src/HOL/Library/Continuity.thy
 changeset 21404 eb85850d3eb7 parent 21312 1d39091a3208 child 22367 6860f09242bf
```     1.1 --- a/src/HOL/Library/Continuity.thy	Fri Nov 17 02:19:55 2006 +0100
1.2 +++ b/src/HOL/Library/Continuity.thy	Fri Nov 17 02:20:03 2006 +0100
1.3 @@ -18,8 +18,8 @@
1.4  "continuous F == !M. chain M \<longrightarrow> F(SUP i. M i) = (SUP i. F(M i))"
1.5
1.6  abbreviation
1.7 - bot :: "'a::order"
1.8 -"bot == Sup {}"
1.9 +  bot :: "'a::order" where
1.10 +  "bot == Sup {}"
1.11
1.12  lemma SUP_nat_conv:
1.13   "(SUP n::nat. M n::'a::comp_lat) = join (M 0) (SUP n. M(Suc n))"
1.14 @@ -91,7 +91,7 @@
1.15  subsection "Chains"
1.16
1.17  definition
1.18 -  up_chain :: "(nat => 'a set) => bool"
1.19 +  up_chain :: "(nat => 'a set) => bool" where
1.20    "up_chain F = (\<forall>i. F i \<subseteq> F (Suc i))"
1.21
1.22  lemma up_chainI: "(!!i. F i \<subseteq> F (Suc i)) ==> up_chain F"
1.23 @@ -113,7 +113,7 @@
1.24
1.25
1.26  definition
1.27 -  down_chain :: "(nat => 'a set) => bool"
1.28 +  down_chain :: "(nat => 'a set) => bool" where
1.29    "down_chain F = (\<forall>i. F (Suc i) \<subseteq> F i)"
1.30
1.31  lemma down_chainI: "(!!i. F (Suc i) \<subseteq> F i) ==> down_chain F"
1.32 @@ -137,7 +137,7 @@
1.33  subsection "Continuity"
1.34
1.35  definition
1.36 -  up_cont :: "('a set => 'a set) => bool"
1.37 +  up_cont :: "('a set => 'a set) => bool" where
1.38    "up_cont f = (\<forall>F. up_chain F --> f (\<Union>(range F)) = \<Union>(f ` range F))"
1.39
1.40  lemma up_contI:
1.41 @@ -164,7 +164,7 @@
1.42
1.43
1.44  definition
1.45 -  down_cont :: "('a set => 'a set) => bool"
1.46 +  down_cont :: "('a set => 'a set) => bool" where
1.47    "down_cont f =
1.48      (\<forall>F. down_chain F --> f (Inter (range F)) = Inter (f ` range F))"
1.49
1.50 @@ -194,7 +194,7 @@
1.51  subsection "Iteration"
1.52
1.53  definition
1.54 -  up_iterate :: "('a set => 'a set) => nat => 'a set"
1.55 +  up_iterate :: "('a set => 'a set) => nat => 'a set" where
1.56    "up_iterate f n = (f^n) {}"
1.57
1.58  lemma up_iterate_0 [simp]: "up_iterate f 0 = {}"
1.59 @@ -246,7 +246,7 @@
1.60
1.61
1.62  definition
1.63 -  down_iterate :: "('a set => 'a set) => nat => 'a set"
1.64 +  down_iterate :: "('a set => 'a set) => nat => 'a set" where
1.65    "down_iterate f n = (f^n) UNIV"
1.66
1.67  lemma down_iterate_0 [simp]: "down_iterate f 0 = UNIV"
```