# Theory Presburger

Up to index of Isabelle/HOL

theory Presburger
imports Groebner_Basis Set_Interval
`(* Title:      HOL/Presburger.thy   Author:     Amine Chaieb, TU Muenchen*)header {* Decision Procedure for Presburger Arithmetic *}theory Presburgerimports Groebner_Basis Set_IntervalbeginML_file "Tools/Qelim/qelim.ML"ML_file "Tools/Qelim/cooper_procedure.ML"subsection{* The @{text "-∞"} and @{text "+∞"} Properties *}lemma minf:  "[|∃(z ::'a::linorder).∀x<z. P x = P' x; ∃z.∀x<z. Q x = Q' x|]      ==> ∃z.∀x<z. (P x ∧ Q x) = (P' x ∧ Q' x)"  "[|∃(z ::'a::linorder).∀x<z. P x = P' x; ∃z.∀x<z. Q x = Q' x|]      ==> ∃z.∀x<z. (P x ∨ Q x) = (P' x ∨ 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) = True"  "∃(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.∀(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"  by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastforce)+) simp_alllemma pinf:  "[|∃(z ::'a::linorder).∀x>z. P x = P' x; ∃z.∀x>z. Q x = Q' x|]      ==> ∃z.∀x>z. (P x ∧ Q x) = (P' x ∧ Q' x)"  "[|∃(z ::'a::linorder).∀x>z. P x = P' x; ∃z.∀x>z. Q x = Q' x|]      ==> ∃z.∀x>z. (P x ∨ Q x) = (P' x ∨ 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"  by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastforce)+) simp_alllemma inf_period:  "[|∀x k. P x = P (x - k*D); ∀x k. Q x = Q (x - k*D)|]     ==> ∀x k. (P x ∧ Q x) = (P (x - k*D) ∧ Q (x - k*D))"  "[|∀x k. P x = P (x - k*D); ∀x k. Q x = Q (x - k*D)|]     ==> ∀x k. (P x ∨ Q x) = (P (x - k*D) ∨ 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 autosubsection{* The A and B sets *}lemma bset:  "[|∀x.(∀j ∈ {1 .. D}. ∀b∈B. x ≠ b + j)--> P x --> P(x - D) ;     ∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> Q x --> Q(x - D)|] ==>   ∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j) --> (P x ∧ Q x) --> (P(x - D) ∧ Q (x - D))"  "[|∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> P x --> P(x - D) ;     ∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> Q x --> Q(x - D)|] ==>   ∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> (P x ∨ Q x) --> (P(x - D) ∨ 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_allnext  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_allnext  assume dp: "D > 0" thus "(∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> (x < t) --> (x - D < t))" by arithnext  assume dp: "D > 0" thus "∀x.(∀j∈{1 .. D}. ∀b∈B. x ≠ b + j)--> (x ≤ t) --> (x - D ≤ t)" by arithnext  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 blastnext  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 blastnext  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 simpnext  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 autoqed blastlemma aset:  "[|∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> P x --> P(x + D) ;     ∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> Q x --> Q(x + D)|] ==>   ∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j) --> (P x ∧ Q x) --> (P(x + D) ∧ Q (x + D))"  "[|∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> P x --> P(x + D) ;     ∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> Q x --> Q(x + D)|] ==>   ∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> (P x ∨ Q x) --> (P(x + D) ∨ 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_allnext  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_allnext  assume dp: "D > 0" thus "(∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> (x > t) --> (x + D > t))" by arithnext  assume dp: "D > 0" thus "∀x.(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> (x ≥ t) --> (x + D ≥ t)" by arithnext  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 blastnext  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 blastnext  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: algebra_simps)}  thus "∀(x::int).(∀j∈{1 .. D}. ∀b∈A. x ≠ b - j)--> (d dvd x+t) --> (d dvd (x + D) + t)" by simpnext  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∈A. x ≠ b - j)--> (¬d dvd x+t) --> (¬d dvd (x + D) + t)" by autoqed blastsubsection{* Cooper's Theorem @{text "-∞"} and @{text "+∞"} 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    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 ..  qedqed autosubsubsection{* The @{text "-∞"} Version*}lemma decr_lemma: "0 < (d::int) ==> x - (abs(x-z)+1) * d < z"by(induct rule: int_gr_induct,simp_all add:int_distrib)lemma incr_lemma: "0 < (d::int) ==> z < x + (abs(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 "ALL x. P x --> P(x - k*d)"using knnegproof (induct rule:int_ge_induct)  case base thus ?case by simpnext  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 ..qedlemma  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 --> (P x = P1 x)"  shows "(EX x. P1 x) --> (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 --> (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 "… = P(?w)" using w P1eqP by blast  finally have "P ?w" using P1 by blast  thus "EX x. P x" ..qedlemma 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 "~(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: " ∀x. P x --> P (x - D)" by auto   from p1 obtain z where z: "ALL 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 blastqedsubsubsection {* The @{text "+∞"} Version*}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: "EX 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 + (abs(x-z)+1) * d"  let ?w = "x - (-(abs(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 "EX x. P x" ..qedlemma incr_mult_lemma:  assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x --> P(x + d)" and knneg: "0 <= k"  shows "ALL x. P x --> P(x + k*d)"using knnegproof (induct rule:int_ge_induct)  case base thus ?case by simpnext  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 add_ac)    ultimately have "P x --> P(x + (i + 1) * d)" by blast}  thus ?case ..qedlemma 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 "~(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: " ∀x. P x --> P (x + D)" by auto   from p1 obtain z where z: "ALL 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 blastqedlemma 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)donetheorem unity_coeff_ex: "(∃(x::'a::{semiring_0,Rings.dvd}). P (l * x)) ≡ (∃x. l dvd (x + 0) ∧ 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 simp  apply (erule dvdE)  apply (rule_tac x = k in exI)  apply simp  donelemma zdvd_mono: assumes not0: "(k::int) ≠ 0"shows "((m::int) dvd t) ≡ (k*m dvd k*t)"   using not0 by (simp add: dvd_def)lemma uminus_dvd_conv: "(d dvd (t::int)) ≡ (-d dvd t)" "(d dvd (t::int)) ≡ (d dvd -t)"  by simp_alltext {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}lemma zdiff_int_split: "P (int (x - y)) =  ((y ≤ x --> P (int x - int y)) ∧ (x < y --> P 0))"  by (cases "y ≤ x") (simp_all add: zdiff_int)text {*  \medskip Specific instances of congruence rules, to prevent  simplifier from looping. *}theorem imp_le_cong:  "[|x = x'; 0 ≤ x' ==> P = P'|] ==> (0 ≤ (x::int) --> P) = (0 ≤ x' --> P')"  by simptheorem conj_le_cong:  "[|x = x'; 0 ≤ x' ==> P = P'|] ==> (0 ≤ (x::int) ∧ P) = (0 ≤ x' ∧ P')"  by (simp cong: conj_cong)ML_file "Tools/Qelim/cooper.ML"setup Cooper.setupmethod_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    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*} "Cooper's algorithm for Presburger arithmetic"declare dvd_eq_mod_eq_0[symmetric, presburger]declare mod_1[presburger] declare mod_0[presburger]declare mod_by_1[presburger]declare mod_self[presburger]declare mod_by_0[presburger]declare 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 div_mod_equality[presburger]declare div_mod_equality2[presburger]declare mod2_Suc_Suc[presburger]lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"by simp_alllemma [presburger, algebra]: "m mod 2 = (1::nat) <-> ¬ 2 dvd m " by presburgerlemma [presburger, algebra]: "m mod 2 = Suc 0 <-> ¬ 2 dvd m " by presburgerlemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) <-> ¬ 2 dvd m " by presburgerlemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 <-> ¬ 2 dvd m " by presburgerlemma [presburger, algebra]: "m mod 2 = (1::int) <-> ¬ 2 dvd m " by presburgerend`