# HG changeset patch
# User paulson
# Date 1086081926 -7200
# Node ID 8d710bece29f50c505ebcc243236e7f75912578f
# Parent  fffab59e0050156271507db40302e1a26f6d32f5
more on bij_betw

diff -r fffab59e0050 -r 8d710bece29f src/HOL/Algebra/Bij.thy
--- a/src/HOL/Algebra/Bij.thy	Tue Jun 01 11:25:01 2004 +0200
+++ b/src/HOL/Algebra/Bij.thy	Tue Jun 01 11:25:26 2004 +0200
@@ -10,7 +10,7 @@
 constdefs
   Bij :: "'a set => ('a => 'a) set"
     --{*Only extensional functions, since otherwise we get too many.*}
-  "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
+  "Bij S == extensional S \<inter> {f. bij_betw f S S}"
 
   BijGroup :: "'a set => ('a => 'a) monoid"
   "BijGroup S ==
@@ -25,53 +25,23 @@
   by (simp add: Bij_def)
 
 lemma Bij_imp_funcset: "f \<in> Bij S ==> f \<in> S -> S"
-  by (auto simp add: Bij_def Pi_def)
-
-lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
-  by (simp add: Bij_def)
-
-lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
-  by (simp add: Bij_def)
-
-lemma BijI: "[| f \<in> extensional(S); f`S = S; inj_on f S |] ==> f \<in> Bij S"
-  by (simp add: Bij_def)
+  by (auto simp add: Bij_def bij_betw_imp_funcset)
 
 
 subsection {*Bijections Form a Group *}
 
 lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
-apply (simp add: Bij_def)
-apply (intro conjI)
-txt{*Proving @{term "restrict (Inv S f) S ` S = S"}*}
- apply (rule equalityI)
-  apply (force simp add: Inv_mem) --{*first inclusion*}
- apply (rule subsetI)   --{*second inclusion*}
- apply (rule_tac x = "f x" in image_eqI)
-  apply (force intro:  simp add: Inv_f_f, blast)
-txt{*Remaining goal: @{term "inj_on (restrict (Inv S f) S) S"}*}
-apply (rule inj_onI)
-apply (auto elim: Inv_injective)
-done
+  by (simp add: Bij_def bij_betw_Inv)
 
 lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
-apply (rule BijI)
-apply (auto simp add: inj_on_def)
-done
+  by (auto simp add: Bij_def bij_betw_def inj_on_def)
 
 lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
-apply (rule BijI)
-  apply (simp add: compose_extensional)
- apply (blast del: equalityI
-              intro: surj_compose dest: Bij_imp_apply Bij_imp_inj_on)
-apply (blast intro: inj_on_compose dest: Bij_imp_apply Bij_imp_inj_on)
-done
+  by (auto simp add: Bij_def bij_betw_compose) 
 
 lemma Bij_compose_restrict_eq:
      "f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
-apply (rule compose_Inv_id)
- apply (simp add: Bij_imp_inj_on)
-apply (simp add: Bij_imp_apply)
-done
+  by (simp add: Bij_def compose_Inv_id)
 
 theorem group_BijGroup: "group (BijGroup S)"
 apply (simp add: BijGroup_def)
@@ -87,16 +57,16 @@
 
 subsection{*Automorphisms Form a Group*}
 
-lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x : S |] ==> Inv S f x : S"
-by (simp add: Bij_def Inv_mem)
+lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x \<in> S |] ==> Inv S f x \<in> S"
+by (simp add: Bij_def bij_betw_def Inv_mem)
 
 lemma Bij_Inv_lemma:
  assumes eq: "!!x y. [|x \<in> S; y \<in> S|] ==> h(g x y) = g (h x) (h y)"
  shows "[| h \<in> Bij S;  g \<in> S \<rightarrow> S \<rightarrow> S;  x \<in> S;  y \<in> S |]
         ==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)"
-apply (simp add: Bij_def)
-apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'", clarify)
- apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
+apply (simp add: Bij_def bij_betw_def)
+apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' & y = h y'", clarify)
+ apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast )
 done
 
 constdefs
@@ -130,7 +100,7 @@
 apply (rule group.subgroupI)
     apply (rule group_BijGroup)
    apply (force simp add: auto_def BijGroup_def)
-  apply (blast intro: dest: id_in_auto)
+  apply (blast dest: id_in_auto)
  apply (simp del: restrict_apply
              add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
 apply (auto simp add: BijGroup_def auto_def Bij_imp_funcset group.hom_compose
diff -r fffab59e0050 -r 8d710bece29f src/HOL/Library/FuncSet.thy
--- a/src/HOL/Library/FuncSet.thy	Tue Jun 01 11:25:01 2004 +0200
+++ b/src/HOL/Library/FuncSet.thy	Tue Jun 01 11:25:26 2004 +0200
@@ -100,10 +100,6 @@
 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
   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)
-
 
 subsection{*Bounded Abstraction: @{term restrict}*}
 
@@ -121,7 +117,7 @@
     "(!!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"
+lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
   by (simp add: inj_on_def restrict_def)
 
 lemma Id_compose:
@@ -132,41 +128,8 @@
     "[|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)
-
-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"
-  by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
-
-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, 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
+lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
+  by (auto simp add: restrict_def) 
 
 
 subsection{*Bijections Between Sets*}
@@ -190,12 +153,55 @@
 apply (simp add: image_def Inv_f_f) 
 done
 
+lemma inj_on_compose:
+    "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
+  by (auto simp add: bij_betw_def inj_on_def compose_eq)
+
 lemma bij_betw_compose:
     "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
 apply (simp add: bij_betw_def compose_eq inj_on_compose)
 apply (auto simp add: compose_def image_def)
 done
 
+lemma bij_betw_restrict_eq [simp]:
+     "bij_betw (restrict f A) A B = bij_betw f A B"
+  by (simp add: bij_betw_def)
+
+
+subsection{*Extensionality*}
+
+lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = arbitrary"
+  by (simp add: extensional_def)
+
+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"
+  by (unfold Inv_def) (fast intro: restrict_in_funcset someI2)
+
+lemma compose_Inv_id:
+    "bij_betw f A B ==> compose A (\<lambda>y\<in>B. Inv A f y) f = (\<lambda>x\<in>A. x)"
+  apply (simp add: bij_betw_def 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
+
 
 subsection{*Cardinality*}