src/HOL/Meson.thy
 author wenzelm Tue Oct 10 19:23:03 2017 +0200 (23 months ago) changeset 66831 29ea2b900a05 parent 62381 a6479cb85944 child 67091 1393c2340eec permissions -rw-r--r--
tuned: each session has at most one defining entry;
```     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
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```    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
```