author huffman Tue, 01 Jul 2008 01:25:40 +0200 changeset 27409 f65a889f97f9 parent 27408 22a515a55bf5 child 27410 22f75653163f
theory of algebraic deflations
 src/HOLCF/Algebraic.thy file | annotate | diff | comparison | revisions
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+++ b/src/HOLCF/Algebraic.thy	Tue Jul 01 01:25:40 2008 +0200
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+(*  Title:      HOLCF/Algebraic.thy
+    ID:         \$Id\$
+    Author:     Brian Huffman
+*)
+
+
+theory Algebraic
+imports Completion Fix Eventual
+begin
+
+declare range_composition [simp del]
+
+subsection {* Constructing finite deflations by iteration *}
+
+lemma finite_deflation_imp_deflation:
+  "finite_deflation d \<Longrightarrow> deflation d"
+unfolding finite_deflation_def by simp
+
+lemma le_Suc_induct:
+  assumes le: "i \<le> j"
+  assumes step: "\<And>i. P i (Suc i)"
+  assumes refl: "\<And>i. P i i"
+  assumes trans: "\<And>i j k. \<lbrakk>P i j; P j k\<rbrakk> \<Longrightarrow> P i k"
+  shows "P i j"
+proof (cases "i = j")
+  assume "i = j"
+  thus "P i j" by (simp add: refl)
+next
+  assume "i \<noteq> j"
+  with le have "i < j" by simp
+  thus "P i j" using step trans by (rule less_Suc_induct)
+qed
+
+text {* A pre-deflation is like a deflation, but not idempotent. *}
+
+locale pre_deflation =
+  fixes f :: "'a \<rightarrow> 'a::cpo"
+  assumes less: "\<And>x. f\<cdot>x \<sqsubseteq> x"
+  assumes finite_range: "finite (range (\<lambda>x. f\<cdot>x))"
+begin
+
+lemma iterate_less: "iterate i\<cdot>f\<cdot>x \<sqsubseteq> x"
+by (induct i, simp_all add: trans_less [OF less])
+
+lemma iterate_fixed: "f\<cdot>x = x \<Longrightarrow> iterate i\<cdot>f\<cdot>x = x"
+by (induct i, simp_all)
+
+lemma antichain_iterate_app: "i \<le> j \<Longrightarrow> iterate j\<cdot>f\<cdot>x \<sqsubseteq> iterate i\<cdot>f\<cdot>x"
+apply (erule le_Suc_induct)
+apply (rule refl_less)
+apply (erule (1) trans_less)
+done
+
+lemma finite_range_iterate_app: "finite (range (\<lambda>i. iterate i\<cdot>f\<cdot>x))"
+proof (rule finite_subset)
+  show "range (\<lambda>i. iterate i\<cdot>f\<cdot>x) \<subseteq> insert x (range (\<lambda>x. f\<cdot>x))"
+    by (clarify, case_tac i, simp_all)
+  show "finite (insert x (range (\<lambda>x. f\<cdot>x)))"
+qed
+
+lemma eventually_constant_iterate_app:
+  "eventually_constant (\<lambda>i. iterate i\<cdot>f\<cdot>x)"
+unfolding eventually_constant_def MOST_nat_le
+proof -
+  let ?Y = "\<lambda>i. iterate i\<cdot>f\<cdot>x"
+  have "\<exists>j. \<forall>k. ?Y j \<sqsubseteq> ?Y k"
+    apply (rule finite_range_has_max)
+    apply (erule antichain_iterate_app)
+    apply (rule finite_range_iterate_app)
+    done
+  then obtain j where j: "\<And>k. ?Y j \<sqsubseteq> ?Y k" by fast
+  show "\<exists>z m. \<forall>n\<ge>m. ?Y n = z"
+  proof (intro exI allI impI)
+    fix k
+    assume "j \<le> k"
+    hence "?Y k \<sqsubseteq> ?Y j" by (rule antichain_iterate_app)
+    also have "?Y j \<sqsubseteq> ?Y k" by (rule j)
+    finally show "?Y k = ?Y j" .
+  qed
+qed
+
+lemma eventually_constant_iterate:
+  "eventually_constant (\<lambda>n. iterate n\<cdot>f)"
+proof -
+  have "\<forall>y\<in>range (\<lambda>x. f\<cdot>x). eventually_constant (\<lambda>i. iterate i\<cdot>f\<cdot>y)"
+  hence "\<forall>y\<in>range (\<lambda>x. f\<cdot>x). MOST i. MOST j. iterate j\<cdot>f\<cdot>y = iterate i\<cdot>f\<cdot>y"
+    unfolding eventually_constant_MOST_MOST .
+  hence "MOST i. MOST j. \<forall>y\<in>range (\<lambda>x. f\<cdot>x). iterate j\<cdot>f\<cdot>y = iterate i\<cdot>f\<cdot>y"
+    by (simp only: MOST_finite_Ball_distrib [OF finite_range])
+  hence "MOST i. MOST j. \<forall>x. iterate j\<cdot>f\<cdot>(f\<cdot>x) = iterate i\<cdot>f\<cdot>(f\<cdot>x)"
+    by simp
+  hence "MOST i. MOST j. \<forall>x. iterate (Suc j)\<cdot>f\<cdot>x = iterate (Suc i)\<cdot>f\<cdot>x"
+    by (simp only: iterate_Suc2)
+  hence "MOST i. MOST j. iterate (Suc j)\<cdot>f = iterate (Suc i)\<cdot>f"
+    by (simp only: expand_cfun_eq)
+  hence "eventually_constant (\<lambda>i. iterate (Suc i)\<cdot>f)"
+    unfolding eventually_constant_MOST_MOST .
+  thus "eventually_constant (\<lambda>i. iterate i\<cdot>f)"
+    by (rule eventually_constant_SucD)
+qed
+
+abbreviation
+  d :: "'a \<rightarrow> 'a"
+where
+  "d \<equiv> eventual (\<lambda>n. iterate n\<cdot>f)"
+
+lemma MOST_d: "MOST n. P (iterate n\<cdot>f) \<Longrightarrow> P d"
+using eventually_constant_iterate by (rule MOST_eventual)
+
+lemma f_d: "f\<cdot>(d\<cdot>x) = d\<cdot>x"
+apply (rule MOST_d)
+apply (subst iterate_Suc [symmetric])
+apply (rule eventually_constant_MOST_Suc_eq)
+apply (rule eventually_constant_iterate_app)
+done
+
+lemma d_fixed_iff: "d\<cdot>x = x \<longleftrightarrow> f\<cdot>x = x"
+proof
+  assume "d\<cdot>x = x"
+  with f_d [where x=x]
+  show "f\<cdot>x = x" by simp
+next
+  assume f: "f\<cdot>x = x"
+  have "\<forall>n. iterate n\<cdot>f\<cdot>x = x"
+    by (rule allI, rule nat.induct, simp, simp add: f)
+  hence "MOST n. iterate n\<cdot>f\<cdot>x = x"
+    by (rule ALL_MOST)
+  thus "d\<cdot>x = x"
+    by (rule MOST_d)
+qed
+
+lemma finite_deflation_d: "finite_deflation d"
+proof
+  fix x :: 'a
+  have "d \<in> range (\<lambda>n. iterate n\<cdot>f)"
+    using eventually_constant_iterate
+    by (rule eventual_mem_range)
+  then obtain n where n: "d = iterate n\<cdot>f" ..
+  have "iterate n\<cdot>f\<cdot>(d\<cdot>x) = d\<cdot>x"
+    using f_d by (rule iterate_fixed)
+  thus "d\<cdot>(d\<cdot>x) = d\<cdot>x"
+next
+  fix x :: 'a
+  show "d\<cdot>x \<sqsubseteq> x"
+    by (rule MOST_d, simp add: iterate_less)
+next
+  from finite_range
+  have "finite {x. f\<cdot>x = x}"
+    by (rule finite_range_imp_finite_fixes)
+  thus "finite {x. d\<cdot>x = x}"
+qed
+
+end
+
+lemma pre_deflation_d_f:
+  includes finite_deflation d
+  assumes f: "\<And>x. f\<cdot>x \<sqsubseteq> x"
+  shows "pre_deflation (d oo f)"
+proof
+  fix x
+  show "\<And>x. (d oo f)\<cdot>x \<sqsubseteq> x"
+    by (simp, rule trans_less [OF d.less f])
+  show "finite (range (\<lambda>x. (d oo f)\<cdot>x))"
+    by (rule finite_subset [OF _ d.finite_range], auto)
+qed
+
+lemma eventual_iterate_oo_fixed_iff:
+  includes finite_deflation d
+  assumes f: "\<And>x. f\<cdot>x \<sqsubseteq> x"
+  shows "eventual (\<lambda>n. iterate n\<cdot>(d oo f))\<cdot>x = x \<longleftrightarrow> d\<cdot>x = x \<and> f\<cdot>x = x"
+proof -
+  let ?e = "d oo f"
+  interpret e: pre_deflation ["d oo f"]
+    using `finite_deflation d` f
+    by (rule pre_deflation_d_f)
+  let ?g = "eventual (\<lambda>n. iterate n\<cdot>?e)"
+  show ?thesis
+    apply (subst e.d_fixed_iff)
+    apply simp
+    apply safe
+    apply (erule subst)
+    apply (rule d.idem)
+    apply (rule antisym_less)
+    apply (rule f)
+    apply (erule subst, rule d.less)
+    apply simp
+    done
+qed
+
+subsection {* Type constructor for finite deflations *}
+
+defaultsort profinite
+
+typedef (open) 'a fin_defl = "{d::'a \<rightarrow> 'a. finite_deflation d}"
+by (fast intro: finite_deflation_approx)
+
+instantiation fin_defl :: (profinite) sq_ord
+begin
+
+definition
+  sq_le_fin_defl_def:
+    "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep_fin_defl x \<sqsubseteq> Rep_fin_defl y"
+
+instance ..
+end
+
+instance fin_defl :: (profinite) po
+by (rule typedef_po [OF type_definition_fin_defl sq_le_fin_defl_def])
+
+lemma finite_deflation_Rep_fin_defl: "finite_deflation (Rep_fin_defl d)"
+using Rep_fin_defl by simp
+
+interpretation Rep_fin_defl: finite_deflation ["Rep_fin_defl d"]
+by (rule finite_deflation_Rep_fin_defl)
+
+lemma fin_defl_lessI:
+  "(\<And>x. Rep_fin_defl a\<cdot>x = x \<Longrightarrow> Rep_fin_defl b\<cdot>x = x) \<Longrightarrow> a \<sqsubseteq> b"
+unfolding sq_le_fin_defl_def
+by (rule Rep_fin_defl.lessI)
+
+lemma fin_defl_lessD:
+  "\<lbrakk>a \<sqsubseteq> b; Rep_fin_defl a\<cdot>x = x\<rbrakk> \<Longrightarrow> Rep_fin_defl b\<cdot>x = x"
+unfolding sq_le_fin_defl_def
+by (rule Rep_fin_defl.lessD)
+
+lemma fin_defl_eqI:
+  "(\<And>x. Rep_fin_defl a\<cdot>x = x \<longleftrightarrow> Rep_fin_defl b\<cdot>x = x) \<Longrightarrow> a = b"
+apply (rule antisym_less)
+apply (rule fin_defl_lessI, simp)
+apply (rule fin_defl_lessI, simp)
+done
+
+lemma Abs_fin_defl_mono:
+  "\<lbrakk>finite_deflation a; finite_deflation b; a \<sqsubseteq> b\<rbrakk>
+    \<Longrightarrow> Abs_fin_defl a \<sqsubseteq> Abs_fin_defl b"
+unfolding sq_le_fin_defl_def
+
+
+subsection {* Take function for finite deflations *}
+
+definition
+  fd_take :: "nat \<Rightarrow> 'a fin_defl \<Rightarrow> 'a fin_defl"
+where
+  "fd_take i d = Abs_fin_defl (eventual (\<lambda>n. iterate n\<cdot>(approx i oo Rep_fin_defl d)))"
+
+lemma Rep_fin_defl_fd_take:
+  "Rep_fin_defl (fd_take i d) =
+    eventual (\<lambda>n. iterate n\<cdot>(approx i oo Rep_fin_defl d))"
+unfolding fd_take_def
+apply (rule Abs_fin_defl_inverse [unfolded mem_Collect_eq])
+apply (rule pre_deflation.finite_deflation_d)
+apply (rule pre_deflation_d_f)
+apply (rule finite_deflation_approx)
+apply (rule Rep_fin_defl.less)
+done
+
+lemma fd_take_fixed_iff:
+  "Rep_fin_defl (fd_take i d)\<cdot>x = x \<longleftrightarrow>
+    approx i\<cdot>x = x \<and> Rep_fin_defl d\<cdot>x = x"
+unfolding Rep_fin_defl_fd_take
+by (rule eventual_iterate_oo_fixed_iff
+    [OF finite_deflation_approx Rep_fin_defl.less])
+
+lemma fd_take_less: "fd_take n d \<sqsubseteq> d"
+apply (rule fin_defl_lessI)
+done
+
+lemma fd_take_idem: "fd_take n (fd_take n d) = fd_take n d"
+apply (rule fin_defl_eqI)
+done
+
+lemma fd_take_mono: "a \<sqsubseteq> b \<Longrightarrow> fd_take n a \<sqsubseteq> fd_take n b"
+apply (rule fin_defl_lessI)
+done
+
+lemma approx_fixed_le_lemma: "\<lbrakk>i \<le> j; approx i\<cdot>x = x\<rbrakk> \<Longrightarrow> approx j\<cdot>x = x"
+by (erule subst, simp add: min_def)
+
+lemma fd_take_chain: "m \<le> n \<Longrightarrow> fd_take m a \<sqsubseteq> fd_take n a"
+apply (rule fin_defl_lessI)
+done
+
+lemma finite_range_fd_take: "finite (range (fd_take n))"
+apply (rule finite_imageD [where f="\<lambda>a. {x. Rep_fin_defl a\<cdot>x = x}"])
+apply (rule finite_subset [where B="Pow {x. approx n\<cdot>x = x}"])
+apply (rule inj_onI, clarify)
+done
+
+lemma fd_take_covers: "\<exists>n. fd_take n a = a"
+apply (rule_tac x=
+  "Max ((\<lambda>x. LEAST n. approx n\<cdot>x = x) ` {x. Rep_fin_defl a\<cdot>x = x})" in exI)
+apply (rule antisym_less)
+apply (rule fd_take_less)
+apply (rule fin_defl_lessI)
+apply (rule approx_fixed_le_lemma)
+apply (rule Max_ge)
+apply (rule finite_imageI)
+apply (rule Rep_fin_defl.finite_fixes)
+apply (rule imageI)
+apply (erule CollectI)
+apply (rule LeastI_ex)
+apply (rule profinite_compact_eq_approx)
+apply (erule subst)
+apply (rule Rep_fin_defl.compact)
+done
+
+interpretation fin_defl: basis_take [sq_le fd_take]
+apply default
+apply (rule fd_take_less)
+apply (rule fd_take_idem)
+apply (erule fd_take_mono)
+apply (rule fd_take_chain, simp)
+apply (rule finite_range_fd_take)
+apply (rule fd_take_covers)
+done
+
+subsection {* Defining algebraic deflations by ideal completion *}
+
+typedef (open) 'a alg_defl =
+  "{S::'a fin_defl set. sq_le.ideal S}"
+by (fast intro: sq_le.ideal_principal)
+
+instantiation alg_defl :: (profinite) sq_ord
+begin
+
+definition
+  "x \<sqsubseteq> y \<longleftrightarrow> Rep_alg_defl x \<subseteq> Rep_alg_defl y"
+
+instance ..
+end
+
+instance alg_defl :: (profinite) po
+by (rule sq_le.typedef_ideal_po
+    [OF type_definition_alg_defl sq_le_alg_defl_def])
+
+instance alg_defl :: (profinite) cpo
+by (rule sq_le.typedef_ideal_cpo
+    [OF type_definition_alg_defl sq_le_alg_defl_def])
+
+lemma Rep_alg_defl_lub:
+  "chain Y \<Longrightarrow> Rep_alg_defl (\<Squnion>i. Y i) = (\<Union>i. Rep_alg_defl (Y i))"
+by (rule sq_le.typedef_ideal_rep_contlub
+    [OF type_definition_alg_defl sq_le_alg_defl_def])
+
+lemma ideal_Rep_alg_defl: "sq_le.ideal (Rep_alg_defl xs)"
+by (rule Rep_alg_defl [unfolded mem_Collect_eq])
+
+definition
+  alg_defl_principal :: "'a fin_defl \<Rightarrow> 'a alg_defl" where
+  "alg_defl_principal t = Abs_alg_defl {u. u \<sqsubseteq> t}"
+
+lemma Rep_alg_defl_principal:
+  "Rep_alg_defl (alg_defl_principal t) = {u. u \<sqsubseteq> t}"
+unfolding alg_defl_principal_def
+
+interpretation alg_defl:
+  ideal_completion [sq_le fd_take alg_defl_principal Rep_alg_defl]
+apply default
+apply (rule ideal_Rep_alg_defl)
+apply (erule Rep_alg_defl_lub)
+apply (rule Rep_alg_defl_principal)
+apply (simp only: sq_le_alg_defl_def)
+done
+
+text {* Algebraic deflations are pointed *}
+
+lemma finite_deflation_UU: "finite_deflation \<bottom>"
+by default simp_all
+
+lemma alg_defl_minimal:
+  "alg_defl_principal (Abs_fin_defl \<bottom>) \<sqsubseteq> x"
+apply (induct x rule: alg_defl.principal_induct, simp)
+apply (rule alg_defl.principal_mono)
+apply (induct_tac a)
+apply (rule Abs_fin_defl_mono)
+apply (rule finite_deflation_UU)
+apply simp
+apply (rule minimal)
+done
+
+instance alg_defl :: (bifinite) pcpo
+by intro_classes (fast intro: alg_defl_minimal)
+
+lemma inst_alg_defl_pcpo: "\<bottom> = alg_defl_principal (Abs_fin_defl \<bottom>)"
+by (rule alg_defl_minimal [THEN UU_I, symmetric])
+
+text {* Algebraic deflations are profinite *}
+
+instantiation alg_defl :: (profinite) profinite
+begin
+
+definition
+  approx_alg_defl_def: "approx = alg_defl.completion_approx"
+
+instance
+apply (intro_classes, unfold approx_alg_defl_def)
+apply (rule alg_defl.chain_completion_approx)
+apply (rule alg_defl.lub_completion_approx)
+apply (rule alg_defl.completion_approx_idem)
+apply (rule alg_defl.finite_fixes_completion_approx)
+done
+
+end
+
+instance alg_defl :: (bifinite) bifinite ..
+
+lemma approx_alg_defl_principal [simp]:
+  "approx n\<cdot>(alg_defl_principal t) = alg_defl_principal (fd_take n t)"
+unfolding approx_alg_defl_def
+by (rule alg_defl.completion_approx_principal)
+
+lemma approx_eq_alg_defl_principal:
+  "\<exists>t\<in>Rep_alg_defl xs. approx n\<cdot>xs = alg_defl_principal (fd_take n t)"
+unfolding approx_alg_defl_def
+by (rule alg_defl.completion_approx_eq_principal)
+
+
+subsection {* Applying algebraic deflations *}
+
+definition
+  cast :: "'a alg_defl \<rightarrow> 'a \<rightarrow> 'a"
+where
+  "cast = alg_defl.basis_fun Rep_fin_defl"
+
+lemma cast_alg_defl_principal:
+  "cast\<cdot>(alg_defl_principal a) = Rep_fin_defl a"
+unfolding cast_def
+apply (rule alg_defl.basis_fun_principal)
+apply (simp only: sq_le_fin_defl_def)
+done
+
+lemma deflation_cast: "deflation (cast\<cdot>d)"
+apply (induct d rule: alg_defl.principal_induct)
+apply (rule finite_deflation_imp_deflation)
+apply (rule finite_deflation_Rep_fin_defl)
+done
+
+lemma finite_deflation_cast:
+  "compact d \<Longrightarrow> finite_deflation (cast\<cdot>d)"
+apply (drule alg_defl.compact_imp_principal, clarify)
+apply (rule finite_deflation_Rep_fin_defl)
+done
+
+interpretation cast: deflation ["cast\<cdot>d"]
+by (rule deflation_cast)
+
+lemma "cast\<cdot>(\<Squnion>i. alg_defl_principal (Abs_fin_defl (approx i)))\<cdot>x = x"
+apply (subst contlub_cfun_arg)
+apply (rule chainI)
+apply (rule alg_defl.principal_mono)
+apply (rule Abs_fin_defl_mono)
+apply (rule finite_deflation_approx)
+apply (rule finite_deflation_approx)
+apply (rule chainE)
+apply (rule chain_approx)
+apply (simp add: cast_alg_defl_principal Abs_fin_defl_inverse finite_deflation_approx)
+done
+
+text {* This lemma says that if we have an ep-pair from
+a bifinite domain into a universal domain, then e oo p
+is an algebraic deflation. *}
+
+lemma
+  includes ep_pair e p
+  constrains e :: "'a::profinite \<rightarrow> 'b::profinite"
+  shows "\<exists>d. cast\<cdot>d = e oo p"
+proof
+  let ?a = "\<lambda>i. e oo approx i oo p"
+  have a: "\<And>i. finite_deflation (?a i)"
+    apply (rule finite_deflation_e_d_p)
+    apply (rule finite_deflation_approx)
+    done
+  let ?d = "\<Squnion>i. alg_defl_principal (Abs_fin_defl (?a i))"
+  show "cast\<cdot>?d = e oo p"
+    apply (subst contlub_cfun_arg)
+    apply (rule chainI)
+    apply (rule alg_defl.principal_mono)
+    apply (rule Abs_fin_defl_mono [OF a a])
+    apply (rule chainE, simp)
+    apply (subst cast_alg_defl_principal)
+    apply (simp add: Abs_fin_defl_inverse a)
+    apply (simp add: expand_cfun_eq lub_distribs)
+    done
+qed
+
+text {* This lemma says that if we have an ep-pair
+from a cpo into a bifinite domain, and e oo p is
+an algebraic deflation, then the cpo is bifinite. *}
+
+lemma
+  includes ep_pair e p
+  constrains e :: "'a::cpo \<rightarrow> 'b::profinite"
+  assumes d: "\<And>x. cast\<cdot>d\<cdot>x = e\<cdot>(p\<cdot>x)"
+  obtains a :: "nat \<Rightarrow> 'a \<rightarrow> 'a" where
+    "\<And>i. finite_deflation (a i)"
+    "(\<Squnion>i. a i) = ID"
+proof
+  let ?a = "\<lambda>i. p oo cast\<cdot>(approx i\<cdot>d) oo e"
+  show "\<And>i. finite_deflation (?a i)"
+    apply (rule finite_deflation_p_d_e)
+    apply (rule finite_deflation_cast)
+    apply (rule compact_approx)
+    apply (rule sq_ord_less_eq_trans [OF _ d])
+    apply (rule monofun_cfun_fun)
+    apply (rule monofun_cfun_arg)
+    apply (rule approx_less)
+    done
+  show "(\<Squnion>i. ?a i) = ID"
+    apply (rule ext_cfun, simp)