(* Title: HOL/HOLCF/Cprod.thy
Author: Franz Regensburger
*)
section \<open>The cpo of cartesian products\<close>
theory Cprod
imports Cfun
begin
default_sort cpo
subsection \<open>Continuous case function for unit type\<close>
definition unit_when :: "'a \<rightarrow> unit \<rightarrow> 'a"
where "unit_when = (\<Lambda> a _. a)"
translations
"\<Lambda>(). t" \<rightleftharpoons> "CONST unit_when\<cdot>t"
lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a"
by (simp add: unit_when_def)
subsection \<open>Continuous version of split function\<close>
definition csplit :: "('a \<rightarrow> 'b \<rightarrow> 'c) \<rightarrow> ('a \<times> 'b) \<rightarrow> 'c"
where "csplit = (\<Lambda> f p. f\<cdot>(fst p)\<cdot>(snd p))"
translations
"\<Lambda>(CONST Pair x y). t" \<rightleftharpoons> "CONST csplit\<cdot>(\<Lambda> x y. t)"
abbreviation cfst :: "'a \<times> 'b \<rightarrow> 'a"
where "cfst \<equiv> Abs_cfun fst"
abbreviation csnd :: "'a \<times> 'b \<rightarrow> 'b"
where "csnd \<equiv> Abs_cfun snd"
subsection \<open>Convert all lemmas to the continuous versions\<close>
lemma csplit1 [simp]: "csplit\<cdot>f\<cdot>\<bottom> = f\<cdot>\<bottom>\<cdot>\<bottom>"
by (simp add: csplit_def)
lemma csplit_Pair [simp]: "csplit\<cdot>f\<cdot>(x, y) = f\<cdot>x\<cdot>y"
by (simp add: csplit_def)
end