(* Title: HOL/Presburger.thy Author: Amine Chaieb, TU Muenchen *) header {* Decision Procedure for Presburger Arithmetic *} theory Presburger imports Groebner_Basis Set_Interval begin ML_file "Tools/Qelim/qelim.ML" ML_file "Tools/Qelim/cooper_procedure.ML" subsection{* The @{text "-\"} and @{text "+\"} Properties *} lemma minf: "\\(z ::'a::linorder).\xz.\x \ \z.\x Q x) = (P' x \ Q' x)" "\\(z ::'a::linorder).\xz.\x \ \z.\x Q x) = (P' x \ 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\(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_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) 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 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 "Tools/Qelim/cooper.ML" setup Cooper.setup 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 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_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