src/HOL/Presburger.thy
changeset 23477 f4b83f03cac9
parent 23472 02099ea56555
child 23685 1b0f4071946c
     1.1 --- a/src/HOL/Presburger.thy	Fri Jun 22 22:41:17 2007 +0200
     1.2 +++ b/src/HOL/Presburger.thy	Sat Jun 23 19:33:22 2007 +0200
     1.3 @@ -60,7 +60,7 @@
     1.4    "\<forall>x k. F = F"
     1.5  by simp_all
     1.6    (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
     1.7 -    simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
     1.8 +    simp add: ring_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_simps)+
     1.9  
    1.10  subsection{* The A and B sets *}
    1.11  lemma bset:
    1.12 @@ -98,7 +98,7 @@
    1.13    {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
    1.14      hence "x -t \<le> D" and "1 \<le> x - t" by simp+
    1.15        hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
    1.16 -      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
    1.17 +      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps)
    1.18        with nob tB have "False" by simp}
    1.19    thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
    1.20  next
    1.21 @@ -106,18 +106,18 @@
    1.22    {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
    1.23      hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
    1.24        hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
    1.25 -      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
    1.26 +      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps)
    1.27        with nob tB have "False" by simp}
    1.28    thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
    1.29  next
    1.30    assume d: "d dvd D"
    1.31    {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
    1.32 -      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
    1.33 +      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_simps)}
    1.34    thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp
    1.35  next
    1.36    assume d: "d dvd D"
    1.37    {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
    1.38 -      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
    1.39 +      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)}
    1.40    thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
    1.41  qed blast
    1.42  
    1.43 @@ -156,26 +156,26 @@
    1.44    {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
    1.45      hence "t - x \<le> D" and "1 \<le> t - x" by simp+
    1.46        hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
    1.47 -      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps) 
    1.48 +      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps) 
    1.49        with nob tA have "False" by simp}
    1.50    thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
    1.51  next
    1.52    assume dp: "D > 0" and tA:"t + 1\<in> A"
    1.53    {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
    1.54 -    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
    1.55 +    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps)
    1.56        hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
    1.57 -      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
    1.58 +      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps)
    1.59        with nob tA have "False" by simp}
    1.60    thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
    1.61  next
    1.62    assume d: "d dvd D"
    1.63    {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
    1.64 -      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
    1.65 +      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)}
    1.66    thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
    1.67  next
    1.68    assume d: "d dvd D"
    1.69    {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
    1.70 -      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
    1.71 +      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)}
    1.72    thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
    1.73  qed blast
    1.74  
    1.75 @@ -302,7 +302,7 @@
    1.76    from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
    1.77    let ?w' = "x + (abs(x-z)+1) * d"
    1.78    let ?w = "x - (-(abs(x-z) + 1))*d"
    1.79 -  have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
    1.80 +  have ww'[simp]: "?w = ?w'" by (simp add: ring_simps)
    1.81    from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
    1.82    hence "P' x = P' ?w" using P1eqP1 by blast
    1.83    also have "\<dots> = P(?w)" using w P1eqP by blast