src/HOL/Library/FuncSet.thy
changeset 13586 0f339348df0e
child 13593 e39f0751e4bf
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/Library/FuncSet.thy	Thu Sep 26 10:51:58 2002 +0200
     1.3 @@ -0,0 +1,177 @@
     1.4 +(*  Title:      HOL/Library/FuncSet.thy
     1.5 +    ID:         $Id$
     1.6 +    Author:     Florian Kammueller and Lawrence C Paulson
     1.7 +*)
     1.8 +
     1.9 +header {*
    1.10 +  \title{Pi and Function Sets}
    1.11 +  \author{Florian Kammueller and Lawrence C Paulson}
    1.12 +*}
    1.13 +
    1.14 +theory FuncSet = Main:
    1.15 +
    1.16 +constdefs
    1.17 +  Pi      :: "['a set, 'a => 'b set] => ('a => 'b) set"
    1.18 +    "Pi A B == {f. \<forall>x. x \<in> A --> f(x) \<in> B(x)}"
    1.19 +
    1.20 +  extensional :: "'a set => ('a => 'b) set"
    1.21 +    "extensional A == {f. \<forall>x. x~:A --> f(x) = arbitrary}"
    1.22 +
    1.23 +  restrict :: "['a => 'b, 'a set] => ('a => 'b)"
    1.24 +    "restrict f A == (%x. if x \<in> A then f x else arbitrary)"
    1.25 +
    1.26 +syntax
    1.27 +  "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
    1.28 +  funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
    1.29 +  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
    1.30 +
    1.31 +syntax (xsymbols)
    1.32 +  "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
    1.33 +  funcset :: "['a set, 'b set] => ('a => 'b) set"  (infixr "\<rightarrow>" 60) 
    1.34 +  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
    1.35 +
    1.36 +translations
    1.37 +  "PI x:A. B" => "Pi A (%x. B)"
    1.38 +  "A -> B"    => "Pi A (_K B)"
    1.39 +  "%x:A. f"  == "restrict (%x. f) A"
    1.40 +
    1.41 +constdefs
    1.42 +  compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
    1.43 +  "compose A g f == \<lambda>x\<in>A. g (f x)"
    1.44 +
    1.45 +
    1.46 +
    1.47 +subsection{*Basic Properties of @{term Pi}*}
    1.48 +
    1.49 +lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
    1.50 +by (simp add: Pi_def)
    1.51 +
    1.52 +lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
    1.53 +by (simp add: Pi_def)
    1.54 +
    1.55 +lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
    1.56 +apply (simp add: Pi_def)
    1.57 +done
    1.58 +
    1.59 +lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
    1.60 +by (simp add: Pi_def)
    1.61 +
    1.62 +lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
    1.63 +apply (simp add: Pi_def)
    1.64 +apply auto
    1.65 +txt{*Converse direction requires Axiom of Choice to exhibit a function
    1.66 +picking an element from each non-empty @{term "B x"}*}
    1.67 +apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec) 
    1.68 +apply (auto );
    1.69 +apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex)
    1.70 +apply (auto ); 
    1.71 +done
    1.72 +
    1.73 +lemma Pi_empty: "Pi {} B = UNIV"
    1.74 +apply (simp add: Pi_def) 
    1.75 +done
    1.76 +
    1.77 +text{*Covariance of Pi-sets in their second argument*}
    1.78 +lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
    1.79 +by (simp add: Pi_def, blast)
    1.80 +
    1.81 +text{*Contravariance of Pi-sets in their first argument*}
    1.82 +lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
    1.83 +by (simp add: Pi_def, blast)
    1.84 +
    1.85 +
    1.86 +subsection{*Composition With a Restricted Domain: @{term compose}*}
    1.87 +
    1.88 +lemma funcset_compose: 
    1.89 +     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
    1.90 +by (simp add: Pi_def compose_def restrict_def)
    1.91 +
    1.92 +lemma compose_assoc:
    1.93 +     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |] 
    1.94 +      ==> compose A h (compose A g f) = compose A (compose B h g) f"
    1.95 +by (simp add: expand_fun_eq Pi_def compose_def restrict_def) 
    1.96 +
    1.97 +lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
    1.98 +apply (simp add: compose_def restrict_def)
    1.99 +done
   1.100 +
   1.101 +lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
   1.102 +apply (auto simp add: image_def compose_eq)
   1.103 +done
   1.104 +
   1.105 +lemma inj_on_compose:
   1.106 +     "[| f ` A = B; inj_on f A; inj_on g B |] ==> inj_on (compose A g f) A"
   1.107 +by (auto simp add: inj_on_def compose_eq)
   1.108 +
   1.109 +
   1.110 +subsection{*Bounded Abstraction: @{term restrict}*}
   1.111 +
   1.112 +lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
   1.113 +by (simp add: Pi_def restrict_def)
   1.114 +
   1.115 +
   1.116 +lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
   1.117 +by (simp add: Pi_def restrict_def)
   1.118 +
   1.119 +lemma restrict_apply [simp]:
   1.120 +     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
   1.121 +by (simp add: restrict_def)
   1.122 +
   1.123 +lemma restrict_ext: 
   1.124 +    "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
   1.125 +by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
   1.126 +
   1.127 +lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"
   1.128 +apply (simp add: inj_on_def restrict_def)
   1.129 +done
   1.130 +
   1.131 +
   1.132 +lemma Id_compose:
   1.133 +     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
   1.134 +by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
   1.135 +
   1.136 +lemma compose_Id:
   1.137 +     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
   1.138 +by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
   1.139 +
   1.140 +
   1.141 +subsection{*Extensionality*}
   1.142 +
   1.143 +lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
   1.144 +apply (simp add: extensional_def)
   1.145 +done
   1.146 +
   1.147 +lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
   1.148 +by (simp add: restrict_def extensional_def)
   1.149 +
   1.150 +lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
   1.151 +by (simp add: compose_def)
   1.152 +
   1.153 +lemma extensionalityI:
   1.154 +     "[| f \<in> extensional A; g \<in> extensional A; 
   1.155 +         !!x. x\<in>A ==> f x = g x |] ==> f = g"
   1.156 +by (force simp add: expand_fun_eq extensional_def)
   1.157 +
   1.158 +lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
   1.159 +apply (unfold Inv_def)
   1.160 +apply (fast intro: restrict_in_funcset someI2)
   1.161 +done
   1.162 +
   1.163 +lemma compose_Inv_id:
   1.164 +     "[| inj_on f A;  f ` A = B |]  
   1.165 +      ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
   1.166 +apply (simp add: compose_def)
   1.167 +apply (rule restrict_ext)
   1.168 +apply auto
   1.169 +apply (erule subst)
   1.170 +apply (simp add: Inv_f_f)
   1.171 +done
   1.172 +
   1.173 +lemma compose_id_Inv:
   1.174 +     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
   1.175 +apply (simp add: compose_def)
   1.176 +apply (rule restrict_ext)
   1.177 +apply (simp add: f_Inv_f)
   1.178 +done
   1.179 +
   1.180 +end