(* Title: HOL/Meson.thy
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory
Author: Tobias Nipkow, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2001 University of Cambridge
*)
section \<open>MESON Proof Method\<close>
theory Meson
imports Nat
begin
subsection \<open>Negation Normal Form\<close>
text \<open>de Morgan laws\<close>
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 \<open>Removal of \<open>\<longrightarrow>\<close> and \<open>\<longleftrightarrow>\<close> (positive and negative occurrences)\<close>
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)"
\<comment> \<open>Much more efficient than \<^prop>\<open>(P \<and> \<not>Q) \<or> (Q \<and> \<not>P)\<close> for computing CNF\<close>
and not_refl_disj_D: "x \ x \ P \ P"
by fast+
subsection \<open>Pulling out the existential quantifiers\<close>
text \<open>Conjunction\<close>
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 \<open>Disjunction\<close>
lemma disj_exD: "\P Q. (\x. P(x)) \ (\x. Q(x)) \ \x. P(x) \ Q(x)"
\<comment> \<open>DO NOT USE with forall-Skolemization: makes fewer schematic variables!!\<close>
\<comment> \<open>With ex-Skolemization, makes fewer Skolem constants\<close>
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\<open>Generation of contrapositives\<close>
text\<open>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.\<close>
lemma make_neg_rule: "\P\Q \ ((\P\P) \ Q)"
by blast
text\<open>Version for Plaisted's "Postive refinement" of the Meson procedure\<close>
lemma make_refined_neg_rule: "\P\Q \ (P \ Q)"
by blast
text\<open>\<^term>\<open>P\<close> should be a literal\<close>
lemma make_pos_rule: "P\Q \ ((P\\P) \ Q)"
by blast
text\<open>Versions of \<open>make_neg_rule\<close> and \<open>make_pos_rule\<close> that don't
insert new assumptions, for ordinary resolution.\<close>
lemmas make_neg_rule' = make_refined_neg_rule
lemma make_pos_rule': "\P\Q; \P\ \ Q"
by blast
text\<open>Generation of a goal clause -- put away the final literal\<close>
lemma make_neg_goal: "\P \ ((\P\P) \ False)"
by blast
lemma make_pos_goal: "P \ ((P\\P) \ False)"
by blast
subsection \<open>Lemmas for Forward Proof\<close>
text\<open>There is a similarity to congruence rules. They are also useful in ordinary proofs.\<close>
(*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
lemma imp_forward2: "\P' \ Q'; P \ 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 \<open>Clausification helper\<close>
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\<open>Combinator translation helpers\<close>
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: "\x. (f x) (g x) \ COMBS f g"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBS_def)
done
lemma abs_I: "\x. x \ COMBI"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBI_def)
done
lemma abs_K: "\x. y \ COMBK y"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBK_def)
done
lemma abs_B: "\x. a (g x) \ COMBB a g"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBB_def)
done
lemma abs_C: "\x. (f x) b \ COMBC f b"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBC_def)
done
subsection \<open>Skolemization helpers\<close>
definition skolem :: "'a \ 'a" where
"skolem = (\x. 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 \<open>Meson package\<close>
ML_file \<open>Tools/Meson/meson.ML\<close>
ML_file \<open>Tools/Meson/meson_clausify.ML\<close>
ML_file \<open>Tools/Meson/meson_tactic.ML\<close>
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
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