diff --git a/src/HOL/Divides.thy b/src/HOL/Divides.thy
--- a/src/HOL/Divides.thy
+++ b/src/HOL/Divides.thy
@@ -1,674 +1,675 @@
 (*  Title:      HOL/Divides.thy
     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     Copyright   1999  University of Cambridge
 *)
 
 section \<open>More on quotient and remainder\<close>
 
 theory Divides
 imports Parity
 begin
 
 subsection \<open>More on division\<close>
 
 subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
 
 lemma unique_quotient_lemma:
   assumes "b * q' + r' \<le> b * q + r" "0 \<le> r'" "r' < b" "r < b" shows "q' \<le> (q::int)"
 proof -
   have "r' + b * (q'-q) \<le> r"
     using assms by (simp add: right_diff_distrib)
   moreover have "0 < b * (1 + q - q') "
     using assms by (simp add: right_diff_distrib distrib_left)
   moreover have "b * q' < b * (1 + q)"
     using assms by (simp add: right_diff_distrib distrib_left)
   ultimately show ?thesis
     using assms by (simp add: mult_less_cancel_left)
 qed
 
 lemma unique_quotient_lemma_neg:
   "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
   using unique_quotient_lemma[where b = "-b" and r = "-r'" and r'="-r"] by auto
 
 lemma zdiv_mono1:
   \<open>a div b \<le> a' div b\<close>
   if \<open>a \<le> a'\<close> \<open>0 < b\<close>
   for a b b' :: int
 proof (rule unique_quotient_lemma)
   show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
     using \<open>a \<le> a'\<close> by auto
 qed (use that in auto)
 
 lemma zdiv_mono1_neg:
   fixes b::int
   assumes "a \<le> a'" "b < 0" shows "a' div b \<le> a div b"
 proof (rule unique_quotient_lemma_neg)
   show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
     using assms(1) by auto
 qed (use assms in auto)
 
 
 subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
 
 lemma q_pos_lemma:
   fixes q'::int
   assumes "0 \<le> b'*q' + r'" "r' < b'" "0 < b'"
   shows "0 \<le> q'"
 proof -
   have "0 < b'* (q' + 1)"
     using assms by (simp add: distrib_left)
   with assms show ?thesis
     by (simp add: zero_less_mult_iff)
 qed
 
 lemma zdiv_mono2_lemma:
   fixes q'::int
   assumes eq: "b*q + r = b'*q' + r'" and le: "0 \<le> b'*q' + r'" and "r' < b'" "0 \<le> r" "0 < b'" "b' \<le> b"
   shows "q \<le> q'"
 proof -
   have "0 \<le> q'"
     using q_pos_lemma le \<open>r' < b'\<close> \<open>0 < b'\<close> by blast
   moreover have "b*q = r' - r + b'*q'"
     using eq by linarith
   ultimately have "b*q < b* (q' + 1)"
     using mult_right_mono assms unfolding distrib_left by fastforce
   with assms show ?thesis
     by (simp add: mult_less_cancel_left_pos)
 qed
 
 lemma zdiv_mono2:
   fixes a::int
   assumes "0 \<le> a" "0 < b'" "b' \<le> b" shows "a div b \<le> a div b'"
 proof (rule zdiv_mono2_lemma)
   have "b \<noteq> 0"
     using assms by linarith
   show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
     by simp
 qed (use assms in auto)
 
 lemma zdiv_mono2_neg_lemma:
     fixes q'::int
     assumes "b*q + r = b'*q' + r'" "b'*q' + r' < 0" "r < b" "0 \<le> r'" "0 < b'" "b' \<le> b"
     shows "q' \<le> q"
 proof -
   have "b'*q' < 0"
     using assms by linarith
   with assms have "q' \<le> 0"
     by (simp add: mult_less_0_iff)
   have "b*q' \<le> b'*q'"
     by (simp add: \<open>q' \<le> 0\<close> assms(6) mult_right_mono_neg)
   then have "b*q' < b* (q + 1)"
     using assms by (simp add: distrib_left)
   then show ?thesis
     using assms by (simp add: mult_less_cancel_left)
 qed
 
 lemma zdiv_mono2_neg:
   fixes a::int
   assumes "a < 0" "0 < b'" "b' \<le> b" shows "a div b' \<le> a div b"
 proof (rule zdiv_mono2_neg_lemma)
   have "b \<noteq> 0"
     using assms by linarith
   show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
     by simp
 qed (use assms in auto)
 
 
 subsubsection \<open>Quotients of Signs\<close>
 
 lemma div_eq_minus1: "0 < b \<Longrightarrow> - 1 div b = - 1" for b :: int
   by (simp add: divide_int_def)
 
 lemma zmod_minus1: "0 < b \<Longrightarrow> - 1 mod b = b - 1" for b :: int
   by (auto simp add: modulo_int_def)
 
 lemma minus_mod_int_eq:
   \<open>- k mod l = l - 1 - (k - 1) mod l\<close> if \<open>l \<ge> 0\<close> for k l :: int
 proof (cases \<open>l = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   with that have \<open>l > 0\<close>
     by simp
   then show ?thesis
   proof (cases \<open>l dvd k\<close>)
     case True
     then obtain j where \<open>k = l * j\<close> ..
     moreover have \<open>(l * j mod l - 1) mod l = l - 1\<close>
       using \<open>l > 0\<close> by (simp add: zmod_minus1)
     then have \<open>(l * j - 1) mod l = l - 1\<close>
       by (simp only: mod_simps)
     ultimately show ?thesis
       by simp
   next
     case False
     moreover have 1: \<open>0 < k mod l\<close>
       using \<open>0 < l\<close> False le_less by fastforce
     moreover have 2: \<open>k mod l < 1 + l\<close>
       using \<open>0 < l\<close> pos_mod_bound[of l k] by linarith
     from 1 2 \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
       by (simp add: zmod_trivial_iff)
     ultimately show ?thesis
       by (simp only: zmod_zminus1_eq_if)
          (simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
   qed
 qed
 
 lemma div_neg_pos_less0:
   fixes a::int
   assumes "a < 0" "0 < b" 
   shows "a div b < 0"
 proof -
   have "a div b \<le> - 1 div b"
     using zdiv_mono1 assms by auto
   also have "... \<le> -1"
     by (simp add: assms(2) div_eq_minus1)
   finally show ?thesis 
     by force
 qed
 
 lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
   by (drule zdiv_mono1_neg, auto)
 
 lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
   by (drule zdiv_mono1, auto)
 
 text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
 conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
 They should all be simp rules unless that causes too much search.\<close>
 
 lemma pos_imp_zdiv_nonneg_iff:
       fixes a::int
       assumes "0 < b" 
       shows "(0 \<le> a div b) = (0 \<le> a)"
 proof
   show "0 \<le> a div b \<Longrightarrow> 0 \<le> a"
     using assms
     by (simp add: linorder_not_less [symmetric]) (blast intro: div_neg_pos_less0)
 next
   assume "0 \<le> a"
   then have "0 div b \<le> a div b"
     using zdiv_mono1 assms by blast
   then show "0 \<le> a div b"
     by auto
 qed
 
 lemma pos_imp_zdiv_pos_iff:
   "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
   using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k] by arith
 
 lemma neg_imp_zdiv_nonneg_iff:
   fixes a::int
   assumes "b < 0" 
   shows "(0 \<le> a div b) = (a \<le> 0)"
   using assms by (simp add: div_minus_minus [of a, symmetric] pos_imp_zdiv_nonneg_iff del: div_minus_minus)
 
 (*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
 lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
   by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
 
 (*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
 lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
   by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
 
 lemma nonneg1_imp_zdiv_pos_iff:
   fixes a::int
   assumes "0 \<le> a" 
   shows "a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b>0"
 proof -
   have "0 < a div b \<Longrightarrow> b \<le> a"
     using div_pos_pos_trivial[of a b] assms by arith
   moreover have "0 < a div b \<Longrightarrow> b > 0"
     using assms div_nonneg_neg_le0[of a b]  by(cases "b=0"; force)
   moreover have "b \<le> a \<and> 0 < b \<Longrightarrow> 0 < a div b"
     using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b] by simp
   ultimately show ?thesis
     by blast
 qed
 
 lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 \<Longrightarrow> m mod k \<le> m"
   by (rule split_zmod[THEN iffD2]) (fastforce dest: q_pos_lemma intro: split_mult_pos_le)
 
 lemma sgn_div_eq_sgn_mult:
   \<open>sgn (k div l) = of_bool (k div l \<noteq> 0) * sgn (k * l)\<close>
   for k l :: int
 proof (cases \<open>k div l = 0\<close>)
   case True
   then show ?thesis
     by simp
 next
   case False
   have \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close>
     by (cases \<open>l = 0\<close>) (simp_all add: pos_imp_zdiv_nonneg_iff)
   then have \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> 0 < \<bar>k\<bar> div \<bar>l\<bar>\<close>
     by (simp add: less_le)
   also have \<open>\<dots> \<longleftrightarrow> \<bar>k\<bar> \<ge> \<bar>l\<bar>\<close>
     using False nonneg1_imp_zdiv_pos_iff by auto
   finally have *: \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> \<bar>l\<bar> \<le> \<bar>k\<bar>\<close> .
   show ?thesis
     using \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close> False
   by (auto simp add: div_eq_div_abs [of k l] div_eq_sgn_abs [of k l]
     sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
 qed
 
 lemma
   fixes a b q r :: int
   assumes \<open>a = b * q + r\<close> \<open>0 \<le> r\<close> \<open>r < b\<close>
   shows int_div_pos_eq:
       \<open>a div b = q\<close> (is ?Q)
     and int_mod_pos_eq:
       \<open>a mod b = r\<close> (is ?R)
 proof -
   from assms have \<open>(a div b, a mod b) = (q, r)\<close>
     by (cases b q r a rule: euclidean_relationI)
       (auto simp add: division_segment_int_def ac_simps dvd_add_left_iff dest: zdvd_imp_le)
   then show ?Q and ?R
     by simp_all
 qed
 
 lemma int_div_neg_eq:
   \<open>a div b = q\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
   using that int_div_pos_eq [of a \<open>- b\<close> \<open>- q\<close> \<open>- r\<close>] by simp_all
 
 lemma int_mod_neg_eq:
   \<open>a mod b = r\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
   using that int_div_neg_eq [of a b q r] by simp
 
 
 subsubsection \<open>Further properties\<close>
 
 lemma div_int_pos_iff:
   "k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
     \<or> k < 0 \<and> l < 0"
   for k l :: int
 proof (cases "k = 0 \<or> l = 0")
   case False
   then have *: "k \<noteq> 0" "l \<noteq> 0"
     by auto
   then have "0 \<le> k div l \<Longrightarrow> \<not> k < 0 \<Longrightarrow> 0 \<le> l"
     by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
   then show ?thesis
    using * by (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
 qed auto
 
 lemma mod_int_pos_iff:
   "k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
   for k l :: int
 proof (cases "l > 0")
   case False
   then show ?thesis 
     by (simp add: dvd_eq_mod_eq_0) (use neg_mod_sign [of l k] in \<open>auto simp add: le_less not_less\<close>)
 qed auto
 
 text \<open>Simplify expressions in which div and mod combine numerical constants\<close>
 
 lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
   unfolding dvd_def by (cases "y=0") (auto simp add: abs_mult)
 
 text\<open>Suggested by Matthias Daum\<close>
 lemma int_power_div_base:
   fixes k :: int
   assumes "0 < m" "0 < k"
   shows "k ^ m div k = (k::int) ^ (m - Suc 0)"
 proof -
   have eq: "k ^ m = k ^ ((m - Suc 0) + Suc 0)"
     by (simp add: assms)
   show ?thesis
     using assms by (simp only: power_add eq) auto
 qed
 
 text\<open>Suggested by Matthias Daum\<close>
 lemma int_div_less_self:
   fixes x::int
   assumes "0 < x" "1 < k"
   shows  "x div k < x"
 proof -
   have "nat x div nat k < nat x"
     by (simp add: assms)
   with assms show ?thesis
     by (simp add: nat_div_distrib [symmetric])
 qed
 
 lemma mod_eq_dvd_iff_nat:
   "m mod q = n mod q \<longleftrightarrow> q dvd m - n" if "m \<ge> n" for m n q :: nat
 proof -
   have "int m mod int q = int n mod int q \<longleftrightarrow> int q dvd int m - int n"
     by (simp add: mod_eq_dvd_iff)
   with that have "int (m mod q) = int (n mod q) \<longleftrightarrow> int q dvd int (m - n)"
     by (simp only: of_nat_mod of_nat_diff)
   then show ?thesis
     by simp
 qed
 
 lemma mod_eq_nat1E:
   fixes m n q :: nat
   assumes "m mod q = n mod q" and "m \<ge> n"
   obtains s where "m = n + q * s"
 proof -
   from assms have "q dvd m - n"
     by (simp add: mod_eq_dvd_iff_nat)
   then obtain s where "m - n = q * s" ..
   with \<open>m \<ge> n\<close> have "m = n + q * s"
     by simp
   with that show thesis .
 qed
 
 lemma mod_eq_nat2E:
   fixes m n q :: nat
   assumes "m mod q = n mod q" and "n \<ge> m"
   obtains s where "n = m + q * s"
   using assms mod_eq_nat1E [of n q m] by (auto simp add: ac_simps)
 
 lemma nat_mod_eq_lemma:
   assumes "(x::nat) mod n = y mod n" and "y \<le> x"
   shows "\<exists>q. x = y + n * q"
   using assms by (rule mod_eq_nat1E) (rule exI)
 
 lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
   (is "?lhs = ?rhs")
 proof
   assume H: "x mod n = y mod n"
   {assume xy: "x \<le> y"
     from H have th: "y mod n = x mod n" by simp
     from nat_mod_eq_lemma[OF th xy] have ?rhs
     proof
       fix q
       assume "y = x + n * q"
       then have "x + n * q = y + n * 0"
         by simp
       then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
         by blast
     qed}
   moreover
   {assume xy: "y \<le> x"
     from nat_mod_eq_lemma[OF H xy] have ?rhs
     proof
       fix q
       assume "x = y + n * q"
       then have "x + n * 0 = y + n * q"
         by simp
       then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
         by blast
     qed}
   ultimately  show ?rhs using linear[of x y] by blast
 next
   assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
   hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
   thus  ?lhs by simp
 qed
 
 
 subsubsection \<open>Uniqueness rules\<close>
 
-inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
-  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
-  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
-  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
-      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
-
-lemma eucl_rel_int_iff:    
-  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
-    k = l * q + r \<and>
-     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
-  by (cases "r = 0")
-    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
-    simp add: ac_simps sgn_1_pos sgn_1_neg)
-
-lemma eucl_rel_int:
-  "eucl_rel_int k l (k div l, k mod l)"
-proof (cases k rule: int_cases3)
-  case zero
-  then show ?thesis
-    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
+lemma euclidean_relation_intI [case_names by0 divides euclidean_relation]:
+  \<open>(k div l, k mod l) = (q, r)\<close>
+    if by0': \<open>l = 0 \<Longrightarrow> q = 0 \<and> r = k\<close>
+    and divides': \<open>l \<noteq> 0 \<Longrightarrow> l dvd k \<Longrightarrow> r = 0 \<and> k = q * l\<close>
+    and euclidean_relation': \<open>l \<noteq> 0 \<Longrightarrow> \<not> l dvd k \<Longrightarrow> sgn r = sgn l
+      \<and> \<bar>r\<bar> < \<bar>l\<bar> \<and> k = q * l + r\<close> for k l :: int
+proof (cases l q r k rule: euclidean_relationI)
+  case by0
+  then show ?case
+    by (rule by0')
 next
-  case (pos n)
-  then show ?thesis
-    using div_mult_mod_eq [of n]
-    by (cases l rule: int_cases3)
-      (auto simp del: of_nat_mult of_nat_add
-        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
-        eucl_rel_int_iff divide_int_def modulo_int_def)
+  case divides
+  then show ?case
+    by (rule divides')
 next
-  case (neg n)
-  then show ?thesis
-    using div_mult_mod_eq [of n]
-    by (cases l rule: int_cases3)
-      (auto simp del: of_nat_mult of_nat_add
-        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
-        eucl_rel_int_iff divide_int_def modulo_int_def)
+  case euclidean_relation
+  with euclidean_relation' have \<open>sgn r = sgn l\<close> \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
+    by simp_all
+  from \<open>sgn r = sgn l\<close> \<open>l \<noteq> 0\<close> have \<open>division_segment r = division_segment l\<close>
+    by (simp add: division_segment_int_def sgn_if split: if_splits)
+  with \<open>\<bar>r\<bar> < \<bar>l\<bar>\<close> \<open>k = q * l + r\<close>
+  show ?case
+    by simp
 qed
 
-lemma unique_quotient:
-  \<open>q = q'\<close> if \<open>eucl_rel_int a b (q, r)\<close> \<open>eucl_rel_int a b (q', r')\<close>
-  apply (rule order_antisym)
-  using that
-   apply (simp_all add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
-     apply (blast intro: order_eq_refl [THEN unique_quotient_lemma] order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
-  done
-
-lemma unique_remainder:
-  \<open>r = r'\<close> if \<open>eucl_rel_int a b (q, r)\<close> \<open>eucl_rel_int a b (q', r')\<close>
-proof -
-  have \<open>q = q'\<close>
-    using that by (blast intro: unique_quotient)
-  then show ?thesis
-    using that by (simp add: eucl_rel_int_iff)
-qed
-
-lemma divmod_int_unique:
-  assumes \<open>eucl_rel_int k l (q, r)\<close>
-  shows div_int_unique: \<open>k div l = q\<close> and mod_int_unique: \<open>k mod l = r\<close>
-  using assms eucl_rel_int [of k l]
-  using unique_quotient [of k l] unique_remainder [of k l]
-  by auto
+lemma euclidean_relation_intI' [case_names by0 pos neg]:
+  \<open>(k div l, k mod l) = (q, r)\<close>
+    if \<open>l = 0 \<Longrightarrow> q = 0 \<and> r = k\<close>
+    and \<open>l > 0 \<Longrightarrow> 0 \<le> r \<and> r < l \<and> k = q * l + r\<close>
+    and \<open>l < 0 \<Longrightarrow> l < r \<and> r \<le> 0 \<and> k = q * l + r\<close> for k l q r :: int
+  by (cases l \<open>0::int\<close> rule: linorder_cases)
+    (auto dest: that)
 
 
 subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
 
 lemma div_pos_geq:
   fixes k l :: int
   assumes "0 < l" and "l \<le> k"
   shows "k div l = (k - l) div l + 1"
 proof -
   have "k = (k - l) + l" by simp
   then obtain j where k: "k = j + l" ..
   with assms show ?thesis by (simp add: div_add_self2)
 qed
 
 lemma mod_pos_geq:
   fixes k l :: int
   assumes "0 < l" and "l \<le> k"
   shows "k mod l = (k - l) mod l"
 proof -
   have "k = (k - l) + l" by simp
   then obtain j where k: "k = j + l" ..
   with assms show ?thesis by simp
 qed
 
-lemma pos_eucl_rel_int_mult_2:
-  assumes "0 \<le> b"
-  assumes "eucl_rel_int a b (q, r)"
-  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
-  using assms unfolding eucl_rel_int_iff by auto
-
-lemma neg_eucl_rel_int_mult_2:
-  assumes "b \<le> 0"
-  assumes "eucl_rel_int (a + 1) b (q, r)"
-  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
-  using assms unfolding eucl_rel_int_iff by auto
-
 text\<open>computing div by shifting\<close>
 
-lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
-  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
-  by (rule div_int_unique)
+lemma pos_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = b div a\<close> (is ?Q)
+  and pos_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)\<close> (is ?R)
+  if \<open>0 \<le> a\<close> for a b :: int
+proof -
+  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = (b div a, 1 + 2 * (b mod a))\<close>
+  proof (cases \<open>2 * a\<close> \<open>b div a\<close> \<open>1 + 2 * (b mod a)\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI')
+    case by0
+    then show ?case
+      by simp
+  next
+    case pos
+    then have \<open>a > 0\<close>
+      by simp
+    moreover have \<open>b mod a < a\<close>
+      using \<open>a > 0\<close> by simp
+    then have \<open>1 + 2 * (b mod a) < 2 * a\<close>
+      by simp
+    moreover have \<open>2 * (b mod a) + a * (2 * (b div a)) = 2 * (b div a * a + b mod a)\<close>
+      by (simp only: algebra_simps)
+    ultimately show ?case
+      by (simp add: algebra_simps)
+  next
+    case neg
+    with that show ?case
+      by simp
+  qed
+  then show ?Q and ?R
+    by simp_all
+qed
 
-lemma neg_zdiv_mult_2:
-  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
-  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
-  by (rule div_int_unique)
+lemma neg_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = (b + 1) div a\<close> (is ?Q)
+  and neg_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1\<close> (is ?R)
+  if \<open>a \<le> 0\<close> for a b :: int
+proof -
+  have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = ((b + 1) div a, 2 * ((b + 1) mod a) - 1)\<close>
+  proof (cases \<open>2 * a\<close> \<open>(b + 1) div a\<close> \<open>2 * ((b + 1) mod a) - 1\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI')
+    case by0
+    then show ?case
+      by simp
+  next
+    case pos
+    with that show ?case
+      by simp
+  next
+    case neg
+    then have \<open>a < 0\<close>
+      by simp
+    moreover have \<open>(b + 1) mod a > a\<close>
+      using \<open>a < 0\<close> by simp
+    then have \<open>2 * ((b + 1) mod a) > 1 + 2 * a\<close>
+      by simp
+    moreover have \<open>((1 + b) mod a) \<le> 0\<close>
+      using \<open>a < 0\<close> by simp
+    then have \<open>2 * ((1 + b) mod a) \<le> 0\<close>
+      by simp
+    moreover have \<open>2 * ((1 + b) mod a) + a * (2 * ((1 + b) div a)) =
+      2 * ((1 + b) div a * a + (1 + b) mod a)\<close>
+      by (simp only: algebra_simps)
+    ultimately show ?case
+      by (simp add: algebra_simps)
+  qed
+  then show ?Q and ?R
+    by simp_all
+qed
 
 lemma zdiv_numeral_Bit0 [simp]:
   "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
     numeral v div (numeral w :: int)"
   unfolding numeral.simps unfolding mult_2 [symmetric]
   by (rule div_mult_mult1, simp)
 
 lemma zdiv_numeral_Bit1 [simp]:
   "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
     (numeral v div (numeral w :: int))"
   unfolding numeral.simps
   unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zdiv_mult_2, simp)
 
-lemma pos_zmod_mult_2:
-  fixes a b :: int
-  assumes "0 \<le> a"
-  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
-  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
-  by (rule mod_int_unique)
-
-lemma neg_zmod_mult_2:
-  fixes a b :: int
-  assumes "a \<le> 0"
-  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
-  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
-  by (rule mod_int_unique)
-
 lemma zmod_numeral_Bit0 [simp]:
   "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
     (2::int) * (numeral v mod numeral w)"
   unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
   unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
 
 lemma zmod_numeral_Bit1 [simp]:
   "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
     2 * (numeral v mod numeral w) + (1::int)"
   unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
   unfolding mult_2 [symmetric] add.commute [of _ 1]
   by (rule pos_zmod_mult_2, simp)
 
   
 code_identifier
   code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
 
 
 subsection \<open>Lemmas of doubtful value\<close>
 
 class unique_euclidean_semiring_numeral = unique_euclidean_semiring_with_nat + linordered_semidom +
   assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
     and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
     and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
     and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
     and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
     and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
     and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
     and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
   assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
 begin
 
 lemma divmod_digit_1:
   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
     and "a mod (2 * b) - b = a mod b" (is "?Q")
 proof -
   from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
     by (auto intro: trans)
   with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
   then have [simp]: "1 \<le> a div b" by (simp add: discrete)
   with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
   define w where "w = a div b mod 2"
   then have w_exhaust: "w = 0 \<or> w = 1" by auto
   have mod_w: "a mod (2 * b) = a mod b + b * w"
     by (simp add: w_def mod_mult2_eq ac_simps)
   from assms w_exhaust have "w = 1"
     using mod_less by (auto simp add: mod_w)
   with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
   have "2 * (a div (2 * b)) = a div b - w"
     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
   with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
   then show ?P and ?Q
     by (simp_all add: div mod add_implies_diff [symmetric])
 qed
 
 lemma divmod_digit_0:
   assumes "0 < b" and "a mod (2 * b) < b"
   shows "2 * (a div (2 * b)) = a div b" (is "?P")
     and "a mod (2 * b) = a mod b" (is "?Q")
 proof -
   define w where "w = a div b mod 2"
   then have w_exhaust: "w = 0 \<or> w = 1" by auto
   have mod_w: "a mod (2 * b) = a mod b + b * w"
     by (simp add: w_def mod_mult2_eq ac_simps)
   moreover have "b \<le> a mod b + b"
   proof -
     from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
     then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
     then show ?thesis by simp
   qed
   moreover note assms w_exhaust
   ultimately have "w = 0" by auto
   with mod_w have mod: "a mod (2 * b) = a mod b" by simp
   have "2 * (a div (2 * b)) = a div b - w"
     by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
   with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
   then show ?P and ?Q
     by (simp_all add: div mod)
 qed
 
 lemma mod_double_modulus:
   assumes "m > 0" "x \<ge> 0"
   shows   "x mod (2 * m) = x mod m \<or> x mod (2 * m) = x mod m + m"
 proof (cases "x mod (2 * m) < m")
   case True
   thus ?thesis using assms using divmod_digit_0(2)[of m x] by auto
 next
   case False
   hence *: "x mod (2 * m) - m = x mod m"
     using assms by (intro divmod_digit_1) auto
   hence "x mod (2 * m) = x mod m + m"
     by (subst * [symmetric], subst le_add_diff_inverse2) (use False in auto)
   thus ?thesis by simp
 qed
 
 end
 
 hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
 
 instance nat :: unique_euclidean_semiring_numeral
   by standard
     (auto simp add: div_greater_zero_iff div_mult2_eq mod_mult2_eq)
 
 instance int :: unique_euclidean_semiring_numeral
   by standard (auto intro: zmod_le_nonneg_dividend simp add:
     pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
 
 lemma div_geq: "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
   by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
 
 lemma mod_geq: "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
   by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
 
 lemma mod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: nat
   using that by (auto simp add: mod_eq_0_iff_dvd)
 
 lemma pos_mod_conj: "0 < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" for a b :: int
   by simp
 
 lemma neg_mod_conj: "b < 0 \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" for a b :: int
   by simp
 
 lemma zmod_eq_0_iff: "m mod d = 0 \<longleftrightarrow> (\<exists>q. m = d * q)" for m d :: int
   by (auto simp add: mod_eq_0_iff_dvd)
 
 (* REVISIT: should this be generalized to all semiring_div types? *)
 lemma zmod_eq_0D [dest!]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
   using that by auto
 
 lemma div_positive_int:
   "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
   using that by (simp add: nonneg1_imp_zdiv_pos_iff)
 
 end
diff --git a/src/HOL/Library/Signed_Division.thy b/src/HOL/Library/Signed_Division.thy
--- a/src/HOL/Library/Signed_Division.thy
+++ b/src/HOL/Library/Signed_Division.thy
@@ -1,206 +1,202 @@
 (*  Author:  Stefan Berghofer et al.
 *)
 
 subsection \<open>Signed division: negative results rounded towards zero rather than minus infinity.\<close>
 
 theory Signed_Division
   imports Main
 begin
 
 class signed_division = comm_semiring_1_cancel +
   fixes signed_divide :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixl \<open>sdiv\<close> 70)
   and signed_modulo :: \<open>'a \<Rightarrow> 'a \<Rightarrow> 'a\<close> (infixl \<open>smod\<close> 70)
   assumes sdiv_mult_smod_eq: \<open>a sdiv b * b + a smod b = a\<close>
 begin
 
 lemma mult_sdiv_smod_eq:
   \<open>b * (a sdiv b) + a smod b = a\<close>
   using sdiv_mult_smod_eq [of a b] by (simp add: ac_simps)
 
 lemma smod_sdiv_mult_eq:
   \<open>a smod b + a sdiv b * b = a\<close>
   using sdiv_mult_smod_eq [of a b] by (simp add: ac_simps)
 
 lemma smod_mult_sdiv_eq:
   \<open>a smod b + b * (a sdiv b) = a\<close>
   using sdiv_mult_smod_eq [of a b] by (simp add: ac_simps)
 
 lemma minus_sdiv_mult_eq_smod:
   \<open>a - a sdiv b * b = a smod b\<close>
   by (rule add_implies_diff [symmetric]) (fact smod_sdiv_mult_eq)
 
 lemma minus_mult_sdiv_eq_smod:
   \<open>a - b * (a sdiv b) = a smod b\<close>
   by (rule add_implies_diff [symmetric]) (fact smod_mult_sdiv_eq)
 
 lemma minus_smod_eq_sdiv_mult:
   \<open>a - a smod b = a sdiv b * b\<close>
   by (rule add_implies_diff [symmetric]) (fact sdiv_mult_smod_eq)
 
 lemma minus_smod_eq_mult_sdiv:
   \<open>a - a smod b = b * (a sdiv b)\<close>
   by (rule add_implies_diff [symmetric]) (fact mult_sdiv_smod_eq)
 
 end
 
 instantiation int :: signed_division
 begin
 
 definition signed_divide_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>k sdiv l = sgn k * sgn l * (\<bar>k\<bar> div \<bar>l\<bar>)\<close> for k l :: int
 
 definition signed_modulo_int :: \<open>int \<Rightarrow> int \<Rightarrow> int\<close>
   where \<open>k smod l = sgn k * (\<bar>k\<bar> mod \<bar>l\<bar>)\<close> for k l :: int
 
 instance by standard
   (simp add: signed_divide_int_def signed_modulo_int_def div_abs_eq mod_abs_eq algebra_simps)
 
 end
 
 lemma divide_int_eq_signed_divide_int:
   \<open>k div l = k sdiv l - of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
   for k l :: int
   by (simp add: div_eq_div_abs [of k l] signed_divide_int_def)
 
 lemma signed_divide_int_eq_divide_int:
   \<open>k sdiv l = k div l + of_bool (l \<noteq> 0 \<and> sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
   for k l :: int
   by (simp add: divide_int_eq_signed_divide_int)
 
 lemma modulo_int_eq_signed_modulo_int:
   \<open>k mod l = k smod l + l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
   for k l :: int
   by (simp add: mod_eq_mod_abs [of k l] signed_modulo_int_def)
 
 lemma signed_modulo_int_eq_modulo_int:
   \<open>k smod l = k mod l - l * of_bool (sgn k \<noteq> sgn l \<and> \<not> l dvd k)\<close>
   for k l :: int
   by (simp add: modulo_int_eq_signed_modulo_int)
 
 lemma sdiv_int_div_0:
   "(x :: int) sdiv 0 = 0"
   by (clarsimp simp: signed_divide_int_def)
 
 lemma sdiv_int_0_div [simp]:
   "0 sdiv (x :: int) = 0"
   by (clarsimp simp: signed_divide_int_def)
 
 lemma smod_int_alt_def:
      "(a::int) smod b = sgn (a) * (abs a mod abs b)"
   by (fact signed_modulo_int_def)
 
 lemma int_sdiv_simps [simp]:
     "(a :: int) sdiv 1 = a"
     "(a :: int) sdiv 0 = 0"
     "(a :: int) sdiv -1 = -a"
   apply (auto simp: signed_divide_int_def sgn_if)
   done
 
 lemma smod_int_mod_0 [simp]:
   "x smod (0 :: int) = x"
   by (clarsimp simp: signed_modulo_int_def abs_mult_sgn ac_simps)
 
 lemma smod_int_0_mod [simp]:
   "0 smod (x :: int) = 0"
   by (clarsimp simp: smod_int_alt_def)
 
 lemma sgn_sdiv_eq_sgn_mult:
   "a sdiv b \<noteq> 0 \<Longrightarrow> sgn ((a :: int) sdiv b) = sgn (a * b)"
   by (auto simp: signed_divide_int_def sgn_div_eq_sgn_mult sgn_mult)
 
 lemma int_sdiv_same_is_1 [simp]:
     "a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = a) = (b = 1)"
   apply (rule iffI)
    apply (clarsimp simp: signed_divide_int_def)
    apply (subgoal_tac "b > 0")
     apply (case_tac "a > 0")
      apply (clarsimp simp: sgn_if)
   apply (simp_all add: not_less algebra_split_simps sgn_if split: if_splits)
   using int_div_less_self [of a b] apply linarith
     apply (metis add.commute add.inverse_inverse group_cancel.rule0 int_div_less_self linorder_neqE_linordered_idom neg_0_le_iff_le not_less verit_comp_simplify1(1) zless_imp_add1_zle)
    apply (metis div_minus_right neg_imp_zdiv_neg_iff neg_le_0_iff_le not_less order.not_eq_order_implies_strict)
   apply (metis abs_le_zero_iff abs_of_nonneg neg_imp_zdiv_nonneg_iff order.not_eq_order_implies_strict)
   done
 
 lemma int_sdiv_negated_is_minus1 [simp]:
     "a \<noteq> 0 \<Longrightarrow> ((a :: int) sdiv b = - a) = (b = -1)"
   apply (clarsimp simp: signed_divide_int_def)
   apply (rule iffI)
    apply (subgoal_tac "b < 0")
     apply (case_tac "a > 0")
      apply (clarsimp simp: sgn_if algebra_split_simps not_less)
      apply (case_tac "sgn (a * b) = -1")
       apply (simp_all add: not_less algebra_split_simps sgn_if split: if_splits)
      apply (metis add.inverse_inverse int_div_less_self int_one_le_iff_zero_less less_le neg_0_less_iff_less)
     apply (metis add.inverse_inverse div_minus_right int_div_less_self int_one_le_iff_zero_less less_le neg_0_less_iff_less)
    apply (metis less_le neg_less_0_iff_less not_less pos_imp_zdiv_neg_iff)
   apply (metis div_minus_right dual_order.eq_iff neg_imp_zdiv_nonneg_iff neg_less_0_iff_less)
   done
 
 lemma sdiv_int_range:
   \<open>a sdiv b \<in> {- \<bar>a\<bar>..\<bar>a\<bar>}\<close> for a b :: int
   using zdiv_mono2 [of \<open>\<bar>a\<bar>\<close> 1 \<open>\<bar>b\<bar>\<close>]
   by (cases \<open>b = 0\<close>; cases \<open>sgn b = sgn a\<close>)
      (auto simp add: signed_divide_int_def pos_imp_zdiv_nonneg_iff
      dest!: sgn_not_eq_imp intro: order_trans [of _ 0])
 
 lemma smod_int_range:
   \<open>a smod b \<in> {- \<bar>b\<bar> + 1..\<bar>b\<bar> - 1}\<close>
   if \<open>b \<noteq> 0\<close> for a b :: int
-  using that
-  apply (cases \<open>b > 0\<close>)
-   apply (insert pos_mod_conj [where a=a and b=b])[1]
-   apply (insert pos_mod_conj [where a="-a" and b=b])[1]
-   apply (auto simp: smod_int_alt_def algebra_simps sgn_if
-              abs_if not_less add1_zle_eq [simplified add.commute])[1]
-    apply (metis add_nonneg_nonneg int_one_le_iff_zero_less le_less less_add_same_cancel2 not_le pos_mod_conj)
-  apply (metis (full_types) add.inverse_inverse eucl_rel_int eucl_rel_int_iff le_less_trans neg_0_le_iff_le)
-  apply (insert neg_mod_conj [where a=a and b="b"])[1]
-  apply (insert neg_mod_conj [where a="-a" and b="b"])[1]
-  apply (clarsimp simp: smod_int_alt_def algebra_simps sgn_if
-            abs_if not_less add1_zle_eq [simplified add.commute])
-  apply (metis neg_0_less_iff_less neg_mod_conj not_le not_less_iff_gr_or_eq order_trans pos_mod_conj)
-  done
+proof -
+  define m n where \<open>m = nat \<bar>a\<bar>\<close> \<open>n = nat \<bar>b\<bar>\<close>
+  then have \<open>\<bar>a\<bar> = int m\<close> \<open>\<bar>b\<bar> = int n\<close>
+    by simp_all
+  with that have \<open>n > 0\<close>
+    by simp
+  with signed_modulo_int_def [of a b] \<open>\<bar>a\<bar> = int m\<close> \<open>\<bar>b\<bar> = int n\<close>
+  show ?thesis
+    by (auto simp add: sgn_if diff_le_eq int_one_le_iff_zero_less simp flip: of_nat_mod of_nat_diff)
+qed
 
 lemma smod_int_compares:
    "\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b < b"
    "\<lbrakk> 0 \<le> a; 0 < b \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
    "\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> -b < (a :: int) smod b"
    "\<lbrakk> a \<le> 0; 0 < b \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
    "\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b < - b"
    "\<lbrakk> 0 \<le> a; b < 0 \<rbrakk> \<Longrightarrow> 0 \<le> (a :: int) smod b"
    "\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> (a :: int) smod b \<le> 0"
    "\<lbrakk> a \<le> 0; b < 0 \<rbrakk> \<Longrightarrow> b \<le> (a :: int) smod b"
   apply (insert smod_int_range [where a=a and b=b])
   apply (auto simp: add1_zle_eq smod_int_alt_def sgn_if)
   done
 
 lemma smod_mod_positive:
     "\<lbrakk> 0 \<le> (a :: int); 0 \<le> b \<rbrakk> \<Longrightarrow> a smod b = a mod b"
   by (clarsimp simp: smod_int_alt_def zsgn_def)
 
 lemma minus_sdiv_eq [simp]:
   \<open>- k sdiv l = - (k sdiv l)\<close> for k l :: int
   by (simp add: signed_divide_int_def)
 
 lemma sdiv_minus_eq [simp]:
   \<open>k sdiv - l = - (k sdiv l)\<close> for k l :: int
   by (simp add: signed_divide_int_def)
 
 lemma sdiv_int_numeral_numeral [simp]:
   \<open>numeral m sdiv numeral n = numeral m div (numeral n :: int)\<close>
   by (simp add: signed_divide_int_def)
 
 lemma minus_smod_eq [simp]:
   \<open>- k smod l = - (k smod l)\<close> for k l :: int
   by (simp add: smod_int_alt_def)
 
 lemma smod_minus_eq [simp]:
   \<open>k smod - l = k smod l\<close> for k l :: int
   by (simp add: smod_int_alt_def)
 
 lemma smod_int_numeral_numeral [simp]:
   \<open>numeral m smod numeral n = numeral m mod (numeral n :: int)\<close>
   by (simp add: smod_int_alt_def) 
 
 end
diff --git a/src/HOL/SMT.thy b/src/HOL/SMT.thy
--- a/src/HOL/SMT.thy
+++ b/src/HOL/SMT.thy
@@ -1,898 +1,922 @@
 (*  Title:      HOL/SMT.thy
     Author:     Sascha Boehme, TU Muenchen
     Author:     Jasmin Blanchette, VU Amsterdam
 *)
 
 section \<open>Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2\<close>
 
 theory SMT
   imports Divides Numeral_Simprocs
   keywords "smt_status" :: diag
 begin
 
 subsection \<open>A skolemization tactic and proof method\<close>
 
 lemma choices:
   "\<And>Q. \<forall>x. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x. Q x (f x) (fa x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x. Q x (f x) (fa x) (fb x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x. Q x (f x) (fa x) (fb x) (fc x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd. Q x y ya yb yc yd \<Longrightarrow>
      \<exists>f fa fb fc fd. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye. Q x y ya yb yc yd ye \<Longrightarrow>
      \<exists>f fa fb fc fd fe. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf \<Longrightarrow>
      \<exists>f fa fb fc fd fe ff. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
   "\<And>Q. \<forall>x. \<exists>y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg \<Longrightarrow>
      \<exists>f fa fb fc fd fe ff fg. \<forall>x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
   by metis+
 
 lemma bchoices:
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya. Q x y ya \<Longrightarrow> \<exists>f fa. \<forall>x \<in> S. Q x (f x) (fa x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb. Q x y ya yb \<Longrightarrow> \<exists>f fa fb. \<forall>x \<in> S. Q x (f x) (fa x) (fb x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc. Q x y ya yb yc \<Longrightarrow> \<exists>f fa fb fc. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd. Q x y ya yb yc yd \<Longrightarrow>
     \<exists>f fa fb fc fd. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye. Q x y ya yb yc yd ye \<Longrightarrow>
     \<exists>f fa fb fc fd fe. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf \<Longrightarrow>
     \<exists>f fa fb fc fd fe ff. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
   "\<And>Q. \<forall>x \<in> S. \<exists>y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg \<Longrightarrow>
     \<exists>f fa fb fc fd fe ff fg. \<forall>x \<in> S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
   by metis+
 
 ML \<open>
 fun moura_tac ctxt =
   Atomize_Elim.atomize_elim_tac ctxt THEN'
   SELECT_GOAL (Clasimp.auto_tac (ctxt addSIs @{thms choice choices bchoice bchoices}) THEN
     ALLGOALS (Metis_Tactic.metis_tac (take 1 ATP_Proof_Reconstruct.partial_type_encs)
         ATP_Proof_Reconstruct.default_metis_lam_trans ctxt [] ORELSE'
       blast_tac ctxt))
 \<close>
 
 method_setup moura = \<open>
   Scan.succeed (SIMPLE_METHOD' o moura_tac)
 \<close> "solve skolemization goals, especially those arising from Z3 proofs"
 
 hide_fact (open) choices bchoices
 
 
 subsection \<open>Triggers for quantifier instantiation\<close>
 
 text \<open>
 Some SMT solvers support patterns as a quantifier instantiation
 heuristics. Patterns may either be positive terms (tagged by "pat")
 triggering quantifier instantiations -- when the solver finds a
 term matching a positive pattern, it instantiates the corresponding
 quantifier accordingly -- or negative terms (tagged by "nopat")
 inhibiting quantifier instantiations. A list of patterns
 of the same kind is called a multipattern, and all patterns in a
 multipattern are considered conjunctively for quantifier instantiation.
 A list of multipatterns is called a trigger, and their multipatterns
 act disjunctively during quantifier instantiation. Each multipattern
 should mention at least all quantified variables of the preceding
 quantifier block.
 \<close>
 
 typedecl 'a symb_list
 
 consts
   Symb_Nil :: "'a symb_list"
   Symb_Cons :: "'a \<Rightarrow> 'a symb_list \<Rightarrow> 'a symb_list"
 
 typedecl pattern
 
 consts
   pat :: "'a \<Rightarrow> pattern"
   nopat :: "'a \<Rightarrow> pattern"
 
 definition trigger :: "pattern symb_list symb_list \<Rightarrow> bool \<Rightarrow> bool" where
   "trigger _ P = P"
 
 
 subsection \<open>Higher-order encoding\<close>
 
 text \<open>
 Application is made explicit for constants occurring with varying
 numbers of arguments. This is achieved by the introduction of the
 following constant.
 \<close>
 
 definition fun_app :: "'a \<Rightarrow> 'a" where "fun_app f = f"
 
 text \<open>
 Some solvers support a theory of arrays which can be used to encode
 higher-order functions. The following set of lemmas specifies the
 properties of such (extensional) arrays.
 \<close>
 
 lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def
 
 
 subsection \<open>Normalization\<close>
 
 lemma case_bool_if[abs_def]: "case_bool x y P = (if P then x else y)"
   by simp
 
 lemmas Ex1_def_raw = Ex1_def[abs_def]
 lemmas Ball_def_raw = Ball_def[abs_def]
 lemmas Bex_def_raw = Bex_def[abs_def]
 lemmas abs_if_raw = abs_if[abs_def]
 lemmas min_def_raw = min_def[abs_def]
 lemmas max_def_raw = max_def[abs_def]
 
 lemma nat_zero_as_int:
   "0 = nat 0"
   by simp
 
 lemma nat_one_as_int:
   "1 = nat 1"
   by simp
 
 lemma nat_numeral_as_int: "numeral = (\<lambda>i. nat (numeral i))" by simp
 lemma nat_less_as_int: "(<) = (\<lambda>a b. int a < int b)" by simp
 lemma nat_leq_as_int: "(\<le>) = (\<lambda>a b. int a \<le> int b)" by simp
 lemma Suc_as_int: "Suc = (\<lambda>a. nat (int a + 1))" by (rule ext) simp
 lemma nat_plus_as_int: "(+) = (\<lambda>a b. nat (int a + int b))" by (rule ext)+ simp
 lemma nat_minus_as_int: "(-) = (\<lambda>a b. nat (int a - int b))" by (rule ext)+ simp
 lemma nat_times_as_int: "(*) = (\<lambda>a b. nat (int a * int b))" by (simp add: nat_mult_distrib)
 lemma nat_div_as_int: "(div) = (\<lambda>a b. nat (int a div int b))" by (simp add: nat_div_distrib)
 lemma nat_mod_as_int: "(mod) = (\<lambda>a b. nat (int a mod int b))" by (simp add: nat_mod_distrib)
 
 lemma int_Suc: "int (Suc n) = int n + 1" by simp
 lemma int_plus: "int (n + m) = int n + int m" by (rule of_nat_add)
 lemma int_minus: "int (n - m) = int (nat (int n - int m))" by auto
 
 lemma nat_int_comparison:
   fixes a b :: nat
   shows "(a = b) = (int a = int b)"
     and "(a < b) = (int a < int b)"
     and "(a \<le> b) = (int a \<le> int b)"
   by simp_all
 
 lemma int_ops:
   fixes a b :: nat
   shows "int 0 = 0"
     and "int 1 = 1"
     and "int (numeral n) = numeral n"
     and "int (Suc a) = int a + 1"
     and "int (a + b) = int a + int b"
     and "int (a - b) = (if int a < int b then 0 else int a - int b)"
     and "int (a * b) = int a * int b"
     and "int (a div b) = int a div int b"
     and "int (a mod b) = int a mod int b"
   by (auto intro: zdiv_int zmod_int)
 
 lemma int_if:
   fixes a b :: nat
   shows "int (if P then a else b) = (if P then int a else int b)"
   by simp
 
 
 subsection \<open>Integer division and modulo for Z3\<close>
 
 text \<open>
 The following Z3-inspired definitions are overspecified for the case where \<open>l = 0\<close>. This
 Schönheitsfehler is corrected in the \<open>div_as_z3div\<close> and \<open>mod_as_z3mod\<close> theorems.
 \<close>
 
 definition z3div :: "int \<Rightarrow> int \<Rightarrow> int" where
   "z3div k l = (if l \<ge> 0 then k div l else - (k div - l))"
 
 definition z3mod :: "int \<Rightarrow> int \<Rightarrow> int" where
   "z3mod k l = k mod (if l \<ge> 0 then l else - l)"
 
 lemma div_as_z3div:
   "\<forall>k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"
   by (simp add: z3div_def)
 
 lemma mod_as_z3mod:
   "\<forall>k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"
   by (simp add: z3mod_def)
 
 
 subsection \<open>Extra theorems for veriT reconstruction\<close>
 
 lemma verit_sko_forall: \<open>(\<forall>x. P x) \<longleftrightarrow> P (SOME x. \<not>P x)\<close>
   using someI[of \<open>\<lambda>x. \<not>P x\<close>]
   by auto
 
 lemma verit_sko_forall': \<open>P (SOME x. \<not>P x) = A \<Longrightarrow> (\<forall>x. P x) = A\<close>
   by (subst verit_sko_forall)
 
 lemma verit_sko_forall'': \<open>B = A \<Longrightarrow> (SOME x. P x) = A \<equiv> (SOME x. P x) = B\<close>
   by auto
 
 lemma verit_sko_forall_indirect: \<open>x = (SOME x. \<not>P x) \<Longrightarrow> (\<forall>x. P x) \<longleftrightarrow> P x\<close>
   using someI[of \<open>\<lambda>x. \<not>P x\<close>]
   by auto
 
 lemma verit_sko_forall_indirect2:
     \<open>x = (SOME x. \<not>P x) \<Longrightarrow> (\<And>x :: 'a. (P x = P' x)) \<Longrightarrow>(\<forall>x. P' x) \<longleftrightarrow> P x\<close>
   using someI[of \<open>\<lambda>x. \<not>P x\<close>]
   by auto
 
 lemma verit_sko_ex: \<open>(\<exists>x. P x) \<longleftrightarrow> P (SOME x. P x)\<close>
   using someI[of \<open>\<lambda>x. P x\<close>]
   by auto
 
 lemma verit_sko_ex': \<open>P (SOME x. P x) = A \<Longrightarrow> (\<exists>x. P x) = A\<close>
   by (subst verit_sko_ex)
 
 lemma verit_sko_ex_indirect: \<open>x = (SOME x. P x) \<Longrightarrow> (\<exists>x. P x) \<longleftrightarrow> P x\<close>
   using someI[of \<open>\<lambda>x. P x\<close>]
   by auto
 
 lemma verit_sko_ex_indirect2: \<open>x = (SOME x. P x) \<Longrightarrow> (\<And>x. P x = P' x) \<Longrightarrow> (\<exists>x. P' x) \<longleftrightarrow> P x\<close>
   using someI[of \<open>\<lambda>x. P x\<close>]
   by auto
 
 lemma verit_Pure_trans:
   \<open>P \<equiv> Q \<Longrightarrow> Q \<Longrightarrow> P\<close>
   by auto
 
 lemma verit_if_cong:
   assumes \<open>b \<equiv> c\<close>
     and \<open>c \<Longrightarrow> x \<equiv> u\<close>
     and \<open>\<not> c \<Longrightarrow> y \<equiv> v\<close>
   shows \<open>(if b then x else y) \<equiv> (if c then u else v)\<close>
   using assms if_cong[of b c x u] by auto
 
 lemma verit_if_weak_cong':
   \<open>b \<equiv> c \<Longrightarrow> (if b then x else y) \<equiv> (if c then x else y)\<close>
   by auto
 
 lemma verit_or_neg:
    \<open>(A \<Longrightarrow> B) \<Longrightarrow> B \<or> \<not>A\<close>
    \<open>(\<not>A \<Longrightarrow> B) \<Longrightarrow> B \<or> A\<close>
   by auto
 
 lemma verit_subst_bool: \<open>P \<Longrightarrow> f True \<Longrightarrow> f P\<close>
   by auto
 
 lemma verit_and_pos:
   \<open>(a \<Longrightarrow> \<not>(b \<and> c) \<or> A) \<Longrightarrow> \<not>(a \<and> b \<and> c) \<or> A\<close>
   \<open>(a \<Longrightarrow> b \<Longrightarrow> A) \<Longrightarrow> \<not>(a \<and> b) \<or> A\<close>
   \<open>(a \<Longrightarrow> A) \<Longrightarrow> \<not>a \<or> A\<close>
   \<open>(\<not>a \<Longrightarrow> A) \<Longrightarrow> a \<or> A\<close>
   by blast+
 
 lemma verit_or_pos:
    \<open>A \<and> A' \<Longrightarrow> (c \<and> A) \<or> (\<not>c \<and> A')\<close>
    \<open>A \<and> A' \<Longrightarrow> (\<not>c \<and> A) \<or> (c \<and> A')\<close>
   by blast+
 
 
 lemma verit_la_generic:
   \<open>(a::int) \<le> x \<or> a = x \<or> a \<ge> x\<close>
   by linarith
 
 lemma verit_bfun_elim:
   \<open>(if b then P True else P False) = P b\<close>
   \<open>(\<forall>b. P' b) = (P' False \<and> P' True)\<close>
   \<open>(\<exists>b. P' b) = (P' False \<or> P' True)\<close>
   by (cases b) (auto simp: all_bool_eq ex_bool_eq)
 
 lemma verit_eq_true_simplify:
   \<open>(P = True) \<equiv> P\<close>
   by auto
 
 lemma verit_and_neg:
   \<open>(a \<Longrightarrow> \<not>b \<or> A) \<Longrightarrow> \<not>(a \<and> b) \<or> A\<close>
   \<open>(a \<Longrightarrow> A) \<Longrightarrow> \<not>a \<or> A\<close>
   \<open>(\<not>a \<Longrightarrow> A) \<Longrightarrow> a \<or> A\<close>
   by blast+
 
 lemma verit_forall_inst:
   \<open>A \<longleftrightarrow> B \<Longrightarrow> \<not>A \<or> B\<close>
   \<open>\<not>A \<longleftrightarrow> B \<Longrightarrow> A \<or> B\<close>
   \<open>A \<longleftrightarrow> B \<Longrightarrow> \<not>B \<or> A\<close>
   \<open>A \<longleftrightarrow> \<not>B \<Longrightarrow> B \<or> A\<close>
   \<open>A \<longrightarrow> B \<Longrightarrow> \<not>A \<or> B\<close>
   \<open>\<not>A \<longrightarrow> B \<Longrightarrow> A \<or> B\<close>
   by blast+
 
 lemma verit_eq_transitive:
   \<open>A = B \<Longrightarrow> B = C \<Longrightarrow> A = C\<close>
   \<open>A = B \<Longrightarrow> C = B \<Longrightarrow> A = C\<close>
   \<open>B = A \<Longrightarrow> B = C \<Longrightarrow> A = C\<close>
   \<open>B = A \<Longrightarrow> C = B \<Longrightarrow> A = C\<close>
   by auto
 
 lemma verit_bool_simplify:
   \<open>\<not>(P \<longrightarrow> Q) \<longleftrightarrow> P \<and> \<not>Q\<close>
   \<open>\<not>(P \<or> Q) \<longleftrightarrow> \<not>P \<and> \<not>Q\<close>
   \<open>\<not>(P \<and> Q) \<longleftrightarrow> \<not>P \<or> \<not>Q\<close>
   \<open>(P \<longrightarrow> (Q \<longrightarrow> R)) \<longleftrightarrow> ((P \<and> Q) \<longrightarrow> R)\<close>
   \<open>((P \<longrightarrow> Q) \<longrightarrow> Q) \<longleftrightarrow> P \<or> Q\<close>
   \<open>(Q \<longleftrightarrow> (P \<or> Q)) \<longleftrightarrow> (P \<longrightarrow> Q)\<close> \<comment> \<open>This rule was inverted\<close>
   \<open>P \<and> (P \<longrightarrow> Q) \<longleftrightarrow> P \<and> Q\<close>
   \<open>(P \<longrightarrow> Q) \<and> P \<longleftrightarrow> P \<and> Q\<close>
  (* \<comment>\<open>Additional rules:\<close>
   *  \<open>((P \<longrightarrow> Q) \<longrightarrow> P) \<longleftrightarrow> P\<close>
   *  \<open>((P \<longrightarrow> Q) \<longrightarrow> Q) \<longleftrightarrow> P \<or> Q\<close>
   *  \<open>(P \<longrightarrow> Q) \<or> P\<close> *)
   unfolding not_imp imp_conjL
   by auto
 
 text \<open>We need the last equation for \<^term>\<open>\<not>(\<forall>a b. \<not>P a b)\<close>\<close>
 lemma verit_connective_def: \<comment> \<open>the definition of XOR is missing
   as the operator is not generated by Isabelle\<close>
   \<open>(A = B) \<longleftrightarrow> ((A \<longrightarrow> B) \<and> (B \<longrightarrow> A))\<close>
   \<open>(If A B C) = ((A \<longrightarrow> B) \<and> (\<not>A \<longrightarrow> C))\<close>
   \<open>(\<exists>x. P x) \<longleftrightarrow> \<not>(\<forall>x. \<not>P x)\<close>
   \<open>\<not>(\<exists>x. P x) \<longleftrightarrow> (\<forall>x. \<not>P x)\<close>
   by auto
 
 lemma verit_ite_simplify:
   \<open>(If True B C) = B\<close>
   \<open>(If False B C) = C\<close>
   \<open>(If A' B B) = B\<close>
   \<open>(If (\<not>A') B C) = (If A' C B)\<close>
   \<open>(If c (If c A B) C) = (If c A C)\<close>
   \<open>(If c C (If c A B)) = (If c C B)\<close>
   \<open>(If A' True False) = A'\<close>
   \<open>(If A' False True) \<longleftrightarrow> \<not>A'\<close>
   \<open>(If A' True B') \<longleftrightarrow> A'\<or>B'\<close>
   \<open>(If A' B' False) \<longleftrightarrow> A'\<and>B'\<close>
   \<open>(If A' False B') \<longleftrightarrow> \<not>A'\<and>B'\<close>
   \<open>(If A' B' True) \<longleftrightarrow> \<not>A'\<or>B'\<close>
   \<open>x \<and> True \<longleftrightarrow> x\<close>
   \<open>x \<or> False \<longleftrightarrow> x\<close>
   for B C :: 'a and A' B' C' :: bool
   by auto
 
 lemma verit_and_simplify1:
   \<open>True \<and> b \<longleftrightarrow> b\<close> \<open>b \<and> True \<longleftrightarrow> b\<close>
   \<open>False \<and> b \<longleftrightarrow> False\<close> \<open>b \<and> False \<longleftrightarrow> False\<close>
   \<open>(c \<and> \<not>c) \<longleftrightarrow> False\<close> \<open>(\<not>c \<and> c) \<longleftrightarrow> False\<close>
   \<open>\<not>\<not>a = a\<close>
   by auto
 
 lemmas verit_and_simplify = conj_ac de_Morgan_conj disj_not1
 
 
 lemma verit_or_simplify_1:
   \<open>False \<or> b \<longleftrightarrow> b\<close> \<open>b \<or> False \<longleftrightarrow> b\<close>
   \<open>b \<or> \<not>b\<close>
   \<open>\<not>b \<or> b\<close>
   by auto
 
 lemmas verit_or_simplify = disj_ac
 
 lemma verit_not_simplify:
   \<open>\<not> \<not>b \<longleftrightarrow> b\<close> \<open>\<not>True \<longleftrightarrow> False\<close> \<open>\<not>False \<longleftrightarrow> True\<close>
   by auto
 
 
 lemma verit_implies_simplify:
   \<open>(\<not>a \<longrightarrow> \<not>b) \<longleftrightarrow> (b \<longrightarrow> a)\<close>
   \<open>(False \<longrightarrow> a) \<longleftrightarrow> True\<close>
   \<open>(a \<longrightarrow> True) \<longleftrightarrow> True\<close>
   \<open>(True \<longrightarrow> a) \<longleftrightarrow> a\<close>
   \<open>(a \<longrightarrow> False) \<longleftrightarrow> \<not>a\<close>
   \<open>(a \<longrightarrow> a) \<longleftrightarrow> True\<close>
   \<open>(\<not>a \<longrightarrow> a) \<longleftrightarrow> a\<close>
   \<open>(a \<longrightarrow> \<not>a) \<longleftrightarrow> \<not>a\<close>
   \<open>((a \<longrightarrow> b) \<longrightarrow> b) \<longleftrightarrow> a \<or> b\<close>
   by auto
 
 lemma verit_equiv_simplify:
   \<open>((\<not>a) = (\<not>b)) \<longleftrightarrow> (a = b)\<close>
   \<open>(a = a) \<longleftrightarrow> True\<close>
   \<open>(a = (\<not>a)) \<longleftrightarrow> False\<close>
   \<open>((\<not>a) = a) \<longleftrightarrow> False\<close>
   \<open>(True = a) \<longleftrightarrow> a\<close>
   \<open>(a = True) \<longleftrightarrow> a\<close>
   \<open>(False = a) \<longleftrightarrow> \<not>a\<close>
   \<open>(a = False) \<longleftrightarrow> \<not>a\<close>
   \<open>\<not>\<not>a \<longleftrightarrow> a\<close>
   \<open>(\<not> False) = True\<close>
   for a b :: bool
   by auto
 
 lemmas verit_eq_simplify =
   semiring_char_0_class.eq_numeral_simps eq_refl zero_neq_one num.simps
   neg_equal_zero equal_neg_zero one_neq_zero neg_equal_iff_equal
 
 lemma verit_minus_simplify:
   \<open>(a :: 'a :: cancel_comm_monoid_add) - a = 0\<close>
   \<open>(a :: 'a :: cancel_comm_monoid_add) - 0 = a\<close>
   \<open>0 - (b :: 'b :: {group_add}) = -b\<close>
   \<open>- (- (b :: 'b :: group_add)) = b\<close>
   by auto
 
 lemma verit_sum_simplify:
   \<open>(a :: 'a :: cancel_comm_monoid_add) + 0 = a\<close>
   by auto
 
 lemmas verit_prod_simplify =
 (* already included:
    mult_zero_class.mult_zero_right
    mult_zero_class.mult_zero_left *)
    mult_1
    mult_1_right
 
 lemma verit_comp_simplify1:
   \<open>(a :: 'a ::order) < a \<longleftrightarrow> False\<close>
   \<open>a \<le> a\<close>
   \<open>\<not>(b' \<le> a') \<longleftrightarrow> (a' :: 'b :: linorder) < b'\<close>
   by auto
 
 lemmas verit_comp_simplify =
   verit_comp_simplify1
   le_numeral_simps
   le_num_simps
   less_numeral_simps
   less_num_simps
   zero_less_one
   zero_le_one
   less_neg_numeral_simps
 
 lemma verit_la_disequality:
   \<open>(a :: 'a ::linorder) = b \<or> \<not>a \<le> b \<or> \<not>b \<le> a\<close>
   by auto
 
 context
 begin
 
 text \<open>For the reconstruction, we need to keep the order of the arguments.\<close>
 
 named_theorems smt_arith_multiplication \<open>Theorems to reconstruct arithmetic theorems.\<close>
 
 named_theorems smt_arith_combine \<open>Theorems to reconstruct arithmetic theorems.\<close>
 
 named_theorems smt_arith_simplify \<open>Theorems to combine theorems in the LA procedure\<close>
 
 lemmas [smt_arith_simplify] =
     div_add dvd_numeral_simp divmod_steps less_num_simps le_num_simps if_True if_False divmod_cancel
     dvd_mult dvd_mult2 less_irrefl prod.case numeral_plus_one divmod_step_def order.refl le_zero_eq
     le_numeral_simps less_numeral_simps mult.right_neutral simp_thms divides_aux_eq
     mult_nonneg_nonneg dvd_imp_mod_0 dvd_add zero_less_one mod_mult_self4 numeral_mod_numeral
     divmod_trivial prod.sel mult.left_neutral div_pos_pos_trivial arith_simps div_add div_mult_self1
     add_le_cancel_left add_le_same_cancel2 not_one_le_zero le_numeral_simps add_le_same_cancel1
     zero_neq_one zero_le_one le_num_simps add_Suc mod_div_trivial nat.distinct mult_minus_right
     add.inverse_inverse distrib_left_numeral mult_num_simps numeral_times_numeral add_num_simps
     divmod_steps rel_simps if_True if_False numeral_div_numeral divmod_cancel prod.case
     add_num_simps one_plus_numeral fst_conv arith_simps sub_num_simps dbl_inc_simps
     dbl_simps mult_1 add_le_cancel_right left_diff_distrib_numeral add_uminus_conv_diff zero_neq_one
     zero_le_one One_nat_def add_Suc mod_div_trivial nat.distinct of_int_1 numerals numeral_One
     of_int_numeral add_uminus_conv_diff zle_diff1_eq add_less_same_cancel2 minus_add_distrib
     add_uminus_conv_diff mult.left_neutral semiring_class.distrib_right
     add_diff_cancel_left' add_diff_eq ring_distribs mult_minus_left minus_diff_eq
 
 lemma [smt_arith_simplify]:
   \<open>\<not> (a' :: 'a :: linorder) < b' \<longleftrightarrow> b' \<le> a'\<close>
   \<open>\<not> (a' :: 'a :: linorder) \<le> b' \<longleftrightarrow> b' < a'\<close>
   \<open>(c::int) mod Numeral1 = 0\<close>
   \<open>(a::nat) mod Numeral1 = 0\<close>
   \<open>(c::int) div Numeral1 = c\<close>
   \<open>a div Numeral1 = a\<close>
   \<open>(c::int) mod 1 = 0\<close>
   \<open>a mod 1 = 0\<close>
   \<open>(c::int) div 1 = c\<close>
   \<open>a div 1 = a\<close>
   \<open>\<not>(a' \<noteq> b') \<longleftrightarrow> a' = b'\<close>
   by auto
 
 
 lemma div_mod_decomp: "A = (A div n) * n + (A mod n)" for A :: nat
   by auto
 
 lemma div_less_mono:
   fixes A B :: nat
   assumes "A < B" "0 < n" and
     mod: "A mod n = 0""B mod n = 0"
   shows "(A div n) < (B div n)"
 proof -
   show ?thesis
     using assms(1)
     apply (subst (asm) div_mod_decomp[of "A" n])
     apply (subst (asm) div_mod_decomp[of "B" n])
     unfolding mod
     by (use assms(2,3) in \<open>auto simp: ac_simps\<close>)
 qed
 
 lemma verit_le_mono_div:
   fixes A B :: nat
   assumes "A < B" "0 < n"
   shows "(A div n) + (if B mod n = 0 then 1 else 0) \<le> (B div n)"
   by (auto simp: ac_simps Suc_leI assms less_mult_imp_div_less div_le_mono less_imp_le_nat)
 
 lemmas [smt_arith_multiplication] =
   verit_le_mono_div[THEN mult_le_mono1, unfolded add_mult_distrib]
   div_le_mono[THEN mult_le_mono2, unfolded add_mult_distrib]
 
 lemma div_mod_decomp_int: "A = (A div n) * n + (A mod n)" for A :: int
   by auto
 
 lemma zdiv_mono_strict:
   fixes A B :: int
   assumes "A < B" "0 < n" and
     mod: "A mod n = 0""B mod n = 0"
   shows "(A div n) < (B div n)"
 proof -
   show ?thesis
     using assms(1)
     apply (subst (asm) div_mod_decomp_int[of A n])
     apply (subst (asm) div_mod_decomp_int[of B n])
     unfolding mod
     by (use assms(2,3) in \<open>auto simp: ac_simps\<close>)
 qed
 
 lemma verit_le_mono_div_int:
-  fixes A B :: int
-  assumes "A < B" "0 < n"
-  shows "(A div n) + (if B mod n = 0 then 1 else 0) \<le> (B div n)"
+  \<open>A div n + (if B mod n = 0 then 1 else 0) \<le> B div n\<close>
+    if \<open>A < B\<close> \<open>0 < n\<close>
+    for A B n :: int
 proof -
-  have f2: "B div n = A div n \<or> A div n < B div n"
-    by (metis (no_types) assms less_le zdiv_mono1)
-  have "B div n \<noteq> A div n \<or> B mod n \<noteq> 0"
-    using assms(1) by (metis Groups.add_ac(2) assms(2) eucl_rel_int eucl_rel_int_iff
-      group_cancel.rule0 le_add_same_cancel2 not_le)
-  then have "B mod n = 0 \<longrightarrow> A div n + (if B mod n = 0 then 1 else 0) \<le> B div n"
-    using f2 by (auto dest: zless_imp_add1_zle)
-  then show ?thesis
-    using assms zdiv_mono1 by auto
+  from \<open>A < B\<close> \<open>0 < n\<close> have \<open>A div n \<le> B div n\<close>
+    by (auto intro: zdiv_mono1)
+  show ?thesis
+  proof (cases \<open>n dvd B\<close>)
+    case False
+    with \<open>A div n \<le> B div n\<close> show ?thesis
+      by auto
+  next
+    case True
+    then obtain C where \<open>B = n * C\<close> ..
+    then have \<open>B div n = C\<close>
+      using \<open>0 < n\<close> by simp
+    from \<open>0 < n\<close> have \<open>A mod n \<ge> 0\<close>
+      by simp
+    have \<open>A div n < C\<close>
+    proof (rule ccontr)
+      assume \<open>\<not> A div n < C\<close>
+      then have \<open>C \<le> A div n\<close>
+        by simp
+      with \<open>B div n = C\<close> \<open>A div n \<le> B div n\<close>
+      have \<open>A div n = C\<close>
+        by simp
+      moreover from \<open>A < B\<close> have \<open>n * (A div n) + A mod n < B\<close>
+        by simp
+      ultimately have \<open>n * C + A mod n < n * C\<close>
+        using \<open>B = n * C\<close> by simp
+      moreover have \<open>A mod n \<ge> 0\<close>
+        using \<open>0 < n\<close> by simp
+      ultimately show False
+        by simp
+    qed
+    with \<open>n dvd B\<close> \<open>B div n = C\<close> show ?thesis
+      by simp
+  qed
 qed
 
-
 lemma verit_less_mono_div_int2:
   fixes A B :: int
   assumes "A \<le> B" "0 < -n"
   shows "(A div n) \<ge> (B div n)"
   using assms(1) assms(2) zdiv_mono1_neg by auto
 
 lemmas [smt_arith_multiplication] =
   verit_le_mono_div_int[THEN mult_left_mono, unfolded int_distrib]
   zdiv_mono1[THEN mult_left_mono, unfolded int_distrib]
 
 lemmas [smt_arith_multiplication] =
   arg_cong[of _ _ \<open>\<lambda>a :: nat. a div n * p\<close> for n p :: nat, THEN sym]
   arg_cong[of _ _ \<open>\<lambda>a :: int. a div n * p\<close> for n p :: int, THEN sym]
 
 lemma [smt_arith_combine]:
   "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c + 2 \<le> b + d"
   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c + 1 \<le> b + d"
   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c + 1 \<le> b + d" for a b c :: int
   by auto
 
 lemma [smt_arith_combine]:
   "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c + 2 \<le> b + d"
   "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c + 1 \<le> b + d"
   "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c + 1 \<le> b + d" for a b c :: nat
   by auto
 
 lemmas [smt_arith_combine] =
   add_strict_mono
   add_less_le_mono
   add_mono
   add_le_less_mono
 
 lemma [smt_arith_combine]:
   \<open>m < n \<Longrightarrow> c = d \<Longrightarrow> m + c < n + d\<close>
   \<open>m \<le> n \<Longrightarrow> c = d \<Longrightarrow> m + c \<le> n + d\<close>
   \<open>c = d \<Longrightarrow> m < n \<Longrightarrow> m + c < n + d\<close>
   \<open>c = d \<Longrightarrow> m \<le> n \<Longrightarrow> m + c \<le> n + d\<close>
   for m :: \<open>'a :: ordered_cancel_ab_semigroup_add\<close>
   by (auto intro: ordered_cancel_ab_semigroup_add_class.add_strict_right_mono
     ordered_ab_semigroup_add_class.add_right_mono)
 
 lemma verit_negate_coefficient:
   \<open>a \<le> (b :: 'a :: {ordered_ab_group_add}) \<Longrightarrow> -a \<ge> -b\<close>
   \<open>a < b \<Longrightarrow> -a > -b\<close>
   \<open>a = b \<Longrightarrow> -a = -b\<close>
   by auto
 
 end
 
 lemma verit_ite_intro:
   \<open>(if a then P (if a then a' else b') else Q) \<longleftrightarrow> (if a then P a' else Q)\<close>
   \<open>(if a then P' else Q' (if a then a' else b')) \<longleftrightarrow> (if a then P' else Q' b')\<close>
   \<open>A = f (if a then R else S) \<longleftrightarrow> (if a then A = f R else A = f S)\<close>
   by auto
 
 lemma verit_ite_if_cong:
   fixes x y :: bool
   assumes "b=c"
     and "c \<equiv> True \<Longrightarrow> x = u"
     and "c \<equiv> False \<Longrightarrow> y = v"
   shows "(if b then x else y) \<equiv> (if c then u else v)"
 proof -
   have H: "(if b then x else y) = (if c then u else v)"
     using assms by (auto split: if_splits)
 
   show "(if b then x else y) \<equiv> (if c then u else v)"
     by (subst H) auto
 qed
 
 
 subsection \<open>Setup\<close>
 
 ML_file \<open>Tools/SMT/smt_util.ML\<close>
 ML_file \<open>Tools/SMT/smt_failure.ML\<close>
 ML_file \<open>Tools/SMT/smt_config.ML\<close>
 ML_file \<open>Tools/SMT/smt_builtin.ML\<close>
 ML_file \<open>Tools/SMT/smt_datatypes.ML\<close>
 ML_file \<open>Tools/SMT/smt_normalize.ML\<close>
 ML_file \<open>Tools/SMT/smt_translate.ML\<close>
 ML_file \<open>Tools/SMT/smtlib.ML\<close>
 ML_file \<open>Tools/SMT/smtlib_interface.ML\<close>
 ML_file \<open>Tools/SMT/smtlib_proof.ML\<close>
 ML_file \<open>Tools/SMT/smtlib_isar.ML\<close>
 ML_file \<open>Tools/SMT/z3_proof.ML\<close>
 ML_file \<open>Tools/SMT/z3_isar.ML\<close>
 ML_file \<open>Tools/SMT/smt_solver.ML\<close>
 ML_file \<open>Tools/SMT/cvc_interface.ML\<close>
 ML_file \<open>Tools/SMT/lethe_proof.ML\<close>
 ML_file \<open>Tools/SMT/lethe_isar.ML\<close>
 ML_file \<open>Tools/SMT/lethe_proof_parse.ML\<close>
 ML_file \<open>Tools/SMT/cvc_proof_parse.ML\<close>
 ML_file \<open>Tools/SMT/conj_disj_perm.ML\<close>
 ML_file \<open>Tools/SMT/smt_replay_methods.ML\<close>
 ML_file \<open>Tools/SMT/smt_replay.ML\<close>
 ML_file \<open>Tools/SMT/smt_replay_arith.ML\<close>
 ML_file \<open>Tools/SMT/z3_interface.ML\<close>
 ML_file \<open>Tools/SMT/z3_replay_rules.ML\<close>
 ML_file \<open>Tools/SMT/z3_replay_methods.ML\<close>
 ML_file \<open>Tools/SMT/z3_replay.ML\<close>
 ML_file \<open>Tools/SMT/lethe_replay_methods.ML\<close>
 ML_file \<open>Tools/SMT/verit_replay_methods.ML\<close>
 ML_file \<open>Tools/SMT/verit_strategies.ML\<close>
 ML_file \<open>Tools/SMT/verit_replay.ML\<close>
 ML_file \<open>Tools/SMT/smt_systems.ML\<close>
 
 
 subsection \<open>Configuration\<close>
 
 text \<open>
 The current configuration can be printed by the command
 \<open>smt_status\<close>, which shows the values of most options.
 \<close>
 
 
 subsection \<open>General configuration options\<close>
 
 text \<open>
 The option \<open>smt_solver\<close> can be used to change the target SMT
 solver. The possible values can be obtained from the \<open>smt_status\<close>
 command.
 \<close>
 
 declare [[smt_solver = z3]]
 
 text \<open>
 Since SMT solvers are potentially nonterminating, there is a timeout
 (given in seconds) to restrict their runtime.
 \<close>
 
 declare [[smt_timeout = 0]]
 
 text \<open>
 SMT solvers apply randomized heuristics. In case a problem is not
 solvable by an SMT solver, changing the following option might help.
 \<close>
 
 declare [[smt_random_seed = 1]]
 
 text \<open>
 In general, the binding to SMT solvers runs as an oracle, i.e, the SMT
 solvers are fully trusted without additional checks. The following
 option can cause the SMT solver to run in proof-producing mode, giving
 a checkable certificate. This is currently implemented only for veriT and
 Z3.
 \<close>
 
 declare [[smt_oracle = false]]
 
 text \<open>
 Each SMT solver provides several command-line options to tweak its
 behaviour. They can be passed to the solver by setting the following
 options.
 \<close>
 
 declare [[cvc4_options = ""]]
 declare [[cvc5_options = ""]]
 declare [[verit_options = ""]]
 declare [[z3_options = ""]]
 
 text \<open>
 The SMT method provides an inference mechanism to detect simple triggers
 in quantified formulas, which might increase the number of problems
 solvable by SMT solvers (note: triggers guide quantifier instantiations
 in the SMT solver). To turn it on, set the following option.
 \<close>
 
 declare [[smt_infer_triggers = false]]
 
 text \<open>
 Enable the following option to use built-in support for datatypes,
 codatatypes, and records in CVC4 and cvc5. Currently, this is implemented
 only in oracle mode.
 \<close>
 
 declare [[cvc_extensions = false]]
 
 text \<open>
 Enable the following option to use built-in support for div/mod, datatypes,
 and records in Z3. Currently, this is implemented only in oracle mode.
 \<close>
 
 declare [[z3_extensions = false]]
 
 
 subsection \<open>Certificates\<close>
 
 text \<open>
 By setting the option \<open>smt_certificates\<close> to the name of a file,
 all following applications of an SMT solver a cached in that file.
 Any further application of the same SMT solver (using the very same
 configuration) re-uses the cached certificate instead of invoking the
 solver. An empty string disables caching certificates.
 
 The filename should be given as an explicit path. It is good
 practice to use the name of the current theory (with ending
 \<open>.certs\<close> instead of \<open>.thy\<close>) as the certificates file.
 Certificate files should be used at most once in a certain theory context,
 to avoid race conditions with other concurrent accesses.
 \<close>
 
 declare [[smt_certificates = ""]]
 
 text \<open>
 The option \<open>smt_read_only_certificates\<close> controls whether only
 stored certificates should be used or invocation of an SMT solver
 is allowed. When set to \<open>true\<close>, no SMT solver will ever be
 invoked and only the existing certificates found in the configured
 cache are used;  when set to \<open>false\<close> and there is no cached
 certificate for some proposition, then the configured SMT solver is
 invoked.
 \<close>
 
 declare [[smt_read_only_certificates = false]]
 
 
 subsection \<open>Tracing\<close>
 
 text \<open>
 The SMT method, when applied, traces important information. To
 make it entirely silent, set the following option to \<open>false\<close>.
 \<close>
 
 declare [[smt_verbose = true]]
 
 text \<open>
 For tracing the generated problem file given to the SMT solver as
 well as the returned result of the solver, the option
 \<open>smt_trace\<close> should be set to \<open>true\<close>.
 \<close>
 
 declare [[smt_trace = false]]
 
 
 subsection \<open>Schematic rules for Z3 proof reconstruction\<close>
 
 text \<open>
 Several prof rules of Z3 are not very well documented. There are two
 lemma groups which can turn failing Z3 proof reconstruction attempts
 into succeeding ones: the facts in \<open>z3_rule\<close> are tried prior to
 any implemented reconstruction procedure for all uncertain Z3 proof
 rules;  the facts in \<open>z3_simp\<close> are only fed to invocations of
 the simplifier when reconstructing theory-specific proof steps.
 \<close>
 
 lemmas [z3_rule] =
   refl eq_commute conj_commute disj_commute simp_thms nnf_simps
   ring_distribs field_simps times_divide_eq_right times_divide_eq_left
   if_True if_False not_not
   NO_MATCH_def
 
 lemma [z3_rule]:
   "(P \<and> Q) = (\<not> (\<not> P \<or> \<not> Q))"
   "(P \<and> Q) = (\<not> (\<not> Q \<or> \<not> P))"
   "(\<not> P \<and> Q) = (\<not> (P \<or> \<not> Q))"
   "(\<not> P \<and> Q) = (\<not> (\<not> Q \<or> P))"
   "(P \<and> \<not> Q) = (\<not> (\<not> P \<or> Q))"
   "(P \<and> \<not> Q) = (\<not> (Q \<or> \<not> P))"
   "(\<not> P \<and> \<not> Q) = (\<not> (P \<or> Q))"
   "(\<not> P \<and> \<not> Q) = (\<not> (Q \<or> P))"
   by auto
 
 lemma [z3_rule]:
   "(P \<longrightarrow> Q) = (Q \<or> \<not> P)"
   "(\<not> P \<longrightarrow> Q) = (P \<or> Q)"
   "(\<not> P \<longrightarrow> Q) = (Q \<or> P)"
   "(True \<longrightarrow> P) = P"
   "(P \<longrightarrow> True) = True"
   "(False \<longrightarrow> P) = True"
   "(P \<longrightarrow> P) = True"
   "(\<not> (A \<longleftrightarrow> \<not> B)) \<longleftrightarrow> (A \<longleftrightarrow> B)"
   by auto
 
 lemma [z3_rule]:
   "((P = Q) \<longrightarrow> R) = (R \<or> (Q = (\<not> P)))"
   by auto
 
 lemma [z3_rule]:
   "(\<not> True) = False"
   "(\<not> False) = True"
   "(x = x) = True"
   "(P = True) = P"
   "(True = P) = P"
   "(P = False) = (\<not> P)"
   "(False = P) = (\<not> P)"
   "((\<not> P) = P) = False"
   "(P = (\<not> P)) = False"
   "((\<not> P) = (\<not> Q)) = (P = Q)"
   "\<not> (P = (\<not> Q)) = (P = Q)"
   "\<not> ((\<not> P) = Q) = (P = Q)"
   "(P \<noteq> Q) = (Q = (\<not> P))"
   "(P = Q) = ((\<not> P \<or> Q) \<and> (P \<or> \<not> Q))"
   "(P \<noteq> Q) = ((\<not> P \<or> \<not> Q) \<and> (P \<or> Q))"
   by auto
 
 lemma [z3_rule]:
   "(if P then P else \<not> P) = True"
   "(if \<not> P then \<not> P else P) = True"
   "(if P then True else False) = P"
   "(if P then False else True) = (\<not> P)"
   "(if P then Q else True) = ((\<not> P) \<or> Q)"
   "(if P then Q else True) = (Q \<or> (\<not> P))"
   "(if P then Q else \<not> Q) = (P = Q)"
   "(if P then Q else \<not> Q) = (Q = P)"
   "(if P then \<not> Q else Q) = (P = (\<not> Q))"
   "(if P then \<not> Q else Q) = ((\<not> Q) = P)"
   "(if \<not> P then x else y) = (if P then y else x)"
   "(if P then (if Q then x else y) else x) = (if P \<and> (\<not> Q) then y else x)"
   "(if P then (if Q then x else y) else x) = (if (\<not> Q) \<and> P then y else x)"
   "(if P then (if Q then x else y) else y) = (if P \<and> Q then x else y)"
   "(if P then (if Q then x else y) else y) = (if Q \<and> P then x else y)"
   "(if P then x else if P then y else z) = (if P then x else z)"
   "(if P then x else if Q then x else y) = (if P \<or> Q then x else y)"
   "(if P then x else if Q then x else y) = (if Q \<or> P then x else y)"
   "(if P then x = y else x = z) = (x = (if P then y else z))"
   "(if P then x = y else y = z) = (y = (if P then x else z))"
   "(if P then x = y else z = y) = (y = (if P then x else z))"
   by auto
 
 lemma [z3_rule]:
   "0 + (x::int) = x"
   "x + 0 = x"
   "x + x = 2 * x"
   "0 * x = 0"
   "1 * x = x"
   "x + y = y + x"
   by auto
 
 lemma [z3_rule]:  (* for def-axiom *)
   "P = Q \<or> P \<or> Q"
   "P = Q \<or> \<not> P \<or> \<not> Q"
   "(\<not> P) = Q \<or> \<not> P \<or> Q"
   "(\<not> P) = Q \<or> P \<or> \<not> Q"
   "P = (\<not> Q) \<or> \<not> P \<or> Q"
   "P = (\<not> Q) \<or> P \<or> \<not> Q"
   "P \<noteq> Q \<or> P \<or> \<not> Q"
   "P \<noteq> Q \<or> \<not> P \<or> Q"
   "P \<noteq> (\<not> Q) \<or> P \<or> Q"
   "(\<not> P) \<noteq> Q \<or> P \<or> Q"
   "P \<or> Q \<or> P \<noteq> (\<not> Q)"
   "P \<or> Q \<or> (\<not> P) \<noteq> Q"
   "P \<or> \<not> Q \<or> P \<noteq> Q"
   "\<not> P \<or> Q \<or> P \<noteq> Q"
   "P \<or> y = (if P then x else y)"
   "P \<or> (if P then x else y) = y"
   "\<not> P \<or> x = (if P then x else y)"
   "\<not> P \<or> (if P then x else y) = x"
   "P \<or> R \<or> \<not> (if P then Q else R)"
   "\<not> P \<or> Q \<or> \<not> (if P then Q else R)"
   "\<not> (if P then Q else R) \<or> \<not> P \<or> Q"
   "\<not> (if P then Q else R) \<or> P \<or> R"
   "(if P then Q else R) \<or> \<not> P \<or> \<not> Q"
   "(if P then Q else R) \<or> P \<or> \<not> R"
   "(if P then \<not> Q else R) \<or> \<not> P \<or> Q"
   "(if P then Q else \<not> R) \<or> P \<or> R"
   by auto
 
 hide_type (open) symb_list pattern
 hide_const (open) Symb_Nil Symb_Cons trigger pat nopat fun_app z3div z3mod
 
 end