# Theory Meson

theory Meson
imports Nat
```(*  Title:      HOL/Meson.thy
Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
Author:     Tobias Nipkow, TU Muenchen
Author:     Jasmin Blanchette, TU Muenchen
*)

section ‹MESON Proof Method›

theory Meson
imports Nat
begin

subsection ‹Negation Normal Form›

text ‹de Morgan laws›

lemma not_conjD: "¬(P∧Q) ⟹ ¬P ∨ ¬Q"
and not_disjD: "¬(P∨Q) ⟹ ¬P ∧ ¬Q"
and not_notD: "¬¬P ⟹ P"
and not_allD: "⋀P. ¬(∀x. P(x)) ⟹ ∃x. ¬P(x)"
and not_exD: "⋀P. ¬(∃x. P(x)) ⟹ ∀x. ¬P(x)"
by fast+

text ‹Removal of ‹⟶› and ‹⟷› (positive and negative occurrences)›

lemma imp_to_disjD: "P⟶Q ⟹ ¬P ∨ Q"
and not_impD: "¬(P⟶Q) ⟹ P ∧ ¬Q"
and iff_to_disjD: "P=Q ⟹ (¬P ∨ Q) ∧ (¬Q ∨ P)"
and not_iffD: "¬(P=Q) ⟹ (P ∨ Q) ∧ (¬P ∨ ¬Q)"
― ‹Much more efficient than @{prop "(P ∧ ¬Q) ∨ (Q ∧ ¬P)"} for computing CNF›
and not_refl_disj_D: "x ≠ x ∨ P ⟹ P"
by fast+

subsection ‹Pulling out the existential quantifiers›

text ‹Conjunction›

lemma conj_exD1: "⋀P Q. (∃x. P(x)) ∧ Q ⟹ ∃x. P(x) ∧ Q"
and conj_exD2: "⋀P Q. P ∧ (∃x. Q(x)) ⟹ ∃x. P ∧ Q(x)"
by fast+

text ‹Disjunction›

lemma disj_exD: "⋀P Q. (∃x. P(x)) ∨ (∃x. Q(x)) ⟹ ∃x. P(x) ∨ Q(x)"
― ‹DO NOT USE with forall-Skolemization: makes fewer schematic variables!!›
― ‹With ex-Skolemization, makes fewer Skolem constants›
and disj_exD1: "⋀P Q. (∃x. P(x)) ∨ Q ⟹ ∃x. P(x) ∨ Q"
and disj_exD2: "⋀P Q. P ∨ (∃x. Q(x)) ⟹ ∃x. P ∨ Q(x)"
by fast+

lemma disj_assoc: "(P∨Q)∨R ⟹ P∨(Q∨R)"
and disj_comm: "P∨Q ⟹ Q∨P"
and disj_FalseD1: "False∨P ⟹ P"
and disj_FalseD2: "P∨False ⟹ P"
by fast+

text‹Generation of contrapositives›

text‹Inserts negated disjunct after removing the negation; P is a literal.
Model elimination requires assuming the negation of every attempted subgoal,
hence the negated disjuncts.›
lemma make_neg_rule: "¬P∨Q ⟹ ((¬P⟹P) ⟹ Q)"
by blast

text‹Version for Plaisted's "Postive refinement" of the Meson procedure›
lemma make_refined_neg_rule: "¬P∨Q ⟹ (P ⟹ Q)"
by blast

text‹@{term P} should be a literal›
lemma make_pos_rule: "P∨Q ⟹ ((P⟹¬P) ⟹ Q)"
by blast

text‹Versions of ‹make_neg_rule› and ‹make_pos_rule› that don't
insert new assumptions, for ordinary resolution.›

lemmas make_neg_rule' = make_refined_neg_rule

lemma make_pos_rule': "⟦P∨Q; ¬P⟧ ⟹ Q"
by blast

text‹Generation of a goal clause -- put away the final literal›

lemma make_neg_goal: "¬P ⟹ ((¬P⟹P) ⟹ False)"
by blast

lemma make_pos_goal: "P ⟹ ((P⟹¬P) ⟹ False)"
by blast

subsection ‹Lemmas for Forward Proof›

text‹There is a similarity to congruence rules. They are also useful in ordinary proofs.›

(*NOTE: could handle conjunctions (faster?) by
nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
lemma conj_forward: "⟦P'∧Q';  P' ⟹ P;  Q' ⟹ Q ⟧ ⟹ P∧Q"
by blast

lemma disj_forward: "⟦P'∨Q';  P' ⟹ P;  Q' ⟹ Q ⟧ ⟹ P∨Q"
by blast

lemma imp_forward: "⟦P' ⟶ Q';  P ⟹ P';  Q' ⟹ Q ⟧ ⟹ P ⟶ Q"
by blast

(*Version of @{text disj_forward} for removal of duplicate literals*)
lemma disj_forward2: "⟦ P'∨Q';  P' ⟹ P;  ⟦Q'; P⟹False⟧ ⟹ Q⟧ ⟹ P∨Q"
apply blast
done

lemma all_forward: "[| ∀x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ∀x. P(x)"
by blast

lemma ex_forward: "[| ∃x. P'(x);  !!x. P'(x) ==> P(x) |] ==> ∃x. P(x)"
by blast

subsection ‹Clausification helper›

lemma TruepropI: "P ≡ Q ⟹ Trueprop P ≡ Trueprop Q"
by simp

lemma ext_cong_neq: "F g ≠ F h ⟹ F g ≠ F h ∧ (∃x. g x ≠ h x)"
apply (erule contrapos_np)
apply clarsimp
apply (rule cong[where f = F])
by auto

text‹Combinator translation helpers›

definition COMBI :: "'a ⇒ 'a" where
"COMBI P = P"

definition COMBK :: "'a ⇒ 'b ⇒ 'a" where
"COMBK P Q = P"

definition COMBB :: "('b => 'c) ⇒ ('a => 'b) ⇒ 'a ⇒ 'c" where
"COMBB P Q R = P (Q R)"

definition COMBC :: "('a ⇒ 'b ⇒ 'c) ⇒ 'b ⇒ 'a ⇒ 'c" where
"COMBC P Q R = P R Q"

definition COMBS :: "('a ⇒ 'b ⇒ 'c) ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'c" where
"COMBS P Q R = P R (Q R)"

lemma abs_S: "λx. (f x) (g x) ≡ COMBS f g"
apply (rule eq_reflection)
apply (rule ext)
done

lemma abs_I: "λx. x ≡ COMBI"
apply (rule eq_reflection)
apply (rule ext)
done

lemma abs_K: "λx. y ≡ COMBK y"
apply (rule eq_reflection)
apply (rule ext)
done

lemma abs_B: "λx. a (g x) ≡ COMBB a g"
apply (rule eq_reflection)
apply (rule ext)
done

lemma abs_C: "λx. (f x) b ≡ COMBC f b"
apply (rule eq_reflection)
apply (rule ext)
done

subsection ‹Skolemization helpers›

definition skolem :: "'a ⇒ 'a" where
"skolem = (λx. x)"

lemma skolem_COMBK_iff: "P ⟷ skolem (COMBK P (i::nat))"
unfolding skolem_def COMBK_def by (rule refl)

lemmas skolem_COMBK_I = iffD1 [OF skolem_COMBK_iff]
lemmas skolem_COMBK_D = iffD2 [OF skolem_COMBK_iff]

subsection ‹Meson package›

ML_file "Tools/Meson/meson.ML"
ML_file "Tools/Meson/meson_clausify.ML"
ML_file "Tools/Meson/meson_tactic.ML"

hide_const (open) COMBI COMBK COMBB COMBC COMBS skolem
hide_fact (open) not_conjD not_disjD not_notD not_allD not_exD imp_to_disjD
not_impD iff_to_disjD not_iffD not_refl_disj_D conj_exD1 conj_exD2 disj_exD
disj_exD1 disj_exD2 disj_assoc disj_comm disj_FalseD1 disj_FalseD2 TruepropI
ext_cong_neq COMBI_def COMBK_def COMBB_def COMBC_def COMBS_def abs_I abs_K
abs_B abs_C abs_S skolem_def skolem_COMBK_iff skolem_COMBK_I skolem_COMBK_D

end
```