author  nipkow 
Fri, 20 Feb 2009 20:50:49 +0100  
changeset 30027  ab40c5e007e0 
parent 29707  01cae7ad8576 
child 30031  bd786c37af84 
permissions  rwrr 
23465  1 
(* Title: HOL/Presburger.thy 
2 
Author: Amine Chaieb, TU Muenchen 

3 
*) 

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23472  5 
header {* Decision Procedure for Presburger Arithmetic *} 
6 

23465  7 
theory Presburger 
28402  8 
imports Groebner_Basis SetInterval 
23465  9 
uses 
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"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 

15 

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setup CooperData.setup 

17 

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

19 

24404  20 

23465  21 
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> 

23 
\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)" 

24 
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 

25 
\<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" 

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

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

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

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

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

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

32 
"\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (d dvd x + s) = (d dvd x + s)" 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

33 
"\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 
23465  34 
"\<exists>z.\<forall>x<z. F = F" 
35 
by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all 

36 

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lemma pinf: 

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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> 

39 
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)" 

40 
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 

41 
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)" 

42 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False" 

43 
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True" 

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

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

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

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

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

48 
"\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (d dvd x + s) = (d dvd x + s)" 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

49 
"\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" 
23465  50 
"\<exists>z.\<forall>x>z. F = F" 
51 
by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all 

52 

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lemma inf_period: 

54 
"\<lbrakk>\<forall>x k. P x = P (x  k*D); \<forall>x k. Q x = Q (x  k*D)\<rbrakk> 

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

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

57 
\<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x  k*D) \<or> Q (x  k*D))" 

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

58 
"(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x  k*D) + t)" 
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

59 
"(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x  k*D) + t)" 
23465  60 
"\<forall>x k. F = F" 
29667  61 
apply (auto elim!: dvdE simp add: algebra_simps) 
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

62 
unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric] 
27668  63 
unfolding dvd_def mult_commute [of d] 
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by auto 

23465  65 

23472  66 
subsection{* The A and B sets *} 
23465  67 
lemma bset: 
68 
"\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x  D) ; 

69 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x  D)\<rbrakk> \<Longrightarrow> 

70 
\<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))" 

71 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x  D) ; 

72 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x  D)\<rbrakk> \<Longrightarrow> 

73 
\<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))" 

74 
"\<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))" 

75 
"\<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))" 

76 
"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))" 

77 
"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))" 

78 
"\<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))" 

79 
"\<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))" 

80 
"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))" 

81 
"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))" 

82 
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F" 

83 
proof (blast, blast) 

84 
assume dp: "D > 0" and tB: "t  1\<in> B" 

85 
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x  D = t))" 

27668  86 
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t  1"]) 
87 
apply algebra using dp tB by simp_all 

23465  88 
next 
89 
assume dp: "D > 0" and tB: "t \<in> B" 

90 
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))" 

91 
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 

27668  92 
apply algebra 
23465  93 
using dp tB by simp_all 
94 
next 

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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 

96 
next 

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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 

98 
next 

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

100 
{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" 

101 
hence "x t \<le> D" and "1 \<le> x  t" by simp+ 

102 
hence "\<exists>j \<in> {1 .. D}. x  t = j" by auto 

29667  103 
hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps) 
23465  104 
with nob tB have "False" by simp} 
105 
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 

106 
next 

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

108 
{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" 

109 
hence "x  (t  1) \<le> D" and "1 \<le> x  (t  1)" by simp+ 

110 
hence "\<exists>j \<in> {1 .. D}. x  (t  1) = j" by auto 

29667  111 
hence "\<exists>j \<in> {1 .. D}. x = (t  1) + j" by (simp add: algebra_simps) 
23465  112 
with nob tB have "False" by simp} 
113 
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 

114 
next 

115 
assume d: "d dvd D" 

27668  116 
{fix x assume H: "d dvd x + t" with d have "d dvd (x  D) + t" by algebra} 
23465  117 
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 
118 
next 

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assume d: "d dvd D" 

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

120 
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x  D) + t" 
29667  121 
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)} 
23465  122 
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 
123 
qed blast 

124 

125 
lemma aset: 

126 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 

127 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 

128 
\<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))" 

129 
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; 

130 
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 

131 
\<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))" 

132 
"\<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))" 

133 
"\<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))" 

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 < t) \<longrightarrow> (x + D < t))" 

135 
"\<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))" 

136 
"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))" 

137 
"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))" 

138 
"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))" 

139 
"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))" 

140 
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j) \<longrightarrow> F \<longrightarrow> F" 

141 
proof (blast, blast) 

142 
assume dp: "D > 0" and tA: "t + 1 \<in> A" 

143 
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 

144 
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) 

145 
using dp tA by simp_all 

146 
next 

147 
assume dp: "D > 0" and tA: "t \<in> A" 

148 
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))" 

149 
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) 

150 
using dp tA by simp_all 

151 
next 

152 
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 

153 
next 

154 
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 

155 
next 

156 
assume dp: "D > 0" and tA:"t \<in> A" 

157 
{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" 

158 
hence "t  x \<le> D" and "1 \<le> t  x" by simp+ 

159 
hence "\<exists>j \<in> {1 .. D}. t  x = j" by auto 

29667  160 
hence "\<exists>j \<in> {1 .. D}. x = t  j" by (auto simp add: algebra_simps) 
23465  161 
with nob tA have "False" by simp} 
162 
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 

163 
next 

164 
assume dp: "D > 0" and tA:"t + 1\<in> A" 

165 
{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  166 
hence "(t + 1)  x \<le> D" and "1 \<le> (t + 1)  x" by (simp_all add: algebra_simps) 
23465  167 
hence "\<exists>j \<in> {1 .. D}. (t + 1)  x = j" by auto 
29667  168 
hence "\<exists>j \<in> {1 .. D}. x = (t + 1)  j" by (auto simp add: algebra_simps) 
23465  169 
with nob tA have "False" by simp} 
170 
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 

171 
next 

172 
assume d: "d dvd D" 

173 
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" 

29667  174 
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)} 
23465  175 
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 
176 
next 

177 
assume d: "d dvd D" 

178 
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" 

29667  179 
by (clarsimp simp add: dvd_def,erule_tac x= "ka  k" in allE,simp add: algebra_simps)} 
23465  180 
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 
181 
qed blast 

182 

183 
subsection{* Cooper's Theorem @{text "\<infinity>"} and @{text "+\<infinity>"} Version *} 

184 

185 
subsubsection{* First some trivial facts about periodic sets or predicates *} 

186 
lemma periodic_finite_ex: 

187 
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x  k*d)" 

188 
shows "(EX x. P x) = (EX j : {1..d}. P j)" 

189 
(is "?LHS = ?RHS") 

190 
proof 

191 
assume ?LHS 

192 
then obtain x where P: "P x" .. 

193 
have "x mod d = x  (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) 

194 
hence Pmod: "P x = P(x mod d)" using modd by simp 

195 
show ?RHS 

196 
proof (cases) 

197 
assume "x mod d = 0" 

198 
hence "P 0" using P Pmod by simp 

199 
moreover have "P 0 = P(0  (1)*d)" using modd by blast 

200 
ultimately have "P d" by simp 

201 
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) 

202 
ultimately show ?RHS .. 

203 
next 

204 
assume not0: "x mod d \<noteq> 0" 

205 
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) 

206 
moreover have "x mod d : {1..d}" 

207 
proof  

208 
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) 

209 
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) 

210 
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) 

211 
qed 

212 
ultimately show ?RHS .. 

213 
qed 

214 
qed auto 

215 

216 
subsubsection{* The @{text "\<infinity>"} Version*} 

217 

218 
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x  (abs(xz)+1) * d < z" 

219 
by(induct rule: int_gr_induct,simp_all add:int_distrib) 

220 

221 
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(xz)+1) * d" 

222 
by(induct rule: int_gr_induct, simp_all add:int_distrib) 

223 

224 
theorem int_induct[case_names base step1 step2]: 

225 
assumes 

226 
base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and 

227 
step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i  1)" 

228 
shows "P i" 

229 
proof  

230 
have "i \<le> k \<or> i\<ge> k" by arith 

231 
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 

232 
qed 

233 

234 
lemma decr_mult_lemma: 

235 
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x  d)" and knneg: "0 <= k" 

236 
shows "ALL x. P x \<longrightarrow> P(x  k*d)" 

237 
using knneg 

238 
proof (induct rule:int_ge_induct) 

239 
case base thus ?case by simp 

240 
next 

241 
case (step i) 

242 
{fix x 

243 
have "P x \<longrightarrow> P (x  i * d)" using step.hyps by blast 

244 
also have "\<dots> \<longrightarrow> P(x  (i + 1) * d)" using minus[THEN spec, of "x  i * d"] 

245 
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) 

246 
ultimately have "P x \<longrightarrow> P(x  (i + 1) * d)" by blast} 

247 
thus ?case .. 

248 
qed 

249 

250 
lemma minusinfinity: 

251 
assumes dpos: "0 < d" and 

252 
P1eqP1: "ALL x k. P1 x = P1(x  k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" 

253 
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" 

254 
proof 

255 
assume eP1: "EX x. P1 x" 

256 
then obtain x where P1: "P1 x" .. 

257 
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. 

258 
let ?w = "x  (abs(xz)+1) * d" 

259 
from dpos have w: "?w < z" by(rule decr_lemma) 

260 
have "P1 x = P1 ?w" using P1eqP1 by blast 

261 
also have "\<dots> = P(?w)" using w P1eqP by blast 

262 
finally have "P ?w" using P1 by blast 

263 
thus "EX x. P x" .. 

264 
qed 

265 

266 
lemma cpmi: 

267 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" 

268 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) > P (x) > P (x  D)" 

269 
and pd: "\<forall> x k. P' x = P' (xk*D)" 

270 
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j)  (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 

271 
(is "?L = (?R1 \<or> ?R2)") 

272 
proof 

273 
{assume "?R2" hence "?L" by blast} 

274 
moreover 

275 
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 

276 
moreover 

277 
{ fix x 

278 
assume P: "P x" and H: "\<not> ?R2" 

279 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" 

280 
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto 

281 
with nb P have "P (y  D)" by auto } 

282 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) > P (x) > P (x  D)" by blast 

283 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x  D)" by auto 

284 
from p1 obtain z where z: "ALL x. x < z > (P x = P' x)" by blast 

285 
let ?y = "x  (\<bar>x  z\<bar> + 1)*D" 

286 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 

287 
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp 

288 
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 

289 
with periodic_finite_ex[OF dp pd] 

290 
have "?R1" by blast} 

291 
ultimately show ?thesis by blast 

292 
qed 

293 

294 
subsubsection {* The @{text "+\<infinity>"} Version*} 

295 

296 
lemma plusinfinity: 

297 
assumes dpos: "(0::int) < d" and 

298 
P1eqP1: "\<forall>x k. P' x = P'(x  k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" 

299 
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" 

300 
proof 

301 
assume eP1: "EX x. P' x" 

302 
then obtain x where P1: "P' x" .. 

303 
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. 

304 
let ?w' = "x + (abs(xz)+1) * d" 

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

29667  306 
have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps) 
23465  307 
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) 
308 
hence "P' x = P' ?w" using P1eqP1 by blast 

309 
also have "\<dots> = P(?w)" using w P1eqP by blast 

310 
finally have "P ?w" using P1 by blast 

311 
thus "EX x. P x" .. 

312 
qed 

313 

314 
lemma incr_mult_lemma: 

315 
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" 

316 
shows "ALL x. P x \<longrightarrow> P(x + k*d)" 

317 
using knneg 

318 
proof (induct rule:int_ge_induct) 

319 
case base thus ?case by simp 

320 
next 

321 
case (step i) 

322 
{fix x 

323 
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast 

324 
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] 

325 
by (simp add:int_distrib zadd_ac) 

326 
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} 

327 
thus ?case .. 

328 
qed 

329 

330 
lemma cppi: 

331 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" 

332 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b  j) > P (x) > P (x + D)" 

333 
and pd: "\<forall> x k. P' x= P' (xk*D)" 

334 
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)") 

335 
proof 

336 
{assume "?R2" hence "?L" by blast} 

337 
moreover 

338 
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 

339 
moreover 

340 
{ fix x 

341 
assume P: "P x" and H: "\<not> ?R2" 

342 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b  j))" and P: "P y" 

343 
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b  j)" by auto 

344 
with nb P have "P (y + D)" by auto } 

345 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(bj)) > P (x) > P (x + D)" by blast 

346 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto 

347 
from p1 obtain z where z: "ALL x. x > z > (P x = P' x)" by blast 

348 
let ?y = "x + (\<bar>x  z\<bar> + 1)*D" 

349 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 

350 
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp 

351 
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 

352 
with periodic_finite_ex[OF dp pd] 

353 
have "?R1" by blast} 

354 
ultimately show ?thesis by blast 

355 
qed 

356 

357 
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" 

358 
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) 

359 
apply(fastsimp) 

360 
done 

361 

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parents:
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changeset

362 
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Ring_and_Field.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" 
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parents:
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363 
apply (rule eq_reflection [symmetric]) 
23465  364 
apply (rule iffI) 
365 
defer 

366 
apply (erule exE) 

367 
apply (rule_tac x = "l * x" in exI) 

368 
apply (simp add: dvd_def) 

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369 
apply (rule_tac x = x in exI, simp) 
23465  370 
apply (erule exE) 
371 
apply (erule conjE) 

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parents:
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changeset

372 
apply simp 
23465  373 
apply (erule dvdE) 
374 
apply (rule_tac x = k in exI) 

375 
apply simp 

376 
done 

377 

378 
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" 

379 
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 

380 
using not0 by (simp add: dvd_def) 

381 

382 
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd t)" 

383 
by simp_all 

384 
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} 

385 
lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))" 

386 
by (simp split add: split_nat) 

387 

388 
lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))" 

389 
apply (auto split add: split_nat) 

390 
apply (rule_tac x="int x" in exI, simp) 

391 
apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp) 

392 
done 

393 

394 
lemma zdiff_int_split: "P (int (x  y)) = 

395 
((y \<le> x \<longrightarrow> P (int x  int y)) \<and> (x < y \<longrightarrow> P 0))" 

396 
by (case_tac "y \<le> x", simp_all add: zdiff_int) 

397 

26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset

398 
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
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset

399 
by simp 
23465  400 
lemma number_of2: "(0::int) <= Numeral0" by simp 
401 
lemma Suc_plus1: "Suc n = n + 1" by simp 

402 

403 
text {* 

404 
\medskip Specific instances of congruence rules, to prevent 

405 
simplifier from looping. *} 

406 

407 
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp 

408 

409 
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 

410 
by (simp cong: conj_cong) 

411 
lemma int_eq_number_of_eq: 

412 
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" 

28967
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28402
diff
changeset

413 
by (rule eq_number_of_eq) 
23465  414 

415 
lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m" 

416 
unfolding dvd_eq_mod_eq_0[symmetric] .. 

417 

418 
lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m" 

419 
unfolding zdvd_iff_zmod_eq_0[symmetric] .. 

27651
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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

420 
declare mod_1[presburger] 
23465  421 
declare mod_0[presburger] 
422 
declare zmod_1[presburger] 

423 
declare zmod_zero[presburger] 

424 
declare zmod_self[presburger] 

425 
declare mod_self[presburger] 

27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

426 
declare mod_by_0[presburger] 
30027  427 
declare mod_div_trivial[presburger] 
23465  428 
declare div_mod_equality2[presburger] 
429 
declare div_mod_equality[presburger] 

430 
declare mod_div_equality2[presburger] 

431 
declare mod_div_equality[presburger] 

432 
declare mod_mult_self1[presburger] 

433 
declare mod_mult_self2[presburger] 

434 
declare zdiv_zmod_equality2[presburger] 

435 
declare zdiv_zmod_equality[presburger] 

436 
declare mod2_Suc_Suc[presburger] 

437 
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" 

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moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset

438 
by simp_all 
23465  439 

440 
use "Tools/Qelim/cooper.ML" 

28290  441 
oracle linzqe_oracle = Coopereif.cooper_oracle 
23465  442 

443 
use "Tools/Qelim/presburger.ML" 

444 

24075  445 
declaration {* fn _ => 
446 
arith_tactic_add 

24094  447 
(mk_arith_tactic "presburger" (fn ctxt => fn i => fn st => 
23465  448 
(warning "Trying Presburger arithmetic ..."; 
24094  449 
Presburger.cooper_tac true [] [] ctxt i st))) 
23465  450 
*} 
451 

452 
method_setup presburger = {* 

453 
let 

454 
fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 

455 
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () 

456 
val addN = "add" 

457 
val delN = "del" 

458 
val elimN = "elim" 

459 
val any_keyword = keyword addN  keyword delN  simple_keyword elimN 

460 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 

461 
in 

462 
fn src => Method.syntax 

463 
((Scan.optional (simple_keyword elimN >> K false) true)  

464 
(Scan.optional (keyword addN  thms) [])  

465 
(Scan.optional (keyword delN  thms) [])) src 

466 
#> (fn (((elim, add_ths), del_ths),ctxt) => 

467 
Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) 

468 
end 

469 
*} "Cooper's algorithm for Presburger arithmetic" 

470 

27668  471 
lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
472 
lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 

473 
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 

474 
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 

475 
lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger 

23465  476 

477 

23685  478 
lemma zdvd_period: 
479 
fixes a d :: int 

480 
assumes advdd: "a dvd d" 

481 
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" 

27668  482 
using advdd 
483 
apply  

484 
apply (rule iffI) 

485 
by algebra+ 

23685  486 

23465  487 
end 