(* Title: HOLCF/Discrete.thy ID: $Id: Discrete.thy,v 1.10 2005/06/03 21:30:31 huffman Exp $ Author: Tobias Nipkow Discrete CPOs. *) header {* Discrete cpo types *} theory Discrete imports Cont Datatype begin datatype 'a discr = Discr "'a :: type" subsection {* Type @{typ "'a discr"} is a partial order *} instance discr :: (type) sq_ord .. defs (overloaded) less_discr_def: "((op <<)::('a::type)discr=>'a discr=>bool) == op =" lemma discr_less_eq [iff]: "((x::('a::type)discr) << y) = (x = y)" by (unfold less_discr_def) (rule refl) instance discr :: (type) po proof fix x y z :: "'a discr" show "x << x" by simp { assume "x << y" and "y << x" thus "x = y" by simp } { assume "x << y" and "y << z" thus "x << z" by simp } qed subsection {* Type @{typ "'a discr"} is a cpo *} lemma discr_chain0: "!!S::nat=>('a::type)discr. chain S ==> S i = S 0" apply (unfold chain_def) apply (induct_tac "i") apply (rule refl) apply (erule subst) apply (rule sym) apply fast done lemma discr_chain_range0 [simp]: "!!S::nat=>('a::type)discr. chain(S) ==> range(S) = {S 0}" by (fast elim: discr_chain0) lemma discr_cpo: "!!S. chain S ==> ? x::('a::type)discr. range(S) <<| x" by (unfold is_lub_def is_ub_def) simp instance discr :: (type) cpo by intro_classes (rule discr_cpo) subsection {* @{term undiscr} *} constdefs undiscr :: "('a::type)discr => 'a" "undiscr x == (case x of Discr y => y)" lemma undiscr_Discr [simp]: "undiscr(Discr x) = x" by (simp add: undiscr_def) lemma discr_chain_f_range0: "!!S::nat=>('a::type)discr. chain(S) ==> range(%i. f(S i)) = {f(S 0)}" by (fast dest: discr_chain0 elim: arg_cong) lemma cont_discr [iff]: "cont(%x::('a::type)discr. f x)" apply (unfold cont_def is_lub_def is_ub_def) apply (simp add: discr_chain_f_range0) done end
lemma discr_less_eq:
x << y = (x = y)
lemma discr_chain0:
chain S ==> S i = S 0
lemma discr_chain_range0:
chain S ==> range S = {S 0}
lemma discr_cpo:
chain S ==> ∃x. range S <<| x
lemma undiscr_Discr:
undiscr (Discr x) = x
lemma discr_chain_f_range0:
chain S ==> range (%i. f (S i)) = {f (S 0)}
lemma cont_discr:
cont f