src/HOL/Library/Order_Continuity.thy
changeset 60614 e39e6881985c
parent 60500 903bb1495239
child 60636 ee18efe9b246
     1.1 --- a/src/HOL/Library/Order_Continuity.thy	Tue Jun 30 13:29:30 2015 +0200
     1.2 +++ b/src/HOL/Library/Order_Continuity.thy	Tue Jun 30 13:30:04 2015 +0200
     1.3 @@ -37,14 +37,13 @@
     1.4  subsection \<open>Continuity for complete lattices\<close>
     1.5  
     1.6  definition
     1.7 -  sup_continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
     1.8 +  sup_continuous :: "('a::complete_lattice \<Rightarrow> 'b::complete_lattice) \<Rightarrow> bool" where
     1.9    "sup_continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. mono M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
    1.10  
    1.11  lemma sup_continuousD: "sup_continuous F \<Longrightarrow> mono M \<Longrightarrow> F (SUP i::nat. M i) = (SUP i. F (M i))"
    1.12    by (auto simp: sup_continuous_def)
    1.13  
    1.14  lemma sup_continuous_mono:
    1.15 -  fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
    1.16    assumes [simp]: "sup_continuous F" shows "mono F"
    1.17  proof
    1.18    fix A B :: "'a" assume [simp]: "A \<le> B"
    1.19 @@ -56,6 +55,25 @@
    1.20      by (simp add: SUP_nat_binary le_iff_sup)
    1.21  qed
    1.22  
    1.23 +lemma sup_continuous_intros:
    1.24 +  shows sup_continuous_const: "sup_continuous (\<lambda>x. c)"
    1.25 +    and sup_continuous_id: "sup_continuous (\<lambda>x. x)"
    1.26 +    and sup_continuous_apply: "sup_continuous (\<lambda>f. f x)"
    1.27 +    and sup_continuous_fun: "(\<And>s. sup_continuous (\<lambda>x. P x s)) \<Longrightarrow> sup_continuous P"
    1.28 + by (auto simp: sup_continuous_def)
    1.29 +
    1.30 +lemma sup_continuous_compose:
    1.31 +  assumes f: "sup_continuous f" and g: "sup_continuous g"
    1.32 +  shows "sup_continuous (\<lambda>x. f (g x))"
    1.33 +  unfolding sup_continuous_def
    1.34 +proof safe
    1.35 +  fix M :: "nat \<Rightarrow> 'c" assume "mono M"
    1.36 +  moreover then have "mono (\<lambda>i. g (M i))"
    1.37 +    using sup_continuous_mono[OF g] by (auto simp: mono_def)
    1.38 +  ultimately show "f (g (SUPREMUM UNIV M)) = (SUP i. f (g (M i)))"
    1.39 +    by (auto simp: sup_continuous_def g[THEN sup_continuousD] f[THEN sup_continuousD])
    1.40 +qed
    1.41 +
    1.42  lemma sup_continuous_lfp:
    1.43    assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)" (is "lfp F = ?U")
    1.44  proof (rule antisym)
    1.45 @@ -105,14 +123,13 @@
    1.46  qed
    1.47  
    1.48  definition
    1.49 -  inf_continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
    1.50 +  inf_continuous :: "('a::complete_lattice \<Rightarrow> 'b::complete_lattice) \<Rightarrow> bool" where
    1.51    "inf_continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. antimono M \<longrightarrow> F (INF i. M i) = (INF i. F (M i)))"
    1.52  
    1.53  lemma inf_continuousD: "inf_continuous F \<Longrightarrow> antimono M \<Longrightarrow> F (INF i::nat. M i) = (INF i. F (M i))"
    1.54    by (auto simp: inf_continuous_def)
    1.55  
    1.56  lemma inf_continuous_mono:
    1.57 -  fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
    1.58    assumes [simp]: "inf_continuous F" shows "mono F"
    1.59  proof
    1.60    fix A B :: "'a" assume [simp]: "A \<le> B"
    1.61 @@ -124,6 +141,25 @@
    1.62      by (simp add: INF_nat_binary le_iff_inf inf_commute)
    1.63  qed
    1.64  
    1.65 +lemma inf_continuous_intros:
    1.66 +  shows inf_continuous_const: "inf_continuous (\<lambda>x. c)"
    1.67 +    and inf_continuous_id: "inf_continuous (\<lambda>x. x)"
    1.68 +    and inf_continuous_apply: "inf_continuous (\<lambda>f. f x)"
    1.69 +    and inf_continuous_fun: "(\<And>s. inf_continuous (\<lambda>x. P x s)) \<Longrightarrow> inf_continuous P"
    1.70 + by (auto simp: inf_continuous_def)
    1.71 +
    1.72 +lemma inf_continuous_compose:
    1.73 +  assumes f: "inf_continuous f" and g: "inf_continuous g"
    1.74 +  shows "inf_continuous (\<lambda>x. f (g x))"
    1.75 +  unfolding inf_continuous_def
    1.76 +proof safe
    1.77 +  fix M :: "nat \<Rightarrow> 'c" assume "antimono M"
    1.78 +  moreover then have "antimono (\<lambda>i. g (M i))"
    1.79 +    using inf_continuous_mono[OF g] by (auto simp: mono_def antimono_def)
    1.80 +  ultimately show "f (g (INFIMUM UNIV M)) = (INF i. f (g (M i)))"
    1.81 +    by (auto simp: inf_continuous_def g[THEN inf_continuousD] f[THEN inf_continuousD])
    1.82 +qed
    1.83 +
    1.84  lemma inf_continuous_gfp:
    1.85    assumes "inf_continuous F" shows "gfp F = (INF i. (F ^^ i) top)" (is "gfp F = ?U")
    1.86  proof (rule antisym)