src/HOL/Library/FuncSet.thy
 author paulson Fri Sep 27 16:44:50 2002 +0200 (2002-09-27) changeset 13595 7e6cdcd113a2 parent 13593 e39f0751e4bf child 14565 c6dc17aab88a permissions -rw-r--r--
Proof tidying
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammueller and Lawrence C Paulson
```
```     4 *)
```
```     5
```
```     6 header {*
```
```     7   \title{Pi and Function Sets}
```
```     8   \author{Florian Kammueller and Lawrence C Paulson}
```
```     9 *}
```
```    10
```
```    11 theory FuncSet = Main:
```
```    12
```
```    13 constdefs
```
```    14   Pi      :: "['a set, 'a => 'b set] => ('a => 'b) set"
```
```    15     "Pi A B == {f. \<forall>x. x \<in> A --> f(x) \<in> B(x)}"
```
```    16
```
```    17   extensional :: "'a set => ('a => 'b) set"
```
```    18     "extensional A == {f. \<forall>x. x~:A --> f(x) = arbitrary}"
```
```    19
```
```    20   restrict :: "['a => 'b, 'a set] => ('a => 'b)"
```
```    21     "restrict f A == (%x. if x \<in> A then f x else arbitrary)"
```
```    22
```
```    23 syntax
```
```    24   "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
```
```    25   funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
```
```    26   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
```
```    27
```
```    28 syntax (xsymbols)
```
```    29   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
```
```    30   funcset :: "['a set, 'b set] => ('a => 'b) set"  (infixr "\<rightarrow>" 60)
```
```    31   "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
```
```    32
```
```    33 translations
```
```    34   "PI x:A. B" => "Pi A (%x. B)"
```
```    35   "A -> B"    => "Pi A (_K B)"
```
```    36   "%x:A. f"  == "restrict (%x. f) A"
```
```    37
```
```    38 constdefs
```
```    39   compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
```
```    40   "compose A g f == \<lambda>x\<in>A. g (f x)"
```
```    41
```
```    42 print_translation {* [("Pi", dependent_tr' ("@Pi", "funcset"))] *}
```
```    43
```
```    44
```
```    45 subsection{*Basic Properties of @{term Pi}*}
```
```    46
```
```    47 lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    48 by (simp add: Pi_def)
```
```    49
```
```    50 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    51 by (simp add: Pi_def)
```
```    52
```
```    53 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    54 by (simp add: Pi_def)
```
```    55
```
```    56 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    57 by (simp add: Pi_def)
```
```    58
```
```    59 lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    60 apply (simp add: Pi_def, auto)
```
```    61 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    62 picking an element from each non-empty @{term "B x"}*}
```
```    63 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    64 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    65 done
```
```    66
```
```    67 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    68 by (simp add: Pi_def)
```
```    69
```
```    70 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```    71 by (simp add: Pi_def)
```
```    72
```
```    73 text{*Covariance of Pi-sets in their second argument*}
```
```    74 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```    75 by (simp add: Pi_def, blast)
```
```    76
```
```    77 text{*Contravariance of Pi-sets in their first argument*}
```
```    78 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```    79 by (simp add: Pi_def, blast)
```
```    80
```
```    81
```
```    82 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```    83
```
```    84 lemma funcset_compose:
```
```    85      "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```    86 by (simp add: Pi_def compose_def restrict_def)
```
```    87
```
```    88 lemma compose_assoc:
```
```    89      "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```    90       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```    91 by (simp add: expand_fun_eq Pi_def compose_def restrict_def)
```
```    92
```
```    93 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```    94 by (simp add: compose_def restrict_def)
```
```    95
```
```    96 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```    97 by (auto simp add: image_def compose_eq)
```
```    98
```
```    99 lemma inj_on_compose:
```
```   100      "[| f ` A = B; inj_on f A; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   101 by (auto simp add: inj_on_def compose_eq)
```
```   102
```
```   103
```
```   104 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   105
```
```   106 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   107 by (simp add: Pi_def restrict_def)
```
```   108
```
```   109 lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   110 by (simp add: Pi_def restrict_def)
```
```   111
```
```   112 lemma restrict_apply [simp]:
```
```   113      "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
```
```   114 by (simp add: restrict_def)
```
```   115
```
```   116 lemma restrict_ext:
```
```   117     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   118 by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
```
```   119
```
```   120 lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"
```
```   121 by (simp add: inj_on_def restrict_def)
```
```   122
```
```   123
```
```   124 lemma Id_compose:
```
```   125      "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   126 by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   127
```
```   128 lemma compose_Id:
```
```   129      "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   130 by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
```
```   131
```
```   132
```
```   133 subsection{*Extensionality*}
```
```   134
```
```   135 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
```
```   136 by (simp add: extensional_def)
```
```   137
```
```   138 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   139 by (simp add: restrict_def extensional_def)
```
```   140
```
```   141 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   142 by (simp add: compose_def)
```
```   143
```
```   144 lemma extensionalityI:
```
```   145      "[| f \<in> extensional A; g \<in> extensional A;
```
```   146          !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   147 by (force simp add: expand_fun_eq extensional_def)
```
```   148
```
```   149 lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
```
```   150 apply (unfold Inv_def)
```
```   151 apply (fast intro: restrict_in_funcset someI2)
```
```   152 done
```
```   153
```
```   154 lemma compose_Inv_id:
```
```   155      "[| inj_on f A;  f ` A = B |]
```
```   156       ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
```
```   157 apply (simp add: compose_def)
```
```   158 apply (rule restrict_ext, auto)
```
```   159 apply (erule subst)
```
```   160 apply (simp add: Inv_f_f)
```
```   161 done
```
```   162
```
```   163 lemma compose_id_Inv:
```
```   164      "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
```
```   165 apply (simp add: compose_def)
```
```   166 apply (rule restrict_ext)
```
```   167 apply (simp add: f_Inv_f)
```
```   168 done
```
```   169
```
```   170 end
```