author  haftmann 
Mon, 08 Feb 2010 17:12:38 +0100  
changeset 35050  9f841f20dca6 
parent 33318  ddd97d9dfbfb 
child 35216  7641e8d831d2 
permissions  rwrr 
23465  1 
(* Title: HOL/Presburger.thy 
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Author: Amine Chaieb, TU Muenchen 

3 
*) 

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23472  5 
header {* Decision Procedure for Presburger Arithmetic *} 
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23465  7 
theory Presburger 
28402  8 
imports Groebner_Basis SetInterval 
23465  9 
uses 
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"Tools/Qelim/qelim.ML" 
23465  11 
"Tools/Qelim/cooper_data.ML" 
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"Tools/Qelim/generated_cooper.ML" 

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("Tools/Qelim/cooper.ML") 

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("Tools/Qelim/presburger.ML") 

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begin 

16 

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setup CooperData.setup 

18 

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subsection{* The @{text "\<infinity>"} and @{text "+\<infinity>"} Properties *} 

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24404  21 

23465  22 
lemma minf: 
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"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 

24 
\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)" 

25 
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 

26 
\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)" 

27 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False" 

28 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True" 

29 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True" 

30 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True" 

31 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False" 

32 
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False" 

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"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)" 
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"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 
23465  35 
"\<exists>z.\<forall>x<z. F = F" 
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by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all 

37 

38 
lemma pinf: 

39 
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 

40 
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)" 

41 
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 

42 
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)" 

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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False" 

44 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True" 

45 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False" 

46 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False" 

47 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True" 

48 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True" 

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"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)" 
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"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 
23465  51 
"\<exists>z.\<forall>x>z. F = F" 
52 
by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all 

53 

54 
lemma inf_period: 

55 
"\<lbrakk>\<forall>x k. P x = P (x  k*D); \<forall>x k. Q x = Q (x  k*D)\<rbrakk> 

56 
\<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x  k*D) \<and> Q (x  k*D))" 

57 
"\<lbrakk>\<forall>x k. P x = P (x  k*D); \<forall>x k. Q x = Q (x  k*D)\<rbrakk> 

58 
\<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x  k*D) \<or> Q (x  k*D))" 

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"(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x  k*D) + t)" 
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"(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x  k*D) + t)" 
23465  61 
"\<forall>x k. F = F" 
29667  62 
apply (auto elim!: dvdE simp add: algebra_simps) 
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unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric] 
27668  64 
unfolding dvd_def mult_commute [of d] 
65 
by auto 

23465  66 

23472  67 
subsection{* The A and B sets *} 
23465  68 
lemma bset: 
69 
"\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x  D) ; 

70 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x  D)\<rbrakk> \<Longrightarrow> 

71 
\<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))" 

72 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x  D) ; 

73 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x  D)\<rbrakk> \<Longrightarrow> 

74 
\<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))" 

75 
"\<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))" 

76 
"\<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))" 

77 
"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))" 

78 
"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))" 

79 
"\<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))" 

80 
"\<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))" 

81 
"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))" 

82 
"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))" 

83 
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F" 

84 
proof (blast, blast) 

85 
assume dp: "D > 0" and tB: "t  1\<in> B" 

86 
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x  D = t))" 

27668  87 
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t  1"]) 
88 
apply algebra using dp tB by simp_all 

23465  89 
next 
90 
assume dp: "D > 0" and tB: "t \<in> B" 

91 
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))" 

92 
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 

27668  93 
apply algebra 
23465  94 
using dp tB by simp_all 
95 
next 

96 
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 

97 
next 

98 
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 

99 
next 

100 
assume dp: "D > 0" and tB:"t \<in> B" 

101 
{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" 

102 
hence "x t \<le> D" and "1 \<le> x  t" by simp+ 

103 
hence "\<exists>j \<in> {1 .. D}. x  t = j" by auto 

29667  104 
hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps) 
23465  105 
with nob tB have "False" by simp} 
106 
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 

107 
next 

108 
assume dp: "D > 0" and tB:"t  1\<in> B" 

109 
{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" 

110 
hence "x  (t  1) \<le> D" and "1 \<le> x  (t  1)" by simp+ 

111 
hence "\<exists>j \<in> {1 .. D}. x  (t  1) = j" by auto 

29667  112 
hence "\<exists>j \<in> {1 .. D}. x = (t  1) + j" by (simp add: algebra_simps) 
23465  113 
with nob tB have "False" by simp} 
114 
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 

115 
next 

116 
assume d: "d dvd D" 

27668  117 
{fix x assume H: "d dvd x + t" with d have "d dvd (x  D) + t" by algebra} 
23465  118 
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 
119 
next 

120 
assume d: "d dvd D" 

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{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x  D) + t" 
29667  122 
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)} 
23465  123 
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 
124 
qed blast 

125 

126 
lemma aset: 

127 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 

128 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 

129 
\<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))" 

130 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 

131 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 

132 
\<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))" 

133 
"\<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))" 

134 
"\<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))" 

135 
"\<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))" 

136 
"\<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))" 

137 
"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))" 

138 
"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))" 

139 
"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))" 

140 
"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))" 

141 
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j) \<longrightarrow> F \<longrightarrow> F" 

142 
proof (blast, blast) 

143 
assume dp: "D > 0" and tA: "t + 1 \<in> A" 

144 
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 

145 
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) 

146 
using dp tA by simp_all 

147 
next 

148 
assume dp: "D > 0" and tA: "t \<in> A" 

149 
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))" 

150 
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 

151 
using dp tA by simp_all 

152 
next 

153 
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 

154 
next 

155 
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 

156 
next 

157 
assume dp: "D > 0" and tA:"t \<in> A" 

158 
{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" 

159 
hence "t  x \<le> D" and "1 \<le> t  x" by simp+ 

160 
hence "\<exists>j \<in> {1 .. D}. t  x = j" by auto 

29667  161 
hence "\<exists>j \<in> {1 .. D}. x = t  j" by (auto simp add: algebra_simps) 
23465  162 
with nob tA have "False" by simp} 
163 
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 

164 
next 

165 
assume dp: "D > 0" and tA:"t + 1\<in> A" 

166 
{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" 

29667  167 
hence "(t + 1)  x \<le> D" and "1 \<le> (t + 1)  x" by (simp_all add: algebra_simps) 
23465  168 
hence "\<exists>j \<in> {1 .. D}. (t + 1)  x = j" by auto 
29667  169 
hence "\<exists>j \<in> {1 .. D}. x = (t + 1)  j" by (auto simp add: algebra_simps) 
23465  170 
with nob tA have "False" by simp} 
171 
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 

172 
next 

173 
assume d: "d dvd D" 

174 
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" 

29667  175 
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)} 
23465  176 
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 
177 
next 

178 
assume d: "d dvd D" 

179 
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" 

29667  180 
by (clarsimp simp add: dvd_def,erule_tac x= "ka  k" in allE,simp add: algebra_simps)} 
23465  181 
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 
182 
qed blast 

183 

184 
subsection{* Cooper's Theorem @{text "\<infinity>"} and @{text "+\<infinity>"} Version *} 

185 

186 
subsubsection{* First some trivial facts about periodic sets or predicates *} 

187 
lemma periodic_finite_ex: 

188 
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x  k*d)" 

189 
shows "(EX x. P x) = (EX j : {1..d}. P j)" 

190 
(is "?LHS = ?RHS") 

191 
proof 

192 
assume ?LHS 

193 
then obtain x where P: "P x" .. 

194 
have "x mod d = x  (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) 

195 
hence Pmod: "P x = P(x mod d)" using modd by simp 

196 
show ?RHS 

197 
proof (cases) 

198 
assume "x mod d = 0" 

199 
hence "P 0" using P Pmod by simp 

200 
moreover have "P 0 = P(0  (1)*d)" using modd by blast 

201 
ultimately have "P d" by simp 

202 
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) 

203 
ultimately show ?RHS .. 

204 
next 

205 
assume not0: "x mod d \<noteq> 0" 

206 
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) 

207 
moreover have "x mod d : {1..d}" 

208 
proof  

209 
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) 

210 
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) 

211 
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) 

212 
qed 

213 
ultimately show ?RHS .. 

214 
qed 

215 
qed auto 

216 

217 
subsubsection{* The @{text "\<infinity>"} Version*} 

218 

219 
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x  (abs(xz)+1) * d < z" 

220 
by(induct rule: int_gr_induct,simp_all add:int_distrib) 

221 

222 
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(xz)+1) * d" 

223 
by(induct rule: int_gr_induct, simp_all add:int_distrib) 

224 

225 
theorem int_induct[case_names base step1 step2]: 

226 
assumes 

227 
base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and 

228 
step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i  1)" 

229 
shows "P i" 

230 
proof  

231 
have "i \<le> k \<or> i\<ge> k" by arith 

232 
thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast 

233 
qed 

234 

235 
lemma decr_mult_lemma: 

236 
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x  d)" and knneg: "0 <= k" 

237 
shows "ALL x. P x \<longrightarrow> P(x  k*d)" 

238 
using knneg 

239 
proof (induct rule:int_ge_induct) 

240 
case base thus ?case by simp 

241 
next 

242 
case (step i) 

243 
{fix x 

244 
have "P x \<longrightarrow> P (x  i * d)" using step.hyps by blast 

245 
also have "\<dots> \<longrightarrow> P(x  (i + 1) * d)" using minus[THEN spec, of "x  i * d"] 

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246 
by (simp add: algebra_simps) 
23465  247 
ultimately have "P x \<longrightarrow> P(x  (i + 1) * d)" by blast} 
248 
thus ?case .. 

249 
qed 

250 

251 
lemma minusinfinity: 

252 
assumes dpos: "0 < d" and 

253 
P1eqP1: "ALL x k. P1 x = P1(x  k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" 

254 
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" 

255 
proof 

256 
assume eP1: "EX x. P1 x" 

257 
then obtain x where P1: "P1 x" .. 

258 
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. 

259 
let ?w = "x  (abs(xz)+1) * d" 

260 
from dpos have w: "?w < z" by(rule decr_lemma) 

261 
have "P1 x = P1 ?w" using P1eqP1 by blast 

262 
also have "\<dots> = P(?w)" using w P1eqP by blast 

263 
finally have "P ?w" using P1 by blast 

264 
thus "EX x. P x" .. 

265 
qed 

266 

267 
lemma cpmi: 

268 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" 

269 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) > P (x) > P (x  D)" 

270 
and pd: "\<forall> x k. P' x = P' (xk*D)" 

271 
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j)  (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 

272 
(is "?L = (?R1 \<or> ?R2)") 

273 
proof 

274 
{assume "?R2" hence "?L" by blast} 

275 
moreover 

276 
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 

277 
moreover 

278 
{ fix x 

279 
assume P: "P x" and H: "\<not> ?R2" 

280 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" 

281 
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto 

282 
with nb P have "P (y  D)" by auto } 

283 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) > P (x) > P (x  D)" by blast 

284 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x  D)" by auto 

285 
from p1 obtain z where z: "ALL x. x < z > (P x = P' x)" by blast 

286 
let ?y = "x  (\<bar>x  z\<bar> + 1)*D" 

287 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 

288 
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp 

289 
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 

290 
with periodic_finite_ex[OF dp pd] 

291 
have "?R1" by blast} 

292 
ultimately show ?thesis by blast 

293 
qed 

294 

295 
subsubsection {* The @{text "+\<infinity>"} Version*} 

296 

297 
lemma plusinfinity: 

298 
assumes dpos: "(0::int) < d" and 

299 
P1eqP1: "\<forall>x k. P' x = P'(x  k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" 

300 
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" 

301 
proof 

302 
assume eP1: "EX x. P' x" 

303 
then obtain x where P1: "P' x" .. 

304 
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. 

305 
let ?w' = "x + (abs(xz)+1) * d" 

306 
let ?w = "x  ((abs(xz) + 1))*d" 

29667  307 
have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps) 
23465  308 
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) 
309 
hence "P' x = P' ?w" using P1eqP1 by blast 

310 
also have "\<dots> = P(?w)" using w P1eqP by blast 

311 
finally have "P ?w" using P1 by blast 

312 
thus "EX x. P x" .. 

313 
qed 

314 

315 
lemma incr_mult_lemma: 

316 
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" 

317 
shows "ALL x. P x \<longrightarrow> P(x + k*d)" 

318 
using knneg 

319 
proof (induct rule:int_ge_induct) 

320 
case base thus ?case by simp 

321 
next 

322 
case (step i) 

323 
{fix x 

324 
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast 

325 
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] 

326 
by (simp add:int_distrib zadd_ac) 

327 
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} 

328 
thus ?case .. 

329 
qed 

330 

331 
lemma cppi: 

332 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" 

333 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b  j) > P (x) > P (x + D)" 

334 
and pd: "\<forall> x k. P' x= P' (xk*D)" 

335 
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j)  (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b  j)))" (is "?L = (?R1 \<or> ?R2)") 

336 
proof 

337 
{assume "?R2" hence "?L" by blast} 

338 
moreover 

339 
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 

340 
moreover 

341 
{ fix x 

342 
assume P: "P x" and H: "\<not> ?R2" 

343 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b  j))" and P: "P y" 

344 
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b  j)" by auto 

345 
with nb P have "P (y + D)" by auto } 

346 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(bj)) > P (x) > P (x + D)" by blast 

347 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto 

348 
from p1 obtain z where z: "ALL x. x > z > (P x = P' x)" by blast 

349 
let ?y = "x + (\<bar>x  z\<bar> + 1)*D" 

350 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 

351 
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp 

352 
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 

353 
with periodic_finite_ex[OF dp pd] 

354 
have "?R1" by blast} 

355 
ultimately show ?thesis by blast 

356 
qed 

357 

358 
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" 

359 
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) 

360 
apply(fastsimp) 

361 
done 

362 

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363 
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" 
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364 
apply (rule eq_reflection [symmetric]) 
23465  365 
apply (rule iffI) 
366 
defer 

367 
apply (erule exE) 

368 
apply (rule_tac x = "l * x" in exI) 

369 
apply (simp add: dvd_def) 

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370 
apply (rule_tac x = x in exI, simp) 
23465  371 
apply (erule exE) 
372 
apply (erule conjE) 

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373 
apply simp 
23465  374 
apply (erule dvdE) 
375 
apply (rule_tac x = k in exI) 

376 
apply simp 

377 
done 

378 

379 
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" 

380 
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 

381 
using not0 by (simp add: dvd_def) 

382 

383 
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd t)" 

384 
by simp_all 

32553  385 

23465  386 
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} 
32553  387 

23465  388 
lemma zdiff_int_split: "P (int (x  y)) = 
389 
((y \<le> x \<longrightarrow> P (int x  int y)) \<and> (x < y \<longrightarrow> P 0))" 

390 
by (case_tac "y \<le> x", simp_all add: zdiff_int) 

391 

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392 
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (Int.Bit0 n) \<and> (0::int) <= number_of (Int.Bit1 n)" 
3c243098b64a
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393 
by simp 
23465  394 
lemma number_of2: "(0::int) <= Numeral0" by simp 
395 

396 
text {* 

397 
\medskip Specific instances of congruence rules, to prevent 

398 
simplifier from looping. *} 

399 

400 
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp 

401 

402 
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 

403 
by (simp cong: conj_cong) 

404 
lemma int_eq_number_of_eq: 

405 
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" 

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3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
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28402
diff
changeset

406 
by (rule eq_number_of_eq) 
23465  407 

30031  408 
declare dvd_eq_mod_eq_0[symmetric, presburger] 
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changeset

409 
declare mod_1[presburger] 
23465  410 
declare mod_0[presburger] 
30031  411 
declare mod_by_1[presburger] 
23465  412 
declare zmod_zero[presburger] 
413 
declare zmod_self[presburger] 

414 
declare mod_self[presburger] 

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haftmann
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27540
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changeset

415 
declare mod_by_0[presburger] 
30027  416 
declare mod_div_trivial[presburger] 
23465  417 
declare div_mod_equality2[presburger] 
418 
declare div_mod_equality[presburger] 

419 
declare mod_div_equality2[presburger] 

420 
declare mod_div_equality[presburger] 

421 
declare mod_mult_self1[presburger] 

422 
declare mod_mult_self2[presburger] 

423 
declare zdiv_zmod_equality2[presburger] 

424 
declare zdiv_zmod_equality[presburger] 

425 
declare mod2_Suc_Suc[presburger] 

426 
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" 

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427 
by simp_all 
23465  428 

429 
use "Tools/Qelim/cooper.ML" 

28290  430 
oracle linzqe_oracle = Coopereif.cooper_oracle 
23465  431 

432 
use "Tools/Qelim/presburger.ML" 

433 

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434 
setup {* Arith_Data.add_tactic "Presburger arithmetic" (K (Presburger.cooper_tac true [] [])) *} 
23465  435 

436 
method_setup presburger = {* 

437 
let 

438 
fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 

439 
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () 

440 
val addN = "add" 

441 
val delN = "del" 

442 
val elimN = "elim" 

443 
val any_keyword = keyword addN  keyword delN  simple_keyword elimN 

444 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 

445 
in 

30549  446 
Scan.optional (simple_keyword elimN >> K false) true  
447 
Scan.optional (keyword addN  thms) []  

448 
Scan.optional (keyword delN  thms) [] >> 

449 
(fn ((elim, add_ths), del_ths) => fn ctxt => 

450 
SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) 

23465  451 
end 
452 
*} "Cooper's algorithm for Presburger arithmetic" 

453 

27668  454 
lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
455 
lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 

456 
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 

457 
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 

458 
lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger 

23465  459 

460 

23685  461 
lemma zdvd_period: 
462 
fixes a d :: int 

463 
assumes advdd: "a dvd d" 

464 
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" 

27668  465 
using advdd 
466 
apply  

467 
apply (rule iffI) 

468 
by algebra+ 

23685  469 

23465  470 
end 