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products/sources/formale Sprachen/Isabelle/HOL/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 27 kB

Quelle Presburger.thy Sprache: Isabelle

(* Title:      HOL/Presburger.thy
   Author:     Amine Chaieb, TU Muenchen
*)

section \<open>Decision Procedure for Presburger Arithmetic\<close>

theory Presburger
imports Groebner_Basis Set_Interval
begin

ML_file \<open>Tools/Qelim/qelim.ML\<close>
ML_file \<open>Tools/Qelim/cooper_procedure.ML\<close>

subsection\<open>The \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Properties\<close>

lemma minf:
  "\\(z ::'a::linorder).\xz.\x
     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
  "\\(z ::'a::linorder).\xz.\x
     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
  "\(z ::'a::{linorder}).\x
  "\(z ::'a::{linorder}).\x t) = True"
  "\(z ::'a::{linorder}).\x
  "\(z ::'a::{linorder}).\x t) = True"
  "\(z ::'a::{linorder}).\x t) = False"
  "\(z ::'a::{linorder}).\x t) = False"
  "\z.\(x::'b::{linorder,plus,Rings.dvd})
  "\z.\(x::'b::{linorder,plus,Rings.dvd}) d dvd x + s) = (\ d dvd x + s)"
  "\z.\x
proof safe
  fix z1 z2
  assume "\xx
  then have "\x < min z1 z2. (P x \ Q x) = (P' x \ Q' x)"
    by simp
  then show "\z. \x Q x) = (P' x \ Q' x)"
    by blast
next
  fix z1 z2
  assume "\xx
  then have "\x < min z1 z2. (P x \ Q x) = (P' x \ Q' x)"
    by simp
  then show "\z. \x Q x) = (P' x \ Q' x)"
    by blast
next
  have "\x t"
    by fastforce
  then show "\z. \x t) = True"
    by auto
next
  have "\x t < x"
    by fastforce
  then show "\z. \x
    by auto
next
  have "\x t \ x"
    by fastforce
  then show "\z. \x x) = False"
    by auto
qed auto

lemma pinf:
  "\\(z ::'a::linorder).\x>z. P x = P' x; \z.\x>z. Q x = Q' x\
     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
  "\\(z ::'a::linorder).\x>z. P x = P' x; \z.\x>z. Q x = Q' x\
     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
  "\(z ::'a::{linorder}).\x>z.(x = t) = False"
  "\(z ::'a::{linorder}).\x>z.(x \ t) = True"
  "\(z ::'a::{linorder}).\x>z.(x < t) = False"
  "\(z ::'a::{linorder}).\x>z.(x \ t) = False"
  "\(z ::'a::{linorder}).\x>z.(x > t) = True"
  "\(z ::'a::{linorder}).\x>z.(x \ t) = True"
  "\z.\(x::'b::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)"
  "\z.\(x::'b::{linorder,plus,Rings.dvd})>z. (\ d dvd x + s) = (\ d dvd x + s)"
  "\z.\x>z. F = F"
proof safe
  fix z1 z2
  assume "\x>z1. P x = P' x" and "\x>z2. Q x = Q' x"
  then have "\x > max z1 z2. (P x \ Q x) = (P' x \ Q' x)"
    by simp
  then show "\z. \x>z. (P x \ Q x) = (P' x \ Q' x)"
    by blast
next
  fix z1 z2
  assume "\x>z1. P x = P' x" and "\x>z2. Q x = Q' x"
  then have "\x > max z1 z2. (P x \ Q x) = (P' x \ Q' x)"
    by simp
  then show "\z. \x>z. (P x \ Q x) = (P' x \ Q' x)"
    by blast
next
  have "\x>t. \ x < t"
    by fastforce
  then show "\z. \x>z. x < t = False"
    by blast
next
  have "\x>t. \ x \ t"
    by fastforce
  then show "\z. \x>z. x \ t = False"
    by blast
next
  have "\x>t. t \ x"
    by fastforce
  then show "\z. \x>z. t \ x = True"
    by blast
qed auto

lemma inf_period:
  "\\x k. P x = P (x - k*D); \x k. Q x = Q (x - k*D)\
    \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
  "\\x k. P x = P (x - k*D); \x k. Q x = Q (x - k*D)\
    \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
  "(d::'a::{comm_ring,Rings.dvd}) dvd D \ \x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
  "(d::'a::{comm_ring,Rings.dvd}) dvd D \ \x k. (\d dvd x + t) = (\d dvd (x - k*D) + t)"
  "\x k. F = F"
apply (auto elim!: dvdE simp add: algebra_simps)
unfolding mult.assoc [symmetric] distrib_right [symmetric] left_diff_distrib [symmetric]
unfolding dvd_def mult.commute [of d]
by auto

subsection\<open>The A and B sets\<close>
lemma bset:
  "\\x.(\j \ {1 .. D}. \b\B. x \ b + j)\ P x \ 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))"
  "\\x.(\j\{1 .. D}. \b\B. x \ b + j)\ P x \ 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))"
  "\D>0; t - 1\ B\ \ (\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x = t) \ (x - D = t))"
  "\D>0 ; t \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))"
  "D>0 \ (\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x < t) \ (x - D < t))"
  "D>0 \ (\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))"
  "\D>0 ; t \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x > t) \ (x - D > t))"
  "\D>0 ; t - 1 \ B\ \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))"
  "d dvd D \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (d dvd x+t) \ (d dvd (x - D) + t))"
  "d dvd D \(\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (\d dvd x+t) \ (\ d dvd (x - D) + t))"
  "\x.(\j\{1 .. D}. \b\B. x \ b + j) \ F \ F"
proof (blast, blast)
  assume dp: "D > 0" and tB: "t - 1\ B"
  show "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x = t) \ (x - D = t))"
    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
    apply algebra using dp tB by simp_all
next
  assume dp: "D > 0" and tB: "t \ B"
  show "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t))"
    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
    apply algebra
    using dp tB by simp_all
next
  assume dp: "D > 0" thus "(\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x < t) \ (x - D < t))" by arith
next
  assume dp: "D > 0" thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ t)" by arith
next
  assume dp: "D > 0" and tB:"t \ B"
  {fix x assume nob: "\j\{1 .. D}. \b\B. x \ b + j" and g: "x > t" and ng: "\ (x - D) > t"
    hence "x -t \ D" and "1 \ x - t" by simp+
      hence "\j \ {1 .. D}. x - t = j" by auto
      hence "\j \ {1 .. D}. x = t + j" by (simp add: algebra_simps)
      with nob tB have "False" by simp}
  thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x > t) \ (x - D > t)" by blast
next
  assume dp: "D > 0" and tB:"t - 1\ B"
  {fix x assume nob: "\j\{1 .. D}. \b\B. x \ b + j" and g: "x \ t" and ng: "\ (x - D) \ t"
    hence "x - (t - 1) \ D" and "1 \ x - (t - 1)" by simp+
      hence "\j \ {1 .. D}. x - (t - 1) = j" by auto
      hence "\j \ {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps)
      with nob tB have "False" by simp}
  thus "\x.(\j\{1 .. D}. \b\B. x \ b + j)\ (x \ t) \ (x - D \ 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 algebra}
  thus "\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (d dvd x+t) \ (d dvd (x - D) + t)" by simp
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,erule_tac x= "ka + k" in allE,simp add: algebra_simps)}
  thus "\(x::int).(\j\{1 .. D}. \b\B. x \ b + j)\ (\d dvd x+t) \ (\d dvd (x - D) + t)" by auto
qed blast

lemma aset:
  "\\x.(\j\{1 .. D}. \b\A. x \ b - j)\ P x \ 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))"
  "\\x.(\j\{1 .. D}. \b\A. x \ b - j)\ P x \ 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))"
  "\D>0; t + 1\ A\ \ (\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x = t) \ (x + D = t))"
  "\D>0 ; t \ A\ \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))"
  "\D>0; t\ A\ \(\(x::int). (\j\{1 .. D}. \b\A. x \ b - j)\ (x < t) \ (x + D < t))"
  "\D>0; t + 1 \ A\ \ (\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))"
  "D>0 \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x > t) \ (x + D > t))"
  "D>0 \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t))"
  "d dvd D \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (d dvd x+t) \ (d dvd (x + D) + t))"
  "d dvd D \(\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (\d dvd x+t) \ (\ d dvd (x + D) + t))"
  "\x.(\j\{1 .. D}. \b\A. x \ b - j) \ F \ F"
proof (blast, blast)
  assume dp: "D > 0" and tA: "t + 1 \ A"
  show "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x = t) \ (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 \ A"
  show "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ 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 "(\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x > t) \ (x + D > t))" by arith
next
  assume dp: "D > 0" thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t)" by arith
next
  assume dp: "D > 0" and tA:"t \ A"
  {fix x assume nob: "\j\{1 .. D}. \b\A. x \ b - j" and g: "x < t" and ng: "\ (x + D) < t"
    hence "t - x \ D" and "1 \ t - x" by simp+
      hence "\j \ {1 .. D}. t - x = j" by auto
      hence "\j \ {1 .. D}. x = t - j" by (auto simp add: algebra_simps)
      with nob tA have "False" by simp}
  thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x < t) \ (x + D < t)" by blast
next
  assume dp: "D > 0" and tA:"t + 1\ A"
  {fix x assume nob: "\j\{1 .. D}. \b\A. x \ b - j" and g: "x \ t" and ng: "\ (x + D) \ t"
    hence "(t + 1) - x \ D" and "1 \ (t + 1) - x" by (simp_all add: algebra_simps)
      hence "\j \ {1 .. D}. (t + 1) - x = j" by auto
      hence "\j \ {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps)
      with nob tA have "False" by simp}
  thus "\x.(\j\{1 .. D}. \b\A. x \ b - j)\ (x \ t) \ (x + D \ t)" by blast
next
  assume d: "d dvd D"
  have "\x. d dvd x + t \ d dvd x + D + t"
  proof -
    fix x
    assume H: "d dvd x + t"
    then obtain ka where "x + t = d * ka"
      unfolding dvd_def by blast
    moreover from d obtain k where *:"D = d * k"
      unfolding dvd_def by blast
    ultimately have "x + d * k + t = d * (ka + k)"
      by (simp add: algebra_simps)
    then show "d dvd (x + D) + t"
      using * unfolding dvd_def by blast
  qed
  thus "\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (d dvd x+t) \ (d dvd (x + D) + t)" by simp
next
  assume d: "d dvd D"
  {fix x assume H: "\(d dvd x + t)" with d have "\d dvd (x + D) + t"
      using dvd_add_left_iff[OF d, of "x+t"] by (simp add: algebra_simps)}
  thus "\(x::int).(\j\{1 .. D}. \b\A. x \ b - j)\ (\d dvd x+t) \ (\d dvd (x + D) + t)" by auto
qed blast

subsection\<open>Cooper's Theorem \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Version\<close>

subsubsection\<open>First some trivial facts about periodic sets or predicates\<close>
lemma periodic_finite_ex:
  assumes dpos: "(0::int) < d" and modd: "\x k. P x = P(x - k*d)"
  shows "(\x. P x) = (\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:mult_div_mod_eq [symmetric] ac_simps 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
    ultimately show ?RHS ..
  next
    assume not0: "x mod d \ 0"
    have "P(x mod d)" using dpos P Pmod by simp
    moreover have "x mod d \ {1..d}"
    proof -
      from dpos have "0 \ 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
    qed
    ultimately show ?RHS ..
  qed
qed auto

subsubsection\<open>The \<open>-\<infinity>\<close> Version\<close>

lemma decr_lemma: "0 < (d::int) \ x - (\x - z\ + 1) * d < z"
  by (induct rule: int_gr_induct) (simp_all add: int_distrib)

lemma incr_lemma: "0 < (d::int) \ z < x + (\x - z\ + 1) * d"
  by (induct rule: int_gr_induct) (simp_all add: int_distrib)

lemma decr_mult_lemma:
  assumes dpos: "(0::int) < d" and minus: "\x. P x \ P(x - d)" and knneg: "0 <= k"
  shows "\x. P x \ 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 \ P (x - i * d)" using step.hyps by blast
    also have "\ \ P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
      by (simp add: algebra_simps)
    ultimately have "P x \ P(x - (i + 1) * d)" by blast}
  thus ?case ..
qed

lemma  minusinfinity:
  assumes dpos: "0 < d" and
    P1eqP1: "\x k. P1 x = P1(x - k*d)" and ePeqP1: "\z::int. \x. x < z \ (P x = P1 x)"
  shows "(\x. P1 x) \ (\x. P x)"
proof
  assume eP1: "\x. P1 x"
  then obtain x where P1: "P1 x" ..
  from ePeqP1 obtain z where P1eqP: "\x. x < z \ (P x = P1 x)" ..
  let ?w = "x - (\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 "\ = P(?w)" using w P1eqP by blast
  finally have "P ?w" using P1 by blast
  thus "\x. P x" ..
qed

lemma cpmi:
  assumes dp: "0 < D" and p1:"\z. \ x< z. P x = P' x"
  and nb:"\x.(\ j\ {1..D}. \(b::int) \ B. x \ b+j) \ P (x) \ P (x - D)"
  and pd: "\ x k. P' x = P' (x-k*D)"
  shows "(\x. P x) = ((\j \ {1..D} . P' j) \ (\j \ {1..D}. \ b \ B. P (b+j)))"
         (is "?L = (?R1 \ ?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: "\ ?R2"
   {fix y assume "\ (\j\{1..D}. \b\B. P (b + j))" and P: "P y"
     hence "\(\(j::int) \ {1..D}. \(b::int) \ B. y = b+j)" by auto
     with nb P  have "P (y - D)" by auto }
   hence "\x. \(\(j::int) \ {1..D}. \(b::int) \ B. P(b+j)) \ P (x) \ P (x - D)" by blast
   with H P have th: " \x. P x \ P (x - D)" by auto
   from p1 obtain z where z: "\x. x < z \ (P x = P' x)" by blast
   let ?y = "x - (\x - z\ + 1)*D"
   have zp: "0 <= (\x - z\ + 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 \<open>The \<open>+\<infinity>\<close> Version\<close>

lemma  plusinfinity:
  assumes dpos: "(0::int) < d" and
    P1eqP1: "\x k. P' x = P'(x - k*d)" and ePeqP1: "\ z. \ x>z. P x = P' x"
  shows "(\ x. P' x) \ (\ x. P x)"
proof
  assume eP1: "\x. P' x"
  then obtain x where P1: "P' x" ..
  from ePeqP1 obtain z where P1eqP: "\x>z. P x = P' x" ..
  let ?w' = "x + (\x - z\ + 1) * d"
  let ?w = "x - (- (\x - z\ + 1)) * d"
  have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps)
  from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
  hence "P' x = P' ?w" using P1eqP1 by blast
  also have "\ = P(?w)" using w P1eqP by blast
  finally have "P ?w" using P1 by blast
  thus "\x. P x" ..
qed

lemma incr_mult_lemma:
  assumes dpos: "(0::int) < d" and plus: "\x::int. P x \ P(x + d)" and knneg: "0 <= k"
  shows "\x. P x \ 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 \ P (x + i * d)" using step.hyps by blast
    also have "\ \ P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
      by (simp add:int_distrib ac_simps)
    ultimately have "P x \ P(x + (i + 1) * d)" by blast}
  thus ?case ..
qed

lemma cppi:
  assumes dp: "0 < D" and p1:"\z. \ x> z. P x = P' x"
  and nb:"\x.(\ j\ {1..D}. \(b::int) \ A. x \ b - j) \ P (x) \ P (x + D)"
  and pd: "\ x k. P' x= P' (x-k*D)"
  shows "(\x. P x) = ((\j \ {1..D} . P' j) \ (\ j \ {1..D}. \ b\ A. P (b - j)))" (is "?L = (?R1 \ ?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: "\ ?R2"
   {fix y assume "\ (\j\{1..D}. \b\A. P (b - j))" and P: "P y"
     hence "\(\(j::int) \ {1..D}. \(b::int) \ A. y = b - j)" by auto
     with nb P  have "P (y + D)" by auto }
   hence "\x. \(\(j::int) \ {1..D}. \(b::int) \ A. P(b-j)) \ P (x) \ P (x + D)" by blast
   with H P have th: " \x. P x \ P (x + D)" by auto
   from p1 obtain z where z: "\x. x > z \ (P x = P' x)" by blast
   let ?y = "x + (\x - z\ + 1)*D"
   have zp: "0 <= (\x - z\ + 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(fastforce)
done

theorem unity_coeff_ex: "(\(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \ (\x. l dvd (x + 0) \ P x)"
  unfolding dvd_def by (rule eq_reflection, rule iffI) auto

lemma zdvd_mono:
  fixes k m t :: int
  assumes "k \ 0"
  shows "m dvd t \ k * m dvd k * t"
  using assms by simp

lemma uminus_dvd_conv:
  fixes d t :: int
  shows "d dvd t \ - d dvd t" and "d dvd t \ d dvd - t"
  by simp_all

text \<open>\bigskip Theorems for transforming predicates on nat to predicates on \<open>int\<close>\<close>

lemma zdiff_int_split: "P (int (x - y)) =
  ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
  by (cases "y \ x") (simp_all add: of_nat_diff)

text \<open>
  \medskip Specific instances of congruence rules, to prevent
  simplifier from looping.\<close>

theorem imp_le_cong:
  "\x = x'; 0 \ x' \ P = P'\ \ (0 \ (x::int) \ P) = (0 \ x' \ P')"
  by simp

theorem conj_le_cong:
  "\x = x'; 0 \ x' \ P = P'\ \ (0 \ (x::int) \ P) = (0 \ x' \ P')"
  by (simp cong: conj_cong)

ML_file \<open>Tools/Qelim/cooper.ML\<close>

method_setup presburger = \<open>
  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.repeats (Scan.unless any_keyword Attrib.multi_thm)
  in
    Scan.optional (simple_keyword elimN >> K false) true --
    Scan.optional (keyword addN |-- thms) [] --
    Scan.optional (keyword delN |-- thms) [] >>
    (fn ((elim, add_ths), del_ths) => fn ctxt =>
      SIMPLE_METHOD' (Cooper.tac elim add_ths del_ths ctxt))
  end
\<close> "Cooper's algorithm for Presburger arithmetic"

declare mod_eq_0_iff_dvd [presburger]
declare mod_by_Suc_0 [presburger]
declare mod_0 [presburger]
declare mod_by_1 [presburger]
declare mod_self [presburger]
declare div_by_0 [presburger]
declare mod_by_0 [presburger]
declare mod_div_trivial [presburger]
declare mult_div_mod_eq [presburger]
declare div_mult_mod_eq [presburger]
declare mod_mult_self1 [presburger]
declare mod_mult_self2 [presburger]
declare mod2_Suc_Suc [presburger]
declare not_mod_2_eq_0_eq_1 [presburger]
declare nat_zero_less_power_iff [presburger]

lemma [presburger, algebra]: "m mod 2 = (1::nat) \ \ 2 dvd m " by presburger
lemma [presburger, algebra]: "m mod 2 = Suc 0 \ \ 2 dvd m " by presburger
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \ \ 2 dvd m " by presburger
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \ \ 2 dvd m " by presburger
lemma [presburger, algebra]: "m mod 2 = (1::int) \ \ 2 dvd m " by presburger

context semiring_parity
begin

declare even_mult_iff [presburger]

declare even_power [presburger]

lemma [presburger]:
  "even (a + b) \ even a \ even b \ odd a \ odd b"
  by auto

end

context ring_parity
begin

declare even_minus [presburger]

end

context linordered_idom
begin

declare zero_le_power_eq [presburger]

declare zero_less_power_eq [presburger]

declare power_less_zero_eq [presburger]

declare power_le_zero_eq [presburger]

end

declare even_Suc [presburger]

lemma [presburger]:
  "Suc n div Suc (Suc 0) = n div Suc (Suc 0) \ even n"
  by presburger

declare even_diff_nat [presburger]

lemma [presburger]:
  fixes k :: int
  shows "(k + 1) div 2 = k div 2 \ even k"
  by presburger

lemma [presburger]:
  fixes k :: int
  shows "(k + 1) div 2 = k div 2 + 1 \ odd k"
  by presburger

lemma [presburger]:
  "even n \ even (int n)"
  by simp


subsection \<open>Nice facts about division by \<^term>\<open>4\<close>\<close>

lemma even_even_mod_4_iff:
  "even (n::nat) \ even (n mod 4)"
  by presburger

lemma odd_mod_4_div_2:
  "n mod 4 = (3::nat) \ odd ((n - Suc 0) div 2)"
  by presburger

lemma even_mod_4_div_2:
  "n mod 4 = Suc 0 \ even ((n - Suc 0) div 2)"
  by presburger

end

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