Put in minimal simpset to avoid excessive simplification,
just as in revision 1.9 of HOL/indrule.ML
(* Title: Order.thy
ID: $Id$
Author: Markus Wenzel, TU Muenchen
Basic theory of orders (quasi orders, partial orders, linear orders)
and limits (inf, sup, min, max).
*)
Order = HOL + Set +
(** class definitions **)
(* syntax for orders *)
axclass
le < term
consts
"[=" :: "['a::le, 'a] => bool" (infixl 50)
(* quasi orders *)
axclass
quasi_order < le
le_refl "x [= x"
le_trans "x [= y & y [= z --> x [= z"
(* partial orders *)
axclass
order < quasi_order
le_antisym "x [= y & y [= x --> x = y"
(* linear orders *)
axclass
lin_order < order
le_lin "x [= y | y [= x"
(** limits **)
(* infima and suprema (on orders) *)
consts
(*binary*)
is_inf :: "['a::order, 'a, 'a] => bool"
is_sup :: "['a::order, 'a, 'a] => bool"
(*general*)
is_Inf :: "['a::order set, 'a] => bool"
is_Sup :: "['a::order set, 'a] => bool"
defs
is_inf_def "is_inf x y inf ==
inf [= x & inf [= y &
(ALL lb. lb [= x & lb [= y --> lb [= inf)"
is_sup_def "is_sup x y sup ==
x [= sup & y [= sup &
(ALL ub. x [= ub & y [= ub --> sup [= ub)"
is_Inf_def "is_Inf A inf ==
(ALL x:A. inf [= x) &
(ALL lb. (ALL x:A. lb [= x) --> lb [= inf)"
is_Sup_def "is_Sup A sup ==
(ALL x:A. x [= sup) &
(ALL ub. (ALL x:A. x [= ub) --> sup [= ub)"
(* binary minima and maxima (on lin_orders) *)
constdefs
minimum :: "['a::lin_order, 'a] => 'a"
"minimum x y == (if x [= y then x else y)"
maximum :: "['a::lin_order, 'a] => 'a"
"maximum x y == (if x [= y then y else x)"
end