(* Title: HOL/Ord.ML
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
The type class for ordered types
*)
(** mono **)
val [prem] = goalw Ord.thy [mono_def]
"[| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)";
by (REPEAT (ares_tac [allI, impI, prem] 1));
qed "monoI";
val [major,minor] = goalw Ord.thy [mono_def]
"[| mono(f); A <= B |] ==> f(A) <= f(B)";
by (rtac (major RS spec RS spec RS mp) 1);
by (rtac minor 1);
qed "monoD";
section "Orders";
AddIffs [order_refl];
goal Ord.thy "~ x < (x::'a::order)";
by (simp_tac (simpset() addsimps [order_less_le]) 1);
qed "order_less_irrefl";
AddIffs [order_less_irrefl];
goal thy "(x::'a::order) <= y = (x < y | x = y)";
by (simp_tac (simpset() addsimps [order_less_le]) 1);
by (Blast_tac 1);
qed "order_le_less";
(** min **)
goalw thy [min_def] "!!least. (!!x. least <= x) ==> min least x = least";
by (split_tac [expand_if] 1);
by (Asm_simp_tac 1);
qed "min_leastL";
val prems = goalw thy [min_def]
"(!!x::'a::order. least <= x) ==> min x least = least";
by (cut_facts_tac prems 1);
by (split_tac [expand_if] 1);
by (Asm_simp_tac 1);
by (blast_tac (claset() addIs [order_antisym]) 1);
qed "min_leastR";