src/FOL/IFOL.thy
author wenzelm
Sat Feb 28 14:02:12 2009 +0100 (2009-02-28)
changeset 30160 5f7b17941730
parent 28856 5e009a80fe6d
child 30165 6ee87f67d9cd
permissions -rw-r--r--
moved some generic tools to src/Tools/ -- src/Provers is essentially obsolete;
clasohm@1268
     1
(*  Title:      FOL/IFOL.thy
wenzelm@11677
     2
    Author:     Lawrence C Paulson and Markus Wenzel
wenzelm@11677
     3
*)
lcp@35
     4
wenzelm@11677
     5
header {* Intuitionistic first-order logic *}
lcp@35
     6
paulson@15481
     7
theory IFOL
paulson@15481
     8
imports Pure
wenzelm@23155
     9
uses
wenzelm@23155
    10
  "~~/src/Provers/splitter.ML"
wenzelm@23155
    11
  "~~/src/Provers/hypsubst.ML"
wenzelm@23171
    12
  "~~/src/Tools/IsaPlanner/zipper.ML"
wenzelm@23171
    13
  "~~/src/Tools/IsaPlanner/isand.ML"
wenzelm@23171
    14
  "~~/src/Tools/IsaPlanner/rw_tools.ML"
wenzelm@23171
    15
  "~~/src/Tools/IsaPlanner/rw_inst.ML"
wenzelm@30160
    16
  "~~/src/Tools/eqsubst.ML"
wenzelm@23155
    17
  "~~/src/Provers/quantifier1.ML"
wenzelm@30160
    18
  "~~/src/Tools/project_rule.ML"
krauss@26580
    19
  "~~/src/Tools/atomize_elim.ML"
wenzelm@23155
    20
  ("fologic.ML")
wenzelm@23155
    21
  ("hypsubstdata.ML")
wenzelm@23155
    22
  ("intprover.ML")
paulson@15481
    23
begin
wenzelm@7355
    24
clasohm@0
    25
wenzelm@11677
    26
subsection {* Syntax and axiomatic basis *}
wenzelm@11677
    27
wenzelm@26956
    28
setup PureThy.old_appl_syntax_setup
wenzelm@26956
    29
wenzelm@3906
    30
global
wenzelm@3906
    31
wenzelm@14854
    32
classes "term"
wenzelm@7355
    33
defaultsort "term"
clasohm@0
    34
wenzelm@7355
    35
typedecl o
wenzelm@79
    36
wenzelm@11747
    37
judgment
wenzelm@11747
    38
  Trueprop      :: "o => prop"                  ("(_)" 5)
clasohm@0
    39
wenzelm@11747
    40
consts
wenzelm@7355
    41
  True          :: o
wenzelm@7355
    42
  False         :: o
wenzelm@79
    43
wenzelm@79
    44
  (* Connectives *)
wenzelm@79
    45
wenzelm@17276
    46
  "op ="        :: "['a, 'a] => o"              (infixl "=" 50)
lcp@35
    47
wenzelm@7355
    48
  Not           :: "o => o"                     ("~ _" [40] 40)
wenzelm@17276
    49
  "op &"        :: "[o, o] => o"                (infixr "&" 35)
wenzelm@17276
    50
  "op |"        :: "[o, o] => o"                (infixr "|" 30)
wenzelm@17276
    51
  "op -->"      :: "[o, o] => o"                (infixr "-->" 25)
wenzelm@17276
    52
  "op <->"      :: "[o, o] => o"                (infixr "<->" 25)
wenzelm@79
    53
wenzelm@79
    54
  (* Quantifiers *)
wenzelm@79
    55
wenzelm@7355
    56
  All           :: "('a => o) => o"             (binder "ALL " 10)
wenzelm@7355
    57
  Ex            :: "('a => o) => o"             (binder "EX " 10)
wenzelm@7355
    58
  Ex1           :: "('a => o) => o"             (binder "EX! " 10)
wenzelm@79
    59
clasohm@0
    60
wenzelm@19363
    61
abbreviation
wenzelm@21404
    62
  not_equal :: "['a, 'a] => o"  (infixl "~=" 50) where
wenzelm@19120
    63
  "x ~= y == ~ (x = y)"
wenzelm@79
    64
wenzelm@21210
    65
notation (xsymbols)
wenzelm@19656
    66
  not_equal  (infixl "\<noteq>" 50)
wenzelm@19363
    67
wenzelm@21210
    68
notation (HTML output)
wenzelm@19656
    69
  not_equal  (infixl "\<noteq>" 50)
wenzelm@19363
    70
wenzelm@21524
    71
notation (xsymbols)
wenzelm@21539
    72
  Not       ("\<not> _" [40] 40) and
wenzelm@21539
    73
  "op &"    (infixr "\<and>" 35) and
wenzelm@21539
    74
  "op |"    (infixr "\<or>" 30) and
wenzelm@21539
    75
  All       (binder "\<forall>" 10) and
wenzelm@21539
    76
  Ex        (binder "\<exists>" 10) and
wenzelm@21539
    77
  Ex1       (binder "\<exists>!" 10) and
wenzelm@21524
    78
  "op -->"  (infixr "\<longrightarrow>" 25) and
wenzelm@21524
    79
  "op <->"  (infixr "\<longleftrightarrow>" 25)
lcp@35
    80
wenzelm@21524
    81
notation (HTML output)
wenzelm@21539
    82
  Not       ("\<not> _" [40] 40) and
wenzelm@21539
    83
  "op &"    (infixr "\<and>" 35) and
wenzelm@21539
    84
  "op |"    (infixr "\<or>" 30) and
wenzelm@21539
    85
  All       (binder "\<forall>" 10) and
wenzelm@21539
    86
  Ex        (binder "\<exists>" 10) and
wenzelm@21539
    87
  Ex1       (binder "\<exists>!" 10)
wenzelm@6340
    88
wenzelm@3932
    89
local
wenzelm@3906
    90
paulson@14236
    91
finalconsts
paulson@14236
    92
  False All Ex
paulson@14236
    93
  "op ="
paulson@14236
    94
  "op &"
paulson@14236
    95
  "op |"
paulson@14236
    96
  "op -->"
paulson@14236
    97
wenzelm@7355
    98
axioms
clasohm@0
    99
wenzelm@79
   100
  (* Equality *)
clasohm@0
   101
wenzelm@7355
   102
  refl:         "a=a"
haftmann@28681
   103
  subst:        "a=b \<Longrightarrow> P(a) \<Longrightarrow> P(b)"
clasohm@0
   104
wenzelm@79
   105
  (* Propositional logic *)
clasohm@0
   106
wenzelm@7355
   107
  conjI:        "[| P;  Q |] ==> P&Q"
wenzelm@7355
   108
  conjunct1:    "P&Q ==> P"
wenzelm@7355
   109
  conjunct2:    "P&Q ==> Q"
clasohm@0
   110
wenzelm@7355
   111
  disjI1:       "P ==> P|Q"
wenzelm@7355
   112
  disjI2:       "Q ==> P|Q"
wenzelm@7355
   113
  disjE:        "[| P|Q;  P ==> R;  Q ==> R |] ==> R"
clasohm@0
   114
wenzelm@7355
   115
  impI:         "(P ==> Q) ==> P-->Q"
wenzelm@7355
   116
  mp:           "[| P-->Q;  P |] ==> Q"
clasohm@0
   117
wenzelm@7355
   118
  FalseE:       "False ==> P"
wenzelm@7355
   119
wenzelm@79
   120
  (* Quantifiers *)
clasohm@0
   121
wenzelm@7355
   122
  allI:         "(!!x. P(x)) ==> (ALL x. P(x))"
wenzelm@7355
   123
  spec:         "(ALL x. P(x)) ==> P(x)"
clasohm@0
   124
wenzelm@7355
   125
  exI:          "P(x) ==> (EX x. P(x))"
wenzelm@7355
   126
  exE:          "[| EX x. P(x);  !!x. P(x) ==> R |] ==> R"
clasohm@0
   127
haftmann@28681
   128
haftmann@28681
   129
axioms
haftmann@28681
   130
haftmann@28681
   131
  (* Reflection, admissible *)
clasohm@0
   132
wenzelm@7355
   133
  eq_reflection:  "(x=y)   ==> (x==y)"
wenzelm@7355
   134
  iff_reflection: "(P<->Q) ==> (P==Q)"
clasohm@0
   135
wenzelm@4092
   136
wenzelm@19756
   137
lemmas strip = impI allI
wenzelm@19756
   138
wenzelm@19756
   139
paulson@14236
   140
defs
paulson@14236
   141
  (* Definitions *)
paulson@14236
   142
paulson@14236
   143
  True_def:     "True  == False-->False"
paulson@14236
   144
  not_def:      "~P    == P-->False"
paulson@14236
   145
  iff_def:      "P<->Q == (P-->Q) & (Q-->P)"
paulson@14236
   146
paulson@14236
   147
  (* Unique existence *)
paulson@14236
   148
paulson@14236
   149
  ex1_def:      "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)"
paulson@14236
   150
paulson@13779
   151
wenzelm@11677
   152
subsection {* Lemmas and proof tools *}
wenzelm@11677
   153
wenzelm@21539
   154
lemma TrueI: True
wenzelm@21539
   155
  unfolding True_def by (rule impI)
wenzelm@21539
   156
wenzelm@21539
   157
wenzelm@21539
   158
(*** Sequent-style elimination rules for & --> and ALL ***)
wenzelm@21539
   159
wenzelm@21539
   160
lemma conjE:
wenzelm@21539
   161
  assumes major: "P & Q"
wenzelm@21539
   162
    and r: "[| P; Q |] ==> R"
wenzelm@21539
   163
  shows R
wenzelm@21539
   164
  apply (rule r)
wenzelm@21539
   165
   apply (rule major [THEN conjunct1])
wenzelm@21539
   166
  apply (rule major [THEN conjunct2])
wenzelm@21539
   167
  done
wenzelm@21539
   168
wenzelm@21539
   169
lemma impE:
wenzelm@21539
   170
  assumes major: "P --> Q"
wenzelm@21539
   171
    and P
wenzelm@21539
   172
  and r: "Q ==> R"
wenzelm@21539
   173
  shows R
wenzelm@21539
   174
  apply (rule r)
wenzelm@21539
   175
  apply (rule major [THEN mp])
wenzelm@21539
   176
  apply (rule `P`)
wenzelm@21539
   177
  done
wenzelm@21539
   178
wenzelm@21539
   179
lemma allE:
wenzelm@21539
   180
  assumes major: "ALL x. P(x)"
wenzelm@21539
   181
    and r: "P(x) ==> R"
wenzelm@21539
   182
  shows R
wenzelm@21539
   183
  apply (rule r)
wenzelm@21539
   184
  apply (rule major [THEN spec])
wenzelm@21539
   185
  done
wenzelm@21539
   186
wenzelm@21539
   187
(*Duplicates the quantifier; for use with eresolve_tac*)
wenzelm@21539
   188
lemma all_dupE:
wenzelm@21539
   189
  assumes major: "ALL x. P(x)"
wenzelm@21539
   190
    and r: "[| P(x); ALL x. P(x) |] ==> R"
wenzelm@21539
   191
  shows R
wenzelm@21539
   192
  apply (rule r)
wenzelm@21539
   193
   apply (rule major [THEN spec])
wenzelm@21539
   194
  apply (rule major)
wenzelm@21539
   195
  done
wenzelm@21539
   196
wenzelm@21539
   197
wenzelm@21539
   198
(*** Negation rules, which translate between ~P and P-->False ***)
wenzelm@21539
   199
wenzelm@21539
   200
lemma notI: "(P ==> False) ==> ~P"
wenzelm@21539
   201
  unfolding not_def by (erule impI)
wenzelm@21539
   202
wenzelm@21539
   203
lemma notE: "[| ~P;  P |] ==> R"
wenzelm@21539
   204
  unfolding not_def by (erule mp [THEN FalseE])
wenzelm@21539
   205
wenzelm@21539
   206
lemma rev_notE: "[| P; ~P |] ==> R"
wenzelm@21539
   207
  by (erule notE)
wenzelm@21539
   208
wenzelm@21539
   209
(*This is useful with the special implication rules for each kind of P. *)
wenzelm@21539
   210
lemma not_to_imp:
wenzelm@21539
   211
  assumes "~P"
wenzelm@21539
   212
    and r: "P --> False ==> Q"
wenzelm@21539
   213
  shows Q
wenzelm@21539
   214
  apply (rule r)
wenzelm@21539
   215
  apply (rule impI)
wenzelm@21539
   216
  apply (erule notE [OF `~P`])
wenzelm@21539
   217
  done
wenzelm@21539
   218
wenzelm@21539
   219
(* For substitution into an assumption P, reduce Q to P-->Q, substitute into
wenzelm@27150
   220
   this implication, then apply impI to move P back into the assumptions.*)
wenzelm@21539
   221
lemma rev_mp: "[| P;  P --> Q |] ==> Q"
wenzelm@21539
   222
  by (erule mp)
wenzelm@21539
   223
wenzelm@21539
   224
(*Contrapositive of an inference rule*)
wenzelm@21539
   225
lemma contrapos:
wenzelm@21539
   226
  assumes major: "~Q"
wenzelm@21539
   227
    and minor: "P ==> Q"
wenzelm@21539
   228
  shows "~P"
wenzelm@21539
   229
  apply (rule major [THEN notE, THEN notI])
wenzelm@21539
   230
  apply (erule minor)
wenzelm@21539
   231
  done
wenzelm@21539
   232
wenzelm@21539
   233
wenzelm@21539
   234
(*** Modus Ponens Tactics ***)
wenzelm@21539
   235
wenzelm@21539
   236
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
wenzelm@21539
   237
ML {*
wenzelm@22139
   238
  fun mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i  THEN  assume_tac i
wenzelm@22139
   239
  fun eq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i  THEN  eq_assume_tac i
wenzelm@21539
   240
*}
wenzelm@21539
   241
wenzelm@21539
   242
wenzelm@21539
   243
(*** If-and-only-if ***)
wenzelm@21539
   244
wenzelm@21539
   245
lemma iffI: "[| P ==> Q; Q ==> P |] ==> P<->Q"
wenzelm@21539
   246
  apply (unfold iff_def)
wenzelm@21539
   247
  apply (rule conjI)
wenzelm@21539
   248
   apply (erule impI)
wenzelm@21539
   249
  apply (erule impI)
wenzelm@21539
   250
  done
wenzelm@21539
   251
wenzelm@21539
   252
wenzelm@21539
   253
(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
wenzelm@21539
   254
lemma iffE:
wenzelm@21539
   255
  assumes major: "P <-> Q"
wenzelm@21539
   256
    and r: "P-->Q ==> Q-->P ==> R"
wenzelm@21539
   257
  shows R
wenzelm@21539
   258
  apply (insert major, unfold iff_def)
wenzelm@21539
   259
  apply (erule conjE)
wenzelm@21539
   260
  apply (erule r)
wenzelm@21539
   261
  apply assumption
wenzelm@21539
   262
  done
wenzelm@21539
   263
wenzelm@21539
   264
(* Destruct rules for <-> similar to Modus Ponens *)
wenzelm@21539
   265
wenzelm@21539
   266
lemma iffD1: "[| P <-> Q;  P |] ==> Q"
wenzelm@21539
   267
  apply (unfold iff_def)
wenzelm@21539
   268
  apply (erule conjunct1 [THEN mp])
wenzelm@21539
   269
  apply assumption
wenzelm@21539
   270
  done
wenzelm@21539
   271
wenzelm@21539
   272
lemma iffD2: "[| P <-> Q;  Q |] ==> P"
wenzelm@21539
   273
  apply (unfold iff_def)
wenzelm@21539
   274
  apply (erule conjunct2 [THEN mp])
wenzelm@21539
   275
  apply assumption
wenzelm@21539
   276
  done
wenzelm@21539
   277
wenzelm@21539
   278
lemma rev_iffD1: "[| P; P <-> Q |] ==> Q"
wenzelm@21539
   279
  apply (erule iffD1)
wenzelm@21539
   280
  apply assumption
wenzelm@21539
   281
  done
wenzelm@21539
   282
wenzelm@21539
   283
lemma rev_iffD2: "[| Q; P <-> Q |] ==> P"
wenzelm@21539
   284
  apply (erule iffD2)
wenzelm@21539
   285
  apply assumption
wenzelm@21539
   286
  done
wenzelm@21539
   287
wenzelm@21539
   288
lemma iff_refl: "P <-> P"
wenzelm@21539
   289
  by (rule iffI)
wenzelm@21539
   290
wenzelm@21539
   291
lemma iff_sym: "Q <-> P ==> P <-> Q"
wenzelm@21539
   292
  apply (erule iffE)
wenzelm@21539
   293
  apply (rule iffI)
wenzelm@21539
   294
  apply (assumption | erule mp)+
wenzelm@21539
   295
  done
wenzelm@21539
   296
wenzelm@21539
   297
lemma iff_trans: "[| P <-> Q;  Q<-> R |] ==> P <-> R"
wenzelm@21539
   298
  apply (rule iffI)
wenzelm@21539
   299
  apply (assumption | erule iffE | erule (1) notE impE)+
wenzelm@21539
   300
  done
wenzelm@21539
   301
wenzelm@21539
   302
wenzelm@21539
   303
(*** Unique existence.  NOTE THAT the following 2 quantifications
wenzelm@21539
   304
   EX!x such that [EX!y such that P(x,y)]     (sequential)
wenzelm@21539
   305
   EX!x,y such that P(x,y)                    (simultaneous)
wenzelm@21539
   306
 do NOT mean the same thing.  The parser treats EX!x y.P(x,y) as sequential.
wenzelm@21539
   307
***)
wenzelm@21539
   308
wenzelm@21539
   309
lemma ex1I:
wenzelm@23393
   310
  "P(a) \<Longrightarrow> (!!x. P(x) ==> x=a) \<Longrightarrow> EX! x. P(x)"
wenzelm@21539
   311
  apply (unfold ex1_def)
wenzelm@23393
   312
  apply (assumption | rule exI conjI allI impI)+
wenzelm@21539
   313
  done
wenzelm@21539
   314
wenzelm@21539
   315
(*Sometimes easier to use: the premises have no shared variables.  Safe!*)
wenzelm@21539
   316
lemma ex_ex1I:
wenzelm@23393
   317
  "EX x. P(x) \<Longrightarrow> (!!x y. [| P(x); P(y) |] ==> x=y) \<Longrightarrow> EX! x. P(x)"
wenzelm@23393
   318
  apply (erule exE)
wenzelm@23393
   319
  apply (rule ex1I)
wenzelm@23393
   320
   apply assumption
wenzelm@23393
   321
  apply assumption
wenzelm@21539
   322
  done
wenzelm@21539
   323
wenzelm@21539
   324
lemma ex1E:
wenzelm@23393
   325
  "EX! x. P(x) \<Longrightarrow> (!!x. [| P(x);  ALL y. P(y) --> y=x |] ==> R) \<Longrightarrow> R"
wenzelm@23393
   326
  apply (unfold ex1_def)
wenzelm@21539
   327
  apply (assumption | erule exE conjE)+
wenzelm@21539
   328
  done
wenzelm@21539
   329
wenzelm@21539
   330
wenzelm@21539
   331
(*** <-> congruence rules for simplification ***)
wenzelm@21539
   332
wenzelm@21539
   333
(*Use iffE on a premise.  For conj_cong, imp_cong, all_cong, ex_cong*)
wenzelm@21539
   334
ML {*
wenzelm@22139
   335
  fun iff_tac prems i =
wenzelm@22139
   336
    resolve_tac (prems RL @{thms iffE}) i THEN
wenzelm@22139
   337
    REPEAT1 (eresolve_tac [@{thm asm_rl}, @{thm mp}] i)
wenzelm@21539
   338
*}
wenzelm@21539
   339
wenzelm@21539
   340
lemma conj_cong:
wenzelm@21539
   341
  assumes "P <-> P'"
wenzelm@21539
   342
    and "P' ==> Q <-> Q'"
wenzelm@21539
   343
  shows "(P&Q) <-> (P'&Q')"
wenzelm@21539
   344
  apply (insert assms)
wenzelm@21539
   345
  apply (assumption | rule iffI conjI | erule iffE conjE mp |
wenzelm@21539
   346
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   347
  done
wenzelm@21539
   348
wenzelm@21539
   349
(*Reversed congruence rule!   Used in ZF/Order*)
wenzelm@21539
   350
lemma conj_cong2:
wenzelm@21539
   351
  assumes "P <-> P'"
wenzelm@21539
   352
    and "P' ==> Q <-> Q'"
wenzelm@21539
   353
  shows "(Q&P) <-> (Q'&P')"
wenzelm@21539
   354
  apply (insert assms)
wenzelm@21539
   355
  apply (assumption | rule iffI conjI | erule iffE conjE mp |
wenzelm@21539
   356
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   357
  done
wenzelm@21539
   358
wenzelm@21539
   359
lemma disj_cong:
wenzelm@21539
   360
  assumes "P <-> P'" and "Q <-> Q'"
wenzelm@21539
   361
  shows "(P|Q) <-> (P'|Q')"
wenzelm@21539
   362
  apply (insert assms)
wenzelm@21539
   363
  apply (erule iffE disjE disjI1 disjI2 | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   364
  done
wenzelm@21539
   365
wenzelm@21539
   366
lemma imp_cong:
wenzelm@21539
   367
  assumes "P <-> P'"
wenzelm@21539
   368
    and "P' ==> Q <-> Q'"
wenzelm@21539
   369
  shows "(P-->Q) <-> (P'-->Q')"
wenzelm@21539
   370
  apply (insert assms)
wenzelm@21539
   371
  apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE |
wenzelm@21539
   372
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   373
  done
wenzelm@21539
   374
wenzelm@21539
   375
lemma iff_cong: "[| P <-> P'; Q <-> Q' |] ==> (P<->Q) <-> (P'<->Q')"
wenzelm@21539
   376
  apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
wenzelm@21539
   377
  done
wenzelm@21539
   378
wenzelm@21539
   379
lemma not_cong: "P <-> P' ==> ~P <-> ~P'"
wenzelm@21539
   380
  apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
wenzelm@21539
   381
  done
wenzelm@21539
   382
wenzelm@21539
   383
lemma all_cong:
wenzelm@21539
   384
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   385
  shows "(ALL x. P(x)) <-> (ALL x. Q(x))"
wenzelm@21539
   386
  apply (assumption | rule iffI allI | erule (1) notE impE | erule allE |
wenzelm@21539
   387
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   388
  done
wenzelm@21539
   389
wenzelm@21539
   390
lemma ex_cong:
wenzelm@21539
   391
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   392
  shows "(EX x. P(x)) <-> (EX x. Q(x))"
wenzelm@21539
   393
  apply (erule exE | assumption | rule iffI exI | erule (1) notE impE |
wenzelm@21539
   394
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   395
  done
wenzelm@21539
   396
wenzelm@21539
   397
lemma ex1_cong:
wenzelm@21539
   398
  assumes "!!x. P(x) <-> Q(x)"
wenzelm@21539
   399
  shows "(EX! x. P(x)) <-> (EX! x. Q(x))"
wenzelm@21539
   400
  apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE |
wenzelm@21539
   401
    tactic {* iff_tac (thms "assms") 1 *})+
wenzelm@21539
   402
  done
wenzelm@21539
   403
wenzelm@21539
   404
(*** Equality rules ***)
wenzelm@21539
   405
wenzelm@21539
   406
lemma sym: "a=b ==> b=a"
wenzelm@21539
   407
  apply (erule subst)
wenzelm@21539
   408
  apply (rule refl)
wenzelm@21539
   409
  done
wenzelm@21539
   410
wenzelm@21539
   411
lemma trans: "[| a=b;  b=c |] ==> a=c"
wenzelm@21539
   412
  apply (erule subst, assumption)
wenzelm@21539
   413
  done
wenzelm@21539
   414
wenzelm@21539
   415
(**  **)
wenzelm@21539
   416
lemma not_sym: "b ~= a ==> a ~= b"
wenzelm@21539
   417
  apply (erule contrapos)
wenzelm@21539
   418
  apply (erule sym)
wenzelm@21539
   419
  done
wenzelm@21539
   420
  
wenzelm@21539
   421
(* Two theorms for rewriting only one instance of a definition:
wenzelm@21539
   422
   the first for definitions of formulae and the second for terms *)
wenzelm@21539
   423
wenzelm@21539
   424
lemma def_imp_iff: "(A == B) ==> A <-> B"
wenzelm@21539
   425
  apply unfold
wenzelm@21539
   426
  apply (rule iff_refl)
wenzelm@21539
   427
  done
wenzelm@21539
   428
wenzelm@21539
   429
lemma meta_eq_to_obj_eq: "(A == B) ==> A = B"
wenzelm@21539
   430
  apply unfold
wenzelm@21539
   431
  apply (rule refl)
wenzelm@21539
   432
  done
wenzelm@21539
   433
wenzelm@21539
   434
lemma meta_eq_to_iff: "x==y ==> x<->y"
wenzelm@21539
   435
  by unfold (rule iff_refl)
wenzelm@21539
   436
wenzelm@21539
   437
(*substitution*)
wenzelm@21539
   438
lemma ssubst: "[| b = a; P(a) |] ==> P(b)"
wenzelm@21539
   439
  apply (drule sym)
wenzelm@21539
   440
  apply (erule (1) subst)
wenzelm@21539
   441
  done
wenzelm@21539
   442
wenzelm@21539
   443
(*A special case of ex1E that would otherwise need quantifier expansion*)
wenzelm@21539
   444
lemma ex1_equalsE:
wenzelm@21539
   445
    "[| EX! x. P(x);  P(a);  P(b) |] ==> a=b"
wenzelm@21539
   446
  apply (erule ex1E)
wenzelm@21539
   447
  apply (rule trans)
wenzelm@21539
   448
   apply (rule_tac [2] sym)
wenzelm@21539
   449
   apply (assumption | erule spec [THEN mp])+
wenzelm@21539
   450
  done
wenzelm@21539
   451
wenzelm@21539
   452
(** Polymorphic congruence rules **)
wenzelm@21539
   453
wenzelm@21539
   454
lemma subst_context: "[| a=b |]  ==>  t(a)=t(b)"
wenzelm@21539
   455
  apply (erule ssubst)
wenzelm@21539
   456
  apply (rule refl)
wenzelm@21539
   457
  done
wenzelm@21539
   458
wenzelm@21539
   459
lemma subst_context2: "[| a=b;  c=d |]  ==>  t(a,c)=t(b,d)"
wenzelm@21539
   460
  apply (erule ssubst)+
wenzelm@21539
   461
  apply (rule refl)
wenzelm@21539
   462
  done
wenzelm@21539
   463
wenzelm@21539
   464
lemma subst_context3: "[| a=b;  c=d;  e=f |]  ==>  t(a,c,e)=t(b,d,f)"
wenzelm@21539
   465
  apply (erule ssubst)+
wenzelm@21539
   466
  apply (rule refl)
wenzelm@21539
   467
  done
wenzelm@21539
   468
wenzelm@21539
   469
(*Useful with eresolve_tac for proving equalties from known equalities.
wenzelm@21539
   470
        a = b
wenzelm@21539
   471
        |   |
wenzelm@21539
   472
        c = d   *)
wenzelm@21539
   473
lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
wenzelm@21539
   474
  apply (rule trans)
wenzelm@21539
   475
   apply (rule trans)
wenzelm@21539
   476
    apply (rule sym)
wenzelm@21539
   477
    apply assumption+
wenzelm@21539
   478
  done
wenzelm@21539
   479
wenzelm@21539
   480
(*Dual of box_equals: for proving equalities backwards*)
wenzelm@21539
   481
lemma simp_equals: "[| a=c;  b=d;  c=d |] ==> a=b"
wenzelm@21539
   482
  apply (rule trans)
wenzelm@21539
   483
   apply (rule trans)
wenzelm@21539
   484
    apply assumption+
wenzelm@21539
   485
  apply (erule sym)
wenzelm@21539
   486
  done
wenzelm@21539
   487
wenzelm@21539
   488
(** Congruence rules for predicate letters **)
wenzelm@21539
   489
wenzelm@21539
   490
lemma pred1_cong: "a=a' ==> P(a) <-> P(a')"
wenzelm@21539
   491
  apply (rule iffI)
wenzelm@21539
   492
   apply (erule (1) subst)
wenzelm@21539
   493
  apply (erule (1) ssubst)
wenzelm@21539
   494
  done
wenzelm@21539
   495
wenzelm@21539
   496
lemma pred2_cong: "[| a=a';  b=b' |] ==> P(a,b) <-> P(a',b')"
wenzelm@21539
   497
  apply (rule iffI)
wenzelm@21539
   498
   apply (erule subst)+
wenzelm@21539
   499
   apply assumption
wenzelm@21539
   500
  apply (erule ssubst)+
wenzelm@21539
   501
  apply assumption
wenzelm@21539
   502
  done
wenzelm@21539
   503
wenzelm@21539
   504
lemma pred3_cong: "[| a=a';  b=b';  c=c' |] ==> P(a,b,c) <-> P(a',b',c')"
wenzelm@21539
   505
  apply (rule iffI)
wenzelm@21539
   506
   apply (erule subst)+
wenzelm@21539
   507
   apply assumption
wenzelm@21539
   508
  apply (erule ssubst)+
wenzelm@21539
   509
  apply assumption
wenzelm@21539
   510
  done
wenzelm@21539
   511
wenzelm@21539
   512
(*special case for the equality predicate!*)
wenzelm@21539
   513
lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'"
wenzelm@21539
   514
  apply (erule (1) pred2_cong)
wenzelm@21539
   515
  done
wenzelm@21539
   516
wenzelm@21539
   517
wenzelm@21539
   518
(*** Simplifications of assumed implications.
wenzelm@21539
   519
     Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
wenzelm@21539
   520
     used with mp_tac (restricted to atomic formulae) is COMPLETE for 
wenzelm@21539
   521
     intuitionistic propositional logic.  See
wenzelm@21539
   522
   R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
wenzelm@21539
   523
    (preprint, University of St Andrews, 1991)  ***)
wenzelm@21539
   524
wenzelm@21539
   525
lemma conj_impE:
wenzelm@21539
   526
  assumes major: "(P&Q)-->S"
wenzelm@21539
   527
    and r: "P-->(Q-->S) ==> R"
wenzelm@21539
   528
  shows R
wenzelm@21539
   529
  by (assumption | rule conjI impI major [THEN mp] r)+
wenzelm@21539
   530
wenzelm@21539
   531
lemma disj_impE:
wenzelm@21539
   532
  assumes major: "(P|Q)-->S"
wenzelm@21539
   533
    and r: "[| P-->S; Q-->S |] ==> R"
wenzelm@21539
   534
  shows R
wenzelm@21539
   535
  by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
wenzelm@21539
   536
wenzelm@21539
   537
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   538
  Still UNSAFE since Q must be provable -- backtracking needed.  *)
wenzelm@21539
   539
lemma imp_impE:
wenzelm@21539
   540
  assumes major: "(P-->Q)-->S"
wenzelm@21539
   541
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   542
    and r2: "S ==> R"
wenzelm@21539
   543
  shows R
wenzelm@21539
   544
  by (assumption | rule impI major [THEN mp] r1 r2)+
wenzelm@21539
   545
wenzelm@21539
   546
(*Simplifies the implication.  Classical version is stronger. 
wenzelm@21539
   547
  Still UNSAFE since ~P must be provable -- backtracking needed.  *)
wenzelm@21539
   548
lemma not_impE:
wenzelm@23393
   549
  "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R"
wenzelm@23393
   550
  apply (drule mp)
wenzelm@23393
   551
   apply (rule notI)
wenzelm@23393
   552
   apply assumption
wenzelm@23393
   553
  apply assumption
wenzelm@21539
   554
  done
wenzelm@21539
   555
wenzelm@21539
   556
(*Simplifies the implication.   UNSAFE.  *)
wenzelm@21539
   557
lemma iff_impE:
wenzelm@21539
   558
  assumes major: "(P<->Q)-->S"
wenzelm@21539
   559
    and r1: "[| P; Q-->S |] ==> Q"
wenzelm@21539
   560
    and r2: "[| Q; P-->S |] ==> P"
wenzelm@21539
   561
    and r3: "S ==> R"
wenzelm@21539
   562
  shows R
wenzelm@21539
   563
  apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
wenzelm@21539
   564
  done
wenzelm@21539
   565
wenzelm@21539
   566
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
wenzelm@21539
   567
lemma all_impE:
wenzelm@21539
   568
  assumes major: "(ALL x. P(x))-->S"
wenzelm@21539
   569
    and r1: "!!x. P(x)"
wenzelm@21539
   570
    and r2: "S ==> R"
wenzelm@21539
   571
  shows R
wenzelm@23393
   572
  apply (rule allI impI major [THEN mp] r1 r2)+
wenzelm@21539
   573
  done
wenzelm@21539
   574
wenzelm@21539
   575
(*Unsafe: (EX x.P(x))-->S  is equivalent to  ALL x.P(x)-->S.  *)
wenzelm@21539
   576
lemma ex_impE:
wenzelm@21539
   577
  assumes major: "(EX x. P(x))-->S"
wenzelm@21539
   578
    and r: "P(x)-->S ==> R"
wenzelm@21539
   579
  shows R
wenzelm@21539
   580
  apply (assumption | rule exI impI major [THEN mp] r)+
wenzelm@21539
   581
  done
wenzelm@21539
   582
wenzelm@21539
   583
(*** Courtesy of Krzysztof Grabczewski ***)
wenzelm@21539
   584
wenzelm@21539
   585
lemma disj_imp_disj:
wenzelm@23393
   586
  "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> R|S"
wenzelm@23393
   587
  apply (erule disjE)
wenzelm@21539
   588
  apply (rule disjI1) apply assumption
wenzelm@21539
   589
  apply (rule disjI2) apply assumption
wenzelm@21539
   590
  done
wenzelm@11734
   591
wenzelm@18481
   592
ML {*
wenzelm@18481
   593
structure ProjectRule = ProjectRuleFun
wenzelm@18481
   594
(struct
wenzelm@22139
   595
  val conjunct1 = @{thm conjunct1}
wenzelm@22139
   596
  val conjunct2 = @{thm conjunct2}
wenzelm@22139
   597
  val mp = @{thm mp}
wenzelm@18481
   598
end)
wenzelm@18481
   599
*}
wenzelm@18481
   600
wenzelm@7355
   601
use "fologic.ML"
wenzelm@21539
   602
wenzelm@21539
   603
lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" .
wenzelm@21539
   604
wenzelm@9886
   605
use "hypsubstdata.ML"
wenzelm@9886
   606
setup hypsubst_setup
wenzelm@7355
   607
use "intprover.ML"
wenzelm@7355
   608
wenzelm@4092
   609
wenzelm@12875
   610
subsection {* Intuitionistic Reasoning *}
wenzelm@12368
   611
wenzelm@12349
   612
lemma impE':
wenzelm@12937
   613
  assumes 1: "P --> Q"
wenzelm@12937
   614
    and 2: "Q ==> R"
wenzelm@12937
   615
    and 3: "P --> Q ==> P"
wenzelm@12937
   616
  shows R
wenzelm@12349
   617
proof -
wenzelm@12349
   618
  from 3 and 1 have P .
wenzelm@12368
   619
  with 1 have Q by (rule impE)
wenzelm@12349
   620
  with 2 show R .
wenzelm@12349
   621
qed
wenzelm@12349
   622
wenzelm@12349
   623
lemma allE':
wenzelm@12937
   624
  assumes 1: "ALL x. P(x)"
wenzelm@12937
   625
    and 2: "P(x) ==> ALL x. P(x) ==> Q"
wenzelm@12937
   626
  shows Q
wenzelm@12349
   627
proof -
wenzelm@12349
   628
  from 1 have "P(x)" by (rule spec)
wenzelm@12349
   629
  from this and 1 show Q by (rule 2)
wenzelm@12349
   630
qed
wenzelm@12349
   631
wenzelm@12937
   632
lemma notE':
wenzelm@12937
   633
  assumes 1: "~ P"
wenzelm@12937
   634
    and 2: "~ P ==> P"
wenzelm@12937
   635
  shows R
wenzelm@12349
   636
proof -
wenzelm@12349
   637
  from 2 and 1 have P .
wenzelm@12349
   638
  with 1 show R by (rule notE)
wenzelm@12349
   639
qed
wenzelm@12349
   640
wenzelm@12349
   641
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12349
   642
  and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12349
   643
  and [Pure.elim 2] = allE notE' impE'
wenzelm@12349
   644
  and [Pure.intro] = exI disjI2 disjI1
wenzelm@12349
   645
wenzelm@18708
   646
setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac ORELSE' tac) *}
wenzelm@12349
   647
wenzelm@12349
   648
wenzelm@12368
   649
lemma iff_not_sym: "~ (Q <-> P) ==> ~ (P <-> Q)"
nipkow@17591
   650
  by iprover
wenzelm@12368
   651
wenzelm@12368
   652
lemmas [sym] = sym iff_sym not_sym iff_not_sym
wenzelm@12368
   653
  and [Pure.elim?] = iffD1 iffD2 impE
wenzelm@12368
   654
wenzelm@12368
   655
paulson@13435
   656
lemma eq_commute: "a=b <-> b=a"
paulson@13435
   657
apply (rule iffI) 
paulson@13435
   658
apply (erule sym)+
paulson@13435
   659
done
paulson@13435
   660
paulson@13435
   661
wenzelm@11677
   662
subsection {* Atomizing meta-level rules *}
wenzelm@11677
   663
wenzelm@11747
   664
lemma atomize_all [atomize]: "(!!x. P(x)) == Trueprop (ALL x. P(x))"
wenzelm@11976
   665
proof
wenzelm@11677
   666
  assume "!!x. P(x)"
wenzelm@22931
   667
  then show "ALL x. P(x)" ..
wenzelm@11677
   668
next
wenzelm@11677
   669
  assume "ALL x. P(x)"
wenzelm@22931
   670
  then show "!!x. P(x)" ..
wenzelm@11677
   671
qed
wenzelm@11677
   672
wenzelm@11747
   673
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@11976
   674
proof
wenzelm@12368
   675
  assume "A ==> B"
wenzelm@22931
   676
  then show "A --> B" ..
wenzelm@11677
   677
next
wenzelm@11677
   678
  assume "A --> B" and A
wenzelm@22931
   679
  then show B by (rule mp)
wenzelm@11677
   680
qed
wenzelm@11677
   681
wenzelm@11747
   682
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@11976
   683
proof
wenzelm@11677
   684
  assume "x == y"
wenzelm@22931
   685
  show "x = y" unfolding `x == y` by (rule refl)
wenzelm@11677
   686
next
wenzelm@11677
   687
  assume "x = y"
wenzelm@22931
   688
  then show "x == y" by (rule eq_reflection)
wenzelm@11677
   689
qed
wenzelm@11677
   690
wenzelm@18813
   691
lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)"
wenzelm@18813
   692
proof
wenzelm@18813
   693
  assume "A == B"
wenzelm@22931
   694
  show "A <-> B" unfolding `A == B` by (rule iff_refl)
wenzelm@18813
   695
next
wenzelm@18813
   696
  assume "A <-> B"
wenzelm@22931
   697
  then show "A == B" by (rule iff_reflection)
wenzelm@18813
   698
qed
wenzelm@18813
   699
wenzelm@28856
   700
lemma atomize_conj [atomize]: "(A &&& B) == Trueprop (A & B)"
wenzelm@11976
   701
proof
wenzelm@28856
   702
  assume conj: "A &&& B"
wenzelm@19120
   703
  show "A & B"
wenzelm@19120
   704
  proof (rule conjI)
wenzelm@19120
   705
    from conj show A by (rule conjunctionD1)
wenzelm@19120
   706
    from conj show B by (rule conjunctionD2)
wenzelm@19120
   707
  qed
wenzelm@11953
   708
next
wenzelm@19120
   709
  assume conj: "A & B"
wenzelm@28856
   710
  show "A &&& B"
wenzelm@19120
   711
  proof -
wenzelm@19120
   712
    from conj show A ..
wenzelm@19120
   713
    from conj show B ..
wenzelm@11953
   714
  qed
wenzelm@11953
   715
qed
wenzelm@11953
   716
wenzelm@12368
   717
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@18861
   718
  and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
wenzelm@11771
   719
wenzelm@11848
   720
krauss@26580
   721
subsection {* Atomizing elimination rules *}
krauss@26580
   722
krauss@26580
   723
setup AtomizeElim.setup
krauss@26580
   724
krauss@26580
   725
lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX x. P(x)) ==> Q)"
krauss@26580
   726
by rule iprover+
krauss@26580
   727
krauss@26580
   728
lemma atomize_conjL[atomize_elim]: "(A ==> B ==> C) == (A & B ==> C)"
krauss@26580
   729
by rule iprover+
krauss@26580
   730
krauss@26580
   731
lemma atomize_disjL[atomize_elim]: "((A ==> C) ==> (B ==> C) ==> C) == ((A | B ==> C) ==> C)"
krauss@26580
   732
by rule iprover+
krauss@26580
   733
krauss@26580
   734
lemma atomize_elimL[atomize_elim]: "(!!B. (A ==> B) ==> B) == Trueprop(A)" ..
krauss@26580
   735
krauss@26580
   736
wenzelm@11848
   737
subsection {* Calculational rules *}
wenzelm@11848
   738
wenzelm@11848
   739
lemma forw_subst: "a = b ==> P(b) ==> P(a)"
wenzelm@11848
   740
  by (rule ssubst)
wenzelm@11848
   741
wenzelm@11848
   742
lemma back_subst: "P(a) ==> a = b ==> P(b)"
wenzelm@11848
   743
  by (rule subst)
wenzelm@11848
   744
wenzelm@11848
   745
text {*
wenzelm@11848
   746
  Note that this list of rules is in reverse order of priorities.
wenzelm@11848
   747
*}
wenzelm@11848
   748
wenzelm@12019
   749
lemmas basic_trans_rules [trans] =
wenzelm@11848
   750
  forw_subst
wenzelm@11848
   751
  back_subst
wenzelm@11848
   752
  rev_mp
wenzelm@11848
   753
  mp
wenzelm@11848
   754
  trans
wenzelm@11848
   755
paulson@13779
   756
subsection {* ``Let'' declarations *}
paulson@13779
   757
paulson@13779
   758
nonterminals letbinds letbind
paulson@13779
   759
paulson@13779
   760
constdefs
wenzelm@14854
   761
  Let :: "['a::{}, 'a => 'b] => ('b::{})"
paulson@13779
   762
    "Let(s, f) == f(s)"
paulson@13779
   763
paulson@13779
   764
syntax
paulson@13779
   765
  "_bind"       :: "[pttrn, 'a] => letbind"           ("(2_ =/ _)" 10)
paulson@13779
   766
  ""            :: "letbind => letbinds"              ("_")
paulson@13779
   767
  "_binds"      :: "[letbind, letbinds] => letbinds"  ("_;/ _")
paulson@13779
   768
  "_Let"        :: "[letbinds, 'a] => 'a"             ("(let (_)/ in (_))" 10)
paulson@13779
   769
paulson@13779
   770
translations
paulson@13779
   771
  "_Let(_binds(b, bs), e)"  == "_Let(b, _Let(bs, e))"
paulson@13779
   772
  "let x = a in e"          == "Let(a, %x. e)"
paulson@13779
   773
paulson@13779
   774
paulson@13779
   775
lemma LetI: 
wenzelm@21539
   776
  assumes "!!x. x=t ==> P(u(x))"
wenzelm@21539
   777
  shows "P(let x=t in u(x))"
wenzelm@21539
   778
  apply (unfold Let_def)
wenzelm@21539
   779
  apply (rule refl [THEN assms])
wenzelm@21539
   780
  done
wenzelm@21539
   781
wenzelm@21539
   782
wenzelm@26286
   783
subsection {* Intuitionistic simplification rules *}
wenzelm@26286
   784
wenzelm@26286
   785
lemma conj_simps:
wenzelm@26286
   786
  "P & True <-> P"
wenzelm@26286
   787
  "True & P <-> P"
wenzelm@26286
   788
  "P & False <-> False"
wenzelm@26286
   789
  "False & P <-> False"
wenzelm@26286
   790
  "P & P <-> P"
wenzelm@26286
   791
  "P & P & Q <-> P & Q"
wenzelm@26286
   792
  "P & ~P <-> False"
wenzelm@26286
   793
  "~P & P <-> False"
wenzelm@26286
   794
  "(P & Q) & R <-> P & (Q & R)"
wenzelm@26286
   795
  by iprover+
wenzelm@26286
   796
wenzelm@26286
   797
lemma disj_simps:
wenzelm@26286
   798
  "P | True <-> True"
wenzelm@26286
   799
  "True | P <-> True"
wenzelm@26286
   800
  "P | False <-> P"
wenzelm@26286
   801
  "False | P <-> P"
wenzelm@26286
   802
  "P | P <-> P"
wenzelm@26286
   803
  "P | P | Q <-> P | Q"
wenzelm@26286
   804
  "(P | Q) | R <-> P | (Q | R)"
wenzelm@26286
   805
  by iprover+
wenzelm@26286
   806
wenzelm@26286
   807
lemma not_simps:
wenzelm@26286
   808
  "~(P|Q)  <-> ~P & ~Q"
wenzelm@26286
   809
  "~ False <-> True"
wenzelm@26286
   810
  "~ True <-> False"
wenzelm@26286
   811
  by iprover+
wenzelm@26286
   812
wenzelm@26286
   813
lemma imp_simps:
wenzelm@26286
   814
  "(P --> False) <-> ~P"
wenzelm@26286
   815
  "(P --> True) <-> True"
wenzelm@26286
   816
  "(False --> P) <-> True"
wenzelm@26286
   817
  "(True --> P) <-> P"
wenzelm@26286
   818
  "(P --> P) <-> True"
wenzelm@26286
   819
  "(P --> ~P) <-> ~P"
wenzelm@26286
   820
  by iprover+
wenzelm@26286
   821
wenzelm@26286
   822
lemma iff_simps:
wenzelm@26286
   823
  "(True <-> P) <-> P"
wenzelm@26286
   824
  "(P <-> True) <-> P"
wenzelm@26286
   825
  "(P <-> P) <-> True"
wenzelm@26286
   826
  "(False <-> P) <-> ~P"
wenzelm@26286
   827
  "(P <-> False) <-> ~P"
wenzelm@26286
   828
  by iprover+
wenzelm@26286
   829
wenzelm@26286
   830
(*The x=t versions are needed for the simplification procedures*)
wenzelm@26286
   831
lemma quant_simps:
wenzelm@26286
   832
  "!!P. (ALL x. P) <-> P"
wenzelm@26286
   833
  "(ALL x. x=t --> P(x)) <-> P(t)"
wenzelm@26286
   834
  "(ALL x. t=x --> P(x)) <-> P(t)"
wenzelm@26286
   835
  "!!P. (EX x. P) <-> P"
wenzelm@26286
   836
  "EX x. x=t"
wenzelm@26286
   837
  "EX x. t=x"
wenzelm@26286
   838
  "(EX x. x=t & P(x)) <-> P(t)"
wenzelm@26286
   839
  "(EX x. t=x & P(x)) <-> P(t)"
wenzelm@26286
   840
  by iprover+
wenzelm@26286
   841
wenzelm@26286
   842
(*These are NOT supplied by default!*)
wenzelm@26286
   843
lemma distrib_simps:
wenzelm@26286
   844
  "P & (Q | R) <-> P&Q | P&R"
wenzelm@26286
   845
  "(Q | R) & P <-> Q&P | R&P"
wenzelm@26286
   846
  "(P | Q --> R) <-> (P --> R) & (Q --> R)"
wenzelm@26286
   847
  by iprover+
wenzelm@26286
   848
wenzelm@26286
   849
wenzelm@26286
   850
text {* Conversion into rewrite rules *}
wenzelm@26286
   851
wenzelm@26286
   852
lemma P_iff_F: "~P ==> (P <-> False)" by iprover
wenzelm@26286
   853
lemma iff_reflection_F: "~P ==> (P == False)" by (rule P_iff_F [THEN iff_reflection])
wenzelm@26286
   854
wenzelm@26286
   855
lemma P_iff_T: "P ==> (P <-> True)" by iprover
wenzelm@26286
   856
lemma iff_reflection_T: "P ==> (P == True)" by (rule P_iff_T [THEN iff_reflection])
wenzelm@26286
   857
wenzelm@26286
   858
wenzelm@26286
   859
text {* More rewrite rules *}
wenzelm@26286
   860
wenzelm@26286
   861
lemma conj_commute: "P&Q <-> Q&P" by iprover
wenzelm@26286
   862
lemma conj_left_commute: "P&(Q&R) <-> Q&(P&R)" by iprover
wenzelm@26286
   863
lemmas conj_comms = conj_commute conj_left_commute
wenzelm@26286
   864
wenzelm@26286
   865
lemma disj_commute: "P|Q <-> Q|P" by iprover
wenzelm@26286
   866
lemma disj_left_commute: "P|(Q|R) <-> Q|(P|R)" by iprover
wenzelm@26286
   867
lemmas disj_comms = disj_commute disj_left_commute
wenzelm@26286
   868
wenzelm@26286
   869
lemma conj_disj_distribL: "P&(Q|R) <-> (P&Q | P&R)" by iprover
wenzelm@26286
   870
lemma conj_disj_distribR: "(P|Q)&R <-> (P&R | Q&R)" by iprover
wenzelm@26286
   871
wenzelm@26286
   872
lemma disj_conj_distribL: "P|(Q&R) <-> (P|Q) & (P|R)" by iprover
wenzelm@26286
   873
lemma disj_conj_distribR: "(P&Q)|R <-> (P|R) & (Q|R)" by iprover
wenzelm@26286
   874
wenzelm@26286
   875
lemma imp_conj_distrib: "(P --> (Q&R)) <-> (P-->Q) & (P-->R)" by iprover
wenzelm@26286
   876
lemma imp_conj: "((P&Q)-->R)   <-> (P --> (Q --> R))" by iprover
wenzelm@26286
   877
lemma imp_disj: "(P|Q --> R)   <-> (P-->R) & (Q-->R)" by iprover
wenzelm@26286
   878
wenzelm@26286
   879
lemma de_Morgan_disj: "(~(P | Q)) <-> (~P & ~Q)" by iprover
wenzelm@26286
   880
wenzelm@26286
   881
lemma not_ex: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover
wenzelm@26286
   882
lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover
wenzelm@26286
   883
wenzelm@26286
   884
lemma ex_disj_distrib:
wenzelm@26286
   885
  "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover
wenzelm@26286
   886
wenzelm@26286
   887
lemma all_conj_distrib:
wenzelm@26286
   888
  "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover
wenzelm@26286
   889
wenzelm@26286
   890
wenzelm@26286
   891
subsection {* Legacy ML bindings *}
paulson@13779
   892
wenzelm@21539
   893
ML {*
wenzelm@22139
   894
val refl = @{thm refl}
wenzelm@22139
   895
val trans = @{thm trans}
wenzelm@22139
   896
val sym = @{thm sym}
wenzelm@22139
   897
val subst = @{thm subst}
wenzelm@22139
   898
val ssubst = @{thm ssubst}
wenzelm@22139
   899
val conjI = @{thm conjI}
wenzelm@22139
   900
val conjE = @{thm conjE}
wenzelm@22139
   901
val conjunct1 = @{thm conjunct1}
wenzelm@22139
   902
val conjunct2 = @{thm conjunct2}
wenzelm@22139
   903
val disjI1 = @{thm disjI1}
wenzelm@22139
   904
val disjI2 = @{thm disjI2}
wenzelm@22139
   905
val disjE = @{thm disjE}
wenzelm@22139
   906
val impI = @{thm impI}
wenzelm@22139
   907
val impE = @{thm impE}
wenzelm@22139
   908
val mp = @{thm mp}
wenzelm@22139
   909
val rev_mp = @{thm rev_mp}
wenzelm@22139
   910
val TrueI = @{thm TrueI}
wenzelm@22139
   911
val FalseE = @{thm FalseE}
wenzelm@22139
   912
val iff_refl = @{thm iff_refl}
wenzelm@22139
   913
val iff_trans = @{thm iff_trans}
wenzelm@22139
   914
val iffI = @{thm iffI}
wenzelm@22139
   915
val iffE = @{thm iffE}
wenzelm@22139
   916
val iffD1 = @{thm iffD1}
wenzelm@22139
   917
val iffD2 = @{thm iffD2}
wenzelm@22139
   918
val notI = @{thm notI}
wenzelm@22139
   919
val notE = @{thm notE}
wenzelm@22139
   920
val allI = @{thm allI}
wenzelm@22139
   921
val allE = @{thm allE}
wenzelm@22139
   922
val spec = @{thm spec}
wenzelm@22139
   923
val exI = @{thm exI}
wenzelm@22139
   924
val exE = @{thm exE}
wenzelm@22139
   925
val eq_reflection = @{thm eq_reflection}
wenzelm@22139
   926
val iff_reflection = @{thm iff_reflection}
wenzelm@22139
   927
val meta_eq_to_obj_eq = @{thm meta_eq_to_obj_eq}
wenzelm@22139
   928
val meta_eq_to_iff = @{thm meta_eq_to_iff}
paulson@13779
   929
*}
paulson@13779
   930
wenzelm@4854
   931
end