src/HOLCF/Bifinite.thy
author huffman
Tue, 02 Mar 2010 17:21:10 -0800
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child 37678 0040bafffdef
permissions -rw-r--r--
proper names for types cfun, sprod, ssum
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(*  Title:      HOLCF/Bifinite.thy
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    Author:     Brian Huffman
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*)
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header {* Bifinite domains and approximation *}
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theory Bifinite
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imports Deflation
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begin
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subsection {* Omega-profinite and bifinite domains *}
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class profinite =
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  fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"
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  assumes chain_approx [simp]: "chain approx"
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  assumes lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x"
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  assumes approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
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  assumes finite_fixes_approx: "finite {x. approx i\<cdot>x = x}"
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class bifinite = profinite + pcpo
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lemma approx_below: "approx i\<cdot>x \<sqsubseteq> x"
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proof -
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  have "chain (\<lambda>i. approx i\<cdot>x)" by simp
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  hence "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by (rule is_ub_thelub)
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  thus "approx i\<cdot>x \<sqsubseteq> x" by simp
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qed
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lemma finite_deflation_approx: "finite_deflation (approx i)"
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proof
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  fix x :: 'a
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  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
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    by (rule approx_idem)
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  show "approx i\<cdot>x \<sqsubseteq> x"
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    by (rule approx_below)
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  show "finite {x. approx i\<cdot>x = x}"
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    by (rule finite_fixes_approx)
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qed
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interpretation approx: finite_deflation "approx i"
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by (rule finite_deflation_approx)
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lemma (in deflation) deflation: "deflation d" ..
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lemma deflation_approx: "deflation (approx i)"
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by (rule approx.deflation)
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lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"
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by (rule ext_cfun, simp add: contlub_cfun_fun)
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lemma approx_strict [simp]: "approx i\<cdot>\<bottom> = \<bottom>"
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by (rule UU_I, rule approx_below)
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lemma approx_approx1:
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  "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>x"
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apply (rule deflation_below_comp1 [OF deflation_approx deflation_approx])
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apply (erule chain_mono [OF chain_approx])
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done
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lemma approx_approx2:
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  "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>x"
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apply (rule deflation_below_comp2 [OF deflation_approx deflation_approx])
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apply (erule chain_mono [OF chain_approx])
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done
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lemma approx_approx [simp]:
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  "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>x"
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apply (rule_tac x=i and y=j in linorder_le_cases)
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apply (simp add: approx_approx1 min_def)
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apply (simp add: approx_approx2 min_def)
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done
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lemma finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"
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by (rule approx.finite_image)
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lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
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by (rule approx.finite_range)
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lemma compact_approx [simp]: "compact (approx n\<cdot>x)"
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by (rule approx.compact)
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lemma profinite_compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
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by (rule admD2, simp_all)
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lemma profinite_compact_iff: "compact x \<longleftrightarrow> (\<exists>n. approx n\<cdot>x = x)"
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 apply (rule iffI)
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  apply (erule profinite_compact_eq_approx)
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 apply (erule exE)
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 apply (erule subst)
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 apply (rule compact_approx)
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done
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lemma approx_induct:
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  assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)"
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  shows "P x"
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proof -
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  have "P (\<Squnion>n. approx n\<cdot>x)"
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    by (rule admD [OF adm], simp, simp add: P)
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  thus "P x" by simp
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qed
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lemma profinite_below_ext: "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y"
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apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp)
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apply (rule lub_mono, simp, simp, simp)
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done
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subsection {* Instance for product type *}
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definition
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  cprod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<times> 'c \<rightarrow> 'b \<times> 'd"
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where
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  "cprod_map = (\<Lambda> f g p. (f\<cdot>(fst p), g\<cdot>(snd p)))"
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lemma cprod_map_Pair [simp]: "cprod_map\<cdot>f\<cdot>g\<cdot>(x, y) = (f\<cdot>x, g\<cdot>y)"
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unfolding cprod_map_def by simp
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lemma cprod_map_ID: "cprod_map\<cdot>ID\<cdot>ID = ID"
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unfolding expand_cfun_eq by auto
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lemma cprod_map_map:
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  "cprod_map\<cdot>f1\<cdot>g1\<cdot>(cprod_map\<cdot>f2\<cdot>g2\<cdot>p) =
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    cprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
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by (induct p) simp
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lemma ep_pair_cprod_map:
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  assumes "ep_pair e1 p1" and "ep_pair e2 p2"
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  shows "ep_pair (cprod_map\<cdot>e1\<cdot>e2) (cprod_map\<cdot>p1\<cdot>p2)"
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proof
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  interpret e1p1: ep_pair e1 p1 by fact
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  interpret e2p2: ep_pair e2 p2 by fact
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  fix x show "cprod_map\<cdot>p1\<cdot>p2\<cdot>(cprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
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    by (induct x) simp
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  fix y show "cprod_map\<cdot>e1\<cdot>e2\<cdot>(cprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
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    by (induct y) (simp add: e1p1.e_p_below e2p2.e_p_below)
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qed
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lemma deflation_cprod_map:
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  assumes "deflation d1" and "deflation d2"
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   139
  shows "deflation (cprod_map\<cdot>d1\<cdot>d2)"
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huffman
parents: 31113
diff changeset
   140
proof
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huffman
parents: 31113
diff changeset
   141
  interpret d1: deflation d1 by fact
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huffman
parents: 31113
diff changeset
   142
  interpret d2: deflation d2 by fact
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huffman
parents: 31113
diff changeset
   143
  fix x
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huffman
parents: 31113
diff changeset
   144
  show "cprod_map\<cdot>d1\<cdot>d2\<cdot>(cprod_map\<cdot>d1\<cdot>d2\<cdot>x) = cprod_map\<cdot>d1\<cdot>d2\<cdot>x"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   145
    by (induct x) (simp add: d1.idem d2.idem)
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huffman
parents: 31113
diff changeset
   146
  show "cprod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   147
    by (induct x) (simp add: d1.below d2.below)
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huffman
parents: 31113
diff changeset
   148
qed
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   149
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
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diff changeset
   150
lemma finite_deflation_cprod_map:
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huffman
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diff changeset
   151
  assumes "finite_deflation d1" and "finite_deflation d2"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   152
  shows "finite_deflation (cprod_map\<cdot>d1\<cdot>d2)"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   153
proof (intro finite_deflation.intro finite_deflation_axioms.intro)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   154
  interpret d1: finite_deflation d1 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   155
  interpret d2: finite_deflation d2 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   156
  have "deflation d1" and "deflation d2" by fact+
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   157
  thus "deflation (cprod_map\<cdot>d1\<cdot>d2)" by (rule deflation_cprod_map)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   158
  have "{p. cprod_map\<cdot>d1\<cdot>d2\<cdot>p = p} \<subseteq> {x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   159
    by clarsimp
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   160
  thus "finite {p. cprod_map\<cdot>d1\<cdot>d2\<cdot>p = p}"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   161
    by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   162
qed
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huffman
parents: 31113
diff changeset
   163
31113
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   164
instantiation "*" :: (profinite, profinite) profinite
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
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diff changeset
   165
begin
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   166
33504
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diff changeset
   167
definition
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
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diff changeset
   168
  approx_prod_def:
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
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diff changeset
   169
    "approx = (\<lambda>n. cprod_map\<cdot>(approx n)\<cdot>(approx n))"
31113
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   170
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   171
instance proof
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   172
  fix i :: nat and x :: "'a \<times> 'b"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   173
  show "chain (approx :: nat \<Rightarrow> 'a \<times> 'b \<rightarrow> 'a \<times> 'b)"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   174
    unfolding approx_prod_def by simp
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   175
  show "(\<Squnion>i. approx i\<cdot>x) = x"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   176
    unfolding approx_prod_def cprod_map_def
31113
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   177
    by (simp add: lub_distribs thelub_Pair)
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   178
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   179
    unfolding approx_prod_def cprod_map_def by simp
31113
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   180
  have "{x::'a \<times> 'b. approx i\<cdot>x = x} \<subseteq>
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   181
        {x::'a. approx i\<cdot>x = x} \<times> {x::'b. approx i\<cdot>x = x}"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   182
    unfolding approx_prod_def by clarsimp
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   183
  thus "finite {x::'a \<times> 'b. approx i\<cdot>x = x}"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   184
    by (rule finite_subset,
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   185
        intro finite_cartesian_product finite_fixes_approx)
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   186
qed
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   187
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   188
end
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   189
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   190
instance "*" :: (bifinite, bifinite) bifinite ..
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   191
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   192
lemma approx_Pair [simp]:
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   193
  "approx i\<cdot>(x, y) = (approx i\<cdot>x, approx i\<cdot>y)"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   194
unfolding approx_prod_def by simp
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   195
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   196
lemma fst_approx: "fst (approx i\<cdot>p) = approx i\<cdot>(fst p)"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   197
by (induct p, simp)
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   198
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   199
lemma snd_approx: "snd (approx i\<cdot>p) = approx i\<cdot>(snd p)"
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   200
by (induct p, simp)
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   201
15cf300a742f move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman
parents: 31076
diff changeset
   202
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   203
subsection {* Instance for continuous function space *}
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   204
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
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diff changeset
   205
definition
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   206
  cfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'd)"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   207
where
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   208
  "cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   209
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   210
lemma cfun_map_beta [simp]: "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   211
unfolding cfun_map_def by simp
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   212
33808
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33587
diff changeset
   213
lemma cfun_map_ID: "cfun_map\<cdot>ID\<cdot>ID = ID"
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33587
diff changeset
   214
unfolding expand_cfun_eq by simp
31169fdc5ae7 add map_ID lemmas
huffman
parents: 33587
diff changeset
   215
33587
54f98d225163 add map_map lemmas
huffman
parents: 33504
diff changeset
   216
lemma cfun_map_map:
54f98d225163 add map_map lemmas
huffman
parents: 33504
diff changeset
   217
  "cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
54f98d225163 add map_map lemmas
huffman
parents: 33504
diff changeset
   218
    cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
54f98d225163 add map_map lemmas
huffman
parents: 33504
diff changeset
   219
by (rule ext_cfun) simp
54f98d225163 add map_map lemmas
huffman
parents: 33504
diff changeset
   220
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   221
lemma ep_pair_cfun_map:
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   222
  assumes "ep_pair e1 p1" and "ep_pair e2 p2"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   223
  shows "ep_pair (cfun_map\<cdot>p1\<cdot>e2) (cfun_map\<cdot>e1\<cdot>p2)"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   224
proof
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   225
  interpret e1p1: ep_pair e1 p1 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   226
  interpret e2p2: ep_pair e2 p2 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   227
  fix f show "cfun_map\<cdot>e1\<cdot>p2\<cdot>(cfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   228
    by (simp add: expand_cfun_eq)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   229
  fix g show "cfun_map\<cdot>p1\<cdot>e2\<cdot>(cfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   230
    apply (rule below_cfun_ext, simp)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   231
    apply (rule below_trans [OF e2p2.e_p_below])
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   232
    apply (rule monofun_cfun_arg)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   233
    apply (rule e1p1.e_p_below)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   234
    done
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   235
qed
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   236
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   237
lemma deflation_cfun_map:
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   238
  assumes "deflation d1" and "deflation d2"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   239
  shows "deflation (cfun_map\<cdot>d1\<cdot>d2)"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   240
proof
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   241
  interpret d1: deflation d1 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   242
  interpret d2: deflation d2 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   243
  fix f
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   244
  show "cfun_map\<cdot>d1\<cdot>d2\<cdot>(cfun_map\<cdot>d1\<cdot>d2\<cdot>f) = cfun_map\<cdot>d1\<cdot>d2\<cdot>f"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   245
    by (simp add: expand_cfun_eq d1.idem d2.idem)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   246
  show "cfun_map\<cdot>d1\<cdot>d2\<cdot>f \<sqsubseteq> f"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   247
    apply (rule below_cfun_ext, simp)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   248
    apply (rule below_trans [OF d2.below])
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   249
    apply (rule monofun_cfun_arg)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   250
    apply (rule d1.below)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   251
    done
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   252
qed
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   253
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   254
lemma finite_range_cfun_map:
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   255
  assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   256
  assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   257
  shows "finite (range (\<lambda>f. cfun_map\<cdot>a\<cdot>b\<cdot>f))"  (is "finite (range ?h)")
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   258
proof (rule finite_imageD)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   259
  let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   260
  show "finite (?f ` range ?h)"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   261
  proof (rule finite_subset)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   262
    let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   263
    show "?f ` range ?h \<subseteq> ?B"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   264
      by clarsimp
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   265
    show "finite ?B"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   266
      by (simp add: a b)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   267
  qed
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   268
  show "inj_on ?f (range ?h)"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   269
  proof (rule inj_onI, rule ext_cfun, clarsimp)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   270
    fix x f g
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
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   271
    assume "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) = range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   272
    hence "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   273
      by (rule equalityD1)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   274
    hence "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   275
      by (simp add: subset_eq)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   276
    then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   277
      by (rule rangeE)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   278
    thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   279
      by clarsimp
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   280
  qed
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   281
qed
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   282
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   283
lemma finite_deflation_cfun_map:
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   284
  assumes "finite_deflation d1" and "finite_deflation d2"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   285
  shows "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   286
proof (intro finite_deflation.intro finite_deflation_axioms.intro)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   287
  interpret d1: finite_deflation d1 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   288
  interpret d2: finite_deflation d2 by fact
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   289
  have "deflation d1" and "deflation d2" by fact+
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   290
  thus "deflation (cfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_cfun_map)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   291
  have "finite (range (\<lambda>f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f))"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   292
    using d1.finite_range d2.finite_range
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   293
    by (rule finite_range_cfun_map)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   294
  thus "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   295
    by (rule finite_range_imp_finite_fixes)
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   296
qed
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   297
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 33808
diff changeset
   298
instantiation cfun :: (profinite, profinite) profinite
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 26407
diff changeset
   299
begin
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   300
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 26407
diff changeset
   301
definition
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   302
  approx_cfun_def:
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   303
    "approx = (\<lambda>n. cfun_map\<cdot>(approx n)\<cdot>(approx n))"
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   304
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   305
instance proof
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   306
  show "chain (approx :: nat \<Rightarrow> ('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow> 'b))"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   307
    unfolding approx_cfun_def by simp
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   308
next
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   309
  fix x :: "'a \<rightarrow> 'b"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   310
  show "(\<Squnion>i. approx i\<cdot>x) = x"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   311
    unfolding approx_cfun_def cfun_map_def
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   312
    by (simp add: lub_distribs eta_cfun)
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   313
next
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   314
  fix i :: nat and x :: "'a \<rightarrow> 'b"
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   315
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   316
    unfolding approx_cfun_def cfun_map_def by simp
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   317
next
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   318
  fix i :: nat
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   319
  show "finite {x::'a \<rightarrow> 'b. approx i\<cdot>x = x}"
33504
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   320
    unfolding approx_cfun_def
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   321
    by (intro finite_deflation.finite_fixes
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   322
              finite_deflation_cfun_map
b4210cc3ac97 map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents: 31113
diff changeset
   323
              finite_deflation_approx)
27402
253a06dfadce reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents: 27310
diff changeset
   324
qed
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   325
26962
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 26407
diff changeset
   326
end
c8b20f615d6c use new class package for classes profinite, bifinite; remove approx class
huffman
parents: 26407
diff changeset
   327
35525
fa231b86cb1e proper names for types cfun, sprod, ssum
huffman
parents: 33808
diff changeset
   328
instance cfun :: (profinite, bifinite) bifinite ..
25909
6b96b9392873 add class bifinite_cpo for possibly-unpointed bifinite domains
huffman
parents: 25903
diff changeset
   329
25903
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   330
lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))"
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   331
by (simp add: approx_cfun_def)
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   332
5e59af604d4f new theory of bifinite domains
huffman
parents:
diff changeset
   333
end