src/HOL/Presburger.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62348 9a5f43dac883 child 63961 2fd9656c4c82 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     1 (* Title:      HOL/Presburger.thy
```
```     2    Author:     Amine Chaieb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Decision Procedure for Presburger Arithmetic\<close>
```
```     6
```
```     7 theory Presburger
```
```     8 imports Groebner_Basis Set_Interval
```
```     9 keywords "try0" :: diag
```
```    10 begin
```
```    11
```
```    12 ML_file "Tools/Qelim/qelim.ML"
```
```    13 ML_file "Tools/Qelim/cooper_procedure.ML"
```
```    14
```
```    15 subsection\<open>The \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Properties\<close>
```
```    16
```
```    17 lemma 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
```
```    33 lemma 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
```
```    49 lemma 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"
```
```    57 apply (auto elim!: dvdE simp add: algebra_simps)
```
```    58 unfolding mult.assoc [symmetric] distrib_right [symmetric] left_diff_distrib [symmetric]
```
```    59 unfolding dvd_def mult.commute [of d]
```
```    60 by auto
```
```    61
```
```    62 subsection\<open>The A and B sets\<close>
```
```    63 lemma 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"
```
```    79 proof (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
```
```    84 next
```
```    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
```
```    90 next
```
```    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
```
```    92 next
```
```    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
```
```    94 next
```
```    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
```
```   102 next
```
```   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
```
```   110 next
```
```   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
```
```   114 next
```
```   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
```
```   119 qed blast
```
```   120
```
```   121 lemma 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"
```
```   137 proof (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
```
```   142 next
```
```   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
```
```   147 next
```
```   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
```
```   149 next
```
```   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
```
```   151 next
```
```   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
```
```   159 next
```
```   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
```
```   167 next
```
```   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
```
```   172 next
```
```   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
```
```   177 qed blast
```
```   178
```
```   179 subsection\<open>Cooper's Theorem \<open>-\<infinity>\<close> and \<open>+\<infinity>\<close> Version\<close>
```
```   180
```
```   181 subsubsection\<open>First some trivial facts about periodic sets or predicates\<close>
```
```   182 lemma periodic_finite_ex:
```
```   183   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
```
```   184   shows "(EX x. P x) = (EX j : {1..d}. P j)"
```
```   185   (is "?LHS = ?RHS")
```
```   186 proof
```
```   187   assume ?LHS
```
```   188   then obtain x where P: "P x" ..
```
```   189   have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality 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 : {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 : {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
```
```   210 qed auto
```
```   211
```
```   212 subsubsection\<open>The \<open>-\<infinity>\<close> Version\<close>
```
```   213
```
```   214 lemma 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
```
```   217 lemma 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
```
```   220 lemma 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 "ALL x. P x \<longrightarrow> P(x - k*d)"
```
```   223 using knneg
```
```   224 proof (induct rule:int_ge_induct)
```
```   225   case base thus ?case by simp
```
```   226 next
```
```   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 ..
```
```   234 qed
```
```   235
```
```   236 lemma  minusinfinity:
```
```   237   assumes dpos: "0 < d" and
```
```   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)"
```
```   239   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
```
```   240 proof
```
```   241   assume eP1: "EX x. P1 x"
```
```   242   then obtain x where P1: "P1 x" ..
```
```   243   from ePeqP1 obtain z where P1eqP: "ALL 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 "EX x. P x" ..
```
```   250 qed
```
```   251
```
```   252 lemma 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) --> P (x) --> 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) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
```
```   257          (is "?L = (?R1 \<or> ?R2)")
```
```   258 proof-
```
```   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 "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
```
```   267      with nb P  have "P (y - D)" by auto }
```
```   268    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> 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: "ALL x. x < z --> (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
```
```   278 qed
```
```   279
```
```   280 subsubsection \<open>The \<open>+\<infinity>\<close> Version\<close>
```
```   281
```
```   282 lemma  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)"
```
```   286 proof
```
```   287   assume eP1: "EX 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 "EX x. P x" ..
```
```   298 qed
```
```   299
```
```   300 lemma incr_mult_lemma:
```
```   301   assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
```
```   302   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
```
```   303 using knneg
```
```   304 proof (induct rule:int_ge_induct)
```
```   305   case base thus ?case by simp
```
```   306 next
```
```   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 ..
```
```   314 qed
```
```   315
```
```   316 lemma 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) --> P (x) --> 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) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
```
```   321 proof-
```
```   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 "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
```
```   330      with nb P  have "P (y + D)" by auto }
```
```   331    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> 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: "ALL x. x > z --> (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
```
```   341 qed
```
```   342
```
```   343 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
```
```   344 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   345 apply(fastforce)
```
```   346 done
```
```   347
```
```   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)"
```
```   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
```
```   364 lemma 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
```
```   370 lemma 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
```
```   375 text \<open>\bigskip Theorems for transforming predicates on nat to predicates on \<open>int\<close>\<close>
```
```   376
```
```   377 lemma 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
```
```   381 text \<open>
```
```   382   \medskip Specific instances of congruence rules, to prevent
```
```   383   simplifier from looping.\<close>
```
```   384
```
```   385 theorem 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
```
```   389 theorem 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
```
```   393 ML_file "Tools/Qelim/cooper.ML"
```
```   394
```
```   395 method_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
```
```   413 declare dvd_eq_mod_eq_0 [symmetric, presburger]
```
```   414 declare mod_1 [presburger]
```
```   415 declare mod_0 [presburger]
```
```   416 declare mod_by_1 [presburger]
```
```   417 declare mod_self [presburger]
```
```   418 declare div_by_0 [presburger]
```
```   419 declare mod_by_0 [presburger]
```
```   420 declare mod_div_trivial [presburger]
```
```   421 declare div_mod_equality2 [presburger]
```
```   422 declare div_mod_equality [presburger]
```
```   423 declare mod_div_equality2 [presburger]
```
```   424 declare mod_div_equality [presburger]
```
```   425 declare mod_mult_self1 [presburger]
```
```   426 declare mod_mult_self2 [presburger]
```
```   427 declare mod2_Suc_Suc[presburger]
```
```   428 declare not_mod_2_eq_0_eq_1 [presburger]
```
```   429 declare nat_zero_less_power_iff [presburger]
```
```   430
```
```   431 lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   432 lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   433 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   434 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   435 lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   436
```
```   437 context semiring_parity
```
```   438 begin
```
```   439
```
```   440 declare even_times_iff [presburger]
```
```   441
```
```   442 declare even_power [presburger]
```
```   443
```
```   444 lemma [presburger]:
```
```   445   "even (a + b) \<longleftrightarrow> even a \<and> even b \<or> odd a \<and> odd b"
```
```   446   by auto
```
```   447
```
```   448 end
```
```   449
```
```   450 context ring_parity
```
```   451 begin
```
```   452
```
```   453 declare even_minus [presburger]
```
```   454
```
```   455 end
```
```   456
```
```   457 context linordered_idom
```
```   458 begin
```
```   459
```
```   460 declare zero_le_power_eq [presburger]
```
```   461
```
```   462 declare zero_less_power_eq [presburger]
```
```   463
```
```   464 declare power_less_zero_eq [presburger]
```
```   465
```
```   466 declare power_le_zero_eq [presburger]
```
```   467
```
```   468 end
```
```   469
```
```   470 declare even_Suc [presburger]
```
```   471
```
```   472 lemma [presburger]:
```
```   473   "Suc n div Suc (Suc 0) = n div Suc (Suc 0) \<longleftrightarrow> even n"
```
```   474   by presburger
```
```   475
```
```   476 declare even_diff_nat [presburger]
```
```   477
```
```   478 lemma [presburger]:
```
```   479   fixes k :: int
```
```   480   shows "(k + 1) div 2 = k div 2 \<longleftrightarrow> even k"
```
```   481   by presburger
```
```   482
```
```   483 lemma [presburger]:
```
```   484   fixes k :: int
```
```   485   shows "(k + 1) div 2 = k div 2 + 1 \<longleftrightarrow> odd k"
```
```   486   by presburger
```
```   487
```
```   488 lemma [presburger]:
```
```   489   "even n \<longleftrightarrow> even (int n)"
```
```   490   using even_int_iff [of n] by simp
```
```   491
```
```   492
```
```   493 subsection \<open>Nice facts about division by @{term 4}\<close>
```
```   494
```
```   495 lemma even_even_mod_4_iff:
```
```   496   "even (n::nat) \<longleftrightarrow> even (n mod 4)"
```
```   497   by presburger
```
```   498
```
```   499 lemma odd_mod_4_div_2:
```
```   500   "n mod 4 = (3::nat) \<Longrightarrow> odd ((n - 1) div 2)"
```
```   501   by presburger
```
```   502
```
```   503 lemma even_mod_4_div_2:
```
```   504   "n mod 4 = (1::nat) \<Longrightarrow> even ((n - 1) div 2)"
```
```   505   by presburger
```
```   506
```
```   507
```
```   508 subsection \<open>Try0\<close>
```
```   509
```
```   510 ML_file "Tools/try0.ML"
```
```   511
```
```   512 end
```