1 (* Title: HOL/Presburger.thy
3 Author: Amine Chaieb, TU Muenchen
6 header {* Decision Procedure for Presburger Arithmetic *}
9 imports Arith_Tools SetInterval
11 "Tools/Qelim/cooper_data.ML"
12 "Tools/Qelim/generated_cooper.ML"
13 "Tools/Qelim/qelim.ML"
14 ("Tools/Qelim/cooper.ML")
15 ("Tools/Qelim/presburger.ML")
18 setup CooperData.setup
20 subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
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 \<and> Q x) = (P' x \<and> Q' x)"
26 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
27 \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
28 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
29 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
30 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
31 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
32 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
33 "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
34 "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (d dvd x + s) = (d dvd x + s)"
35 "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
36 "\<exists>z.\<forall>x<z. F = F"
37 by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
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 \<and> Q x) = (P' x \<and> Q' x)"
42 "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
43 \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
44 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
45 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
46 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
47 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
48 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
49 "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
50 "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (d dvd x + s) = (d dvd x + s)"
51 "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
52 "\<exists>z.\<forall>x>z. F = F"
53 by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
56 "\<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 \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
58 "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
59 \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
60 "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
61 "(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)"
63 apply (auto elim!: dvdE simp add: ring_simps)
64 unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
65 unfolding dvd_def mult_commute [of d]
68 subsection{* The A and B sets *}
70 "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
71 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
72 \<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))"
73 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
74 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
75 \<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))"
76 "\<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))"
77 "\<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))"
78 "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))"
79 "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))"
80 "\<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))"
81 "\<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))"
82 "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))"
83 "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))"
84 "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
86 assume dp: "D > 0" and tB: "t - 1\<in> B"
87 show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
88 apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
89 apply algebra using dp tB by simp_all
91 assume dp: "D > 0" and tB: "t \<in> B"
92 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))"
93 apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
95 using dp tB by simp_all
97 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
99 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
101 assume dp: "D > 0" and tB:"t \<in> B"
102 {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"
103 hence "x -t \<le> D" and "1 \<le> x - t" by simp+
104 hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
105 hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps)
106 with nob tB have "False" by simp}
107 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
109 assume dp: "D > 0" and tB:"t - 1\<in> B"
110 {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"
111 hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
112 hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
113 hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps)
114 with nob tB have "False" by simp}
115 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
118 {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra}
119 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
122 {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t"
123 by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)}
124 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
128 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
129 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
130 \<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))"
131 "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
132 \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
133 \<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))"
134 "\<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))"
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 \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
136 "\<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))"
137 "\<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))"
138 "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))"
139 "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))"
140 "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))"
141 "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))"
142 "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
144 assume dp: "D > 0" and tA: "t + 1 \<in> A"
145 show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
146 apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
147 using dp tA by simp_all
149 assume dp: "D > 0" and tA: "t \<in> A"
150 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))"
151 apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
152 using dp tA by simp_all
154 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
156 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
158 assume dp: "D > 0" and tA:"t \<in> A"
159 {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"
160 hence "t - x \<le> D" and "1 \<le> t - x" by simp+
161 hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
162 hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps)
163 with nob tA have "False" by simp}
164 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
166 assume dp: "D > 0" and tA:"t + 1\<in> A"
167 {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"
168 hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps)
169 hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
170 hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps)
171 with nob tA have "False" by simp}
172 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
175 {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
176 by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)}
177 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
180 {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
181 by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)}
182 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
185 subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
187 subsubsection{* First some trivial facts about periodic sets or predicates *}
188 lemma periodic_finite_ex:
189 assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
190 shows "(EX x. P x) = (EX j : {1..d}. P j)"
194 then obtain x where P: "P x" ..
195 have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
196 hence Pmod: "P x = P(x mod d)" using modd by simp
200 hence "P 0" using P Pmod by simp
201 moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
202 ultimately have "P d" by simp
203 moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
204 ultimately show ?RHS ..
206 assume not0: "x mod d \<noteq> 0"
207 have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
208 moreover have "x mod d : {1..d}"
210 from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
211 moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
212 ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
214 ultimately show ?RHS ..
218 subsubsection{* The @{text "-\<infinity>"} Version*}
220 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
221 by(induct rule: int_gr_induct,simp_all add:int_distrib)
223 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
224 by(induct rule: int_gr_induct, simp_all add:int_distrib)
226 theorem int_induct[case_names base step1 step2]:
228 base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
229 step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
232 have "i \<le> k \<or> i\<ge> k" by arith
233 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
236 lemma decr_mult_lemma:
237 assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
238 shows "ALL x. P x \<longrightarrow> P(x - k*d)"
240 proof (induct rule:int_ge_induct)
241 case base thus ?case by simp
245 have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
246 also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
247 by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
248 ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
253 assumes dpos: "0 < d" and
254 P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
255 shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
257 assume eP1: "EX x. P1 x"
258 then obtain x where P1: "P1 x" ..
259 from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
260 let ?w = "x - (abs(x-z)+1) * d"
261 from dpos have w: "?w < z" by(rule decr_lemma)
262 have "P1 x = P1 ?w" using P1eqP1 by blast
263 also have "\<dots> = P(?w)" using w P1eqP by blast
264 finally have "P ?w" using P1 by blast
269 assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
270 and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
271 and pd: "\<forall> x k. P' x = P' (x-k*D)"
272 shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
273 (is "?L = (?R1 \<or> ?R2)")
275 {assume "?R2" hence "?L" by blast}
277 {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
280 assume P: "P x" and H: "\<not> ?R2"
281 {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
282 hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
283 with nb P have "P (y - D)" by auto }
284 hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
285 with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
286 from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
287 let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
288 have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
289 from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
290 from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
291 with periodic_finite_ex[OF dp pd]
293 ultimately show ?thesis by blast
296 subsubsection {* The @{text "+\<infinity>"} Version*}
299 assumes dpos: "(0::int) < d" and
300 P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
301 shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
303 assume eP1: "EX x. P' x"
304 then obtain x where P1: "P' x" ..
305 from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
306 let ?w' = "x + (abs(x-z)+1) * d"
307 let ?w = "x - (-(abs(x-z) + 1))*d"
308 have ww'[simp]: "?w = ?w'" by (simp add: ring_simps)
309 from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
310 hence "P' x = P' ?w" using P1eqP1 by blast
311 also have "\<dots> = P(?w)" using w P1eqP by blast
312 finally have "P ?w" using P1 by blast
316 lemma incr_mult_lemma:
317 assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
318 shows "ALL x. P x \<longrightarrow> P(x + k*d)"
320 proof (induct rule:int_ge_induct)
321 case base thus ?case by simp
325 have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
326 also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
327 by (simp add:int_distrib zadd_ac)
328 ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
333 assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
334 and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
335 and pd: "\<forall> x k. P' x= P' (x-k*D)"
336 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)")
338 {assume "?R2" hence "?L" by blast}
340 {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
343 assume P: "P x" and H: "\<not> ?R2"
344 {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
345 hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
346 with nb P have "P (y + D)" by auto }
347 hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
348 with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
349 from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
350 let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
351 have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
352 from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
353 from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
354 with periodic_finite_ex[OF dp pd]
356 ultimately show ?thesis by blast
359 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
360 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
364 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)"
365 apply (rule eq_reflection [symmetric])
369 apply (rule_tac x = "l * x" in exI)
370 apply (simp add: dvd_def)
371 apply (rule_tac x = x in exI, simp)
376 apply (rule_tac x = k in exI)
380 lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
381 shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
382 using not0 by (simp add: dvd_def)
384 lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
386 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
387 lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
388 by (simp split add: split_nat)
390 lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
391 apply (auto split add: split_nat)
392 apply (rule_tac x="int x" in exI, simp)
393 apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
396 lemma zdiff_int_split: "P (int (x - y)) =
397 ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
398 by (case_tac "y \<le> x", simp_all add: zdiff_int)
400 lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (Int.Bit0 n) \<and> (0::int) <= number_of (Int.Bit1 n)"
402 lemma number_of2: "(0::int) <= Numeral0" by simp
403 lemma Suc_plus1: "Suc n = n + 1" by simp
406 \medskip Specific instances of congruence rules, to prevent
407 simplifier from looping. *}
409 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
411 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
412 by (simp cong: conj_cong)
413 lemma int_eq_number_of_eq:
414 "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
417 lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
418 unfolding dvd_eq_mod_eq_0[symmetric] ..
420 lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
421 unfolding zdvd_iff_zmod_eq_0[symmetric] ..
422 declare mod_1[presburger]
423 declare mod_0[presburger]
424 declare zmod_1[presburger]
425 declare zmod_zero[presburger]
426 declare zmod_self[presburger]
427 declare mod_self[presburger]
428 declare mod_by_0[presburger]
429 declare nat_mod_div_trivial[presburger]
430 declare div_mod_equality2[presburger]
431 declare div_mod_equality[presburger]
432 declare mod_div_equality2[presburger]
433 declare mod_div_equality[presburger]
434 declare mod_mult_self1[presburger]
435 declare mod_mult_self2[presburger]
436 declare zdiv_zmod_equality2[presburger]
437 declare zdiv_zmod_equality[presburger]
438 declare mod2_Suc_Suc[presburger]
439 lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
442 use "Tools/Qelim/cooper.ML"
443 oracle linzqe_oracle = Coopereif.cooper_oracle
445 use "Tools/Qelim/presburger.ML"
447 declaration {* fn _ =>
449 (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st =>
450 (warning "Trying Presburger arithmetic ...";
451 Presburger.cooper_tac true [] [] ctxt i st)))
454 method_setup presburger = {*
456 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
457 fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
461 val any_keyword = keyword addN || keyword delN || simple_keyword elimN
462 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
464 fn src => Method.syntax
465 ((Scan.optional (simple_keyword elimN >> K false) true) --
466 (Scan.optional (keyword addN |-- thms) []) --
467 (Scan.optional (keyword delN |-- thms) [])) src
468 #> (fn (((elim, add_ths), del_ths),ctxt) =>
469 Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
471 *} "Cooper's algorithm for Presburger arithmetic"
473 lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
474 lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
475 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
476 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
477 lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
482 assumes advdd: "a dvd d"
483 shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"