--- a/src/ZF/Finite.thy Thu Mar 15 15:54:22 2012 +0000
+++ b/src/ZF/Finite.thy Thu Mar 15 16:35:02 2012 +0000
@@ -2,7 +2,7 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
-prove: b: Fin(A) ==> inj(b,b) \<subseteq> surj(b,b)
+prove: b \<in> Fin(A) ==> inj(b,b) \<subseteq> surj(b,b)
*)
header{*Finite Powerset Operator and Finite Function Space*}
@@ -25,7 +25,7 @@
domains "Fin(A)" \<subseteq> "Pow(A)"
intros
emptyI: "0 \<in> Fin(A)"
- consI: "[| a: A; b: Fin(A) |] ==> cons(a,b) \<in> Fin(A)"
+ consI: "[| a \<in> A; b \<in> Fin(A) |] ==> cons(a,b) \<in> Fin(A)"
type_intros empty_subsetI cons_subsetI PowI
type_elims PowD [elim_format]
@@ -33,7 +33,7 @@
domains "FiniteFun(A,B)" \<subseteq> "Fin(A*B)"
intros
emptyI: "0 \<in> A -||> B"
- consI: "[| a: A; b: B; h: A -||> B; a \<notin> domain(h) |]
+ consI: "[| a \<in> A; b \<in> B; h \<in> A -||> B; a \<notin> domain(h) |]
==> cons(<a,b>,h) \<in> A -||> B"
type_intros Fin.intros
@@ -54,12 +54,12 @@
(*Discharging @{term"x\<notin>y"} entails extra work*)
lemma Fin_induct [case_names 0 cons, induct set: Fin]:
- "[| b: Fin(A);
+ "[| b \<in> Fin(A);
P(0);
- !!x y. [| x: A; y: Fin(A); x\<notin>y; P(y) |] ==> P(cons(x,y))
+ !!x y. [| x \<in> A; y \<in> Fin(A); x\<notin>y; P(y) |] ==> P(cons(x,y))
|] ==> P(b)"
apply (erule Fin.induct, simp)
-apply (case_tac "a:b")
+apply (case_tac "a \<in> b")
apply (erule cons_absorb [THEN ssubst], assumption) (*backtracking!*)
apply simp
done
@@ -72,7 +72,7 @@
by (blast intro: Fin.emptyI dest: FinD)
(*The union of two finite sets is finite.*)
-lemma Fin_UnI [simp]: "[| b: Fin(A); c: Fin(A) |] ==> b \<union> c \<in> Fin(A)"
+lemma Fin_UnI [simp]: "[| b \<in> Fin(A); c \<in> Fin(A) |] ==> b \<union> c \<in> Fin(A)"
apply (erule Fin_induct)
apply (simp_all add: Un_cons)
done
@@ -83,25 +83,25 @@
by (erule Fin_induct, simp_all)
(*Every subset of a finite set is finite.*)
-lemma Fin_subset_lemma [rule_format]: "b: Fin(A) ==> \<forall>z. z<=b \<longrightarrow> z: Fin(A)"
+lemma Fin_subset_lemma [rule_format]: "b \<in> Fin(A) ==> \<forall>z. z<=b \<longrightarrow> z \<in> Fin(A)"
apply (erule Fin_induct)
apply (simp add: subset_empty_iff)
apply (simp add: subset_cons_iff distrib_simps, safe)
apply (erule_tac b = z in cons_Diff [THEN subst], simp)
done
-lemma Fin_subset: "[| c<=b; b: Fin(A) |] ==> c: Fin(A)"
+lemma Fin_subset: "[| c<=b; b \<in> Fin(A) |] ==> c \<in> Fin(A)"
by (blast intro: Fin_subset_lemma)
-lemma Fin_IntI1 [intro,simp]: "b: Fin(A) ==> b \<inter> c \<in> Fin(A)"
+lemma Fin_IntI1 [intro,simp]: "b \<in> Fin(A) ==> b \<inter> c \<in> Fin(A)"
by (blast intro: Fin_subset)
-lemma Fin_IntI2 [intro,simp]: "c: Fin(A) ==> b \<inter> c \<in> Fin(A)"
+lemma Fin_IntI2 [intro,simp]: "c \<in> Fin(A) ==> b \<inter> c \<in> Fin(A)"
by (blast intro: Fin_subset)
lemma Fin_0_induct_lemma [rule_format]:
- "[| c: Fin(A); b: Fin(A); P(b);
- !!x y. [| x: A; y: Fin(A); x:y; P(y) |] ==> P(y-{x})
+ "[| c \<in> Fin(A); b \<in> Fin(A); P(b);
+ !!x y. [| x \<in> A; y \<in> Fin(A); x \<in> y; P(y) |] ==> P(y-{x})
|] ==> c<=b \<longrightarrow> P(b-c)"
apply (erule Fin_induct, simp)
apply (subst Diff_cons)
@@ -109,16 +109,16 @@
done
lemma Fin_0_induct:
- "[| b: Fin(A);
+ "[| b \<in> Fin(A);
P(b);
- !!x y. [| x: A; y: Fin(A); x:y; P(y) |] ==> P(y-{x})
+ !!x y. [| x \<in> A; y \<in> Fin(A); x \<in> y; P(y) |] ==> P(y-{x})
|] ==> P(0)"
apply (rule Diff_cancel [THEN subst])
apply (blast intro: Fin_0_induct_lemma)
done
(*Functions from a finite ordinal*)
-lemma nat_fun_subset_Fin: "n: nat ==> n->A \<subseteq> Fin(nat*A)"
+lemma nat_fun_subset_Fin: "n \<in> nat ==> n->A \<subseteq> Fin(nat*A)"
apply (induct_tac "n")
apply (simp add: subset_iff)
apply (simp add: succ_def mem_not_refl [THEN cons_fun_eq])
@@ -139,19 +139,19 @@
lemma FiniteFun_mono1: "A<=B ==> A -||> A \<subseteq> B -||> B"
by (blast dest: FiniteFun_mono)
-lemma FiniteFun_is_fun: "h: A -||>B ==> h: domain(h) -> B"
+lemma FiniteFun_is_fun: "h \<in> A -||>B ==> h \<in> domain(h) -> B"
apply (erule FiniteFun.induct, simp)
apply (simp add: fun_extend3)
done
-lemma FiniteFun_domain_Fin: "h: A -||>B ==> domain(h) \<in> Fin(A)"
+lemma FiniteFun_domain_Fin: "h \<in> A -||>B ==> domain(h) \<in> Fin(A)"
by (erule FiniteFun.induct, simp, simp)
lemmas FiniteFun_apply_type = FiniteFun_is_fun [THEN apply_type]
(*Every subset of a finite function is a finite function.*)
lemma FiniteFun_subset_lemma [rule_format]:
- "b: A-||>B ==> \<forall>z. z<=b \<longrightarrow> z: A-||>B"
+ "b \<in> A-||>B ==> \<forall>z. z<=b \<longrightarrow> z \<in> A-||>B"
apply (erule FiniteFun.induct)
apply (simp add: subset_empty_iff FiniteFun.intros)
apply (simp add: subset_cons_iff distrib_simps, safe)
@@ -160,15 +160,15 @@
apply (fast intro!: FiniteFun.intros)
done
-lemma FiniteFun_subset: "[| c<=b; b: A-||>B |] ==> c: A-||>B"
+lemma FiniteFun_subset: "[| c<=b; b \<in> A-||>B |] ==> c \<in> A-||>B"
by (blast intro: FiniteFun_subset_lemma)
(** Some further results by Sidi O. Ehmety **)
-lemma fun_FiniteFunI [rule_format]: "A:Fin(X) ==> \<forall>f. f:A->B \<longrightarrow> f:A-||>B"
+lemma fun_FiniteFunI [rule_format]: "A \<in> Fin(X) ==> \<forall>f. f \<in> A->B \<longrightarrow> f \<in> A-||>B"
apply (erule Fin.induct)
apply (simp add: FiniteFun.intros, clarify)
-apply (case_tac "a:b")
+apply (case_tac "a \<in> b")
apply (simp add: cons_absorb)
apply (subgoal_tac "restrict (f,b) \<in> b -||> B")
prefer 2 apply (blast intro: restrict_type2)
@@ -178,11 +178,11 @@
FiniteFun_mono [THEN [2] rev_subsetD])
done
-lemma lam_FiniteFun: "A: Fin(X) ==> (\<lambda>x\<in>A. b(x)) \<in> A -||> {b(x). x:A}"
+lemma lam_FiniteFun: "A \<in> Fin(X) ==> (\<lambda>x\<in>A. b(x)) \<in> A -||> {b(x). x \<in> A}"
by (blast intro: fun_FiniteFunI lam_funtype)
lemma FiniteFun_Collect_iff:
- "f \<in> FiniteFun(A, {y:B. P(y)})
+ "f \<in> FiniteFun(A, {y \<in> B. P(y)})
\<longleftrightarrow> f \<in> FiniteFun(A,B) & (\<forall>x\<in>domain(f). P(f`x))"
apply auto
apply (blast intro: FiniteFun_mono [THEN [2] rev_subsetD])