src/HOL/HOLCF/Sfun.thy
author kuncar
Fri, 09 Dec 2011 18:07:04 +0100
changeset 45802 b16f976db515
parent 42151 4da4fc77664b
child 49759 ecf1d5d87e5e
permissions -rw-r--r--
Quotient_Info stores only relation maps
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(*  Title:      HOL/HOLCF/Sfun.thy
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    Author:     Brian Huffman
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*)
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header {* The Strict Function Type *}
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theory Sfun
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imports Cfun
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begin
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pcpodef (open) ('a, 'b) sfun (infixr "->!" 0)
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  = "{f :: 'a \<rightarrow> 'b. f\<cdot>\<bottom> = \<bottom>}"
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by simp_all
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type_notation (xsymbols)
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  sfun  (infixr "\<rightarrow>!" 0)
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text {* TODO: Define nice syntax for abstraction, application. *}
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definition
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  sfun_abs :: "('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow>! 'b)"
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where
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  "sfun_abs = (\<Lambda> f. Abs_sfun (strictify\<cdot>f))"
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definition
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  sfun_rep :: "('a \<rightarrow>! 'b) \<rightarrow> 'a \<rightarrow> 'b"
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where
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  "sfun_rep = (\<Lambda> f. Rep_sfun f)"
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lemma sfun_rep_beta: "sfun_rep\<cdot>f = Rep_sfun f"
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  unfolding sfun_rep_def by (simp add: cont_Rep_sfun)
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lemma sfun_rep_strict1 [simp]: "sfun_rep\<cdot>\<bottom> = \<bottom>"
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  unfolding sfun_rep_beta by (rule Rep_sfun_strict)
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lemma sfun_rep_strict2 [simp]: "sfun_rep\<cdot>f\<cdot>\<bottom> = \<bottom>"
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  unfolding sfun_rep_beta by (rule Rep_sfun [simplified])
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lemma strictify_cancel: "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> strictify\<cdot>f = f"
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  by (simp add: cfun_eq_iff strictify_conv_if)
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lemma sfun_abs_sfun_rep [simp]: "sfun_abs\<cdot>(sfun_rep\<cdot>f) = f"
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  unfolding sfun_abs_def sfun_rep_def
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  apply (simp add: cont_Abs_sfun cont_Rep_sfun)
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  apply (simp add: Rep_sfun_inject [symmetric] Abs_sfun_inverse)
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  apply (simp add: cfun_eq_iff strictify_conv_if)
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  apply (simp add: Rep_sfun [simplified])
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  done
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lemma sfun_rep_sfun_abs [simp]: "sfun_rep\<cdot>(sfun_abs\<cdot>f) = strictify\<cdot>f"
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  unfolding sfun_abs_def sfun_rep_def
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  apply (simp add: cont_Abs_sfun cont_Rep_sfun)
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  apply (simp add: Abs_sfun_inverse)
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  done
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lemma sfun_eq_iff: "f = g \<longleftrightarrow> sfun_rep\<cdot>f = sfun_rep\<cdot>g"
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by (simp add: sfun_rep_def cont_Rep_sfun Rep_sfun_inject)
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lemma sfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> sfun_rep\<cdot>f \<sqsubseteq> sfun_rep\<cdot>g"
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by (simp add: sfun_rep_def cont_Rep_sfun below_sfun_def)
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end