# HG changeset patch
# User wenzelm
# Date 1182451727 -7200
# Node ID 8f8835aac29909e711f68aff6d2ec08a94ab9f06
# Parent  bc2563c37b1aef9d5d831989f1dffb87f62753bc
moved Presburger setup back to Presburger.thy;

diff -r bc2563c37b1a -r 8f8835aac299 src/HOL/Arith_Tools.thy
--- a/src/HOL/Arith_Tools.thy	Thu Jun 21 20:07:26 2007 +0200
+++ b/src/HOL/Arith_Tools.thy	Thu Jun 21 20:48:47 2007 +0200
@@ -7,16 +7,12 @@
 header {* Setup of arithmetic tools *}
 
 theory Arith_Tools
-imports Groebner_Basis Dense_Linear_Order SetInterval
+imports Groebner_Basis Dense_Linear_Order
 uses
   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
   "~~/src/Provers/Arith/extract_common_term.ML"
   "int_factor_simprocs.ML"
   "nat_simprocs.ML"
-  "Tools/Presburger/cooper_data.ML"
-  "Tools/Presburger/generated_cooper.ML"
-  ("Tools/Presburger/cooper.ML")
-  ("Tools/Presburger/presburger.ML") 
 begin
 
 subsection {* Simprocs for the Naturals *}
@@ -941,681 +937,4 @@
 end
 *}
 
-
-subsection {* Decision Procedure for Presburger Arithmetic *}
-
-setup CooperData.setup
-
-subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
-
-lemma minf:
-  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
-     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
-  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
-     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
-  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
-  "\<exists>z.\<forall>x<z. F = F"
-  by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
-
-lemma pinf:
-  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
-     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
-  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
-     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
-  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
-  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
-  "\<exists>z.\<forall>x>z. F = F"
-  by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
-
-lemma inf_period:
-  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
-    \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
-  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
-    \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
-  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
-  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
-  "\<forall>x k. F = F"
-by simp_all
-  (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
-    simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
-
-section{* The A and B sets *}
-lemma bset:
-  "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
-     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
-  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
-  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
-     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
-  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
-  "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
-  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
-  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
-  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
-  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
-  "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
-  "d dvd D \<Longrightarrow>(\<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))"
-  "d dvd D \<Longrightarrow>(\<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))"
-  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
-proof (blast, blast)
-  assume dp: "D > 0" and tB: "t - 1\<in> B"
-  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 
-    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
-    using dp tB by simp_all
-next
-  assume dp: "D > 0" and tB: "t \<in> B"
-  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))" 
-    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
-    using dp tB by simp_all
-next
-  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
-next
-  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
-next
-  assume dp: "D > 0" and tB:"t \<in> B"
-  {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"
-    hence "x -t \<le> D" and "1 \<le> x - t" by simp+
-      hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
-      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
-      with nob tB have "False" by simp}
-  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
-next
-  assume dp: "D > 0" and tB:"t - 1\<in> B"
-  {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"
-    hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
-      hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
-      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
-      with nob tB have "False" by simp}
-  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
-next
-  assume d: "d dvd D"
-  {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
-      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
-  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
-next
-  assume d: "d dvd D"
-  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
-      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
-  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
-qed blast
-
-lemma aset:
-  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
-     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
-  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
-  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
-     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
-  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
-  "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
-  "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
-  "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
-  "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
-  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
-  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
-  "d dvd D \<Longrightarrow>(\<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))"
-  "d dvd D \<Longrightarrow>(\<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))"
-  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
-proof (blast, blast)
-  assume dp: "D > 0" and tA: "t + 1 \<in> A"
-  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 
-    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
-    using dp tA by simp_all
-next
-  assume dp: "D > 0" and tA: "t \<in> A"
-  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))" 
-    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
-    using dp tA by simp_all
-next
-  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
-next
-  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
-next
-  assume dp: "D > 0" and tA:"t \<in> A"
-  {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"
-    hence "t - x \<le> D" and "1 \<le> t - x" by simp+
-      hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
-      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps) 
-      with nob tA have "False" by simp}
-  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
-next
-  assume dp: "D > 0" and tA:"t + 1\<in> A"
-  {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"
-    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
-      hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
-      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
-      with nob tA have "False" by simp}
-  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
-next
-  assume d: "d dvd D"
-  {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
-      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
-  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
-next
-  assume d: "d dvd D"
-  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
-      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
-  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
-qed blast
-
-subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
-
-subsubsection{* First some trivial facts about periodic sets or predicates *}
-lemma periodic_finite_ex:
-  assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
-  shows "(EX x. P x) = (EX j : {1..d}. P j)"
-  (is "?LHS = ?RHS")
-proof
-  assume ?LHS
-  then obtain x where P: "P x" ..
-  have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
-  hence Pmod: "P x = P(x mod d)" using modd by simp
-  show ?RHS
-  proof (cases)
-    assume "x mod d = 0"
-    hence "P 0" using P Pmod by simp
-    moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
-    ultimately have "P d" by simp
-    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
-    ultimately show ?RHS ..
-  next
-    assume not0: "x mod d \<noteq> 0"
-    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
-    moreover have "x mod d : {1..d}"
-    proof -
-      from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
-      moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
-      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
-    qed
-    ultimately show ?RHS ..
-  qed
-qed auto
-
-subsubsection{* The @{text "-\<infinity>"} Version*}
-
-lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
-by(induct rule: int_gr_induct,simp_all add:int_distrib)
-
-lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
-by(induct rule: int_gr_induct, simp_all add:int_distrib)
-
-theorem int_induct[case_names base step1 step2]:
-  assumes 
-  base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
-  step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
-  shows "P i"
-proof -
-  have "i \<le> k \<or> i\<ge> k" by arith
-  thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
-qed
-
-lemma decr_mult_lemma:
-  assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
-  shows "ALL x. P x \<longrightarrow> P(x - k*d)"
-using knneg
-proof (induct rule:int_ge_induct)
-  case base thus ?case by simp
-next
-  case (step i)
-  {fix x
-    have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
-    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
-      by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
-    ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
-  thus ?case ..
-qed
-
-lemma  minusinfinity:
-  assumes dpos: "0 < d" and
-    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
-  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
-proof
-  assume eP1: "EX x. P1 x"
-  then obtain x where P1: "P1 x" ..
-  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
-  let ?w = "x - (abs(x-z)+1) * d"
-  from dpos have w: "?w < z" by(rule decr_lemma)
-  have "P1 x = P1 ?w" using P1eqP1 by blast
-  also have "\<dots> = P(?w)" using w P1eqP by blast
-  finally have "P ?w" using P1 by blast
-  thus "EX x. P x" ..
-qed
-
-lemma cpmi: 
-  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
-  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
-  and pd: "\<forall> x k. P' x = P' (x-k*D)"
-  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 
-         (is "?L = (?R1 \<or> ?R2)")
-proof-
- {assume "?R2" hence "?L"  by blast}
- moreover
- {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
- moreover 
- { fix x
-   assume P: "P x" and H: "\<not> ?R2"
-   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
-     hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
-     with nb P  have "P (y - D)" by auto }
-   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
-   with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
-   from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
-   let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
-   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
-   from dp have yz: "?y < z" using decr_lemma[OF dp] by simp   
-   from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
-   with periodic_finite_ex[OF dp pd]
-   have "?R1" by blast}
- ultimately show ?thesis by blast
-qed
-
-subsubsection {* The @{text "+\<infinity>"} Version*}
-
-lemma  plusinfinity:
-  assumes dpos: "(0::int) < d" and
-    P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
-  shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
-proof
-  assume eP1: "EX x. P' x"
-  then obtain x where P1: "P' x" ..
-  from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
-  let ?w' = "x + (abs(x-z)+1) * d"
-  let ?w = "x - (-(abs(x-z) + 1))*d"
-  have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
-  from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
-  hence "P' x = P' ?w" using P1eqP1 by blast
-  also have "\<dots> = P(?w)" using w P1eqP by blast
-  finally have "P ?w" using P1 by blast
-  thus "EX x. P x" ..
-qed
-
-lemma incr_mult_lemma:
-  assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
-  shows "ALL x. P x \<longrightarrow> P(x + k*d)"
-using knneg
-proof (induct rule:int_ge_induct)
-  case base thus ?case by simp
-next
-  case (step i)
-  {fix x
-    have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
-    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
-      by (simp add:int_distrib zadd_ac)
-    ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
-  thus ?case ..
-qed
-
-lemma cppi: 
-  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
-  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
-  and pd: "\<forall> x k. P' x= P' (x-k*D)"
-  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
-proof-
- {assume "?R2" hence "?L"  by blast}
- moreover
- {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
- moreover 
- { fix x
-   assume P: "P x" and H: "\<not> ?R2"
-   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
-     hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
-     with nb P  have "P (y + D)" by auto }
-   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
-   with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
-   from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
-   let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
-   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
-   from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
-   from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
-   with periodic_finite_ex[OF dp pd]
-   have "?R1" by blast}
- ultimately show ?thesis by blast
-qed
-
-lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
-apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
-apply(fastsimp)
-done
-
-theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
-  apply (rule eq_reflection[symmetric])
-  apply (rule iffI)
-  defer
-  apply (erule exE)
-  apply (rule_tac x = "l * x" in exI)
-  apply (simp add: dvd_def)
-  apply (rule_tac x="x" in exI, simp)
-  apply (erule exE)
-  apply (erule conjE)
-  apply (erule dvdE)
-  apply (rule_tac x = k in exI)
-  apply simp
-  done
-
-lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
-shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
-  using not0 by (simp add: dvd_def)
-
-lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
-  by simp_all
-text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
-lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
-  by (simp split add: split_nat)
-
-lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
-  apply (auto split add: split_nat)
-  apply (rule_tac x="int x" in exI, simp)
-  apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
-  done
-
-lemma zdiff_int_split: "P (int (x - y)) =
-  ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
-  by (case_tac "y \<le> x", simp_all add: zdiff_int)
-
-lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
-lemma number_of2: "(0::int) <= Numeral0" by simp
-lemma Suc_plus1: "Suc n = n + 1" by simp
-
-text {*
-  \medskip Specific instances of congruence rules, to prevent
-  simplifier from looping. *}
-
-theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
-
-theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
-  by (simp cong: conj_cong)
-lemma int_eq_number_of_eq:
-  "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
-  by simp
-
-lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
-unfolding dvd_eq_mod_eq_0[symmetric] ..
-
-lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
-unfolding zdvd_iff_zmod_eq_0[symmetric] ..
-declare mod_1[presburger]
-declare mod_0[presburger]
-declare zmod_1[presburger]
-declare zmod_zero[presburger]
-declare zmod_self[presburger]
-declare mod_self[presburger]
-declare DIVISION_BY_ZERO_MOD[presburger]
-declare nat_mod_div_trivial[presburger]
-declare div_mod_equality2[presburger]
-declare div_mod_equality[presburger]
-declare mod_div_equality2[presburger]
-declare mod_div_equality[presburger]
-declare mod_mult_self1[presburger]
-declare mod_mult_self2[presburger]
-declare zdiv_zmod_equality2[presburger]
-declare zdiv_zmod_equality[presburger]
-declare mod2_Suc_Suc[presburger]
-lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
-using IntDiv.DIVISION_BY_ZERO by blast+
-
-use "Tools/Presburger/cooper.ML"
-oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
-
-use "Tools/Presburger/presburger.ML"
-
-setup {* 
-  arith_tactic_add 
-    (mk_arith_tactic "presburger" (fn i => fn st =>
-       (warning "Trying Presburger arithmetic ...";   
-    Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st)))
-  (* FIXME!!!!!!! get the right context!!*)	
-*}
-
-method_setup presburger = {*
-let
- fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
- fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
- val addN = "add"
- val delN = "del"
- val elimN = "elim"
- val any_keyword = keyword addN || keyword delN || simple_keyword elimN
- val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-in
-  fn src => Method.syntax 
-   ((Scan.optional (simple_keyword elimN >> K false) true) -- 
-    (Scan.optional (keyword addN |-- thms) []) -- 
-    (Scan.optional (keyword delN |-- thms) [])) src 
-  #> (fn (((elim, add_ths), del_ths),ctxt) => 
-         Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
 end
-*} "Cooper's algorithm for Presburger arithmetic"
-
-lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
-lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
-
-
-subsection {* Code generator setup *}
-
-text {*
-  Presburger arithmetic is convenient to prove some
-  of the following code lemmas on integer numerals:
-*}
-
-lemma eq_Pls_Pls:
-  "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
-
-lemma eq_Pls_Min:
-  "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
-  unfolding Pls_def Numeral.Min_def by presburger
-
-lemma eq_Pls_Bit0:
-  "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
-  unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Pls_Bit1:
-  "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
-  unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Min_Pls:
-  "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
-  unfolding Pls_def Numeral.Min_def by presburger
-
-lemma eq_Min_Min:
-  "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
-
-lemma eq_Min_Bit0:
-  "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
-  unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Min_Bit1:
-  "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
-  unfolding Numeral.Min_def Bit_def bit.cases by presburger
-
-lemma eq_Bit0_Pls:
-  "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
-  unfolding Pls_def Bit_def bit.cases by presburger
-
-lemma eq_Bit1_Pls:
-  "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
-  unfolding Pls_def Bit_def bit.cases  by presburger
-
-lemma eq_Bit0_Min:
-  "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
-  unfolding Numeral.Min_def Bit_def bit.cases  by presburger
-
-lemma eq_Bit1_Min:
-  "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
-  unfolding Numeral.Min_def Bit_def bit.cases  by presburger
-
-lemma eq_Bit_Bit:
-  "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
-    v1 = v2 \<and> k1 = k2" 
-  unfolding Bit_def
-  apply (cases v1)
-  apply (cases v2)
-  apply auto
-  apply presburger
-  apply (cases v2)
-  apply auto
-  apply presburger
-  apply (cases v2)
-  apply auto
-  done
-
-lemma eq_number_of:
-  "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l" 
-  unfolding number_of_is_id ..
-
-
-lemma less_eq_Pls_Pls:
-  "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Pls_Min:
-  "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
-  unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_eq_Pls_Bit:
-  "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
-  unfolding Pls_def Bit_def by (cases v) auto
-
-lemma less_eq_Min_Pls:
-  "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
-  unfolding Pls_def Numeral.Min_def by presburger
-
-lemma less_eq_Min_Min:
-  "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
-
-lemma less_eq_Min_Bit0:
-  "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
-  unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_eq_Min_Bit1:
-  "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
-  unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_eq_Bit0_Pls:
-  "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
-  unfolding Pls_def Bit_def by simp
-
-lemma less_eq_Bit1_Pls:
-  "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
-  unfolding Pls_def Bit_def by auto
-
-lemma less_eq_Bit_Min:
-  "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
-  unfolding Numeral.Min_def Bit_def by (cases v) auto
-
-lemma less_eq_Bit0_Bit:
-  "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
-  unfolding Bit_def bit.cases by (cases v) auto
-
-lemma less_eq_Bit_Bit1:
-  "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
-  unfolding Bit_def bit.cases by (cases v) auto
-
-lemma less_eq_Bit1_Bit0:
-  "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
-  unfolding Bit_def by (auto split: bit.split)
-
-lemma less_eq_number_of:
-  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
-  unfolding number_of_is_id ..
-
-
-lemma less_Pls_Pls:
-  "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp 
-
-lemma less_Pls_Min:
-  "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
-  unfolding Pls_def Numeral.Min_def  by presburger 
-
-lemma less_Pls_Bit0:
-  "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
-  unfolding Pls_def Bit_def by auto
-
-lemma less_Pls_Bit1:
-  "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
-  unfolding Pls_def Bit_def by auto
-
-lemma less_Min_Pls:
-  "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
-  unfolding Pls_def Numeral.Min_def by presburger 
-
-lemma less_Min_Min:
-  "Numeral.Min < Numeral.Min \<longleftrightarrow> False"  by simp
-
-lemma less_Min_Bit:
-  "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
-  unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
-
-lemma less_Bit_Pls:
-  "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
-  unfolding Pls_def Bit_def by (auto split: bit.split)
-
-lemma less_Bit0_Min:
-  "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
-  unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_Bit1_Min:
-  "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
-  unfolding Numeral.Min_def Bit_def by auto
-
-lemma less_Bit_Bit0:
-  "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
-  unfolding Bit_def by (auto split: bit.split)
-
-lemma less_Bit1_Bit:
-  "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
-  unfolding Bit_def by (auto split: bit.split)
-
-lemma less_Bit0_Bit1:
-  "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
-  unfolding Bit_def bit.cases  by arith
-
-lemma less_number_of:
-  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
-  unfolding number_of_is_id ..
-
-lemmas pred_succ_numeral_code [code func] =
-  arith_simps(5-12)
-
-lemmas plus_numeral_code [code func] =
-  arith_simps(13-17)
-  arith_simps(26-27)
-  arith_extra_simps(1) [where 'a = int]
-
-lemmas minus_numeral_code [code func] =
-  arith_simps(18-21)
-  arith_extra_simps(2) [where 'a = int]
-  arith_extra_simps(5) [where 'a = int]
-
-lemmas times_numeral_code [code func] =
-  arith_simps(22-25)
-  arith_extra_simps(4) [where 'a = int]
-
-lemmas eq_numeral_code [code func] =
-  eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
-  eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
-  eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
-  eq_number_of
-
-lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
-  less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
-  less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
-  less_eq_number_of
-
-lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
-  less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
-  less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
-  less_number_of
-
-end
diff -r bc2563c37b1a -r 8f8835aac299 src/HOL/PreList.thy
--- a/src/HOL/PreList.thy	Thu Jun 21 20:07:26 2007 +0200
+++ b/src/HOL/PreList.thy	Thu Jun 21 20:48:47 2007 +0200
@@ -7,7 +7,7 @@
 header {* A Basis for Building the Theory of Lists *}
 
 theory PreList
-imports Wellfounded_Relations Arith_Tools Relation_Power SAT
+imports Wellfounded_Relations Presburger Relation_Power SAT
   FunDef Recdef Extraction
 begin
 
diff -r bc2563c37b1a -r 8f8835aac299 src/HOL/Presburger.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Presburger.thy	Thu Jun 21 20:48:47 2007 +0200
@@ -0,0 +1,691 @@
+(* Title:      HOL/Presburger.thy
+   ID:         $Id$
+   Author:     Amine Chaieb, TU Muenchen
+*)
+
+theory Presburger
+imports Arith_Tools SetInterval
+uses
+  "Tools/Qelim/cooper_data.ML"
+  "Tools/Qelim/generated_cooper.ML"
+  ("Tools/Qelim/cooper.ML")
+  ("Tools/Qelim/presburger.ML")
+begin
+
+subsection {* Decision Procedure for Presburger Arithmetic *}
+
+setup CooperData.setup
+
+subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
+
+lemma minf:
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
+     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
+     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>x<z. F = F"
+  by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
+
+lemma pinf:
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
+     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
+  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
+     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
+  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
+  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
+  "\<exists>z.\<forall>x>z. F = F"
+  by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
+
+lemma inf_period:
+  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
+    \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
+  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
+    \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
+  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
+  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
+  "\<forall>x k. F = F"
+by simp_all
+  (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
+    simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
+
+section{* The A and B sets *}
+lemma bset:
+  "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
+  "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
+  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
+  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
+  "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
+  "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
+  "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
+  "d dvd D \<Longrightarrow>(\<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))"
+  "d dvd D \<Longrightarrow>(\<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))"
+  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+  assume dp: "D > 0" and tB: "t - 1\<in> B"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 
+    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
+    using dp tB by simp_all
+next
+  assume dp: "D > 0" and tB: "t \<in> B"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))" 
+    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+    using dp tB by simp_all
+next
+  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
+next
+  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
+next
+  assume dp: "D > 0" and tB:"t \<in> B"
+  {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"
+    hence "x -t \<le> D" and "1 \<le> x - t" by simp+
+      hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
+      with nob tB have "False" by simp}
+  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
+next
+  assume dp: "D > 0" and tB:"t - 1\<in> B"
+  {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"
+    hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
+      hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
+      with nob tB have "False" by simp}
+  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
+next
+  assume d: "d dvd D"
+  {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
+      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
+  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
+next
+  assume d: "d dvd D"
+  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
+      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
+  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
+qed blast
+
+lemma aset:
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
+  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
+     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
+  \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
+  "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
+  "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
+  "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
+  "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
+  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
+  "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
+  "d dvd D \<Longrightarrow>(\<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))"
+  "d dvd D \<Longrightarrow>(\<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))"
+  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
+proof (blast, blast)
+  assume dp: "D > 0" and tA: "t + 1 \<in> A"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 
+    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
+    using dp tA by simp_all
+next
+  assume dp: "D > 0" and tA: "t \<in> A"
+  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))" 
+    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
+    using dp tA by simp_all
+next
+  assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
+next
+  assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
+next
+  assume dp: "D > 0" and tA:"t \<in> A"
+  {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"
+    hence "t - x \<le> D" and "1 \<le> t - x" by simp+
+      hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps) 
+      with nob tA have "False" by simp}
+  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
+next
+  assume dp: "D > 0" and tA:"t + 1\<in> A"
+  {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"
+    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
+      hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
+      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
+      with nob tA have "False" by simp}
+  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
+next
+  assume d: "d dvd D"
+  {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
+      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
+  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
+next
+  assume d: "d dvd D"
+  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
+      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
+  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
+qed blast
+
+subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
+
+subsubsection{* First some trivial facts about periodic sets or predicates *}
+lemma periodic_finite_ex:
+  assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
+  shows "(EX x. P x) = (EX j : {1..d}. P j)"
+  (is "?LHS = ?RHS")
+proof
+  assume ?LHS
+  then obtain x where P: "P x" ..
+  have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
+  hence Pmod: "P x = P(x mod d)" using modd by simp
+  show ?RHS
+  proof (cases)
+    assume "x mod d = 0"
+    hence "P 0" using P Pmod by simp
+    moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
+    ultimately have "P d" by simp
+    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
+    ultimately show ?RHS ..
+  next
+    assume not0: "x mod d \<noteq> 0"
+    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
+    moreover have "x mod d : {1..d}"
+    proof -
+      from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
+      moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
+      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
+    qed
+    ultimately show ?RHS ..
+  qed
+qed auto
+
+subsubsection{* The @{text "-\<infinity>"} Version*}
+
+lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
+by(induct rule: int_gr_induct,simp_all add:int_distrib)
+
+lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
+by(induct rule: int_gr_induct, simp_all add:int_distrib)
+
+theorem int_induct[case_names base step1 step2]:
+  assumes 
+  base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
+  step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
+  shows "P i"
+proof -
+  have "i \<le> k \<or> i\<ge> k" by arith
+  thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
+qed
+
+lemma decr_mult_lemma:
+  assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
+  shows "ALL x. P x \<longrightarrow> P(x - k*d)"
+using knneg
+proof (induct rule:int_ge_induct)
+  case base thus ?case by simp
+next
+  case (step i)
+  {fix x
+    have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
+    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
+      by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
+    ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
+  thus ?case ..
+qed
+
+lemma  minusinfinity:
+  assumes dpos: "0 < d" and
+    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
+  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
+proof
+  assume eP1: "EX x. P1 x"
+  then obtain x where P1: "P1 x" ..
+  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
+  let ?w = "x - (abs(x-z)+1) * d"
+  from dpos have w: "?w < z" by(rule decr_lemma)
+  have "P1 x = P1 ?w" using P1eqP1 by blast
+  also have "\<dots> = P(?w)" using w P1eqP by blast
+  finally have "P ?w" using P1 by blast
+  thus "EX x. P x" ..
+qed
+
+lemma cpmi: 
+  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
+  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
+  and pd: "\<forall> x k. P' x = P' (x-k*D)"
+  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 
+         (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L"  by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover 
+ { fix x
+   assume P: "P x" and H: "\<not> ?R2"
+   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
+     hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
+     with nb P  have "P (y - D)" by auto }
+   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
+   with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
+   from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
+   let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
+   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+   from dp have yz: "?y < z" using decr_lemma[OF dp] by simp   
+   from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+   with periodic_finite_ex[OF dp pd]
+   have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed
+
+subsubsection {* The @{text "+\<infinity>"} Version*}
+
+lemma  plusinfinity:
+  assumes dpos: "(0::int) < d" and
+    P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
+  shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
+proof
+  assume eP1: "EX x. P' x"
+  then obtain x where P1: "P' x" ..
+  from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
+  let ?w' = "x + (abs(x-z)+1) * d"
+  let ?w = "x - (-(abs(x-z) + 1))*d"
+  have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
+  from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
+  hence "P' x = P' ?w" using P1eqP1 by blast
+  also have "\<dots> = P(?w)" using w P1eqP by blast
+  finally have "P ?w" using P1 by blast
+  thus "EX x. P x" ..
+qed
+
+lemma incr_mult_lemma:
+  assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
+  shows "ALL x. P x \<longrightarrow> P(x + k*d)"
+using knneg
+proof (induct rule:int_ge_induct)
+  case base thus ?case by simp
+next
+  case (step i)
+  {fix x
+    have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
+    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
+      by (simp add:int_distrib zadd_ac)
+    ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
+  thus ?case ..
+qed
+
+lemma cppi: 
+  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
+  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
+  and pd: "\<forall> x k. P' x= P' (x-k*D)"
+  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
+proof-
+ {assume "?R2" hence "?L"  by blast}
+ moreover
+ {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
+ moreover 
+ { fix x
+   assume P: "P x" and H: "\<not> ?R2"
+   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
+     hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
+     with nb P  have "P (y + D)" by auto }
+   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
+   with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
+   from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
+   let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
+   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
+   from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
+   from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
+   with periodic_finite_ex[OF dp pd]
+   have "?R1" by blast}
+ ultimately show ?thesis by blast
+qed
+
+lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
+apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
+apply(fastsimp)
+done
+
+theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
+  apply (rule eq_reflection[symmetric])
+  apply (rule iffI)
+  defer
+  apply (erule exE)
+  apply (rule_tac x = "l * x" in exI)
+  apply (simp add: dvd_def)
+  apply (rule_tac x="x" in exI, simp)
+  apply (erule exE)
+  apply (erule conjE)
+  apply (erule dvdE)
+  apply (rule_tac x = k in exI)
+  apply simp
+  done
+
+lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
+shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
+  using not0 by (simp add: dvd_def)
+
+lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
+  by simp_all
+text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
+lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
+  by (simp split add: split_nat)
+
+lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
+  apply (auto split add: split_nat)
+  apply (rule_tac x="int x" in exI, simp)
+  apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
+  done
+
+lemma zdiff_int_split: "P (int (x - y)) =
+  ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
+  by (case_tac "y \<le> x", simp_all add: zdiff_int)
+
+lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
+lemma number_of2: "(0::int) <= Numeral0" by simp
+lemma Suc_plus1: "Suc n = n + 1" by simp
+
+text {*
+  \medskip Specific instances of congruence rules, to prevent
+  simplifier from looping. *}
+
+theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
+
+theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
+  by (simp cong: conj_cong)
+lemma int_eq_number_of_eq:
+  "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
+  by simp
+
+lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
+unfolding dvd_eq_mod_eq_0[symmetric] ..
+
+lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
+unfolding zdvd_iff_zmod_eq_0[symmetric] ..
+declare mod_1[presburger]
+declare mod_0[presburger]
+declare zmod_1[presburger]
+declare zmod_zero[presburger]
+declare zmod_self[presburger]
+declare mod_self[presburger]
+declare DIVISION_BY_ZERO_MOD[presburger]
+declare nat_mod_div_trivial[presburger]
+declare div_mod_equality2[presburger]
+declare div_mod_equality[presburger]
+declare mod_div_equality2[presburger]
+declare mod_div_equality[presburger]
+declare mod_mult_self1[presburger]
+declare mod_mult_self2[presburger]
+declare zdiv_zmod_equality2[presburger]
+declare zdiv_zmod_equality[presburger]
+declare mod2_Suc_Suc[presburger]
+lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
+using IntDiv.DIVISION_BY_ZERO by blast+
+
+use "Tools/Qelim/cooper.ML"
+oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
+
+use "Tools/Qelim/presburger.ML"
+
+setup {* 
+  arith_tactic_add 
+    (mk_arith_tactic "presburger" (fn i => fn st =>
+       (warning "Trying Presburger arithmetic ...";   
+    Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st)))
+  (* FIXME!!!!!!! get the right context!!*)	
+*}
+
+method_setup presburger = {*
+let
+ fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
+ fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
+ val addN = "add"
+ val delN = "del"
+ val elimN = "elim"
+ val any_keyword = keyword addN || keyword delN || simple_keyword elimN
+ val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+in
+  fn src => Method.syntax 
+   ((Scan.optional (simple_keyword elimN >> K false) true) -- 
+    (Scan.optional (keyword addN |-- thms) []) -- 
+    (Scan.optional (keyword delN |-- thms) [])) src 
+  #> (fn (((elim, add_ths), del_ths),ctxt) => 
+         Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
+end
+*} "Cooper's algorithm for Presburger arithmetic"
+
+lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
+lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
+
+
+subsection {* Code generator setup *}
+
+text {*
+  Presburger arithmetic is convenient to prove some
+  of the following code lemmas on integer numerals:
+*}
+
+lemma eq_Pls_Pls:
+  "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
+
+lemma eq_Pls_Min:
+  "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
+  unfolding Pls_def Numeral.Min_def by presburger
+
+lemma eq_Pls_Bit0:
+  "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
+  unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Pls_Bit1:
+  "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
+  unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Min_Pls:
+  "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
+  unfolding Pls_def Numeral.Min_def by presburger
+
+lemma eq_Min_Min:
+  "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
+
+lemma eq_Min_Bit0:
+  "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
+  unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Min_Bit1:
+  "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
+  unfolding Numeral.Min_def Bit_def bit.cases by presburger
+
+lemma eq_Bit0_Pls:
+  "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
+  unfolding Pls_def Bit_def bit.cases by presburger
+
+lemma eq_Bit1_Pls:
+  "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
+  unfolding Pls_def Bit_def bit.cases  by presburger
+
+lemma eq_Bit0_Min:
+  "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
+  unfolding Numeral.Min_def Bit_def bit.cases  by presburger
+
+lemma eq_Bit1_Min:
+  "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
+  unfolding Numeral.Min_def Bit_def bit.cases  by presburger
+
+lemma eq_Bit_Bit:
+  "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
+    v1 = v2 \<and> k1 = k2" 
+  unfolding Bit_def
+  apply (cases v1)
+  apply (cases v2)
+  apply auto
+  apply presburger
+  apply (cases v2)
+  apply auto
+  apply presburger
+  apply (cases v2)
+  apply auto
+  done
+
+lemma eq_number_of:
+  "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l" 
+  unfolding number_of_is_id ..
+
+
+lemma less_eq_Pls_Pls:
+  "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
+
+lemma less_eq_Pls_Min:
+  "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
+  unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_eq_Pls_Bit:
+  "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
+  unfolding Pls_def Bit_def by (cases v) auto
+
+lemma less_eq_Min_Pls:
+  "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
+  unfolding Pls_def Numeral.Min_def by presburger
+
+lemma less_eq_Min_Min:
+  "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
+
+lemma less_eq_Min_Bit0:
+  "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
+  unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_eq_Min_Bit1:
+  "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
+  unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_eq_Bit0_Pls:
+  "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
+  unfolding Pls_def Bit_def by simp
+
+lemma less_eq_Bit1_Pls:
+  "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+  unfolding Pls_def Bit_def by auto
+
+lemma less_eq_Bit_Min:
+  "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
+  unfolding Numeral.Min_def Bit_def by (cases v) auto
+
+lemma less_eq_Bit0_Bit:
+  "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
+  unfolding Bit_def bit.cases by (cases v) auto
+
+lemma less_eq_Bit_Bit1:
+  "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+  unfolding Bit_def bit.cases by (cases v) auto
+
+lemma less_eq_Bit1_Bit0:
+  "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+  unfolding Bit_def by (auto split: bit.split)
+
+lemma less_eq_number_of:
+  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
+  unfolding number_of_is_id ..
+
+
+lemma less_Pls_Pls:
+  "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp 
+
+lemma less_Pls_Min:
+  "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
+  unfolding Pls_def Numeral.Min_def  by presburger 
+
+lemma less_Pls_Bit0:
+  "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
+  unfolding Pls_def Bit_def by auto
+
+lemma less_Pls_Bit1:
+  "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
+  unfolding Pls_def Bit_def by auto
+
+lemma less_Min_Pls:
+  "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
+  unfolding Pls_def Numeral.Min_def by presburger 
+
+lemma less_Min_Min:
+  "Numeral.Min < Numeral.Min \<longleftrightarrow> False"  by simp
+
+lemma less_Min_Bit:
+  "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
+  unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit_Pls:
+  "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
+  unfolding Pls_def Bit_def by (auto split: bit.split)
+
+lemma less_Bit0_Min:
+  "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
+  unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_Bit1_Min:
+  "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
+  unfolding Numeral.Min_def Bit_def by auto
+
+lemma less_Bit_Bit0:
+  "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
+  unfolding Bit_def by (auto split: bit.split)
+
+lemma less_Bit1_Bit:
+  "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
+  unfolding Bit_def by (auto split: bit.split)
+
+lemma less_Bit0_Bit1:
+  "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
+  unfolding Bit_def bit.cases  by arith
+
+lemma less_number_of:
+  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
+  unfolding number_of_is_id ..
+
+lemmas pred_succ_numeral_code [code func] =
+  arith_simps(5-12)
+
+lemmas plus_numeral_code [code func] =
+  arith_simps(13-17)
+  arith_simps(26-27)
+  arith_extra_simps(1) [where 'a = int]
+
+lemmas minus_numeral_code [code func] =
+  arith_simps(18-21)
+  arith_extra_simps(2) [where 'a = int]
+  arith_extra_simps(5) [where 'a = int]
+
+lemmas times_numeral_code [code func] =
+  arith_simps(22-25)
+  arith_extra_simps(4) [where 'a = int]
+
+lemmas eq_numeral_code [code func] =
+  eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
+  eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
+  eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
+  eq_number_of
+
+lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
+  less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
+  less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
+  less_eq_number_of
+
+lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
+  less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
+  less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
+  less_number_of
+
+end