(* Title: HOL/HOLCF/Library/Int_Discrete.thy
Author: Brian Huffman
*)
header {* Discrete cpo instance for integers *}
theory Int_Discrete
imports HOLCF
begin
text {* Discrete cpo instance for @{typ int}. *}
instantiation int :: discrete_cpo
begin
definition below_int_def:
"(x::int) \<sqsubseteq> y \<longleftrightarrow> x = y"
instance proof
qed (rule below_int_def)
end
text {*
TODO: implement a command to automate discrete predomain instances.
*}
instantiation int :: predomain
begin
definition
"(liftemb :: int u \<rightarrow> udom u) \<equiv> liftemb oo u_map\<cdot>(\<Lambda> x. Discr x)"
definition
"(liftprj :: udom u \<rightarrow> int u) \<equiv> u_map\<cdot>(\<Lambda> y. undiscr y) oo liftprj"
definition
"liftdefl \<equiv> (\<lambda>(t::int itself). LIFTDEFL(int discr))"
instance proof
show "ep_pair liftemb (liftprj :: udom u \<rightarrow> int u)"
unfolding liftemb_int_def liftprj_int_def
apply (rule ep_pair_comp)
apply (rule ep_pair_u_map)
apply (simp add: ep_pair.intro)
apply (rule predomain_ep)
done
show "cast\<cdot>LIFTDEFL(int) = liftemb oo (liftprj :: udom u \<rightarrow> int u)"
unfolding liftemb_int_def liftprj_int_def liftdefl_int_def
apply (simp add: cast_liftdefl cfcomp1 u_map_map)
apply (simp add: ID_def [symmetric] u_map_ID)
done
qed
end
end