# HG changeset patch
# User nipkow
# Date 1245697173 -7200
# Node ID 78529fc872b10998f50632193c3101b35682f062
# Parent  a61475260e479ac6a87b580abee02ecbc0ce4751# Parent  b5260f5272a482f609629272fc08adfa8ba85967
merged

diff -r a61475260e47 -r 78529fc872b1 src/HOL/Algebra/Bij.thy
--- a/src/HOL/Algebra/Bij.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/Algebra/Bij.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -50,7 +50,7 @@
     apply (simp add: compose_Bij)
    apply (simp add: id_Bij)
   apply (simp add: compose_Bij)
-  apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
+  apply (blast intro: compose_assoc [symmetric] dest: Bij_imp_funcset)
  apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
 apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
 done
diff -r a61475260e47 -r 78529fc872b1 src/HOL/Algebra/Group.thy
--- a/src/HOL/Algebra/Group.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/Algebra/Group.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -542,10 +542,8 @@
       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"
 
 lemma (in group) hom_compose:
-     "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
-apply (auto simp add: hom_def funcset_compose) 
-apply (simp add: compose_def Pi_def)
-done
+  "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
+by (fastsimp simp add: hom_def compose_def)
 
 constdefs
   iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
@@ -568,7 +566,7 @@
 
 lemma DirProd_commute_iso:
   shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
-by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
+by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)
 
 lemma DirProd_assoc_iso:
   shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"
@@ -592,7 +590,7 @@
   "x \<in> carrier G ==> h x \<in> carrier H"
 proof -
   assume "x \<in> carrier G"
-  with homh [unfolded hom_def] show ?thesis by (auto simp add: Pi_def)
+  with homh [unfolded hom_def] show ?thesis by auto
 qed
 
 lemma (in group_hom) one_closed [simp]:
diff -r a61475260e47 -r 78529fc872b1 src/HOL/Algebra/Sylow.thy
--- a/src/HOL/Algebra/Sylow.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/Algebra/Sylow.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -371,4 +371,3 @@
 done
 
 end
-
diff -r a61475260e47 -r 78529fc872b1 src/HOL/Library/FuncSet.thy
--- a/src/HOL/Library/FuncSet.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/Library/FuncSet.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -51,7 +51,7 @@
 
 subsection{*Basic Properties of @{term Pi}*}
 
-lemma Pi_I: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
+lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
   by (simp add: Pi_def)
 
 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
@@ -63,13 +63,17 @@
 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
   by (simp add: Pi_def)
 
+lemma ballE [elim]:
+  "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
+by(auto simp: Pi_def)
+
 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
   by (simp add: Pi_def)
 
 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
-  by (auto simp add: Pi_def)
+by auto
 
-lemma Pi_eq_empty: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
+lemma Pi_eq_empty[simp]: "((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"}*}
@@ -78,36 +82,36 @@
 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)
 (*
 lemma funcset_id [simp]: "(%x. x): A -> A"
   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 auto
 
 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 auto
 
 
 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)
+  "[| 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)
+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)
@@ -118,7 +122,7 @@
 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"
+lemma restrictI[intro!]: "(!!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]:
@@ -127,7 +131,7 @@
 
 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 restrict_def)
 
 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
   by (simp add: inj_on_def restrict_def)
@@ -150,68 +154,66 @@
 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
 
 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
-  by (auto simp add: bij_betw_def inj_on_Inv Pi_def)
+by (auto simp add: bij_betw_def inj_on_Inv)
 
 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)
+  "[| 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
+  "[| 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)
+  "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 = undefined"
-  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;
+  "[| 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)
+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)
+by (unfold Inv_def) (fast intro: 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
+  "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
+  "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*}
 
 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
-  apply (rule card_inj_on_le)
-    apply (auto simp add: Pi_def)
-  done
+by (rule card_inj_on_le) auto
 
 lemma card_bij:
-     "[|f \<in> A\<rightarrow>B; inj_on f A;
-        g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
-  by (blast intro: card_inj order_antisym)
+  "[|f \<in> A\<rightarrow>B; inj_on f A;
+     g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
+by (blast intro: card_inj order_antisym)
 
 end
diff -r a61475260e47 -r 78529fc872b1 src/HOL/MetisExamples/Abstraction.thy
--- a/src/HOL/MetisExamples/Abstraction.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/MetisExamples/Abstraction.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -201,7 +201,7 @@
    "(cl,f) \<in> CLF ==> 
     CLF \<subseteq> (SIGMA cl': CL. {f. f \<in> pset cl' \<rightarrow> pset cl'}) ==> 
     f \<in> pset cl \<rightarrow> pset cl"
-by auto
+by fast
 (*??no longer terminates, with combinators
 by (metis Collect_mem_eq SigmaD2 subsetD)
 *)
diff -r a61475260e47 -r 78529fc872b1 src/HOL/ex/Tarski.thy
--- a/src/HOL/ex/Tarski.thy	Mon Jun 22 20:31:08 2009 +0200
+++ b/src/HOL/ex/Tarski.thy	Mon Jun 22 20:59:33 2009 +0200
@@ -824,11 +824,6 @@
 apply (simp add: intY1_def interval_def  intY1_elem)
 done
 
-lemma (in Tarski) intY1_func: "(%x: intY1. f x) \<in> intY1 -> intY1"
-apply (rule restrictI)
-apply (erule intY1_f_closed)
-done
-
 lemma (in Tarski) intY1_mono:
      "monotone (%x: intY1. f x) intY1 (induced intY1 r)"
 apply (auto simp add: monotone_def induced_def intY1_f_closed)
@@ -853,7 +848,7 @@
 apply (rule CLF.glbH_is_fixp [OF CLF.intro, unfolded CLF_set_def, of "\<lparr>pset = intY1, order = induced intY1 r\<rparr>", simplified])
 apply auto
 apply (rule intY1_is_cl)
-apply (rule intY1_func)
+apply (erule intY1_f_closed)
 apply (rule intY1_mono)
 done