(*  Title:      HOLCF/Cprod1.ML

    ID:         $Id$

    Author:     Franz Regensburger

    Copyright   1993  Technische Universitaet Muenchen

Partial ordering for cartesian product of HOL theory Product_Type.thy

*)

(* ------------------------------------------------------------------------ *)

(* less_cprod is a partial order on 'a * 'b                                 *)

(* ------------------------------------------------------------------------ *)

(*###TO Product_Type_lemmas.ML *)

Goal "[|fst x = fst y; snd x = snd y|] ==> x = y";

by (subgoal_tac "(fst x,snd x)=(fst y,snd y)" 1);

by (rotate_tac ~1 1);

by (asm_full_simp_tac(HOL_ss addsimps[surjective_pairing RS sym])1);

by (asm_simp_tac (simpset_of (theory "Product_Type")) 1);

qed "Sel_injective_cprod";

Goalw [less_cprod_def] "(p::'a*'b) << p";

by (Simp_tac 1);

qed "refl_less_cprod";

Goalw [less_cprod_def] "[|(p1::'a * 'b) << p2;p2 << p1|] ==> p1=p2";

by (rtac Sel_injective_cprod 1);

by (fast_tac (HOL_cs addIs [antisym_less]) 1);

by (fast_tac (HOL_cs addIs [antisym_less]) 1);

qed "antisym_less_cprod";

Goalw [less_cprod_def]

        "[|(p1::'a*'b) << p2;p2 << p3|] ==> p1 << p3";

by (rtac conjI 1);

by (fast_tac (HOL_cs addIs [trans_less]) 1);

by (fast_tac (HOL_cs addIs [trans_less]) 1);

qed "trans_less_cprod";