(* 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: "\<not>(P\<and>Q) \<Longrightarrow> \<not>P \<or> \<not>Q"
and not_disjD: "\<not>(P\<or>Q) \<Longrightarrow> \<not>P \<and> \<not>Q"
and not_notD: "\<not>\<not>P \<Longrightarrow> P"
and not_allD: "\<And>P. \<not>(\<forall>x. P(x)) \<Longrightarrow> \<exists>x. \<not>P(x)"
and not_exD: "\<And>P. \<not>(\<exists>x. P(x)) \<Longrightarrow> \<forall>x. \<not>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\<longrightarrow>Q \<Longrightarrow> \<not>P \<or> Q"
and not_impD: "\<not>(P\<longrightarrow>Q) \<Longrightarrow> P \<and> \<not>Q"
and iff_to_disjD: "P=Q \<Longrightarrow> (\<not>P \<or> Q) \<and> (\<not>Q \<or> P)"
and not_iffD: "\<not>(P=Q) \<Longrightarrow> (P \<or> Q) \<and> (\<not>P \<or> \<not>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 \<noteq> x \<or> P \<Longrightarrow> P"
by fast+
subsection \<open>Pulling out the existential quantifiers\<close>
text \<open>Conjunction\<close>
lemma conj_exD1: "\<And>P Q. (\<exists>x. P(x)) \<and> Q \<Longrightarrow> \<exists>x. P(x) \<and> Q"
and conj_exD2: "\<And>P Q. P \<and> (\<exists>x. Q(x)) \<Longrightarrow> \<exists>x. P \<and> Q(x)"
by fast+
text \<open>Disjunction\<close>
lemma disj_exD: "\<And>P Q. (\<exists>x. P(x)) \<or> (\<exists>x. Q(x)) \<Longrightarrow> \<exists>x. P(x) \<or> 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: "\<And>P Q. (\<exists>x. P(x)) \<or> Q \<Longrightarrow> \<exists>x. P(x) \<or> Q"
and disj_exD2: "\<And>P Q. P \<or> (\<exists>x. Q(x)) \<Longrightarrow> \<exists>x. P \<or> Q(x)"
by fast+
lemma disj_assoc: "(P\<or>Q)\<or>R \<Longrightarrow> P\<or>(Q\<or>R)"
and disj_comm: "P\<or>Q \<Longrightarrow> Q\<or>P"
and disj_FalseD1: "False\<or>P \<Longrightarrow> P"
and disj_FalseD2: "P\<or>False \<Longrightarrow> 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: "\<not>P\<or>Q \<Longrightarrow> ((\<not>P\<Longrightarrow>P) \<Longrightarrow> Q)"
by blast
text\<open>Version for Plaisted's "Postive refinement" of the Meson procedure\<close>
lemma make_refined_neg_rule: "\<not>P\<or>Q \<Longrightarrow> (P \<Longrightarrow> Q)"
by blast
text\<open>\<^term>\<open>P\<close> should be a literal\<close>
lemma make_pos_rule: "P\<or>Q \<Longrightarrow> ((P\<Longrightarrow>\<not>P) \<Longrightarrow> 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': "\<lbrakk>P\<or>Q; \<not>P\<rbrakk> \<Longrightarrow> Q"
by blast
text\<open>Generation of a goal clause -- put away the final literal\<close>
lemma make_neg_goal: "\<not>P \<Longrightarrow> ((\<not>P\<Longrightarrow>P) \<Longrightarrow> False)"
by blast
lemma make_pos_goal: "P \<Longrightarrow> ((P\<Longrightarrow>\<not>P) \<Longrightarrow> 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: "\<lbrakk>P'\<and>Q'; P' \<Longrightarrow> P; Q' \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> P\<and>Q"
by blast
lemma disj_forward: "\<lbrakk>P'\<or>Q'; P' \<Longrightarrow> P; Q' \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> P\<or>Q"
by blast
lemma imp_forward: "\<lbrakk>P' \<longrightarrow> Q'; P \<Longrightarrow> P'; Q' \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> P \<longrightarrow> Q"
by blast
lemma imp_forward2: "\<lbrakk>P' \<longrightarrow> Q'; P \<Longrightarrow> P'; P' \<Longrightarrow> Q' \<Longrightarrow> Q \<rbrakk> \<Longrightarrow> P \<longrightarrow> Q"
by blast
(*Version of @{text disj_forward} for removal of duplicate literals*)
lemma disj_forward2: "\<lbrakk> P'\<or>Q'; P' \<Longrightarrow> P; \<lbrakk>Q'; P\<Longrightarrow>False\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> P\<or>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 \<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