Theory Sfun

(*  Title:      HOL/HOLCF/Sfun.thy
    Author:     Brian Huffman
*)

section ‹The Strict Function Type›

theory Sfun
  imports Cfun
begin

pcpodef ('a::pcpo, 'b::pcpo) sfun (infixr ‹→!› 0) = "{f :: 'a → 'b. f⋅⊥ = ⊥}"
  by simp_all

type_notation (ASCII)
  sfun  (infixr ‹->!› 0)

text ‹TODO: Define nice syntax for abstraction, application.›

definition sfun_abs :: "('a::pcpo → 'b::pcpo) → ('a →! 'b)"
  where "sfun_abs = (Λ f. Abs_sfun (strictify⋅f))"

definition sfun_rep :: "('a::pcpo →! 'b::pcpo) → 'a → 'b"
  where "sfun_rep = (Λ f. Rep_sfun f)"

lemma sfun_rep_beta: "sfun_rep⋅f = Rep_sfun f"
  by (simp add: sfun_rep_def cont_Rep_sfun)

lemma sfun_rep_strict1 [simp]: "sfun_rep⋅⊥ = ⊥"
  unfolding sfun_rep_beta by (rule Rep_sfun_strict)

lemma sfun_rep_strict2 [simp]: "sfun_rep⋅f⋅⊥ = ⊥"
  unfolding sfun_rep_beta by (rule Rep_sfun [simplified])

lemma strictify_cancel: "f⋅⊥ = ⊥ ⟹ strictify⋅f = f"
  by (simp add: cfun_eq_iff strictify_conv_if)

lemma sfun_abs_sfun_rep [simp]: "sfun_abs⋅(sfun_rep⋅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⋅(sfun_abs⋅f) = strictify⋅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 ⟷ sfun_rep⋅f = sfun_rep⋅g"
  by (simp add: sfun_rep_def cont_Rep_sfun Rep_sfun_inject)

lemma sfun_below_iff: "f ⊑ g ⟷ sfun_rep⋅f ⊑ sfun_rep⋅g"
  by (simp add: sfun_rep_def cont_Rep_sfun below_sfun_def)

end