src/HOL/HOL.thy
author schirmer
Mon Jan 26 10:34:02 2004 +0100 (2004-01-26)
changeset 14361 ad2f5da643b4
parent 14357 e49d5d5ae66a
child 14365 3d4df8c166ae
permissions -rw-r--r--
* Support for raw latex output in control symbols: \<^raw...>
* Symbols may only start with one backslash: \<...>. \\<...> is no longer
accepted by the scanner.
- Adapted some Isar-theories to fit to this policy
clasohm@923
     1
(*  Title:      HOL/HOL.thy
clasohm@923
     2
    ID:         $Id$
wenzelm@11750
     3
    Author:     Tobias Nipkow, Markus Wenzel, and Larry Paulson
wenzelm@12386
     4
    License:    GPL (GNU GENERAL PUBLIC LICENSE)
wenzelm@11750
     5
*)
clasohm@923
     6
wenzelm@11750
     7
header {* The basis of Higher-Order Logic *}
clasohm@923
     8
wenzelm@7357
     9
theory HOL = CPure
paulson@11451
    10
files ("HOL_lemmas.ML") ("cladata.ML") ("blastdata.ML") ("simpdata.ML"):
clasohm@923
    11
wenzelm@2260
    12
wenzelm@11750
    13
subsection {* Primitive logic *}
wenzelm@11750
    14
wenzelm@11750
    15
subsubsection {* Core syntax *}
wenzelm@2260
    16
wenzelm@12338
    17
classes type < logic
wenzelm@12338
    18
defaultsort type
wenzelm@3947
    19
wenzelm@12338
    20
global
clasohm@923
    21
wenzelm@7357
    22
typedecl bool
clasohm@923
    23
clasohm@923
    24
arities
wenzelm@12338
    25
  bool :: type
wenzelm@12338
    26
  fun :: (type, type) type
clasohm@923
    27
wenzelm@11750
    28
judgment
wenzelm@11750
    29
  Trueprop      :: "bool => prop"                   ("(_)" 5)
clasohm@923
    30
wenzelm@11750
    31
consts
wenzelm@7357
    32
  Not           :: "bool => bool"                   ("~ _" [40] 40)
wenzelm@7357
    33
  True          :: bool
wenzelm@7357
    34
  False         :: bool
wenzelm@7357
    35
  If            :: "[bool, 'a, 'a] => 'a"           ("(if (_)/ then (_)/ else (_))" 10)
wenzelm@3947
    36
  arbitrary     :: 'a
clasohm@923
    37
wenzelm@11432
    38
  The           :: "('a => bool) => 'a"
wenzelm@7357
    39
  All           :: "('a => bool) => bool"           (binder "ALL " 10)
wenzelm@7357
    40
  Ex            :: "('a => bool) => bool"           (binder "EX " 10)
wenzelm@7357
    41
  Ex1           :: "('a => bool) => bool"           (binder "EX! " 10)
wenzelm@7357
    42
  Let           :: "['a, 'a => 'b] => 'b"
clasohm@923
    43
wenzelm@7357
    44
  "="           :: "['a, 'a] => bool"               (infixl 50)
wenzelm@7357
    45
  &             :: "[bool, bool] => bool"           (infixr 35)
wenzelm@7357
    46
  "|"           :: "[bool, bool] => bool"           (infixr 30)
wenzelm@7357
    47
  -->           :: "[bool, bool] => bool"           (infixr 25)
clasohm@923
    48
wenzelm@10432
    49
local
wenzelm@10432
    50
wenzelm@2260
    51
wenzelm@11750
    52
subsubsection {* Additional concrete syntax *}
wenzelm@2260
    53
wenzelm@4868
    54
nonterminals
clasohm@923
    55
  letbinds  letbind
clasohm@923
    56
  case_syn  cases_syn
clasohm@923
    57
clasohm@923
    58
syntax
wenzelm@12650
    59
  "_not_equal"  :: "['a, 'a] => bool"                    (infixl "~=" 50)
wenzelm@11432
    60
  "_The"        :: "[pttrn, bool] => 'a"                 ("(3THE _./ _)" [0, 10] 10)
clasohm@923
    61
wenzelm@7357
    62
  "_bind"       :: "[pttrn, 'a] => letbind"              ("(2_ =/ _)" 10)
wenzelm@7357
    63
  ""            :: "letbind => letbinds"                 ("_")
wenzelm@7357
    64
  "_binds"      :: "[letbind, letbinds] => letbinds"     ("_;/ _")
wenzelm@7357
    65
  "_Let"        :: "[letbinds, 'a] => 'a"                ("(let (_)/ in (_))" 10)
clasohm@923
    66
wenzelm@9060
    67
  "_case_syntax":: "['a, cases_syn] => 'b"               ("(case _ of/ _)" 10)
wenzelm@9060
    68
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ =>/ _)" 10)
wenzelm@7357
    69
  ""            :: "case_syn => cases_syn"               ("_")
wenzelm@9060
    70
  "_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ | _")
clasohm@923
    71
clasohm@923
    72
translations
wenzelm@7238
    73
  "x ~= y"                == "~ (x = y)"
nipkow@13764
    74
  "THE x. P"              == "The (%x. P)"
clasohm@923
    75
  "_Let (_binds b bs) e"  == "_Let b (_Let bs e)"
nipkow@1114
    76
  "let x = a in e"        == "Let a (%x. e)"
clasohm@923
    77
nipkow@13763
    78
print_translation {*
nipkow@13763
    79
(* To avoid eta-contraction of body: *)
nipkow@13763
    80
[("The", fn [Abs abs] =>
nipkow@13763
    81
     let val (x,t) = atomic_abs_tr' abs
nipkow@13763
    82
     in Syntax.const "_The" $ x $ t end)]
nipkow@13763
    83
*}
nipkow@13763
    84
wenzelm@12633
    85
syntax (output)
wenzelm@11687
    86
  "="           :: "['a, 'a] => bool"                    (infix 50)
wenzelm@12650
    87
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "~=" 50)
wenzelm@2260
    88
wenzelm@12114
    89
syntax (xsymbols)
wenzelm@11687
    90
  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
wenzelm@11687
    91
  "op &"        :: "[bool, bool] => bool"                (infixr "\<and>" 35)
wenzelm@11687
    92
  "op |"        :: "[bool, bool] => bool"                (infixr "\<or>" 30)
wenzelm@12114
    93
  "op -->"      :: "[bool, bool] => bool"                (infixr "\<longrightarrow>" 25)
wenzelm@12650
    94
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
wenzelm@11687
    95
  "ALL "        :: "[idts, bool] => bool"                ("(3\<forall>_./ _)" [0, 10] 10)
wenzelm@11687
    96
  "EX "         :: "[idts, bool] => bool"                ("(3\<exists>_./ _)" [0, 10] 10)
wenzelm@11687
    97
  "EX! "        :: "[idts, bool] => bool"                ("(3\<exists>!_./ _)" [0, 10] 10)
wenzelm@11687
    98
  "_case1"      :: "['a, 'b] => case_syn"                ("(2_ \<Rightarrow>/ _)" 10)
schirmer@14361
    99
(*"_case2"      :: "[case_syn, cases_syn] => cases_syn"  ("_/ \<orelse> _")*)
wenzelm@2372
   100
wenzelm@12114
   101
syntax (xsymbols output)
wenzelm@12650
   102
  "_not_equal"  :: "['a, 'a] => bool"                    (infix "\<noteq>" 50)
wenzelm@3820
   103
wenzelm@6340
   104
syntax (HTML output)
wenzelm@11687
   105
  Not           :: "bool => bool"                        ("\<not> _" [40] 40)
wenzelm@6340
   106
wenzelm@7238
   107
syntax (HOL)
wenzelm@7357
   108
  "ALL "        :: "[idts, bool] => bool"                ("(3! _./ _)" [0, 10] 10)
wenzelm@7357
   109
  "EX "         :: "[idts, bool] => bool"                ("(3? _./ _)" [0, 10] 10)
wenzelm@7357
   110
  "EX! "        :: "[idts, bool] => bool"                ("(3?! _./ _)" [0, 10] 10)
wenzelm@7238
   111
wenzelm@7238
   112
wenzelm@11750
   113
subsubsection {* Axioms and basic definitions *}
wenzelm@2260
   114
wenzelm@7357
   115
axioms
wenzelm@7357
   116
  eq_reflection: "(x=y) ==> (x==y)"
clasohm@923
   117
wenzelm@7357
   118
  refl:         "t = (t::'a)"
wenzelm@7357
   119
  subst:        "[| s = t; P(s) |] ==> P(t::'a)"
paulson@6289
   120
wenzelm@7357
   121
  ext:          "(!!x::'a. (f x ::'b) = g x) ==> (%x. f x) = (%x. g x)"
wenzelm@11750
   122
    -- {* Extensionality is built into the meta-logic, and this rule expresses *}
wenzelm@11750
   123
    -- {* a related property.  It is an eta-expanded version of the traditional *}
wenzelm@11750
   124
    -- {* rule, and similar to the ABS rule of HOL *}
paulson@6289
   125
wenzelm@11432
   126
  the_eq_trivial: "(THE x. x = a) = (a::'a)"
clasohm@923
   127
wenzelm@7357
   128
  impI:         "(P ==> Q) ==> P-->Q"
wenzelm@7357
   129
  mp:           "[| P-->Q;  P |] ==> Q"
clasohm@923
   130
clasohm@923
   131
defs
wenzelm@7357
   132
  True_def:     "True      == ((%x::bool. x) = (%x. x))"
wenzelm@7357
   133
  All_def:      "All(P)    == (P = (%x. True))"
paulson@11451
   134
  Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"
wenzelm@7357
   135
  False_def:    "False     == (!P. P)"
wenzelm@7357
   136
  not_def:      "~ P       == P-->False"
wenzelm@7357
   137
  and_def:      "P & Q     == !R. (P-->Q-->R) --> R"
wenzelm@7357
   138
  or_def:       "P | Q     == !R. (P-->R) --> (Q-->R) --> R"
wenzelm@7357
   139
  Ex1_def:      "Ex1(P)    == ? x. P(x) & (! y. P(y) --> y=x)"
clasohm@923
   140
wenzelm@7357
   141
axioms
wenzelm@7357
   142
  iff:          "(P-->Q) --> (Q-->P) --> (P=Q)"
wenzelm@7357
   143
  True_or_False:  "(P=True) | (P=False)"
clasohm@923
   144
clasohm@923
   145
defs
wenzelm@7357
   146
  Let_def:      "Let s f == f(s)"
paulson@11451
   147
  if_def:       "If P x y == THE z::'a. (P=True --> z=x) & (P=False --> z=y)"
wenzelm@5069
   148
skalberg@14223
   149
finalconsts
skalberg@14223
   150
  "op ="
skalberg@14223
   151
  "op -->"
skalberg@14223
   152
  The
skalberg@14223
   153
  arbitrary
nipkow@3320
   154
wenzelm@11750
   155
subsubsection {* Generic algebraic operations *}
wenzelm@4868
   156
wenzelm@12338
   157
axclass zero < type
wenzelm@12338
   158
axclass one < type
wenzelm@12338
   159
axclass plus < type
wenzelm@12338
   160
axclass minus < type
wenzelm@12338
   161
axclass times < type
wenzelm@12338
   162
axclass inverse < type
wenzelm@11750
   163
wenzelm@11750
   164
global
wenzelm@11750
   165
wenzelm@11750
   166
consts
wenzelm@11750
   167
  "0"           :: "'a::zero"                       ("0")
wenzelm@11750
   168
  "1"           :: "'a::one"                        ("1")
wenzelm@11750
   169
  "+"           :: "['a::plus, 'a]  => 'a"          (infixl 65)
wenzelm@11750
   170
  -             :: "['a::minus, 'a] => 'a"          (infixl 65)
wenzelm@11750
   171
  uminus        :: "['a::minus] => 'a"              ("- _" [81] 80)
wenzelm@11750
   172
  *             :: "['a::times, 'a] => 'a"          (infixl 70)
wenzelm@11750
   173
wenzelm@13456
   174
syntax
wenzelm@13456
   175
  "_index1"  :: index    ("\<^sub>1")
wenzelm@13456
   176
translations
wenzelm@13456
   177
  (index) "\<^sub>1" == "_index 1"
wenzelm@13456
   178
wenzelm@11750
   179
local
wenzelm@11750
   180
wenzelm@11750
   181
typed_print_translation {*
wenzelm@11750
   182
  let
wenzelm@11750
   183
    fun tr' c = (c, fn show_sorts => fn T => fn ts =>
wenzelm@11750
   184
      if T = dummyT orelse not (! show_types) andalso can Term.dest_Type T then raise Match
wenzelm@11750
   185
      else Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts T);
wenzelm@11750
   186
  in [tr' "0", tr' "1"] end;
wenzelm@11750
   187
*} -- {* show types that are presumably too general *}
wenzelm@11750
   188
wenzelm@11750
   189
wenzelm@11750
   190
consts
wenzelm@11750
   191
  abs           :: "'a::minus => 'a"
wenzelm@11750
   192
  inverse       :: "'a::inverse => 'a"
wenzelm@11750
   193
  divide        :: "['a::inverse, 'a] => 'a"        (infixl "'/" 70)
wenzelm@11750
   194
wenzelm@11750
   195
syntax (xsymbols)
wenzelm@11750
   196
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
wenzelm@11750
   197
syntax (HTML output)
wenzelm@11750
   198
  abs :: "'a::minus => 'a"    ("\<bar>_\<bar>")
wenzelm@11750
   199
wenzelm@11750
   200
axclass plus_ac0 < plus, zero
wenzelm@11750
   201
  commute: "x + y = y + x"
wenzelm@11750
   202
  assoc:   "(x + y) + z = x + (y + z)"
wenzelm@11750
   203
  zero:    "0 + x = x"
wenzelm@11750
   204
wenzelm@11750
   205
wenzelm@11750
   206
subsection {* Theory and package setup *}
wenzelm@11750
   207
wenzelm@11750
   208
subsubsection {* Basic lemmas *}
wenzelm@4868
   209
nipkow@9736
   210
use "HOL_lemmas.ML"
wenzelm@11687
   211
theorems case_split = case_split_thm [case_names True False]
wenzelm@9869
   212
wenzelm@12386
   213
wenzelm@12386
   214
subsubsection {* Intuitionistic Reasoning *}
wenzelm@12386
   215
wenzelm@12386
   216
lemma impE':
wenzelm@12937
   217
  assumes 1: "P --> Q"
wenzelm@12937
   218
    and 2: "Q ==> R"
wenzelm@12937
   219
    and 3: "P --> Q ==> P"
wenzelm@12937
   220
  shows R
wenzelm@12386
   221
proof -
wenzelm@12386
   222
  from 3 and 1 have P .
wenzelm@12386
   223
  with 1 have Q by (rule impE)
wenzelm@12386
   224
  with 2 show R .
wenzelm@12386
   225
qed
wenzelm@12386
   226
wenzelm@12386
   227
lemma allE':
wenzelm@12937
   228
  assumes 1: "ALL x. P x"
wenzelm@12937
   229
    and 2: "P x ==> ALL x. P x ==> Q"
wenzelm@12937
   230
  shows Q
wenzelm@12386
   231
proof -
wenzelm@12386
   232
  from 1 have "P x" by (rule spec)
wenzelm@12386
   233
  from this and 1 show Q by (rule 2)
wenzelm@12386
   234
qed
wenzelm@12386
   235
wenzelm@12937
   236
lemma notE':
wenzelm@12937
   237
  assumes 1: "~ P"
wenzelm@12937
   238
    and 2: "~ P ==> P"
wenzelm@12937
   239
  shows R
wenzelm@12386
   240
proof -
wenzelm@12386
   241
  from 2 and 1 have P .
wenzelm@12386
   242
  with 1 show R by (rule notE)
wenzelm@12386
   243
qed
wenzelm@12386
   244
wenzelm@12386
   245
lemmas [CPure.elim!] = disjE iffE FalseE conjE exE
wenzelm@12386
   246
  and [CPure.intro!] = iffI conjI impI TrueI notI allI refl
wenzelm@12386
   247
  and [CPure.elim 2] = allE notE' impE'
wenzelm@12386
   248
  and [CPure.intro] = exI disjI2 disjI1
wenzelm@12386
   249
wenzelm@12386
   250
lemmas [trans] = trans
wenzelm@12386
   251
  and [sym] = sym not_sym
wenzelm@12386
   252
  and [CPure.elim?] = iffD1 iffD2 impE
wenzelm@11750
   253
wenzelm@11438
   254
wenzelm@11750
   255
subsubsection {* Atomizing meta-level connectives *}
wenzelm@11750
   256
wenzelm@11750
   257
lemma atomize_all [atomize]: "(!!x. P x) == Trueprop (ALL x. P x)"
wenzelm@12003
   258
proof
wenzelm@9488
   259
  assume "!!x. P x"
wenzelm@10383
   260
  show "ALL x. P x" by (rule allI)
wenzelm@9488
   261
next
wenzelm@9488
   262
  assume "ALL x. P x"
wenzelm@10383
   263
  thus "!!x. P x" by (rule allE)
wenzelm@9488
   264
qed
wenzelm@9488
   265
wenzelm@11750
   266
lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)"
wenzelm@12003
   267
proof
wenzelm@9488
   268
  assume r: "A ==> B"
wenzelm@10383
   269
  show "A --> B" by (rule impI) (rule r)
wenzelm@9488
   270
next
wenzelm@9488
   271
  assume "A --> B" and A
wenzelm@10383
   272
  thus B by (rule mp)
wenzelm@9488
   273
qed
wenzelm@9488
   274
wenzelm@11750
   275
lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)"
wenzelm@12003
   276
proof
wenzelm@10432
   277
  assume "x == y"
wenzelm@10432
   278
  show "x = y" by (unfold prems) (rule refl)
wenzelm@10432
   279
next
wenzelm@10432
   280
  assume "x = y"
wenzelm@10432
   281
  thus "x == y" by (rule eq_reflection)
wenzelm@10432
   282
qed
wenzelm@10432
   283
wenzelm@12023
   284
lemma atomize_conj [atomize]:
wenzelm@12023
   285
  "(!!C. (A ==> B ==> PROP C) ==> PROP C) == Trueprop (A & B)"
wenzelm@12003
   286
proof
wenzelm@11953
   287
  assume "!!C. (A ==> B ==> PROP C) ==> PROP C"
wenzelm@11953
   288
  show "A & B" by (rule conjI)
wenzelm@11953
   289
next
wenzelm@11953
   290
  fix C
wenzelm@11953
   291
  assume "A & B"
wenzelm@11953
   292
  assume "A ==> B ==> PROP C"
wenzelm@11953
   293
  thus "PROP C"
wenzelm@11953
   294
  proof this
wenzelm@11953
   295
    show A by (rule conjunct1)
wenzelm@11953
   296
    show B by (rule conjunct2)
wenzelm@11953
   297
  qed
wenzelm@11953
   298
qed
wenzelm@11953
   299
wenzelm@12386
   300
lemmas [symmetric, rulify] = atomize_all atomize_imp
wenzelm@12386
   301
wenzelm@11750
   302
wenzelm@11750
   303
subsubsection {* Classical Reasoner setup *}
wenzelm@9529
   304
wenzelm@10383
   305
use "cladata.ML"
wenzelm@10383
   306
setup hypsubst_setup
wenzelm@11977
   307
wenzelm@12386
   308
ML_setup {*
wenzelm@12386
   309
  Context.>> (ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac));
wenzelm@12386
   310
*}
wenzelm@11977
   311
wenzelm@10383
   312
setup Classical.setup
wenzelm@10383
   313
setup clasetup
wenzelm@10383
   314
wenzelm@12386
   315
lemmas [intro?] = ext
wenzelm@12386
   316
  and [elim?] = ex1_implies_ex
wenzelm@11977
   317
wenzelm@9869
   318
use "blastdata.ML"
wenzelm@9869
   319
setup Blast.setup
wenzelm@4868
   320
wenzelm@11750
   321
wenzelm@11750
   322
subsubsection {* Simplifier setup *}
wenzelm@11750
   323
wenzelm@12281
   324
lemma meta_eq_to_obj_eq: "x == y ==> x = y"
wenzelm@12281
   325
proof -
wenzelm@12281
   326
  assume r: "x == y"
wenzelm@12281
   327
  show "x = y" by (unfold r) (rule refl)
wenzelm@12281
   328
qed
wenzelm@12281
   329
wenzelm@12281
   330
lemma eta_contract_eq: "(%s. f s) = f" ..
wenzelm@12281
   331
wenzelm@12281
   332
lemma simp_thms:
wenzelm@12937
   333
  shows not_not: "(~ ~ P) = P"
wenzelm@12937
   334
  and
berghofe@12436
   335
    "(P ~= Q) = (P = (~Q))"
berghofe@12436
   336
    "(P | ~P) = True"    "(~P | P) = True"
berghofe@12436
   337
    "((~P) = (~Q)) = (P=Q)"
wenzelm@12281
   338
    "(x = x) = True"
wenzelm@12281
   339
    "(~True) = False"  "(~False) = True"
berghofe@12436
   340
    "(~P) ~= P"  "P ~= (~P)"
wenzelm@12281
   341
    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
wenzelm@12281
   342
    "(True --> P) = P"  "(False --> P) = True"
wenzelm@12281
   343
    "(P --> True) = True"  "(P --> P) = True"
wenzelm@12281
   344
    "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
wenzelm@12281
   345
    "(P & True) = P"  "(True & P) = P"
wenzelm@12281
   346
    "(P & False) = False"  "(False & P) = False"
wenzelm@12281
   347
    "(P & P) = P"  "(P & (P & Q)) = (P & Q)"
wenzelm@12281
   348
    "(P & ~P) = False"    "(~P & P) = False"
wenzelm@12281
   349
    "(P | True) = True"  "(True | P) = True"
wenzelm@12281
   350
    "(P | False) = P"  "(False | P) = P"
berghofe@12436
   351
    "(P | P) = P"  "(P | (P | Q)) = (P | Q)" and
wenzelm@12281
   352
    "(ALL x. P) = P"  "(EX x. P) = P"  "EX x. x=t"  "EX x. t=x"
wenzelm@12281
   353
    -- {* needed for the one-point-rule quantifier simplification procs *}
wenzelm@12281
   354
    -- {* essential for termination!! *} and
wenzelm@12281
   355
    "!!P. (EX x. x=t & P(x)) = P(t)"
wenzelm@12281
   356
    "!!P. (EX x. t=x & P(x)) = P(t)"
wenzelm@12281
   357
    "!!P. (ALL x. x=t --> P(x)) = P(t)"
wenzelm@12937
   358
    "!!P. (ALL x. t=x --> P(x)) = P(t)"
berghofe@12436
   359
  by (blast, blast, blast, blast, blast, rules+)
wenzelm@13421
   360
wenzelm@12281
   361
lemma imp_cong: "(P = P') ==> (P' ==> (Q = Q')) ==> ((P --> Q) = (P' --> Q'))"
wenzelm@12354
   362
  by rules
wenzelm@12281
   363
wenzelm@12281
   364
lemma ex_simps:
wenzelm@12281
   365
  "!!P Q. (EX x. P x & Q)   = ((EX x. P x) & Q)"
wenzelm@12281
   366
  "!!P Q. (EX x. P & Q x)   = (P & (EX x. Q x))"
wenzelm@12281
   367
  "!!P Q. (EX x. P x | Q)   = ((EX x. P x) | Q)"
wenzelm@12281
   368
  "!!P Q. (EX x. P | Q x)   = (P | (EX x. Q x))"
wenzelm@12281
   369
  "!!P Q. (EX x. P x --> Q) = ((ALL x. P x) --> Q)"
wenzelm@12281
   370
  "!!P Q. (EX x. P --> Q x) = (P --> (EX x. Q x))"
wenzelm@12281
   371
  -- {* Miniscoping: pushing in existential quantifiers. *}
berghofe@12436
   372
  by (rules | blast)+
wenzelm@12281
   373
wenzelm@12281
   374
lemma all_simps:
wenzelm@12281
   375
  "!!P Q. (ALL x. P x & Q)   = ((ALL x. P x) & Q)"
wenzelm@12281
   376
  "!!P Q. (ALL x. P & Q x)   = (P & (ALL x. Q x))"
wenzelm@12281
   377
  "!!P Q. (ALL x. P x | Q)   = ((ALL x. P x) | Q)"
wenzelm@12281
   378
  "!!P Q. (ALL x. P | Q x)   = (P | (ALL x. Q x))"
wenzelm@12281
   379
  "!!P Q. (ALL x. P x --> Q) = ((EX x. P x) --> Q)"
wenzelm@12281
   380
  "!!P Q. (ALL x. P --> Q x) = (P --> (ALL x. Q x))"
wenzelm@12281
   381
  -- {* Miniscoping: pushing in universal quantifiers. *}
berghofe@12436
   382
  by (rules | blast)+
wenzelm@12281
   383
paulson@14201
   384
lemma disj_absorb: "(A | A) = A"
paulson@14201
   385
  by blast
paulson@14201
   386
paulson@14201
   387
lemma disj_left_absorb: "(A | (A | B)) = (A | B)"
paulson@14201
   388
  by blast
paulson@14201
   389
paulson@14201
   390
lemma conj_absorb: "(A & A) = A"
paulson@14201
   391
  by blast
paulson@14201
   392
paulson@14201
   393
lemma conj_left_absorb: "(A & (A & B)) = (A & B)"
paulson@14201
   394
  by blast
paulson@14201
   395
wenzelm@12281
   396
lemma eq_ac:
wenzelm@12937
   397
  shows eq_commute: "(a=b) = (b=a)"
wenzelm@12937
   398
    and eq_left_commute: "(P=(Q=R)) = (Q=(P=R))"
wenzelm@12937
   399
    and eq_assoc: "((P=Q)=R) = (P=(Q=R))" by (rules, blast+)
berghofe@12436
   400
lemma neq_commute: "(a~=b) = (b~=a)" by rules
wenzelm@12281
   401
wenzelm@12281
   402
lemma conj_comms:
wenzelm@12937
   403
  shows conj_commute: "(P&Q) = (Q&P)"
wenzelm@12937
   404
    and conj_left_commute: "(P&(Q&R)) = (Q&(P&R))" by rules+
berghofe@12436
   405
lemma conj_assoc: "((P&Q)&R) = (P&(Q&R))" by rules
wenzelm@12281
   406
wenzelm@12281
   407
lemma disj_comms:
wenzelm@12937
   408
  shows disj_commute: "(P|Q) = (Q|P)"
wenzelm@12937
   409
    and disj_left_commute: "(P|(Q|R)) = (Q|(P|R))" by rules+
berghofe@12436
   410
lemma disj_assoc: "((P|Q)|R) = (P|(Q|R))" by rules
wenzelm@12281
   411
berghofe@12436
   412
lemma conj_disj_distribL: "(P&(Q|R)) = (P&Q | P&R)" by rules
berghofe@12436
   413
lemma conj_disj_distribR: "((P|Q)&R) = (P&R | Q&R)" by rules
wenzelm@12281
   414
berghofe@12436
   415
lemma disj_conj_distribL: "(P|(Q&R)) = ((P|Q) & (P|R))" by rules
berghofe@12436
   416
lemma disj_conj_distribR: "((P&Q)|R) = ((P|R) & (Q|R))" by rules
wenzelm@12281
   417
berghofe@12436
   418
lemma imp_conjR: "(P --> (Q&R)) = ((P-->Q) & (P-->R))" by rules
berghofe@12436
   419
lemma imp_conjL: "((P&Q) -->R)  = (P --> (Q --> R))" by rules
berghofe@12436
   420
lemma imp_disjL: "((P|Q) --> R) = ((P-->R)&(Q-->R))" by rules
wenzelm@12281
   421
wenzelm@12281
   422
text {* These two are specialized, but @{text imp_disj_not1} is useful in @{text "Auth/Yahalom"}. *}
wenzelm@12281
   423
lemma imp_disj_not1: "(P --> Q | R) = (~Q --> P --> R)" by blast
wenzelm@12281
   424
lemma imp_disj_not2: "(P --> Q | R) = (~R --> P --> Q)" by blast
wenzelm@12281
   425
wenzelm@12281
   426
lemma imp_disj1: "((P-->Q)|R) = (P--> Q|R)" by blast
wenzelm@12281
   427
lemma imp_disj2: "(Q|(P-->R)) = (P--> Q|R)" by blast
wenzelm@12281
   428
berghofe@12436
   429
lemma de_Morgan_disj: "(~(P | Q)) = (~P & ~Q)" by rules
wenzelm@12281
   430
lemma de_Morgan_conj: "(~(P & Q)) = (~P | ~Q)" by blast
wenzelm@12281
   431
lemma not_imp: "(~(P --> Q)) = (P & ~Q)" by blast
wenzelm@12281
   432
lemma not_iff: "(P~=Q) = (P = (~Q))" by blast
wenzelm@12281
   433
lemma disj_not1: "(~P | Q) = (P --> Q)" by blast
wenzelm@12281
   434
lemma disj_not2: "(P | ~Q) = (Q --> P)"  -- {* changes orientation :-( *}
wenzelm@12281
   435
  by blast
wenzelm@12281
   436
lemma imp_conv_disj: "(P --> Q) = ((~P) | Q)" by blast
wenzelm@12281
   437
berghofe@12436
   438
lemma iff_conv_conj_imp: "(P = Q) = ((P --> Q) & (Q --> P))" by rules
wenzelm@12281
   439
wenzelm@12281
   440
wenzelm@12281
   441
lemma cases_simp: "((P --> Q) & (~P --> Q)) = Q"
wenzelm@12281
   442
  -- {* Avoids duplication of subgoals after @{text split_if}, when the true and false *}
wenzelm@12281
   443
  -- {* cases boil down to the same thing. *}
wenzelm@12281
   444
  by blast
wenzelm@12281
   445
wenzelm@12281
   446
lemma not_all: "(~ (! x. P(x))) = (? x.~P(x))" by blast
wenzelm@12281
   447
lemma imp_all: "((! x. P x) --> Q) = (? x. P x --> Q)" by blast
berghofe@12436
   448
lemma not_ex: "(~ (? x. P(x))) = (! x.~P(x))" by rules
berghofe@12436
   449
lemma imp_ex: "((? x. P x) --> Q) = (! x. P x --> Q)" by rules
wenzelm@12281
   450
berghofe@12436
   451
lemma ex_disj_distrib: "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))" by rules
berghofe@12436
   452
lemma all_conj_distrib: "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))" by rules
wenzelm@12281
   453
wenzelm@12281
   454
text {*
wenzelm@12281
   455
  \medskip The @{text "&"} congruence rule: not included by default!
wenzelm@12281
   456
  May slow rewrite proofs down by as much as 50\% *}
wenzelm@12281
   457
wenzelm@12281
   458
lemma conj_cong:
wenzelm@12281
   459
    "(P = P') ==> (P' ==> (Q = Q')) ==> ((P & Q) = (P' & Q'))"
wenzelm@12354
   460
  by rules
wenzelm@12281
   461
wenzelm@12281
   462
lemma rev_conj_cong:
wenzelm@12281
   463
    "(Q = Q') ==> (Q' ==> (P = P')) ==> ((P & Q) = (P' & Q'))"
wenzelm@12354
   464
  by rules
wenzelm@12281
   465
wenzelm@12281
   466
text {* The @{text "|"} congruence rule: not included by default! *}
wenzelm@12281
   467
wenzelm@12281
   468
lemma disj_cong:
wenzelm@12281
   469
    "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
wenzelm@12281
   470
  by blast
wenzelm@12281
   471
wenzelm@12281
   472
lemma eq_sym_conv: "(x = y) = (y = x)"
wenzelm@12354
   473
  by rules
wenzelm@12281
   474
wenzelm@12281
   475
wenzelm@12281
   476
text {* \medskip if-then-else rules *}
wenzelm@12281
   477
wenzelm@12281
   478
lemma if_True: "(if True then x else y) = x"
wenzelm@12281
   479
  by (unfold if_def) blast
wenzelm@12281
   480
wenzelm@12281
   481
lemma if_False: "(if False then x else y) = y"
wenzelm@12281
   482
  by (unfold if_def) blast
wenzelm@12281
   483
wenzelm@12281
   484
lemma if_P: "P ==> (if P then x else y) = x"
wenzelm@12281
   485
  by (unfold if_def) blast
wenzelm@12281
   486
wenzelm@12281
   487
lemma if_not_P: "~P ==> (if P then x else y) = y"
wenzelm@12281
   488
  by (unfold if_def) blast
wenzelm@12281
   489
wenzelm@12281
   490
lemma split_if: "P (if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
wenzelm@12281
   491
  apply (rule case_split [of Q])
wenzelm@12281
   492
   apply (subst if_P)
paulson@14208
   493
    prefer 3 apply (subst if_not_P, blast+)
wenzelm@12281
   494
  done
wenzelm@12281
   495
wenzelm@12281
   496
lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
paulson@14208
   497
by (subst split_if, blast)
wenzelm@12281
   498
wenzelm@12281
   499
lemmas if_splits = split_if split_if_asm
wenzelm@12281
   500
wenzelm@12281
   501
lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
wenzelm@12281
   502
  by (rule split_if)
wenzelm@12281
   503
wenzelm@12281
   504
lemma if_cancel: "(if c then x else x) = x"
paulson@14208
   505
by (subst split_if, blast)
wenzelm@12281
   506
wenzelm@12281
   507
lemma if_eq_cancel: "(if x = y then y else x) = x"
paulson@14208
   508
by (subst split_if, blast)
wenzelm@12281
   509
wenzelm@12281
   510
lemma if_bool_eq_conj: "(if P then Q else R) = ((P-->Q) & (~P-->R))"
wenzelm@12281
   511
  -- {* This form is useful for expanding @{text if}s on the RIGHT of the @{text "==>"} symbol. *}
wenzelm@12281
   512
  by (rule split_if)
wenzelm@12281
   513
wenzelm@12281
   514
lemma if_bool_eq_disj: "(if P then Q else R) = ((P&Q) | (~P&R))"
wenzelm@12281
   515
  -- {* And this form is useful for expanding @{text if}s on the LEFT. *}
paulson@14208
   516
  apply (subst split_if, blast)
wenzelm@12281
   517
  done
wenzelm@12281
   518
berghofe@12436
   519
lemma Eq_TrueI: "P ==> P == True" by (unfold atomize_eq) rules
berghofe@12436
   520
lemma Eq_FalseI: "~P ==> P == False" by (unfold atomize_eq) rules
wenzelm@12281
   521
paulson@14201
   522
subsubsection {* Actual Installation of the Simplifier *}
paulson@14201
   523
wenzelm@9869
   524
use "simpdata.ML"
wenzelm@9869
   525
setup Simplifier.setup
wenzelm@9869
   526
setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
wenzelm@9869
   527
setup Splitter.setup setup Clasimp.setup
wenzelm@9869
   528
paulson@14201
   529
declare disj_absorb [simp] conj_absorb [simp] 
paulson@14201
   530
nipkow@13723
   531
lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
nipkow@13723
   532
by blast+
nipkow@13723
   533
berghofe@13638
   534
theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
berghofe@13638
   535
  apply (rule iffI)
berghofe@13638
   536
  apply (rule_tac a = "%x. THE y. P x y" in ex1I)
berghofe@13638
   537
  apply (fast dest!: theI')
berghofe@13638
   538
  apply (fast intro: ext the1_equality [symmetric])
berghofe@13638
   539
  apply (erule ex1E)
berghofe@13638
   540
  apply (rule allI)
berghofe@13638
   541
  apply (rule ex1I)
berghofe@13638
   542
  apply (erule spec)
berghofe@13638
   543
  apply (erule_tac x = "%z. if z = x then y else f z" in allE)
berghofe@13638
   544
  apply (erule impE)
berghofe@13638
   545
  apply (rule allI)
berghofe@13638
   546
  apply (rule_tac P = "xa = x" in case_split_thm)
paulson@14208
   547
  apply (drule_tac [3] x = x in fun_cong, simp_all)
berghofe@13638
   548
  done
berghofe@13638
   549
nipkow@13438
   550
text{*Needs only HOL-lemmas:*}
nipkow@13438
   551
lemma mk_left_commute:
nipkow@13438
   552
  assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
nipkow@13438
   553
          c: "\<And>x y. f x y = f y x"
nipkow@13438
   554
  shows "f x (f y z) = f y (f x z)"
nipkow@13438
   555
by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
nipkow@13438
   556
wenzelm@11750
   557
wenzelm@11824
   558
subsubsection {* Generic cases and induction *}
wenzelm@11824
   559
wenzelm@11824
   560
constdefs
wenzelm@11989
   561
  induct_forall :: "('a => bool) => bool"
wenzelm@11989
   562
  "induct_forall P == \<forall>x. P x"
wenzelm@11989
   563
  induct_implies :: "bool => bool => bool"
wenzelm@11989
   564
  "induct_implies A B == A --> B"
wenzelm@11989
   565
  induct_equal :: "'a => 'a => bool"
wenzelm@11989
   566
  "induct_equal x y == x = y"
wenzelm@11989
   567
  induct_conj :: "bool => bool => bool"
wenzelm@11989
   568
  "induct_conj A B == A & B"
wenzelm@11824
   569
wenzelm@11989
   570
lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
wenzelm@11989
   571
  by (simp only: atomize_all induct_forall_def)
wenzelm@11824
   572
wenzelm@11989
   573
lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
wenzelm@11989
   574
  by (simp only: atomize_imp induct_implies_def)
wenzelm@11824
   575
wenzelm@11989
   576
lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
wenzelm@11989
   577
  by (simp only: atomize_eq induct_equal_def)
wenzelm@11824
   578
wenzelm@11989
   579
lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
wenzelm@11989
   580
    induct_conj (induct_forall A) (induct_forall B)"
wenzelm@12354
   581
  by (unfold induct_forall_def induct_conj_def) rules
wenzelm@11824
   582
wenzelm@11989
   583
lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
wenzelm@11989
   584
    induct_conj (induct_implies C A) (induct_implies C B)"
wenzelm@12354
   585
  by (unfold induct_implies_def induct_conj_def) rules
wenzelm@11989
   586
berghofe@13598
   587
lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
berghofe@13598
   588
proof
berghofe@13598
   589
  assume r: "induct_conj A B ==> PROP C" and A B
berghofe@13598
   590
  show "PROP C" by (rule r) (simp! add: induct_conj_def)
berghofe@13598
   591
next
berghofe@13598
   592
  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
berghofe@13598
   593
  show "PROP C" by (rule r) (simp! add: induct_conj_def)+
berghofe@13598
   594
qed
wenzelm@11824
   595
wenzelm@11989
   596
lemma induct_impliesI: "(A ==> B) ==> induct_implies A B"
wenzelm@11989
   597
  by (simp add: induct_implies_def)
wenzelm@11824
   598
wenzelm@12161
   599
lemmas induct_atomize = atomize_conj induct_forall_eq induct_implies_eq induct_equal_eq
wenzelm@12161
   600
lemmas induct_rulify1 [symmetric, standard] = induct_forall_eq induct_implies_eq induct_equal_eq
wenzelm@12161
   601
lemmas induct_rulify2 = induct_forall_def induct_implies_def induct_equal_def induct_conj_def
wenzelm@11989
   602
lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
wenzelm@11824
   603
wenzelm@11989
   604
hide const induct_forall induct_implies induct_equal induct_conj
wenzelm@11824
   605
wenzelm@11824
   606
wenzelm@11824
   607
text {* Method setup. *}
wenzelm@11824
   608
wenzelm@11824
   609
ML {*
wenzelm@11824
   610
  structure InductMethod = InductMethodFun
wenzelm@11824
   611
  (struct
wenzelm@11824
   612
    val dest_concls = HOLogic.dest_concls;
wenzelm@11824
   613
    val cases_default = thm "case_split";
wenzelm@11989
   614
    val local_impI = thm "induct_impliesI";
wenzelm@11824
   615
    val conjI = thm "conjI";
wenzelm@11989
   616
    val atomize = thms "induct_atomize";
wenzelm@11989
   617
    val rulify1 = thms "induct_rulify1";
wenzelm@11989
   618
    val rulify2 = thms "induct_rulify2";
wenzelm@12240
   619
    val localize = [Thm.symmetric (thm "induct_implies_def")];
wenzelm@11824
   620
  end);
wenzelm@11824
   621
*}
wenzelm@11824
   622
wenzelm@11824
   623
setup InductMethod.setup
wenzelm@11824
   624
wenzelm@11824
   625
wenzelm@11750
   626
subsection {* Order signatures and orders *}
wenzelm@11750
   627
wenzelm@11750
   628
axclass
wenzelm@12338
   629
  ord < type
wenzelm@11750
   630
wenzelm@11750
   631
syntax
wenzelm@11750
   632
  "op <"        :: "['a::ord, 'a] => bool"             ("op <")
wenzelm@11750
   633
  "op <="       :: "['a::ord, 'a] => bool"             ("op <=")
wenzelm@11750
   634
wenzelm@11750
   635
global
wenzelm@11750
   636
wenzelm@11750
   637
consts
wenzelm@11750
   638
  "op <"        :: "['a::ord, 'a] => bool"             ("(_/ < _)"  [50, 51] 50)
wenzelm@11750
   639
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ <= _)" [50, 51] 50)
wenzelm@11750
   640
wenzelm@11750
   641
local
wenzelm@11750
   642
wenzelm@12114
   643
syntax (xsymbols)
wenzelm@11750
   644
  "op <="       :: "['a::ord, 'a] => bool"             ("op \<le>")
wenzelm@11750
   645
  "op <="       :: "['a::ord, 'a] => bool"             ("(_/ \<le> _)"  [50, 51] 50)
wenzelm@11750
   646
wenzelm@11750
   647
paulson@14295
   648
lemma Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
paulson@14295
   649
by blast
paulson@14295
   650
wenzelm@11750
   651
subsubsection {* Monotonicity *}
wenzelm@11750
   652
wenzelm@13412
   653
locale mono =
wenzelm@13412
   654
  fixes f
wenzelm@13412
   655
  assumes mono: "A <= B ==> f A <= f B"
wenzelm@11750
   656
wenzelm@13421
   657
lemmas monoI [intro?] = mono.intro
wenzelm@13412
   658
  and monoD [dest?] = mono.mono
wenzelm@11750
   659
wenzelm@11750
   660
constdefs
wenzelm@11750
   661
  min :: "['a::ord, 'a] => 'a"
wenzelm@11750
   662
  "min a b == (if a <= b then a else b)"
wenzelm@11750
   663
  max :: "['a::ord, 'a] => 'a"
wenzelm@11750
   664
  "max a b == (if a <= b then b else a)"
wenzelm@11750
   665
wenzelm@11750
   666
lemma min_leastL: "(!!x. least <= x) ==> min least x = least"
wenzelm@11750
   667
  by (simp add: min_def)
wenzelm@11750
   668
wenzelm@11750
   669
lemma min_of_mono:
wenzelm@11750
   670
    "ALL x y. (f x <= f y) = (x <= y) ==> min (f m) (f n) = f (min m n)"
wenzelm@11750
   671
  by (simp add: min_def)
wenzelm@11750
   672
wenzelm@11750
   673
lemma max_leastL: "(!!x. least <= x) ==> max least x = x"
wenzelm@11750
   674
  by (simp add: max_def)
wenzelm@11750
   675
wenzelm@11750
   676
lemma max_of_mono:
wenzelm@11750
   677
    "ALL x y. (f x <= f y) = (x <= y) ==> max (f m) (f n) = f (max m n)"
wenzelm@11750
   678
  by (simp add: max_def)
wenzelm@11750
   679
wenzelm@11750
   680
wenzelm@11750
   681
subsubsection "Orders"
wenzelm@11750
   682
wenzelm@11750
   683
axclass order < ord
wenzelm@11750
   684
  order_refl [iff]: "x <= x"
wenzelm@11750
   685
  order_trans: "x <= y ==> y <= z ==> x <= z"
wenzelm@11750
   686
  order_antisym: "x <= y ==> y <= x ==> x = y"
wenzelm@11750
   687
  order_less_le: "(x < y) = (x <= y & x ~= y)"
wenzelm@11750
   688
wenzelm@11750
   689
wenzelm@11750
   690
text {* Reflexivity. *}
wenzelm@11750
   691
wenzelm@11750
   692
lemma order_eq_refl: "!!x::'a::order. x = y ==> x <= y"
wenzelm@11750
   693
    -- {* This form is useful with the classical reasoner. *}
wenzelm@11750
   694
  apply (erule ssubst)
wenzelm@11750
   695
  apply (rule order_refl)
wenzelm@11750
   696
  done
wenzelm@11750
   697
nipkow@13553
   698
lemma order_less_irrefl [iff]: "~ x < (x::'a::order)"
wenzelm@11750
   699
  by (simp add: order_less_le)
wenzelm@11750
   700
wenzelm@11750
   701
lemma order_le_less: "((x::'a::order) <= y) = (x < y | x = y)"
wenzelm@11750
   702
    -- {* NOT suitable for iff, since it can cause PROOF FAILED. *}
paulson@14208
   703
  apply (simp add: order_less_le, blast)
wenzelm@11750
   704
  done
wenzelm@11750
   705
wenzelm@11750
   706
lemmas order_le_imp_less_or_eq = order_le_less [THEN iffD1, standard]
wenzelm@11750
   707
wenzelm@11750
   708
lemma order_less_imp_le: "!!x::'a::order. x < y ==> x <= y"
wenzelm@11750
   709
  by (simp add: order_less_le)
wenzelm@11750
   710
wenzelm@11750
   711
wenzelm@11750
   712
text {* Asymmetry. *}
wenzelm@11750
   713
wenzelm@11750
   714
lemma order_less_not_sym: "(x::'a::order) < y ==> ~ (y < x)"
wenzelm@11750
   715
  by (simp add: order_less_le order_antisym)
wenzelm@11750
   716
wenzelm@11750
   717
lemma order_less_asym: "x < (y::'a::order) ==> (~P ==> y < x) ==> P"
wenzelm@11750
   718
  apply (drule order_less_not_sym)
paulson@14208
   719
  apply (erule contrapos_np, simp)
wenzelm@11750
   720
  done
wenzelm@11750
   721
paulson@14295
   722
lemma order_eq_iff: "!!x::'a::order. (x = y) = (x \<le> y & y \<le> x)"  
paulson@14295
   723
by (blast intro: order_antisym)
paulson@14295
   724
wenzelm@11750
   725
wenzelm@11750
   726
text {* Transitivity. *}
wenzelm@11750
   727
wenzelm@11750
   728
lemma order_less_trans: "!!x::'a::order. [| x < y; y < z |] ==> x < z"
wenzelm@11750
   729
  apply (simp add: order_less_le)
wenzelm@11750
   730
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   731
  done
wenzelm@11750
   732
wenzelm@11750
   733
lemma order_le_less_trans: "!!x::'a::order. [| x <= y; y < z |] ==> x < z"
wenzelm@11750
   734
  apply (simp add: order_less_le)
wenzelm@11750
   735
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   736
  done
wenzelm@11750
   737
wenzelm@11750
   738
lemma order_less_le_trans: "!!x::'a::order. [| x < y; y <= z |] ==> x < z"
wenzelm@11750
   739
  apply (simp add: order_less_le)
wenzelm@11750
   740
  apply (blast intro: order_trans order_antisym)
wenzelm@11750
   741
  done
wenzelm@11750
   742
wenzelm@11750
   743
wenzelm@11750
   744
text {* Useful for simplification, but too risky to include by default. *}
wenzelm@11750
   745
wenzelm@11750
   746
lemma order_less_imp_not_less: "(x::'a::order) < y ==>  (~ y < x) = True"
wenzelm@11750
   747
  by (blast elim: order_less_asym)
wenzelm@11750
   748
wenzelm@11750
   749
lemma order_less_imp_triv: "(x::'a::order) < y ==>  (y < x --> P) = True"
wenzelm@11750
   750
  by (blast elim: order_less_asym)
wenzelm@11750
   751
wenzelm@11750
   752
lemma order_less_imp_not_eq: "(x::'a::order) < y ==>  (x = y) = False"
wenzelm@11750
   753
  by auto
wenzelm@11750
   754
wenzelm@11750
   755
lemma order_less_imp_not_eq2: "(x::'a::order) < y ==>  (y = x) = False"
wenzelm@11750
   756
  by auto
wenzelm@11750
   757
wenzelm@11750
   758
wenzelm@11750
   759
text {* Other operators. *}
wenzelm@11750
   760
wenzelm@11750
   761
lemma min_leastR: "(!!x::'a::order. least <= x) ==> min x least = least"
wenzelm@11750
   762
  apply (simp add: min_def)
wenzelm@11750
   763
  apply (blast intro: order_antisym)
wenzelm@11750
   764
  done
wenzelm@11750
   765
wenzelm@11750
   766
lemma max_leastR: "(!!x::'a::order. least <= x) ==> max x least = x"
wenzelm@11750
   767
  apply (simp add: max_def)
wenzelm@11750
   768
  apply (blast intro: order_antisym)
wenzelm@11750
   769
  done
wenzelm@11750
   770
wenzelm@11750
   771
wenzelm@11750
   772
subsubsection {* Least value operator *}
wenzelm@11750
   773
wenzelm@11750
   774
constdefs
wenzelm@11750
   775
  Least :: "('a::ord => bool) => 'a"               (binder "LEAST " 10)
wenzelm@11750
   776
  "Least P == THE x. P x & (ALL y. P y --> x <= y)"
wenzelm@11750
   777
    -- {* We can no longer use LeastM because the latter requires Hilbert-AC. *}
wenzelm@11750
   778
wenzelm@11750
   779
lemma LeastI2:
wenzelm@11750
   780
  "[| P (x::'a::order);
wenzelm@11750
   781
      !!y. P y ==> x <= y;
wenzelm@11750
   782
      !!x. [| P x; ALL y. P y --> x \<le> y |] ==> Q x |]
wenzelm@12281
   783
   ==> Q (Least P)"
wenzelm@11750
   784
  apply (unfold Least_def)
wenzelm@11750
   785
  apply (rule theI2)
wenzelm@11750
   786
    apply (blast intro: order_antisym)+
wenzelm@11750
   787
  done
wenzelm@11750
   788
wenzelm@11750
   789
lemma Least_equality:
wenzelm@12281
   790
    "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k"
wenzelm@11750
   791
  apply (simp add: Least_def)
wenzelm@11750
   792
  apply (rule the_equality)
wenzelm@11750
   793
  apply (auto intro!: order_antisym)
wenzelm@11750
   794
  done
wenzelm@11750
   795
wenzelm@11750
   796
wenzelm@11750
   797
subsubsection "Linear / total orders"
wenzelm@11750
   798
wenzelm@11750
   799
axclass linorder < order
wenzelm@11750
   800
  linorder_linear: "x <= y | y <= x"
wenzelm@11750
   801
wenzelm@11750
   802
lemma linorder_less_linear: "!!x::'a::linorder. x<y | x=y | y<x"
wenzelm@11750
   803
  apply (simp add: order_less_le)
paulson@14208
   804
  apply (insert linorder_linear, blast)
wenzelm@11750
   805
  done
wenzelm@11750
   806
wenzelm@11750
   807
lemma linorder_cases [case_names less equal greater]:
wenzelm@11750
   808
    "((x::'a::linorder) < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P"
paulson@14208
   809
  apply (insert linorder_less_linear, blast)
wenzelm@11750
   810
  done
wenzelm@11750
   811
wenzelm@11750
   812
lemma linorder_not_less: "!!x::'a::linorder. (~ x < y) = (y <= x)"
wenzelm@11750
   813
  apply (simp add: order_less_le)
wenzelm@11750
   814
  apply (insert linorder_linear)
wenzelm@11750
   815
  apply (blast intro: order_antisym)
wenzelm@11750
   816
  done
wenzelm@11750
   817
wenzelm@11750
   818
lemma linorder_not_le: "!!x::'a::linorder. (~ x <= y) = (y < x)"
wenzelm@11750
   819
  apply (simp add: order_less_le)
wenzelm@11750
   820
  apply (insert linorder_linear)
wenzelm@11750
   821
  apply (blast intro: order_antisym)
wenzelm@11750
   822
  done
wenzelm@11750
   823
wenzelm@11750
   824
lemma linorder_neq_iff: "!!x::'a::linorder. (x ~= y) = (x<y | y<x)"
paulson@14208
   825
by (cut_tac x = x and y = y in linorder_less_linear, auto)
wenzelm@11750
   826
wenzelm@11750
   827
lemma linorder_neqE: "x ~= (y::'a::linorder) ==> (x < y ==> R) ==> (y < x ==> R) ==> R"
paulson@14208
   828
by (simp add: linorder_neq_iff, blast)
wenzelm@11750
   829
wenzelm@11750
   830
wenzelm@11750
   831
subsubsection "Min and max on (linear) orders"
wenzelm@11750
   832
wenzelm@11750
   833
lemma min_same [simp]: "min (x::'a::order) x = x"
wenzelm@11750
   834
  by (simp add: min_def)
wenzelm@11750
   835
wenzelm@11750
   836
lemma max_same [simp]: "max (x::'a::order) x = x"
wenzelm@11750
   837
  by (simp add: max_def)
wenzelm@11750
   838
wenzelm@11750
   839
lemma le_max_iff_disj: "!!z::'a::linorder. (z <= max x y) = (z <= x | z <= y)"
wenzelm@11750
   840
  apply (simp add: max_def)
wenzelm@11750
   841
  apply (insert linorder_linear)
wenzelm@11750
   842
  apply (blast intro: order_trans)
wenzelm@11750
   843
  done
wenzelm@11750
   844
wenzelm@11750
   845
lemma le_maxI1: "(x::'a::linorder) <= max x y"
wenzelm@11750
   846
  by (simp add: le_max_iff_disj)
wenzelm@11750
   847
wenzelm@11750
   848
lemma le_maxI2: "(y::'a::linorder) <= max x y"
wenzelm@11750
   849
    -- {* CANNOT use with @{text "[intro!]"} because blast will give PROOF FAILED. *}
wenzelm@11750
   850
  by (simp add: le_max_iff_disj)
wenzelm@11750
   851
wenzelm@11750
   852
lemma less_max_iff_disj: "!!z::'a::linorder. (z < max x y) = (z < x | z < y)"
wenzelm@11750
   853
  apply (simp add: max_def order_le_less)
wenzelm@11750
   854
  apply (insert linorder_less_linear)
wenzelm@11750
   855
  apply (blast intro: order_less_trans)
wenzelm@11750
   856
  done
wenzelm@11750
   857
wenzelm@11750
   858
lemma max_le_iff_conj [simp]:
wenzelm@11750
   859
    "!!z::'a::linorder. (max x y <= z) = (x <= z & y <= z)"
wenzelm@11750
   860
  apply (simp add: max_def)
wenzelm@11750
   861
  apply (insert linorder_linear)
wenzelm@11750
   862
  apply (blast intro: order_trans)
wenzelm@11750
   863
  done
wenzelm@11750
   864
wenzelm@11750
   865
lemma max_less_iff_conj [simp]:
wenzelm@11750
   866
    "!!z::'a::linorder. (max x y < z) = (x < z & y < z)"
wenzelm@11750
   867
  apply (simp add: order_le_less max_def)
wenzelm@11750
   868
  apply (insert linorder_less_linear)
wenzelm@11750
   869
  apply (blast intro: order_less_trans)
wenzelm@11750
   870
  done
wenzelm@11750
   871
wenzelm@11750
   872
lemma le_min_iff_conj [simp]:
wenzelm@11750
   873
    "!!z::'a::linorder. (z <= min x y) = (z <= x & z <= y)"
wenzelm@12892
   874
    -- {* @{text "[iff]"} screws up a @{text blast} in MiniML *}
wenzelm@11750
   875
  apply (simp add: min_def)
wenzelm@11750
   876
  apply (insert linorder_linear)
wenzelm@11750
   877
  apply (blast intro: order_trans)
wenzelm@11750
   878
  done
wenzelm@11750
   879
wenzelm@11750
   880
lemma min_less_iff_conj [simp]:
wenzelm@11750
   881
    "!!z::'a::linorder. (z < min x y) = (z < x & z < y)"
wenzelm@11750
   882
  apply (simp add: order_le_less min_def)
wenzelm@11750
   883
  apply (insert linorder_less_linear)
wenzelm@11750
   884
  apply (blast intro: order_less_trans)
wenzelm@11750
   885
  done
wenzelm@11750
   886
wenzelm@11750
   887
lemma min_le_iff_disj: "!!z::'a::linorder. (min x y <= z) = (x <= z | y <= z)"
wenzelm@11750
   888
  apply (simp add: min_def)
wenzelm@11750
   889
  apply (insert linorder_linear)
wenzelm@11750
   890
  apply (blast intro: order_trans)
wenzelm@11750
   891
  done
wenzelm@11750
   892
wenzelm@11750
   893
lemma min_less_iff_disj: "!!z::'a::linorder. (min x y < z) = (x < z | y < z)"
wenzelm@11750
   894
  apply (simp add: min_def order_le_less)
wenzelm@11750
   895
  apply (insert linorder_less_linear)
wenzelm@11750
   896
  apply (blast intro: order_less_trans)
wenzelm@11750
   897
  done
wenzelm@11750
   898
nipkow@13438
   899
lemma max_assoc: "!!x::'a::linorder. max (max x y) z = max x (max y z)"
nipkow@13438
   900
apply(simp add:max_def)
nipkow@13438
   901
apply(rule conjI)
nipkow@13438
   902
apply(blast intro:order_trans)
nipkow@13438
   903
apply(simp add:linorder_not_le)
nipkow@13438
   904
apply(blast dest: order_less_trans order_le_less_trans)
nipkow@13438
   905
done
nipkow@13438
   906
nipkow@13438
   907
lemma max_commute: "!!x::'a::linorder. max x y = max y x"
nipkow@13438
   908
apply(simp add:max_def)
nipkow@13438
   909
apply(rule conjI)
nipkow@13438
   910
apply(blast intro:order_antisym)
nipkow@13438
   911
apply(simp add:linorder_not_le)
nipkow@13438
   912
apply(blast dest: order_less_trans)
nipkow@13438
   913
done
nipkow@13438
   914
nipkow@13438
   915
lemmas max_ac = max_assoc max_commute
nipkow@13438
   916
                mk_left_commute[of max,OF max_assoc max_commute]
nipkow@13438
   917
nipkow@13438
   918
lemma min_assoc: "!!x::'a::linorder. min (min x y) z = min x (min y z)"
nipkow@13438
   919
apply(simp add:min_def)
nipkow@13438
   920
apply(rule conjI)
nipkow@13438
   921
apply(blast intro:order_trans)
nipkow@13438
   922
apply(simp add:linorder_not_le)
nipkow@13438
   923
apply(blast dest: order_less_trans order_le_less_trans)
nipkow@13438
   924
done
nipkow@13438
   925
nipkow@13438
   926
lemma min_commute: "!!x::'a::linorder. min x y = min y x"
nipkow@13438
   927
apply(simp add:min_def)
nipkow@13438
   928
apply(rule conjI)
nipkow@13438
   929
apply(blast intro:order_antisym)
nipkow@13438
   930
apply(simp add:linorder_not_le)
nipkow@13438
   931
apply(blast dest: order_less_trans)
nipkow@13438
   932
done
nipkow@13438
   933
nipkow@13438
   934
lemmas min_ac = min_assoc min_commute
nipkow@13438
   935
                mk_left_commute[of min,OF min_assoc min_commute]
nipkow@13438
   936
wenzelm@11750
   937
lemma split_min:
wenzelm@11750
   938
    "P (min (i::'a::linorder) j) = ((i <= j --> P(i)) & (~ i <= j --> P(j)))"
wenzelm@11750
   939
  by (simp add: min_def)
wenzelm@11750
   940
wenzelm@11750
   941
lemma split_max:
wenzelm@11750
   942
    "P (max (i::'a::linorder) j) = ((i <= j --> P(j)) & (~ i <= j --> P(i)))"
wenzelm@11750
   943
  by (simp add: max_def)
wenzelm@11750
   944
wenzelm@11750
   945
wenzelm@11750
   946
subsubsection "Bounded quantifiers"
wenzelm@11750
   947
wenzelm@11750
   948
syntax
wenzelm@11750
   949
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   950
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3EX _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   951
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
   952
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3EX _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
   953
wenzelm@12114
   954
syntax (xsymbols)
wenzelm@11750
   955
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3\<forall>_<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   956
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3\<exists>_<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   957
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@11750
   958
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
wenzelm@11750
   959
wenzelm@11750
   960
syntax (HOL)
wenzelm@11750
   961
  "_lessAll" :: "[idt, 'a, bool] => bool"   ("(3! _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   962
  "_lessEx"  :: "[idt, 'a, bool] => bool"   ("(3? _<_./ _)"  [0, 0, 10] 10)
wenzelm@11750
   963
  "_leAll"   :: "[idt, 'a, bool] => bool"   ("(3! _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
   964
  "_leEx"    :: "[idt, 'a, bool] => bool"   ("(3? _<=_./ _)" [0, 0, 10] 10)
wenzelm@11750
   965
wenzelm@11750
   966
translations
wenzelm@11750
   967
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
wenzelm@11750
   968
 "EX x<y. P"    =>  "EX x. x < y  & P"
wenzelm@11750
   969
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
wenzelm@11750
   970
 "EX x<=y. P"   =>  "EX x. x <= y & P"
wenzelm@11750
   971
kleing@14357
   972
print_translation {*
kleing@14357
   973
let
kleing@14357
   974
  fun all_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
   975
               Const("op -->",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
   976
  (if v=v' then Syntax.const "_lessAll" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
   977
kleing@14357
   978
  | all_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
   979
               Const("op -->",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
   980
  (if v=v' then Syntax.const "_leAll" $ Syntax.mark_bound v' $ n $ P else raise Match);
kleing@14357
   981
kleing@14357
   982
  fun ex_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
   983
               Const("op &",_) $ (Const ("op <",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
   984
  (if v=v' then Syntax.const "_lessEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
   985
kleing@14357
   986
  | ex_tr' [Const ("_bound",_) $ Free (v,_), 
kleing@14357
   987
               Const("op &",_) $ (Const ("op <=",_) $ (Const ("_bound",_) $ Free (v',_)) $ n ) $ P] = 
kleing@14357
   988
  (if v=v' then Syntax.const "_leEx" $ Syntax.mark_bound v' $ n $ P else raise Match)
kleing@14357
   989
in
kleing@14357
   990
[("ALL ", all_tr'), ("EX ", ex_tr')]
clasohm@923
   991
end
kleing@14357
   992
*}
kleing@14357
   993
kleing@14357
   994
end