(* Title: HOL/HOLCF/Sfun.thy
Author: Brian Huffman
*)
section \<open>The Strict Function Type\<close>
theory Sfun
imports Cfun
begin
pcpodef ('a::pcpo, 'b::pcpo) sfun (infixr \<open>\<rightarrow>!\<close> 0) = "{f :: 'a \<rightarrow> 'b. f\<cdot>\<bottom> = \<bottom>}"
by simp_all
type_notation (ASCII)
sfun (infixr \<open>->!\<close> 0)
text \<open>TODO: Define nice syntax for abstraction, application.\<close>
definition sfun_abs :: "('a::pcpo \<rightarrow> 'b::pcpo) \<rightarrow> ('a \<rightarrow>! 'b)"
where "sfun_abs = (\<Lambda> f. Abs_sfun (strictify\<cdot>f))"
definition sfun_rep :: "('a::pcpo \<rightarrow>! 'b::pcpo) \<rightarrow> 'a \<rightarrow> 'b"
where "sfun_rep = (\<Lambda> f. Rep_sfun f)"
lemma sfun_rep_beta: "sfun_rep\<cdot>f = Rep_sfun f"
by (simp add: sfun_rep_def cont_Rep_sfun)
lemma sfun_rep_strict1 [simp]: "sfun_rep\<cdot>\<bottom> = \<bottom>"
unfolding sfun_rep_beta by (rule Rep_sfun_strict)
lemma sfun_rep_strict2 [simp]: "sfun_rep\<cdot>f\<cdot>\<bottom> = \<bottom>"
unfolding sfun_rep_beta by (rule Rep_sfun [simplified])
lemma strictify_cancel: "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> strictify\<cdot>f = f"
by (simp add: cfun_eq_iff strictify_conv_if)
lemma sfun_abs_sfun_rep [simp]: "sfun_abs\<cdot>(sfun_rep\<cdot>f) = f"
unfolding sfun_abs_def sfun_rep_def
apply (simp add: cont_Abs_sfun cont_Rep_sfun)
apply (simp add: Rep_sfun_inject [symmetric] Abs_sfun_inverse)
apply (simp add: cfun_eq_iff strictify_conv_if)
apply (simp add: Rep_sfun [simplified])
done
lemma sfun_rep_sfun_abs [simp]: "sfun_rep\<cdot>(sfun_abs\<cdot>f) = strictify\<cdot>f"
unfolding sfun_abs_def sfun_rep_def
apply (simp add: cont_Abs_sfun cont_Rep_sfun)
apply (simp add: Abs_sfun_inverse)
done
lemma sfun_eq_iff: "f = g \<longleftrightarrow> sfun_rep\<cdot>f = sfun_rep\<cdot>g"
by (simp add: sfun_rep_def cont_Rep_sfun Rep_sfun_inject)
lemma sfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> sfun_rep\<cdot>f \<sqsubseteq> sfun_rep\<cdot>g"
by (simp add: sfun_rep_def cont_Rep_sfun below_sfun_def)
end