src/ZF/UNITY/UNITYMisc.ML

Converting ZF/UNITY to Isar

(*  Title:      HOL/UNITY/UNITYMisc.ML
    ID:         $Id$
    Author:     Sidi O Ehmety, Computer Laboratory
    Copyright   2001  University of Cambridge

Some miscellaneous and add-hoc set theory concepts.

*)

Goalw [greaterThan_def]
"i:greaterThan(k,A) <-> i:A & k<i";
by Auto_tac;
qed "greaterThan_iff";

(** Ad-hoc set-theory rules **)

Goal "Union(B) Int A = (UN b:B. b Int A)";
by Auto_tac;
qed "Int_Union_Union";

Goal "A Int Union(B) = (UN b:B. A Int b)";
by Auto_tac;
qed "Int_Union_Union2";

Goal "A Un B - (A - B) = B";
by (Blast_tac 1);
qed "Un_Diff_Diff";
AddIffs [Un_Diff_Diff];


(** Needed in State theory for the current definition of variables 
where they are indexed by lists  **)

Goal "i:list(nat) ==> i:univ(0)";
by (dres_inst_tac [("B", "0")] list_into_univ 1);
by (blast_tac (claset() addIs [nat_into_univ]) 1);
by (assume_tac 1);
qed "list_nat_into_univ";


(** Simplication rules for Collect; ????
  At least change to "{x:A. P(x)} Int A = {x : A Int B. P(x)} **)

Goal "{x:A. P(x)} Int A = {x:A. P(x)}";
by Auto_tac;
qed "Collect_Int2";

Goal "A Int {x:A. P(x)} = {x:A. P(x)}";
by Auto_tac;
qed "Collect_Int3";

(*????????????????*)
AddIffs [Collect_Int2, Collect_Int3];


(*for main ZF????*)

Goal "{x:A. P(x) | Q(x)} = Collect(A, P) Un Collect(A, Q)";
by Auto_tac;
qed "Collect_disj_eq";

Goal "{x:A. P(x) & Q(x)} = Collect(A, P) Int Collect(A, Q)";
by Auto_tac;
qed "Collect_conj_eq";

Goal "Union(B) Int A = (UN C:B. C Int A)"; 
by (Blast_tac 1);
qed "Int_Union2";

Goal "A Int succ(k) = (if k : A then cons(k, A Int k) else A Int k)";
by Auto_tac;
qed "Int_succ_right";