# HG changeset patch
# User nipkow
# Date 1245497694 -7200
# Node ID d74830dc3e4ab51ad3e3e609aebe2a516834687f
# Parent  b9299916d618f07cc50e01a913938edad4afba16
added lemmas; tuned

diff -r b9299916d618 -r d74830dc3e4a src/HOL/Decision_Procs/Cooper.thy
--- a/src/HOL/Decision_Procs/Cooper.thy	Sat Jun 20 01:53:39 2009 +0200
+++ b/src/HOL/Decision_Procs/Cooper.thy	Sat Jun 20 13:34:54 2009 +0200
@@ -1106,18 +1106,18 @@
   let ?d = "\<delta> (And p q)"
   from prems int_lcm_pos have dp: "?d >0" by simp
   have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
-  hence th: "d\<delta> p ?d" using delta_mono prems(3-4) by(simp del:int_lcm_dvd1)
+  hence th: "d\<delta> p ?d" using delta_mono prems(3-4) by(simp only: iszlfm.simps)
   have "\<delta> q dvd \<delta> (And p q)" using prems by simp
-  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:int_lcm_dvd2)
+  hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps)
   from th th' dp show ?case by simp
 next
   case (2 p q)  
   let ?d = "\<delta> (And p q)"
   from prems int_lcm_pos have dp: "?d >0" by simp
   have "\<delta> p dvd \<delta> (And p q)" using prems by simp
-  hence th: "d\<delta> p ?d" using delta_mono prems by(simp del:int_lcm_dvd1)
+  hence th: "d\<delta> p ?d" using delta_mono prems by(simp only: iszlfm.simps)
   have "\<delta> q dvd \<delta> (And p q)" using prems by simp
-  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:int_lcm_dvd2)
+  hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps)
   from th th' dp show ?case by simp
 qed simp_all
 
diff -r b9299916d618 -r d74830dc3e4a src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy	Sat Jun 20 01:53:39 2009 +0200
+++ b/src/HOL/Decision_Procs/MIR.thy	Sat Jun 20 13:34:54 2009 +0200
@@ -2129,18 +2129,18 @@
   from prems int_lcm_pos have dp: "?d >0" by simp
   have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp 
    hence th: "d\<delta> p ?d" 
-     using delta_mono prems by (auto simp del: int_lcm_dvd1)
+     using delta_mono prems by(simp only: iszlfm.simps) blast
   have "\<delta> q dvd \<delta> (And p q)" using prems  by simp 
-  hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: int_lcm_dvd2)
+  hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps) blast
   from th th' dp show ?case by simp 
 next
   case (2 p q)  
   let ?d = "\<delta> (And p q)"
   from prems int_lcm_pos have dp: "?d >0" by simp
   have "\<delta> p dvd \<delta> (And p q)" using prems by simp hence th: "d\<delta> p ?d" using delta_mono prems 
-    by (auto simp del: int_lcm_dvd1)
-  have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by (auto simp del: int_lcm_dvd2)
-  from th th' dp show ?case by simp 
+    by(simp only: iszlfm.simps) blast
+  have "\<delta> q dvd \<delta> (And p q)" using prems by simp hence th': "d\<delta> q ?d" using delta_mono prems by(simp only: iszlfm.simps) blast
+  from th th' dp show ?case by simp
 qed simp_all
 
 
diff -r b9299916d618 -r d74830dc3e4a src/HOL/GCD.thy
--- a/src/HOL/GCD.thy	Sat Jun 20 01:53:39 2009 +0200
+++ b/src/HOL/GCD.thy	Sat Jun 20 13:34:54 2009 +0200
@@ -283,6 +283,18 @@
   apply auto
 done
 
+lemma dvd_gcd_D1_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd m"
+by(metis nat_gcd_dvd1 dvd_trans)
+
+lemma dvd_gcd_D2_nat: "k dvd gcd m n \<Longrightarrow> (k::nat) dvd n"
+by(metis nat_gcd_dvd2 dvd_trans)
+
+lemma dvd_gcd_D1_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd m"
+by(metis int_gcd_dvd1 dvd_trans)
+
+lemma dvd_gcd_D2_int: "i dvd gcd m n \<Longrightarrow> (i::int) dvd n"
+by(metis int_gcd_dvd2 dvd_trans)
+
 lemma nat_gcd_le1 [simp]: "a \<noteq> 0 \<Longrightarrow> gcd (a::nat) b \<le> a"
   by (rule dvd_imp_le, auto)
 
@@ -1386,7 +1398,7 @@
   using prems apply auto
 done
 
-lemma nat_lcm_dvd1 [iff]: "(m::nat) dvd lcm m n"
+lemma nat_lcm_dvd1: "(m::nat) dvd lcm m n"
 proof (cases m)
   case 0 then show ?thesis by simp
 next
@@ -1407,7 +1419,7 @@
   qed
 qed
 
-lemma int_lcm_dvd1 [iff]: "(m::int) dvd lcm m n"
+lemma int_lcm_dvd1: "(m::int) dvd lcm m n"
   apply (subst int_lcm_abs)
   apply (rule dvd_trans)
   prefer 2
@@ -1415,27 +1427,27 @@
   apply auto
 done
 
-lemma nat_lcm_dvd2 [iff]: "(n::nat) dvd lcm m n"
+lemma nat_lcm_dvd2: "(n::nat) dvd lcm m n"
   by (subst nat_lcm_commute, rule nat_lcm_dvd1)
 
-lemma int_lcm_dvd2 [iff]: "(n::int) dvd lcm m n"
+lemma int_lcm_dvd2: "(n::int) dvd lcm m n"
   by (subst int_lcm_commute, rule int_lcm_dvd1)
 
-lemma dvd_lcm_if_dvd1_nat: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
+lemma dvd_lcm_I1_nat[simp]: "(k::nat) dvd m \<Longrightarrow> k dvd lcm m n"
 by(metis nat_lcm_dvd1 dvd_trans)
 
-lemma dvd_lcm_if_dvd2_nat: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
+lemma dvd_lcm_I2_nat[simp]: "(k::nat) dvd n \<Longrightarrow> k dvd lcm m n"
 by(metis nat_lcm_dvd2 dvd_trans)
 
-lemma dvd_lcm_if_dvd1_int: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
+lemma dvd_lcm_I1_int[simp]: "(i::int) dvd m \<Longrightarrow> i dvd lcm m n"
 by(metis int_lcm_dvd1 dvd_trans)
 
-lemma dvd_lcm_if_dvd2_int: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
+lemma dvd_lcm_I2_int[simp]: "(i::int) dvd n \<Longrightarrow> i dvd lcm m n"
 by(metis int_lcm_dvd2 dvd_trans)
 
 lemma nat_lcm_unique: "(a::nat) dvd d \<and> b dvd d \<and>
     (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
-  by (auto intro: dvd_anti_sym nat_lcm_least)
+  by (auto intro: dvd_anti_sym nat_lcm_least nat_lcm_dvd1 nat_lcm_dvd2)
 
 lemma int_lcm_unique: "d >= 0 \<and> (a::int) dvd d \<and> b dvd d \<and>
     (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"