src/HOL/Fun.thy
author wenzelm
Fri Apr 16 21:28:09 2010 +0200 (2010-04-16)
changeset 36176 3fe7e97ccca8
parent 35584 768f8d92b767
child 37751 89e16802b6cc
permissions -rw-r--r--
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
clasohm@1475
     1
(*  Title:      HOL/Fun.thy
clasohm@1475
     2
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
clasohm@923
     3
    Copyright   1994  University of Cambridge
huffman@18154
     4
*)
clasohm@923
     5
huffman@18154
     6
header {* Notions about functions *}
clasohm@923
     7
paulson@15510
     8
theory Fun
haftmann@32139
     9
imports Complete_Lattice
nipkow@15131
    10
begin
nipkow@2912
    11
haftmann@26147
    12
text{*As a simplification rule, it replaces all function equalities by
haftmann@26147
    13
  first-order equalities.*}
haftmann@26147
    14
lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
haftmann@26147
    15
apply (rule iffI)
haftmann@26147
    16
apply (simp (no_asm_simp))
haftmann@26147
    17
apply (rule ext)
haftmann@26147
    18
apply (simp (no_asm_simp))
haftmann@26147
    19
done
oheimb@5305
    20
haftmann@26147
    21
lemma apply_inverse:
haftmann@26357
    22
  "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
haftmann@26147
    23
  by auto
nipkow@2912
    24
wenzelm@12258
    25
haftmann@26147
    26
subsection {* The Identity Function @{text id} *}
paulson@6171
    27
haftmann@22744
    28
definition
haftmann@22744
    29
  id :: "'a \<Rightarrow> 'a"
haftmann@22744
    30
where
haftmann@22744
    31
  "id = (\<lambda>x. x)"
nipkow@13910
    32
haftmann@26147
    33
lemma id_apply [simp]: "id x = x"
haftmann@26147
    34
  by (simp add: id_def)
haftmann@26147
    35
haftmann@26147
    36
lemma image_ident [simp]: "(%x. x) ` Y = Y"
haftmann@26147
    37
by blast
haftmann@26147
    38
haftmann@26147
    39
lemma image_id [simp]: "id ` Y = Y"
haftmann@26147
    40
by (simp add: id_def)
haftmann@26147
    41
haftmann@26147
    42
lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
haftmann@26147
    43
by blast
haftmann@26147
    44
haftmann@26147
    45
lemma vimage_id [simp]: "id -` A = A"
haftmann@26147
    46
by (simp add: id_def)
haftmann@26147
    47
haftmann@26147
    48
haftmann@26147
    49
subsection {* The Composition Operator @{text "f \<circ> g"} *}
haftmann@26147
    50
haftmann@22744
    51
definition
haftmann@22744
    52
  comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
haftmann@22744
    53
where
haftmann@22744
    54
  "f o g = (\<lambda>x. f (g x))"
oheimb@11123
    55
wenzelm@21210
    56
notation (xsymbols)
wenzelm@19656
    57
  comp  (infixl "\<circ>" 55)
wenzelm@19656
    58
wenzelm@21210
    59
notation (HTML output)
wenzelm@19656
    60
  comp  (infixl "\<circ>" 55)
wenzelm@19656
    61
paulson@13585
    62
text{*compatibility*}
paulson@13585
    63
lemmas o_def = comp_def
nipkow@2912
    64
paulson@13585
    65
lemma o_apply [simp]: "(f o g) x = f (g x)"
paulson@13585
    66
by (simp add: comp_def)
paulson@13585
    67
paulson@13585
    68
lemma o_assoc: "f o (g o h) = f o g o h"
paulson@13585
    69
by (simp add: comp_def)
paulson@13585
    70
paulson@13585
    71
lemma id_o [simp]: "id o g = g"
paulson@13585
    72
by (simp add: comp_def)
paulson@13585
    73
paulson@13585
    74
lemma o_id [simp]: "f o id = f"
paulson@13585
    75
by (simp add: comp_def)
paulson@13585
    76
haftmann@34150
    77
lemma o_eq_dest:
haftmann@34150
    78
  "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
haftmann@34150
    79
  by (simp only: o_def) (fact fun_cong)
haftmann@34150
    80
haftmann@34150
    81
lemma o_eq_elim:
haftmann@34150
    82
  "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
haftmann@34150
    83
  by (erule meta_mp) (fact o_eq_dest) 
haftmann@34150
    84
paulson@13585
    85
lemma image_compose: "(f o g) ` r = f`(g`r)"
paulson@13585
    86
by (simp add: comp_def, blast)
paulson@13585
    87
paulson@33044
    88
lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
paulson@33044
    89
  by auto
paulson@33044
    90
paulson@13585
    91
lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
paulson@13585
    92
by (unfold comp_def, blast)
paulson@13585
    93
paulson@13585
    94
haftmann@26588
    95
subsection {* The Forward Composition Operator @{text fcomp} *}
haftmann@26357
    96
haftmann@26357
    97
definition
haftmann@26357
    98
  fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
haftmann@26357
    99
where
haftmann@26357
   100
  "f o> g = (\<lambda>x. g (f x))"
haftmann@26357
   101
haftmann@26357
   102
lemma fcomp_apply:  "(f o> g) x = g (f x)"
haftmann@26357
   103
  by (simp add: fcomp_def)
haftmann@26357
   104
haftmann@26357
   105
lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
haftmann@26357
   106
  by (simp add: fcomp_def)
haftmann@26357
   107
haftmann@26357
   108
lemma id_fcomp [simp]: "id o> g = g"
haftmann@26357
   109
  by (simp add: fcomp_def)
haftmann@26357
   110
haftmann@26357
   111
lemma fcomp_id [simp]: "f o> id = f"
haftmann@26357
   112
  by (simp add: fcomp_def)
haftmann@26357
   113
haftmann@31202
   114
code_const fcomp
haftmann@31202
   115
  (Eval infixl 1 "#>")
haftmann@31202
   116
haftmann@26588
   117
no_notation fcomp (infixl "o>" 60)
haftmann@26588
   118
haftmann@26357
   119
haftmann@26147
   120
subsection {* Injectivity and Surjectivity *}
haftmann@26147
   121
haftmann@35416
   122
definition
haftmann@35416
   123
  inj_on :: "['a => 'b, 'a set] => bool" where
haftmann@35416
   124
  -- "injective"
haftmann@26147
   125
  "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
haftmann@26147
   126
haftmann@26147
   127
text{*A common special case: functions injective over the entire domain type.*}
haftmann@26147
   128
haftmann@26147
   129
abbreviation
haftmann@26147
   130
  "inj f == inj_on f UNIV"
paulson@13585
   131
haftmann@26147
   132
definition
haftmann@26147
   133
  bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
haftmann@28562
   134
  [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
haftmann@26147
   135
haftmann@35416
   136
definition
haftmann@35416
   137
  surj :: "('a => 'b) => bool" where
haftmann@35416
   138
  -- "surjective"
haftmann@26147
   139
  "surj f == ! y. ? x. y=f(x)"
paulson@13585
   140
haftmann@35416
   141
definition
haftmann@35416
   142
  bij :: "('a => 'b) => bool" where
haftmann@35416
   143
  -- "bijective"
haftmann@26147
   144
  "bij f == inj f & surj f"
haftmann@26147
   145
haftmann@26147
   146
lemma injI:
haftmann@26147
   147
  assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
haftmann@26147
   148
  shows "inj f"
haftmann@26147
   149
  using assms unfolding inj_on_def by auto
paulson@13585
   150
haftmann@31775
   151
text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
paulson@13585
   152
lemma datatype_injI:
paulson@13585
   153
    "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
paulson@13585
   154
by (simp add: inj_on_def)
paulson@13585
   155
berghofe@13637
   156
theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
berghofe@13637
   157
  by (unfold inj_on_def, blast)
berghofe@13637
   158
paulson@13585
   159
lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
paulson@13585
   160
by (simp add: inj_on_def)
paulson@13585
   161
nipkow@32988
   162
lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
paulson@13585
   163
by (force simp add: inj_on_def)
paulson@13585
   164
nipkow@32988
   165
lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
nipkow@32988
   166
by (simp add: inj_on_eq_iff)
nipkow@32988
   167
haftmann@26147
   168
lemma inj_on_id[simp]: "inj_on id A"
haftmann@26147
   169
  by (simp add: inj_on_def) 
paulson@13585
   170
haftmann@26147
   171
lemma inj_on_id2[simp]: "inj_on (%x. x) A"
haftmann@26147
   172
by (simp add: inj_on_def) 
haftmann@26147
   173
haftmann@26147
   174
lemma surj_id[simp]: "surj id"
haftmann@26147
   175
by (simp add: surj_def) 
haftmann@26147
   176
haftmann@26147
   177
lemma bij_id[simp]: "bij id"
krauss@34209
   178
by (simp add: bij_def)
paulson@13585
   179
paulson@13585
   180
lemma inj_onI:
paulson@13585
   181
    "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
paulson@13585
   182
by (simp add: inj_on_def)
paulson@13585
   183
paulson@13585
   184
lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
paulson@13585
   185
by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
paulson@13585
   186
paulson@13585
   187
lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
paulson@13585
   188
by (unfold inj_on_def, blast)
paulson@13585
   189
paulson@13585
   190
lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
paulson@13585
   191
by (blast dest!: inj_onD)
paulson@13585
   192
paulson@13585
   193
lemma comp_inj_on:
paulson@13585
   194
     "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
paulson@13585
   195
by (simp add: comp_def inj_on_def)
paulson@13585
   196
nipkow@15303
   197
lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
nipkow@15303
   198
apply(simp add:inj_on_def image_def)
nipkow@15303
   199
apply blast
nipkow@15303
   200
done
nipkow@15303
   201
nipkow@15439
   202
lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
nipkow@15439
   203
  inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
nipkow@15439
   204
apply(unfold inj_on_def)
nipkow@15439
   205
apply blast
nipkow@15439
   206
done
nipkow@15439
   207
paulson@13585
   208
lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
paulson@13585
   209
by (unfold inj_on_def, blast)
wenzelm@12258
   210
paulson@13585
   211
lemma inj_singleton: "inj (%s. {s})"
paulson@13585
   212
by (simp add: inj_on_def)
paulson@13585
   213
nipkow@15111
   214
lemma inj_on_empty[iff]: "inj_on f {}"
nipkow@15111
   215
by(simp add: inj_on_def)
nipkow@15111
   216
nipkow@15303
   217
lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
paulson@13585
   218
by (unfold inj_on_def, blast)
paulson@13585
   219
nipkow@15111
   220
lemma inj_on_Un:
nipkow@15111
   221
 "inj_on f (A Un B) =
nipkow@15111
   222
  (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
nipkow@15111
   223
apply(unfold inj_on_def)
nipkow@15111
   224
apply (blast intro:sym)
nipkow@15111
   225
done
nipkow@15111
   226
nipkow@15111
   227
lemma inj_on_insert[iff]:
nipkow@15111
   228
  "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
nipkow@15111
   229
apply(unfold inj_on_def)
nipkow@15111
   230
apply (blast intro:sym)
nipkow@15111
   231
done
nipkow@15111
   232
nipkow@15111
   233
lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
nipkow@15111
   234
apply(unfold inj_on_def)
nipkow@15111
   235
apply (blast)
nipkow@15111
   236
done
nipkow@15111
   237
paulson@13585
   238
lemma surjI: "(!! x. g(f x) = x) ==> surj g"
paulson@13585
   239
apply (simp add: surj_def)
paulson@13585
   240
apply (blast intro: sym)
paulson@13585
   241
done
paulson@13585
   242
paulson@13585
   243
lemma surj_range: "surj f ==> range f = UNIV"
paulson@13585
   244
by (auto simp add: surj_def)
paulson@13585
   245
paulson@13585
   246
lemma surjD: "surj f ==> EX x. y = f x"
paulson@13585
   247
by (simp add: surj_def)
paulson@13585
   248
paulson@13585
   249
lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
paulson@13585
   250
by (simp add: surj_def, blast)
paulson@13585
   251
paulson@13585
   252
lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
paulson@13585
   253
apply (simp add: comp_def surj_def, clarify)
paulson@13585
   254
apply (drule_tac x = y in spec, clarify)
paulson@13585
   255
apply (drule_tac x = x in spec, blast)
paulson@13585
   256
done
paulson@13585
   257
paulson@13585
   258
lemma bijI: "[| inj f; surj f |] ==> bij f"
paulson@13585
   259
by (simp add: bij_def)
paulson@13585
   260
paulson@13585
   261
lemma bij_is_inj: "bij f ==> inj f"
paulson@13585
   262
by (simp add: bij_def)
paulson@13585
   263
paulson@13585
   264
lemma bij_is_surj: "bij f ==> surj f"
paulson@13585
   265
by (simp add: bij_def)
paulson@13585
   266
nipkow@26105
   267
lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
nipkow@26105
   268
by (simp add: bij_betw_def)
nipkow@26105
   269
nipkow@32337
   270
lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
nipkow@32337
   271
by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
nipkow@32337
   272
nipkow@31438
   273
lemma bij_betw_trans:
nipkow@31438
   274
  "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
nipkow@31438
   275
by(auto simp add:bij_betw_def comp_inj_on)
nipkow@31438
   276
nipkow@26105
   277
lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
nipkow@26105
   278
proof -
nipkow@26105
   279
  have i: "inj_on f A" and s: "f ` A = B"
nipkow@26105
   280
    using assms by(auto simp:bij_betw_def)
nipkow@26105
   281
  let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
nipkow@26105
   282
  { fix a b assume P: "?P b a"
nipkow@26105
   283
    hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
nipkow@26105
   284
    hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
nipkow@26105
   285
    hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
nipkow@26105
   286
  } note g = this
nipkow@26105
   287
  have "inj_on ?g B"
nipkow@26105
   288
  proof(rule inj_onI)
nipkow@26105
   289
    fix x y assume "x:B" "y:B" "?g x = ?g y"
nipkow@26105
   290
    from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
nipkow@26105
   291
    from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
nipkow@26105
   292
    from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
nipkow@26105
   293
  qed
nipkow@26105
   294
  moreover have "?g ` B = A"
nipkow@26105
   295
  proof(auto simp:image_def)
nipkow@26105
   296
    fix b assume "b:B"
nipkow@26105
   297
    with s obtain a where P: "?P b a" unfolding image_def by blast
nipkow@26105
   298
    thus "?g b \<in> A" using g[OF P] by auto
nipkow@26105
   299
  next
nipkow@26105
   300
    fix a assume "a:A"
nipkow@26105
   301
    then obtain b where P: "?P b a" using s unfolding image_def by blast
nipkow@26105
   302
    then have "b:B" using s unfolding image_def by blast
nipkow@26105
   303
    with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
nipkow@26105
   304
  qed
nipkow@26105
   305
  ultimately show ?thesis by(auto simp:bij_betw_def)
nipkow@26105
   306
qed
nipkow@26105
   307
paulson@13585
   308
lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
paulson@13585
   309
by (simp add: surj_range)
paulson@13585
   310
paulson@13585
   311
lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
paulson@13585
   312
by (simp add: inj_on_def, blast)
paulson@13585
   313
paulson@13585
   314
lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
paulson@13585
   315
apply (unfold surj_def)
paulson@13585
   316
apply (blast intro: sym)
paulson@13585
   317
done
paulson@13585
   318
paulson@13585
   319
lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
paulson@13585
   320
by (unfold inj_on_def, blast)
paulson@13585
   321
paulson@13585
   322
lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
paulson@13585
   323
apply (unfold bij_def)
paulson@13585
   324
apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
paulson@13585
   325
done
paulson@13585
   326
nipkow@31438
   327
lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
nipkow@31438
   328
by(blast dest: inj_onD)
nipkow@31438
   329
paulson@13585
   330
lemma inj_on_image_Int:
paulson@13585
   331
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   332
apply (simp add: inj_on_def, blast)
paulson@13585
   333
done
paulson@13585
   334
paulson@13585
   335
lemma inj_on_image_set_diff:
paulson@13585
   336
   "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
paulson@13585
   337
apply (simp add: inj_on_def, blast)
paulson@13585
   338
done
paulson@13585
   339
paulson@13585
   340
lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
paulson@13585
   341
by (simp add: inj_on_def, blast)
paulson@13585
   342
paulson@13585
   343
lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
paulson@13585
   344
by (simp add: inj_on_def, blast)
paulson@13585
   345
paulson@13585
   346
lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
paulson@13585
   347
by (blast dest: injD)
paulson@13585
   348
paulson@13585
   349
lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
paulson@13585
   350
by (simp add: inj_on_def, blast)
paulson@13585
   351
paulson@13585
   352
lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
paulson@13585
   353
by (blast dest: injD)
paulson@13585
   354
paulson@13585
   355
(*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
paulson@13585
   356
lemma image_INT:
paulson@13585
   357
   "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
paulson@13585
   358
    ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   359
apply (simp add: inj_on_def, blast)
paulson@13585
   360
done
paulson@13585
   361
paulson@13585
   362
(*Compare with image_INT: no use of inj_on, and if f is surjective then
paulson@13585
   363
  it doesn't matter whether A is empty*)
paulson@13585
   364
lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
paulson@13585
   365
apply (simp add: bij_def)
paulson@13585
   366
apply (simp add: inj_on_def surj_def, blast)
paulson@13585
   367
done
paulson@13585
   368
paulson@13585
   369
lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
paulson@13585
   370
by (auto simp add: surj_def)
paulson@13585
   371
paulson@13585
   372
lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
paulson@13585
   373
by (auto simp add: inj_on_def)
paulson@5852
   374
paulson@13585
   375
lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
paulson@13585
   376
apply (simp add: bij_def)
paulson@13585
   377
apply (rule equalityI)
paulson@13585
   378
apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
paulson@13585
   379
done
paulson@13585
   380
hoelzl@35584
   381
lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
hoelzl@35580
   382
  by (auto intro!: inj_onI)
paulson@13585
   383
hoelzl@35584
   384
lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
hoelzl@35584
   385
  by (auto intro!: inj_onI dest: strict_mono_eq)
hoelzl@35584
   386
paulson@13585
   387
subsection{*Function Updating*}
paulson@13585
   388
haftmann@35416
   389
definition
haftmann@35416
   390
  fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
haftmann@26147
   391
  "fun_upd f a b == % x. if x=a then b else f x"
haftmann@26147
   392
haftmann@26147
   393
nonterminals
haftmann@26147
   394
  updbinds updbind
haftmann@26147
   395
syntax
haftmann@26147
   396
  "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
haftmann@26147
   397
  ""         :: "updbind => updbinds"             ("_")
haftmann@26147
   398
  "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
wenzelm@35115
   399
  "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
haftmann@26147
   400
haftmann@26147
   401
translations
wenzelm@35115
   402
  "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
wenzelm@35115
   403
  "f(x:=y)" == "CONST fun_upd f x y"
haftmann@26147
   404
haftmann@26147
   405
(* Hint: to define the sum of two functions (or maps), use sum_case.
haftmann@26147
   406
         A nice infix syntax could be defined (in Datatype.thy or below) by
wenzelm@35115
   407
notation
wenzelm@35115
   408
  sum_case  (infixr "'(+')"80)
haftmann@26147
   409
*)
haftmann@26147
   410
paulson@13585
   411
lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
paulson@13585
   412
apply (simp add: fun_upd_def, safe)
paulson@13585
   413
apply (erule subst)
paulson@13585
   414
apply (rule_tac [2] ext, auto)
paulson@13585
   415
done
paulson@13585
   416
paulson@13585
   417
(* f x = y ==> f(x:=y) = f *)
paulson@13585
   418
lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
paulson@13585
   419
paulson@13585
   420
(* f(x := f x) = f *)
paulson@17084
   421
lemmas fun_upd_triv = refl [THEN fun_upd_idem]
paulson@17084
   422
declare fun_upd_triv [iff]
paulson@13585
   423
paulson@13585
   424
lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
paulson@17084
   425
by (simp add: fun_upd_def)
paulson@13585
   426
paulson@13585
   427
(* fun_upd_apply supersedes these two,   but they are useful
paulson@13585
   428
   if fun_upd_apply is intentionally removed from the simpset *)
paulson@13585
   429
lemma fun_upd_same: "(f(x:=y)) x = y"
paulson@13585
   430
by simp
paulson@13585
   431
paulson@13585
   432
lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
paulson@13585
   433
by simp
paulson@13585
   434
paulson@13585
   435
lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
paulson@13585
   436
by (simp add: expand_fun_eq)
paulson@13585
   437
paulson@13585
   438
lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
paulson@13585
   439
by (rule ext, auto)
paulson@13585
   440
nipkow@15303
   441
lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
krauss@34209
   442
by (fastsimp simp:inj_on_def image_def)
nipkow@15303
   443
paulson@15510
   444
lemma fun_upd_image:
paulson@15510
   445
     "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
paulson@15510
   446
by auto
paulson@15510
   447
nipkow@31080
   448
lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
krauss@34209
   449
by (auto intro: ext)
nipkow@31080
   450
haftmann@26147
   451
haftmann@26147
   452
subsection {* @{text override_on} *}
haftmann@26147
   453
haftmann@26147
   454
definition
haftmann@26147
   455
  override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
haftmann@26147
   456
where
haftmann@26147
   457
  "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
nipkow@13910
   458
nipkow@15691
   459
lemma override_on_emptyset[simp]: "override_on f g {} = f"
nipkow@15691
   460
by(simp add:override_on_def)
nipkow@13910
   461
nipkow@15691
   462
lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
nipkow@15691
   463
by(simp add:override_on_def)
nipkow@13910
   464
nipkow@15691
   465
lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
nipkow@15691
   466
by(simp add:override_on_def)
nipkow@13910
   467
haftmann@26147
   468
haftmann@26147
   469
subsection {* @{text swap} *}
paulson@15510
   470
haftmann@22744
   471
definition
haftmann@22744
   472
  swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@22744
   473
where
haftmann@22744
   474
  "swap a b f = f (a := f b, b:= f a)"
paulson@15510
   475
huffman@34101
   476
lemma swap_self [simp]: "swap a a f = f"
nipkow@15691
   477
by (simp add: swap_def)
paulson@15510
   478
paulson@15510
   479
lemma swap_commute: "swap a b f = swap b a f"
paulson@15510
   480
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   481
paulson@15510
   482
lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
paulson@15510
   483
by (rule ext, simp add: fun_upd_def swap_def)
paulson@15510
   484
huffman@34145
   485
lemma swap_triple:
huffman@34145
   486
  assumes "a \<noteq> c" and "b \<noteq> c"
huffman@34145
   487
  shows "swap a b (swap b c (swap a b f)) = swap a c f"
huffman@34145
   488
  using assms by (simp add: expand_fun_eq swap_def)
huffman@34145
   489
huffman@34101
   490
lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
huffman@34101
   491
by (rule ext, simp add: fun_upd_def swap_def)
huffman@34101
   492
paulson@15510
   493
lemma inj_on_imp_inj_on_swap:
haftmann@22744
   494
  "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
paulson@15510
   495
by (simp add: inj_on_def swap_def, blast)
paulson@15510
   496
paulson@15510
   497
lemma inj_on_swap_iff [simp]:
paulson@15510
   498
  assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
paulson@15510
   499
proof 
paulson@15510
   500
  assume "inj_on (swap a b f) A"
paulson@15510
   501
  with A have "inj_on (swap a b (swap a b f)) A" 
nipkow@17589
   502
    by (iprover intro: inj_on_imp_inj_on_swap) 
paulson@15510
   503
  thus "inj_on f A" by simp 
paulson@15510
   504
next
paulson@15510
   505
  assume "inj_on f A"
krauss@34209
   506
  with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
paulson@15510
   507
qed
paulson@15510
   508
paulson@15510
   509
lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
paulson@15510
   510
apply (simp add: surj_def swap_def, clarify)
wenzelm@27125
   511
apply (case_tac "y = f b", blast)
wenzelm@27125
   512
apply (case_tac "y = f a", auto)
paulson@15510
   513
done
paulson@15510
   514
paulson@15510
   515
lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
paulson@15510
   516
proof 
paulson@15510
   517
  assume "surj (swap a b f)"
paulson@15510
   518
  hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
paulson@15510
   519
  thus "surj f" by simp 
paulson@15510
   520
next
paulson@15510
   521
  assume "surj f"
paulson@15510
   522
  thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
paulson@15510
   523
qed
paulson@15510
   524
paulson@15510
   525
lemma bij_swap_iff: "bij (swap a b f) = bij f"
paulson@15510
   526
by (simp add: bij_def)
haftmann@21547
   527
wenzelm@36176
   528
hide_const (open) swap
haftmann@21547
   529
haftmann@31949
   530
haftmann@31949
   531
subsection {* Inversion of injective functions *}
haftmann@31949
   532
nipkow@33057
   533
definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
nipkow@33057
   534
"the_inv_into A f == %x. THE y. y : A & f y = x"
nipkow@32961
   535
nipkow@33057
   536
lemma the_inv_into_f_f:
nipkow@33057
   537
  "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
nipkow@33057
   538
apply (simp add: the_inv_into_def inj_on_def)
krauss@34209
   539
apply blast
nipkow@32961
   540
done
nipkow@32961
   541
nipkow@33057
   542
lemma f_the_inv_into_f:
nipkow@33057
   543
  "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
nipkow@33057
   544
apply (simp add: the_inv_into_def)
nipkow@32961
   545
apply (rule the1I2)
nipkow@32961
   546
 apply(blast dest: inj_onD)
nipkow@32961
   547
apply blast
nipkow@32961
   548
done
nipkow@32961
   549
nipkow@33057
   550
lemma the_inv_into_into:
nipkow@33057
   551
  "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
nipkow@33057
   552
apply (simp add: the_inv_into_def)
nipkow@32961
   553
apply (rule the1I2)
nipkow@32961
   554
 apply(blast dest: inj_onD)
nipkow@32961
   555
apply blast
nipkow@32961
   556
done
nipkow@32961
   557
nipkow@33057
   558
lemma the_inv_into_onto[simp]:
nipkow@33057
   559
  "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
nipkow@33057
   560
by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
nipkow@32961
   561
nipkow@33057
   562
lemma the_inv_into_f_eq:
nipkow@33057
   563
  "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
nipkow@32961
   564
  apply (erule subst)
nipkow@33057
   565
  apply (erule the_inv_into_f_f, assumption)
nipkow@32961
   566
  done
nipkow@32961
   567
nipkow@33057
   568
lemma the_inv_into_comp:
nipkow@32961
   569
  "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
nipkow@33057
   570
  the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
nipkow@33057
   571
apply (rule the_inv_into_f_eq)
nipkow@32961
   572
  apply (fast intro: comp_inj_on)
nipkow@33057
   573
 apply (simp add: f_the_inv_into_f the_inv_into_into)
nipkow@33057
   574
apply (simp add: the_inv_into_into)
nipkow@32961
   575
done
nipkow@32961
   576
nipkow@33057
   577
lemma inj_on_the_inv_into:
nipkow@33057
   578
  "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
nipkow@33057
   579
by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
nipkow@32961
   580
nipkow@33057
   581
lemma bij_betw_the_inv_into:
nipkow@33057
   582
  "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
nipkow@33057
   583
by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
nipkow@32961
   584
berghofe@32998
   585
abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
nipkow@33057
   586
  "the_inv f \<equiv> the_inv_into UNIV f"
berghofe@32998
   587
berghofe@32998
   588
lemma the_inv_f_f:
berghofe@32998
   589
  assumes "inj f"
berghofe@32998
   590
  shows "the_inv f (f x) = x" using assms UNIV_I
nipkow@33057
   591
  by (rule the_inv_into_f_f)
berghofe@32998
   592
haftmann@31949
   593
haftmann@22845
   594
subsection {* Proof tool setup *} 
haftmann@22845
   595
haftmann@22845
   596
text {* simplifies terms of the form
haftmann@22845
   597
  f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
haftmann@22845
   598
wenzelm@24017
   599
simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
haftmann@22845
   600
let
haftmann@22845
   601
  fun gen_fun_upd NONE T _ _ = NONE
wenzelm@24017
   602
    | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
haftmann@22845
   603
  fun dest_fun_T1 (Type (_, T :: Ts)) = T
haftmann@22845
   604
  fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
haftmann@22845
   605
    let
haftmann@22845
   606
      fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
haftmann@22845
   607
            if v aconv x then SOME g else gen_fun_upd (find g) T v w
haftmann@22845
   608
        | find t = NONE
haftmann@22845
   609
    in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
wenzelm@24017
   610
wenzelm@24017
   611
  fun proc ss ct =
wenzelm@24017
   612
    let
wenzelm@24017
   613
      val ctxt = Simplifier.the_context ss
wenzelm@24017
   614
      val t = Thm.term_of ct
wenzelm@24017
   615
    in
wenzelm@24017
   616
      case find_double t of
wenzelm@24017
   617
        (T, NONE) => NONE
wenzelm@24017
   618
      | (T, SOME rhs) =>
wenzelm@27330
   619
          SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
wenzelm@24017
   620
            (fn _ =>
wenzelm@24017
   621
              rtac eq_reflection 1 THEN
wenzelm@24017
   622
              rtac ext 1 THEN
wenzelm@24017
   623
              simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
wenzelm@24017
   624
    end
wenzelm@24017
   625
in proc end
haftmann@22845
   626
*}
haftmann@22845
   627
haftmann@22845
   628
haftmann@21870
   629
subsection {* Code generator setup *}
haftmann@21870
   630
berghofe@25886
   631
types_code
berghofe@25886
   632
  "fun"  ("(_ ->/ _)")
berghofe@25886
   633
attach (term_of) {*
berghofe@25886
   634
fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
berghofe@25886
   635
*}
berghofe@25886
   636
attach (test) {*
berghofe@25886
   637
fun gen_fun_type aF aT bG bT i =
berghofe@25886
   638
  let
wenzelm@32740
   639
    val tab = Unsynchronized.ref [];
berghofe@25886
   640
    fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
berghofe@25886
   641
      (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
berghofe@25886
   642
  in
berghofe@25886
   643
    (fn x =>
berghofe@25886
   644
       case AList.lookup op = (!tab) x of
berghofe@25886
   645
         NONE =>
berghofe@25886
   646
           let val p as (y, _) = bG i
berghofe@25886
   647
           in (tab := (x, p) :: !tab; y) end
berghofe@25886
   648
       | SOME (y, _) => y,
berghofe@28711
   649
     fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
berghofe@25886
   650
  end;
berghofe@25886
   651
*}
berghofe@25886
   652
haftmann@21870
   653
code_const "op \<circ>"
haftmann@21870
   654
  (SML infixl 5 "o")
haftmann@21870
   655
  (Haskell infixr 9 ".")
haftmann@21870
   656
haftmann@21906
   657
code_const "id"
haftmann@21906
   658
  (Haskell "id")
haftmann@21906
   659
nipkow@2912
   660
end