src/HOL/Meson.thy
author wenzelm
Fri Mar 07 22:30:58 2014 +0100 (2014-03-07)
changeset 55990 41c6b99c5fb7
parent 54553 2b0da4c1dd40
child 58817 4cd778c91fdc
permissions -rw-r--r--
more antiquotations;
     1 (*  Title:      HOL/Meson.thy
     2     Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
     3     Author:     Tobias Nipkow, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Copyright   2001  University of Cambridge
     6 *)
     7 
     8 header {* MESON Proof Method *}
     9 
    10 theory Meson
    11 imports Nat
    12 begin
    13 
    14 subsection {* Negation Normal Form *}
    15 
    16 text {* de Morgan laws *}
    17 
    18 lemma not_conjD: "~(P&Q) ==> ~P | ~Q"
    19   and not_disjD: "~(P|Q) ==> ~P & ~Q"
    20   and not_notD: "~~P ==> P"
    21   and not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
    22   and not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
    23   by fast+
    24 
    25 text {* Removal of @{text "-->"} and @{text "<->"} (positive and
    26 negative occurrences) *}
    27 
    28 lemma imp_to_disjD: "P-->Q ==> ~P | Q"
    29   and not_impD: "~(P-->Q) ==> P & ~Q"
    30   and iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
    31   and not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
    32     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
    33   and not_refl_disj_D: "x ~= x | P ==> P"
    34   by fast+
    35 
    36 
    37 subsection {* Pulling out the existential quantifiers *}
    38 
    39 text {* Conjunction *}
    40 
    41 lemma conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
    42   and conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
    43   by fast+
    44 
    45 
    46 text {* Disjunction *}
    47 
    48 lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
    49   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
    50   -- {* With ex-Skolemization, makes fewer Skolem constants *}
    51   and disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
    52   and disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
    53   by fast+
    54 
    55 lemma disj_assoc: "(P|Q)|R ==> P|(Q|R)"
    56   and disj_comm: "P|Q ==> Q|P"
    57   and disj_FalseD1: "False|P ==> P"
    58   and disj_FalseD2: "P|False ==> P"
    59   by fast+
    60 
    61 
    62 text{* Generation of contrapositives *}
    63 
    64 text{*Inserts negated disjunct after removing the negation; P is a literal.
    65   Model elimination requires assuming the negation of every attempted subgoal,
    66   hence the negated disjuncts.*}
    67 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
    68 by blast
    69 
    70 text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
    71 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
    72 by blast
    73 
    74 text{*@{term P} should be a literal*}
    75 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
    76 by blast
    77 
    78 text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
    79 insert new assumptions, for ordinary resolution.*}
    80 
    81 lemmas make_neg_rule' = make_refined_neg_rule
    82 
    83 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
    84 by blast
    85 
    86 text{* Generation of a goal clause -- put away the final literal *}
    87 
    88 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
    89 by blast
    90 
    91 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
    92 by blast
    93 
    94 
    95 subsection {* Lemmas for Forward Proof *}
    96 
    97 text{*There is a similarity to congruence rules*}
    98 
    99 (*NOTE: could handle conjunctions (faster?) by
   100     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
   101 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
   102 by blast
   103 
   104 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
   105 by blast
   106 
   107 (*Version of @{text disj_forward} for removal of duplicate literals*)
   108 lemma disj_forward2:
   109     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
   110 apply blast 
   111 done
   112 
   113 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
   114 by blast
   115 
   116 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
   117 by blast
   118 
   119 
   120 subsection {* Clausification helper *}
   121 
   122 lemma TruepropI: "P \<equiv> Q \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
   123 by simp
   124 
   125 lemma ext_cong_neq: "F g \<noteq> F h \<Longrightarrow> F g \<noteq> F h \<and> (\<exists>x. g x \<noteq> h x)"
   126 apply (erule contrapos_np)
   127 apply clarsimp
   128 apply (rule cong[where f = F])
   129 by auto
   130 
   131 
   132 text{* Combinator translation helpers *}
   133 
   134 definition COMBI :: "'a \<Rightarrow> 'a" where
   135 "COMBI P = P"
   136 
   137 definition COMBK :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" where
   138 "COMBK P Q = P"
   139 
   140 definition COMBB :: "('b => 'c) \<Rightarrow> ('a => 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
   141 "COMBB P Q R = P (Q R)"
   142 
   143 definition COMBC :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
   144 "COMBC P Q R = P R Q"
   145 
   146 definition COMBS :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
   147 "COMBS P Q R = P R (Q R)"
   148 
   149 lemma abs_S: "\<lambda>x. (f x) (g x) \<equiv> COMBS f g"
   150 apply (rule eq_reflection)
   151 apply (rule ext) 
   152 apply (simp add: COMBS_def) 
   153 done
   154 
   155 lemma abs_I: "\<lambda>x. x \<equiv> COMBI"
   156 apply (rule eq_reflection)
   157 apply (rule ext) 
   158 apply (simp add: COMBI_def) 
   159 done
   160 
   161 lemma abs_K: "\<lambda>x. y \<equiv> COMBK y"
   162 apply (rule eq_reflection)
   163 apply (rule ext) 
   164 apply (simp add: COMBK_def) 
   165 done
   166 
   167 lemma abs_B: "\<lambda>x. a (g x) \<equiv> COMBB a g"
   168 apply (rule eq_reflection)
   169 apply (rule ext) 
   170 apply (simp add: COMBB_def) 
   171 done
   172 
   173 lemma abs_C: "\<lambda>x. (f x) b \<equiv> COMBC f b"
   174 apply (rule eq_reflection)
   175 apply (rule ext) 
   176 apply (simp add: COMBC_def) 
   177 done
   178 
   179 
   180 subsection {* Skolemization helpers *}
   181 
   182 definition skolem :: "'a \<Rightarrow> 'a" where
   183 "skolem = (\<lambda>x. x)"
   184 
   185 lemma skolem_COMBK_iff: "P \<longleftrightarrow> skolem (COMBK P (i\<Colon>nat))"
   186 unfolding skolem_def COMBK_def by (rule refl)
   187 
   188 lemmas skolem_COMBK_I = iffD1 [OF skolem_COMBK_iff]
   189 lemmas skolem_COMBK_D = iffD2 [OF skolem_COMBK_iff]
   190 
   191 
   192 subsection {* Meson package *}
   193 
   194 ML_file "Tools/Meson/meson.ML"
   195 ML_file "Tools/Meson/meson_clausify.ML"
   196 ML_file "Tools/Meson/meson_tactic.ML"
   197 
   198 setup {* Meson_Tactic.setup *}
   199 
   200 hide_const (open) COMBI COMBK COMBB COMBC COMBS skolem
   201 hide_fact (open) not_conjD not_disjD not_notD not_allD not_exD imp_to_disjD
   202     not_impD iff_to_disjD not_iffD not_refl_disj_D conj_exD1 conj_exD2 disj_exD
   203     disj_exD1 disj_exD2 disj_assoc disj_comm disj_FalseD1 disj_FalseD2 TruepropI
   204     ext_cong_neq COMBI_def COMBK_def COMBB_def COMBC_def COMBS_def abs_I abs_K
   205     abs_B abs_C abs_S skolem_def skolem_COMBK_iff skolem_COMBK_I skolem_COMBK_D
   206 
   207 end