src/ZF/Finite.thy

 (*  Title:      ZF/Finite.thy
     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*}
 
 theory Finite imports Inductive_ZF Epsilon Nat_ZF begin

 
 inductive
   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]
 
 inductive
   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
 
 
 subsection {* Finite Powerset Operator *}

 
 (** Induction on finite sets **)
 
 (*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
 
 

 
 lemma Fin_0: "Fin(0) = {0}"
 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
 
 
 (*The union of a set of finite sets is finite.*)
 lemma Fin_UnionI: "C \<in> Fin(Fin(A)) ==> \<Union>(C) \<in> Fin(A)"
 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)
 apply (simp add: cons_subset_iff Diff_subset [THEN Fin_subset])
 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])
 apply (fast intro!: Fin.consI)
 done

 done
 
 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)
 apply (erule_tac b = z in cons_Diff [THEN subst])
 apply (drule spec [THEN mp], assumption)
 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)
 apply (subst fun_cons_restrict_eq, assumption)
 apply (simp add: restrict_def lam_def)
 apply (blast intro: apply_funtype FiniteFun.intros
                     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])
 apply (blast dest: Pair_mem_PiD FiniteFun_is_fun)
 apply (rule_tac A1="domain(f)" in