src/HOL/Presburger.thy
author huffman
Sat Jun 06 09:11:12 2009 -0700 (2009-06-06)
changeset 31488 5691ccb8d6b5
parent 30686 47a32dd1b86e
child 31790 05c92381363c
permissions -rw-r--r--
generalize tendsto to class topological_space
     1 (* Title:      HOL/Presburger.thy
     2    Author:     Amine Chaieb, TU Muenchen
     3 *)
     4 
     5 header {* Decision Procedure for Presburger Arithmetic *}
     6 
     7 theory Presburger
     8 imports Groebner_Basis SetInterval
     9 uses
    10   "Tools/Qelim/qelim.ML"
    11   "Tools/Qelim/cooper_data.ML"
    12   "Tools/Qelim/generated_cooper.ML"
    13   ("Tools/Qelim/cooper.ML")
    14   ("Tools/Qelim/presburger.ML")
    15 begin
    16 
    17 setup CooperData.setup
    18 
    19 subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
    20 
    21 
    22 lemma minf:
    23   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
    24      \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
    25   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
    26      \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
    27   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
    28   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
    29   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
    30   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
    31   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
    32   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
    33   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (d dvd x + s) = (d dvd x + s)"
    34   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
    35   "\<exists>z.\<forall>x<z. F = F"
    36   by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
    37 
    38 lemma pinf:
    39   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
    40      \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
    41   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
    42      \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
    43   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
    44   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
    45   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
    46   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
    47   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
    48   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
    49   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (d dvd x + s) = (d dvd x + s)"
    50   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
    51   "\<exists>z.\<forall>x>z. F = F"
    52   by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
    53 
    54 lemma inf_period:
    55   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
    56     \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
    57   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
    58     \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
    59   "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
    60   "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
    61   "\<forall>x k. F = F"
    62 apply (auto elim!: dvdE simp add: algebra_simps)
    63 unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
    64 unfolding dvd_def mult_commute [of d] 
    65 by auto
    66 
    67 subsection{* The A and B sets *}
    68 lemma bset:
    69   "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
    70      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
    71   \<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))"
    72   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
    73      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
    74   \<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))"
    75   "\<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))"
    76   "\<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))"
    77   "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))"
    78   "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))"
    79   "\<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))"
    80   "\<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))"
    81   "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))"
    82   "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))"
    83   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
    84 proof (blast, blast)
    85   assume dp: "D > 0" and tB: "t - 1\<in> B"
    86   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 
    87     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) 
    88     apply algebra using dp tB by simp_all
    89 next
    90   assume dp: "D > 0" and tB: "t \<in> B"
    91   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))" 
    92     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
    93     apply algebra
    94     using dp tB by simp_all
    95 next
    96   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
    97 next
    98   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
    99 next
   100   assume dp: "D > 0" and tB:"t \<in> B"
   101   {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"
   102     hence "x -t \<le> D" and "1 \<le> x - t" by simp+
   103       hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
   104       hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps)
   105       with nob tB have "False" by simp}
   106   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
   107 next
   108   assume dp: "D > 0" and tB:"t - 1\<in> B"
   109   {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"
   110     hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
   111       hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
   112       hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps)
   113       with nob tB have "False" by simp}
   114   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
   115 next
   116   assume d: "d dvd D"
   117   {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra}
   118   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
   119 next
   120   assume d: "d dvd D"
   121   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t"
   122       by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)}
   123   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
   124 qed blast
   125 
   126 lemma aset:
   127   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
   128      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
   129   \<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))"
   130   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
   131      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
   132   \<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))"
   133   "\<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))"
   134   "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
   135   "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
   136   "\<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))"
   137   "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))"
   138   "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))"
   139   "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))"
   140   "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))"
   141   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
   142 proof (blast, blast)
   143   assume dp: "D > 0" and tA: "t + 1 \<in> A"
   144   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 
   145     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
   146     using dp tA by simp_all
   147 next
   148   assume dp: "D > 0" and tA: "t \<in> A"
   149   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))" 
   150     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
   151     using dp tA by simp_all
   152 next
   153   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
   154 next
   155   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
   156 next
   157   assume dp: "D > 0" and tA:"t \<in> A"
   158   {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"
   159     hence "t - x \<le> D" and "1 \<le> t - x" by simp+
   160       hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
   161       hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: algebra_simps) 
   162       with nob tA have "False" by simp}
   163   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
   164 next
   165   assume dp: "D > 0" and tA:"t + 1\<in> A"
   166   {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"
   167     hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: algebra_simps)
   168       hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
   169       hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps)
   170       with nob tA have "False" by simp}
   171   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
   172 next
   173   assume d: "d dvd D"
   174   {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
   175       by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)}
   176   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
   177 next
   178   assume d: "d dvd D"
   179   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
   180       by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: algebra_simps)}
   181   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
   182 qed blast
   183 
   184 subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
   185 
   186 subsubsection{* First some trivial facts about periodic sets or predicates *}
   187 lemma periodic_finite_ex:
   188   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   189   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   190   (is "?LHS = ?RHS")
   191 proof
   192   assume ?LHS
   193   then obtain x where P: "P x" ..
   194   have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   195   hence Pmod: "P x = P(x mod d)" using modd by simp
   196   show ?RHS
   197   proof (cases)
   198     assume "x mod d = 0"
   199     hence "P 0" using P Pmod by simp
   200     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   201     ultimately have "P d" by simp
   202     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   203     ultimately show ?RHS ..
   204   next
   205     assume not0: "x mod d \<noteq> 0"
   206     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   207     moreover have "x mod d : {1..d}"
   208     proof -
   209       from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
   210       moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
   211       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   212     qed
   213     ultimately show ?RHS ..
   214   qed
   215 qed auto
   216 
   217 subsubsection{* The @{text "-\<infinity>"} Version*}
   218 
   219 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   220 by(induct rule: int_gr_induct,simp_all add:int_distrib)
   221 
   222 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   223 by(induct rule: int_gr_induct, simp_all add:int_distrib)
   224 
   225 theorem int_induct[case_names base step1 step2]:
   226   assumes 
   227   base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
   228   step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
   229   shows "P i"
   230 proof -
   231   have "i \<le> k \<or> i\<ge> k" by arith
   232   thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
   233 qed
   234 
   235 lemma decr_mult_lemma:
   236   assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
   237   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   238 using knneg
   239 proof (induct rule:int_ge_induct)
   240   case base thus ?case by simp
   241 next
   242   case (step i)
   243   {fix x
   244     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   245     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
   246       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
   247     ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
   248   thus ?case ..
   249 qed
   250 
   251 lemma  minusinfinity:
   252   assumes dpos: "0 < d" and
   253     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   254   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   255 proof
   256   assume eP1: "EX x. P1 x"
   257   then obtain x where P1: "P1 x" ..
   258   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   259   let ?w = "x - (abs(x-z)+1) * d"
   260   from dpos have w: "?w < z" by(rule decr_lemma)
   261   have "P1 x = P1 ?w" using P1eqP1 by blast
   262   also have "\<dots> = P(?w)" using w P1eqP by blast
   263   finally have "P ?w" using P1 by blast
   264   thus "EX x. P x" ..
   265 qed
   266 
   267 lemma cpmi: 
   268   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
   269   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
   270   and pd: "\<forall> x k. P' x = P' (x-k*D)"
   271   shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 
   272          (is "?L = (?R1 \<or> ?R2)")
   273 proof-
   274  {assume "?R2" hence "?L"  by blast}
   275  moreover
   276  {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
   277  moreover 
   278  { fix x
   279    assume P: "P x" and H: "\<not> ?R2"
   280    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
   281      hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
   282      with nb P  have "P (y - D)" by auto }
   283    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
   284    with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
   285    from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
   286    let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
   287    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
   288    from dp have yz: "?y < z" using decr_lemma[OF dp] by simp   
   289    from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
   290    with periodic_finite_ex[OF dp pd]
   291    have "?R1" by blast}
   292  ultimately show ?thesis by blast
   293 qed
   294 
   295 subsubsection {* The @{text "+\<infinity>"} Version*}
   296 
   297 lemma  plusinfinity:
   298   assumes dpos: "(0::int) < d" and
   299     P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
   300   shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
   301 proof
   302   assume eP1: "EX x. P' x"
   303   then obtain x where P1: "P' x" ..
   304   from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
   305   let ?w' = "x + (abs(x-z)+1) * d"
   306   let ?w = "x - (-(abs(x-z) + 1))*d"
   307   have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps)
   308   from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
   309   hence "P' x = P' ?w" using P1eqP1 by blast
   310   also have "\<dots> = P(?w)" using w P1eqP by blast
   311   finally have "P ?w" using P1 by blast
   312   thus "EX x. P x" ..
   313 qed
   314 
   315 lemma incr_mult_lemma:
   316   assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
   317   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   318 using knneg
   319 proof (induct rule:int_ge_induct)
   320   case base thus ?case by simp
   321 next
   322   case (step i)
   323   {fix x
   324     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   325     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
   326       by (simp add:int_distrib zadd_ac)
   327     ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
   328   thus ?case ..
   329 qed
   330 
   331 lemma cppi: 
   332   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
   333   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
   334   and pd: "\<forall> x k. P' x= P' (x-k*D)"
   335   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)")
   336 proof-
   337  {assume "?R2" hence "?L"  by blast}
   338  moreover
   339  {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
   340  moreover 
   341  { fix x
   342    assume P: "P x" and H: "\<not> ?R2"
   343    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
   344      hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
   345      with nb P  have "P (y + D)" by auto }
   346    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
   347    with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
   348    from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
   349    let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
   350    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
   351    from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
   352    from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
   353    with periodic_finite_ex[OF dp pd]
   354    have "?R1" by blast}
   355  ultimately show ?thesis by blast
   356 qed
   357 
   358 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   359 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   360 apply(fastsimp)
   361 done
   362 
   363 theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Ring_and_Field.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
   364   apply (rule eq_reflection [symmetric])
   365   apply (rule iffI)
   366   defer
   367   apply (erule exE)
   368   apply (rule_tac x = "l * x" in exI)
   369   apply (simp add: dvd_def)
   370   apply (rule_tac x = x in exI, simp)
   371   apply (erule exE)
   372   apply (erule conjE)
   373   apply simp
   374   apply (erule dvdE)
   375   apply (rule_tac x = k in exI)
   376   apply simp
   377   done
   378 
   379 lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
   380 shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
   381   using not0 by (simp add: dvd_def)
   382 
   383 lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
   384   by simp_all
   385 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
   386 lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   387   by (simp split add: split_nat)
   388 
   389 lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   390   apply (auto split add: split_nat)
   391   apply (rule_tac x="int x" in exI, simp)
   392   apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
   393   done
   394 
   395 lemma zdiff_int_split: "P (int (x - y)) =
   396   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   397   by (case_tac "y \<le> x", simp_all add: zdiff_int)
   398 
   399 lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (Int.Bit0 n) \<and> (0::int) <= number_of (Int.Bit1 n)"
   400 by simp
   401 lemma number_of2: "(0::int) <= Numeral0" by simp
   402 lemma Suc_plus1: "Suc n = n + 1" by simp
   403 
   404 text {*
   405   \medskip Specific instances of congruence rules, to prevent
   406   simplifier from looping. *}
   407 
   408 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
   409 
   410 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
   411   by (simp cong: conj_cong)
   412 lemma int_eq_number_of_eq:
   413   "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
   414   by (rule eq_number_of_eq)
   415 
   416 declare dvd_eq_mod_eq_0[symmetric, presburger]
   417 declare mod_1[presburger] 
   418 declare mod_0[presburger]
   419 declare mod_by_1[presburger]
   420 declare zmod_zero[presburger]
   421 declare zmod_self[presburger]
   422 declare mod_self[presburger]
   423 declare mod_by_0[presburger]
   424 declare mod_div_trivial[presburger]
   425 declare div_mod_equality2[presburger]
   426 declare div_mod_equality[presburger]
   427 declare mod_div_equality2[presburger]
   428 declare mod_div_equality[presburger]
   429 declare mod_mult_self1[presburger]
   430 declare mod_mult_self2[presburger]
   431 declare zdiv_zmod_equality2[presburger]
   432 declare zdiv_zmod_equality[presburger]
   433 declare mod2_Suc_Suc[presburger]
   434 lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
   435 by simp_all
   436 
   437 use "Tools/Qelim/cooper.ML"
   438 oracle linzqe_oracle = Coopereif.cooper_oracle
   439 
   440 use "Tools/Qelim/presburger.ML"
   441 
   442 setup {* Arith_Data.add_tactic "Presburger arithmetic" (K (Presburger.cooper_tac true [] [])) *}
   443 
   444 method_setup presburger = {*
   445 let
   446  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
   447  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
   448  val addN = "add"
   449  val delN = "del"
   450  val elimN = "elim"
   451  val any_keyword = keyword addN || keyword delN || simple_keyword elimN
   452  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
   453 in
   454   Scan.optional (simple_keyword elimN >> K false) true --
   455   Scan.optional (keyword addN |-- thms) [] --
   456   Scan.optional (keyword delN |-- thms) [] >>
   457   (fn ((elim, add_ths), del_ths) => fn ctxt =>
   458     SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
   459 end
   460 *} "Cooper's algorithm for Presburger arithmetic"
   461 
   462 lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   463 lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
   464 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   465 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
   466 lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   467 
   468 
   469 lemma zdvd_period:
   470   fixes a d :: int
   471   assumes advdd: "a dvd d"
   472   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
   473   using advdd
   474   apply -
   475   apply (rule iffI)
   476   by algebra+
   477 
   478 end