src/HOL/Meson.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 61941 31f2105521ee
child 62381 a6479cb85944
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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(*  Title:      HOL/Meson.thy
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    Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2001  University of Cambridge
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*)
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section \<open>MESON Proof Method\<close>
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theory Meson
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imports Nat
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begin
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subsection \<open>Negation Normal Form\<close>
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text \<open>de Morgan laws\<close>
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lemma not_conjD: "~(P&Q) ==> ~P | ~Q"
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  and not_disjD: "~(P|Q) ==> ~P & ~Q"
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  and not_notD: "~~P ==> P"
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  and not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
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  and not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
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  by fast+
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text \<open>Removal of \<open>\<longrightarrow>\<close> and \<open>\<longleftrightarrow>\<close> (positive and negative occurrences)\<close>
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lemma imp_to_disjD: "P-->Q ==> ~P | Q"
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  and not_impD: "~(P-->Q) ==> P & ~Q"
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  and iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
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  and not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
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    \<comment> \<open>Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF\<close>
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  and not_refl_disj_D: "x ~= x | P ==> P"
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  by fast+
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subsection \<open>Pulling out the existential quantifiers\<close>
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text \<open>Conjunction\<close>
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lemma conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
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  and conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
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  by fast+
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text \<open>Disjunction\<close>
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lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
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  \<comment> \<open>DO NOT USE with forall-Skolemization: makes fewer schematic variables!!\<close>
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  \<comment> \<open>With ex-Skolemization, makes fewer Skolem constants\<close>
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  and disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
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  and disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
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  by fast+
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lemma disj_assoc: "(P|Q)|R ==> P|(Q|R)"
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  and disj_comm: "P|Q ==> Q|P"
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  and disj_FalseD1: "False|P ==> P"
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  and disj_FalseD2: "P|False ==> P"
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  by fast+
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text\<open>Generation of contrapositives\<close>
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text\<open>Inserts negated disjunct after removing the negation; P is a literal.
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  Model elimination requires assuming the negation of every attempted subgoal,
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  hence the negated disjuncts.\<close>
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lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
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by blast
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text\<open>Version for Plaisted's "Postive refinement" of the Meson procedure\<close>
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lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
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by blast
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text\<open>@{term P} should be a literal\<close>
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lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
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by blast
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text\<open>Versions of \<open>make_neg_rule\<close> and \<open>make_pos_rule\<close> that don't
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insert new assumptions, for ordinary resolution.\<close>
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lemmas make_neg_rule' = make_refined_neg_rule
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lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
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by blast
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text\<open>Generation of a goal clause -- put away the final literal\<close>
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lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
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by blast
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lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
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by blast
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subsection \<open>Lemmas for Forward Proof\<close>
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text\<open>There is a similarity to congruence rules\<close>
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(*NOTE: could handle conjunctions (faster?) by
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    nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
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lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
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by blast
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lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
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by blast
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(*Version of @{text disj_forward} for removal of duplicate literals*)
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lemma disj_forward2:
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    "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
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apply blast 
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done
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lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
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by blast
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lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
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by blast
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subsection \<open>Clausification helper\<close>
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lemma TruepropI: "P \<equiv> Q \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
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by simp
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lemma ext_cong_neq: "F g \<noteq> F h \<Longrightarrow> F g \<noteq> F h \<and> (\<exists>x. g x \<noteq> h x)"
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apply (erule contrapos_np)
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apply clarsimp
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apply (rule cong[where f = F])
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by auto
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text\<open>Combinator translation helpers\<close>
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definition COMBI :: "'a \<Rightarrow> 'a" where
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"COMBI P = P"
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definition COMBK :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" where
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"COMBK P Q = P"
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definition COMBB :: "('b => 'c) \<Rightarrow> ('a => 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
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"COMBB P Q R = P (Q R)"
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definition COMBC :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
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"COMBC P Q R = P R Q"
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definition COMBS :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
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"COMBS P Q R = P R (Q R)"
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lemma abs_S: "\<lambda>x. (f x) (g x) \<equiv> COMBS f g"
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apply (rule eq_reflection)
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apply (rule ext) 
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apply (simp add: COMBS_def) 
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done
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lemma abs_I: "\<lambda>x. x \<equiv> COMBI"
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apply (rule eq_reflection)
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apply (rule ext) 
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apply (simp add: COMBI_def) 
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done
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lemma abs_K: "\<lambda>x. y \<equiv> COMBK y"
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apply (rule eq_reflection)
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apply (rule ext) 
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apply (simp add: COMBK_def) 
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done
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lemma abs_B: "\<lambda>x. a (g x) \<equiv> COMBB a g"
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apply (rule eq_reflection)
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apply (rule ext) 
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apply (simp add: COMBB_def) 
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done
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lemma abs_C: "\<lambda>x. (f x) b \<equiv> COMBC f b"
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apply (rule eq_reflection)
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apply (rule ext) 
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apply (simp add: COMBC_def) 
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done
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subsection \<open>Skolemization helpers\<close>
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definition skolem :: "'a \<Rightarrow> 'a" where
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"skolem = (\<lambda>x. x)"
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lemma skolem_COMBK_iff: "P \<longleftrightarrow> skolem (COMBK P (i::nat))"
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unfolding skolem_def COMBK_def by (rule refl)
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lemmas skolem_COMBK_I = iffD1 [OF skolem_COMBK_iff]
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lemmas skolem_COMBK_D = iffD2 [OF skolem_COMBK_iff]
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subsection \<open>Meson package\<close>
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ML_file "Tools/Meson/meson.ML"
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ML_file "Tools/Meson/meson_clausify.ML"
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ML_file "Tools/Meson/meson_tactic.ML"
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hide_const (open) COMBI COMBK COMBB COMBC COMBS skolem
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hide_fact (open) not_conjD not_disjD not_notD not_allD not_exD imp_to_disjD
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    not_impD iff_to_disjD not_iffD not_refl_disj_D conj_exD1 conj_exD2 disj_exD
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    disj_exD1 disj_exD2 disj_assoc disj_comm disj_FalseD1 disj_FalseD2 TruepropI
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    ext_cong_neq COMBI_def COMBK_def COMBB_def COMBC_def COMBS_def abs_I abs_K
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    abs_B abs_C abs_S skolem_def skolem_COMBK_iff skolem_COMBK_I skolem_COMBK_D
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end