(* Title: HOLCF/porder.thy
ID: $Id$
Author: Franz Regensburger
Copyright 1993 Technische Universitaet Muenchen
Conservative extension of theory Porder0 by constant definitions
*)
Porder = Porder0 +
consts
"<|" :: "['a set,'a::po] => bool" (infixl 55)
"<<|" :: "['a set,'a::po] => bool" (infixl 55)
lub :: "'a set => 'a::po"
is_tord :: "'a::po set => bool"
is_chain :: "(nat=>'a::po) => bool"
max_in_chain :: "[nat,nat=>'a::po]=>bool"
finite_chain :: "(nat=>'a::po)=>bool"
syntax
"@LUB" :: "(('b::term) => 'a) => 'a" (binder "LUB " 10)
translations
"LUB x. t" == "lub(range(%x.t))"
syntax (symbols)
"LUB " :: "[idts, 'a] => 'a" ("(3\\<Squnion>_./ _)"[0,10] 10)
defs
(* class definitions *)
is_ub "S <| x == ! y.y:S --> y<<x"
is_lub "S <<| x == S <| x & (! u. S <| u --> x << u)"
(* Arbitrary chains are total orders *)
is_tord "is_tord S == ! x y. x:S & y:S --> (x<<y | y<<x)"
(* Here we use countable chains and I prefer to code them as functions! *)
is_chain "is_chain F == (! i.F(i) << F(Suc(i)))"
(* finite chains, needed for monotony of continouous functions *)
max_in_chain_def "max_in_chain i C == ! j. i <= j --> C(i) = C(j)"
finite_chain_def "finite_chain C == is_chain(C) & (? i. max_in_chain i C)"
rules
lub "lub S = (@x. S <<| x)"
end