1(* Title: HOL/Presburger.thy 2 Author: Amine Chaieb, TU Muenchen 3*) 4 5section \<open>Decision Procedure for Presburger Arithmetic\<close> 6 7theory Presburger 8imports Groebner_Basis Set_Interval 9keywords "try0" :: diag 10begin 11 12ML_file \<open>Tools/Qelim/qelim.ML\<close> 13ML_file \<open>Tools/Qelim/cooper_procedure.ML\<close> 14 15subsection\<open>The \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Properties\<close> 16 17lemma minf: 18 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 19 \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)" 20 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 21 \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)" 22 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False" 23 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True" 24 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True" 25 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True" 26 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False" 27 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False" 28 "\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)" 29 "\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 30 "\<exists>z.\<forall>x<z. F = F" 31 by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastforce)+) simp_all 32 33lemma pinf: 34 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 35 \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)" 36 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 37 \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)" 38 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False" 39 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True" 40 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False" 41 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False" 42 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True" 43 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True" 44 "\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)" 45 "\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 46 "\<exists>z.\<forall>x>z. F = F" 47 by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastforce)+) simp_all 48 49lemma inf_period: 50 "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 51 \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))" 52 "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 53 \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))" 54 "(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)" 55 "(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)" 56 "\<forall>x k. F = F" 57apply (auto elim!: dvdE simp add: algebra_simps) 58unfolding mult.assoc [symmetric] distrib_right [symmetric] left_diff_distrib [symmetric] 59unfolding dvd_def mult.commute [of d] 60by auto 61 62subsection\<open>The A and B sets\<close> 63lemma bset: 64 "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ; 65 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 66 \<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))" 67 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ; 68 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 69 \<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))" 70 "\<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))" 71 "\<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))" 72 "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))" 73 "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))" 74 "\<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))" 75 "\<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))" 76 "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))" 77 "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))" 78 "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F" 79proof (blast, blast) 80 assume dp: "D > 0" and tB: "t - 1\<in> B" 81 show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 82 apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) 83 apply algebra using dp tB by simp_all 84next 85 assume dp: "D > 0" and tB: "t \<in> B" 86 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))" 87 apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 88 apply algebra 89 using dp tB by simp_all 90next 91 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 92next 93 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 94next 95 assume dp: "D > 0" and tB:"t \<in> B" 96 {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" 97 hence "x -t \<le> D" and "1 \<le> x - t" by simp+ 98 hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto 99 hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps) 100 with nob tB have "False" by simp} 101 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 102next 103 assume dp: "D > 0" and tB:"t - 1\<in> B" 104 {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" 105 hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+ 106 hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto 107 hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps) 108 with nob tB have "False" by simp} 109 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 110next 111 assume d: "d dvd D" 112 {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra} 113 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 114next 115 assume d: "d dvd D" 116 {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t" 117 by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)} 118 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 119qed blast 120 121lemma aset: 122 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 123 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 124 \<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))" 125 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 126 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 127 \<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))" 128 "\<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))" 129 "\<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))" 130 "\<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))" 131 "\<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))" 132 "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))" 133 "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))" 134 "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))" 135 "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))" 136 "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F" 137proof (blast, blast) 138 assume dp: "D > 0" and tA: "t + 1 \<in> A" 139 show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 140 apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) 141 using dp tA by simp_all 142next 143 assume dp: "D > 0" and tA: "t \<in> A" 144 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))" 145 apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 146 using dp tA by simp_all 147next 148 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 149next 150 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 151next 152 assume dp: "D > 0" and tA:"t \<in> A" 153 {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" 154 hence "t - x \<le> D" and "1 \<le> t - x" by simp+ 155 hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto 156 hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: algebra_simps) 157 with nob tA have "False" by simp} 158 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 159next 160 assume dp: "D > 0" and tA:"t + 1\<in> A" 161 {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" 162 hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: algebra_simps) 163 hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto 164 hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps) 165 with nob tA have "False" by simp} 166 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 167next 168 assume d: "d dvd D" 169 {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" 170 by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)} 171 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 172next 173 assume d: "d dvd D" 174 {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" 175 by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: algebra_simps)} 176 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 177qed blast 178 179subsection\<open>Cooper's Theorem \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Version\<close> 180 181subsubsection\<open>First some trivial facts about periodic sets or predicates\<close> 182lemma periodic_finite_ex: 183 assumes dpos: "(0::int) < d" and modd: "\<forall>x k. P x = P(x - k*d)" 184 shows "(\<exists>x. P x) = (\<exists>j \<in> {1..d}. P j)" 185 (is "?LHS = ?RHS") 186proof 187 assume ?LHS 188 then obtain x where P: "P x" .. 189 have "x mod d = x - (x div d)*d" by(simp add:mult_div_mod_eq [symmetric] ac_simps eq_diff_eq) 190 hence Pmod: "P x = P(x mod d)" using modd by simp 191 show ?RHS 192 proof (cases) 193 assume "x mod d = 0" 194 hence "P 0" using P Pmod by simp 195 moreover have "P 0 = P(0 - (-1)*d)" using modd by blast 196 ultimately have "P d" by simp 197 moreover have "d \<in> {1..d}" using dpos by simp 198 ultimately show ?RHS .. 199 next 200 assume not0: "x mod d \<noteq> 0" 201 have "P(x mod d)" using dpos P Pmod by simp 202 moreover have "x mod d \<in> {1..d}" 203 proof - 204 from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) 205 moreover from dpos have "x mod d < d" by(rule pos_mod_bound) 206 ultimately show ?thesis using not0 by simp 207 qed 208 ultimately show ?RHS .. 209 qed 210qed auto 211 212subsubsection\<open>The \<open>-\<infinity>\<close> Version\<close> 213 214lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (\<bar>x - z\<bar> + 1) * d < z" 215 by (induct rule: int_gr_induct) (simp_all add: int_distrib) 216 217lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (\<bar>x - z\<bar> + 1) * d" 218 by (induct rule: int_gr_induct) (simp_all add: int_distrib) 219 220lemma decr_mult_lemma: 221 assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k" 222 shows "\<forall>x. P x \<longrightarrow> P(x - k*d)" 223using knneg 224proof (induct rule:int_ge_induct) 225 case base thus ?case by simp 226next 227 case (step i) 228 {fix x 229 have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast 230 also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"] 231 by (simp add: algebra_simps) 232 ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast} 233 thus ?case .. 234qed 235 236lemma minusinfinity: 237 assumes dpos: "0 < d" and 238 P1eqP1: "\<forall>x k. P1 x = P1(x - k*d)" and ePeqP1: "\<exists>z::int. \<forall>x. x < z \<longrightarrow> (P x = P1 x)" 239 shows "(\<exists>x. P1 x) \<longrightarrow> (\<exists>x. P x)" 240proof 241 assume eP1: "\<exists>x. P1 x" 242 then obtain x where P1: "P1 x" .. 243 from ePeqP1 obtain z where P1eqP: "\<forall>x. x < z \<longrightarrow> (P x = P1 x)" .. 244 let ?w = "x - (\<bar>x - z\<bar> + 1) * d" 245 from dpos have w: "?w < z" by(rule decr_lemma) 246 have "P1 x = P1 ?w" using P1eqP1 by blast 247 also have "\<dots> = P(?w)" using w P1eqP by blast 248 finally have "P ?w" using P1 by blast 249 thus "\<exists>x. P x" .. 250qed 251 252lemma cpmi: 253 assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" 254 and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) \<longrightarrow> P (x) \<longrightarrow> P (x - D)" 255 and pd: "\<forall> x k. P' x = P' (x-k*D)" 256 shows "(\<exists>x. P x) = ((\<exists>j \<in> {1..D} . P' j) \<or> (\<exists>j \<in> {1..D}. \<exists> b \<in> B. P (b+j)))" 257 (is "?L = (?R1 \<or> ?R2)") 258proof- 259 {assume "?R2" hence "?L" by blast} 260 moreover 261 {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 262 moreover 263 { fix x 264 assume P: "P x" and H: "\<not> ?R2" 265 {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" 266 hence "\<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. y = b+j)" by auto 267 with nb P have "P (y - D)" by auto } 268 hence "\<forall>x. \<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. P(b+j)) \<longrightarrow> P (x) \<longrightarrow> P (x - D)" by blast 269 with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto 270 from p1 obtain z where z: "\<forall>x. x < z \<longrightarrow> (P x = P' x)" by blast 271 let ?y = "x - (\<bar>x - z\<bar> + 1)*D" 272 have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith 273 from dp have yz: "?y < z" using decr_lemma[OF dp] by simp 274 from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 275 with periodic_finite_ex[OF dp pd] 276 have "?R1" by blast} 277 ultimately show ?thesis by blast 278qed 279 280subsubsection \<open>The \<open>+\<infinity>\<close> Version\<close> 281 282lemma plusinfinity: 283 assumes dpos: "(0::int) < d" and 284 P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" 285 shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" 286proof 287 assume eP1: "\<exists>x. P' x" 288 then obtain x where P1: "P' x" .. 289 from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. 290 let ?w' = "x + (\<bar>x - z\<bar> + 1) * d" 291 let ?w = "x - (- (\<bar>x - z\<bar> + 1)) * d" 292 have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps) 293 from dpos have w: "?w > z" by(simp only: ww' incr_lemma) 294 hence "P' x = P' ?w" using P1eqP1 by blast 295 also have "\<dots> = P(?w)" using w P1eqP by blast 296 finally have "P ?w" using P1 by blast 297 thus "\<exists>x. P x" .. 298qed 299 300lemma incr_mult_lemma: 301 assumes dpos: "(0::int) < d" and plus: "\<forall>x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" 302 shows "\<forall>x. P x \<longrightarrow> P(x + k*d)" 303using knneg 304proof (induct rule:int_ge_induct) 305 case base thus ?case by simp 306next 307 case (step i) 308 {fix x 309 have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast 310 also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] 311 by (simp add:int_distrib ac_simps) 312 ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} 313 thus ?case .. 314qed 315 316lemma cppi: 317 assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" 318 and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) \<longrightarrow> P (x) \<longrightarrow> P (x + D)" 319 and pd: "\<forall> x k. P' x= P' (x-k*D)" 320 shows "(\<exists>x. P x) = ((\<exists>j \<in> {1..D} . P' j) \<or> (\<exists> j \<in> {1..D}. \<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)") 321proof- 322 {assume "?R2" hence "?L" by blast} 323 moreover 324 {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 325 moreover 326 { fix x 327 assume P: "P x" and H: "\<not> ?R2" 328 {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y" 329 hence "\<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> A. y = b - j)" by auto 330 with nb P have "P (y + D)" by auto } 331 hence "\<forall>x. \<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> A. P(b-j)) \<longrightarrow> P (x) \<longrightarrow> P (x + D)" by blast 332 with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto 333 from p1 obtain z where z: "\<forall>x. x > z \<longrightarrow> (P x = P' x)" by blast 334 let ?y = "x + (\<bar>x - z\<bar> + 1)*D" 335 have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith 336 from dp have yz: "?y > z" using incr_lemma[OF dp] by simp 337 from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 338 with periodic_finite_ex[OF dp pd] 339 have "?R1" by blast} 340 ultimately show ?thesis by blast 341qed 342 343lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" 344apply(simp add:atLeastAtMost_def atLeast_def atMost_def) 345apply(fastforce) 346done 347 348theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" 349 apply (rule eq_reflection [symmetric]) 350 apply (rule iffI) 351 defer 352 apply (erule exE) 353 apply (rule_tac x = "l * x" in exI) 354 apply (simp add: dvd_def) 355 apply (rule_tac x = x in exI, simp) 356 apply (erule exE) 357 apply (erule conjE) 358 apply simp 359 apply (erule dvdE) 360 apply (rule_tac x = k in exI) 361 apply simp 362 done 363 364lemma zdvd_mono: 365 fixes k m t :: int 366 assumes "k \<noteq> 0" 367 shows "m dvd t \<equiv> k * m dvd k * t" 368 using assms by simp 369 370lemma uminus_dvd_conv: 371 fixes d t :: int 372 shows "d dvd t \<equiv> - d dvd t" and "d dvd t \<equiv> d dvd - t" 373 by simp_all 374 375text \<open>\bigskip Theorems for transforming predicates on nat to predicates on \<open>int\<close>\<close> 376 377lemma zdiff_int_split: "P (int (x - y)) = 378 ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))" 379 by (cases "y \<le> x") (simp_all add: of_nat_diff) 380 381text \<open> 382 \medskip Specific instances of congruence rules, to prevent 383 simplifier from looping.\<close> 384 385theorem imp_le_cong: 386 "\<lbrakk>x = x'; 0 \<le> x' \<Longrightarrow> P = P'\<rbrakk> \<Longrightarrow> (0 \<le> (x::int) \<longrightarrow> P) = (0 \<le> x' \<longrightarrow> P')" 387 by simp 388 389theorem conj_le_cong: 390 "\<lbrakk>x = x'; 0 \<le> x' \<Longrightarrow> P = P'\<rbrakk> \<Longrightarrow> (0 \<le> (x::int) \<and> P) = (0 \<le> x' \<and> P')" 391 by (simp cong: conj_cong) 392 393ML_file \<open>Tools/Qelim/cooper.ML\<close> 394 395method_setup presburger = \<open> 396 let 397 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () 398 fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () 399 val addN = "add" 400 val delN = "del" 401 val elimN = "elim" 402 val any_keyword = keyword addN || keyword delN || simple_keyword elimN 403 val thms = Scan.repeats (Scan.unless any_keyword Attrib.multi_thm) 404 in 405 Scan.optional (simple_keyword elimN >> K false) true -- 406 Scan.optional (keyword addN |-- thms) [] -- 407 Scan.optional (keyword delN |-- thms) [] >> 408 (fn ((elim, add_ths), del_ths) => fn ctxt => 409 SIMPLE_METHOD' (Cooper.tac elim add_ths del_ths ctxt)) 410 end 411\<close> "Cooper's algorithm for Presburger arithmetic" 412 413declare mod_eq_0_iff_dvd [presburger] 414declare mod_by_Suc_0 [presburger] 415declare mod_0 [presburger] 416declare mod_by_1 [presburger] 417declare mod_self [presburger] 418declare div_by_0 [presburger] 419declare mod_by_0 [presburger] 420declare mod_div_trivial [presburger] 421declare mult_div_mod_eq [presburger] 422declare div_mult_mod_eq [presburger] 423declare mod_mult_self1 [presburger] 424declare mod_mult_self2 [presburger] 425declare mod2_Suc_Suc [presburger] 426declare not_mod_2_eq_0_eq_1 [presburger] 427declare nat_zero_less_power_iff [presburger] 428 429lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 430lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 431lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 432lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 433lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger 434 435context semiring_parity 436begin 437 438declare even_mult_iff [presburger] 439 440declare even_power [presburger] 441 442lemma [presburger]: 443 "even (a + b) \<longleftrightarrow> even a \<and> even b \<or> odd a \<and> odd b" 444 by auto 445 446end 447 448context ring_parity 449begin 450 451declare even_minus [presburger] 452 453end 454 455context linordered_idom 456begin 457 458declare zero_le_power_eq [presburger] 459 460declare zero_less_power_eq [presburger] 461 462declare power_less_zero_eq [presburger] 463 464declare power_le_zero_eq [presburger] 465 466end 467 468declare even_Suc [presburger] 469 470lemma [presburger]: 471 "Suc n div Suc (Suc 0) = n div Suc (Suc 0) \<longleftrightarrow> even n" 472 by presburger 473 474declare even_diff_nat [presburger] 475 476lemma [presburger]: 477 fixes k :: int 478 shows "(k + 1) div 2 = k div 2 \<longleftrightarrow> even k" 479 by presburger 480 481lemma [presburger]: 482 fixes k :: int 483 shows "(k + 1) div 2 = k div 2 + 1 \<longleftrightarrow> odd k" 484 by presburger 485 486lemma [presburger]: 487 "even n \<longleftrightarrow> even (int n)" 488 by simp 489 490 491subsection \<open>Nice facts about division by \<^term>\<open>4\<close>\<close> 492 493lemma even_even_mod_4_iff: 494 "even (n::nat) \<longleftrightarrow> even (n mod 4)" 495 by presburger 496 497lemma odd_mod_4_div_2: 498 "n mod 4 = (3::nat) \<Longrightarrow> odd ((n - Suc 0) div 2)" 499 by presburger 500 501lemma even_mod_4_div_2: 502 "n mod 4 = Suc 0 \<Longrightarrow> even ((n - Suc 0) div 2)" 503 by presburger 504 505 506subsection \<open>Try0\<close> 507 508ML_file \<open>Tools/try0.ML\<close> 509 510end 511