(* Title: HOL/HOLCF/Library/Bool_Discrete.thy
Author: Brian Huffman
*)
section \<open>Discrete cpo instance for booleans\<close>
theory Bool_Discrete
imports HOLCF
begin
text \<open>Discrete cpo instance for @{typ bool}.\<close>
instantiation bool :: discrete_cpo
begin
definition below_bool_def:
"(x::bool) \<sqsubseteq> y \<longleftrightarrow> x = y"
instance proof
qed (rule below_bool_def)
end
text \<open>
TODO: implement a command to automate discrete predomain instances.
\<close>
instantiation bool :: predomain
begin
definition
"(liftemb :: bool u \<rightarrow> udom u) \<equiv> liftemb oo u_map\<cdot>(\<Lambda> x. Discr x)"
definition
"(liftprj :: udom u \<rightarrow> bool u) \<equiv> u_map\<cdot>(\<Lambda> y. undiscr y) oo liftprj"
definition
"liftdefl \<equiv> (\<lambda>(t::bool itself). LIFTDEFL(bool discr))"
instance proof
show "ep_pair liftemb (liftprj :: udom u \<rightarrow> bool u)"
unfolding liftemb_bool_def liftprj_bool_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(bool) = liftemb oo (liftprj :: udom u \<rightarrow> bool u)"
unfolding liftemb_bool_def liftprj_bool_def liftdefl_bool_def
apply (simp add: cast_liftdefl cfcomp1 u_map_map)
apply (simp add: ID_def [symmetric] u_map_ID)
done
qed
end
lemma cont2cont_if [simp, cont2cont]:
assumes b: "cont b" and f: "cont f" and g: "cont g"
shows "cont (\<lambda>x. if b x then f x else g x)"
by (rule cont_apply [OF b cont_discrete_cpo], simp add: f g)
lemma cont2cont_eq [simp, cont2cont]:
fixes f g :: "'a::cpo \<Rightarrow> 'b::discrete_cpo"
assumes f: "cont f" and g: "cont g"
shows "cont (\<lambda>x. f x = g x)"
apply (rule cont_apply [OF f cont_discrete_cpo])
apply (rule cont_apply [OF g cont_discrete_cpo])
apply (rule cont_const)
done
end