src/HOL/Library/Continuity.thy
changeset 11351 c5c403d30c77
child 11355 778c369559d9
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Continuity.thy	Thu May 31 17:06:00 2001 +0200
@@ -0,0 +1,219 @@
+(*  Title:      HOL/Library/Continuity.thy
+    ID:         $$
+    Author: 	David von Oheimb, TU Muenchen
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+*)
+
+header {*
+  \title{Continuity and interations (of set transformers)}
+  \author{David von Oheimb}
+*}
+
+theory Continuity = Relation_Power:
+
+
+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)
+by blast
+
+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_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
+
+
+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)
+by blast
+
+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_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
+
+
+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