--- a/src/HOL/Set.thy Sun Mar 28 12:49:14 2010 -0700
+++ b/src/HOL/Set.thy Sun Mar 28 12:50:38 2010 -0700
@@ -507,7 +507,6 @@
apply (rule Collect_mem_eq)
done
-(* Due to Brian Huffman *)
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))"
by(auto intro:set_ext)
@@ -1002,25 +1001,25 @@
text {* \medskip Finite Union -- the least upper bound of two sets. *}
lemma Un_upper1: "A \<subseteq> A \<union> B"
- by blast
+ by (fact sup_ge1)
lemma Un_upper2: "B \<subseteq> A \<union> B"
- by blast
+ by (fact sup_ge2)
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C"
- by blast
+ by (fact sup_least)
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *}
lemma Int_lower1: "A \<inter> B \<subseteq> A"
- by blast
+ by (fact inf_le1)
lemma Int_lower2: "A \<inter> B \<subseteq> B"
- by blast
+ by (fact inf_le2)
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B"
- by blast
+ by (fact inf_greatest)
text {* \medskip Set difference. *}
@@ -1166,34 +1165,34 @@
text {* \medskip @{text Int} *}
lemma Int_absorb [simp]: "A \<inter> A = A"
- by blast
+ by (fact inf_idem)
lemma Int_left_absorb: "A \<inter> (A \<inter> B) = A \<inter> B"
- by blast
+ by (fact inf_left_idem)
lemma Int_commute: "A \<inter> B = B \<inter> A"
- by blast
+ by (fact inf_commute)
lemma Int_left_commute: "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
- by blast
+ by (fact inf_left_commute)
lemma Int_assoc: "(A \<inter> B) \<inter> C = A \<inter> (B \<inter> C)"
- by blast
+ by (fact inf_assoc)
lemmas Int_ac = Int_assoc Int_left_absorb Int_commute Int_left_commute
-- {* Intersection is an AC-operator *}
lemma Int_absorb1: "B \<subseteq> A ==> A \<inter> B = B"
- by blast
+ by (fact inf_absorb2)
lemma Int_absorb2: "A \<subseteq> B ==> A \<inter> B = A"
- by blast
+ by (fact inf_absorb1)
lemma Int_empty_left [simp]: "{} \<inter> B = {}"
- by blast
+ by (fact inf_bot_left)
lemma Int_empty_right [simp]: "A \<inter> {} = {}"
- by blast
+ by (fact inf_bot_right)
lemma disjoint_eq_subset_Compl: "(A \<inter> B = {}) = (A \<subseteq> -B)"
by blast
@@ -1202,22 +1201,22 @@
by blast
lemma Int_UNIV_left [simp]: "UNIV \<inter> B = B"
- by blast
+ by (fact inf_top_left)
lemma Int_UNIV_right [simp]: "A \<inter> UNIV = A"
- by blast
+ by (fact inf_top_right)
lemma Int_Un_distrib: "A \<inter> (B \<union> C) = (A \<inter> B) \<union> (A \<inter> C)"
- by blast
+ by (fact inf_sup_distrib1)
lemma Int_Un_distrib2: "(B \<union> C) \<inter> A = (B \<inter> A) \<union> (C \<inter> A)"
- by blast
+ by (fact inf_sup_distrib2)
lemma Int_UNIV [simp,no_atp]: "(A \<inter> B = UNIV) = (A = UNIV & B = UNIV)"
- by blast
+ by (fact inf_eq_top_iff)
lemma Int_subset_iff [simp]: "(C \<subseteq> A \<inter> B) = (C \<subseteq> A & C \<subseteq> B)"
- by blast
+ by (fact le_inf_iff)
lemma Int_Collect: "(x \<in> A \<inter> {x. P x}) = (x \<in> A & P x)"
by blast
@@ -1226,40 +1225,40 @@
text {* \medskip @{text Un}. *}
lemma Un_absorb [simp]: "A \<union> A = A"
- by blast
+ by (fact sup_idem)
lemma Un_left_absorb: "A \<union> (A \<union> B) = A \<union> B"
- by blast
+ by (fact sup_left_idem)
lemma Un_commute: "A \<union> B = B \<union> A"
- by blast
+ by (fact sup_commute)
lemma Un_left_commute: "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
- by blast
+ by (fact sup_left_commute)
lemma Un_assoc: "(A \<union> B) \<union> C = A \<union> (B \<union> C)"
- by blast
+ by (fact sup_assoc)
lemmas Un_ac = Un_assoc Un_left_absorb Un_commute Un_left_commute
-- {* Union is an AC-operator *}
lemma Un_absorb1: "A \<subseteq> B ==> A \<union> B = B"
- by blast
+ by (fact sup_absorb2)
lemma Un_absorb2: "B \<subseteq> A ==> A \<union> B = A"
- by blast
+ by (fact sup_absorb1)
lemma Un_empty_left [simp]: "{} \<union> B = B"
- by blast
+ by (fact sup_bot_left)
lemma Un_empty_right [simp]: "A \<union> {} = A"
- by blast
+ by (fact sup_bot_right)
lemma Un_UNIV_left [simp]: "UNIV \<union> B = UNIV"
- by blast
+ by (fact sup_top_left)
lemma Un_UNIV_right [simp]: "A \<union> UNIV = UNIV"
- by blast
+ by (fact sup_top_right)
lemma Un_insert_left [simp]: "(insert a B) \<union> C = insert a (B \<union> C)"
by blast
@@ -1292,23 +1291,23 @@
by auto
lemma Un_Int_distrib: "A \<union> (B \<inter> C) = (A \<union> B) \<inter> (A \<union> C)"
- by blast
+ by (fact sup_inf_distrib1)
lemma Un_Int_distrib2: "(B \<inter> C) \<union> A = (B \<union> A) \<inter> (C \<union> A)"
- by blast
+ by (fact sup_inf_distrib2)
lemma Un_Int_crazy:
"(A \<inter> B) \<union> (B \<inter> C) \<union> (C \<inter> A) = (A \<union> B) \<inter> (B \<union> C) \<inter> (C \<union> A)"
by blast
lemma subset_Un_eq: "(A \<subseteq> B) = (A \<union> B = B)"
- by blast
+ by (fact le_iff_sup)
lemma Un_empty [iff]: "(A \<union> B = {}) = (A = {} & B = {})"
- by blast
+ by (fact sup_eq_bot_iff)
lemma Un_subset_iff [simp]: "(A \<union> B \<subseteq> C) = (A \<subseteq> C & B \<subseteq> C)"
- by blast
+ by (fact le_sup_iff)
lemma Un_Diff_Int: "(A - B) \<union> (A \<inter> B) = A"
by blast
@@ -1320,25 +1319,25 @@
text {* \medskip Set complement *}
lemma Compl_disjoint [simp]: "A \<inter> -A = {}"
- by blast
+ by (fact inf_compl_bot)
lemma Compl_disjoint2 [simp]: "-A \<inter> A = {}"
- by blast
+ by (fact compl_inf_bot)
lemma Compl_partition: "A \<union> -A = UNIV"
- by blast
+ by (fact sup_compl_top)
lemma Compl_partition2: "-A \<union> A = UNIV"
- by blast
+ by (fact compl_sup_top)
lemma double_complement [simp]: "- (-A) = (A::'a set)"
- by blast
+ by (fact double_compl)
lemma Compl_Un [simp]: "-(A \<union> B) = (-A) \<inter> (-B)"
- by blast
+ by (fact compl_sup)
lemma Compl_Int [simp]: "-(A \<inter> B) = (-A) \<union> (-B)"
- by blast
+ by (fact compl_inf)
lemma subset_Compl_self_eq: "(A \<subseteq> -A) = (A = {})"
by blast
@@ -1348,16 +1347,16 @@
by blast
lemma Compl_UNIV_eq [simp]: "-UNIV = {}"
- by blast
+ by (fact compl_top_eq)
lemma Compl_empty_eq [simp]: "-{} = UNIV"
- by blast
+ by (fact compl_bot_eq)
lemma Compl_subset_Compl_iff [iff]: "(-A \<subseteq> -B) = (B \<subseteq> A)"
- by blast
+ by (fact compl_le_compl_iff)
lemma Compl_eq_Compl_iff [iff]: "(-A = -B) = (A = (B::'a set))"
- by blast
+ by (fact compl_eq_compl_iff)
text {* \medskip Bounded quantifiers.
@@ -1531,16 +1530,16 @@
by blast
lemma Un_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<union> B \<subseteq> C \<union> D"
- by blast
+ by (fact sup_mono)
lemma Int_mono: "A \<subseteq> C ==> B \<subseteq> D ==> A \<inter> B \<subseteq> C \<inter> D"
- by blast
+ by (fact inf_mono)
lemma Diff_mono: "A \<subseteq> C ==> D \<subseteq> B ==> A - B \<subseteq> C - D"
by blast
lemma Compl_anti_mono: "A \<subseteq> B ==> -B \<subseteq> -A"
- by blast
+ by (fact compl_mono)
text {* \medskip Monotonicity of implications. *}