src/HOL/Library/FuncSet.thy
author nipkow
Tue Sep 07 10:05:19 2010 +0200 (2010-09-07)
changeset 39198 f967a16dfcdd
parent 38656 d5d342611edb
child 39302 d7728f65b353
permissions -rw-r--r--
expand_fun_eq -> ext_iff
expand_set_eq -> set_ext_iff
Naming in line now with multisets
paulson@13586
     1
(*  Title:      HOL/Library/FuncSet.thy
paulson@13586
     2
    Author:     Florian Kammueller and Lawrence C Paulson
paulson@13586
     3
*)
paulson@13586
     4
wenzelm@14706
     5
header {* Pi and Function Sets *}
paulson@13586
     6
nipkow@15131
     7
theory FuncSet
haftmann@30663
     8
imports Hilbert_Choice Main
nipkow@15131
     9
begin
paulson@13586
    10
wenzelm@19736
    11
definition
wenzelm@21404
    12
  Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
wenzelm@19736
    13
  "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
paulson@13586
    14
wenzelm@21404
    15
definition
wenzelm@21404
    16
  extensional :: "'a set => ('a => 'b) set" where
haftmann@28524
    17
  "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
paulson@13586
    18
wenzelm@21404
    19
definition
wenzelm@21404
    20
  "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
haftmann@28524
    21
  "restrict f A = (%x. if x \<in> A then f x else undefined)"
paulson@13586
    22
wenzelm@19536
    23
abbreviation
wenzelm@21404
    24
  funcset :: "['a set, 'b set] => ('a => 'b) set"
wenzelm@21404
    25
    (infixr "->" 60) where
wenzelm@19536
    26
  "A -> B == Pi A (%_. B)"
wenzelm@19536
    27
wenzelm@21210
    28
notation (xsymbols)
wenzelm@19656
    29
  funcset  (infixr "\<rightarrow>" 60)
wenzelm@19536
    30
paulson@13586
    31
syntax
wenzelm@19736
    32
  "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
wenzelm@19736
    33
  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
paulson@13586
    34
paulson@13586
    35
syntax (xsymbols)
wenzelm@19736
    36
  "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
wenzelm@19736
    37
  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
paulson@13586
    38
kleing@14565
    39
syntax (HTML output)
wenzelm@19736
    40
  "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
wenzelm@19736
    41
  "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
kleing@14565
    42
paulson@13586
    43
translations
wenzelm@20770
    44
  "PI x:A. B" == "CONST Pi A (%x. B)"
wenzelm@20770
    45
  "%x:A. f" == "CONST restrict (%x. f) A"
paulson@13586
    46
wenzelm@19736
    47
definition
wenzelm@21404
    48
  "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
wenzelm@19736
    49
  "compose A g f = (\<lambda>x\<in>A. g (f x))"
paulson@13586
    50
paulson@13586
    51
paulson@13586
    52
subsection{*Basic Properties of @{term Pi}*}
paulson@13586
    53
nipkow@31754
    54
lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
wenzelm@14706
    55
  by (simp add: Pi_def)
paulson@13586
    56
nipkow@31731
    57
lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
nipkow@31731
    58
by(simp add:Pi_def)
nipkow@31731
    59
paulson@13586
    60
lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
wenzelm@14706
    61
  by (simp add: Pi_def)
paulson@13586
    62
paulson@13586
    63
lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
wenzelm@14706
    64
  by (simp add: Pi_def)
paulson@13586
    65
nipkow@31759
    66
lemma PiE [elim]:
nipkow@31754
    67
  "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
nipkow@31754
    68
by(auto simp: Pi_def)
nipkow@31754
    69
hoelzl@38656
    70
lemma Pi_cong:
hoelzl@38656
    71
  "(\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
hoelzl@38656
    72
  by (auto simp: Pi_def)
hoelzl@38656
    73
haftmann@31769
    74
lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
haftmann@31769
    75
  by (auto intro: Pi_I)
haftmann@31769
    76
paulson@13586
    77
lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
wenzelm@14706
    78
  by (simp add: Pi_def)
paulson@13586
    79
paulson@14762
    80
lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
nipkow@31754
    81
by auto
paulson@14762
    82
nipkow@31754
    83
lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
paulson@13593
    84
apply (simp add: Pi_def, auto)
paulson@13586
    85
txt{*Converse direction requires Axiom of Choice to exhibit a function
paulson@13586
    86
picking an element from each non-empty @{term "B x"}*}
paulson@13593
    87
apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
wenzelm@14706
    88
apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
paulson@13586
    89
done
paulson@13586
    90
paulson@13593
    91
lemma Pi_empty [simp]: "Pi {} B = UNIV"
nipkow@31754
    92
by (simp add: Pi_def)
paulson@13593
    93
paulson@13593
    94
lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
nipkow@31754
    95
by (simp add: Pi_def)
nipkow@31727
    96
(*
nipkow@31727
    97
lemma funcset_id [simp]: "(%x. x): A -> A"
nipkow@31727
    98
  by (simp add: Pi_def)
nipkow@31727
    99
*)
paulson@13586
   100
text{*Covariance of Pi-sets in their second argument*}
paulson@13586
   101
lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
nipkow@31754
   102
by auto
paulson@13586
   103
paulson@13586
   104
text{*Contravariance of Pi-sets in their first argument*}
paulson@13586
   105
lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
nipkow@31754
   106
by auto
paulson@13586
   107
paulson@33271
   108
lemma prod_final:
paulson@33271
   109
  assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
paulson@33271
   110
  shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
paulson@33271
   111
proof (rule Pi_I) 
paulson@33271
   112
  fix z
paulson@33271
   113
  assume z: "z \<in> A" 
paulson@33271
   114
  have "f z = (fst (f z), snd (f z))" 
paulson@33271
   115
    by simp
paulson@33271
   116
  also have "...  \<in> B z \<times> C z"
paulson@33271
   117
    by (metis SigmaI PiE o_apply 1 2 z) 
paulson@33271
   118
  finally show "f z \<in> B z \<times> C z" .
paulson@33271
   119
qed
paulson@33271
   120
paulson@13586
   121
paulson@13586
   122
subsection{*Composition With a Restricted Domain: @{term compose}*}
paulson@13586
   123
wenzelm@14706
   124
lemma funcset_compose:
nipkow@31754
   125
  "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
nipkow@31754
   126
by (simp add: Pi_def compose_def restrict_def)
paulson@13586
   127
paulson@13586
   128
lemma compose_assoc:
wenzelm@14706
   129
    "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
paulson@13586
   130
      ==> compose A h (compose A g f) = compose A (compose B h g) f"
nipkow@39198
   131
by (simp add: ext_iff Pi_def compose_def restrict_def)
paulson@13586
   132
paulson@13586
   133
lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
nipkow@31754
   134
by (simp add: compose_def restrict_def)
paulson@13586
   135
paulson@13586
   136
lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
wenzelm@14706
   137
  by (auto simp add: image_def compose_eq)
paulson@13586
   138
paulson@13586
   139
paulson@13586
   140
subsection{*Bounded Abstraction: @{term restrict}*}
paulson@13586
   141
paulson@13586
   142
lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
wenzelm@14706
   143
  by (simp add: Pi_def restrict_def)
paulson@13586
   144
nipkow@31754
   145
lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
wenzelm@14706
   146
  by (simp add: Pi_def restrict_def)
paulson@13586
   147
paulson@13586
   148
lemma restrict_apply [simp]:
haftmann@28524
   149
    "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
wenzelm@14706
   150
  by (simp add: restrict_def)
paulson@13586
   151
wenzelm@14706
   152
lemma restrict_ext:
paulson@13586
   153
    "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
nipkow@39198
   154
  by (simp add: ext_iff Pi_def restrict_def)
paulson@13586
   155
paulson@14853
   156
lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
wenzelm@14706
   157
  by (simp add: inj_on_def restrict_def)
paulson@13586
   158
paulson@13586
   159
lemma Id_compose:
wenzelm@14706
   160
    "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
nipkow@39198
   161
  by (auto simp add: ext_iff compose_def extensional_def Pi_def)
paulson@13586
   162
paulson@13586
   163
lemma compose_Id:
wenzelm@14706
   164
    "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
nipkow@39198
   165
  by (auto simp add: ext_iff compose_def extensional_def Pi_def)
paulson@13586
   166
paulson@14853
   167
lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
wenzelm@19736
   168
  by (auto simp add: restrict_def)
paulson@13586
   169
paulson@14745
   170
paulson@14762
   171
subsection{*Bijections Between Sets*}
paulson@14762
   172
nipkow@26106
   173
text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
paulson@14762
   174
the theorems belong here, or need at least @{term Hilbert_Choice}.*}
paulson@14762
   175
paulson@14762
   176
lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
nipkow@32988
   177
by (auto simp add: bij_betw_def)
paulson@14762
   178
paulson@14853
   179
lemma inj_on_compose:
nipkow@31754
   180
  "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
nipkow@31754
   181
by (auto simp add: bij_betw_def inj_on_def compose_eq)
paulson@14853
   182
paulson@14762
   183
lemma bij_betw_compose:
nipkow@31754
   184
  "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
nipkow@31754
   185
apply (simp add: bij_betw_def compose_eq inj_on_compose)
nipkow@31754
   186
apply (auto simp add: compose_def image_def)
nipkow@31754
   187
done
paulson@14762
   188
paulson@14853
   189
lemma bij_betw_restrict_eq [simp]:
nipkow@31754
   190
  "bij_betw (restrict f A) A B = bij_betw f A B"
nipkow@31754
   191
by (simp add: bij_betw_def)
paulson@14853
   192
paulson@14853
   193
paulson@14853
   194
subsection{*Extensionality*}
paulson@14853
   195
haftmann@28524
   196
lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
nipkow@31754
   197
by (simp add: extensional_def)
paulson@14853
   198
paulson@14853
   199
lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
nipkow@31754
   200
by (simp add: restrict_def extensional_def)
paulson@14853
   201
paulson@14853
   202
lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
nipkow@31754
   203
by (simp add: compose_def)
paulson@14853
   204
paulson@14853
   205
lemma extensionalityI:
nipkow@31754
   206
  "[| f \<in> extensional A; g \<in> extensional A;
paulson@14853
   207
      !!x. x\<in>A ==> f x = g x |] ==> f = g"
nipkow@39198
   208
by (force simp add: ext_iff extensional_def)
paulson@14853
   209
nipkow@33057
   210
lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
nipkow@33057
   211
by (unfold inv_into_def) (fast intro: someI2)
paulson@14853
   212
nipkow@33057
   213
lemma compose_inv_into_id:
nipkow@33057
   214
  "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
nipkow@31754
   215
apply (simp add: bij_betw_def compose_def)
nipkow@31754
   216
apply (rule restrict_ext, auto)
nipkow@31754
   217
done
paulson@14853
   218
nipkow@33057
   219
lemma compose_id_inv_into:
nipkow@33057
   220
  "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
nipkow@31754
   221
apply (simp add: compose_def)
nipkow@31754
   222
apply (rule restrict_ext)
nipkow@33057
   223
apply (simp add: f_inv_into_f)
nipkow@31754
   224
done
paulson@14853
   225
paulson@14762
   226
paulson@14745
   227
subsection{*Cardinality*}
paulson@14745
   228
paulson@14745
   229
lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
nipkow@31754
   230
by (rule card_inj_on_le) auto
paulson@14745
   231
paulson@14745
   232
lemma card_bij:
nipkow@31754
   233
  "[|f \<in> A\<rightarrow>B; inj_on f A;
nipkow@31754
   234
     g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
nipkow@31754
   235
by (blast intro: card_inj order_antisym)
paulson@14745
   236
paulson@13586
   237
end