--- a/src/HOLCF/ex/Strict_Fun.thy Mon May 24 11:29:49 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,239 +0,0 @@
-(* Title: HOLCF/ex/Strict_Fun.thy
- Author: Brian Huffman
-*)
-
-header {* The Strict Function Type *}
-
-theory Strict_Fun
-imports HOLCF
-begin
-
-pcpodef (open) ('a, 'b) sfun (infixr "->!" 0)
- = "{f :: 'a \<rightarrow> 'b. f\<cdot>\<bottom> = \<bottom>}"
-by simp_all
-
-type_notation (xsymbols)
- sfun (infixr "\<rightarrow>!" 0)
-
-text {* TODO: Define nice syntax for abstraction, application. *}
-
-definition
- sfun_abs :: "('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow>! 'b)"
-where
- "sfun_abs = (\<Lambda> f. Abs_sfun (strictify\<cdot>f))"
-
-definition
- sfun_rep :: "('a \<rightarrow>! 'b) \<rightarrow> 'a \<rightarrow> 'b"
-where
- "sfun_rep = (\<Lambda> f. Rep_sfun f)"
-
-lemma sfun_rep_beta: "sfun_rep\<cdot>f = Rep_sfun f"
- unfolding sfun_rep_def by (simp add: 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: expand_cfun_eq strictify_conv_if)
-
-lemma sfun_abs_sfun_rep: "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: expand_cfun_eq 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 ep_pair_sfun: "ep_pair sfun_rep sfun_abs"
-apply default
-apply (rule sfun_abs_sfun_rep)
-apply (simp add: expand_cfun_below strictify_conv_if)
-done
-
-interpretation sfun: ep_pair sfun_rep sfun_abs
- by (rule ep_pair_sfun)
-
-subsection {* Map functional for strict function space *}
-
-definition
- sfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow>! 'c) \<rightarrow> ('b \<rightarrow>! 'd)"
-where
- "sfun_map = (\<Lambda> a b. sfun_abs oo cfun_map\<cdot>a\<cdot>b oo sfun_rep)"
-
-lemma sfun_map_ID: "sfun_map\<cdot>ID\<cdot>ID = ID"
- unfolding sfun_map_def
- by (simp add: cfun_map_ID expand_cfun_eq)
-
-lemma sfun_map_map:
- assumes "f2\<cdot>\<bottom> = \<bottom>" and "g2\<cdot>\<bottom> = \<bottom>" shows
- "sfun_map\<cdot>f1\<cdot>g1\<cdot>(sfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
- sfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
-unfolding sfun_map_def
-by (simp add: expand_cfun_eq strictify_cancel assms cfun_map_map)
-
-lemma ep_pair_sfun_map:
- assumes 1: "ep_pair e1 p1"
- assumes 2: "ep_pair e2 p2"
- shows "ep_pair (sfun_map\<cdot>p1\<cdot>e2) (sfun_map\<cdot>e1\<cdot>p2)"
-proof
- interpret e1p1: pcpo_ep_pair e1 p1
- unfolding pcpo_ep_pair_def by fact
- interpret e2p2: pcpo_ep_pair e2 p2
- unfolding pcpo_ep_pair_def by fact
- fix f show "sfun_map\<cdot>e1\<cdot>p2\<cdot>(sfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
- unfolding sfun_map_def
- apply (simp add: sfun.e_eq_iff [symmetric] strictify_cancel)
- apply (rule ep_pair.e_inverse)
- apply (rule ep_pair_cfun_map [OF 1 2])
- done
- fix g show "sfun_map\<cdot>p1\<cdot>e2\<cdot>(sfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
- unfolding sfun_map_def
- apply (simp add: sfun.e_below_iff [symmetric] strictify_cancel)
- apply (rule ep_pair.e_p_below)
- apply (rule ep_pair_cfun_map [OF 1 2])
- done
-qed
-
-lemma deflation_sfun_map:
- assumes 1: "deflation d1"
- assumes 2: "deflation d2"
- shows "deflation (sfun_map\<cdot>d1\<cdot>d2)"
-apply (simp add: sfun_map_def)
-apply (rule deflation.intro)
-apply simp
-apply (subst strictify_cancel)
-apply (simp add: cfun_map_def deflation_strict 1 2)
-apply (simp add: cfun_map_def deflation.idem 1 2)
-apply (simp add: sfun.e_below_iff [symmetric])
-apply (subst strictify_cancel)
-apply (simp add: cfun_map_def deflation_strict 1 2)
-apply (rule deflation.below)
-apply (rule deflation_cfun_map [OF 1 2])
-done
-
-lemma finite_deflation_sfun_map:
- assumes 1: "finite_deflation d1"
- assumes 2: "finite_deflation d2"
- shows "finite_deflation (sfun_map\<cdot>d1\<cdot>d2)"
-proof (intro finite_deflation.intro finite_deflation_axioms.intro)
- interpret d1: finite_deflation d1 by fact
- interpret d2: finite_deflation d2 by fact
- have "deflation d1" and "deflation d2" by fact+
- thus "deflation (sfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_sfun_map)
- from 1 2 have "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
- by (rule finite_deflation_cfun_map)
- then have "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
- by (rule finite_deflation.finite_fixes)
- moreover have "inj (\<lambda>f. sfun_rep\<cdot>f)"
- by (rule inj_onI, simp)
- ultimately have "finite ((\<lambda>f. sfun_rep\<cdot>f) -` {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f})"
- by (rule finite_vimageI)
- then show "finite {f. sfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
- unfolding sfun_map_def sfun.e_eq_iff [symmetric]
- by (simp add: strictify_cancel
- deflation_strict `deflation d1` `deflation d2`)
-qed
-
-subsection {* Strict function space is bifinite *}
-
-instantiation sfun :: (bifinite, bifinite) bifinite
-begin
-
-definition
- "approx = (\<lambda>i. sfun_map\<cdot>(approx i)\<cdot>(approx i))"
-
-instance proof
- show "chain (approx :: nat \<Rightarrow> ('a \<rightarrow>! 'b) \<rightarrow> ('a \<rightarrow>! 'b))"
- unfolding approx_sfun_def by simp
-next
- fix x :: "'a \<rightarrow>! 'b"
- show "(\<Squnion>i. approx i\<cdot>x) = x"
- unfolding approx_sfun_def
- by (simp add: lub_distribs sfun_map_ID [unfolded ID_def])
-next
- fix i :: nat and x :: "'a \<rightarrow>! 'b"
- show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
- unfolding approx_sfun_def
- by (intro deflation.idem deflation_sfun_map deflation_approx)
-next
- fix i :: nat
- show "finite {x::'a \<rightarrow>! 'b. approx i\<cdot>x = x}"
- unfolding approx_sfun_def
- by (intro finite_deflation.finite_fixes
- finite_deflation_sfun_map
- finite_deflation_approx)
-qed
-
-end
-
-subsection {* Strict function space is representable *}
-
-instantiation sfun :: (rep, rep) rep
-begin
-
-definition
- "emb = udom_emb oo sfun_map\<cdot>prj\<cdot>emb"
-
-definition
- "prj = sfun_map\<cdot>emb\<cdot>prj oo udom_prj"
-
-instance
-apply (default, unfold emb_sfun_def prj_sfun_def)
-apply (rule ep_pair_comp)
-apply (rule ep_pair_sfun_map)
-apply (rule ep_pair_emb_prj)
-apply (rule ep_pair_emb_prj)
-apply (rule ep_pair_udom)
-done
-
-end
-
-text {*
- A deflation constructor lets us configure the domain package to work
- with the strict function space type constructor.
-*}
-
-definition
- sfun_defl :: "TypeRep \<rightarrow> TypeRep \<rightarrow> TypeRep"
-where
- "sfun_defl = TypeRep_fun2 sfun_map"
-
-lemma cast_sfun_defl:
- "cast\<cdot>(sfun_defl\<cdot>A\<cdot>B) = udom_emb oo sfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj"
-unfolding sfun_defl_def
-apply (rule cast_TypeRep_fun2)
-apply (erule (1) finite_deflation_sfun_map)
-done
-
-lemma REP_sfun: "REP('a::rep \<rightarrow>! 'b::rep) = sfun_defl\<cdot>REP('a)\<cdot>REP('b)"
-apply (rule cast_eq_imp_eq, rule ext_cfun)
-apply (simp add: cast_REP cast_sfun_defl)
-apply (simp only: prj_sfun_def emb_sfun_def)
-apply (simp add: sfun_map_def cfun_map_def strictify_cancel)
-done
-
-lemma isodefl_sfun:
- "isodefl d1 t1 \<Longrightarrow> isodefl d2 t2 \<Longrightarrow>
- isodefl (sfun_map\<cdot>d1\<cdot>d2) (sfun_defl\<cdot>t1\<cdot>t2)"
-apply (rule isodeflI)
-apply (simp add: cast_sfun_defl cast_isodefl)
-apply (simp add: emb_sfun_def prj_sfun_def)
-apply (simp add: sfun_map_map deflation_strict [OF isodefl_imp_deflation])
-done
-
-setup {*
- Domain_Isomorphism.add_type_constructor
- (@{type_name "sfun"}, @{term sfun_defl}, @{const_name sfun_map}, @{thm REP_sfun},
- @{thm isodefl_sfun}, @{thm sfun_map_ID}, @{thm deflation_sfun_map})
-*}
-
-end