# HG changeset patch
# User haftmann
# Date 1311278184 -7200
# Node ID 3406cd754dd28a783d3a3c867312ae417e4a3ef5
# Parent  78a0a2ad91a387d9f1ab5d09f1a5f42e722c0c46# Parent  481566bc20e43078519660de5212f8843dd8482b
merged

diff -r 78a0a2ad91a3 -r 3406cd754dd2 NEWS
--- a/NEWS	Thu Jul 21 08:33:57 2011 +0200
+++ b/NEWS	Thu Jul 21 21:56:24 2011 +0200
@@ -63,15 +63,16 @@
 * Classes bot and top require underlying partial order rather than preorder:
 uniqueness of bot and top is guaranteed.  INCOMPATIBILITY.
 
-* Class 'complete_lattice': generalized a couple of lemmas from sets;
-generalized theorems INF_cong and SUP_cong.  More consistent and less
-misunderstandable names:
+* Class complete_lattice: generalized a couple of lemmas from sets;
+generalized theorems INF_cong and SUP_cong.  New type classes for complete
+boolean algebras and complete linear orderes.  Lemmas Inf_less_iff,
+less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder.
+More consistent and less misunderstandable names:
   INFI_def ~> INF_def
   SUPR_def ~> SUP_def
   le_SUPI ~> le_SUP_I
   le_SUPI2 ~> le_SUP_I2
   le_INFI ~> le_INF_I
-  INF_subset ~> INF_superset_mono
   INFI_bool_eq ~> INF_bool_eq
   SUPR_bool_eq ~> SUP_bool_eq
   INFI_apply ~> INF_apply
diff -r 78a0a2ad91a3 -r 3406cd754dd2 src/HOL/Complete_Lattice.thy
--- a/src/HOL/Complete_Lattice.thy	Thu Jul 21 08:33:57 2011 +0200
+++ b/src/HOL/Complete_Lattice.thy	Thu Jul 21 21:56:24 2011 +0200
@@ -292,12 +292,13 @@
   by (force intro!: Sup_mono simp: SUP_def)
 
 lemma INF_superset_mono:
-  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
-  by (rule INF_mono) auto
+  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
+  -- {* The last inclusion is POSITIVE! *}
+  by (blast intro: INF_mono dest: subsetD)
 
 lemma SUP_subset_mono:
-  "A \<subseteq> B \<Longrightarrow> SUPR A f \<sqsubseteq> SUPR B f"
-  by (rule SUP_mono) auto
+  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
+  by (blast intro: SUP_mono dest: subsetD)
 
 lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
   by (iprover intro: INF_leI le_INF_I order_trans antisym)
@@ -371,38 +372,8 @@
   "(\<Squnion>b. A b) = A True \<squnion> A False"
   by (simp add: UNIV_bool SUP_empty SUP_insert sup_commute)
 
-lemma INF_mono':
-  "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
-  -- {* The last inclusion is POSITIVE! *}
-  by (rule INF_mono) auto
-
-lemma SUP_mono':
-  "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Squnion>x\<in>A. f x) \<sqsubseteq> (\<Squnion>x\<in>B. g x)"
-  -- {* The last inclusion is POSITIVE! *}
-  by (blast intro: SUP_mono dest: subsetD)
-
 end
 
-lemma Inf_less_iff:
-  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
-  shows "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
-  unfolding not_le [symmetric] le_Inf_iff by auto
-
-lemma less_Sup_iff:
-  fixes a :: "'a\<Colon>{complete_lattice,linorder}"
-  shows "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
-  unfolding not_le [symmetric] Sup_le_iff by auto
-
-lemma INF_less_iff:
-  fixes a :: "'a::{complete_lattice,linorder}"
-  shows "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
-  unfolding INF_def Inf_less_iff by auto
-
-lemma less_SUP_iff:
-  fixes a :: "'a::{complete_lattice,linorder}"
-  shows "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
-  unfolding SUP_def less_Sup_iff by auto
-
 class complete_boolean_algebra = boolean_algebra + complete_lattice
 begin
 
@@ -430,6 +401,27 @@
 
 end
 
+class complete_linorder = linorder + complete_lattice
+begin
+
+lemma Inf_less_iff:
+  "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
+  unfolding not_le [symmetric] le_Inf_iff by auto
+
+lemma less_Sup_iff:
+  "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
+  unfolding not_le [symmetric] Sup_le_iff by auto
+
+lemma INF_less_iff:
+  "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
+  unfolding INF_def Inf_less_iff by auto
+
+lemma less_SUP_iff:
+  "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
+  unfolding SUP_def less_Sup_iff by auto
+
+end
+
 
 subsection {* @{typ bool} and @{typ "_ \<Rightarrow> _"} as complete lattice *}
 
@@ -688,7 +680,7 @@
 lemma INT_anti_mono:
   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow> (\<Inter>x\<in>B. f x) \<subseteq> (\<Inter>x\<in>A. g x)"
   -- {* The last inclusion is POSITIVE! *}
-  by (fact INF_mono')
+  by (fact INF_superset_mono)
 
 lemma Pow_INT_eq: "Pow (\<Inter>x\<in>A. B x) = (\<Inter>x\<in>A. Pow (B x))"
   by blast
@@ -922,7 +914,7 @@
 lemma UN_mono:
   "A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x) \<Longrightarrow>
     (\<Union>x\<in>A. f x) \<subseteq> (\<Union>x\<in>B. g x)"
-  by (fact SUP_mono')
+  by (fact SUP_subset_mono)
 
 lemma vimage_Union: "f -` (\<Union>A) = (\<Union>X\<in>A. f -` X)"
   by blast
@@ -1083,7 +1075,11 @@
 lemmas (in complete_lattice) le_SUPI = le_SUP_I
 lemmas (in complete_lattice) le_SUPI2 = le_SUP_I2
 lemmas (in complete_lattice) le_INFI = le_INF_I
-lemmas (in complete_lattice) INF_subset = INF_superset_mono 
+
+lemma (in complete_lattice) INF_subset:
+  "B \<subseteq> A \<Longrightarrow> INFI A f \<sqsubseteq> INFI B f"
+  by (rule INF_superset_mono) auto
+
 lemmas INFI_apply = INF_apply
 lemmas SUPR_apply = SUP_apply
 
diff -r 78a0a2ad91a3 -r 3406cd754dd2 src/HOL/Library/Extended_Real.thy
--- a/src/HOL/Library/Extended_Real.thy	Thu Jul 21 08:33:57 2011 +0200
+++ b/src/HOL/Library/Extended_Real.thy	Thu Jul 21 21:56:24 2011 +0200
@@ -8,7 +8,7 @@
 header {* Extended real number line *}
 
 theory Extended_Real
-  imports Complex_Main Extended_Nat
+imports Complex_Main Extended_Nat
 begin
 
 text {*
@@ -1244,8 +1244,11 @@
     with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
       unfolding ereal_Sup_uminus_image_eq by force }
 qed
+
 end
 
+instance ereal :: complete_linorder ..
+
 lemma ereal_SUPR_uminus:
   fixes f :: "'a => ereal"
   shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
@@ -1335,12 +1338,11 @@
     by (cases e) auto
 qed
 
-lemma Sup_eq_top_iff:
-  fixes A :: "'a::{complete_lattice, linorder} set"
-  shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
+lemma (in complete_linorder) Sup_eq_top_iff: -- "FIXME move"
+  "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
 proof
   assume *: "Sup A = top"
-  show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
+  show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding * [symmetric]
   proof (intro allI impI)
     fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
       unfolding less_Sup_iff by auto
@@ -1350,7 +1352,7 @@
   show "Sup A = top"
   proof (rule ccontr)
     assume "Sup A \<noteq> top"
-    with top_greatest[of "Sup A"]
+    with top_greatest [of "Sup A"]
     have "Sup A < top" unfolding le_less by auto
     then have "Sup A < Sup A"
       using * unfolding less_Sup_iff by auto
@@ -1358,8 +1360,8 @@
   qed
 qed
 
-lemma SUP_eq_top_iff:
-  fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
+lemma (in complete_linorder) SUP_eq_top_iff: -- "FIXME move"
+  fixes f :: "'b \<Rightarrow> 'a"
   shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
   unfolding SUPR_def Sup_eq_top_iff by auto
 
@@ -2182,12 +2184,12 @@
   "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
 
 lemma Liminf_Sup:
-  fixes f :: "'a => 'b::{complete_lattice, linorder}"
+  fixes f :: "'a => 'b::complete_linorder"
   shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
   by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
 
 lemma Limsup_Inf:
-  fixes f :: "'a => 'b::{complete_lattice, linorder}"
+  fixes f :: "'a => 'b::complete_linorder"
   shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
   by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
 
@@ -2208,7 +2210,7 @@
   using assms by (auto intro!: Greatest_equality)
 
 lemma Limsup_const:
-  fixes c :: "'a::{complete_lattice, linorder}"
+  fixes c :: "'a::complete_linorder"
   assumes ntriv: "\<not> trivial_limit net"
   shows "Limsup net (\<lambda>x. c) = c"
   unfolding Limsup_Inf
@@ -2222,7 +2224,7 @@
 qed auto
 
 lemma Liminf_const:
-  fixes c :: "'a::{complete_lattice, linorder}"
+  fixes c :: "'a::complete_linorder"
   assumes ntriv: "\<not> trivial_limit net"
   shows "Liminf net (\<lambda>x. c) = c"
   unfolding Liminf_Sup
@@ -2235,18 +2237,17 @@
   qed
 qed auto
 
-lemma mono_set:
-  fixes S :: "('a::order) set"
-  shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
+lemma (in order) mono_set:
+  "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
   by (auto simp: mono_def mem_def)
 
-lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
-lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
-lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
-lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
+lemma (in order) mono_greaterThan [intro, simp]: "mono {B<..}" unfolding mono_set by auto
+lemma (in order) mono_atLeast [intro, simp]: "mono {B..}" unfolding mono_set by auto
+lemma (in order) mono_UNIV [intro, simp]: "mono UNIV" unfolding mono_set by auto
+lemma (in order) mono_empty [intro, simp]: "mono {}" unfolding mono_set by auto
 
-lemma mono_set_iff:
-  fixes S :: "'a::{linorder,complete_lattice} set"
+lemma (in complete_linorder) mono_set_iff:
+  fixes S :: "'a set"
   defines "a \<equiv> Inf S"
   shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
 proof
@@ -2257,13 +2258,13 @@
     assume "a \<in> S"
     show ?c
       using mono[OF _ `a \<in> S`]
-      by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
+      by (auto intro: Inf_lower simp: a_def)
   next
     assume "a \<notin> S"
     have "S = {a <..}"
     proof safe
       fix x assume "x \<in> S"
-      then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
+      then have "a \<le> x" unfolding a_def by (rule Inf_lower)
       then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
     next
       fix x assume "a < x"
diff -r 78a0a2ad91a3 -r 3406cd754dd2 src/HOL/Probability/Lebesgue_Integration.thy
--- a/src/HOL/Probability/Lebesgue_Integration.thy	Thu Jul 21 08:33:57 2011 +0200
+++ b/src/HOL/Probability/Lebesgue_Integration.thy	Thu Jul 21 21:56:24 2011 +0200
@@ -6,7 +6,7 @@
 header {*Lebesgue Integration*}
 
 theory Lebesgue_Integration
-imports Measure Borel_Space
+  imports Measure Borel_Space
 begin
 
 lemma real_ereal_1[simp]: "real (1::ereal) = 1"