src/HOLCF/Discrete.thy

 (*  Title:      HOLCF/Discrete.thy
     ID:         $Id$
     Author:     Tobias Nipkow
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
 
 Discrete CPOs.
 *)
 
-Discrete = Discrete1 +
+theory Discrete
+imports Cont Datatype
+begin
 
-instance discr :: (type)cpo   (discr_cpo)
+datatype 'a discr = Discr "'a :: type"
+
+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)"
+apply (unfold less_discr_def)
+apply (rule refl)
+done
+
+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
+
+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: 
+ "!!S::nat=>('a::type)discr. chain(S) ==> range(S) = {S 0}"
+apply (fast elim: discr_chain0)
+done
+declare discr_chain_range0 [simp]
+
+lemma discr_cpo: 
+ "!!S. chain S ==> ? x::('a::type)discr. range(S) <<| x"
+apply (unfold is_lub_def is_ub_def)
+apply (simp (no_asm_simp))
+done
+
+instance discr :: (type)cpo
+by (intro_classes, rule discr_cpo)
 
 constdefs
-   undiscr :: ('a::type)discr => 'a
+   undiscr :: "('a::type)discr => 'a"
   "undiscr x == (case x of Discr y => y)"
 
+lemma undiscr_Discr [simp]: "undiscr(Discr x) = x"
+apply (unfold undiscr_def)
+apply (simp (no_asm))
+done
+
+lemma discr_chain_f_range0:
+ "!!S::nat=>('a::type)discr. chain(S) ==> range(%i. f(S i)) = {f(S 0)}"
+apply (fast dest: discr_chain0 elim: arg_cong)
+done
+
+lemma cont_discr [iff]: "cont(%x::('a::type)discr. f x)"
+apply (unfold cont is_lub_def is_ub_def)
+apply (simp (no_asm) add: discr_chain_f_range0)
+done
+
 end