(*  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 {* 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. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"

  and not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"

  by fast+

text {* Removal of @{text "-->"} and @{text "<->"} (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. (\<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 {* Disjunction *}

lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>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. (\<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{* 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 @{text make_neg_rule} and @{text 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*}

(*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

(*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 {* Clausification helper *}

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{* Combinator translation helpers *}

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 {* Skolemization helpers *}

definition skolem :: "'a \<Rightarrow> 'a" where

"skolem = (\<lambda>x. x)"

lemma skolem_COMBK_iff: "P \<longleftrightarrow> skolem (COMBK P (i\<Colon>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