src/HOL/Library/FuncSet.thy
 author nipkow Sun Oct 18 12:07:25 2009 +0200 (2009-10-18) changeset 32988 d1d4d7a08a66 parent 31770 ba52fcfaec28 child 33057 764547b68538 permissions -rw-r--r--
Inv -> inv_onto, inv abbr. inv_onto UNIV.
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson
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```     3 *)
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```     4
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```     5 header {* Pi and Function Sets *}
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```     6
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```     7 theory FuncSet
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```     8 imports Hilbert_Choice Main
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```     9 begin
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```    10
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```    11 definition
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```    12   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
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```    13   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
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```    14
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```    15 definition
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```    16   extensional :: "'a set => ('a => 'b) set" where
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```    17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
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```    18
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```    19 definition
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```    20   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
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```    21   "restrict f A = (%x. if x \<in> A then f x else undefined)"
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```    22
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```    23 abbreviation
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```    24   funcset :: "['a set, 'b set] => ('a => 'b) set"
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```    25     (infixr "->" 60) where
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```    26   "A -> B == Pi A (%_. B)"
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```    27
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```    28 notation (xsymbols)
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```    29   funcset  (infixr "\<rightarrow>" 60)
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```    30
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```    31 syntax
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```    32   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    33   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    34
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```    35 syntax (xsymbols)
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```    36   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    37   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    38
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```    39 syntax (HTML output)
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```    40   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    41   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    42
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```    43 translations
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```    44   "PI x:A. B" == "CONST Pi A (%x. B)"
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```    45   "%x:A. f" == "CONST restrict (%x. f) A"
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```    46
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```    47 definition
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```    48   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
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```    49   "compose A g f = (\<lambda>x\<in>A. g (f x))"
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```    50
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```    51
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```    52 subsection{*Basic Properties of @{term Pi}*}
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```    53
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```    54 lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    55   by (simp add: Pi_def)
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```    56
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```    57 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
```
```    58 by(simp add:Pi_def)
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```    59
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```    60 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    61   by (simp add: Pi_def)
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```    62
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```    63 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    64   by (simp add: Pi_def)
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```    65
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```    66 lemma PiE [elim]:
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```    67   "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
```
```    68 by(auto simp: Pi_def)
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```    69
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```    70 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
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```    71   by (auto intro: Pi_I)
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```    72
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```    73 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
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```    74   by (simp add: Pi_def)
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```    75
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```    76 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    77 by auto
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```    78
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```    79 lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    80 apply (simp add: Pi_def, auto)
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```    81 txt{*Converse direction requires Axiom of Choice to exhibit a function
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```    82 picking an element from each non-empty @{term "B x"}*}
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```    83 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
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```    84 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
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```    85 done
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```    86
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```    87 lemma Pi_empty [simp]: "Pi {} B = UNIV"
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```    88 by (simp add: Pi_def)
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```    89
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```    90 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
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```    91 by (simp add: Pi_def)
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```    92 (*
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```    93 lemma funcset_id [simp]: "(%x. x): A -> A"
```
```    94   by (simp add: Pi_def)
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```    95 *)
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```    96 text{*Covariance of Pi-sets in their second argument*}
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```    97 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```    98 by auto
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```    99
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```   100 text{*Contravariance of Pi-sets in their first argument*}
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```   101 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
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```   102 by auto
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```   103
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```   104
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```   105 subsection{*Composition With a Restricted Domain: @{term compose}*}
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```   106
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```   107 lemma funcset_compose:
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```   108   "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```   109 by (simp add: Pi_def compose_def restrict_def)
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```   110
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```   111 lemma compose_assoc:
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```   112     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
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```   113       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   114 by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
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```   115
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```   116 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
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```   117 by (simp add: compose_def restrict_def)
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```   118
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```   119 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
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```   120   by (auto simp add: image_def compose_eq)
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```   121
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```   122
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```   123 subsection{*Bounded Abstraction: @{term restrict}*}
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```   124
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```   125 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
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```   126   by (simp add: Pi_def restrict_def)
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```   127
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```   128 lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   129   by (simp add: Pi_def restrict_def)
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```   130
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```   131 lemma restrict_apply [simp]:
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```   132     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
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```   133   by (simp add: restrict_def)
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```   134
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```   135 lemma restrict_ext:
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```   136     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   137   by (simp add: expand_fun_eq Pi_def restrict_def)
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```   138
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```   139 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
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```   140   by (simp add: inj_on_def restrict_def)
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```   141
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```   142 lemma Id_compose:
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```   143     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
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```   144   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
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```   145
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```   146 lemma compose_Id:
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```   147     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
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```   148   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
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```   149
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```   150 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
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```   151   by (auto simp add: restrict_def)
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```   152
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```   153
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```   154 subsection{*Bijections Between Sets*}
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```   155
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```   156 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
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```   157 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
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```   158
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```   159 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
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```   160 by (auto simp add: bij_betw_def)
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```   161
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```   162 lemma inj_on_compose:
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```   163   "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
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```   164 by (auto simp add: bij_betw_def inj_on_def compose_eq)
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```   165
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```   166 lemma bij_betw_compose:
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```   167   "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   168 apply (simp add: bij_betw_def compose_eq inj_on_compose)
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```   169 apply (auto simp add: compose_def image_def)
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```   170 done
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```   171
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```   172 lemma bij_betw_restrict_eq [simp]:
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```   173   "bij_betw (restrict f A) A B = bij_betw f A B"
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```   174 by (simp add: bij_betw_def)
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```   175
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```   176
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```   177 subsection{*Extensionality*}
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```   178
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```   179 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
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```   180 by (simp add: extensional_def)
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```   181
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```   182 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
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```   183 by (simp add: restrict_def extensional_def)
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```   184
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```   185 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
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```   186 by (simp add: compose_def)
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```   187
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```   188 lemma extensionalityI:
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```   189   "[| f \<in> extensional A; g \<in> extensional A;
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```   190       !!x. x\<in>A ==> f x = g x |] ==> f = g"
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```   191 by (force simp add: expand_fun_eq extensional_def)
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```   192
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```   193 lemma inv_onto_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_onto A f x) : B -> A"
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```   194 by (unfold inv_onto_def) (fast intro: someI2)
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```   195
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```   196 lemma compose_inv_onto_id:
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```   197   "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_onto A f y) f = (\<lambda>x\<in>A. x)"
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```   198 apply (simp add: bij_betw_def compose_def)
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```   199 apply (rule restrict_ext, auto)
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```   200 done
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```   201
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```   202 lemma compose_id_inv_onto:
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```   203   "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_onto A f y) = (\<lambda>x\<in>B. x)"
```
```   204 apply (simp add: compose_def)
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```   205 apply (rule restrict_ext)
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```   206 apply (simp add: f_inv_onto_f)
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```   207 done
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```   208
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```   209
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```   210 subsection{*Cardinality*}
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```   211
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```   212 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
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```   213 by (rule card_inj_on_le) auto
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```   214
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```   215 lemma card_bij:
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```   216   "[|f \<in> A\<rightarrow>B; inj_on f A;
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```   217      g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   218 by (blast intro: card_inj order_antisym)
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```   219
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```   220 end
```