Theory Continuity

(*  Title:      HOL/Library/Continuity.thy
    ID:         $Id: Continuity.thy,v 1.7 2004/08/18 09:10:01 nipkow Exp $
    Author:     David von Oheimb, TU Muenchen
*)

header {* Continuity and iterations (of set transformers) *}

theory Continuity
imports Main
begin

subsection "Chains"

constdefs
  up_chain :: "(nat => 'a set) => bool"
  "up_chain F == ∀i. F i ⊆ F (Suc i)"

lemma up_chainI: "(!!i. F i ⊆ F (Suc i)) ==> up_chain F"
  by (simp add: up_chain_def)

lemma up_chainD: "up_chain F ==> F i ⊆ F (Suc i)"
  by (simp add: up_chain_def)

lemma up_chain_less_mono [rule_format]:
    "up_chain F ==> x < y --> F x ⊆ F y"
  apply (induct_tac y)
  apply (blast dest: up_chainD elim: less_SucE)+
  done

lemma up_chain_mono: "up_chain F ==> x ≤ y ==> F x ⊆ F y"
  apply (drule le_imp_less_or_eq)
  apply (blast dest: up_chain_less_mono)
  done


constdefs
  down_chain :: "(nat => 'a set) => bool"
  "down_chain F == ∀i. F (Suc i) ⊆ F i"

lemma down_chainI: "(!!i. F (Suc i) ⊆ F i) ==> down_chain F"
  by (simp add: down_chain_def)

lemma down_chainD: "down_chain F ==> F (Suc i) ⊆ F i"
  by (simp add: down_chain_def)

lemma down_chain_less_mono [rule_format]:
    "down_chain F ==> x < y --> F y ⊆ F x"
  apply (induct_tac y)
  apply (blast dest: down_chainD elim: less_SucE)+
  done

lemma down_chain_mono: "down_chain F ==> x ≤ y ==> F y ⊆ F x"
  apply (drule le_imp_less_or_eq)
  apply (blast dest: down_chain_less_mono)
  done


subsection "Continuity"

constdefs
  up_cont :: "('a set => 'a set) => bool"
  "up_cont f == ∀F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F)"

lemma up_contI:
    "(!!F. up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)) ==> up_cont f"
  apply (unfold up_cont_def)
  apply blast
  done

lemma up_contD:
    "up_cont f ==> up_chain F ==> f (\<Union>(range F)) = \<Union>(f ` range F)"
  apply (unfold up_cont_def)
  apply auto
  done


lemma up_cont_mono: "up_cont f ==> mono f"
  apply (rule monoI)
  apply (drule_tac F = "λi. if i = 0 then A else B" in up_contD)
   apply (rule up_chainI)
   apply  simp+
  apply (drule Un_absorb1)
  apply (auto simp add: nat_not_singleton)
  done


constdefs
  down_cont :: "('a set => 'a set) => bool"
  "down_cont f ==
    ∀F. down_chain F --> f (Inter (range F)) = Inter (f ` range F)"

lemma down_contI:
  "(!!F. down_chain F ==> f (Inter (range F)) = Inter (f ` range F)) ==>
    down_cont f"
  apply (unfold down_cont_def)
  apply blast
  done

lemma down_contD: "down_cont f ==> down_chain F ==>
    f (Inter (range F)) = Inter (f ` range F)"
  apply (unfold down_cont_def)
  apply auto
  done

lemma down_cont_mono: "down_cont f ==> mono f"
  apply (rule monoI)
  apply (drule_tac F = "λi. if i = 0 then B else A" in down_contD)
   apply (rule down_chainI)
   apply simp+
  apply (drule Int_absorb1)
  apply (auto simp add: nat_not_singleton)
  done


subsection "Iteration"

constdefs
  up_iterate :: "('a set => 'a set) => nat => 'a set"
  "up_iterate f n == (f^n) {}"

lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
  by (simp add: up_iterate_def)

lemma up_iterate_Suc [simp]: "up_iterate f (Suc i) = f (up_iterate f i)"
  by (simp add: up_iterate_def)

lemma up_iterate_chain: "mono F ==> up_chain (up_iterate F)"
  apply (rule up_chainI)
  apply (induct_tac i)
   apply simp+
  apply (erule (1) monoD)
  done

lemma UNION_up_iterate_is_fp:
  "up_cont F ==>
    F (UNION UNIV (up_iterate F)) = UNION UNIV (up_iterate F)"
  apply (frule up_cont_mono [THEN up_iterate_chain])
  apply (drule (1) up_contD)
  apply simp
  apply (auto simp del: up_iterate_Suc simp add: up_iterate_Suc [symmetric])
  apply (case_tac xa)
   apply auto
  done

lemma UNION_up_iterate_lowerbound:
    "mono F ==> F P = P ==> UNION UNIV (up_iterate F) ⊆ P"
  apply (subgoal_tac "(!!i. up_iterate F i ⊆ P)")
   apply fast
  apply (induct_tac i)
  prefer 2 apply (drule (1) monoD)
   apply auto
  done

lemma UNION_up_iterate_is_lfp:
    "up_cont F ==> lfp F = UNION UNIV (up_iterate F)"
  apply (rule set_eq_subset [THEN iffD2])
  apply (rule conjI)
   prefer 2
   apply (drule up_cont_mono)
   apply (rule UNION_up_iterate_lowerbound)
    apply assumption
   apply (erule lfp_unfold [symmetric])
  apply (rule lfp_lowerbound)
  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
  apply (erule UNION_up_iterate_is_fp [symmetric])
  done


constdefs
  down_iterate :: "('a set => 'a set) => nat => 'a set"
  "down_iterate f n == (f^n) UNIV"

lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
  by (simp add: down_iterate_def)

lemma down_iterate_Suc [simp]:
    "down_iterate f (Suc i) = f (down_iterate f i)"
  by (simp add: down_iterate_def)

lemma down_iterate_chain: "mono F ==> down_chain (down_iterate F)"
  apply (rule down_chainI)
  apply (induct_tac i)
   apply simp+
  apply (erule (1) monoD)
  done

lemma INTER_down_iterate_is_fp:
  "down_cont F ==>
    F (INTER UNIV (down_iterate F)) = INTER UNIV (down_iterate F)"
  apply (frule down_cont_mono [THEN down_iterate_chain])
  apply (drule (1) down_contD)
  apply simp
  apply (auto simp del: down_iterate_Suc simp add: down_iterate_Suc [symmetric])
  apply (case_tac xa)
   apply auto
  done

lemma INTER_down_iterate_upperbound:
    "mono F ==> F P = P ==> P ⊆ INTER UNIV (down_iterate F)"
  apply (subgoal_tac "(!!i. P ⊆ down_iterate F i)")
   apply fast
  apply (induct_tac i)
  prefer 2 apply (drule (1) monoD)
   apply auto
  done

lemma INTER_down_iterate_is_gfp:
    "down_cont F ==> gfp F = INTER UNIV (down_iterate F)"
  apply (rule set_eq_subset [THEN iffD2])
  apply (rule conjI)
   apply (drule down_cont_mono)
   apply (rule INTER_down_iterate_upperbound)
    apply assumption
   apply (erule gfp_unfold [symmetric])
  apply (rule gfp_upperbound)
  apply (rule set_eq_subset [THEN iffD1, THEN conjunct2])
  apply (erule INTER_down_iterate_is_fp)
  done

end