_to_path y 0 t"
+    have ex:"{0..t} \<subseteq> existence_ivl0 y"
+      using ivl_subset_existence_ivl yt(3) by blast
+    then have y0:"flow0 y 0 = y"
+      using mem_existence_ivl_iv_defined(2) yt(3) by auto
+    then have tpos: "t > 0" using yt(2)  \<open>side1 \<inter> side2 = {}\<close>
+      using yt(1) yt(4)
+      by (metis closure_iff_nhds_not_empty less_eq_real_def order_refl that(2)) 
+    from flow_to_path_path[OF yt(2) ex]
+    have a1: "path fp" unfolding fp_def .
+    have "y \<in> closure side2" using yt(1)
+      by (simp add: assms closure_def)
+    then have a2: "pathstart fp \<in> closure side2" unfolding fp_def using y0 by auto
+    have a3:"pathfinish fp \<notin> side2" using yt(4) \<open>side1 \<inter> side2 = {}\<close>
+      unfolding fp_def apply auto
+      using closure_iff_nhds_not_empty that(2) by blast
+    from subpath_to_frontier_strong[OF a1 a3]
+    obtain u where u:"0 \<le> u" "u \<le> 1"
+      "fp u \<notin> interior side2"
+      "u = 0 \<or>
+      (\<forall>x. 0 \<le> x \<and> x < 1 \<longrightarrow>
+        subpath 0 u fp x \<in> interior side2) \<and> fp u \<in> closure side2" by blast
+    have p1:"path_image (subpath 0 u fp) =  flow0 y ` {0 ..  u*t}"
+      unfolding fp_def subpath0_flow_to_path using path_image_flow_to_path
+      by (simp add: u(1) yt(2))
+    have p2:"fp u = flow0 y (u*t)" unfolding fp_def flow_to_path_unfold by simp
+    have inout:"interior side2 = side2" using \<open>open side2\<close>
+      by (simp add: interior_eq)
+    then have iemp: "side2 \<inter> path_image J = {}"
+      using \<open>frontier side2 = path_image J\<close>
+      by (metis frontier_disjoint_eq inf_sup_aci(1) interior_eq)
+    have "u \<noteq> 0" using inout u(3) y0 p2 yt(1) by force
+    then have c1:"u * t > 0" using tpos u y0  \<open>side1 \<inter> side2 = {}\<close>
+      using frontier_disjoint_eq io(5) yt(1) zero_less_mult_iff by fastforce
+    have uim:"fp u \<in> path_image J" using u \<open>u \<noteq> 0\<close>
+      using \<open>frontier side2 = path_image J\<close>
+      by (metis ComplI IntI closure_subset frontier_closures inout subsetD) 
+    have c2:"u * t \<le> t"  using u(1-2) tpos by auto
+    have"(flow_to_path y 0 (u * t) ` {0..<1} \<subseteq> side2)"
+      using \<open>u \<noteq> 0\<close> u inout unfolding fp_def subpath0_flow_to_path by auto
+    then have c3:"\<forall>s \<in>{0..<u*t}. flow0 y s \<in> side2" by auto
+    have c4: "flow0 y (u*t) \<in> path_image J"
+      using uim path_image_join_subset
+      by (simp add: p2)
+    have "flow0 y (u*t) \<notin> path_image J1 \<or> flow0 y (u*t) = flow0 x t1"
+    proof clarsimp
+      assume "flow0 y (u*t) \<in> path_image J1"
+      then obtain s where s: "t1 \<le> s" "s \<le> t2" "flow0 x s = flow0 y (u*t)"
+        using J1_def \<open>t1 \<le> t2\<close> by auto
+      have "s = t1"
+      proof (rule ccontr)
+        assume "s \<noteq> t1"
+        then have st1:"s > t1" using s(1) by linarith
+        define sc where "sc = min (s-t1) (u*t)"
+        have scd: "s-sc \<in> {t1..t2}" unfolding sc_def
+          using c1 s(1) s(2) by auto
+        then have *:"flow0 x (s-sc) \<in> path_image J1" unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
+          by blast
+        have "flow0 x (s-sc) = flow0 (flow0 x s) (-sc)"
+          by (smt exist atLeastAtMost_iff existence_ivl_trans' flow_trans s(1) s(2) scd subsetD)
+        then have **:"flow0 (flow0 y (u*t)) (-sc)  \<in> path_image J1"
+          using s(3) * by auto
+        have b:"u*t - sc \<in> {0..<u*t}" unfolding sc_def by (simp add: st1 c1 s(1))
+        then have "u*t - sc \<in> existence_ivl0 y"
+          using c2 ex by auto 
+        then have "flow0 y (u*t - sc) \<in> path_image J1" using **
+          by (smt atLeastAtMost_iff diff_existence_ivl_trans ex flow_trans mult_left_le_one_le mult_nonneg_nonneg subset_eq u(1) u(2) yt(2))
+        thus False using b c3 iemp piJ by blast
+      qed
+      thus "flow0 y (u * t) = flow0 x t1" using s by simp
+    qed
+    thus "\<exists>T>0. T \<le> t \<and> (\<forall>s\<in>{0..<T}. flow0 y s \<in> side2) \<and>
+          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
+      using c1 c2 c3 c4 unfolding piD
+      by (metis DiffE UnE ends_in_segment(1) half_open_segment_closed_segmentI insertCI open_segment_def x1neqx2)
+  qed
+  have outside_in: "\<And>y t. y \<in> outer \<Longrightarrow>
+    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
+    flow0 y t \<in> closure inner \<Longrightarrow>
+    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> outer) \<and>
+          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
+    by (rule swap_side; (rule io | assumption))
+  have inside_out: "\<And>y t. y \<in> inner \<Longrightarrow>
+    0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 y \<Longrightarrow>
+    flow0 y t \<in> closure outer \<Longrightarrow>
+    \<exists>T. 0 < T \<and> T \<le> t \<and> (\<forall>s \<in>{0..<T}. flow0 y s \<in> inner) \<and>
+          flow0 y T \<in> {flow0 x t1--<flow0 x t2}"
+    by (rule swap_side; (rule io2 io | assumption))
+
+  from leaves_transversal_segmentE[OF assms(1)]
+  obtain d n where d: "d > (0::real)"
+    and n: "n = a - b \<or> n = b - a"
+    and d_ex: "\<And>x. x \<in> {a -- b} \<Longrightarrow> {-d..d} \<subseteq> existence_ivl0 x"
+    and d_above: "\<And>x s. x \<in> {a -- b} \<Longrightarrow> 0 < s \<Longrightarrow> s \<le> d \<Longrightarrow> (flow0 x s - x) \<bullet> rot n > 0"
+    and d_below: "\<And>x s. x \<in> {a -- b} \<Longrightarrow> -d \<le> s \<Longrightarrow> s < 0 \<Longrightarrow> (flow0 x s - x) \<bullet> rot n < 0"
+    by blast
+
+  have ortho: "(a - b) \<bullet> rot n = 0"
+    using n by (auto simp: algebra_simps)
+
+(* These "rectangles" are either fully inside or fully outside
+           |-----------------------|
+           |           r1          | (flow d)
+    a --- (t1) --- rp --- (t2) --- b
+    |          r2           | (flow -d)
+    |-----------------------|
+   *)
+  define r1 where "r1 = (\<lambda>(x, y). flow0 x y)`({flow0 x t1<--<b} \<times> {0<..<d}) "
+  have r1a1: "path_connected {flow0 x t1 <--<b}" by simp
+  have r1a2: "path_connected {0<..<d}" by simp
+  have "{flow0 x t1<--<b} \<subseteq> {a--b}"
+    by (simp add: open_closed_segment subset_oc_segment x1)
+  then have r1a3: "y \<in> {flow0 x t1<--<b} \<Longrightarrow> {0<..<d} \<subseteq> existence_ivl0 y" for y
+    using d_ex[of y]
+    by force
+  from flow0_path_connected[OF r1a1 r1a2 r1a3]
+  have pcr1:"path_connected r1" unfolding r1_def by auto
+  have pir1J1: "r1 \<inter> path_image J1 = {}"
+    unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
+  proof (rule ccontr)
+    assume "r1 \<inter> flow0 x ` {t1..t2} \<noteq> {}"
+    then obtain xx tt ss where
+      eq: "flow0 xx tt = flow0 x ss"
+      and xx: "xx \<in> {flow0 x t1<--<b}"
+      and ss: "t1 \<le> ss" "ss \<le> t2"
+      and tt: "0 < tt" "tt < d"
+      unfolding r1_def
+      by force
+    have "xx \<in> {a -- b}"
+      using sub1b
+      apply (rule set_mp)
+      using xx by (simp add: open_closed_segment)
+    then have [simp]: "xx \<in> X" using \<open>transversal_segment a b\<close> by (auto simp: transversal_segment_def)
+    from ss have ss_ex: "ss \<in> existence_ivl0 x" using exist
+      by auto
+    from d_ex[OF \<open>xx \<in> {a -- b}\<close>] tt
+    have tt_ex: "tt \<in> existence_ivl0 xx" by auto
+    then have neg_tt_ex: "- tt \<in> existence_ivl0 (flow0 xx tt)"
+      by (rule existence_ivl_reverse[simplified])
+    from eq have "flow0 (flow0 xx tt) (-tt) = flow0 (flow0 x ss) (-tt)"
+      by simp
+    then have "xx = flow0 x (ss - tt)"
+      apply (subst (asm) flow_trans[symmetric])
+        apply (rule tt_ex)
+       apply (rule neg_tt_ex)
+      apply (subst (asm) flow_trans[symmetric])
+        apply (rule ss_ex)
+       apply (subst eq[symmetric])
+       apply (rule neg_tt_ex)
+      by simp
+    moreover
+    define e where "e = ss - t1"
+    consider "e > tt" | "e \<le> tt" by arith
+    then show False
+    proof cases
+      case 1
+      have "flow0 (flow0 x ss) (-tt) \<notin> {a<--<b}"
+        apply (subst flow_trans[symmetric])
+          apply fact
+        subgoal using neg_tt_ex eq by simp
+        apply (rule t1t2)
+        using 1 ss tt
+        unfolding e_def
+        by auto
+      moreover have "flow0 (flow0 x ss) (-tt) \<in> {a<--<b}"
+        unfolding eq[symmetric] using tt_ex xx
+        apply (subst flow_trans[symmetric])
+          apply (auto simp add: neg_tt_ex)
+        by (metis (no_types, hide_lams) sub1b subset_eq subset_open_segment)
+      ultimately show ?thesis by simp
+    next
+      case 2
+      have les: "0 \<le> tt - e" "tt - e \<le> d"
+        using tt ss 2 e_def
+        by auto
+      have xxtte: "flow0 xx (tt - e) = flow0 x t1"
+        apply (simp add: e_def)
+        by (smt \<open>0 \<le> tt - e\<close> \<open>{- d..d} \<subseteq> existence_ivl0 xx\<close> atLeastAtMost_iff e_def eq
+            local.existence_ivl_reverse local.existence_ivl_trans local.flow_trans ss(1) ss_ex subset_iff tt(2))
+      show False
+      proof (cases "tt = e")
+        case True
+        with xxtte have "xx = flow0 x t1"
+          by (simp add: )
+        with xx show ?thesis
+          apply auto
+          by (auto simp: open_segment_def)
+      next
+        case False
+        with les have "0 < tt - e" by (simp)
+        from d_above[OF \<open>xx \<in> {a -- b}\<close> this \<open>tt - e \<le> d\<close>]
+        have "flow0 xx (tt - e) \<notin> {a -- b}"
+          apply (simp add: in_closed_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
+              not_le )
+          by (smt \<open>xx \<in> {a--b}\<close> inner_minus_right inner_rot_neg_move_base inner_rot_pos_move_base n rot_diff_commute)
+        with xxtte show ?thesis
+          using \<open>flow0 x t1 \<in> {a<--<flow0 x t2}\<close> suba2o by auto
+      qed
+    qed
+  qed
+    (* for sufficiently small d, the flow does not return to the line *)
+  moreover
+  have pir1J2: "r1 \<inter> path_image J2 = {}"
+  proof -
+    have "r1 \<subseteq> {x. (x - a) \<bullet> rot n > 0}"
+      unfolding r1_def
+    proof safe
+      fix aa ba
+      assume "aa \<in> {flow0 x t1<--<b}" "ba \<in> {0<..<d}"
+      with sub1b show "0 < (flow0 aa ba - a) \<bullet> rot n"
+        using segment_open_subset_closed[of "flow0 x t1" b]
+        by (intro inner_pos_move_base[OF ortho d_above]) auto
+    qed
+    also have "\<dots> \<inter> {a -- b} = {}"
+      using in_segment_inner_rot in_segment_inner_rot2 n by auto
+    finally show ?thesis
+      unfolding J2_def path_image_linepath
+      using t12sub open_closed_segment
+      by (force simp: closed_segment_commute)
+  qed
+  ultimately have pir1:"r1 \<inter> (path_image J) = {}" unfolding J_def
+    by (metis disjoint_iff_not_equal not_in_path_image_join)
+
+  define r2 where "r2 =(\<lambda>(x, y). flow0 x y)`({a <--< flow0 x t2} \<times> {-d<..<0})"
+  have r2a1:"path_connected {a <--< flow0 x t2}" by simp
+  have r2a2:"path_connected {-d<..<0}" by simp
+  have "{a <--< flow0 x t2} \<subseteq> {a -- b}"
+    by (meson ends_in_segment(1) open_closed_segment subset_co_segment subset_oc_segment t12sub)
+  then have r2a3: "y \<in> {a <--< flow0 x t2} \<Longrightarrow> {-d<..<0} \<subseteq> existence_ivl0 y" for y
+    using d_ex[of y]
+    by force
+  from flow0_path_connected[OF r2a1 r2a2 r2a3]
+  have pcr2:"path_connected r2" unfolding r2_def by auto
+  have pir2J2: "r2 \<inter> path_image J1 = {}"
+    unfolding J1_def path_image_flow_to_path[OF \<open>t1 \<le> t2\<close>]
+  proof (rule ccontr)
+    assume "r2 \<inter> flow0 x ` {t1..t2} \<noteq> {}"
+    then obtain xx tt ss where
+      eq: "flow0 xx tt = flow0 x ss"
+      and xx: "xx \<in> {a<--<flow0 x t2}"
+      and ss: "t1 \<le> ss" "ss \<le> t2"
+      and tt: "-d < tt" "tt < 0"
+      unfolding r2_def
+      by force
+    have "xx \<in> {a -- b}"
+      using suba2
+      apply (rule set_mp)
+      using xx by (simp add: open_closed_segment)
+    then have [simp]: "xx \<in> X" using \<open>transversal_segment a b\<close> by (auto simp: transversal_segment_def)
+    from ss have ss_ex: "ss \<in> existence_ivl0 x" using exist
+      by auto
+    from d_ex[OF \<open>xx \<in> {a -- b}\<close>] tt
+    have tt_ex: "tt \<in> existence_ivl0 xx" by auto
+    then have neg_tt_ex: "- tt \<in> existence_ivl0 (flow0 xx tt)"
+      by (rule existence_ivl_reverse[simplified])
+    from eq have "flow0 (flow0 xx tt) (-tt) = flow0 (flow0 x ss) (-tt)"
+      by simp
+    then have "xx = flow0 x (ss - tt)"
+      apply (subst (asm) flow_trans[symmetric])
+        apply (rule tt_ex)
+       apply (rule neg_tt_ex)
+      apply (subst (asm) flow_trans[symmetric])
+        apply (rule ss_ex)
+       apply (subst eq[symmetric])
+       apply (rule neg_tt_ex)
+      by simp
+    moreover
+    define e where "e = t2 - ss"
+    consider "e > - tt" | "e \<le> -tt" by arith
+    then show False
+    proof cases
+      case 1
+      have "flow0 (flow0 x ss) (-tt) \<notin> {a<--<b}"
+        apply (subst flow_trans[symmetric])
+          apply fact
+        subgoal using neg_tt_ex eq by simp
+        apply (rule t1t2)
+        using 1 ss tt
+        unfolding e_def
+        by auto
+      moreover have "flow0 (flow0 x ss) (-tt) \<in> {a<--<b}"
+        unfolding eq[symmetric] using tt_ex xx
+        apply (subst flow_trans[symmetric])
+          apply (auto simp add: neg_tt_ex)
+        by (metis (no_types, hide_lams) suba2 subset_eq subset_open_segment)
+      ultimately show ?thesis by simp
+    next
+      case 2
+      have les: "tt + e \<le> 0" "-d \<le> tt + e"
+        using tt ss 2 e_def
+        by auto
+      have xxtte: "flow0 xx (tt + e) = flow0 x t2"
+        apply (simp add: e_def)
+        by (smt atLeastAtMost_iff calculation eq exist local.existence_ivl_trans' local.flow_trans neg_tt_ex ss_ex subset_iff \<open>t1 \<le> t2\<close>)
+      show False
+      proof (cases "tt=-e")
+        case True
+        with xxtte have "xx = flow0 x t2"
+          by (simp add: )
+        with xx show ?thesis
+          apply auto
+          by (auto simp: open_segment_def)
+      next
+        case False
+        with les have "tt+e < 0" by simp
+        from d_below[OF \<open>xx \<in> {a -- b}\<close>  \<open>-d \<le> tt + e\<close> this]
+        have "flow0 xx (tt + e) \<notin> {a -- b}"
+          apply (simp add: in_closed_segment_iff_rot[OF \<open>a \<noteq> b\<close>]
+              not_le )
+          by (smt \<open>xx \<in> {a--b}\<close> inner_minus_right inner_rot_neg_move_base inner_rot_pos_move_base n rot_diff_commute)
+        with xxtte show ?thesis
+          using \<open>flow0 x t2 \<in> {a--b}\<close> by simp
+      qed
+    qed
+  qed
+  moreover
+  have pir2J2: "r2 \<inter> path_image J2 = {}"
+  proof -
+    have "r2 \<subseteq> {x. (x - a) \<bullet> rot n < 0}"
+      unfolding r2_def
+    proof safe
+      fix aa ba
+      assume "aa \<in> {a<--<flow0 x t2}" "ba \<in> {-d<..<0}"
+      with suba2 show "0 > (flow0 aa ba - a) \<bullet> rot n"
+        using segment_open_subset_closed[of a "flow0 x t2"]
+        by (intro inner_neg_move_base[OF ortho d_below]) auto
+    qed
+    also have "\<dots> \<inter> {a -- b} = {}"
+      using in_segment_inner_rot in_segment_inner_rot2 n by auto
+    finally show ?thesis
+      unfolding J2_def path_image_linepath
+      using t12sub open_closed_segment
+      by (force simp: closed_segment_commute)
+  qed
+  ultimately have pir2:"r2 \<inter> (path_image J) = {}"
+    unfolding J_def
+    by (metis disjoint_iff_not_equal not_in_path_image_join)
+
+  define rp where "rp = midpoint (flow0 x t1) (flow0 x t2)"
+  have rpi: "rp \<in> {flow0 x t1<--<flow0 x t2}" unfolding rp_def
+    by (simp add: x1neqx2)
+  have "rp \<in> {a -- b}"
+    using rpi suba2o subl by blast
+  then have [simp]: "rp \<in> X"
+    using \<open>{a--b} \<subseteq> X\<close> by blast
+
+(* The fundamental case distinction *)
+  have *: "pathfinish J1 = flow0 x t2"
+    "pathstart J1 = flow0 x t1"
+    "rp \<in> {flow0 x t2<--<flow0 x t1}"
+    using rpi
+    by (auto simp: open_segment_commute J1_def)
+  have "{y. 0 < (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1)} = {y. 0 < (y - rp) \<bullet> rot (flow0 x t2 - flow0 x t1)}"
+    by (smt Collect_cong in_open_segment_rotD inner_diff_left nrm_dot rpi)
+  also have "... =  {y. 0 > (y - rp) \<bullet> rot (flow0 x t1 - flow0 x t2)}"
+    by (smt Collect_cong inner_minus_left nrm_reverse)
+  also have " ... = {y. 0 > (y - rp) \<bullet> rot (a - b) }"
+    by (metis rot_same_dir(2) x1 x2)
+  finally have side1: "{y. 0 < (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1)} = {y. 0 > (y - rp) \<bullet> rot (a - b) }"
+    (is "_ = ?lower1") .
+  have "{y. (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0} = {y. (y - rp) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0}"
+    by (smt Collect_cong in_open_segment_rotD inner_diff_left nrm_dot rpi)
+  also have "... =  {y. (y - rp) \<bullet> rot (flow0 x t1 - flow0 x t2) > 0}"
+    by (smt Collect_cong inner_minus_left nrm_reverse)
+  also have " ... = {y. 0 < (y - rp) \<bullet> rot (a - b) }"
+    by (metis rot_same_dir(1) x1 x2)
+  finally have side2: "{y. (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1) < 0} = {y. 0 < (y - rp) \<bullet> rot (a - b) }"
+    (is "_ = ?upper1") .
+  from linepath_ball_inside_outside[OF \<open>simple_path J\<close>[unfolded J_def J2_def] *,
+      folded J2_def J_def, unfolded side1 side2]
+  obtain e where e0: "0 < e"
+    "ball rp e \<inter> path_image J1 = {}"
+    "ball rp e \<inter> ?lower1 \<subseteq> inner \<and>
+        ball rp e \<inter> ?upper1 \<subseteq> outer \<or>
+        ball rp e \<inter> ?upper1 \<subseteq> inner \<and>
+        ball rp e \<inter> ?lower1 \<subseteq> outer"
+    by (auto simp: inner_def outer_def)
+
+  let ?lower = "{y. 0 > (y - rp) \<bullet> rot n }"
+  let ?upper = "{y. 0 < (y - rp) \<bullet> rot n }"
+  have "?lower1 = {y. 0 < (y - rp) \<bullet> rot n } \<and> ?upper1 = {y. 0 > (y - rp) \<bullet> rot n } \<or>
+      ?lower1 = {y. 0 > (y - rp) \<bullet> rot n } \<and> ?upper1 =  {y. 0 < (y - rp) \<bullet> rot n }"
+    using n rot_diff_commute[of a b]
+    by auto
+  from this e0 have e: "0 < e"
+    "ball rp e \<inter> path_image J1 = {}"
+    "ball rp e \<inter> ?lower \<subseteq> inner \<and>
+        ball rp e \<inter> ?upper \<subseteq> outer \<or>
+        ball rp e \<inter> ?upper \<subseteq> inner \<and>
+        ball rp e \<inter> ?lower \<subseteq> outer"
+    by auto
+
+  have "\<forall>\<^sub>F t in at_right 0. t < d"
+    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
+  then have evr: "\<forall>\<^sub>F t in at_right 0. 0 < (flow0 rp t - rp) \<bullet> rot n"
+    unfolding eventually_at_filter
+    by eventually_elim (auto intro!: \<open>rp \<in> {a--b}\<close> d_above)
+  have "\<forall>\<^sub>F t in at_left 0. t > -d"
+    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
+  then have evl: "\<forall>\<^sub>F t in at_left 0. 0 > (flow0 rp t - rp) \<bullet> rot n"
+    unfolding eventually_at_filter
+    by eventually_elim (auto intro!: \<open>rp \<in> {a--b}\<close> d_below)
+  have "\<forall>\<^sub>F t in at 0. flow0 rp t \<in> ball rp e"
+    unfolding mem_ball
+    apply (subst dist_commute)
+    apply (rule tendstoD)
+    by (auto intro!: tendsto_eq_intros \<open>0 < e\<close>)
+  then have evl2: "(\<forall>\<^sub>F t in at_left 0. flow0 rp t \<in> ball rp e)"
+    and evr2: "(\<forall>\<^sub>F t in at_right 0. flow0 rp t \<in> ball rp e)"
+    unfolding eventually_at_split by auto
+  have evl3: "(\<forall>\<^sub>F t in at_left 0. t > -d)"
+    and evr3: "(\<forall>\<^sub>F t in at_right 0. t < d)"
+    by (auto intro!: order_tendstoD \<open>0 < d\<close>)
+  have evl4: "(\<forall>\<^sub>F t in at_left 0. t < 0)"
+    and evr4: "(\<forall>\<^sub>F t in at_right 0. t > 0)"
+    by (auto simp: eventually_at_filter)
+  from evl evl2 evl3 evl4
+  have "\<forall>\<^sub>F t in at_left 0. (flow0 rp t - rp) \<bullet> rot n < 0 \<and> flow0 rp t \<in> ball rp e \<and> t > -d \<and> t < 0"
+    by eventually_elim auto
+  from eventually_happens[OF this]
+  obtain dl where dl: "(flow0 rp dl - rp) \<bullet> rot n < 0" "flow0 rp dl \<in> ball rp e" "- d < dl" "dl < 0"
+    by auto
+  from evr evr2 evr3 evr4
+  have "\<forall>\<^sub>F t in at_right 0. (flow0 rp t - rp) \<bullet> rot n > 0 \<and> flow0 rp t \<in> ball rp e \<and> t < d \<and> t > 0"
+    by eventually_elim auto
+  from eventually_happens[OF this]
+  obtain dr where dr: "(flow0 rp dr - rp) \<bullet> rot n > 0" "flow0 rp dr \<in> ball rp e" "d > dr" "dr > 0"
+    by auto
+
+  have "rp \<in> {flow0 x t1<--<b}" using rpi subr by auto
+  then have rpr1:"flow0 rp (dr) \<in> r1" unfolding r1_def using \<open>d > dr\<close> \<open>dr > 0\<close>
+    by auto
+  have "rp \<in> {a<--<flow0 x t2}" using rpi subl by auto
+  then have rpr2:"flow0 rp (dl) \<in> r2" unfolding r2_def using \<open>-d < dl\<close> \<open>dl < 0\<close>
+    by auto
+
+  from e(3) dr dl
+  have "flow0 rp (dr) \<in> outer \<and> flow0 rp (dl) \<in> inner \<or> flow0 rp (dr) \<in> inner \<and> flow0 rp (dl) \<in> outer"
+    by auto
+  moreover {
+    assume "flow0 rp dr \<in> outer" "flow0 rp dl \<in> inner"
+    then have
+      r1o: "r1 \<inter> outer \<noteq> {}" and
+      r2i: "r2 \<inter> inner \<noteq> {}" using rpr1 rpr2 by auto
+    from path_connected_not_frontier_subset[OF pcr1 r1o]
+    have "r1 \<subseteq> outer" using pir1 by (simp add: io(12))
+    from path_connected_not_frontier_subset[OF pcr2 r2i]
+    have "r2 \<subseteq> inner" using pir2 by (simp add: io(11))
+    have "(\<lambda>(x, y). flow0 x y)`({flow0 x t2} \<times> {0<..<d}) \<subseteq> r1" unfolding r1_def
+      by (auto intro!:image_mono simp add: x2)
+    then have *:"\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 (flow0 x t2) t \<in> outer"
+      by (smt \<open>r1 \<subseteq> outer\<close> greaterThanLessThan_iff mem_Sigma_iff pair_imageI r1_def subset_eq x2)
+
+    then have t2o: "\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 x (t2 + t) \<in> outer"
+      using r1a3[OF x2] exist flow_trans
+      by (metis (no_types, hide_lams) closed_segment_commute ends_in_segment(1) local.existence_ivl_trans' local.flow_undefined0 real_Icc_closed_segment subset_eq \<open>t1 \<le> t2\<close>)
+
+(* Construct a sequence of times converging to these points in r2 \<subseteq> inner *)
+    have inner: "{a <--< flow0 x t2} \<subseteq> closure inner"
+    proof (rule subsetI)
+      fix y
+      assume y: "y \<in> {a <--< flow0 x t2}"
+      have [simp]: "y \<in> X"
+        using y suba12_open suba2o \<open>{a -- b} \<subseteq> X\<close>
+        by auto
+      have "(\<forall>n. flow0 y (- d / real (Suc (Suc n))) \<in> inner)"
+        using y
+        using suba12_open \<open>0 < d\<close> suba2o \<open>{a -- b} \<subseteq> X\<close>
+        by (auto intro!: set_mp[OF \<open>r2 \<subseteq> inner\<close>] image_eqI[where x="(y, -d/Suc (Suc n))" for n]
+            simp: r2_def divide_simps) 
+      moreover
+      have d_over_0: "(\<lambda>s. - d / real (Suc (Suc s))) \<longlonglongrightarrow> 0"
+        by (rule real_tendsto_divide_at_top)
+          (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
+      have "(\<lambda>n. flow0 y (- d / real (Suc (Suc n)))) \<longlonglongrightarrow> y"
+        apply (rule tendsto_eq_intros)
+           apply (rule tendsto_intros)
+          apply (rule d_over_0)
+        by auto
+      ultimately show "y \<in> closure inner"
+        unfolding closure_sequential
+        by (intro exI[where x="\<lambda>n. flow0 y (-d/Suc (Suc n))"]) (rule conjI)
+    qed
+    then have "{a <--< flow0 x t1} \<subseteq> closure inner"
+      using suba12_open by blast
+    then have "{flow0 x t1 -- flow0 x t2} \<subseteq> closure inner"
+      by (metis (no_types, lifting) closure_closure closure_mono closure_open_segment dual_order.trans inner subl x1neqx2)
+    have outer:"\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<in> outer"
+    proof (rule ccontr)
+      fix t
+      assume t: "t > t2" "t \<in> existence_ivl0 x" "flow0 x t \<notin> outer"
+      have "0 \<le> t- (t2+d)" using t2o t  by smt 
+      then have a2:"0 \<le> t - (t2+dr)" using d \<open>0 < dr\<close> \<open>dr < d\<close> by linarith
+      have t2d2_ex: "t2 + dr \<in> existence_ivl0 x"
+        using \<open>t1 \<le> t2\<close> exist d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a--b}\<close> \<open>0 < d\<close> \<open>0 < dr\<close> \<open>dr < d\<close> 
+        by (intro existence_ivl_trans) auto
+      then have a3: "t - (t2 + dr) \<in> existence_ivl0 (flow0 x (t2 + dr))"
+        using t(2)
+        by (intro diff_existence_ivl_trans) auto
+      then have "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) = flow0 x t"
+        by (subst flow_trans[symmetric]) (auto simp: t2d2_ex)
+      moreover have "flow0 x t \<in> closure inner" using t(3) io
+        by (metis ComplI Un_iff closure_Un_frontier)
+      ultimately have a4: "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) \<in> closure inner" by auto
+      have a1: "flow0 x (t2+dr) \<in> outer"
+        by (simp add: d t2o \<open>0 < dr\<close> \<open>dr < d\<close>)
+      from outside_in[OF a1 a2 a3 a4]
+      obtain T where T: "T > 0" "T \<le> t - (t2 + dr)"
+        "(\<forall>s\<in>{0..<T}. flow0 (flow0 x (t2 + dr)) s \<in> outer)"
+        "flow0 (flow0 x (t2 + dr)) T \<in> {flow0 x t1 --< flow0 x t2}" by blast
+      define y where "y = flow0 (flow0 x (t2 + dr)) T"
+      have "y \<in> {a <--< flow0 x t2}" unfolding y_def using T(4)
+        using subl2 by blast 
+      then have "(\<lambda>(x, y). flow0 x y)`({y} \<times> {-d<..<0}) \<subseteq> r2" unfolding r2_def
+        by (auto intro!:image_mono)
+      then have *:"\<And>t. -d < t \<Longrightarrow> t < 0 \<Longrightarrow> flow0 y t \<in> r2"
+        by (simp add: pair_imageI subsetD)
+      have "max (-T/2) dl < 0" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
+      moreover have "-d < max (-T/2) dl" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
+      ultimately have inner: "flow0 y (max (-T/2) dl) \<in> inner" using * \<open>r2 \<subseteq> inner\<close> by blast 
+      have "0\<le>(T+(max (-T/2) dl))" using T(1) by linarith
+      moreover have "(T+(max (-T/2) dl)) < T" using T(1) d \<open>0 > dl\<close> \<open>dl > -d\<close> by linarith
+      ultimately have outer: " flow0 (flow0 x (t2 + dr)) (T+(max (-T/2) dl)) \<in> outer"
+        using T by auto
+      have T_ex: "T \<in> existence_ivl0 (flow0 x (t2 + dr))"
+        apply (subst flow_trans)
+        using exist \<open>t1 \<le> t2\<close>
+        using d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a -- b}\<close> \<open>d > 0\<close> T \<open>0 < dr\<close> \<open>dr < d\<close>
+          apply (auto simp: )
+        apply (rule set_rev_mp[where A="{0 .. t - (t2 + dr)}"], force)
+        apply (rule ivl_subset_existence_ivl)
+        apply (rule existence_ivl_trans')
+         apply (rule existence_ivl_trans')
+        by (auto simp: t)
+      have T_ex2: "dr + T \<in> existence_ivl0 (flow0 x t2)"
+        by (smt T_ex ends_in_segment(2) exist local.existence_ivl_trans local.existence_ivl_trans' real_Icc_closed_segment subset_eq t2d2_ex \<open>t1 \<le> t2\<close>)
+      thus False using T \<open>t1 \<le> t2\<close> exist
+        by (smt T_ex diff_existence_ivl_trans disjoint_iff_not_equal inner io(9) local.flow_trans local.flow_undefined0 outer y_def)
+    qed
+    have "closure inner \<inter> outer = {}"
+      by (simp add: inf_sup_aci(1) io(5) io(9) open_Int_closure_eq_empty) 
+    then have "flow0 x t \<notin> {a<--<flow0 x t2}"
+      using \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close> inner outer by blast
+  }
+  moreover {
+    assume "flow0 rp dr \<in> inner" "flow0 rp dl \<in> outer"
+    then have
+      r1i: "r1 \<inter> inner \<noteq> {}" and
+      r2o: "r2 \<inter> outer \<noteq> {}" using rpr1 rpr2 by auto
+    from path_connected_not_frontier_subset[OF pcr1 r1i]
+    have "r1 \<subseteq> inner" using pir1 by (simp add: io(11))
+    from path_connected_not_frontier_subset[OF pcr2 r2o]
+    have "r2 \<subseteq> outer" using pir2 by (simp add: io(12))
+
+    have "(\<lambda>(x, y). flow0 x y)`({flow0 x t2} \<times> {0<..<d}) \<subseteq> r1" unfolding r1_def
+      by (auto intro!:image_mono simp add: x2)
+    then have
+      *:"\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 (flow0 x t2) t \<in> inner"
+      by (smt \<open>r1 \<subseteq> inner\<close> greaterThanLessThan_iff mem_Sigma_iff pair_imageI r1_def subset_eq x2)
+
+    then have t2o: "\<And>t. 0 < t \<Longrightarrow> t < d \<Longrightarrow> flow0 x (t2 + t) \<in> inner"
+      using r1a3[OF x2] exist flow_trans
+      by (metis (no_types, hide_lams) closed_segment_commute ends_in_segment(1) local.existence_ivl_trans' local.flow_undefined0 real_Icc_closed_segment subset_eq \<open>t1 \<le> t2\<close>)
+
+(* Construct a sequence of times converging to these points in r2 \<subseteq> outer *)
+    have outer: "{a <--< flow0 x t2} \<subseteq> closure outer"
+    proof (rule subsetI)
+      fix y
+      assume y: "y \<in> {a <--< flow0 x t2}"
+      have [simp]: "y \<in> X"
+        using y suba12_open suba2o \<open>{a -- b} \<subseteq> X\<close>
+        by auto
+      have "(\<forall>n. flow0 y (- d / real (Suc (Suc n))) \<in> outer)"
+        using y
+        using suba12_open \<open>0 < d\<close> suba2o \<open>{a -- b} \<subseteq> X\<close>
+        by (auto intro!: set_mp[OF \<open>r2 \<subseteq> outer\<close>] image_eqI[where x="(y, -d/Suc (Suc n))" for n]
+            simp: r2_def divide_simps) 
+      moreover
+      have d_over_0: "(\<lambda>s. - d / real (Suc (Suc s))) \<longlonglongrightarrow> 0"
+        by (rule real_tendsto_divide_at_top)
+          (auto intro!: filterlim_tendsto_add_at_top filterlim_real_sequentially)
+      have "(\<lambda>n. flow0 y (- d / real (Suc (Suc n)))) \<longlonglongrightarrow> y"
+        apply (rule tendsto_eq_intros)
+           apply (rule tendsto_intros)
+          apply (rule d_over_0)
+        by auto
+      ultimately show "y \<in> closure outer"
+        unfolding closure_sequential
+        by (intro exI[where x="\<lambda>n. flow0 y (-d/Suc (Suc n))"]) (rule conjI)
+    qed
+    then have "{a <--< flow0 x t1} \<subseteq> closure outer"
+      using suba12_open by blast
+    then have "{flow0 x t1 -- flow0 x t2} \<subseteq> closure outer"
+      by (metis (no_types, lifting) closure_closure closure_mono closure_open_segment dual_order.trans outer subl x1neqx2)
+
+    have inner:"\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<in> inner"
+    proof (rule ccontr)
+      fix t
+      assume t: "t > t2" "t \<in> existence_ivl0 x" "flow0 x t \<notin> inner"
+      have "0 \<le> t- (t2+d)" using t2o t by smt 
+      then have a2:"0 \<le> t - (t2+dr)" using d \<open>0 < dr\<close> \<open>dr < d\<close> by linarith
+      have t2d2_ex: "t2 + dr \<in> existence_ivl0 x"
+        using \<open>t1 \<le> t2\<close> exist d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a--b}\<close> \<open>0 < d\<close> \<open>0 < dr\<close> \<open>dr < d\<close> 
+        by (intro existence_ivl_trans) auto
+      then have a3: "t - (t2 + dr) \<in> existence_ivl0 (flow0 x (t2 + dr))"
+        using t(2)
+        by (intro diff_existence_ivl_trans) auto
+      then have "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) = flow0 x t"
+        by (subst flow_trans[symmetric]) (auto simp: t2d2_ex)
+      moreover have "flow0 x t \<in> closure outer" using t(3) io
+        by (metis ComplI Un_iff closure_Un_frontier)
+      ultimately have a4: "flow0 (flow0 x (t2 + dr)) (t - (t2 + dr)) \<in> closure outer" by auto
+      have a1: "flow0 x (t2+dr) \<in> inner"
+        by (simp add: d t2o \<open>0 < dr\<close> \<open>dr < d\<close>)
+      from inside_out[OF a1 a2 a3 a4]
+      obtain T where T: "T > 0" "T \<le> t - (t2 + dr)"
+        "(\<forall>s\<in>{0..<T}. flow0 (flow0 x (t2 + dr)) s \<in> inner)"
+        "flow0 (flow0 x (t2 + dr)) T \<in> {flow0 x t1 --< flow0 x t2}" by blast
+      define y where "y = flow0 (flow0 x (t2 + dr)) T"
+      have "y \<in> {a <--< flow0 x t2}" unfolding y_def using T(4)
+        using subl2 by blast 
+      then have "(\<lambda>(x, y). flow0 x y)`({y} \<times> {-d<..<0}) \<subseteq> r2" unfolding r2_def
+        by (auto intro!:image_mono)
+      then have *:"\<And>t. -d < t \<Longrightarrow> t < 0 \<Longrightarrow> flow0 y t \<in> r2"
+        by (simp add: pair_imageI subsetD)
+      have "max (-T/2) dl < 0" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
+      moreover have "-d < max (-T/2) dl" using d T \<open>0 > dl\<close> \<open>dl > -d\<close> by auto
+      ultimately have outer: "flow0 y (max (-T/2) dl) \<in> outer" using * \<open>r2 \<subseteq> outer\<close> by blast 
+      have "0\<le>(T+(max (-T/2) dl))" using T(1) by linarith
+      moreover have "(T+(max (-T/2) dl)) < T" using T(1) d \<open>0 > dl\<close> \<open>dl > -d\<close> by linarith
+      ultimately have inner: " flow0 (flow0 x (t2 + dr)) (T+(max (-T/2) dl)) \<in> inner"
+        using T by auto
+      have T_ex: "T \<in> existence_ivl0 (flow0 x (t2 + dr))"
+        apply (subst flow_trans)
+        using exist \<open>t1 \<le> t2\<close>
+        using d_ex[of "flow0 x t2"] \<open>flow0 x t2 \<in> {a -- b}\<close> \<open>d > 0\<close> T \<open>0 < dr\<close> \<open>dr < d\<close>
+          apply (auto simp: )
+        apply (rule set_rev_mp[where A="{0 .. t - (t2 + dr)}"], force)
+        apply (rule ivl_subset_existence_ivl)
+        apply (rule existence_ivl_trans')
+         apply (rule existence_ivl_trans')
+        by (auto simp: t)
+      have T_ex2: "dr + T \<in> existence_ivl0 (flow0 x t2)"
+        by (smt T_ex ends_in_segment(2) exist local.existence_ivl_trans local.existence_ivl_trans' real_Icc_closed_segment subset_eq t2d2_ex \<open>t1 \<le> t2\<close>)
+      thus False using T \<open>t1 \<le> t2\<close> exist
+        by (smt T_ex diff_existence_ivl_trans disjoint_iff_not_equal inner io(9) local.flow_trans local.flow_undefined0 outer y_def)
+    qed
+    have "closure outer \<inter> inner = {}"
+      by (metis inf_sup_aci(1) io(2) io2(1) open_Int_closure_eq_empty)
+    then have "flow0 x t \<notin> {a<--<flow0 x t2}"
+      using \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close> inner outer by blast
+  }
+  ultimately show
+    "flow0 x t \<notin> {a<--<flow0 x t2}" by auto
+qed
+
+lemma open_segment_trichotomy:
+  fixes x y a b::'a
+  assumes x:"x \<in> {a<--<b}"
+  assumes y:"y \<in> {a<--<b}"
+  shows "x = y \<or> y \<in> {x<--<b} \<or> y \<in> {a<--<x}"
+proof -
+  from Un_open_segment[OF y]
+  have "{a<--<y} \<union> {y} \<union> {y<--<b} = {a<--<b}" .
+  then have "x \<in> {a<--<y} \<or> x = y \<or> x \<in> {y <--<b}" using x by blast
+  moreover {
+    assume "x \<in> {a<--<y}"
+    then have "y \<in> {x<--<b}" using open_segment_subsegment
+      using open_segment_commute y by blast
+  }
+  moreover {
+    assume "x \<in> {y<--<b}"
+    from open_segment_subsegment[OF y this]
+    have "y \<in> {a<--<x}" .
+  }
+  ultimately show ?thesis by blast
+qed
+
+sublocale rev: c1_on_open_R2 "-f" "-f'" rewrites "-(-f) = f" and "-(-f') = f'"
+  by unfold_locales (auto simp: dim2)
+
+lemma rev_transversal_segment: "rev.transversal_segment a b = transversal_segment a b"
+  by (auto simp: transversal_segment_def rev.transversal_segment_def)
+
+lemma flow0_transversal_segment_monotone_step_reverse:
+  assumes "transversal_segment a b"
+  assumes "t1 \<le> t2"
+  assumes "{t1..t2} \<subseteq> existence_ivl0 x"
+  assumes x1: "flow0 x t1 \<in> {a<--<b}"
+  assumes x2: "flow0 x t2 \<in> {a<--<flow0 x t1}"
+  assumes "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
+  assumes "t < t1" "t \<in> existence_ivl0 x"
+  shows "flow0 x t \<notin> {a<--<flow0 x t1}"
+proof -
+  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
+  note t1t2 = \<open>\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}\<close>
+  from \<open>transversal_segment a b\<close> have [simp]: "a \<noteq> b" by (simp add: transversal_segment_def)
+  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
+    by (auto simp: in_open_segment_iff_line)
+  from x2 obtain i2 where i2: "flow0 x t2 = line a b i2" "0 < i2" "i2 < i1"
+    by (auto simp: i1 open_segment_line_iff)
+
+  have t2_exist[simp]: "t2 \<in> existence_ivl0 x"
+    using \<open>t1 \<le> t2\<close> exist by auto
+  have t2_mem: "flow0 x t2 \<in> {a<--<b}"
+    and x1_mem: "flow0 x t1 \<in> {flow0 x t2<--<b}"
+    using i1 i2
+    by (auto simp: line_in_subsegment line_line1)
+
+  have transversal': "rev.transversal_segment a b"
+    using \<open>transversal_segment a b\<close> unfolding rev_transversal_segment .
+  have time': "0 \<le> t2 - t1" using \<open>t1 \<le> t2\<close> by simp
+  have [simp, intro]: "flow0 x t2 \<in> X"
+    using exist \<open>t1 \<le> t2\<close>
+    by auto
+  have exivl': "{0..t2 - t1} \<subseteq> rev.existence_ivl0 (flow0 x t2)"
+    using exist \<open>t1 \<le> t2\<close>
+    by (force simp add: rev_existence_ivl_eq0 intro!: existence_ivl_trans')
+  have step': "rev.flow0 (flow0 x t2) (t2-t) \<notin>  {a<--<rev.flow0 (flow0 x t2) (t2 - t1)}"
+    apply (rule rev.flow0_transversal_segment_monotone_step[OF transversal' time' exivl'])
+    using exist \<open>t1 \<le> t2\<close> x1 x2 t2_mem x1_mem t1t2 \<open>t < t1\<close> \<open>t \<in> existence_ivl0 x\<close>
+        apply (auto simp: rev_existence_ivl_eq0 rev_eq_flow existence_ivl_trans' flow_trans[symmetric])
+    by (subst (asm) flow_trans[symmetric]) (auto intro!: existence_ivl_trans')
+  then show ?thesis
+    unfolding rev_eq_flow
+    using \<open>t1 \<le> t2\<close> exist \<open>t < t1\<close> \<open>t \<in> existence_ivl0 x\<close>
+    by (auto simp: flow_trans[symmetric] existence_ivl_trans')
+qed
+
+lemma flow0_transversal_segment_monotone_step_reverse2:
+  assumes transversal: "transversal_segment a b"
+  assumes time: "t1 \<le> t2"
+  assumes exist: "{t1..t2} \<subseteq> existence_ivl0 x"
+  assumes t1: "flow0 x t1 \<in> {a<--<b}"
+  assumes t2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
+  assumes t1t2: "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
+  assumes t: "t < t1" "t \<in> existence_ivl0 x"
+  shows "flow0 x t \<notin> {flow0 x t1<--<b}"
+  using flow0_transversal_segment_monotone_step_reverse[of b a, OF _ time exist, of t]
+    assms
+  by (auto simp: open_segment_commute transversal_segment_commute)
+
+lemma flow0_transversal_segment_monotone_step2:
+  assumes transversal: "transversal_segment a b"
+  assumes time: "t1 \<le> t2"
+  assumes exist: "{t1..t2} \<subseteq> existence_ivl0 x"
+  assumes t1: "flow0 x t1 \<in> {a<--<b}"
+  assumes t2: "flow0 x t2 \<in> {a<--<flow0 x t1}"
+  assumes t1t2: "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"
+  shows "\<And>t. t > t2 \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<notin> {flow0 x t2<--<b}"
+  using flow0_transversal_segment_monotone_step[of b a, OF _ time exist]
+    assms
+  by (auto simp: transversal_segment_commute open_segment_commute)
+
+lemma flow0_transversal_segment_monotone:
+  assumes "transversal_segment a b"
+  assumes "t1 \<le> t2"
+  assumes "{t1..t2} \<subseteq> existence_ivl0 x"
+  assumes x1: "flow0 x t1 \<in> {a<--<b}"
+  assumes x2: "flow0 x t2 \<in> {flow0 x t1<--<b}"
+  assumes "t > t2" "t \<in> existence_ivl0 x"
+  shows "flow0 x t \<notin> {a<--<flow0 x t2}"
+proof -
+  note exist = \<open>{t1..t2} \<subseteq> existence_ivl0 x\<close>
+  note t = \<open>t > t2\<close> \<open>t \<in> existence_ivl0 x\<close>
+  have x1neqx2: "flow0 x t1 \<noteq> flow0 x t2"
+    using open_segment_def x2 by force 
+  then have t1neqt2: "t1 \<noteq> t2" by auto
+  with \<open>t1 \<le> t2\<close> have "t1 < t2" by simp
+
+  from \<open>transversal_segment a b\<close> have [simp]: "a \<noteq> b" by (simp add: transversal_segment_def)
+  from x1 obtain i1 where i1: "flow0 x t1 = line a b i1" "0 < i1" "i1 < 1"
+    by (auto simp: in_open_segment_iff_line)
+  from x2 i1 obtain i2 where i2: "flow0 x t2 = line a b i2" "i1 < i2" "i2 < 1"
+    by (auto simp: line_open_segment_iff)
+  have t2_in: "flow0 x t2 \<in> {a<--<b}"
+    using i1 i2
+    by simp
+
+  let ?T = "{s \<in> {t1..t2}. flow0 x s \<in> {a--b}}"
+  let ?T' = "{s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}}"
+  from flow_transversal_segment_finite_intersections[OF \<open>transversal_segment a b\<close> \<open>t1 \<le> t2\<close> exist]
+  have "finite ?T" .
+  then have "finite ?T'" by (rule finite_subset[rotated]) (auto simp: open_closed_segment)
+  have "?T' \<noteq> {}"
+    by (auto intro!: exI[where x=t1] \<open>t1 < t2\<close> x1)
+  note tm_defined = \<open>finite ?T'\<close> \<open>?T' \<noteq> {}\<close>
+  define tm where "tm = Max ?T'"
+  have "tm \<in> ?T'"
+    unfolding tm_def
+    using tm_defined by (rule Max_in)
+  have tm_in: "flow0 x tm \<in> {a<--<b}"
+    using \<open>tm \<in> ?T'\<close>
+    by auto
+  have tm: "t1 \<le> tm" "tm < t2" "tm \<le> t2"
+    using \<open>tm \<in> ?T'\<close> by auto
+  have tm_Max: "t \<le> tm" if "t \<in> ?T'" for t
+    unfolding tm_def
+    using tm_defined(1) that
+    by (rule Max_ge)
+
+  have tm_exclude: "flow0 x t \<notin> {a<--<b}" if "t \<in> {tm<..<t2}" for t
+    using \<open>tm \<in> ?T'\<close> tm_Max that
+    by auto (meson approximation_preproc_push_neg(2) dual_order.strict_trans2 le_cases)
+  have "{tm..t2} \<subseteq> existence_ivl0 x"
+    using exist tm by auto
+
+  from open_segment_trichotomy[OF tm_in t2_in]
+  consider
+    "flow0 x t2 \<in> {flow0 x tm<--<b}" |
+    "flow0 x t2 \<in> {a<--<flow0 x tm}" |
+    "flow0 x tm = flow0 x t2"
+    by blast
+  then show "flow0 x t \<notin> {a<--<flow0 x t2}"
+  proof cases
+    case 1
+    from flow0_transversal_segment_monotone_step[OF \<open>transversal_segment a b\<close> \<open>tm \<le> t2\<close>
+        \<open>{tm..t2} \<subseteq> existence_ivl0 x\<close> tm_in 1 tm_exclude t]
+    show ?thesis .
+  next
+    case 2
+    have "t1 \<noteq> tm"
+      using 2 x2 i1 i2
+      by (auto simp: line_in_subsegment line_in_subsegment2)
+    then have "t1 < tm" using \<open>t1 \<le> tm\<close> by simp
+    from flow0_transversal_segment_monotone_step_reverse[OF \<open>transversal_segment a b\<close> \<open>tm \<le> t2\<close>
+        \<open>{tm..t2} \<subseteq> existence_ivl0 x\<close> tm_in 2 tm_exclude \<open>t1 < tm\<close>] exist \<open>t1 \<le> t2\<close>
+    have "flow0 x t1 \<notin> {a<--<flow0 x tm}" by auto
+    then have False
+      using x1 x2 2 i1 i2
+      apply (auto simp: line_in_subsegment line_in_subsegment2)
+      by (smt greaterThanLessThan_iff in_open_segment_iff_line line_in_subsegment2 tm_in)
+    then show ?thesis by simp
+  next
+    case 3
+    have "t1 \<noteq> tm"
+      using 3 x2
+      by (auto simp: open_segment_def)
+    then have "t1 < tm" using \<open>t1 \<le> tm\<close> by simp
+    have "range (flow0 x) = flow0 x ` {tm..t2}"
+      apply (rule recurrence_time_restricts_compact_flow'[OF \<open>tm < t2\<close> _ _ 3])
+      using exist \<open>t1 \<le> t2\<close> \<open>t1 < tm\<close> \<open>tm < t2\<close>
+      by auto
+    also have "\<dots> = flow0 x ` (insert t2 {tm<..<t2})"
+      using \<open>tm \<le> t2\<close> 3
+      apply (auto simp: )
+      by (smt greaterThanLessThan_iff image_eqI)
+    finally have "flow0 x t1 \<in> flow0 x ` (insert t2 {tm<..<t2})"
+      by auto
+    then have "flow0 x t1 \<in> flow0 x ` {tm<..<t2}" using x1neqx2
+      by auto
+    moreover have "\<dots> \<inter> {a<--<b} = {}"
+      using tm_exclude
+      by auto
+    ultimately have False using x1 by auto
+    then show ?thesis by blast
+  qed
+qed
+
+subsection \<open>Straightening\<close>
+
+text \<open> This lemma uses the implicit function theorem \<close>
+lemma cross_time_continuous:
+  assumes "transversal_segment a b"
+  assumes "x \<in> {a<--<b}"
+  assumes "e > 0"
+  obtains d t where "d > 0" "continuous_on (ball x d) t"
+    "\<And>y. y \<in> ball x d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}"
+    "\<And>y. y \<in> ball x d \<Longrightarrow> \<bar>t y\<bar> < e"
+    "continuous_on (ball x d) t"
+    "t x = 0"
+proof -
+  have "x \<in> X" using assms segment_open_subset_closed[of a b]
+    by (auto simp: transversal_segment_def)
+  have "a \<noteq> b" using assms by auto
+  define s where "s x = (x - a) \<bullet> rot (b - a)" for x
+  have "s x = 0"
+    unfolding s_def
+    by (subst in_segment_inner_rot) (auto intro!: assms open_closed_segment)
+  have Ds: "(s has_derivative blinfun_inner_left (rot (b - a))) (at x)"
+    (is "(_ has_derivative blinfun_apply (?Ds x)) _")
+    for x
+    unfolding s_def
+    by (auto intro!: derivative_eq_intros)
+  have Dsc: "isCont ?Ds x" by (auto intro!: continuous_intros)
+  have nz: "?Ds x (f x) \<noteq> 0"
+    using assms  apply auto
+    unfolding transversal_segment_def
+    by (smt inner_minus_left nrm_reverse open_closed_segment)
+
+  from flow_implicit_function_at[OF \<open>x \<in> X\<close>, of s, OF \<open>s x = 0\<close> Ds Dsc nz \<open>e > 0\<close>]
+  obtain t d1 where "0 < d1"
+    and t0: "t x = 0"
+    and d1: "(\<And>y. y \<in> cball x d1 \<Longrightarrow> s (flow0 y (t y)) = 0)"
+    "(\<And>y. y \<in> cball x d1 \<Longrightarrow> \<bar>t y\<bar> < e)"
+    "(\<And>y. y \<in> cball x d1 \<Longrightarrow> t y \<in> existence_ivl0 y)"
+    and tc: "continuous_on (cball x d1) t"
+    and t': "(t has_derivative
+        (- blinfun_inner_left (rot (b - a)) /\<^sub>R (blinfun_inner_left (rot (b - a))) (f x)))
+      (at x)"
+    by metis
+  from tc
+  have "t \<midarrow>x\<rightarrow> 0"
+    using \<open>0 < d1\<close>
+    by (auto simp: continuous_on_def at_within_interior t0 dest!: bspec[where x=x])
+  then have ftc: "((\<lambda>y. flow0 y (t y)) \<longlongrightarrow> x) (at x)"
+    by (auto intro!: tendsto_eq_intros simp: \<open>x \<in> X\<close>)
+
+
+
+  define e2 where "e2 = min (dist a x) (dist b x)"
+  have "e2 > 0"
+    using assms
+    by (auto simp: e2_def open_segment_def)
+
+  from tendstoD[OF ftc this] have "\<forall>\<^sub>F y in at x. dist (flow0 y (t y)) x < e2" .
+  moreover
+  let ?S = "{x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
+  have "open ?S" "x \<in> ?S"
+    using \<open>x \<in> {a<--<b}\<close>
+    by (auto simp add: open_segment_line_hyperplanes \<open>a \<noteq> b\<close>
+        intro!: open_Collect_conj open_halfspace_component_gt open_halfspace_component_lt)
+  from topological_tendstoD[OF ftc this] have "\<forall>\<^sub>F y in at x. flow0 y (t y) \<in> ?S" .
+  ultimately
+  have "\<forall>\<^sub>F y in at x. flow0 y (t y) \<in> ball x e2 \<inter> ?S" by eventually_elim (auto simp: dist_commute)
+  then obtain d2 where "0 < d2" and "\<And>y. x \<noteq> y \<Longrightarrow> dist y x < d2 \<Longrightarrow> flow0 y (t y) \<in> ball x e2 \<inter> ?S"
+    by (force simp: eventually_at)
+  then have d2: "dist y x < d2 \<Longrightarrow> flow0 y (t y) \<in> ball x e2 \<inter> ?S" for y
+    using \<open>0 < e2\<close> \<open>x \<in> X\<close> t0 \<open>x \<in> ?S\<close>
+    by (cases "y = x") auto
+
+  define d where "d = min d1 d2"
+  have "d > 0" using \<open>0 < d1\<close> \<open>0 < d2\<close> by (simp add: d_def)
+  moreover have "continuous_on (ball x d) t"
+    by (auto intro!:continuous_on_subset[OF tc] simp add: d_def)
+  moreover
+  have "ball x e2 \<inter> ?S \<inter> {x. s x = 0} \<subseteq> {a<--<b}"
+    by (auto simp add: in_open_segment_iff_rot \<open>a \<noteq> b\<close>) (auto simp: s_def e2_def in_segment)
+  then have "\<And>y. y \<in> ball x d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b}"
+    apply (rule set_mp)
+    using d1 d2 \<open>0 < d2\<close>
+    by (auto simp: d_def e2_def dist_commute)
+  moreover have "\<And>y. y \<in> ball x d \<Longrightarrow> \<bar>t y\<bar> < e"
+    using d1 by (auto simp: d_def)
+  moreover have "continuous_on (ball x d) t"
+    using tc by (rule continuous_on_subset) (auto simp: d_def)
+  moreover have "t x = 0" by (simp add: t0)
+  ultimately show ?thesis ..
+qed
+
+lemma \<omega>_limit_crossings:
+  assumes "transversal_segment a b"
+  assumes pos_ex: "{0..} \<subseteq> existence_ivl0 x"
+  assumes "\<omega>_limit_point x p"
+  assumes "p \<in> {a<--<b}"
+  obtains s where 
+    "s \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
+proof -
+  from assms have "p \<in> X" by (auto simp: transversal_segment_def open_closed_segment)
+  from assms(3)
+  obtain t where
+    "t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>" "(flow0 x \<circ> t) \<longlonglongrightarrow> p"
+    by (auto simp: \<omega>_limit_point_def)
+  note t = \<open>t \<longlonglongrightarrow>\<^bsub>\<^esub> \<infinity>\<close> \<open>(flow0 x \<circ> t) \<longlonglongrightarrow> p\<close>
+  note [tendsto_intros] = t(2)
+  from cross_time_continuous[OF assms(1,4) zero_less_one\<comment> \<open>TODO ??\<close>]
+  obtain \<tau> \<delta>
+    where "0 < \<delta>" "continuous_on (ball p \<delta>) \<tau>" 
+      "\<tau> p = 0" "(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> \<bar>\<tau> y\<bar> < 1)"
+      "(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> flow0 y (\<tau> y) \<in> {a<--<b})"
+    by metis
+  note \<tau> =
+    \<open>(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> flow0 y (\<tau> y) \<in> {a<--<b})\<close>
+    \<open>(\<And>y. y \<in> ball p \<delta> \<Longrightarrow> \<bar>\<tau> y\<bar> < 1)\<close>
+    \<open>continuous_on (ball p \<delta>) \<tau>\<close> \<open>\<tau> p = 0\<close>
+  define s where "s n = t n + \<tau> (flow0 x (t n))" for n
+  have ev_in_ball: "\<forall>\<^sub>F n in at_top. flow0 x (t n) \<in> ball p \<delta>"
+    apply (simp add: )
+    apply (subst dist_commute)
+    apply (rule tendstoD)
+     apply (rule t[unfolded o_def])
+    apply (rule \<open>0 < \<delta>\<close>)
+    done
+  have "filterlim s at_top sequentially"
+  proof (rule filterlim_at_top_mono)
+    show "filterlim (\<lambda>n. -1 + t n) at_top sequentially"
+      by (rule filterlim_tendsto_add_at_top) (auto intro!: filterlim_tendsto_add_at_top t)
+    from ev_in_ball show "\<forall>\<^sub>F x in sequentially. -1 + t x \<le> s x"
+      apply eventually_elim
+      using \<tau>
+      by (force simp : s_def)
+  qed
+  moreover
+  have \<tau>_cont: "\<tau> \<midarrow>p\<rightarrow> \<tau> p"
+    using \<tau>(3) \<open>0 < \<delta>\<close>
+    by (auto simp: continuous_on_def at_within_ball dest!: bspec[where x=p])
+  note [tendsto_intros] = tendsto_compose_at[OF _ this, simplified]
+  have ev1: "\<forall>\<^sub>F n in sequentially. t n > 1"
+    using filterlim_at_top_dense t(1) by auto
+  then have ev_eq: "\<forall>\<^sub>F n in sequentially. flow0 ((flow0 x o t) n) ((\<tau> o (flow0 x o t)) n) = (flow0 x o s) n"
+    using ev_in_ball
+    apply (eventually_elim)
+    apply (drule \<tau>(2))
+    unfolding o_def
+    apply (subst flow_trans[symmetric])
+    using pos_ex
+      apply (auto simp: s_def)
+    apply (rule existence_ivl_trans')
+    by auto
+  then
+  have "\<forall>\<^sub>F n in sequentially.
+  (flow0 x o s) n = flow0 ((flow0 x o t) n) ((\<tau> o (flow0 x o t)) n)"
+    by (simp add: eventually_mono)
+  from \<open>(flow0 x \<circ> t) \<longlonglongrightarrow> p\<close> and \<open>\<tau> \<midarrow>p\<rightarrow> \<tau> p\<close>
+  have
+    "(\<lambda>n. flow0 ((flow0 x \<circ> t) n) ((\<tau> \<circ> (flow0 x \<circ> t)) n))
+  \<longlonglongrightarrow>
+  flow0 p (\<tau> p)"
+    using \<open>\<tau> p = 0\<close> \<tau>_cont \<open>p \<in> X\<close>
+    by (intro tendsto_eq_intros) auto
+  then have "(flow0 x o s) \<longlonglongrightarrow> flow0 p (\<tau> p)"
+    using ev_eq by (rule Lim_transform_eventually)
+  then have "(flow0 x o s) \<longlonglongrightarrow> p"
+    using \<open>p \<in> X\<close> \<open>\<tau> p = 0\<close>
+    by simp
+  moreover
+  {
+    have "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b}"
+      using ev_eq ev_in_ball
+      apply eventually_elim
+      apply (drule sym)
+      apply simp
+      apply (rule \<tau>) by simp
+    moreover have "\<forall>\<^sub>F n in sequentially. s n \<in> existence_ivl0 x"
+      using ev_in_ball ev1
+      apply (eventually_elim)
+      apply (drule \<tau>(2))
+      using pos_ex
+      by (auto simp: s_def)
+    ultimately have "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
+      by eventually_elim auto
+  }
+  ultimately show ?thesis ..
+qed
+
+(* Obvious but frequently used step *)
+lemma filterlim_at_top_tendstoE:
+  assumes "e > 0"
+  assumes "filterlim s at_top sequentially"
+  assumes "(flow0 x \<circ> s) \<longlonglongrightarrow> u"
+  assumes "\<forall>\<^sub>F n in sequentially. P (s n)"
+  obtains m where "m > b" "P m" "dist (flow0 x m) u < e"
+proof -
+  from assms(2) have "\<forall>\<^sub>F n in sequentially. b < s n"
+    by (simp add: filterlim_at_top_dense)
+  moreover have "\<forall>\<^sub>F n in sequentially. norm ((flow0 x \<circ> s) n - u) < e"
+    using assms(3)[THEN tendstoD, OF assms(1)] by (simp add: dist_norm)
+  moreover note assms(4)
+  ultimately have "\<forall>\<^sub>F n in sequentially. b < s n \<and> norm ((flow0 x \<circ> s) n - u) < e \<and> P (s n)"
+    by eventually_elim auto
+  then obtain m where "m > b" "P m" "dist (flow0 x m) u < e"
+    by (auto simp add: eventually_sequentially dist_norm)
+  then show ?thesis ..
+qed
+
+lemma open_segment_separate_left:
+  fixes u v x a b::'a
+  assumes u:"u \<in> {a <--< b}"
+  assumes v:"v \<in> {u <--< b}"
+  assumes x: "dist x u < dist u v" "x \<in> {a <--< b}"
+  shows "x \<in> {a <--< v}"
+proof -
+  have "v \<noteq> x"
+    by (smt dist_commute x(1)) 
+  moreover have "x \<notin> {v<--<b}"
+    by (smt dist_commute dist_in_open_segment open_segment_subsegment v x(1))
+  moreover have "v \<in> {a<--<b}" using v
+    by (metis ends_in_segment(1) segment_open_subset_closed subset_eq subset_segment(4) u)
+  ultimately show ?thesis using open_segment_trichotomy[OF _ x(2)]
+    by blast
+qed
+
+lemma open_segment_separate_right:
+  fixes u v x a b::'a
+  assumes u:"u \<in> {a <--< b}"
+  assumes v:"v \<in> {a <--< u}"
+  assumes x: "dist x u < dist u v" "x \<in> {a <--< b}"
+  shows "x \<in> {v <--< b}"
+proof -
+  have "v \<noteq> x"
+    by (smt dist_commute x(1))
+  moreover have "x \<notin> {a<--<v}"
+    by (smt dist_commute dist_in_open_segment open_segment_commute open_segment_subsegment v x(1))
+  moreover have "v \<in> {a<--<b}" using v
+    by (metis ends_in_segment(1) segment_open_subset_closed subset_eq subset_segment(4) u)
+  ultimately show ?thesis using open_segment_trichotomy[OF _ x(2)]
+    by blast
+qed
+
+lemma no_two_\<omega>_limit_points:
+  assumes transversal: "transversal_segment a b"
+  assumes ex_pos: "{0..} \<subseteq> existence_ivl0 x"
+  assumes u: "\<omega>_limit_point x u" "u \<in> {a<--<b}"
+  assumes v: "\<omega>_limit_point x v" "v \<in> {a<--<b}"
+  assumes uv: "v \<in> {u<--<b}"
+  shows False
+proof -
+  have unotv: "u \<noteq> v" using uv
+    using dist_in_open_segment by blast 
+  define duv where "duv = dist u v / 2"
+  have duv: "duv > 0" unfolding duv_def using unotv by simp
+  from \<omega>_limit_crossings[OF transversal ex_pos u]
+  obtain su where su: "filterlim su at_top sequentially"
+    "(flow0 x \<circ> su) \<longlonglongrightarrow> u"
+    "\<forall>\<^sub>F n in sequentially. flow0 x (su n) \<in> {a<--<b} \<and> su n \<in> existence_ivl0 x" by blast
+  from \<omega>_limit_crossings[OF transversal ex_pos v]
+  obtain sv where sv: "filterlim sv at_top sequentially"
+    "(flow0 x \<circ> sv) \<longlonglongrightarrow> v"
+    "\<forall>\<^sub>F n in sequentially. flow0 x (sv n) \<in> {a<--<b} \<and> sv n \<in> existence_ivl0 x" by blast
+  from filterlim_at_top_tendstoE[OF duv su]
+  obtain su1 where su1:"su1 > 0" "flow0 x su1 \<in> {a<--<b}"
+    "su1 \<in> existence_ivl0 x" "dist (flow0 x su1) u < duv" by auto
+  from filterlim_at_top_tendstoE[OF duv sv, of su1]
+  obtain su2 where su2:"su2 > su1" "flow0 x su2 \<in> {a<--<b}"
+    "su2 \<in> existence_ivl0 x" "dist (flow0 x su2) v < duv" by auto
+  from filterlim_at_top_tendstoE[OF duv su, of su2]
+  obtain su3 where su3:"su3 > su2" "flow0 x su3 \<in> {a<--<b}"
+    "su3 \<in> existence_ivl0 x" "dist (flow0 x su3) u < duv" by auto
+  have *: "su1 \<le> su2" "{su1..su2} \<subseteq> existence_ivl0 x" using su1 su2
+     apply linarith
+    by (metis atLeastatMost_empty_iff empty_iff mvar.closed_segment_subset_domain real_Icc_closed_segment su1(3) su2(3) subset_eq)
+
+(* by construction *)
+  have d1: "dist (flow0 x su1) v \<ge> (dist u v)/2" using su1(4) duv unfolding duv_def
+    by (smt dist_triangle_half_r)
+  have "dist (flow0 x su1) u < dist u v" using su1(4) duv unfolding duv_def by linarith
+  from open_segment_separate_left[OF u(2) uv this su1(2)]
+  have su1l:"flow0 x su1 \<in> {a<--<v}" .
+  have "dist (flow0 x su2) v < dist v (flow0 x su1)" using d1
+    by (smt dist_commute duv_def su2(4))
+  from open_segment_separate_right[OF v(2) su1l this su2(2)]
+  have su2l:"flow0 x su2 \<in> {flow0 x su1<--<b}" .
+  then have su2ll:"flow0 x su2 \<in> {u<--<b}"
+    by (smt dist_commute dist_pos_lt duv_def open_segment_subsegment pos_half_less open_segment_separate_right su2(2) su2(4) u(2) uv v(2) unotv)
+
+  have "dist (flow0 x su2) u \<ge> (dist u v)/2" using su2(4) duv unfolding duv_def
+    by (smt dist_triangle_half_r)
+  then have "dist (flow0 x su3) u < dist u (flow0 x su2)"
+    by (smt dist_commute duv_def su3(4)) 
+  from open_segment_separate_left[OF u(2) su2ll this su3(2)]
+  have su3l:"flow0 x su3 \<in> {a<--<flow0 x su2}" .
+
+  from flow0_transversal_segment_monotone[OF transversal * su1(2) su2l su3(1) su3(3)]
+  have "flow0 x su3 \<notin> {a <--<flow0 x su2}" .
+  thus False using su3l by auto
+qed
+
+
+subsection \<open>Unique Intersection\<close>
+
+text \<open>Perko Section 3.7 Remark 2\<close>
+lemma unique_transversal_segment_intersection:
+  assumes "transversal_segment a b"
+  assumes "{0..} \<subseteq> existence_ivl0 x"
+  assumes "u \<in> \<omega>_limit_set x \<inter> {a<--<b}"
+  shows "\<omega>_limit_set x \<inter> {a<--<b} = {u}"
+proof (rule ccontr)
+  assume "\<omega>_limit_set x \<inter> {a<--<b} \<noteq> {u}"
+  then
+  obtain v where uv: "u \<noteq> v"
+    and v: "\<omega>_limit_point x v" "v \<in> {a<--<b}" using assms unfolding \<omega>_limit_set_def
+    by fastforce
+  have u:"\<omega>_limit_point x u" "u \<in> {a<--<b}" using assms unfolding \<omega>_limit_set_def
+    by auto
+  show False using no_two_\<omega>_limit_points[OF \<open>transversal_segment a b\<close>]
+    by (smt dist_commute dist_in_open_segment open_segment_trichotomy u uv v assms)
+qed
+
+text \<open>Adapted from Perko Section 3.7 Lemma 4 (+ Chicone )\<close>
+lemma periodic_imp_\<omega>_limit_set:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes "periodic_orbit y"
+    "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x"
+  shows "flow0 y `UNIV = \<omega>_limit_set x"
+proof (rule ccontr)
+  note y = \<open>periodic_orbit y\<close> \<open>flow0 y ` UNIV \<subseteq> \<omega>_limit_set x\<close>
+  from trapped_sol_right[OF assms(1-4)] 
+  have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
+  assume "flow0 y `UNIV \<noteq> \<omega>_limit_set x"
+  obtain p where p: "p \<in> \<omega>_limit_set x" "p \<notin> flow0 y ` UNIV"
+    using y(2) apply auto
+    using \<open>range (flow0 y) \<noteq> \<omega>_limit_set x\<close> by blast
+  from \<omega>_limit_set_in_compact_connected[OF assms(1-4)] have
+    wcon: "connected (\<omega>_limit_set x)" .
+  from \<omega>_limit_set_invariant have
+    "invariant (\<omega>_limit_set x)" .
+  from \<omega>_limit_set_in_compact_compact[OF assms(1-4)] have
+    "compact (\<omega>_limit_set x)" .
+  then have sc: "seq_compact (\<omega>_limit_set x)"
+    using compact_imp_seq_compact by blast
+  have y1:"closed (flow0 y ` UNIV)"
+    using closed_orbit_\<omega>_limit_set periodic_orbit_def \<omega>_limit_set_closed y(1) by auto
+  have y2: "flow0 y ` UNIV \<noteq> {}" by simp 
+  let ?py = "infdist p (range (flow0 y))"
+  have "0 < ?py"
+    using y1 y2 p(2)
+    by (rule infdist_pos_not_in_closed)
+  have "\<forall>n::nat. \<exists>z. z \<in> \<omega>_limit_set x - flow0 y ` UNIV \<and>
+    infdist z (flow0 y ` UNIV) < ?py/2^n"
+  proof (rule ccontr)
+    assume " \<not> (\<forall>n. \<exists>z. z \<in> \<omega>_limit_set x - range (flow0 y) \<and>
+                infdist z (range (flow0 y))
+                < infdist p (range (flow0 y)) / 2 ^ n) "
+    then obtain n where n: "(\<forall>z \<in> \<omega>_limit_set x - range (flow0 y).
+      infdist z (range (flow0 y)) \<ge> ?py / 2 ^ n)"
+      using not_less by blast
+    define A where "A = flow0 y ` UNIV"
+    define B where "B = {z. infdist z (range (flow0 y)) \<ge> ?py / 2 ^ n}"
+    have Ac:"closed A" unfolding A_def using y1 by auto
+    have Bc:"closed B" unfolding B_def using infdist_closed by auto
+    have "A \<inter> B = {}"
+    proof (rule ccontr)
+      assume "A \<inter> B \<noteq> {}"
+      then obtain q where q: "q \<in> A" "q \<in> B" by blast
+      have qz:"infdist q (range(flow0 y)) = 0" using q(1) unfolding A_def
+        by simp
+      note \<open>0 < ?py\<close>
+      moreover have "2 ^ n > (0::real)" by auto
+      ultimately have "infdist p (range (flow0 y)) / 2 ^ n > (0::real)"
+        by simp
+      then have qnz: "infdist q(range (flow0 y)) > 0" using q(2) unfolding B_def
+        by auto
+      show False using qz qnz by auto
+    qed
+    then have a1:"A \<inter> B \<inter> \<omega>_limit_set x = {}" by auto
+    have "\<omega>_limit_set x - range(flow0 y) \<subseteq> B" using n B_def by blast
+    then have a2:"\<omega>_limit_set x \<subseteq> A \<union> B" using A_def by auto
+    from connected_closedD[OF wcon a1 a2 Ac Bc]
+    have "A \<inter> \<omega>_limit_set x = {} \<or> B \<inter> \<omega>_limit_set x = {}" .
+    moreover {
+      assume "A \<inter> \<omega>_limit_set x = {}"
+      then have False unfolding A_def using y(2) by blast
+    }
+    moreover {
+      assume "B \<inter> \<omega>_limit_set x = {}"
+      then have False unfolding B_def using p
+        using A_def B_def a2 by blast
+    }
+    ultimately show False by blast
+  qed
+  then obtain s where s: "\<forall>n::nat. (s::nat \<Rightarrow> _) n \<in> \<omega>_limit_set x - flow0 y ` UNIV \<and>
+                      infdist (s n) (flow0 y ` UNIV) < ?py/2^n"
+    by metis
+  then have "\<forall>n. s n \<in> \<omega>_limit_set x" by blast
+  from seq_compactE[OF sc this]
+  obtain l r where lr: "l \<in> \<omega>_limit_set x" "strict_mono r" "(s \<circ> r) \<longlonglongrightarrow> l" by blast
+  have "\<And>n. infdist (s n) (range (flow0 y)) \<le> ?py / 2 ^ n" using s
+    using less_eq_real_def by blast
+  then have "\<And>n. norm(infdist (s n) (range (flow0 y))) \<le> ?py / 2 ^ n"
+    by (auto simp add: infdist_nonneg)
+  from LIMSEQ_norm_0_pow[OF \<open>0 < ?py\<close> _ this]
+  have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> s) \<longlonglongrightarrow> 0"
+    by (auto simp add:o_def)
+  from LIMSEQ_subseq_LIMSEQ[OF this lr(2)]
+  have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> (s \<circ> r)) \<longlonglongrightarrow> 0" by (simp add: o_assoc)
+  moreover have "((\<lambda>z. infdist z (flow0 y ` UNIV)) \<circ> (s \<circ> r)) \<longlonglongrightarrow> infdist l (flow0 y ` UNIV)"
+    by (auto intro!: tendsto_eq_intros tendsto_compose_at[OF lr(3)])
+  ultimately have "infdist l (flow0 y `UNIV) = 0" using LIMSEQ_unique by auto
+  then have lu: "l \<in> flow0 y ` UNIV" using in_closed_iff_infdist_zero[OF y1 y2] by auto
+  then have l1:"l \<in> X"
+    using closed_orbit_global_existence periodic_orbit_def y(1) by auto
+      (* TODO: factor out as periodic_orbitE *)
+  have l2:"f l \<noteq> 0"
+    by (smt \<open>l \<in> X\<close> \<open>l \<in> range (flow0 y)\<close> closed_orbit_global_existence fixed_point_imp_closed_orbit_period_zero(2) fixpoint_sol(2) image_iff local.flows_reverse periodic_orbit_def y(1))
+  from transversal_segment_exists[OF l1 l2]
+  obtain a b where ab: "transversal_segment a b" "l \<in> {a<--<b}" by blast
+  then have "l \<in> \<omega>_limit_set x \<inter> {a<--<b}" using lr by auto
+  from unique_transversal_segment_intersection[OF ab(1) ex_pos this]
+  have luniq: "\<omega>_limit_set x \<inter> {a<--<b} = {l} " .
+  from cross_time_continuous[OF ab, of 1]
+  obtain d t where dt: "0 < d"
+    "(\<And>y. y \<in> ball l d \<Longrightarrow> flow0 y (t y) \<in> {a<--<b})"
+    "(\<And>y. y \<in> ball l d \<Longrightarrow> \<bar>t y\<bar> < 1)"
+    "continuous_on (ball l d) t" "t l = 0"
+    by auto
+  obtain n where "(s \<circ> r) n \<in> ball l d" using lr(3) dt(1) unfolding LIMSEQ_iff_nz
+    by (metis dist_commute mem_ball order_refl)
+  then have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> {a<--<b}" using dt by auto
+  moreover have sr: "(s \<circ> r) n \<in> \<omega>_limit_set x" "(s \<circ> r) n \<notin> flow0 y ` UNIV"
+    using s by auto
+  moreover have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> \<omega>_limit_set x"
+    using \<open>invariant (\<omega>_limit_set x)\<close> calculation unfolding invariant_def trapped_def
+    by (smt \<omega>_limit_set_in_compact_subset \<open>invariant (\<omega>_limit_set x)\<close> assms(1-4) invariant_def order_trans range_eqI subsetD trapped_iff_on_existence_ivl0 trapped_sol)
+  ultimately have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) \<in> \<omega>_limit_set x \<inter> {a<--<b}" by auto
+  from unique_transversal_segment_intersection[OF ab(1) ex_pos this]
+  have "flow0 ((s \<circ> r) n) (t ((s \<circ> r) n )) = l" using luniq by auto
+  then have "((s \<circ> r) n) = flow0 l (-(t ((s \<circ> r) n ))) "
+    by (smt UNIV_I \<open>(s \<circ> r) n \<in> \<omega>_limit_set x\<close> flows_reverse \<omega>_limit_set_in_compact_existence assms(1-4)) 
+  thus False using sr(2) lu
+      \<open>flow0 ((s \<circ> r) n) (t ((s \<circ> r) n)) = l\<close> \<open>flow0 ((s \<circ> r) n) (t ((s \<circ> r) n)) \<in> \<omega>_limit_set x\<close>
+      closed_orbit_global_existence image_iff local.flow_trans periodic_orbit_def \<omega>_limit_set_in_compact_existence range_eqI assms y(1)
+    by smt
+qed
+
+end context c1_on_open_R2 begin
+
+lemma \<alpha>_limit_crossings:
+  assumes "transversal_segment a b"
+  assumes pos_ex: "{..0} \<subseteq> existence_ivl0 x"
+  assumes "\<alpha>_limit_point x p"
+  assumes "p \<in> {a<--<b}"
+  obtains s where
+    "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>"
+    "(flow0 x \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>\<^sub>F n in sequentially.
+    flow0 x (s n) \<in> {a<--<b} \<and>
+    s n \<in> existence_ivl0 x"
+proof -
+  from pos_ex have "{0..} \<subseteq> uminus ` existence_ivl0 x" by force
+  from rev.\<omega>_limit_crossings[unfolded rev_transversal_segment rev_existence_ivl_eq0 rev_eq_flow
+      \<alpha>_limit_point_eq_rev[symmetric], OF assms(1) this assms(3,4)]
+  obtain s where "filterlim s at_top sequentially" "((\<lambda>t. flow0 x (- t)) \<circ> s) \<longlonglongrightarrow> p"
+    "\<forall>\<^sub>F n in sequentially. flow0 x (- s n) \<in> {a<--<b} \<and> s n \<in> uminus ` existence_ivl0 x" .
+  then have "filterlim (-s) at_bot sequentially"
+    "(flow0 x \<circ> (-s)) \<longlonglongrightarrow> p"
+    "\<forall>\<^sub>F n in sequentially. flow0 x ((-s) n) \<in> {a<--<b} \<and> (-s) n \<in> existence_ivl0 x"
+    by (auto simp: fun_Compl_def o_def filterlim_uminus_at_top)
+  then show ?thesis ..
+qed
+
+text \<open>If a positive limit point has a regular point in its positive limit set then it is periodic\<close>
+lemma \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes y: "y \<in> \<omega>_limit_set x" "f y \<noteq> 0"
+  assumes z: "z \<in> \<omega>_limit_set y \<union> \<alpha>_limit_set y" "f z \<noteq> 0"
+  shows "periodic_orbit y \<and> flow0 y ` UNIV = \<omega>_limit_set x"
+proof -
+  from trapped_sol_right[OF assms(1-4)] have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
+  from \<omega>_limit_set_in_compact_existence[OF assms(1-4) y(1)]
+  have yex: "existence_ivl0 y = UNIV" .
+  from \<omega>_limit_set_invariant
+  have "invariant (\<omega>_limit_set x)" .
+  then have yinv: "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x" using yex unfolding invariant_def
+    using trapped_iff_on_existence_ivl0 y(1) by blast 
+
+  have zy: "\<omega>_limit_point y z \<or> \<alpha>_limit_point y z"
+    using z unfolding \<omega>_limit_set_def \<alpha>_limit_set_def by auto
+
+  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
+    \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained[OF assms(1-4)]
+  have zx:"z \<in> \<omega>_limit_set x" using zy y
+    using z(1) by blast
+  then have "z \<in> X"
+    by (metis UNIV_I local.existence_ivl_initial_time_iff \<omega>_limit_set_in_compact_existence assms(1-4))
+  from transversal_segment_exists[OF this z(2)]
+  obtain a b where ab: "transversal_segment a b" "z \<in> {a<--<b}" by blast
+
+  from zy
+  obtain t1 t2 where t1: "flow0 y t1 \<in> {a<--<b}" and t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
+  proof
+    assume zy: "\<omega>_limit_point y z"
+    from \<omega>_limit_crossings[OF ab(1) _ zy ab(2), unfolded yex]
+    obtain s where s: "filterlim s at_top sequentially"
+      "(flow0 y \<circ> s) \<longlonglongrightarrow> z"
+      "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b}"
+      by auto
+    from eventually_happens[OF this(3)] obtain t1 where t1: "flow0 y t1 \<in> {a<--<b}" by auto
+    have "\<forall>\<^sub>F n in sequentially. s n > t1"
+      using filterlim_at_top_dense s(1) by auto
+    with s(3) have "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b} \<and> s n > t1"
+      by eventually_elim simp
+    from eventually_happens[OF this] obtain t2 where t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
+      by auto
+    from t1 this show ?thesis ..
+  next
+    assume zy: "\<alpha>_limit_point y z"
+    from \<alpha>_limit_crossings[OF ab(1) _ zy ab(2), unfolded yex]
+    obtain s where s: "filterlim s at_bot sequentially"
+      "(flow0 y \<circ> s) \<longlonglongrightarrow> z"
+      "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b}"
+      by auto
+    from eventually_happens[OF this(3)] obtain t1 where t1: "flow0 y t1 \<in> {a<--<b}" by auto
+    have "\<forall>\<^sub>F n in sequentially. s n < t1"
+      using filterlim_at_bot_dense s(1) by auto
+    with s(3) have "\<forall>\<^sub>F n in sequentially. flow0 y (s n) \<in> {a<--<b} \<and> s n < t1"
+      by eventually_elim simp
+    from eventually_happens[OF this] obtain t2 where t2: "flow0 y t2 \<in> {a<--<b}" and "t1 \<noteq> t2"
+      by auto
+    from t1 this show ?thesis ..
+  qed
+  have "flow0 y t1 \<in> \<omega>_limit_set x \<inter> {a<--<b}" using t1 UNIV_I yinv by auto
+  moreover have "flow0 y t2 \<in> \<omega>_limit_set x \<inter> {a<--<b}" using t2 UNIV_I yinv by auto
+  ultimately have feq:"flow0 y t1 = flow0 y t2"
+    using unique_transversal_segment_intersection[OF \<open>transversal_segment a b\<close> ex_pos]
+    by blast
+  have "t1 \<noteq> t2" "t1 \<in> existence_ivl0 y" "t2 \<in> existence_ivl0 y" using \<open>t1 \<noteq>  t2\<close>
+      apply blast
+     apply (simp add: yex)
+    by (simp add: yex)
+  from periodic_orbitI[OF this feq y(2)]
+  have 1: "periodic_orbit y" .
+  from periodic_imp_\<omega>_limit_set[OF assms(1-4) this yinv]
+  have 2: "flow0 y` UNIV = \<omega>_limit_set x" .
+  show ?thesis using 1 2 by auto
+qed
+
+subsection \<open>Poincare Bendixson Theorems\<close>
+text \<open>Perko Section 3.7 Theorem 1\<close>
+theorem poincare_bendixson:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes "0 \<notin> f ` (\<omega>_limit_set x)"
+  obtains y where "periodic_orbit y"
+    "flow0 y ` UNIV = \<omega>_limit_set x"
+proof -
+  note f = \<open>0 \<notin> f ` (\<omega>_limit_set x)\<close>
+  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-4)]
+  obtain y where y: "y \<in> \<omega>_limit_set x" by fastforce
+  from \<omega>_limit_set_in_compact_existence[OF  assms(1-4) y]
+  have yex: "existence_ivl0 y = UNIV" .
+  from \<omega>_limit_set_invariant
+  have "invariant (\<omega>_limit_set x)" .
+  then have yinv: "flow0 y ` UNIV \<subseteq> \<omega>_limit_set x" using yex unfolding invariant_def
+    using trapped_iff_on_existence_ivl0 y by blast
+  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
+  have "\<omega>_limit_set x \<subseteq> K" .
+  then have "flow0 y ` UNIV \<subseteq> K" using yinv by auto
+  then have yk:"trapped_forward y K"
+    by (simp add: image_subsetI range_subsetD trapped_forward_def)
+  have "y \<in> X"
+    by (simp add: local.mem_existence_ivl_iv_defined(2) yex)
+
+  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-2) this _]
+  obtain z where z: "z \<in> \<omega>_limit_set y" using yk by blast
+  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
+  have zx:"z \<in> \<omega>_limit_set x" using \<open>z \<in> \<omega>_limit_set y\<close> y by auto
+
+  have yreg : "f y \<noteq> 0" using f y
+    by (metis rev_image_eqI)
+  have zreg : "f z \<noteq> 0" using f zx
+    by (metis rev_image_eqI) 
+  from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) y yreg _ zreg] z
+  show ?thesis using that by blast
+qed
+
+lemma fixed_point_in_\<omega>_limit_set_imp_\<omega>_limit_set_singleton_fixed_point:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes fp: "yfp \<in> \<omega>_limit_set x" "f yfp = 0"
+  assumes zpx: "z \<in> \<omega>_limit_set x"
+  assumes finite_fp: "finite {y \<in> K. f y = 0}" (is "finite ?S")
+  shows "(\<exists>p1 \<in> \<omega>_limit_set x. f p1 = 0 \<and> \<omega>_limit_set z = {p1}) \<and>
+    (\<exists>p2 \<in> \<omega>_limit_set x. f p2 = 0 \<and> \<alpha>_limit_set z = {p2})"
+proof -
+  let ?weq = "{y \<in> \<omega>_limit_set x. f y = 0}"
+  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
+  have wxK: "\<omega>_limit_set x \<subseteq> K" .
+  from \<omega>_limit_set_in_compact_\<omega>_limit_set_contained[OF assms(1-4)]
+  have zx: "\<omega>_limit_set z \<subseteq> \<omega>_limit_set x" using zpx by auto
+  have zX: "z \<in> X" using subset_trans[OF wxK assms(2)]
+    by (metis subset_iff zpx)
+  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
+  have "?weq \<subseteq> ?S"
+    by (smt Collect_mono_iff Int_iff inf.absorb_iff1)
+  then have "finite ?weq" using \<open>finite ?S\<close>
+    by (blast intro: rev_finite_subset) 
+
+  consider "f z = 0" | "f z \<noteq> 0" by auto
+  then show ?thesis
+  proof cases
+    assume "f z = 0"
+    from fixed_point_imp_\<omega>_limit_set[OF zX this]
+      fixed_point_imp_\<alpha>_limit_set[OF zX this]
+    show ?thesis
+      by (metis (mono_tags) \<open>f z = 0\<close> zpx)
+  next
+    assume "f z \<noteq> 0"
+    have zweq: "\<omega>_limit_set z \<subseteq> ?weq"
+      apply clarsimp
+    proof (rule ccontr)
+      fix k assume k: "k \<in> \<omega>_limit_set z" "\<not> (k \<in> \<omega>_limit_set x \<and> f k = 0)"
+      then have "f k \<noteq> 0" using zx k by auto
+      from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) zpx \<open>f z \<noteq> 0\<close> _ this] k(1)
+      have "periodic_orbit z" "range(flow0 z) = \<omega>_limit_set x" by auto
+      then have "0 \<notin> f ` (\<omega>_limit_set x)"
+        by (metis image_iff periodic_orbit_imp_flow0_regular)
+      thus False using fp
+        by (metis (mono_tags, lifting) empty_Collect_eq image_eqI)
+    qed
+    have zweq0: "\<alpha>_limit_set z \<subseteq> ?weq"
+      apply clarsimp
+    proof (rule ccontr)
+      fix k assume k: "k \<in> \<alpha>_limit_set z" "\<not> (k \<in> \<omega>_limit_set x \<and> f k = 0)"
+      then have "f k \<noteq> 0" using zx k
+          \<omega>_limit_set_in_compact_\<alpha>_limit_set_contained[OF assms(1-4), of z] zpx
+        by auto
+      from \<omega>_limit_point_\<omega>_limit_set_regular_imp_periodic[OF assms(1-4) zpx \<open>f z \<noteq> 0\<close> _ this] k(1)
+      have "periodic_orbit z" "range(flow0 z) = \<omega>_limit_set x" by auto
+      then have "0 \<notin> f ` (\<omega>_limit_set x)"
+        by (metis image_iff periodic_orbit_imp_flow0_regular)
+      thus False using fp
+        by (metis (mono_tags, lifting) empty_Collect_eq image_eqI)
+    qed
+    from \<omega>_limit_set_in_compact_existence[OF assms(1-4) zpx]
+    have zex: "existence_ivl0 z = UNIV" .
+    from \<omega>_limit_set_invariant
+    have "invariant (\<omega>_limit_set x)" .
+    then have zinv: "flow0 z ` UNIV \<subseteq> \<omega>_limit_set x" using zex unfolding invariant_def
+      using trapped_iff_on_existence_ivl0 zpx by blast
+    then have "flow0 z ` UNIV \<subseteq> K" using wxK by auto
+    then have a2: "trapped_forward z K" "trapped_backward z K"
+      using trapped_def trapped_iff_on_existence_ivl0 apply fastforce
+      using \<open>range (flow0 z) \<subseteq> K\<close> trapped_def trapped_iff_on_existence_ivl0 by blast
+    have a3: "finite (\<omega>_limit_set z)"
+      by (metis \<open>finite ?weq\<close> finite_subset zweq)
+    from finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-2) zX a2(1) a3]
+    obtain p1 where p1: "\<omega>_limit_set z = {p1}" "f p1 = 0" by blast
+    then have "p1 \<in> ?weq" using zweq by blast
+    moreover
+    have "finite (\<alpha>_limit_set z)"
+      by (metis \<open>finite ?weq\<close> finite_subset zweq0)
+    from finite_\<alpha>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-2) zX a2(2) this]
+    obtain p2 where p2: "\<alpha>_limit_set z = {p2}" "f p2 = 0" by blast
+    then have "p2 \<in> ?weq" using zweq0 by blast
+    ultimately show ?thesis
+      by (simp add: p1 p2)
+  qed
+qed
+
+end context c1_on_open_R2 begin
+
+text \<open>Perko Section 3.7 Theorem 2\<close>
+theorem poincare_bendixson_general:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> X" "trapped_forward x K"
+  assumes "S = {y \<in> K. f y = 0}" "finite S"
+  shows
+    "(\<exists>y \<in> S. \<omega>_limit_set x = {y}) \<or>
+  (\<exists>y. periodic_orbit y \<and>
+    flow0 y ` UNIV = \<omega>_limit_set x) \<or>
+  (\<exists>P R. \<omega>_limit_set x = P \<union> R \<and>
+    P \<subseteq> S \<and> 0 \<notin> f ` R \<and> R \<noteq> {} \<and>
+    (\<forall>z \<in> R.
+      (\<exists>p1 \<in> P. \<omega>_limit_set z = {p1}) \<and>
+      (\<exists>p2 \<in> P. \<alpha>_limit_set z = {p2})))"
+proof -
+  note S = \<open>S = {y \<in> K. f y = 0}\<close>
+  let ?wreg = "{y \<in> \<omega>_limit_set x. f y \<noteq> 0}"
+  let ?weq = "{y \<in> \<omega>_limit_set x. f y = 0}"
+  have wreqweq: "?wreg \<union> ?weq = \<omega>_limit_set x"
+    by (smt Collect_cong Collect_disj_eq mem_Collect_eq \<omega>_limit_set_def)
+
+  from trapped_sol_right[OF assms(1-4)] have ex_pos: "{0..} \<subseteq> existence_ivl0 x" by blast
+  from \<omega>_limit_set_in_compact_subset[OF assms(1-4)]
+  have wxK: "\<omega>_limit_set x \<subseteq> K" .
+  then have "?weq \<subseteq> S" using S
+    by (smt Collect_mono_iff Int_iff inf.absorb_iff1)
+  then have "finite ?weq" using \<open>finite S\<close>
+    by (metis rev_finite_subset) 
+  from \<omega>_limit_set_invariant
+  have xinv: "invariant (\<omega>_limit_set x)" .
+
+  from \<omega>_limit_set_in_compact_nonempty[OF assms(1-4)] wreqweq
+  consider "?wreg = {}" |
+    "?weq = {}" |
+    "?weq \<noteq> {}" "?wreg \<noteq> {}" by auto
+  then show ?thesis
+  proof cases
+    (* If w has no regular points then it is equal to a single unique fixed point *)
+    assume "?wreg = {}"
+    then have "finite (\<omega>_limit_set x)"
+      by (metis (mono_tags, lifting) \<open>{y \<in> \<omega>_limit_set x. f y = 0} \<subseteq> S\<close> \<open>finite S\<close> rev_finite_subset sup_bot.left_neutral wreqweq)
+    from finite_\<omega>_limit_set_in_compact_imp_unique_fixed_point[OF assms(1-4) this]
+    obtain y where y: "\<omega>_limit_set x = {y}" "f y = 0" by blast
+    then have "y \<in> S"
+      by (metis Un_empty_left \<open>?weq \<subseteq> S\<close> \<open>?wreg = {}\<close> insert_subset wreqweq)
+    then show ?thesis using y by auto
+  next
+    (* If w has no fixed points, then the Poincare Bendixson theorem applies *)
+    assume "?weq = {}"
+    then have " 0 \<notin> f ` \<omega>_limit_set x"
+      by (smt empty_Collect_eq imageE)
+    from poincare_bendixson[OF assms(1-4) this]
+    have "(\<exists>y. periodic_orbit y \<and> flow0 y ` UNIV = \<omega>_limit_set x)"
+      by metis
+    then show ?thesis by blast
+  next
+    (* Otherwise, all points in the limit set converge to a finite subset of the fixed points *)
+    assume "?weq \<noteq> {}" "?wreg \<noteq> {}"
+    then obtain yfp where yfp: "yfp \<in> \<omega>_limit_set x" "f yfp = 0" by auto
+    have "0 \<notin> f ` ?wreg" by auto
+    have "(\<exists>p1\<in>\<omega>_limit_set x. f p1 = 0 \<and> \<omega>_limit_set z = {p1}) \<and>
+      (\<exists>p2\<in>\<omega>_limit_set x. f p2 = 0 \<and> \<alpha>_limit_set z = {p2})"
+      if zpx: "z \<in> \<omega>_limit_set x" for z
+      using fixed_point_in_\<omega>_limit_set_imp_\<omega>_limit_set_singleton_fixed_point[
+          OF assms(1-4) yfp zpx \<open>finite S\<close>[unfolded S]] by auto
+    then have "\<omega>_limit_set x = ?weq \<union> ?wreg \<and>
+        ?weq \<subseteq> S \<and> 0 \<notin> f ` ?wreg \<and> ?wreg \<noteq> {} \<and>
+        (\<forall>z \<in> ?wreg.
+         (\<exists>p1 \<in> ?weq. \<omega>_limit_set z = {p1}) \<and>
+         (\<exists>p2 \<in> ?weq. \<alpha>_limit_set z = {p2}))"
+      using wreqweq \<open>?weq \<subseteq> S\<close> \<open>?wreg \<noteq> {}\<close> \<open>0 \<notin> f ` ?wreg\<close>
+      by blast
+    then show ?thesis by blast
+  qed
+qed
+
+corollary poincare_bendixson_applied:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "K \<noteq> {}" "positively_invariant K"
+  assumes "0 \<notin> f ` K"
+  obtains y where "periodic_orbit y" "flow0 y ` UNIV \<subseteq> K"
+proof -
+  from assms(1-4) obtain x where "x \<in> K" "x \<in> X" by auto
+  have *: "trapped_forward x K"
+    using assms(4) \<open>x \<in> K\<close>
+    by (auto simp: positively_invariant_def)
+  have subs: "\<omega>_limit_set x \<subseteq> K"
+    by (rule \<omega>_limit_set_in_compact_subset[OF assms(1-2) \<open>x \<in> X\<close> *])
+  with assms(5) have "0 \<notin> f ` \<omega>_limit_set x" by auto
+  from poincare_bendixson[OF assms(1-2) \<open>x \<in> X\<close> * this]
+  obtain y where "periodic_orbit y" "range (flow0 y) = \<omega>_limit_set x"
+    by force
+  then have "periodic_orbit y" "flow0 y ` UNIV \<subseteq> K" using subs by auto
+  then show ?thesis ..
+qed
+
+(*
+  Limit cycles are periodic orbits that is the \<omega> (or \<alpha>)-limit set of some point not in the cycle.
+  As with periodic_orbit, limit_cycles are defined for a representative point y
+*)
+definition "limit_cycle y \<longleftrightarrow>
+  periodic_orbit y \<and>
+  (\<exists>x. x \<notin> flow0 y ` UNIV \<and>
+  (flow0 y ` UNIV = \<omega>_limit_set x \<or> flow0 y ` UNIV = \<alpha>_limit_set x))"
+
+corollary poincare_bendixson_limit_cycle:
+  assumes "compact K" "K \<subseteq> X"
+  assumes "x \<in> K" "positively_invariant K"
+  assumes "0 \<notin> f ` K"
+  assumes "rev.flow0 x t \<notin> K"
+  obtains y where "limit_cycle y" "flow0 y ` UNIV \<subseteq> K"
+proof -
+  have "x \<in> X" using assms(2-3) by blast
+  have *: "trapped_forward x K"
+    using assms(3-4)
+    by (auto simp: positively_invariant_def)
+  have subs: "\<omega>_limit_set x \<subseteq> K"
+    by (rule \<omega>_limit_set_in_compact_subset[OF assms(1-2) \<open>x \<in> X\<close> *])
+  with assms(5) have "0 \<notin> f ` \<omega>_limit_set x" by auto
+  from poincare_bendixson[OF assms(1-2) \<open>x \<in> X\<close> * this]
+  obtain y where y: "periodic_orbit y" "range (flow0 y) = \<omega>_limit_set x"
+    by force
+  then have c2: "flow0 y ` UNIV \<subseteq> K" using subs by auto
+  have exy: "existence_ivl0 y = UNIV"
+    using closed_orbit_global_existence periodic_orbit_def y(1) by blast
+  have "x \<notin> flow0 y ` UNIV"
+  proof clarsimp
+    fix tt
+    assume "x = flow0 y tt"
+    then have "rev.flow0 (flow0 y tt) t \<notin> K" using assms(6) by auto
+    moreover have "rev.flow0 (flow0 y tt) t \<in> flow0 y ` UNIV" using exy unfolding rev_eq_flow
+      using UNIV_I \<open>x = flow0 y tt\<close> closed_orbit_\<omega>_limit_set closed_orbit_flow0 periodic_orbit_def y by auto
+    ultimately show False using c2 by blast
+  qed
+  then have "limit_cycle y" "flow0 y ` UNIV \<subseteq> K" using y c2 unfolding limit_cycle_def by auto
+  then show ?thesis ..
+qed
+
+end
+
+end
diff --git a/thys/Poincare_Bendixson/ROOT b/thys/Poincare_Bendixson/ROOT
--- a/thys/Poincare_Bendixson/ROOT
+++ b/thys/Poincare_Bendixson/ROOT
@@ -1,9 +1,9 @@
-chapter AFP
-
-session Poincare_Bendixson (AFP) = "HOL-ODE-Numerics" +
-  options [timeout = 600]
-  theories
-    Poincare_Bendixson
-    Examples
-  document_files
-    "root.tex"
+chapter AFP
+
+session Poincare_Bendixson (AFP) = "HOL-ODE-Numerics" +
+  options [timeout = 600]
+  theories
+    Poincare_Bendixson
+    Examples
+  document_files
+    "root.tex"