src/HOL/Library/FuncSet.thy
 author haftmann Wed Apr 22 19:09:21 2009 +0200 (2009-04-22) changeset 30960 fec1a04b7220 parent 30663 0b6aff7451b2 child 31727 2621a957d417 permissions -rw-r--r--
power operation defined generic
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson
```
```     3 *)
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```     4
```
```     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
```
```    11 definition
```
```    12   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
```
```    13   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
```
```    14
```
```    15 definition
```
```    16   extensional :: "'a set => ('a => 'b) set" where
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```    17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
```
```    18
```
```    19 definition
```
```    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)"
```
```    22
```
```    23 abbreviation
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```    24   funcset :: "['a set, 'b set] => ('a => 'b) set"
```
```    25     (infixr "->" 60) where
```
```    26   "A -> B == Pi A (%_. B)"
```
```    27
```
```    28 notation (xsymbols)
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```    29   funcset  (infixr "\<rightarrow>" 60)
```
```    30
```
```    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
```
```    35 syntax (xsymbols)
```
```    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
```
```    39 syntax (HTML output)
```
```    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
```
```    43 translations
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```    44   "PI x:A. B" == "CONST Pi A (%x. B)"
```
```    45   "%x:A. f" == "CONST restrict (%x. f) A"
```
```    46
```
```    47 definition
```
```    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))"
```
```    50
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```    51
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```    52 subsection{*Basic Properties of @{term Pi}*}
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```    53
```
```    54 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    58   by (simp add: Pi_def)
```
```    59
```
```    60 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    61   by (simp add: Pi_def)
```
```    62
```
```    63 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    64   by (simp add: Pi_def)
```
```    65
```
```    66 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    67   by (auto simp add: Pi_def)
```
```    68
```
```    69 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    70 apply (simp add: Pi_def, auto)
```
```    71 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    72 picking an element from each non-empty @{term "B x"}*}
```
```    73 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    74 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    75 done
```
```    76
```
```    77 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    78   by (simp add: Pi_def)
```
```    79
```
```    80 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```    81   by (simp add: Pi_def)
```
```    82
```
```    83 text{*Covariance of Pi-sets in their second argument*}
```
```    84 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```    85   by (simp add: Pi_def, blast)
```
```    86
```
```    87 text{*Contravariance of Pi-sets in their first argument*}
```
```    88 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```    89   by (simp add: Pi_def, blast)
```
```    90
```
```    91
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```    92 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```    93
```
```    94 lemma funcset_compose:
```
```    95     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```    96   by (simp add: Pi_def compose_def restrict_def)
```
```    97
```
```    98 lemma compose_assoc:
```
```    99     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```   100       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   101   by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
```
```   102
```
```   103 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```   104   by (simp add: compose_def restrict_def)
```
```   105
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```   106 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   107   by (auto simp add: image_def compose_eq)
```
```   108
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```   109
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```   110 subsection{*Bounded Abstraction: @{term restrict}*}
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```   111
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```   112 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   113   by (simp add: Pi_def restrict_def)
```
```   114
```
```   115 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   116   by (simp add: Pi_def restrict_def)
```
```   117
```
```   118 lemma restrict_apply [simp]:
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```   119     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
```
```   120   by (simp add: restrict_def)
```
```   121
```
```   122 lemma restrict_ext:
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```   123     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   124   by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
```
```   125
```
```   126 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   127   by (simp add: inj_on_def restrict_def)
```
```   128
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```   129 lemma Id_compose:
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```   130     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   131   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   132
```
```   133 lemma compose_Id:
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```   134     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   135   by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   136
```
```   137 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   138   by (auto simp add: restrict_def)
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```   139
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```   140
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```   141 subsection{*Bijections Between Sets*}
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```   142
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```   143 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
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```   144 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
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```   145
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```   146 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   147   by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
```
```   148
```
```   149 lemma inj_on_compose:
```
```   150     "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   151   by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   152
```
```   153 lemma bij_betw_compose:
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```   154     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   155   apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   156   apply (auto simp add: compose_def image_def)
```
```   157   done
```
```   158
```
```   159 lemma bij_betw_restrict_eq [simp]:
```
```   160      "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   161   by (simp add: bij_betw_def)
```
```   162
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```   163
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```   164 subsection{*Extensionality*}
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```   165
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```   166 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
```
```   167   by (simp add: extensional_def)
```
```   168
```
```   169 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   170   by (simp add: restrict_def extensional_def)
```
```   171
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```   172 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   173   by (simp add: compose_def)
```
```   174
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```   175 lemma extensionalityI:
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```   176     "[| f \<in> extensional A; g \<in> extensional A;
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```   177       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   178   by (force simp add: expand_fun_eq extensional_def)
```
```   179
```
```   180 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
```
```   181   by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
```
```   182
```
```   183 lemma compose_Inv_id:
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```   184     "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
```
```   185   apply (simp add: bij_betw_def compose_def)
```
```   186   apply (rule restrict_ext, auto)
```
```   187   apply (erule subst)
```
```   188   apply (simp add: Inv_f_f)
```
```   189   done
```
```   190
```
```   191 lemma compose_id_Inv:
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```   192     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
```
```   193   apply (simp add: compose_def)
```
```   194   apply (rule restrict_ext)
```
```   195   apply (simp add: f_Inv_f)
```
```   196   done
```
```   197
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```   198
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```   199 subsection{*Cardinality*}
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```   200
```
```   201 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   202   apply (rule card_inj_on_le)
```
```   203     apply (auto simp add: Pi_def)
```
```   204   done
```
```   205
```
```   206 lemma card_bij:
```
```   207      "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   208         g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   209   by (blast intro: card_inj order_antisym)
```
```   210
```
```   211 end
```