author haftmann Wed, 19 Dec 2018 08:16:41 +0000 changeset 69478 c505f251f352 parent 69477 1690ba936016 child 69479 4880575ec8a1
tuned proof
```--- a/src/HOL/Hilbert_Choice.thy	Tue Dec 18 15:50:20 2018 +0100
+++ b/src/HOL/Hilbert_Choice.thy	Wed Dec 19 08:16:41 2018 +0000
@@ -1018,30 +1018,33 @@
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 +1054,10 @@
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 (rule INF_greatest)
+        apply (auto simp add: * **)
+        done
qed

{fix x
@@ -1071,7 +1077,7 @@
also have "... = (\<Squnion>x. \<Sqinter>xa. if xa \<in> A then if x xa \<in> xa then g (x xa) else \<bottom> else \<top>)"