diff --git a/src/HOL/SMT.thy b/src/HOL/SMT.thy
--- a/src/HOL/SMT.thy
+++ b/src/HOL/SMT.thy
@@ -1,894 +1,894 @@
 (*  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_eq 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 divmod_step_eq 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)"
 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
 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/cvc4_interface.ML\<close>
 ML_file \<open>Tools/SMT/cvc4_proof_parse.ML\<close>
 ML_file \<open>Tools/SMT/verit_proof.ML\<close>
 ML_file \<open>Tools/SMT/verit_isar.ML\<close>
 ML_file \<open>Tools/SMT/verit_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/verit_replay_methods.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 only implemented for Z3.
 \<close>
 
 declare [[smt_oracle = false]]
 
 text \<open>
 Each SMT solver provides several commandline options to tweak its
 behaviour. They can be passed to the solver by setting the following
 options.
 \<close>
 
 declare [[cvc4_options = "--full-saturate-quant --inst-when=full-last-call --inst-no-entail --term-db-mode=relevant --multi-trigger-linear"]]
 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. Currently, this is implemented only
 in oracle mode.
 \<close>
 
 declare [[cvc4_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 are should be used or invocation of an SMT solver
+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