(* 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. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
and not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>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 "(P & ~Q) | (Q & ~P)"} 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. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
and conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
by fast+
text \<open>Disjunction\<close>
lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>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. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
and disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>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 P} 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' \<longrightarrow> Q'; P ==> P'; Q' ==> Q |] ==> P \<longrightarrow> 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: "[| \<forall>x. P'(x); !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
by blast
lemma ex_forward: "[| \<exists>x. P'(x); !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
by blast
subsection \<open>Clausification helper\<close>
lemma TruepropI: "P \<equiv> Q \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
by simp
lemma ext_cong_neq: "F g \<noteq> F h \<Longrightarrow> F g \<noteq> F h \<and> (\<exists>x. g x \<noteq> 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 \<Rightarrow> 'a" where
"COMBI P = P"
definition COMBK :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" where
"COMBK P Q = P"
definition COMBB :: "('b => 'c) \<Rightarrow> ('a => 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
"COMBB P Q R = P (Q R)"
definition COMBC :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
"COMBC P Q R = P R Q"
definition COMBS :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
"COMBS P Q R = P R (Q R)"
lemma abs_S: "\<lambda>x. (f x) (g x) \<equiv> COMBS f g"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBS_def)
done
lemma abs_I: "\<lambda>x. x \<equiv> COMBI"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBI_def)
done
lemma abs_K: "\<lambda>x. y \<equiv> COMBK y"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBK_def)
done
lemma abs_B: "\<lambda>x. a (g x) \<equiv> COMBB a g"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBB_def)
done
lemma abs_C: "\<lambda>x. (f x) b \<equiv> COMBC f b"
apply (rule eq_reflection)
apply (rule ext)
apply (simp add: COMBC_def)
done
subsection \<open>Skolemization helpers\<close>
definition skolem :: "'a \<Rightarrow> 'a" where
"skolem = (\<lambda>x. x)"
lemma skolem_COMBK_iff: "P \<longleftrightarrow> 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 "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