src/HOL/Library/Continuity.thy
changeset 11355 778c369559d9
parent 11351 c5c403d30c77
child 11461 ffeac9aa1967
--- a/src/HOL/Library/Continuity.thy	Thu May 31 18:28:23 2001 +0200
+++ b/src/HOL/Library/Continuity.thy	Thu May 31 20:52:51 2001 +0200
@@ -1,219 +1,222 @@
 (*  Title:      HOL/Library/Continuity.thy
-    ID:         $$
-    Author: 	David von Oheimb, TU Muenchen
+    ID:         $Id$
+    Author:     David von Oheimb, TU Muenchen
     License:    GPL (GNU GENERAL PUBLIC LICENSE)
-
 *)
 
 header {*
-  \title{Continuity and interations (of set transformers)}
+  \title{Continuity and iterations (of set transformers)}
   \author{David von Oheimb}
 *}
 
-theory Continuity = Relation_Power:
-
+theory Continuity = Main:
 
 subsection "Chains"
 
 constdefs
-  up_chain      :: "(nat => 'a set) => bool"
- "up_chain F      == !i. F i <= F (Suc i)"
+  up_chain :: "(nat => 'a set) => bool"
+  "up_chain F == \<forall>i. F i \<subseteq> F (Suc i)"
 
-lemma up_chainI: "(!!i. F i <= F (Suc i)) ==> up_chain F"
-by (simp add: up_chain_def);
+lemma up_chainI: "(!!i. F i \<subseteq> 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_chainD: "up_chain F ==> F i \<subseteq> 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_less_mono [rule_format]:
+    "up_chain F ==> x < y --> F x \<subseteq> 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
+lemma up_chain_mono: "up_chain F ==> x \<le> y ==> F x \<subseteq> 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"
+  down_chain :: "(nat => 'a set) => bool"
+  "down_chain F == \<forall>i. F (Suc i) \<subseteq> F i"
 
-lemma down_chainI: "(!!i. F (Suc i) <= F i) ==> down_chain F"
-by (simp add: down_chain_def);
+lemma down_chainI: "(!!i. F (Suc i) \<subseteq> 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_chainD: "down_chain F ==> F (Suc i) \<subseteq> 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_less_mono [rule_format]:
+    "down_chain F ==> x < y --> F y \<subseteq> 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
+lemma down_chain_mono: "down_chain F ==> x \<le> y ==> F y \<subseteq> 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))"
+  "up_cont f == \<forall>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)
-by blast
+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)
-by auto
+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 = "%i. if i = 0 then A else B" in up_contD)
-apply  (rule up_chainI)
-apply  simp+
-apply (drule Un_absorb1)
-apply auto
-done
+  apply (rule monoI)
+  apply (drule_tac F = "\<lambda>i. if i = 0 then A else B" in up_contD)
+   apply (rule up_chainI)
+   apply  simp+
+  apply (drule Un_absorb1)
+  apply auto
+  done
 
 
 constdefs
   down_cont :: "('a set => 'a set) => bool"
- "down_cont f == !F. down_chain F --> f (Inter (range F)) = Inter (f`(range F))"
+  "down_cont f ==
+    \<forall>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)
-by blast
+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)
-by auto
+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 = "%i. if i = 0 then B else A" in down_contD)
-apply  (rule down_chainI)
-apply  simp+
-apply (drule Int_absorb1)
-apply auto
-done
+  apply (rule monoI)
+  apply (drule_tac F = "\<lambda>i. if i = 0 then B else A" in down_contD)
+   apply (rule down_chainI)
+   apply simp+
+  apply (drule Int_absorb1)
+  apply auto
+  done
 
 
 subsection "Iteration"
 
 constdefs
-
   up_iterate :: "('a set => 'a set) => nat => 'a set"
- "up_iterate f n == (f^n) {}"
+  "up_iterate f n == (f^n) {}"
 
 lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
-by (simp add: up_iterate_def)
+  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_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
+  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_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_lowerbound:
+    "mono F ==> F P = P ==> UNION UNIV (up_iterate F) \<subseteq> P"
+  apply (subgoal_tac "(!!i. up_iterate F i \<subseteq> 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
+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"
+  "down_iterate f n == (f^n) UNIV"
 
 lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
-by (simp add: down_iterate_def)
+  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_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
+  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_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_upperbound:
+    "mono F ==> F P = P ==> P \<subseteq> INTER UNIV (down_iterate F)"
+  apply (subgoal_tac "(!!i. P \<subseteq> 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
+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