--- a/src/HOL/Lattices.thy Fri Oct 19 16:20:27 2007 +0200
+++ b/src/HOL/Lattices.thy Fri Oct 19 19:45:29 2007 +0200
@@ -30,9 +30,6 @@
context lower_semilattice
begin
-lemmas antisym_intro [intro!] = antisym
-lemmas (in -) [rule del] = antisym_intro
-
lemma le_infI1[intro]:
assumes "a \<sqsubseteq> x"
shows "a \<sqinter> b \<sqsubseteq> x"
@@ -58,11 +55,11 @@
lemmas (in -) [rule del] = le_infE
lemma le_inf_iff [simp]:
- "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
+ "x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)"
by blast
lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
-by(blast dest:eq_iff[THEN iffD1])
+ by (blast intro: antisym dest: eq_iff [THEN iffD1])
end
@@ -73,9 +70,6 @@
context upper_semilattice
begin
-lemmas antisym_intro [intro!] = antisym
-lemmas (in -) [rule del] = antisym_intro
-
lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
by (rule order_trans) auto
lemmas (in -) [rule del] = le_supI1
@@ -92,13 +86,12 @@
by (blast intro: order_trans)
lemmas (in -) [rule del] = le_supE
-
lemma ge_sup_conv[simp]:
- "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
+ "x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)"
by blast
lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
-by(blast dest:eq_iff[THEN iffD1])
+ by (blast intro: antisym dest: eq_iff [THEN iffD1])
end
@@ -113,25 +106,25 @@
begin
lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
-by blast
+ by (blast intro: antisym)
lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
-by blast
+ by (blast intro: antisym)
lemma inf_idem[simp]: "x \<sqinter> x = x"
-by blast
+ by (blast intro: antisym)
lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
-by blast
+ by (blast intro: antisym)
lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
-by blast
+ by (blast intro: antisym)
lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
-by blast
+ by (blast intro: antisym)
lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
-by blast
+ by (blast intro: antisym)
lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem
@@ -142,25 +135,25 @@
begin
lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
-by blast
+ by (blast intro: antisym)
lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
-by blast
+ by (blast intro: antisym)
lemma sup_idem[simp]: "x \<squnion> x = x"
-by blast
+ by (blast intro: antisym)
lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
-by blast
+ by (blast intro: antisym)
lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
-by blast
+ by (blast intro: antisym)
lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
-by blast
+ by (blast intro: antisym)
lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
-by blast
+ by (blast intro: antisym)
lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem
@@ -170,10 +163,10 @@
begin
lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
-by(blast intro: antisym inf_le1 inf_greatest sup_ge1)
+ by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
-by(blast intro: antisym sup_ge1 sup_least inf_le1)
+ by (blast intro: antisym sup_ge1 sup_least inf_le1)
lemmas ACI = inf_ACI sup_ACI
@@ -182,10 +175,10 @@
text{* Towards distributivity *}
lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)"
-by blast
+ by blast
lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)"
-by blast
+ by blast
text{* If you have one of them, you have them all. *}
@@ -293,10 +286,10 @@
by (rule distrib_lattice_min_max)
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (rule ext)+ auto
+ by (rule ext)+ (auto intro: antisym)
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)"
- by (rule ext)+ auto
+ by (rule ext)+ (auto intro: antisym)
lemmas le_maxI1 = min_max.sup_ge1
lemmas le_maxI2 = min_max.sup_ge2
@@ -313,7 +306,7 @@
undesirable.
*}
-lemmas [rule del] = min_max.antisym_intro min_max.le_infI min_max.le_supI
+lemmas [rule del] = min_max.le_infI min_max.le_supI
min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
min_max.le_infI1 min_max.le_infI2
@@ -330,10 +323,10 @@
begin
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
- by (auto intro: Inf_lower Inf_greatest Sup_upper Sup_least)
+ by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
- by (auto intro: Inf_lower Inf_greatest Sup_upper Sup_least)
+ by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
unfolding Sup_Inf by auto
@@ -453,6 +446,9 @@
end
*}
+context complete_lattice
+begin
+
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
by (auto simp add: SUPR_def intro: Sup_upper)
@@ -466,10 +462,12 @@
by (auto simp add: INFI_def intro: Inf_greatest)
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
- by (auto intro: order_antisym SUP_leI le_SUPI)
+ by (auto intro: antisym SUP_leI le_SUPI)
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
- by (auto intro: order_antisym INF_leI le_INFI)
+ by (auto intro: antisym INF_leI le_INFI)
+
+end
subsection {* Bool as lattice *}