--- a/src/HOL/Algebra/Bij.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/Algebra/Bij.thy Mon Jun 22 20:59:12 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
--- a/src/HOL/Algebra/Group.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/Algebra/Group.thy Mon Jun 22 20:59:12 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]:
--- a/src/HOL/Algebra/Sylow.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/Algebra/Sylow.thy Mon Jun 22 20:59:12 2009 +0200
@@ -371,4 +371,3 @@
done
end
-
--- a/src/HOL/Library/FuncSet.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/Library/FuncSet.thy Mon Jun 22 20:59:12 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
--- a/src/HOL/MetisExamples/Abstraction.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/MetisExamples/Abstraction.thy Mon Jun 22 20:59:12 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)
*)
--- a/src/HOL/ex/Tarski.thy Mon Jun 22 08:17:52 2009 +0200
+++ b/src/HOL/ex/Tarski.thy Mon Jun 22 20:59:12 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