# HG changeset patch
# User wenzelm
# Date 1545221222 -3600
# Node ID 3b89c6b723a29ca99ba8313ad0a40feb176c0352
# Parent  4880575ec8a17a950b17e85b75a3cbe9021e0968# Parent  2cc9212342a728cc77ac6a8e67b5e5b8de9f23a9
merged

diff -r 2cc9212342a7 -r 3b89c6b723a2 src/HOL/Hilbert_Choice.thy
--- a/src/HOL/Hilbert_Choice.thy	Wed Dec 19 12:26:38 2018 +0100
+++ b/src/HOL/Hilbert_Choice.thy	Wed Dec 19 13:07:02 2018 +0100
@@ -909,13 +909,14 @@
 
 context complete_distrib_lattice
 begin
-lemma Sup_Inf: "Sup (Inf ` A) = Inf (Sup ` {f ` A | f . (\<forall> Y \<in> A . f Y \<in> Y)})"
+
+lemma Sup_Inf: "\<Squnion> (Inf ` A) = \<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B})"
 proof (rule antisym)
-  show "\<Squnion>(Inf ` A) \<le> \<Sqinter>(Sup ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
+  show "\<Squnion> (Inf ` A) \<le> \<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B})"
     apply (rule Sup_least, rule INF_greatest)
     using Inf_lower2 Sup_upper by auto
 next
-  show "\<Sqinter>(Sup ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}) \<le> \<Squnion>(Inf ` A)"
+  show "\<Sqinter> (Sup ` {f ` A |f. \<forall>B\<in>A. f B \<in> B}) \<le> \<Squnion> (Inf ` A)"
   proof (simp add:  Inf_Sup, rule SUP_least, simp, safe)
     fix f
     assume "\<forall>Y. (\<exists>f. Y = f ` A \<and> (\<forall>Y\<in>A. f Y \<in> Y)) \<longrightarrow> f Y \<in> Y"
@@ -976,7 +977,7 @@
   using dual_complete_distrib_lattice
   by (rule complete_distrib_lattice.sup_Inf)
 
-lemma INF_SUP: "(INF y. SUP x. ((P x y)::'a)) = (SUP x. INF y. P (x y) y)"
+lemma INF_SUP: "(\<Sqinter>y. \<Squnion>x. P x y) = (\<Squnion>f. \<Sqinter>x. P (f x) x)"
 proof (rule antisym)
   show "(SUP x. INF y. P (x y) y) \<le> (INF y. SUP x. P x y)"
     by (rule SUP_least, rule INF_greatest, rule SUP_upper2, simp_all, rule INF_lower2, simp, blast)
@@ -1018,30 +1019,33 @@
     by simp
 qed
 
-lemma INF_SUP_set: "(\<Sqinter>x\<in>A. \<Squnion>(g ` x)) = (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>(g ` x))"
+lemma INF_SUP_set: "(\<Sqinter>B\<in>A. \<Squnion>(g ` B)) = (\<Squnion>B\<in>{f ` A |f. \<forall>C\<in>A. f C \<in> C}. \<Sqinter>(g ` B))"
 proof (rule antisym)
-  have [simp]: "\<And>f xa. \<forall>Y\<in>A. f Y \<in> Y \<Longrightarrow> xa \<in> A \<Longrightarrow> (\<Sqinter>x\<in>A. g (f x)) \<le> g (f xa)"
-    by (rule INF_lower2, blast+)
-  have B: "\<And>f xa. \<forall>Y\<in>A. f Y \<in> Y \<Longrightarrow> xa \<in> A \<Longrightarrow> f xa \<in> xa"
-    by blast
-  have A: "\<And>f xa. \<forall>Y\<in>A. f Y \<in> Y \<Longrightarrow> xa \<in> A \<Longrightarrow> (\<Sqinter>x\<in>A. g (f x)) \<le> \<Squnion>(g ` xa)"
-    by (rule SUP_upper2, rule B, simp_all, simp)
-  show "(\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a) \<le> (\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a)"
-    apply (rule SUP_least, simp, safe, rule INF_greatest, simp)
-    by (rule A)
+  have "\<Sqinter> ((g \<circ> f) ` A) \<le> \<Squnion> (g ` B)" if "\<And>B. B \<in> A \<Longrightarrow> f B \<in> B" and "B \<in> A"
+    for f and B
+    using that by (auto intro: SUP_upper2 INF_lower2)
+  then show "(\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a) \<le> (\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a)"
+    by (auto intro!: SUP_least INF_greatest)
 next
   show "(\<Sqinter>x\<in>A. \<Squnion>a\<in>x. g a) \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a)"
   proof (cases "{} \<in> A")
     case True
     then show ?thesis 
-      by (rule INF_lower2, simp_all)
+      by (rule INF_lower2) simp_all
   next
     case False
-    have [simp]: "\<And>x xa xb. xb \<in> A \<Longrightarrow> x xb \<in> xb \<Longrightarrow> (\<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>) \<le> g (x xb)"
+    have *: "\<And>f B. B \<in> A \<Longrightarrow> f B \<in> B \<Longrightarrow>
+      (\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>) \<le> g (f B)"
+      by (rule INF_lower2, auto)
+    have **: "\<And>f B. B \<in> A \<Longrightarrow> f B \<notin> B \<Longrightarrow>
+      (\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>) \<le> g (SOME x. x \<in> B)"
       by (rule INF_lower2, auto)
-    have [simp]: " \<And>x xa y. y \<in> A \<Longrightarrow> x y \<notin> y \<Longrightarrow> (\<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>) \<le> g (SOME x. x \<in> y)"
-      by (rule INF_lower2, auto)
-    have [simp]: "\<And>x. (\<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>) \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x\<in>x. g x)"
+    have ****: "\<And>f B. B \<in> A \<Longrightarrow>
+      (\<Sqinter>B. if B \<in> A then if f B \<in> B then g (f B) else \<bottom> else \<top>)
+        \<le> (if f B \<in> B then g (f B) else g (SOME x. x \<in> B))"
+      by (rule INF_lower2) auto
+    have ***: "\<And>x. (\<Sqinter>B. if B \<in> A then if x B \<in> B then g (x B) else \<bottom> else \<top>)
+        \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x\<in>x. g x)"
     proof -
       fix x
       define F where "F = (\<lambda> (y::'b set) . if x y \<in> y then x y else (SOME x . x \<in>y))"
@@ -1051,7 +1055,10 @@
         using B by blast
       show "(\<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>) \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>x\<in>x. g x)"
         using A apply (rule SUP_upper2)
-        by (simp add: F_def, rule INF_greatest, auto)
+        apply (simp add: F_def)
+        apply (rule INF_greatest)
+        apply (auto simp add: * **)
+        done
     qed
 
     {fix x
@@ -1071,16 +1078,16 @@
     also have "... = (\<Squnion>x. \<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>)"
       by (simp add: INF_SUP)
     also have "... \<le> (\<Squnion>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Sqinter>a\<in>x. g a)"
-      by (rule SUP_least, simp)
+      by (rule SUP_least, simp add: ***)
     finally show ?thesis by simp
   qed
 qed
 
-lemma SUP_INF: "(SUP y. INF x. ((P x y)::'a)) = (INF x. SUP y. P (x y) y)"
+lemma SUP_INF: "(\<Squnion>y. \<Sqinter>x. P x y) = (\<Sqinter>x. \<Squnion>y. P (x y) y)"
   using dual_complete_distrib_lattice
   by (rule complete_distrib_lattice.INF_SUP)
 
-lemma SUP_INF_set: "(\<Squnion>x\<in>A. \<Sqinter>(g ` x)) = (\<Sqinter>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Squnion>(g ` x))"
+lemma SUP_INF_set: "(\<Squnion>x\<in>A. \<Sqinter> (g ` x)) = (\<Sqinter>x\<in>{f ` A |f. \<forall>Y\<in>A. f Y \<in> Y}. \<Squnion> (g ` x))"
   using dual_complete_distrib_lattice
   by (rule complete_distrib_lattice.INF_SUP_set)
 
@@ -1096,7 +1103,6 @@
 lemma inf_SUP: "a \<sqinter> (\<Squnion>b\<in>B. f b) = (\<Squnion>b\<in>B. a \<sqinter> f b)"
   by (simp add: inf_Sup)
 
-
 lemma Inf_sup: "\<Sqinter>B \<squnion> a = (\<Sqinter>b\<in>B. b \<squnion> a)"
   by (simp add: sup_Inf sup_commute)