Theory Bifinite

Up to index of Isabelle/HOLCF

theory Bifinite
imports Cfun
begin

(*  Title:      HOLCF/Bifinite.thy
    ID:         $Id: Bifinite.thy,v 1.6 2008/05/19 21:49:20 huffman Exp $
    Author:     Brian Huffman
*)

header {* Bifinite domains and approximation *}

theory Bifinite
imports Cfun
begin

subsection {* Omega-profinite and bifinite domains *}

class profinite = cpo +
  fixes approx :: "nat => 'a -> 'a"
  assumes chain_approx_app: "chain (λi. approx i·x)"
  assumes lub_approx_app [simp]: "(\<Squnion>i. approx i·x) = x"
  assumes approx_idem: "approx i·(approx i·x) = approx i·x"
  assumes finite_fixes_approx: "finite {x. approx i·x = x}"

class bifinite = profinite + pcpo

lemma finite_range_imp_finite_fixes:
  "finite {x. ∃y. x = f y} ==> finite {x. f x = x}"
apply (subgoal_tac "{x. f x = x} ⊆ {x. ∃y. x = f y}")
apply (erule (1) finite_subset)
apply (clarify, erule subst, rule exI, rule refl)
done

lemma chain_approx [simp]:
  "chain (approx :: nat => 'a::profinite -> 'a)"
apply (rule chainI)
apply (rule less_cfun_ext)
apply (rule chainE)
apply (rule chain_approx_app)
done

lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (Λ(x::'a::profinite). x)"
by (rule ext_cfun, simp add: contlub_cfun_fun)

lemma approx_less: "approx i·x \<sqsubseteq> (x::'a::profinite)"
apply (subgoal_tac "approx i·x \<sqsubseteq> (\<Squnion>i. approx i·x)", simp)
apply (rule is_ub_thelub, simp)
done

lemma approx_strict [simp]: "approx i·(⊥::'a::bifinite) = ⊥"
by (rule UU_I, rule approx_less)

lemma approx_approx1:
  "i ≤ j ==> approx i·(approx j·x) = approx i·(x::'a::profinite)"
apply (rule antisym_less)
apply (rule monofun_cfun_arg [OF approx_less])
apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
apply (rule monofun_cfun_arg)
apply (rule monofun_cfun_fun)
apply (erule chain_mono [OF chain_approx])
done

lemma approx_approx2:
  "j ≤ i ==> approx i·(approx j·x) = approx j·(x::'a::profinite)"
apply (rule antisym_less)
apply (rule approx_less)
apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
apply (rule monofun_cfun_fun)
apply (erule chain_mono [OF chain_approx])
done

lemma approx_approx [simp]:
  "approx i·(approx j·x) = approx (min i j)·(x::'a::profinite)"
apply (rule_tac x=i and y=j in linorder_le_cases)
apply (simp add: approx_approx1 min_def)
apply (simp add: approx_approx2 min_def)
done

lemma idem_fixes_eq_range:
  "∀x. f (f x) = f x ==> {x. f x = x} = {y. ∃x. y = f x}"
by (auto simp add: eq_sym_conv)

lemma finite_approx: "finite {y::'a::profinite. ∃x. y = approx n·x}"
using finite_fixes_approx by (simp add: idem_fixes_eq_range)

lemma finite_range_approx:
  "finite (range (λx::'a::profinite. approx n·x))"
by (simp add: image_def finite_approx)

lemma compact_approx [simp]:
  fixes x :: "'a::profinite"
  shows "compact (approx n·x)"
proof (rule compactI2)
  fix Y::"nat => 'a"
  assume Y: "chain Y"
  have "finite_chain (λi. approx n·(Y i))"
  proof (rule finite_range_imp_finch)
    show "chain (λi. approx n·(Y i))"
      using Y by simp
    have "range (λi. approx n·(Y i)) ⊆ {x. approx n·x = x}"
      by clarsimp
    thus "finite (range (λi. approx n·(Y i)))"
      using finite_fixes_approx by (rule finite_subset)
  qed
  hence "∃j. (\<Squnion>i. approx n·(Y i)) = approx n·(Y j)"
    by (simp add: finite_chain_def maxinch_is_thelub Y)
  then obtain j where j: "(\<Squnion>i. approx n·(Y i)) = approx n·(Y j)" ..

  assume "approx n·x \<sqsubseteq> (\<Squnion>i. Y i)"
  hence "approx n·(approx n·x) \<sqsubseteq> approx n·(\<Squnion>i. Y i)"
    by (rule monofun_cfun_arg)
  hence "approx n·x \<sqsubseteq> (\<Squnion>i. approx n·(Y i))"
    by (simp add: contlub_cfun_arg Y)
  hence "approx n·x \<sqsubseteq> approx n·(Y j)"
    using j by simp
  hence "approx n·x \<sqsubseteq> Y j"
    using approx_less by (rule trans_less)
  thus "∃j. approx n·x \<sqsubseteq> Y j" ..
qed

lemma bifinite_compact_eq_approx:
  fixes x :: "'a::profinite"
  assumes x: "compact x"
  shows "∃i. approx i·x = x"
proof -
  have chain: "chain (λi. approx i·x)" by simp
  have less: "x \<sqsubseteq> (\<Squnion>i. approx i·x)" by simp
  obtain i where i: "x \<sqsubseteq> approx i·x"
    using compactD2 [OF x chain less] ..
  with approx_less have "approx i·x = x"
    by (rule antisym_less)
  thus "∃i. approx i·x = x" ..
qed

lemma bifinite_compact_iff:
  "compact (x::'a::profinite) = (∃n. approx n·x = x)"
 apply (rule iffI)
  apply (erule bifinite_compact_eq_approx)
 apply (erule exE)
 apply (erule subst)
 apply (rule compact_approx)
done

lemma approx_induct:
  assumes adm: "adm P" and P: "!!n x. P (approx n·x)"
  shows "P (x::'a::profinite)"
proof -
  have "P (\<Squnion>n. approx n·x)"
    by (rule admD [OF adm], simp, simp add: P)
  thus "P x" by simp
qed

lemma bifinite_less_ext:
  fixes x y :: "'a::profinite"
  shows "(!!i. approx i·x \<sqsubseteq> approx i·y) ==> x \<sqsubseteq> y"
apply (subgoal_tac "(\<Squnion>i. approx i·x) \<sqsubseteq> (\<Squnion>i. approx i·y)", simp)
apply (rule lub_mono, simp, simp, simp)
done

subsection {* Instance for continuous function space *}

lemma finite_range_lemma:
  fixes h :: "'a::cpo -> 'b::cpo"
  fixes k :: "'c::cpo -> 'd::cpo"
  shows "[|finite {y. ∃x. y = h·x}; finite {y. ∃x. y = k·x}|]
    ==> finite {g. ∃f. g = (Λ x. k·(f·(h·x)))}"
 apply (rule_tac f="λg. {(h·x, y) |x y. y = g·x}" in finite_imageD)
  apply (rule_tac B="Pow ({y. ∃x. y = h·x} × {y. ∃x. y = k·x})"
           in finite_subset)
   apply (rule image_subsetI)
   apply (clarsimp, fast)
  apply simp
 apply (rule inj_onI)
 apply (clarsimp simp add: expand_set_eq)
 apply (rule ext_cfun, simp)
 apply (drule_tac x="h·x" in spec)
 apply (drule_tac x="k·(f·(h·x))" in spec)
 apply (drule iffD1, fast)
 apply clarsimp
done

instantiation "->" :: (profinite, profinite) profinite
begin

definition
  approx_cfun_def:
    "approx = (λn. Λ f x. approx n·(f·(approx n·x)))"

instance
 apply (intro_classes, unfold approx_cfun_def)
    apply simp
   apply (simp add: lub_distribs eta_cfun)
  apply simp
 apply simp
 apply (rule finite_range_imp_finite_fixes)
 apply (intro finite_range_lemma finite_approx)
done

end

instance "->" :: (profinite, bifinite) bifinite ..

lemma approx_cfun: "approx n·f·x = approx n·(f·(approx n·x))"
by (simp add: approx_cfun_def)

end

Omega-profinite and bifinite domains

lemma finite_range_imp_finite_fixes:

  finite {x. ∃y. x = f y} ==> finite {x. f x = x}

lemma chain_approx:

  chain approx

lemma lub_approx:

  (LUB i. approx i) = (LAM x. x)

lemma approx_less:

  approx i·x << x

lemma approx_strict:

  approx i·UU = UU

lemma approx_approx1:

  i  j ==> approx i·(approx j·x) = approx i·x

lemma approx_approx2:

  j  i ==> approx i·(approx j·x) = approx j·x

lemma approx_approx:

  approx i·(approx j·x) = approx (min i jx

lemma idem_fixes_eq_range:

  x. f (f x) = f x ==> {x. f x = x} = {y. ∃x. y = f x}

lemma finite_approx:

  finite {y. ∃x. y = approx n·x}

lemma finite_range_approx:

  finite (range (Rep_CFun (approx n)))

lemma compact_approx:

  compact (approx n·x)

lemma bifinite_compact_eq_approx:

  compact x ==> ∃i. approx i·x = x

lemma bifinite_compact_iff:

  compact x = (∃n. approx n·x = x)

lemma approx_induct:

  [| adm P; !!n x. P (approx n·x) |] ==> P x

lemma bifinite_less_ext:

  (!!i. approx i·x << approx i·y) ==> x << y

Instance for continuous function space

lemma finite_range_lemma:

  [| finite {y. ∃x. y = h·x}; finite {y. ∃x. y = k·x} |]
  ==> finite {g. ∃f. g = (LAM x. k·(f·(h·x)))}

lemma approx_cfun:

  approx n·f·x = approx n·(f·(approx n·x))