(* Title:      HOL/Presburger.thy

    ID:         $Id$

    Author:     Amine Chaieb, TU Muenchen

 *)

 header {* Decision Procedure for Presburger Arithmetic *}

 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

 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,Divides.div})<z. (d dvd x + s) = (d dvd x + s)"

   "\<exists>z.\<forall>(x::'a::{linorder,plus,Divides.div})<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,Divides.div})>z. (d dvd x + s) = (d dvd x + s)"

   "\<exists>z.\<forall>(x::'a::{linorder,plus,Divides.div})>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,Divides.div}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"

   "(d::'a::{comm_ring,Divides.div}) 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_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_simps)+

 subsection{* 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_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_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_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_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_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_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_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_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_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_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,Divides.div}). 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"

 declaration {* fn _ =>

   arith_tactic_add

     (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st =>

        (warning "Trying Presburger arithmetic ...";   

     Presburger.cooper_tac true [] [] ctxt i st)))

 *}

 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

 lemma zdvd_period:

   fixes a d :: int

   assumes advdd: "a dvd d"

   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"

 proof-

   {

     fix x k

     from inf_period(3) [OF advdd, rule_format, where x=x and k="-k"]  

     have "a dvd (x + t) \<longleftrightarrow> a dvd (x + k * d + t)" by simp

   }

   hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp

   then show ?thesis by simp

 qed

 subsection {* Code generator setup *}

 text {*

   Presburger arithmetic is convenient to prove some

   of the following code lemmas on integer numerals:

 *}

 lemma eq_Pls_Pls:

   "Int.Pls = Int.Pls \<longleftrightarrow> True" by presburger

 lemma eq_Pls_Min:

   "Int.Pls = Int.Min \<longleftrightarrow> False"

   unfolding Pls_def Int.Min_def by presburger

 lemma eq_Pls_Bit0:

   "Int.Pls = Int.Bit k bit.B0 \<longleftrightarrow> Int.Pls = k"

   unfolding Pls_def Bit_def bit.cases by presburger

 lemma eq_Pls_Bit1:

   "Int.Pls = Int.Bit k bit.B1 \<longleftrightarrow> False"

   unfolding Pls_def Bit_def bit.cases by presburger

 lemma eq_Min_Pls:

   "Int.Min = Int.Pls \<longleftrightarrow> False"

   unfolding Pls_def Int.Min_def by presburger

 lemma eq_Min_Min:

   "Int.Min = Int.Min \<longleftrightarrow> True" by presburger

 lemma eq_Min_Bit0:

   "Int.Min = Int.Bit k bit.B0 \<longleftrightarrow> False"

   unfolding Int.Min_def Bit_def bit.cases by presburger

 lemma eq_Min_Bit1:

   "Int.Min = Int.Bit k bit.B1 \<longleftrightarrow> Int.Min = k"

   unfolding Int.Min_def Bit_def bit.cases by presburger

 lemma eq_Bit0_Pls:

   "Int.Bit k bit.B0 = Int.Pls \<longleftrightarrow> Int.Pls = k"

   unfolding Pls_def Bit_def bit.cases by presburger

 lemma eq_Bit1_Pls:

   "Int.Bit k bit.B1 = Int.Pls \<longleftrightarrow> False"

   unfolding Pls_def Bit_def bit.cases  by presburger

 lemma eq_Bit0_Min:

   "Int.Bit k bit.B0 = Int.Min \<longleftrightarrow> False"

   unfolding Int.Min_def Bit_def bit.cases  by presburger

 lemma eq_Bit1_Min:

   "(Int.Bit k bit.B1) = Int.Min \<longleftrightarrow> Int.Min = k"

   unfolding Int.Min_def Bit_def bit.cases  by presburger

 lemma eq_Bit_Bit:

   "Int.Bit k1 v1 = Int.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:

   "Int.Pls \<le> Int.Pls \<longleftrightarrow> True" by rule+

 lemma less_eq_Pls_Min:

   "Int.Pls \<le> Int.Min \<longleftrightarrow> False"

   unfolding Pls_def Int.Min_def by presburger

 lemma less_eq_Pls_Bit:

   "Int.Pls \<le> Int.Bit k v \<longleftrightarrow> Int.Pls \<le> k"

   unfolding Pls_def Bit_def by (cases v) auto

 lemma less_eq_Min_Pls:

   "Int.Min \<le> Int.Pls \<longleftrightarrow> True"

   unfolding Pls_def Int.Min_def by presburger

 lemma less_eq_Min_Min:

   "Int.Min \<le> Int.Min \<longleftrightarrow> True" by rule+

 lemma less_eq_Min_Bit0:

   "Int.Min \<le> Int.Bit k bit.B0 \<longleftrightarrow> Int.Min < k"

   unfolding Int.Min_def Bit_def by auto

 lemma less_eq_Min_Bit1:

   "Int.Min \<le> Int.Bit k bit.B1 \<longleftrightarrow> Int.Min \<le> k"

   unfolding Int.Min_def Bit_def by auto

 lemma less_eq_Bit0_Pls:

   "Int.Bit k bit.B0 \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"

   unfolding Pls_def Bit_def by simp

 lemma less_eq_Bit1_Pls:

   "Int.Bit k bit.B1 \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"

   unfolding Pls_def Bit_def by auto

 lemma less_eq_Bit_Min:

   "Int.Bit k v \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"

   unfolding Int.Min_def Bit_def by (cases v) auto

 lemma less_eq_Bit0_Bit:

   "Int.Bit k1 bit.B0 \<le> Int.Bit k2 v \<longleftrightarrow> k1 \<le> k2"

   unfolding Bit_def bit.cases by (cases v) auto

 lemma less_eq_Bit_Bit1:

   "Int.Bit k1 v \<le> Int.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"

   unfolding Bit_def bit.cases by (cases v) auto

 lemma less_eq_Bit1_Bit0:

   "Int.Bit k1 bit.B1 \<le> Int.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:

   "Int.Pls < Int.Pls \<longleftrightarrow> False" by simp 

 lemma less_Pls_Min:

   "Int.Pls < Int.Min \<longleftrightarrow> False"

   unfolding Pls_def Int.Min_def  by presburger 

 lemma less_Pls_Bit0:

   "Int.Pls < Int.Bit k bit.B0 \<longleftrightarrow> Int.Pls < k"

   unfolding Pls_def Bit_def by auto

 lemma less_Pls_Bit1:

   "Int.Pls < Int.Bit k bit.B1 \<longleftrightarrow> Int.Pls \<le> k"

   unfolding Pls_def Bit_def by auto

 lemma less_Min_Pls:

   "Int.Min < Int.Pls \<longleftrightarrow> True"

   unfolding Pls_def Int.Min_def by presburger 

 lemma less_Min_Min:

   "Int.Min < Int.Min \<longleftrightarrow> False"  by simp

 lemma less_Min_Bit:

   "Int.Min < Int.Bit k v \<longleftrightarrow> Int.Min < k"

   unfolding Int.Min_def Bit_def by (auto split: bit.split)

 lemma less_Bit_Pls:

   "Int.Bit k v < Int.Pls \<longleftrightarrow> k < Int.Pls"

   unfolding Pls_def Bit_def by (auto split: bit.split)

 lemma less_Bit0_Min:

   "Int.Bit k bit.B0 < Int.Min \<longleftrightarrow> k \<le> Int.Min"

   unfolding Int.Min_def Bit_def by auto

 lemma less_Bit1_Min:

   "Int.Bit k bit.B1 < Int.Min \<longleftrightarrow> k < Int.Min"

   unfolding Int.Min_def Bit_def by auto

 lemma less_Bit_Bit0:

   "Int.Bit k1 v < Int.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"

   unfolding Bit_def by (auto split: bit.split)

 lemma less_Bit1_Bit:

   "Int.Bit k1 bit.B1 < Int.Bit k2 v \<longleftrightarrow> k1 < k2"

   unfolding Bit_def by (auto split: bit.split)

 lemma less_Bit0_Bit1:

   "Int.Bit k1 bit.B0 < Int.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

 context ring_1

 begin

 lemma of_int_num [code func]:

   "of_int k = (if k = 0 then 0 else if k < 0 then

      - of_int (- k) else let

        (l, m) = divAlg (k, 2);

        l' = of_int l

      in if m = 0 then l' + l' else l' + l' + 1)"

 proof -

   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 

     of_int k = of_int (k div 2 * 2 + 1)"

   proof -

     assume "k mod 2 \<noteq> 0"

     then have "k mod 2 = 1" by arith

     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp

     ultimately show ?thesis by auto

   qed

   have aux2: "\<And>x. of_int 2 * x = x + x"

   proof -

     fix x

     have int2: "(2::int) = 1 + 1" by arith

     show "of_int 2 * x = x + x"

     unfolding int2 of_int_add left_distrib by simp

   qed

   have aux3: "\<And>x. x * of_int 2 = x + x"

   proof -

     fix x

     have int2: "(2::int) = 1 + 1" by arith

     show "x * of_int 2 = x + x" 

     unfolding int2 of_int_add right_distrib by simp

   qed

   from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3)

 qed

 end

 end