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