(* Title: ZF/Zorn.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Based upon the article
Abrial & Laffitte,
Towards the Mechanization of the Proofs of Some
Classical Theorems of Set Theory.
Union_in_Pow is proved in ZF.ML
*)
Zorn = OrderArith + AC + "inductive" +
consts
Subset_rel :: "i=>i"
increasing :: "i=>i"
chain, maxchain :: "i=>i"
super :: "[i,i]=>i"
rules
Subset_rel_def "Subset_rel(A) == {z: A*A . EX x y. z=<x,y> & x<=y & x~=y}"
increasing_def "increasing(A) == {f: Pow(A)->Pow(A). ALL x. x<=A --> x<=f`x}"
chain_def "chain(A) == {F: Pow(A). ALL X:F. ALL Y:F. X<=Y | Y<=X}"
super_def "super(A,c) == {d: chain(A). c<=d & c~=d}"
maxchain_def "maxchain(A) == {c: chain(A). super(A,c)=0}"
(** We could make the inductive definition conditional on next: increasing(S)
but instead we make this a side-condition of an introduction rule. Thus
the induction rule lets us assume that condition! Many inductive proofs
are therefore unconditional.
**)
consts
"TFin" :: "[i,i]=>i"
inductive
domains "TFin(S,next)" <= "Pow(S)"
intrs
nextI "[| x : TFin(S,next); next: increasing(S) \
\ |] ==> next`x : TFin(S,next)"
Pow_UnionI "Y : Pow(TFin(S,next)) ==> Union(Y) : TFin(S,next)"
monos "[Pow_mono]"
con_defs "[increasing_def]"
type_intrs "[CollectD1 RS apply_funtype, Union_in_Pow]"
end