Up to index of Isabelle/HOL
theory Presburger(* Title: HOL/Presburger.thy
Author: Amine Chaieb, TU Muenchen
*)
header {* Decision Procedure for Presburger Arithmetic *}
theory Presburger
imports Groebner_Basis SetInterval
uses
"Tools/Qelim/qelim.ML"
"Tools/Qelim/cooper_procedure.ML"
("Tools/Qelim/cooper.ML")
begin
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::'a::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)"
"∃z.∀(x::'a::{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_all
lemma 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::'a::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)"
"∃z.∀(x::'a::{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_all
lemma 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] left_distrib [symmetric] left_diff_distrib [symmetric]
unfolding dvd_def mult_commute [of d]
by auto
subsection{* 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_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) ;
∀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_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"
{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 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∈A. x ≠ b - j)--> (¬d dvd x+t) --> (¬d dvd (x + D) + t)" by auto
qed blast
subsection{* 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 ..
qed
qed auto
subsubsection{* 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 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: "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" ..
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 "~(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 blast
qed
subsubsection {* 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" ..
qed
lemma 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 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 add_ac)
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 "~(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 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)"
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
done
lemma 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_all
text {* \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)
lemma number_of1: "(0::int) <= number_of n ==> (0::int) <= number_of (Int.Bit0 n) ∧ (0::int) <= number_of (Int.Bit1 n)"
by simp
lemma number_of2: "(0::int) <= Numeral0" by simp
text {*
\medskip Specific instances of congruence rules, to prevent
simplifier from looping. *}
theorem imp_le_cong: "(0 <= x ==> P = P') ==> (0 <= (x::int) --> P) = (0 <= x --> P')" by simp
theorem conj_le_cong: "(0 <= x ==> P = P') ==> (0 <= (x::int) ∧ P) = (0 <= x ∧ P')"
by (simp cong: conj_cong)
use "Tools/Qelim/cooper.ML"
setup Cooper.setup
method_setup presburger = "Cooper.method" "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 zmod_zero[presburger]
declare zmod_self[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 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"
by simp_all
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
end