--- a/NEWS Tue Aug 09 07:44:17 2011 +0200
+++ b/NEWS Tue Aug 09 08:06:15 2011 +0200
@@ -67,19 +67,27 @@
generalized theorems INF_cong and SUP_cong. New type classes for complete
boolean algebras and complete linear orders. Lemmas Inf_less_iff,
less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder.
-Changes proposition of lemmas Inf_fun_def, Sup_fun_def, Inf_apply, Sup_apply.
-Redundant lemmas Inf_singleton, Sup_singleton, Inf_binary and Sup_binary have
-been discarded.
+Changed proposition of lemmas Inf_fun_def, Sup_fun_def, Inf_apply, Sup_apply.
+Redundant lemmas Inf_singleton, Sup_singleton, Inf_binary, Sup_binary,
+INF_eq, SUP_eq, INF_UNIV_range, SUP_UNIV_range, Int_eq_Inter,
+INTER_eq_Inter_image, Inter_def, INT_eq, Un_eq_Union, UNION_eq_Union_image,
+Union_def, UN_singleton, UN_eq have been discarded.
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_leI ~> INF_lower
+ INF_leI2 ~> INF_lower2
+ le_INFI ~> INF_greatest
+ le_SUPI ~> SUP_upper
+ le_SUPI2 ~> SUP_upper2
+ SUP_leI ~> SUP_least
INFI_bool_eq ~> INF_bool_eq
SUPR_bool_eq ~> SUP_bool_eq
INFI_apply ~> INF_apply
SUPR_apply ~> SUP_apply
+ INTER_def ~> INTER_eq
+ UNION_def ~> UNION_eq
+
INCOMPATIBILITY.
* Theorem collections ball_simps and bex_simps do not contain theorems referring
--- a/src/HOL/Complete_Lattice.thy Tue Aug 09 07:44:17 2011 +0200
+++ b/src/HOL/Complete_Lattice.thy Tue Aug 09 08:06:15 2011 +0200
@@ -1,4 +1,4 @@
-(* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
+ (* Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel; Florian Haftmann, TU Muenchen *)
header {* Complete lattices, with special focus on sets *}
@@ -102,29 +102,29 @@
by (simp only: dual.INF_def SUP_def)
qed
-lemma INF_leI: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
+lemma INF_lower: "i \<in> A \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> f i"
by (auto simp add: INF_def intro: Inf_lower)
-lemma le_SUP_I: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
+lemma INF_greatest: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
+ by (auto simp add: INF_def intro: Inf_greatest)
+
+lemma SUP_upper: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
by (auto simp add: SUP_def intro: Sup_upper)
-lemma le_INF_I: "(\<And>i. i \<in> A \<Longrightarrow> u \<sqsubseteq> f i) \<Longrightarrow> u \<sqsubseteq> (\<Sqinter>i\<in>A. f i)"
- by (auto simp add: INF_def intro: Inf_greatest)
-
-lemma SUP_leI: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
+lemma SUP_least: "(\<And>i. i \<in> A \<Longrightarrow> f i \<sqsubseteq> u) \<Longrightarrow> (\<Squnion>i\<in>A. f i) \<sqsubseteq> u"
by (auto simp add: SUP_def intro: Sup_least)
lemma Inf_lower2: "u \<in> A \<Longrightarrow> u \<sqsubseteq> v \<Longrightarrow> \<Sqinter>A \<sqsubseteq> v"
using Inf_lower [of u A] by auto
-lemma INF_leI2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
- using INF_leI [of i A f] by auto
+lemma INF_lower2: "i \<in> A \<Longrightarrow> f i \<sqsubseteq> u \<Longrightarrow> (\<Sqinter>i\<in>A. f i) \<sqsubseteq> u"
+ using INF_lower [of i A f] by auto
lemma Sup_upper2: "u \<in> A \<Longrightarrow> v \<sqsubseteq> u \<Longrightarrow> v \<sqsubseteq> \<Squnion>A"
using Sup_upper [of u A] by auto
-lemma le_SUP_I2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
- using le_SUP_I [of i A f] by auto
+lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
+ using SUP_upper [of i A f] by auto
lemma le_Inf_iff (*[simp]*): "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
by (auto intro: Inf_greatest dest: Inf_lower)
@@ -266,21 +266,21 @@
lemma INF_union:
"(\<Sqinter>i \<in> A \<union> B. M i) = (\<Sqinter>i \<in> A. M i) \<sqinter> (\<Sqinter>i\<in>B. M i)"
- by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 le_INF_I INF_leI)
+ by (auto intro!: antisym INF_mono intro: le_infI1 le_infI2 INF_greatest INF_lower)
lemma Sup_union_distrib: "\<Squnion>(A \<union> B) = \<Squnion>A \<squnion> \<Squnion>B"
by (rule antisym) (auto intro: Sup_least Sup_upper le_supI1 le_supI2)
lemma SUP_union:
"(\<Squnion>i \<in> A \<union> B. M i) = (\<Squnion>i \<in> A. M i) \<squnion> (\<Squnion>i\<in>B. M i)"
- by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_leI le_SUP_I)
+ by (auto intro!: antisym SUP_mono intro: le_supI1 le_supI2 SUP_least SUP_upper)
lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
- by (rule antisym) (rule le_INF_I, auto intro: le_infI1 le_infI2 INF_leI INF_mono)
+ by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)"
- by (rule antisym) (auto intro: le_supI1 le_supI2 le_SUP_I SUP_mono,
- rule SUP_leI, auto intro: le_supI1 le_supI2 le_SUP_I SUP_mono)
+ by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
+ rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
lemma Inf_top_conv (*[simp]*) [no_atp]:
"\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
@@ -324,10 +324,10 @@
by (auto simp add: SUP_def Sup_bot_conv)
lemma INF_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Sqinter>i\<in>A. f) = f"
- by (auto intro: antisym INF_leI le_INF_I)
+ by (auto intro: antisym INF_lower INF_greatest)
lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
- by (auto intro: antisym SUP_leI le_SUP_I)
+ by (auto intro: antisym SUP_upper SUP_least)
lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
by (cases "A = {}") (simp_all add: INF_empty)
@@ -336,10 +336,10 @@
by (cases "A = {}") (simp_all add: SUP_empty)
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)
+ by (iprover intro: INF_lower INF_greatest order_trans antisym)
lemma SUP_commute: "(\<Squnion>i\<in>A. \<Squnion>j\<in>B. f i j) = (\<Squnion>j\<in>B. \<Squnion>i\<in>A. f i j)"
- by (iprover intro: SUP_leI le_SUP_I order_trans antisym)
+ by (iprover intro: SUP_upper SUP_least order_trans antisym)
lemma INF_absorb:
assumes "k \<in> I"
@@ -370,7 +370,7 @@
proof -
note `y < (\<Sqinter>i\<in>A. f i)`
also have "(\<Sqinter>i\<in>A. f i) \<le> f i" using `i \<in> A`
- by (rule INF_leI)
+ by (rule INF_lower)
finally show "y < f i" .
qed
@@ -378,7 +378,7 @@
assumes "(\<Squnion>i\<in>A. f i) < y" "i \<in> A" shows "f i < y"
proof -
have "f i \<le> (\<Squnion>i\<in>A. f i)" using `i \<in> A`
- by (rule le_SUP_I)
+ by (rule SUP_upper)
also note `(\<Squnion>i\<in>A. f i) < y`
finally show "f i < y" .
qed
@@ -605,7 +605,7 @@
by (simp add: Sup_fun_def)
instance proof
-qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_leI le_SUP_I le_INF_I SUP_leI)
+qed (auto simp add: le_fun_def Inf_apply Sup_apply intro: INF_lower INF_greatest SUP_upper SUP_least)
end
@@ -759,10 +759,10 @@
by blast
lemma INT_lower: "a \<in> A \<Longrightarrow> (\<Inter>x\<in>A. B x) \<subseteq> B a"
- by (fact INF_leI)
+ by (fact INF_lower)
lemma INT_greatest: "(\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x) \<Longrightarrow> C \<subseteq> (\<Inter>x\<in>A. B x)"
- by (fact le_INF_I)
+ by (fact INF_greatest)
lemma INT_empty: "(\<Inter>x\<in>{}. B x) = UNIV"
by (fact INF_empty)
@@ -947,10 +947,10 @@
by blast
lemma UN_upper: "a \<in> A \<Longrightarrow> B a \<subseteq> (\<Union>x\<in>A. B x)"
- by (fact le_SUP_I)
+ by (fact SUP_upper)
lemma UN_least: "(\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C) \<Longrightarrow> (\<Union>x\<in>A. B x) \<subseteq> C"
- by (fact SUP_leI)
+ by (fact SUP_least)
lemma Collect_bex_eq [no_atp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
by blast
@@ -1166,9 +1166,12 @@
lemmas (in complete_lattice) INFI_def = INF_def
lemmas (in complete_lattice) SUPR_def = SUP_def
-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_leI = INF_lower
+lemmas (in complete_lattice) INF_leI2 = INF_lower2
+lemmas (in complete_lattice) le_INFI = INF_greatest
+lemmas (in complete_lattice) le_SUPI = SUP_upper
+lemmas (in complete_lattice) le_SUPI2 = SUP_upper2
+lemmas (in complete_lattice) SUP_leI = SUP_least
lemmas (in complete_lattice) less_INFD = less_INF_D
lemmas INFI_apply = INF_apply