wenzelm@23465
|
1 |
(* Title: HOL/Presburger.thy
|
wenzelm@23465
|
2 |
Author: Amine Chaieb, TU Muenchen
|
wenzelm@23465
|
3 |
*)
|
wenzelm@23465
|
4 |
|
huffman@23472
|
5 |
header {* Decision Procedure for Presburger Arithmetic *}
|
huffman@23472
|
6 |
|
wenzelm@23465
|
7 |
theory Presburger
|
haftmann@28402
|
8 |
imports Groebner_Basis SetInterval
|
wenzelm@23465
|
9 |
uses
|
haftmann@30656
|
10 |
"Tools/Qelim/qelim.ML"
|
haftmann@36792
|
11 |
"Tools/Qelim/cooper_procedure.ML"
|
wenzelm@23465
|
12 |
("Tools/Qelim/cooper.ML")
|
wenzelm@23465
|
13 |
begin
|
wenzelm@23465
|
14 |
|
wenzelm@23465
|
15 |
subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
|
wenzelm@23465
|
16 |
|
wenzelm@23465
|
17 |
lemma minf:
|
wenzelm@23465
|
18 |
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
|
wenzelm@23465
|
19 |
\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
|
wenzelm@23465
|
20 |
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk>
|
wenzelm@23465
|
21 |
\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
|
wenzelm@23465
|
22 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
|
wenzelm@23465
|
23 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
|
wenzelm@23465
|
24 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
|
wenzelm@23465
|
25 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
|
wenzelm@23465
|
26 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
|
wenzelm@23465
|
27 |
"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
|
wenzelm@46296
|
28 |
"\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)"
|
wenzelm@46296
|
29 |
"\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
|
wenzelm@23465
|
30 |
"\<exists>z.\<forall>x<z. F = F"
|
nipkow@45761
|
31 |
by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastforce)+) simp_all
|
wenzelm@23465
|
32 |
|
wenzelm@23465
|
33 |
lemma pinf:
|
wenzelm@23465
|
34 |
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
|
wenzelm@23465
|
35 |
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
|
wenzelm@23465
|
36 |
"\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk>
|
wenzelm@23465
|
37 |
\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
|
wenzelm@23465
|
38 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
|
wenzelm@23465
|
39 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
|
wenzelm@23465
|
40 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
|
wenzelm@23465
|
41 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
|
wenzelm@23465
|
42 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
|
wenzelm@23465
|
43 |
"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
|
wenzelm@46296
|
44 |
"\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)"
|
wenzelm@46296
|
45 |
"\<exists>z.\<forall>(x::'b::{linorder,plus,Rings.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
|
wenzelm@23465
|
46 |
"\<exists>z.\<forall>x>z. F = F"
|
nipkow@45761
|
47 |
by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastforce)+) simp_all
|
wenzelm@23465
|
48 |
|
wenzelm@23465
|
49 |
lemma inf_period:
|
wenzelm@23465
|
50 |
"\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
|
wenzelm@23465
|
51 |
\<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
|
wenzelm@23465
|
52 |
"\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk>
|
wenzelm@23465
|
53 |
\<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
|
haftmann@35050
|
54 |
"(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
|
haftmann@35050
|
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)"
|
wenzelm@23465
|
56 |
"\<forall>x k. F = F"
|
nipkow@29667
|
57 |
apply (auto elim!: dvdE simp add: algebra_simps)
|
haftmann@27651
|
58 |
unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
|
chaieb@27668
|
59 |
unfolding dvd_def mult_commute [of d]
|
chaieb@27668
|
60 |
by auto
|
wenzelm@23465
|
61 |
|
huffman@23472
|
62 |
subsection{* The A and B sets *}
|
wenzelm@23465
|
63 |
lemma bset:
|
wenzelm@23465
|
64 |
"\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
|
wenzelm@23465
|
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>
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
67 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
|
wenzelm@23465
|
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>
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
78 |
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
|
wenzelm@23465
|
79 |
proof (blast, blast)
|
wenzelm@23465
|
80 |
assume dp: "D > 0" and tB: "t - 1\<in> B"
|
wenzelm@23465
|
81 |
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
|
chaieb@27668
|
82 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
|
chaieb@27668
|
83 |
apply algebra using dp tB by simp_all
|
wenzelm@23465
|
84 |
next
|
wenzelm@23465
|
85 |
assume dp: "D > 0" and tB: "t \<in> B"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
87 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
|
chaieb@27668
|
88 |
apply algebra
|
wenzelm@23465
|
89 |
using dp tB by simp_all
|
wenzelm@23465
|
90 |
next
|
wenzelm@23465
|
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
|
wenzelm@23465
|
92 |
next
|
wenzelm@23465
|
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
|
wenzelm@23465
|
94 |
next
|
wenzelm@23465
|
95 |
assume dp: "D > 0" and tB:"t \<in> B"
|
wenzelm@23465
|
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"
|
wenzelm@23465
|
97 |
hence "x -t \<le> D" and "1 \<le> x - t" by simp+
|
wenzelm@23465
|
98 |
hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
|
nipkow@29667
|
99 |
hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps)
|
wenzelm@23465
|
100 |
with nob tB have "False" by simp}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
102 |
next
|
wenzelm@23465
|
103 |
assume dp: "D > 0" and tB:"t - 1\<in> B"
|
wenzelm@23465
|
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"
|
wenzelm@23465
|
105 |
hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
|
wenzelm@23465
|
106 |
hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
|
nipkow@29667
|
107 |
hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps)
|
wenzelm@23465
|
108 |
with nob tB have "False" by simp}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
110 |
next
|
wenzelm@23465
|
111 |
assume d: "d dvd D"
|
chaieb@27668
|
112 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
114 |
next
|
wenzelm@23465
|
115 |
assume d: "d dvd D"
|
haftmann@27651
|
116 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t"
|
nipkow@29667
|
117 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
119 |
qed blast
|
wenzelm@23465
|
120 |
|
wenzelm@23465
|
121 |
lemma aset:
|
wenzelm@23465
|
122 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
|
wenzelm@23465
|
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>
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
125 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
|
wenzelm@23465
|
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>
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
136 |
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
|
wenzelm@23465
|
137 |
proof (blast, blast)
|
wenzelm@23465
|
138 |
assume dp: "D > 0" and tA: "t + 1 \<in> A"
|
wenzelm@23465
|
139 |
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
|
wenzelm@23465
|
140 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
|
wenzelm@23465
|
141 |
using dp tA by simp_all
|
wenzelm@23465
|
142 |
next
|
wenzelm@23465
|
143 |
assume dp: "D > 0" and tA: "t \<in> A"
|
wenzelm@23465
|
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))"
|
wenzelm@23465
|
145 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
|
wenzelm@23465
|
146 |
using dp tA by simp_all
|
wenzelm@23465
|
147 |
next
|
wenzelm@23465
|
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
|
wenzelm@23465
|
149 |
next
|
wenzelm@23465
|
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
|
wenzelm@23465
|
151 |
next
|
wenzelm@23465
|
152 |
assume dp: "D > 0" and tA:"t \<in> A"
|
wenzelm@23465
|
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"
|
wenzelm@23465
|
154 |
hence "t - x \<le> D" and "1 \<le> t - x" by simp+
|
wenzelm@23465
|
155 |
hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
|
nipkow@29667
|
156 |
hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: algebra_simps)
|
wenzelm@23465
|
157 |
with nob tA have "False" by simp}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
159 |
next
|
wenzelm@23465
|
160 |
assume dp: "D > 0" and tA:"t + 1\<in> A"
|
wenzelm@23465
|
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"
|
nipkow@29667
|
162 |
hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: algebra_simps)
|
wenzelm@23465
|
163 |
hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
|
nipkow@29667
|
164 |
hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps)
|
wenzelm@23465
|
165 |
with nob tA have "False" by simp}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
167 |
next
|
wenzelm@23465
|
168 |
assume d: "d dvd D"
|
wenzelm@23465
|
169 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
|
nipkow@29667
|
170 |
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
172 |
next
|
wenzelm@23465
|
173 |
assume d: "d dvd D"
|
wenzelm@23465
|
174 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
|
nipkow@29667
|
175 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: algebra_simps)}
|
wenzelm@23465
|
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
|
wenzelm@23465
|
177 |
qed blast
|
wenzelm@23465
|
178 |
|
wenzelm@23465
|
179 |
subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
|
wenzelm@23465
|
180 |
|
wenzelm@23465
|
181 |
subsubsection{* First some trivial facts about periodic sets or predicates *}
|
wenzelm@23465
|
182 |
lemma periodic_finite_ex:
|
wenzelm@23465
|
183 |
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
|
wenzelm@23465
|
184 |
shows "(EX x. P x) = (EX j : {1..d}. P j)"
|
wenzelm@23465
|
185 |
(is "?LHS = ?RHS")
|
wenzelm@23465
|
186 |
proof
|
wenzelm@23465
|
187 |
assume ?LHS
|
wenzelm@23465
|
188 |
then obtain x where P: "P x" ..
|
wenzelm@23465
|
189 |
have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
|
wenzelm@23465
|
190 |
hence Pmod: "P x = P(x mod d)" using modd by simp
|
wenzelm@23465
|
191 |
show ?RHS
|
wenzelm@23465
|
192 |
proof (cases)
|
wenzelm@23465
|
193 |
assume "x mod d = 0"
|
wenzelm@23465
|
194 |
hence "P 0" using P Pmod by simp
|
wenzelm@23465
|
195 |
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
|
wenzelm@23465
|
196 |
ultimately have "P d" by simp
|
huffman@35208
|
197 |
moreover have "d : {1..d}" using dpos by simp
|
wenzelm@23465
|
198 |
ultimately show ?RHS ..
|
wenzelm@23465
|
199 |
next
|
wenzelm@23465
|
200 |
assume not0: "x mod d \<noteq> 0"
|
huffman@35208
|
201 |
have "P(x mod d)" using dpos P Pmod by simp
|
wenzelm@23465
|
202 |
moreover have "x mod d : {1..d}"
|
wenzelm@23465
|
203 |
proof -
|
wenzelm@23465
|
204 |
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
|
wenzelm@23465
|
205 |
moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
|
huffman@35208
|
206 |
ultimately show ?thesis using not0 by simp
|
wenzelm@23465
|
207 |
qed
|
wenzelm@23465
|
208 |
ultimately show ?RHS ..
|
wenzelm@23465
|
209 |
qed
|
wenzelm@23465
|
210 |
qed auto
|
wenzelm@23465
|
211 |
|
wenzelm@23465
|
212 |
subsubsection{* The @{text "-\<infinity>"} Version*}
|
wenzelm@23465
|
213 |
|
wenzelm@23465
|
214 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
|
wenzelm@23465
|
215 |
by(induct rule: int_gr_induct,simp_all add:int_distrib)
|
wenzelm@23465
|
216 |
|
wenzelm@23465
|
217 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
|
wenzelm@23465
|
218 |
by(induct rule: int_gr_induct, simp_all add:int_distrib)
|
wenzelm@23465
|
219 |
|
wenzelm@23465
|
220 |
lemma decr_mult_lemma:
|
wenzelm@23465
|
221 |
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
|
wenzelm@23465
|
222 |
shows "ALL x. P x \<longrightarrow> P(x - k*d)"
|
wenzelm@23465
|
223 |
using knneg
|
wenzelm@23465
|
224 |
proof (induct rule:int_ge_induct)
|
wenzelm@23465
|
225 |
case base thus ?case by simp
|
wenzelm@23465
|
226 |
next
|
wenzelm@23465
|
227 |
case (step i)
|
wenzelm@23465
|
228 |
{fix x
|
wenzelm@23465
|
229 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
|
wenzelm@23465
|
230 |
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
|
haftmann@35050
|
231 |
by (simp add: algebra_simps)
|
wenzelm@23465
|
232 |
ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
|
wenzelm@23465
|
233 |
thus ?case ..
|
wenzelm@23465
|
234 |
qed
|
wenzelm@23465
|
235 |
|
wenzelm@23465
|
236 |
lemma minusinfinity:
|
wenzelm@23465
|
237 |
assumes dpos: "0 < d" and
|
wenzelm@23465
|
238 |
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
|
wenzelm@23465
|
239 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
|
wenzelm@23465
|
240 |
proof
|
wenzelm@23465
|
241 |
assume eP1: "EX x. P1 x"
|
wenzelm@23465
|
242 |
then obtain x where P1: "P1 x" ..
|
wenzelm@23465
|
243 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
|
wenzelm@23465
|
244 |
let ?w = "x - (abs(x-z)+1) * d"
|
wenzelm@23465
|
245 |
from dpos have w: "?w < z" by(rule decr_lemma)
|
wenzelm@23465
|
246 |
have "P1 x = P1 ?w" using P1eqP1 by blast
|
wenzelm@23465
|
247 |
also have "\<dots> = P(?w)" using w P1eqP by blast
|
wenzelm@23465
|
248 |
finally have "P ?w" using P1 by blast
|
wenzelm@23465
|
249 |
thus "EX x. P x" ..
|
wenzelm@23465
|
250 |
qed
|
wenzelm@23465
|
251 |
|
wenzelm@23465
|
252 |
lemma cpmi:
|
wenzelm@23465
|
253 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
|
wenzelm@23465
|
254 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
|
wenzelm@23465
|
255 |
and pd: "\<forall> x k. P' x = P' (x-k*D)"
|
wenzelm@23465
|
256 |
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
|
wenzelm@23465
|
257 |
(is "?L = (?R1 \<or> ?R2)")
|
wenzelm@23465
|
258 |
proof-
|
wenzelm@23465
|
259 |
{assume "?R2" hence "?L" by blast}
|
wenzelm@23465
|
260 |
moreover
|
wenzelm@23465
|
261 |
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
|
wenzelm@23465
|
262 |
moreover
|
wenzelm@23465
|
263 |
{ fix x
|
wenzelm@23465
|
264 |
assume P: "P x" and H: "\<not> ?R2"
|
wenzelm@23465
|
265 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
|
wenzelm@23465
|
266 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
|
wenzelm@23465
|
267 |
with nb P have "P (y - D)" by auto }
|
wenzelm@23465
|
268 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
|
wenzelm@23465
|
269 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
|
wenzelm@23465
|
270 |
from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
|
wenzelm@23465
|
271 |
let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
|
wenzelm@23465
|
272 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
|
wenzelm@23465
|
273 |
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
|
wenzelm@23465
|
274 |
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
|
wenzelm@23465
|
275 |
with periodic_finite_ex[OF dp pd]
|
wenzelm@23465
|
276 |
have "?R1" by blast}
|
wenzelm@23465
|
277 |
ultimately show ?thesis by blast
|
wenzelm@23465
|
278 |
qed
|
wenzelm@23465
|
279 |
|
wenzelm@23465
|
280 |
subsubsection {* The @{text "+\<infinity>"} Version*}
|
wenzelm@23465
|
281 |
|
wenzelm@23465
|
282 |
lemma plusinfinity:
|
wenzelm@23465
|
283 |
assumes dpos: "(0::int) < d" and
|
wenzelm@23465
|
284 |
P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
|
wenzelm@23465
|
285 |
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
|
wenzelm@23465
|
286 |
proof
|
wenzelm@23465
|
287 |
assume eP1: "EX x. P' x"
|
wenzelm@23465
|
288 |
then obtain x where P1: "P' x" ..
|
wenzelm@23465
|
289 |
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
|
wenzelm@23465
|
290 |
let ?w' = "x + (abs(x-z)+1) * d"
|
wenzelm@23465
|
291 |
let ?w = "x - (-(abs(x-z) + 1))*d"
|
nipkow@29667
|
292 |
have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps)
|
wenzelm@23465
|
293 |
from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
|
wenzelm@23465
|
294 |
hence "P' x = P' ?w" using P1eqP1 by blast
|
wenzelm@23465
|
295 |
also have "\<dots> = P(?w)" using w P1eqP by blast
|
wenzelm@23465
|
296 |
finally have "P ?w" using P1 by blast
|
wenzelm@23465
|
297 |
thus "EX x. P x" ..
|
wenzelm@23465
|
298 |
qed
|
wenzelm@23465
|
299 |
|
wenzelm@23465
|
300 |
lemma incr_mult_lemma:
|
wenzelm@23465
|
301 |
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
|
wenzelm@23465
|
302 |
shows "ALL x. P x \<longrightarrow> P(x + k*d)"
|
wenzelm@23465
|
303 |
using knneg
|
wenzelm@23465
|
304 |
proof (induct rule:int_ge_induct)
|
wenzelm@23465
|
305 |
case base thus ?case by simp
|
wenzelm@23465
|
306 |
next
|
wenzelm@23465
|
307 |
case (step i)
|
wenzelm@23465
|
308 |
{fix x
|
wenzelm@23465
|
309 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
|
wenzelm@23465
|
310 |
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
|
huffman@45637
|
311 |
by (simp add:int_distrib add_ac)
|
wenzelm@23465
|
312 |
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
|
wenzelm@23465
|
313 |
thus ?case ..
|
wenzelm@23465
|
314 |
qed
|
wenzelm@23465
|
315 |
|
wenzelm@23465
|
316 |
lemma cppi:
|
wenzelm@23465
|
317 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
|
wenzelm@23465
|
318 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
|
wenzelm@23465
|
319 |
and pd: "\<forall> x k. P' x= P' (x-k*D)"
|
wenzelm@23465
|
320 |
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)")
|
wenzelm@23465
|
321 |
proof-
|
wenzelm@23465
|
322 |
{assume "?R2" hence "?L" by blast}
|
wenzelm@23465
|
323 |
moreover
|
wenzelm@23465
|
324 |
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
|
wenzelm@23465
|
325 |
moreover
|
wenzelm@23465
|
326 |
{ fix x
|
wenzelm@23465
|
327 |
assume P: "P x" and H: "\<not> ?R2"
|
wenzelm@23465
|
328 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
|
wenzelm@23465
|
329 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
|
wenzelm@23465
|
330 |
with nb P have "P (y + D)" by auto }
|
wenzelm@23465
|
331 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
|
wenzelm@23465
|
332 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
|
wenzelm@23465
|
333 |
from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
|
wenzelm@23465
|
334 |
let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
|
wenzelm@23465
|
335 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
|
wenzelm@23465
|
336 |
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
|
wenzelm@23465
|
337 |
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
|
wenzelm@23465
|
338 |
with periodic_finite_ex[OF dp pd]
|
wenzelm@23465
|
339 |
have "?R1" by blast}
|
wenzelm@23465
|
340 |
ultimately show ?thesis by blast
|
wenzelm@23465
|
341 |
qed
|
wenzelm@23465
|
342 |
|
wenzelm@23465
|
343 |
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
|
wenzelm@23465
|
344 |
apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
|
nipkow@45761
|
345 |
apply(fastforce)
|
wenzelm@23465
|
346 |
done
|
wenzelm@23465
|
347 |
|
haftmann@35050
|
348 |
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
|
haftmann@27651
|
349 |
apply (rule eq_reflection [symmetric])
|
wenzelm@23465
|
350 |
apply (rule iffI)
|
wenzelm@23465
|
351 |
defer
|
wenzelm@23465
|
352 |
apply (erule exE)
|
wenzelm@23465
|
353 |
apply (rule_tac x = "l * x" in exI)
|
wenzelm@23465
|
354 |
apply (simp add: dvd_def)
|
haftmann@27651
|
355 |
apply (rule_tac x = x in exI, simp)
|
wenzelm@23465
|
356 |
apply (erule exE)
|
wenzelm@23465
|
357 |
apply (erule conjE)
|
haftmann@27651
|
358 |
apply simp
|
wenzelm@23465
|
359 |
apply (erule dvdE)
|
wenzelm@23465
|
360 |
apply (rule_tac x = k in exI)
|
wenzelm@23465
|
361 |
apply simp
|
wenzelm@23465
|
362 |
done
|
wenzelm@23465
|
363 |
|
wenzelm@23465
|
364 |
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
|
wenzelm@23465
|
365 |
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
|
wenzelm@23465
|
366 |
using not0 by (simp add: dvd_def)
|
wenzelm@23465
|
367 |
|
wenzelm@23465
|
368 |
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
|
wenzelm@23465
|
369 |
by simp_all
|
haftmann@32553
|
370 |
|
wenzelm@23465
|
371 |
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
|
haftmann@32553
|
372 |
|
wenzelm@23465
|
373 |
lemma zdiff_int_split: "P (int (x - y)) =
|
wenzelm@23465
|
374 |
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
|
haftmann@36794
|
375 |
by (cases "y \<le> x") (simp_all add: zdiff_int)
|
wenzelm@23465
|
376 |
|
wenzelm@23465
|
377 |
text {*
|
wenzelm@23465
|
378 |
\medskip Specific instances of congruence rules, to prevent
|
wenzelm@23465
|
379 |
simplifier from looping. *}
|
wenzelm@23465
|
380 |
|
huffman@47978
|
381 |
theorem imp_le_cong:
|
huffman@47978
|
382 |
"\<lbrakk>x = x'; 0 \<le> x' \<Longrightarrow> P = P'\<rbrakk> \<Longrightarrow> (0 \<le> (x::int) \<longrightarrow> P) = (0 \<le> x' \<longrightarrow> P')"
|
huffman@47978
|
383 |
by simp
|
wenzelm@23465
|
384 |
|
huffman@47978
|
385 |
theorem conj_le_cong:
|
huffman@47978
|
386 |
"\<lbrakk>x = x'; 0 \<le> x' \<Longrightarrow> P = P'\<rbrakk> \<Longrightarrow> (0 \<le> (x::int) \<and> P) = (0 \<le> x' \<and> P')"
|
wenzelm@23465
|
387 |
by (simp cong: conj_cong)
|
haftmann@36793
|
388 |
|
haftmann@36792
|
389 |
use "Tools/Qelim/cooper.ML"
|
wenzelm@23465
|
390 |
|
haftmann@36793
|
391 |
setup Cooper.setup
|
wenzelm@23465
|
392 |
|
haftmann@36798
|
393 |
method_setup presburger = "Cooper.method" "Cooper's algorithm for Presburger arithmetic"
|
wenzelm@23465
|
394 |
|
haftmann@36792
|
395 |
declare dvd_eq_mod_eq_0[symmetric, presburger]
|
haftmann@36792
|
396 |
declare mod_1[presburger]
|
haftmann@36792
|
397 |
declare mod_0[presburger]
|
haftmann@36792
|
398 |
declare mod_by_1[presburger]
|
haftmann@36792
|
399 |
declare mod_self[presburger]
|
haftmann@36792
|
400 |
declare mod_by_0[presburger]
|
haftmann@36792
|
401 |
declare mod_div_trivial[presburger]
|
haftmann@36792
|
402 |
declare div_mod_equality2[presburger]
|
haftmann@36792
|
403 |
declare div_mod_equality[presburger]
|
haftmann@36792
|
404 |
declare mod_div_equality2[presburger]
|
haftmann@36792
|
405 |
declare mod_div_equality[presburger]
|
haftmann@36792
|
406 |
declare mod_mult_self1[presburger]
|
haftmann@36792
|
407 |
declare mod_mult_self2[presburger]
|
haftmann@36792
|
408 |
declare zdiv_zmod_equality2[presburger]
|
haftmann@36792
|
409 |
declare zdiv_zmod_equality[presburger]
|
haftmann@36792
|
410 |
declare mod2_Suc_Suc[presburger]
|
haftmann@36792
|
411 |
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
|
haftmann@36792
|
412 |
by simp_all
|
haftmann@36792
|
413 |
|
chaieb@27668
|
414 |
lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
|
chaieb@27668
|
415 |
lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
|
chaieb@27668
|
416 |
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
|
chaieb@27668
|
417 |
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
|
chaieb@27668
|
418 |
lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
|
wenzelm@23465
|
419 |
|
wenzelm@23465
|
420 |
end
|