src/HOL/Library/FuncSet.thy
 author chaieb Mon Feb 09 17:21:46 2009 +0000 (2009-02-09) changeset 29847 af32126ee729 parent 28524 644b62cf678f child 30663 0b6aff7451b2 permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/FuncSet.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Florian Kammueller and Lawrence C Paulson
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```     4 *)
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```     5
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```     6 header {* Pi and Function Sets *}
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```     7
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```     8 theory FuncSet
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```     9 imports Plain "~~/src/HOL/Hilbert_Choice"
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```    10 begin
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```    11
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```    12 definition
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```    13   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
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```    14   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    15
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```    16 definition
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```    17   extensional :: "'a set => ('a => 'b) set" where
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```    18   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
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```    19
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```    20 definition
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```    21   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
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```    22   "restrict f A = (%x. if x \<in> A then f x else undefined)"
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```    23
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```    24 abbreviation
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```    25   funcset :: "['a set, 'b set] => ('a => 'b) set"
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```    26     (infixr "->" 60) where
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```    27   "A -> B == Pi A (%_. B)"
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```    28
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```    29 notation (xsymbols)
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```    30   funcset  (infixr "\<rightarrow>" 60)
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```    31
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```    32 syntax
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```    33   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
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```    34   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    35
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```    36 syntax (xsymbols)
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```    37   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    38   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    39
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```    40 syntax (HTML output)
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```    41   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
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```    42   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    43
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```    44 translations
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```    45   "PI x:A. B" == "CONST Pi A (%x. B)"
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```    46   "%x:A. f" == "CONST restrict (%x. f) A"
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```    47
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```    48 definition
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```    49   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
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```    50   "compose A g f = (\<lambda>x\<in>A. g (f x))"
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```    51
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```    52
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```    53 subsection{*Basic Properties of @{term Pi}*}
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```    54
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```    55 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    56   by (simp add: Pi_def)
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```    57
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```    58 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    59   by (simp add: Pi_def)
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```    60
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```    61 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
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```    62   by (simp add: Pi_def)
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```    63
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```    64 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
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```    65   by (simp add: Pi_def)
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```    66
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```    67 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
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```    68   by (auto simp add: Pi_def)
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```    69
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```    70 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
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```    71 apply (simp add: Pi_def, auto)
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```    72 txt{*Converse direction requires Axiom of Choice to exhibit a function
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```    73 picking an element from each non-empty @{term "B x"}*}
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```    74 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
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```    75 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
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```    76 done
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```    77
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```    78 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    79   by (simp add: Pi_def)
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```    80
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```    81 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
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```    82   by (simp add: Pi_def)
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```    83
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```    84 text{*Covariance of Pi-sets in their second argument*}
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```    85 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
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```    86   by (simp add: Pi_def, blast)
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```    87
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```    88 text{*Contravariance of Pi-sets in their first argument*}
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```    89 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
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```    90   by (simp add: Pi_def, blast)
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```    91
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```    92
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```    93 subsection{*Composition With a Restricted Domain: @{term compose}*}
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```    94
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```    95 lemma funcset_compose:
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```    96     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```    97   by (simp add: Pi_def compose_def restrict_def)
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```    98
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```    99 lemma compose_assoc:
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```   100     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
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```   101       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   102   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
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```   103
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```   104 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
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```   105   by (simp add: compose_def restrict_def)
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```   106
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```   107 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
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```   108   by (auto simp add: image_def compose_eq)
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```   109
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```   110
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```   111 subsection{*Bounded Abstraction: @{term restrict}*}
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```   112
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```   113 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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```   114   by (simp add: Pi_def restrict_def)
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```   115
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```   116 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
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```   117   by (simp add: Pi_def restrict_def)
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```   118
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```   119 lemma restrict_apply [simp]:
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```   120     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
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```   121   by (simp add: restrict_def)
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```   122
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```   123 lemma restrict_ext:
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```   124     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
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```   125   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
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```   126
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```   127 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
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```   128   by (simp add: inj_on_def restrict_def)
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```   129
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```   130 lemma Id_compose:
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```   131     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   132   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
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```   133
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```   134 lemma compose_Id:
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```   135     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
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```   136   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
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```   137
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```   138 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
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```   139   by (auto simp add: restrict_def)
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```   140
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```   141
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```   142 subsection{*Bijections Between Sets*}
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```   143
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```   144 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
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```   145 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
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```   146
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```   147 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
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```   148   by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
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```   149
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```   150 lemma inj_on_compose:
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```   151     "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
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```   152   by (auto simp add: bij_betw_def inj_on_def compose_eq)
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```   153
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```   154 lemma bij_betw_compose:
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```   155     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
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```   156   apply (simp add: bij_betw_def compose_eq inj_on_compose)
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```   157   apply (auto simp add: compose_def image_def)
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```   158   done
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```   159
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```   160 lemma bij_betw_restrict_eq [simp]:
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```   161      "bij_betw (restrict f A) A B = bij_betw f A B"
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```   162   by (simp add: bij_betw_def)
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```   163
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```   164
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```   165 subsection{*Extensionality*}
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```   166
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```   167 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
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```   168   by (simp add: extensional_def)
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```   169
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```   170 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
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```   171   by (simp add: restrict_def extensional_def)
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```   172
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```   173 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
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```   174   by (simp add: compose_def)
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```   175
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```   176 lemma extensionalityI:
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```   177     "[| f \<in> extensional A; g \<in> extensional A;
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```   178       !!x. x\<in>A ==> f x = g x |] ==> f = g"
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```   179   by (force simp add: expand_fun_eq extensional_def)
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```   180
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```   181 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
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```   182   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
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```   183
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```   184 lemma compose_Inv_id:
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```   185     "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
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```   186   apply (simp add: bij_betw_def compose_def)
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```   187   apply (rule restrict_ext, auto)
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```   188   apply (erule subst)
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```   189   apply (simp add: Inv_f_f)
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```   190   done
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```   191
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```   192 lemma compose_id_Inv:
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```   193     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
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```   194   apply (simp add: compose_def)
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```   195   apply (rule restrict_ext)
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```   196   apply (simp add: f_Inv_f)
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```   197   done
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```   198
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```   199
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```   200 subsection{*Cardinality*}
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```   201
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```   202 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
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```   203   apply (rule card_inj_on_le)
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```   204     apply (auto simp add: Pi_def)
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```   205   done
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```   206
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```   207 lemma card_bij:
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```   208      "[|f \<in> A\<rightarrow>B; inj_on f A;
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```   209         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   210   by (blast intro: card_inj order_antisym)
```
```   211
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```   212 end
```