src/HOL/Meson.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 60758 d8d85a8172b5 child 61799 4cf66f21b764 permissions -rw-r--r--
eliminated \<Colon>;
```     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 @{text "-->"} and @{text "<->"} (positive and
```
```    26 negative occurrences)\<close>
```
```    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     -- \<open>Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF\<close>
```
```    33   and not_refl_disj_D: "x ~= x | P ==> P"
```
```    34   by fast+
```
```    35
```
```    36
```
```    37 subsection \<open>Pulling out the existential quantifiers\<close>
```
```    38
```
```    39 text \<open>Conjunction\<close>
```
```    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 \<open>Disjunction\<close>
```
```    47
```
```    48 lemma disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
```
```    49   -- \<open>DO NOT USE with forall-Skolemization: makes fewer schematic variables!!\<close>
```
```    50   -- \<open>With ex-Skolemization, makes fewer Skolem constants\<close>
```
```    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\<open>Generation of contrapositives\<close>
```
```    63
```
```    64 text\<open>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.\<close>
```
```    67 lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
```
```    68 by blast
```
```    69
```
```    70 text\<open>Version for Plaisted's "Postive refinement" of the Meson procedure\<close>
```
```    71 lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
```
```    72 by blast
```
```    73
```
```    74 text\<open>@{term P} should be a literal\<close>
```
```    75 lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
```
```    76 by blast
```
```    77
```
```    78 text\<open>Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
```
```    79 insert new assumptions, for ordinary resolution.\<close>
```
```    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\<open>Generation of a goal clause -- put away the final literal\<close>
```
```    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 \<open>Lemmas for Forward Proof\<close>
```
```    96
```
```    97 text\<open>There is a similarity to congruence rules\<close>
```
```    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 \<open>Clausification helper\<close>
```
```   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\<open>Combinator translation helpers\<close>
```
```   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 \<open>Skolemization helpers\<close>
```
```   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::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 \<open>Meson package\<close>
```
```   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 hide_const (open) COMBI COMBK COMBB COMBC COMBS skolem
```
```   199 hide_fact (open) not_conjD not_disjD not_notD not_allD not_exD imp_to_disjD
```
```   200     not_impD iff_to_disjD not_iffD not_refl_disj_D conj_exD1 conj_exD2 disj_exD
```
```   201     disj_exD1 disj_exD2 disj_assoc disj_comm disj_FalseD1 disj_FalseD2 TruepropI
```
```   202     ext_cong_neq COMBI_def COMBK_def COMBB_def COMBC_def COMBS_def abs_I abs_K
```
```   203     abs_B abs_C abs_S skolem_def skolem_COMBK_iff skolem_COMBK_I skolem_COMBK_D
```
```   204
```
```   205 end
```