src/HOL/Meson.thy
author blanchet
Tue Nov 07 15:16:42 2017 +0100 (19 months ago)
changeset 67022 49309fe530fd
parent 62381 a6479cb85944
child 67091 1393c2340eec
permissions -rw-r--r--
more robust parsing for THF proofs (esp. polymorphic Leo-III proofs)
     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 section \<open>MESON Proof Method\<close>
     9 
    10 theory Meson
    11 imports Nat
    12 begin
    13 
    14 subsection \<open>Negation Normal Form\<close>
    15 
    16 text \<open>de Morgan laws\<close>
    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 \<open>Removal of \<open>\<longrightarrow>\<close> and \<open>\<longleftrightarrow>\<close> (positive and negative occurrences)\<close>
    26 
    27 lemma imp_to_disjD: "P-->Q ==> ~P | Q"
    28   and not_impD: "~(P-->Q) ==> P & ~Q"
    29   and iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
    30   and not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
    31     \<comment> \<open>Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF\<close>
    32   and not_refl_disj_D: "x ~= x | P ==> P"
    33   by fast+
    34 
    35 
    36 subsection \<open>Pulling out the existential quantifiers\<close>
    37 
    38 text \<open>Conjunction\<close>
    39 
    40 lemma conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
    41   and conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
    42   by fast+
    43 
    44 
    45 text \<open>Disjunction\<close>
    46 
    47 lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
    48   \<comment> \<open>DO NOT USE with forall-Skolemization: makes fewer schematic variables!!\<close>
    49   \<comment> \<open>With ex-Skolemization, makes fewer Skolem constants\<close>
    50   and disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
    51   and disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
    52   by fast+
    53 
    54 lemma disj_assoc: "(P|Q)|R ==> P|(Q|R)"
    55   and disj_comm: "P|Q ==> Q|P"
    56   and disj_FalseD1: "False|P ==> P"
    57   and disj_FalseD2: "P|False ==> P"
    58   by fast+
    59 
    60 
    61 text\<open>Generation of contrapositives\<close>
    62 
    63 text\<open>Inserts negated disjunct after removing the negation; P is a literal.
    64   Model elimination requires assuming the negation of every attempted subgoal,
    65   hence the negated disjuncts.\<close>
    66 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
    67 by blast
    68 
    69 text\<open>Version for Plaisted's "Postive refinement" of the Meson procedure\<close>
    70 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
    71 by blast
    72 
    73 text\<open>@{term P} should be a literal\<close>
    74 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
    75 by blast
    76 
    77 text\<open>Versions of \<open>make_neg_rule\<close> and \<open>make_pos_rule\<close> that don't
    78 insert new assumptions, for ordinary resolution.\<close>
    79 
    80 lemmas make_neg_rule' = make_refined_neg_rule
    81 
    82 lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
    83 by blast
    84 
    85 text\<open>Generation of a goal clause -- put away the final literal\<close>
    86 
    87 lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
    88 by blast
    89 
    90 lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
    91 by blast
    92 
    93 
    94 subsection \<open>Lemmas for Forward Proof\<close>
    95 
    96 text\<open>There is a similarity to congruence rules. They are also useful in ordinary proofs.\<close>
    97 
    98 (*NOTE: could handle conjunctions (faster?) by
    99     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
   100 lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
   101 by blast
   102 
   103 lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
   104 by blast
   105 
   106 lemma imp_forward: "[| P' \<longrightarrow> Q';  P ==> P';  Q' ==> Q |] ==> P \<longrightarrow> Q"
   107 by blast
   108 
   109 (*Version of @{text disj_forward} for removal of duplicate literals*)
   110 lemma disj_forward2:
   111     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
   112 apply blast 
   113 done
   114 
   115 lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
   116 by blast
   117 
   118 lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
   119 by blast
   120 
   121 
   122 subsection \<open>Clausification helper\<close>
   123 
   124 lemma TruepropI: "P \<equiv> Q \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
   125 by simp
   126 
   127 lemma ext_cong_neq: "F g \<noteq> F h \<Longrightarrow> F g \<noteq> F h \<and> (\<exists>x. g x \<noteq> h x)"
   128 apply (erule contrapos_np)
   129 apply clarsimp
   130 apply (rule cong[where f = F])
   131 by auto
   132 
   133 
   134 text\<open>Combinator translation helpers\<close>
   135 
   136 definition COMBI :: "'a \<Rightarrow> 'a" where
   137 "COMBI P = P"
   138 
   139 definition COMBK :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" where
   140 "COMBK P Q = P"
   141 
   142 definition COMBB :: "('b => 'c) \<Rightarrow> ('a => 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
   143 "COMBB P Q R = P (Q R)"
   144 
   145 definition COMBC :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c" where
   146 "COMBC P Q R = P R Q"
   147 
   148 definition COMBS :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" where
   149 "COMBS P Q R = P R (Q R)"
   150 
   151 lemma abs_S: "\<lambda>x. (f x) (g x) \<equiv> COMBS f g"
   152 apply (rule eq_reflection)
   153 apply (rule ext) 
   154 apply (simp add: COMBS_def) 
   155 done
   156 
   157 lemma abs_I: "\<lambda>x. x \<equiv> COMBI"
   158 apply (rule eq_reflection)
   159 apply (rule ext) 
   160 apply (simp add: COMBI_def) 
   161 done
   162 
   163 lemma abs_K: "\<lambda>x. y \<equiv> COMBK y"
   164 apply (rule eq_reflection)
   165 apply (rule ext) 
   166 apply (simp add: COMBK_def) 
   167 done
   168 
   169 lemma abs_B: "\<lambda>x. a (g x) \<equiv> COMBB a g"
   170 apply (rule eq_reflection)
   171 apply (rule ext) 
   172 apply (simp add: COMBB_def) 
   173 done
   174 
   175 lemma abs_C: "\<lambda>x. (f x) b \<equiv> COMBC f b"
   176 apply (rule eq_reflection)
   177 apply (rule ext) 
   178 apply (simp add: COMBC_def) 
   179 done
   180 
   181 
   182 subsection \<open>Skolemization helpers\<close>
   183 
   184 definition skolem :: "'a \<Rightarrow> 'a" where
   185 "skolem = (\<lambda>x. x)"
   186 
   187 lemma skolem_COMBK_iff: "P \<longleftrightarrow> skolem (COMBK P (i::nat))"
   188 unfolding skolem_def COMBK_def by (rule refl)
   189 
   190 lemmas skolem_COMBK_I = iffD1 [OF skolem_COMBK_iff]
   191 lemmas skolem_COMBK_D = iffD2 [OF skolem_COMBK_iff]
   192 
   193 
   194 subsection \<open>Meson package\<close>
   195 
   196 ML_file "Tools/Meson/meson.ML"
   197 ML_file "Tools/Meson/meson_clausify.ML"
   198 ML_file "Tools/Meson/meson_tactic.ML"
   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