src/HOL/Library/FuncSet.thy

+(*  Title:      HOL/Library/FuncSet.thy
+    ID:         $Id$
+    Author:     Florian Kammueller and Lawrence C Paulson
+*)
+
+header {*
+  \title{Pi and Function Sets}
+  \author{Florian Kammueller and Lawrence C Paulson}
+*}
+
+theory FuncSet = Main:
+
+constdefs
+  Pi      :: "['a set, 'a => 'b set] => ('a => 'b) set"
+    "Pi A B == {f. \<forall>x. x \<in> A --> f(x) \<in> B(x)}"
+
+  extensional :: "'a set => ('a => 'b) set"
+    "extensional A == {f. \<forall>x. x~:A --> f(x) = arbitrary}"
+
+  restrict :: "['a => 'b, 'a set] => ('a => 'b)"
+    "restrict f A == (%x. if x \<in> A then f x else arbitrary)"
+
+syntax
+  "@Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
+  funcset :: "['a set, 'b set] => ('a => 'b) set"      (infixr "->" 60)
+  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
+
+syntax (xsymbols)
+  "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
+  funcset :: "['a set, 'b set] => ('a => 'b) set"  (infixr "\<rightarrow>" 60) 
+  "@lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
+
+translations
+  "PI x:A. B" => "Pi A (%x. B)"
+  "A -> B"    => "Pi A (_K B)"
+  "%x:A. f"  == "restrict (%x. f) A"
+
+constdefs
+  compose :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)"
+  "compose A g f == \<lambda>x\<in>A. g (f x)"
+
+
+
+subsection{*Basic Properties of @{term Pi}*}
+
+lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
+by (simp add: Pi_def)
+
+lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
+by (simp add: Pi_def)
+
+lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
+apply (simp add: Pi_def)
+done
+
+lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
+by (simp add: Pi_def)
+
+lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
+apply (simp add: Pi_def)
+apply auto
+txt{*Converse direction requires Axiom of Choice to exhibit a function
+picking an element from each non-empty @{term "B x"}*}
+apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec) 
+apply (auto );
+apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex)
+apply (auto ); 
+done
+
+lemma Pi_empty: "Pi {} B = UNIV"
+apply (simp add: Pi_def) 
+done
+
+text{*Covariance of Pi-sets in their second argument*}
+lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
+by (simp add: Pi_def, blast)
+
+text{*Contravariance of Pi-sets in their first argument*}
+lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
+by (simp add: Pi_def, blast)
+
+
+subsection{*Composition With a Restricted Domain: @{term compose}*}
+
+lemma funcset_compose: 
+     "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
+by (simp add: Pi_def compose_def restrict_def)
+
+lemma compose_assoc:
+     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |] 
+      ==> compose A h (compose A g f) = compose A (compose B h g) f"
+by (simp add: expand_fun_eq Pi_def compose_def restrict_def) 
+
+lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
+apply (simp add: compose_def restrict_def)
+done
+
+lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
+apply (auto simp add: image_def compose_eq)
+done
+
+lemma inj_on_compose:
+     "[| f ` A = B; inj_on f A; inj_on g B |] ==> inj_on (compose A g f) A"
+by (auto simp add: inj_on_def compose_eq)
+
+
+subsection{*Bounded Abstraction: @{term restrict}*}
+
+lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
+by (simp add: Pi_def restrict_def)
+
+
+lemma restrictI: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
+by (simp add: Pi_def restrict_def)
+
+lemma restrict_apply [simp]:
+     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
+by (simp add: restrict_def)
+
+lemma restrict_ext: 
+    "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
+by (simp add: expand_fun_eq Pi_def Pi_def restrict_def)
+
+lemma inj_on_restrict_eq: "inj_on (restrict f A) A = inj_on f A"
+apply (simp add: inj_on_def restrict_def)
+done
+
+
+lemma Id_compose:
+     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
+by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
+
+lemma compose_Id:
+     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
+by (auto simp add: expand_fun_eq compose_def extensional_def Pi_def)
+
+
+subsection{*Extensionality*}
+
+lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
+apply (simp add: extensional_def)
+done
+
+lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
+by (simp add: restrict_def extensional_def)
+
+lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
+by (simp add: compose_def)
+
+lemma extensionalityI:
+     "[| f \<in> extensional A; g \<in> extensional A; 
+         !!x. x\<in>A ==> f x = g x |] ==> f = g"
+by (force simp add: expand_fun_eq extensional_def)
+
+lemma Inv_funcset: "f ` A = B ==> (\<lambda>x\<in>B. Inv A f x) : B -> A"
+apply (unfold Inv_def)
+apply (fast intro: restrict_in_funcset someI2)
+done
+
+lemma compose_Inv_id:
+     "[| inj_on f A;  f ` A = B |]  
+      ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
+apply (simp add: compose_def)
+apply (rule restrict_ext)
+apply auto
+apply (erule subst)
+apply (simp add: Inv_f_f)
+done
+
+lemma compose_id_Inv:
+     "f ` A = B ==> compose B f (\<lambda>y\<in>B. Inv A f y) = (\<lambda>x\<in>B. x)"
+apply (simp add: compose_def)
+apply (rule restrict_ext)
+apply (simp add: f_Inv_f)
+done
+
+end