section ‹The Strict Function Type›
theory Sfun
imports Cfun
begin
pcpodef ('a, 'b) 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 → 'b) → ('a →! 'b)"
where
"sfun_abs = (Λ f. Abs_sfun (strictify⋅f))"
definition
sfun_rep :: "('a →! 'b) → 'a → 'b"
where
"sfun_rep = (Λ f. Rep_sfun f)"
lemma sfun_rep_beta: "sfun_rep⋅f = Rep_sfun f"
unfolding sfun_rep_def by (simp add: 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