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}
-*}
+header {* Pi and Function Sets *}
 
 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)}"
+  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}"
+  "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)"
+  "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) 
+  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)
 
 syntax (HTML output)
   "@Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
   "@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"
+  "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 set, 'b => 'c, 'a => 'b] => ('a => 'c)"
   "compose A g f == \<lambda>x\<in>A. g (f x)"
 
 print_translation {* [("Pi", dependent_tr' ("@Pi", "funcset"))] *}
 
 
 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)
+  by (simp add: Pi_def)
 
 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
-by (simp add: Pi_def)
+  by (simp add: Pi_def)
 
 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
-by (simp add: Pi_def)
+  by (simp add: Pi_def)
 
 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
-by (simp add: Pi_def)
+  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, 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, auto)
-apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto) 
+apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
 done
 
 lemma Pi_empty [simp]: "Pi {} B = UNIV"
-by (simp add: Pi_def)
+  by (simp add: Pi_def)
 
 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
-by (simp add: Pi_def)
+  by (simp add: Pi_def)
 
 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)
+  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)
+  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 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 |] 
+    "[| 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) 
+  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))"
-by (simp add: compose_def restrict_def)
+  by (simp add: compose_def restrict_def)
 
 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
-by (auto simp add: image_def compose_eq)
+  by (auto simp add: image_def compose_eq)
 
 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)
+    "[| 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)
+  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)
+  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)
+    "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else arbitrary)"
+  by (simp add: restrict_def)
 
-lemma restrict_ext: 
+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)
+  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"
-by (simp add: inj_on_def restrict_def)
-
+  by (simp add: inj_on_def restrict_def)
 
 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)
+    "[|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)
+    "[|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"
-by (simp add: extensional_def)
+  by (simp add: extensional_def)
 
 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
-by (simp add: restrict_def extensional_def)
+  by (simp add: restrict_def extensional_def)
 
 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
-by (simp add: compose_def)
+  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)
+    "[| 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
+  by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
 
 lemma compose_Inv_id:
-     "[| inj_on f A;  f ` A = B |]  
+    "[| 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, auto)
-apply (erule subst)
-apply (simp add: Inv_f_f)
-done
+  apply (simp add: compose_def)
+  apply (rule restrict_ext, 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
+    "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