# Theory SMT

```(*  Title:      HOL/SMT.thy
Author:     Sascha Boehme, TU Muenchen
Author:     Jasmin Blanchette, VU Amsterdam
*)

section ‹Bindings to Satisfiability Modulo Theories (SMT) solvers based on SMT-LIB 2›

theory SMT
imports Divides Numeral_Simprocs
keywords "smt_status" :: diag
begin

subsection ‹A skolemization tactic and proof method›

lemma choices:
"⋀Q. ∀x. ∃y ya. Q x y ya ⟹ ∃f fa. ∀x. Q x (f x) (fa x)"
"⋀Q. ∀x. ∃y ya yb. Q x y ya yb ⟹ ∃f fa fb. ∀x. Q x (f x) (fa x) (fb x)"
"⋀Q. ∀x. ∃y ya yb yc. Q x y ya yb yc ⟹ ∃f fa fb fc. ∀x. Q x (f x) (fa x) (fb x) (fc x)"
"⋀Q. ∀x. ∃y ya yb yc yd. Q x y ya yb yc yd ⟹
∃f fa fb fc fd. ∀x. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
"⋀Q. ∀x. ∃y ya yb yc yd ye. Q x y ya yb yc yd ye ⟹
∃f fa fb fc fd fe. ∀x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
"⋀Q. ∀x. ∃y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf ⟹
∃f fa fb fc fd fe ff. ∀x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
"⋀Q. ∀x. ∃y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg ⟹
∃f fa fb fc fd fe ff fg. ∀x. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
by metis+

lemma bchoices:
"⋀Q. ∀x ∈ S. ∃y ya. Q x y ya ⟹ ∃f fa. ∀x ∈ S. Q x (f x) (fa x)"
"⋀Q. ∀x ∈ S. ∃y ya yb. Q x y ya yb ⟹ ∃f fa fb. ∀x ∈ S. Q x (f x) (fa x) (fb x)"
"⋀Q. ∀x ∈ S. ∃y ya yb yc. Q x y ya yb yc ⟹ ∃f fa fb fc. ∀x ∈ S. Q x (f x) (fa x) (fb x) (fc x)"
"⋀Q. ∀x ∈ S. ∃y ya yb yc yd. Q x y ya yb yc yd ⟹
∃f fa fb fc fd. ∀x ∈ S. Q x (f x) (fa x) (fb x) (fc x) (fd x)"
"⋀Q. ∀x ∈ S. ∃y ya yb yc yd ye. Q x y ya yb yc yd ye ⟹
∃f fa fb fc fd fe. ∀x ∈ S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x)"
"⋀Q. ∀x ∈ S. ∃y ya yb yc yd ye yf. Q x y ya yb yc yd ye yf ⟹
∃f fa fb fc fd fe ff. ∀x ∈ S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x)"
"⋀Q. ∀x ∈ S. ∃y ya yb yc yd ye yf yg. Q x y ya yb yc yd ye yf yg ⟹
∃f fa fb fc fd fe ff fg. ∀x ∈ S. Q x (f x) (fa x) (fb x) (fc x) (fd x) (fe x) (ff x) (fg x)"
by metis+

ML ‹
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))
›

method_setup moura = ‹
Scan.succeed (SIMPLE_METHOD' o moura_tac)
› "solve skolemization goals, especially those arising from Z3 proofs"

hide_fact (open) choices bchoices

subsection ‹Triggers for quantifier instantiation›

text ‹
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.
›

typedecl 'a symb_list

consts
Symb_Nil :: "'a symb_list"
Symb_Cons :: "'a ⇒ 'a symb_list ⇒ 'a symb_list"

typedecl pattern

consts
pat :: "'a ⇒ pattern"
nopat :: "'a ⇒ pattern"

definition trigger :: "pattern symb_list symb_list ⇒ bool ⇒ bool" where
"trigger _ P = P"

subsection ‹Higher-order encoding›

text ‹
Application is made explicit for constants occurring with varying
numbers of arguments. This is achieved by the introduction of the
following constant.
›

definition fun_app :: "'a ⇒ 'a" where "fun_app f = f"

text ‹
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.
›

lemmas array_rules = ext fun_upd_apply fun_upd_same fun_upd_other  fun_upd_upd fun_app_def

subsection ‹Normalization›

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 = (λi. nat (numeral i))" by simp
lemma nat_less_as_int: "(<) = (λa b. int a < int b)" by simp
lemma nat_leq_as_int: "(≤) = (λa b. int a ≤ int b)" by simp
lemma Suc_as_int: "Suc = (λa. nat (int a + 1))" by (rule ext) simp
lemma nat_plus_as_int: "(+) = (λa b. nat (int a + int b))" by (rule ext)+ simp
lemma nat_minus_as_int: "(-) = (λa b. nat (int a - int b))" by (rule ext)+ simp
lemma nat_times_as_int: "(*) = (λa b. nat (int a * int b))" by (simp add: nat_mult_distrib)
lemma nat_div_as_int: "(div) = (λa b. nat (int a div int b))" by (simp add: nat_div_distrib)
lemma nat_mod_as_int: "(mod) = (λa 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 ≤ b) = (int a ≤ 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 ‹Integer division and modulo for Z3›

text ‹
The following Z3-inspired definitions are overspecified for the case where ‹l = 0›. This
Schönheitsfehler is corrected in the ‹div_as_z3div› and ‹mod_as_z3mod› theorems.
›

definition z3div :: "int ⇒ int ⇒ int" where
"z3div k l = (if l ≥ 0 then k div l else - (k div - l))"

definition z3mod :: "int ⇒ int ⇒ int" where
"z3mod k l = k mod (if l ≥ 0 then l else - l)"

lemma div_as_z3div:
"∀k l. k div l = (if l = 0 then 0 else if l > 0 then z3div k l else z3div (- k) (- l))"

lemma mod_as_z3mod:
"∀k l. k mod l = (if l = 0 then k else if l > 0 then z3mod k l else - z3mod (- k) (- l))"

subsection ‹Extra theorems for veriT reconstruction›

lemma verit_sko_forall: ‹(∀x. P x) ⟷ P (SOME x. ¬P x)›
using someI[of ‹λx. ¬P x›]
by auto

lemma verit_sko_forall': ‹P (SOME x. ¬P x) = A ⟹ (∀x. P x) = A›
by (subst verit_sko_forall)

lemma verit_sko_forall'': ‹B = A ⟹ (SOME x. P x) = A ≡ (SOME x. P x) = B›
by auto

lemma verit_sko_forall_indirect: ‹x = (SOME x. ¬P x) ⟹ (∀x. P x) ⟷ P x›
using someI[of ‹λx. ¬P x›]
by auto

lemma verit_sko_forall_indirect2:
‹x = (SOME x. ¬P x) ⟹ (⋀x :: 'a. (P x = P' x)) ⟹(∀x. P' x) ⟷ P x›
using someI[of ‹λx. ¬P x›]
by auto

lemma verit_sko_ex: ‹(∃x. P x) ⟷ P (SOME x. P x)›
using someI[of ‹λx. P x›]
by auto

lemma verit_sko_ex': ‹P (SOME x. P x) = A ⟹ (∃x. P x) = A›
by (subst verit_sko_ex)

lemma verit_sko_ex_indirect: ‹x = (SOME x. P x) ⟹ (∃x. P x) ⟷ P x›
using someI[of ‹λx. P x›]
by auto

lemma verit_sko_ex_indirect2: ‹x = (SOME x. P x) ⟹ (⋀x. P x = P' x) ⟹ (∃x. P' x) ⟷ P x›
using someI[of ‹λx. P x›]
by auto

lemma verit_Pure_trans:
‹P ≡ Q ⟹ Q ⟹ P›
by auto

lemma verit_if_cong:
assumes ‹b ≡ c›
and ‹c ⟹ x ≡ u›
and ‹¬ c ⟹ y ≡ v›
shows ‹(if b then x else y) ≡ (if c then u else v)›
using assms if_cong[of b c x u] by auto

lemma verit_if_weak_cong':
‹b ≡ c ⟹ (if b then x else y) ≡ (if c then x else y)›
by auto

lemma verit_or_neg:
‹(A ⟹ B) ⟹ B ∨ ¬A›
‹(¬A ⟹ B) ⟹ B ∨ A›
by auto

lemma verit_subst_bool: ‹P ⟹ f True ⟹ f P›
by auto

lemma verit_and_pos:
‹(a ⟹ ¬(b ∧ c) ∨ A) ⟹ ¬(a ∧ b ∧ c) ∨ A›
‹(a ⟹ b ⟹ A) ⟹ ¬(a ∧ b) ∨ A›
‹(a ⟹ A) ⟹ ¬a ∨ A›
‹(¬a ⟹ A) ⟹ a ∨ A›
by blast+

lemma verit_or_pos:
‹A ∧ A' ⟹ (c ∧ A) ∨ (¬c ∧ A')›
‹A ∧ A' ⟹ (¬c ∧ A) ∨ (c ∧ A')›
by blast+

lemma verit_la_generic:
‹(a::int) ≤ x ∨ a = x ∨ a ≥ x›
by linarith

lemma verit_bfun_elim:
‹(if b then P True else P False) = P b›
‹(∀b. P' b) = (P' False ∧ P' True)›
‹(∃b. P' b) = (P' False ∨ P' True)›
by (cases b) (auto simp: all_bool_eq ex_bool_eq)

lemma verit_eq_true_simplify:
‹(P = True) ≡ P›
by auto

lemma verit_and_neg:
‹(a ⟹ ¬b ∨ A) ⟹ ¬(a ∧ b) ∨ A›
‹(a ⟹ A) ⟹ ¬a ∨ A›
‹(¬a ⟹ A) ⟹ a ∨ A›
by blast+

lemma verit_forall_inst:
‹A ⟷ B ⟹ ¬A ∨ B›
‹¬A ⟷ B ⟹ A ∨ B›
‹A ⟷ B ⟹ ¬B ∨ A›
‹A ⟷ ¬B ⟹ B ∨ A›
‹A ⟶ B ⟹ ¬A ∨ B›
‹¬A ⟶ B ⟹ A ∨ B›
by blast+

lemma verit_eq_transitive:
‹A = B ⟹ B = C ⟹ A = C›
‹A = B ⟹ C = B ⟹ A = C›
‹B = A ⟹ B = C ⟹ A = C›
‹B = A ⟹ C = B ⟹ A = C›
by auto

lemma verit_bool_simplify:
‹¬(P ⟶ Q) ⟷ P ∧ ¬Q›
‹¬(P ∨ Q) ⟷ ¬P ∧ ¬Q›
‹¬(P ∧ Q) ⟷ ¬P ∨ ¬Q›
‹(P ⟶ (Q ⟶ R)) ⟷ ((P ∧ Q) ⟶ R)›
‹((P ⟶ Q) ⟶ Q) ⟷ P ∨ Q›
‹(Q ⟷ (P ∨ Q)) ⟷ (P ⟶ Q)› ― ‹This rule was inverted›
‹P ∧ (P ⟶ Q) ⟷ P ∧ Q›
‹(P ⟶ Q) ∧ P ⟷ P ∧ Q›
*  ‹((P ⟶ Q) ⟶ P) ⟷ P›
*  ‹((P ⟶ Q) ⟶ Q) ⟷ P ∨ Q›
*  ‹(P ⟶ Q) ∨ P› *)
unfolding not_imp imp_conjL
by auto

text ‹We need the last equation for \<^term>‹¬(∀a b. ¬P a b)››
lemma verit_connective_def: ― ‹the definition of XOR is missing
as the operator is not generated by Isabelle›
‹(A = B) ⟷ ((A ⟶ B) ∧ (B ⟶ A))›
‹(If A B C) = ((A ⟶ B) ∧ (¬A ⟶ C))›
‹(∃x. P x) ⟷ ¬(∀x. ¬P x)›
‹¬(∃x. P x) ⟷ (∀x. ¬P x)›
by auto

lemma verit_ite_simplify:
‹(If True B C) = B›
‹(If False B C) = C›
‹(If A' B B) = B›
‹(If (¬A') B C) = (If A' C B)›
‹(If c (If c A B) C) = (If c A C)›
‹(If c C (If c A B)) = (If c C B)›
‹(If A' True False) = A'›
‹(If A' False True) ⟷ ¬A'›
‹(If A' True B') ⟷ A'∨B'›
‹(If A' B' False) ⟷ A'∧B'›
‹(If A' False B') ⟷ ¬A'∧B'›
‹(If A' B' True) ⟷ ¬A'∨B'›
‹x ∧ True ⟷ x›
‹x ∨ False ⟷ x›
for B C :: 'a and A' B' C' :: bool
by auto

lemma verit_and_simplify1:
‹True ∧ b ⟷ b› ‹b ∧ True ⟷ b›
‹False ∧ b ⟷ False› ‹b ∧ False ⟷ False›
‹(c ∧ ¬c) ⟷ False› ‹(¬c ∧ c) ⟷ False›
‹¬¬a = a›
by auto

lemmas verit_and_simplify = conj_ac de_Morgan_conj disj_not1

lemma verit_or_simplify_1:
‹False ∨ b ⟷ b› ‹b ∨ False ⟷ b›
‹b ∨ ¬b›
‹¬b ∨ b›
by auto

lemmas verit_or_simplify = disj_ac

lemma verit_not_simplify:
‹¬ ¬b ⟷ b› ‹¬True ⟷ False› ‹¬False ⟷ True›
by auto

lemma verit_implies_simplify:
‹(¬a ⟶ ¬b) ⟷ (b ⟶ a)›
‹(False ⟶ a) ⟷ True›
‹(a ⟶ True) ⟷ True›
‹(True ⟶ a) ⟷ a›
‹(a ⟶ False) ⟷ ¬a›
‹(a ⟶ a) ⟷ True›
‹(¬a ⟶ a) ⟷ a›
‹(a ⟶ ¬a) ⟷ ¬a›
‹((a ⟶ b) ⟶ b) ⟷ a ∨ b›
by auto

lemma verit_equiv_simplify:
‹((¬a) = (¬b)) ⟷ (a = b)›
‹(a = a) ⟷ True›
‹(a = (¬a)) ⟷ False›
‹((¬a) = a) ⟷ False›
‹(True = a) ⟷ a›
‹(a = True) ⟷ a›
‹(False = a) ⟷ ¬a›
‹(a = False) ⟷ ¬a›
‹¬¬a ⟷ a›
‹(¬ False) = True›
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:
‹(a :: 'a :: cancel_comm_monoid_add) - a = 0›
‹(a :: 'a :: cancel_comm_monoid_add) - 0 = a›
‹0 - (b :: 'b :: {group_add}) = -b›
‹- (- (b :: 'b :: group_add)) = b›
by auto

lemma verit_sum_simplify:
‹(a :: 'a :: cancel_comm_monoid_add) + 0 = a›
by auto

lemmas verit_prod_simplify =
mult_zero_class.mult_zero_right
mult_zero_class.mult_zero_left *)
mult_1
mult_1_right

lemma verit_comp_simplify1:
‹(a :: 'a ::order) < a ⟷ False›
‹a ≤ a›
‹¬(b' ≤ a') ⟷ (a' :: 'b :: linorder) < b'›
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:
‹(a :: 'a ::linorder) = b ∨ ¬a ≤ b ∨ ¬b ≤ a›
by auto

context
begin

text ‹For the reconstruction, we need to keep the order of the arguments.›

named_theorems smt_arith_multiplication ‹Theorems to reconstruct arithmetic theorems.›

named_theorems smt_arith_combine ‹Theorems to reconstruct arithmetic theorems.›

named_theorems smt_arith_simplify ‹Theorems to combine theorems in the LA procedure›

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
zero_neq_one zero_le_one le_num_simps add_Suc mod_div_trivial nat.distinct mult_minus_right
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
zero_le_one One_nat_def add_Suc mod_div_trivial nat.distinct of_int_1 numerals numeral_One

lemma [smt_arith_simplify]:
‹¬ (a' :: 'a :: linorder) < b' ⟷ b' ≤ a'›
‹¬ (a' :: 'a :: linorder) ≤ b' ⟷ b' < a'›
‹(c::int) mod Numeral1 = 0›
‹(a::nat) mod Numeral1 = 0›
‹(c::int) div Numeral1 = c›
‹a div Numeral1 = a›
‹(c::int) mod 1 = 0›
‹a mod 1 = 0›
‹(c::int) div 1 = c›
‹a div 1 = a›
‹¬(a' ≠ b') ⟷ a' = b'›
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 ‹auto simp: ac_simps›)
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) ≤ (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] =

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 ‹auto simp: ac_simps›)
qed

lemma verit_le_mono_div_int:
‹A div n + (if B mod n = 0 then 1 else 0) ≤ B div n›
if ‹A < B› ‹0 < n›
for A B n :: int
proof -
from ‹A < B› ‹0 < n› have ‹A div n ≤ B div n›
by (auto intro: zdiv_mono1)
show ?thesis
proof (cases ‹n dvd B›)
case False
with ‹A div n ≤ B div n› show ?thesis
by auto
next
case True
then obtain C where ‹B = n * C› ..
then have ‹B div n = C›
using ‹0 < n› by simp
from ‹0 < n› have ‹A mod n ≥ 0›
by simp
have ‹A div n < C›
proof (rule ccontr)
assume ‹¬ A div n < C›
then have ‹C ≤ A div n›
by simp
with ‹B div n = C› ‹A div n ≤ B div n›
have ‹A div n = C›
by simp
moreover from ‹A < B› have ‹n * (A div n) + A mod n < B›
by simp
ultimately have ‹n * C + A mod n < n * C›
using ‹B = n * C› by simp
moreover have ‹A mod n ≥ 0›
using ‹0 < n› by simp
ultimately show False
by simp
qed
with ‹n dvd B› ‹B div n = C› show ?thesis
by simp
qed
qed

lemma verit_less_mono_div_int2:
fixes A B :: int
assumes "A ≤ B" "0 < -n"
shows "(A div n) ≥ (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 _ _ ‹λa :: nat. a div n * p› for n p :: nat, THEN sym]
arg_cong[of _ _ ‹λa :: int. a div n * p› for n p :: int, THEN sym]

lemma [smt_arith_combine]:
"a < b ⟹ c < d ⟹ a + c + 2 ≤ b + d"
"a < b ⟹ c ≤ d ⟹ a + c + 1 ≤ b + d"
"a ≤ b ⟹ c < d ⟹ a + c + 1 ≤ b + d" for a b c :: int
by auto

lemma [smt_arith_combine]:
"a < b ⟹ c < d ⟹ a + c + 2 ≤ b + d"
"a < b ⟹ c ≤ d ⟹ a + c + 1 ≤ b + d"
"a ≤ b ⟹ c < d ⟹ a + c + 1 ≤ b + d" for a b c :: nat
by auto

lemmas [smt_arith_combine] =

lemma [smt_arith_combine]:
‹m < n ⟹ c = d ⟹ m + c < n + d›
‹m ≤ n ⟹ c = d ⟹ m + c ≤ n + d›
‹c = d ⟹ m < n ⟹ m + c < n + d›
‹c = d ⟹ m ≤ n ⟹ m + c ≤ n + d›
for m :: ‹'a :: ordered_cancel_ab_semigroup_add›

lemma verit_negate_coefficient:
‹a ≤ (b :: 'a :: {ordered_ab_group_add}) ⟹ -a ≥ -b›
‹a < b ⟹ -a > -b›
‹a = b ⟹ -a = -b›
by auto

end

lemma verit_ite_intro:
‹(if a then P (if a then a' else b') else Q) ⟷ (if a then P a' else Q)›
‹(if a then P' else Q' (if a then a' else b')) ⟷ (if a then P' else Q' b')›
‹A = f (if a then R else S) ⟷ (if a then A = f R else A = f S)›
by auto

lemma verit_ite_if_cong:
fixes x y :: bool
assumes "b=c"
and "c ≡ True ⟹ x = u"
and "c ≡ False ⟹ y = v"
shows "(if b then x else y) ≡ (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) ≡ (if c then u else v)"
by (subst H) auto
qed

subsection ‹Setup›

ML_file ‹Tools/SMT/smt_util.ML›
ML_file ‹Tools/SMT/smt_failure.ML›
ML_file ‹Tools/SMT/smt_config.ML›
ML_file ‹Tools/SMT/smt_builtin.ML›
ML_file ‹Tools/SMT/smt_datatypes.ML›
ML_file ‹Tools/SMT/smt_normalize.ML›
ML_file ‹Tools/SMT/smt_translate.ML›
ML_file ‹Tools/SMT/smtlib.ML›
ML_file ‹Tools/SMT/smtlib_interface.ML›
ML_file ‹Tools/SMT/smtlib_proof.ML›
ML_file ‹Tools/SMT/smtlib_isar.ML›
ML_file ‹Tools/SMT/z3_proof.ML›
ML_file ‹Tools/SMT/z3_isar.ML›
ML_file ‹Tools/SMT/smt_solver.ML›
ML_file ‹Tools/SMT/cvc_interface.ML›
ML_file ‹Tools/SMT/lethe_proof.ML›
ML_file ‹Tools/SMT/lethe_isar.ML›
ML_file ‹Tools/SMT/lethe_proof_parse.ML›
ML_file ‹Tools/SMT/cvc_proof_parse.ML›
ML_file ‹Tools/SMT/conj_disj_perm.ML›
ML_file ‹Tools/SMT/smt_replay_methods.ML›
ML_file ‹Tools/SMT/smt_replay.ML›
ML_file ‹Tools/SMT/smt_replay_arith.ML›
ML_file ‹Tools/SMT/z3_interface.ML›
ML_file ‹Tools/SMT/z3_replay_rules.ML›
ML_file ‹Tools/SMT/z3_replay_methods.ML›
ML_file ‹Tools/SMT/z3_replay.ML›
ML_file ‹Tools/SMT/lethe_replay_methods.ML›
ML_file ‹Tools/SMT/verit_replay_methods.ML›
ML_file ‹Tools/SMT/verit_strategies.ML›
ML_file ‹Tools/SMT/verit_replay.ML›
ML_file ‹Tools/SMT/smt_systems.ML›

subsection ‹Configuration›

text ‹
The current configuration can be printed by the command
‹smt_status›, which shows the values of most options.
›

subsection ‹General configuration options›

text ‹
The option ‹smt_solver› can be used to change the target SMT
solver. The possible values can be obtained from the ‹smt_status›
command.
›

declare [[smt_solver = z3]]

text ‹
Since SMT solvers are potentially nonterminating, there is a timeout
(given in seconds) to restrict their runtime.
›

declare [[smt_timeout = 0]]

text ‹
SMT solvers apply randomized heuristics. In case a problem is not
solvable by an SMT solver, changing the following option might help.
›

declare [[smt_random_seed = 1]]

text ‹
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.
›

declare [[smt_oracle = false]]

text ‹
Each SMT solver provides several command-line options to tweak its
behaviour. They can be passed to the solver by setting the following
options.
›

declare [[cvc4_options = ""]]
declare [[cvc5_options = ""]]
declare [[verit_options = ""]]
declare [[z3_options = ""]]

text ‹
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.
›

declare [[smt_infer_triggers = false]]

text ‹
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.
›

declare [[cvc_extensions = false]]

text ‹
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.
›

declare [[z3_extensions = false]]

subsection ‹Certificates›

text ‹
By setting the option ‹smt_certificates› 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
‹.certs› instead of ‹.thy›) 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.
›

declare [[smt_certificates = ""]]

text ‹
The option ‹smt_read_only_certificates› controls whether only
stored certificates should be used or invocation of an SMT solver
is allowed. When set to ‹true›, no SMT solver will ever be
invoked and only the existing certificates found in the configured
cache are used;  when set to ‹false› and there is no cached
certificate for some proposition, then the configured SMT solver is
invoked.
›

subsection ‹Tracing›

text ‹
The SMT method, when applied, traces important information. To
make it entirely silent, set the following option to ‹false›.
›

declare [[smt_verbose = true]]

text ‹
For tracing the generated problem file given to the SMT solver as
well as the returned result of the solver, the option
‹smt_trace› should be set to ‹true›.
›

declare [[smt_trace = false]]

subsection ‹Schematic rules for Z3 proof reconstruction›

text ‹
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 ‹z3_rule› are tried prior to
any implemented reconstruction procedure for all uncertain Z3 proof
rules;  the facts in ‹z3_simp› are only fed to invocations of
the simplifier when reconstructing theory-specific proof steps.
›

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 ∧ Q) = (¬ (¬ P ∨ ¬ Q))"
"(P ∧ Q) = (¬ (¬ Q ∨ ¬ P))"
"(¬ P ∧ Q) = (¬ (P ∨ ¬ Q))"
"(¬ P ∧ Q) = (¬ (¬ Q ∨ P))"
"(P ∧ ¬ Q) = (¬ (¬ P ∨ Q))"
"(P ∧ ¬ Q) = (¬ (Q ∨ ¬ P))"
"(¬ P ∧ ¬ Q) = (¬ (P ∨ Q))"
"(¬ P ∧ ¬ Q) = (¬ (Q ∨ P))"
by auto

lemma [z3_rule]:
"(P ⟶ Q) = (Q ∨ ¬ P)"
"(¬ P ⟶ Q) = (P ∨ Q)"
"(¬ P ⟶ Q) = (Q ∨ P)"
"(True ⟶ P) = P"
"(P ⟶ True) = True"
"(False ⟶ P) = True"
"(P ⟶ P) = True"
"(¬ (A ⟷ ¬ B)) ⟷ (A ⟷ B)"
by auto

lemma [z3_rule]:
"((P = Q) ⟶ R) = (R ∨ (Q = (¬ P)))"
by auto

lemma [z3_rule]:
"(¬ True) = False"
"(¬ False) = True"
"(x = x) = True"
"(P = True) = P"
"(True = P) = P"
"(P = False) = (¬ P)"
"(False = P) = (¬ P)"
"((¬ P) = P) = False"
"(P = (¬ P)) = False"
"((¬ P) = (¬ Q)) = (P = Q)"
"¬ (P = (¬ Q)) = (P = Q)"
"¬ ((¬ P) = Q) = (P = Q)"
"(P ≠ Q) = (Q = (¬ P))"
"(P = Q) = ((¬ P ∨ Q) ∧ (P ∨ ¬ Q))"
"(P ≠ Q) = ((¬ P ∨ ¬ Q) ∧ (P ∨ Q))"
by auto

lemma [z3_rule]:
"(if P then P else ¬ P) = True"
"(if ¬ P then ¬ P else P) = True"
"(if P then True else False) = P"
"(if P then False else True) = (¬ P)"
"(if P then Q else True) = ((¬ P) ∨ Q)"
"(if P then Q else True) = (Q ∨ (¬ P))"
"(if P then Q else ¬ Q) = (P = Q)"
"(if P then Q else ¬ Q) = (Q = P)"
"(if P then ¬ Q else Q) = (P = (¬ Q))"
"(if P then ¬ Q else Q) = ((¬ Q) = P)"
"(if ¬ P then x else y) = (if P then y else x)"
"(if P then (if Q then x else y) else x) = (if P ∧ (¬ Q) then y else x)"
"(if P then (if Q then x else y) else x) = (if (¬ Q) ∧ P then y else x)"
"(if P then (if Q then x else y) else y) = (if P ∧ Q then x else y)"
"(if P then (if Q then x else y) else y) = (if Q ∧ 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 ∨ Q then x else y)"
"(if P then x else if Q then x else y) = (if Q ∨ 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 ∨ P ∨ Q"
"P = Q ∨ ¬ P ∨ ¬ Q"
"(¬ P) = Q ∨ ¬ P ∨ Q"
"(¬ P) = Q ∨ P ∨ ¬ Q"
"P = (¬ Q) ∨ ¬ P ∨ Q"
"P = (¬ Q) ∨ P ∨ ¬ Q"
"P ≠ Q ∨ P ∨ ¬ Q"
"P ≠ Q ∨ ¬ P ∨ Q"
"P ≠ (¬ Q) ∨ P ∨ Q"
"(¬ P) ≠ Q ∨ P ∨ Q"
"P ∨ Q ∨ P ≠ (¬ Q)"
"P ∨ Q ∨ (¬ P) ≠ Q"
"P ∨ ¬ Q ∨ P ≠ Q"
"¬ P ∨ Q ∨ P ≠ Q"
"P ∨ y = (if P then x else y)"
"P ∨ (if P then x else y) = y"
"¬ P ∨ x = (if P then x else y)"
"¬ P ∨ (if P then x else y) = x"
"P ∨ R ∨ ¬ (if P then Q else R)"
"¬ P ∨ Q ∨ ¬ (if P then Q else R)"
"¬ (if P then Q else R) ∨ ¬ P ∨ Q"
"¬ (if P then Q else R) ∨ P ∨ R"
"(if P then Q else R) ∨ ¬ P ∨ ¬ Q"
"(if P then Q else R) ∨ P ∨ ¬ R"
"(if P then ¬ Q else R) ∨ ¬ P ∨ Q"
"(if P then Q else ¬ R) ∨ P ∨ R"
by auto

hide_type (open) symb_list pattern
hide_const (open) Symb_Nil Symb_Cons trigger pat nopat fun_app z3div z3mod

end
```