src/HOL/Presburger.thy
author wenzelm
Thu Sep 18 19:39:44 2008 +0200 (2008-09-18)
changeset 28290 4cc2b6046258
parent 27668 6eb20b2cecf8
child 28402 09e4aa3ddc25
permissions -rw-r--r--
simplified oracle interface;
     1 (* Title:      HOL/Presburger.thy
     2    ID:         $Id$
     3    Author:     Amine Chaieb, TU Muenchen
     4 *)
     5 
     6 header {* Decision Procedure for Presburger Arithmetic *}
     7 
     8 theory Presburger
     9 imports Arith_Tools SetInterval
    10 uses
    11   "Tools/Qelim/cooper_data.ML"
    12   "Tools/Qelim/generated_cooper.ML"
    13   "Tools/Qelim/qelim.ML"
    14   ("Tools/Qelim/cooper.ML")
    15   ("Tools/Qelim/presburger.ML")
    16 begin
    17 
    18 setup CooperData.setup
    19 
    20 subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
    21 
    22 
    23 lemma minf:
    24   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
    25      \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
    26   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
    27      \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
    28   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
    29   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
    30   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
    31   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
    32   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
    33   "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
    34   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (d dvd x + s) = (d dvd x + s)"
    35   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
    36   "\<exists>z.\<forall>x<z. F = F"
    37   by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
    38 
    39 lemma pinf:
    40   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
    41      \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
    42   "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
    43      \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
    44   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
    45   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
    46   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
    47   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
    48   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
    49   "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
    50   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (d dvd x + s) = (d dvd x + s)"
    51   "\<exists>z.\<forall>(x::'a::{linorder,plus,Ring_and_Field.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
    52   "\<exists>z.\<forall>x>z. F = F"
    53   by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
    54 
    55 lemma inf_period:
    56   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
    57     \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
    58   "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
    59     \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
    60   "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
    61   "(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
    62   "\<forall>x k. F = F"
    63 apply (auto elim!: dvdE simp add: ring_simps)
    64 unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
    65 unfolding dvd_def mult_commute [of d] 
    66 by auto
    67 
    68 subsection{* The A and B sets *}
    69 lemma bset:
    70   "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
    71      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
    72   \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))"
    73   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
    74      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
    75   \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))"
    76   "\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
    77   "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))"
    78   "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))"
    79   "D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))"
    80   "\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))"
    81   "\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))"
    82   "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t))"
    83   "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x - D) + t))"
    84   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
    85 proof (blast, blast)
    86   assume dp: "D > 0" and tB: "t - 1\<in> B"
    87   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 
    88     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) 
    89     apply algebra using dp tB by simp_all
    90 next
    91   assume dp: "D > 0" and tB: "t \<in> B"
    92   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))" 
    93     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
    94     apply algebra
    95     using dp tB by simp_all
    96 next
    97   assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith
    98 next
    99   assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
   100 next
   101   assume dp: "D > 0" and tB:"t \<in> B"
   102   {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
   103     hence "x -t \<le> D" and "1 \<le> x - t" by simp+
   104       hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
   105       hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps)
   106       with nob tB have "False" by simp}
   107   thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
   108 next
   109   assume dp: "D > 0" and tB:"t - 1\<in> B"
   110   {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
   111     hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
   112       hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
   113       hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps)
   114       with nob tB have "False" by simp}
   115   thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
   116 next
   117   assume d: "d dvd D"
   118   {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra}
   119   thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp
   120 next
   121   assume d: "d dvd D"
   122   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t"
   123       by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)}
   124   thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
   125 qed blast
   126 
   127 lemma aset:
   128   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
   129      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
   130   \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x + D) \<and> Q (x + D))"
   131   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
   132      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
   133   \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x + D) \<or> Q (x + D))"
   134   "\<lbrakk>D>0; t + 1\<in> A\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
   135   "\<lbrakk>D>0 ; t \<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))"
   136   "\<lbrakk>D>0; t\<in> A\<rbrakk> \<Longrightarrow>(\<forall>(x::int). (\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t))"
   137   "\<lbrakk>D>0; t + 1 \<in> A\<rbrakk> \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t))"
   138   "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))"
   139   "D>0 \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t))"
   140   "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t))"
   141   "d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x + D) + t))"
   142   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
   143 proof (blast, blast)
   144   assume dp: "D > 0" and tA: "t + 1 \<in> A"
   145   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 
   146     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
   147     using dp tA by simp_all
   148 next
   149   assume dp: "D > 0" and tA: "t \<in> A"
   150   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x + D \<noteq> t))" 
   151     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
   152     using dp tA by simp_all
   153 next
   154   assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x > t) \<longrightarrow> (x + D > t))" by arith
   155 next
   156   assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
   157 next
   158   assume dp: "D > 0" and tA:"t \<in> A"
   159   {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
   160     hence "t - x \<le> D" and "1 \<le> t - x" by simp+
   161       hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
   162       hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps) 
   163       with nob tA have "False" by simp}
   164   thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
   165 next
   166   assume dp: "D > 0" and tA:"t + 1\<in> A"
   167   {fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
   168     hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps)
   169       hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
   170       hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps)
   171       with nob tA have "False" by simp}
   172   thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
   173 next
   174   assume d: "d dvd D"
   175   {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
   176       by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)}
   177   thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
   178 next
   179   assume d: "d dvd D"
   180   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
   181       by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)}
   182   thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
   183 qed blast
   184 
   185 subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
   186 
   187 subsubsection{* First some trivial facts about periodic sets or predicates *}
   188 lemma periodic_finite_ex:
   189   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
   190   shows "(EX x. P x) = (EX j : {1..d}. P j)"
   191   (is "?LHS = ?RHS")
   192 proof
   193   assume ?LHS
   194   then obtain x where P: "P x" ..
   195   have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
   196   hence Pmod: "P x = P(x mod d)" using modd by simp
   197   show ?RHS
   198   proof (cases)
   199     assume "x mod d = 0"
   200     hence "P 0" using P Pmod by simp
   201     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
   202     ultimately have "P d" by simp
   203     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
   204     ultimately show ?RHS ..
   205   next
   206     assume not0: "x mod d \<noteq> 0"
   207     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
   208     moreover have "x mod d : {1..d}"
   209     proof -
   210       from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
   211       moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
   212       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
   213     qed
   214     ultimately show ?RHS ..
   215   qed
   216 qed auto
   217 
   218 subsubsection{* The @{text "-\<infinity>"} Version*}
   219 
   220 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
   221 by(induct rule: int_gr_induct,simp_all add:int_distrib)
   222 
   223 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
   224 by(induct rule: int_gr_induct, simp_all add:int_distrib)
   225 
   226 theorem int_induct[case_names base step1 step2]:
   227   assumes 
   228   base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
   229   step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
   230   shows "P i"
   231 proof -
   232   have "i \<le> k \<or> i\<ge> k" by arith
   233   thus ?thesis using prems int_ge_induct[where P="P" and k="k" and i="i"] int_le_induct[where P="P" and k="k" and i="i"] by blast
   234 qed
   235 
   236 lemma decr_mult_lemma:
   237   assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
   238   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
   239 using knneg
   240 proof (induct rule:int_ge_induct)
   241   case base thus ?case by simp
   242 next
   243   case (step i)
   244   {fix x
   245     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
   246     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
   247       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
   248     ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
   249   thus ?case ..
   250 qed
   251 
   252 lemma  minusinfinity:
   253   assumes dpos: "0 < d" and
   254     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
   255   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
   256 proof
   257   assume eP1: "EX x. P1 x"
   258   then obtain x where P1: "P1 x" ..
   259   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
   260   let ?w = "x - (abs(x-z)+1) * d"
   261   from dpos have w: "?w < z" by(rule decr_lemma)
   262   have "P1 x = P1 ?w" using P1eqP1 by blast
   263   also have "\<dots> = P(?w)" using w P1eqP by blast
   264   finally have "P ?w" using P1 by blast
   265   thus "EX x. P x" ..
   266 qed
   267 
   268 lemma cpmi: 
   269   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
   270   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
   271   and pd: "\<forall> x k. P' x = P' (x-k*D)"
   272   shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 
   273          (is "?L = (?R1 \<or> ?R2)")
   274 proof-
   275  {assume "?R2" hence "?L"  by blast}
   276  moreover
   277  {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
   278  moreover 
   279  { fix x
   280    assume P: "P x" and H: "\<not> ?R2"
   281    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
   282      hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
   283      with nb P  have "P (y - D)" by auto }
   284    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
   285    with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
   286    from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
   287    let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
   288    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
   289    from dp have yz: "?y < z" using decr_lemma[OF dp] by simp   
   290    from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
   291    with periodic_finite_ex[OF dp pd]
   292    have "?R1" by blast}
   293  ultimately show ?thesis by blast
   294 qed
   295 
   296 subsubsection {* The @{text "+\<infinity>"} Version*}
   297 
   298 lemma  plusinfinity:
   299   assumes dpos: "(0::int) < d" and
   300     P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
   301   shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
   302 proof
   303   assume eP1: "EX x. P' x"
   304   then obtain x where P1: "P' x" ..
   305   from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
   306   let ?w' = "x + (abs(x-z)+1) * d"
   307   let ?w = "x - (-(abs(x-z) + 1))*d"
   308   have ww'[simp]: "?w = ?w'" by (simp add: ring_simps)
   309   from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
   310   hence "P' x = P' ?w" using P1eqP1 by blast
   311   also have "\<dots> = P(?w)" using w P1eqP by blast
   312   finally have "P ?w" using P1 by blast
   313   thus "EX x. P x" ..
   314 qed
   315 
   316 lemma incr_mult_lemma:
   317   assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
   318   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
   319 using knneg
   320 proof (induct rule:int_ge_induct)
   321   case base thus ?case by simp
   322 next
   323   case (step i)
   324   {fix x
   325     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
   326     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
   327       by (simp add:int_distrib zadd_ac)
   328     ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
   329   thus ?case ..
   330 qed
   331 
   332 lemma cppi: 
   333   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
   334   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
   335   and pd: "\<forall> x k. P' x= P' (x-k*D)"
   336   shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> A. P (b - j)))" (is "?L = (?R1 \<or> ?R2)")
   337 proof-
   338  {assume "?R2" hence "?L"  by blast}
   339  moreover
   340  {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
   341  moreover 
   342  { fix x
   343    assume P: "P x" and H: "\<not> ?R2"
   344    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
   345      hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
   346      with nb P  have "P (y + D)" by auto }
   347    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
   348    with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
   349    from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
   350    let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
   351    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
   352    from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
   353    from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
   354    with periodic_finite_ex[OF dp pd]
   355    have "?R1" by blast}
   356  ultimately show ?thesis by blast
   357 qed
   358 
   359 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
   360 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
   361 apply(fastsimp)
   362 done
   363 
   364 theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Ring_and_Field.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
   365   apply (rule eq_reflection [symmetric])
   366   apply (rule iffI)
   367   defer
   368   apply (erule exE)
   369   apply (rule_tac x = "l * x" in exI)
   370   apply (simp add: dvd_def)
   371   apply (rule_tac x = x in exI, simp)
   372   apply (erule exE)
   373   apply (erule conjE)
   374   apply simp
   375   apply (erule dvdE)
   376   apply (rule_tac x = k in exI)
   377   apply simp
   378   done
   379 
   380 lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
   381 shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
   382   using not0 by (simp add: dvd_def)
   383 
   384 lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
   385   by simp_all
   386 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
   387 lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
   388   by (simp split add: split_nat)
   389 
   390 lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
   391   apply (auto split add: split_nat)
   392   apply (rule_tac x="int x" in exI, simp)
   393   apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
   394   done
   395 
   396 lemma zdiff_int_split: "P (int (x - y)) =
   397   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
   398   by (case_tac "y \<le> x", simp_all add: zdiff_int)
   399 
   400 lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (Int.Bit0 n) \<and> (0::int) <= number_of (Int.Bit1 n)"
   401 by simp
   402 lemma number_of2: "(0::int) <= Numeral0" by simp
   403 lemma Suc_plus1: "Suc n = n + 1" by simp
   404 
   405 text {*
   406   \medskip Specific instances of congruence rules, to prevent
   407   simplifier from looping. *}
   408 
   409 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
   410 
   411 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
   412   by (simp cong: conj_cong)
   413 lemma int_eq_number_of_eq:
   414   "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
   415   by simp
   416 
   417 lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
   418 unfolding dvd_eq_mod_eq_0[symmetric] ..
   419 
   420 lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
   421 unfolding zdvd_iff_zmod_eq_0[symmetric] ..
   422 declare mod_1[presburger] 
   423 declare mod_0[presburger]
   424 declare zmod_1[presburger]
   425 declare zmod_zero[presburger]
   426 declare zmod_self[presburger]
   427 declare mod_self[presburger]
   428 declare mod_by_0[presburger]
   429 declare nat_mod_div_trivial[presburger]
   430 declare div_mod_equality2[presburger]
   431 declare div_mod_equality[presburger]
   432 declare mod_div_equality2[presburger]
   433 declare mod_div_equality[presburger]
   434 declare mod_mult_self1[presburger]
   435 declare mod_mult_self2[presburger]
   436 declare zdiv_zmod_equality2[presburger]
   437 declare zdiv_zmod_equality[presburger]
   438 declare mod2_Suc_Suc[presburger]
   439 lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
   440 by simp_all
   441 
   442 use "Tools/Qelim/cooper.ML"
   443 oracle linzqe_oracle = Coopereif.cooper_oracle
   444 
   445 use "Tools/Qelim/presburger.ML"
   446 
   447 declaration {* fn _ =>
   448   arith_tactic_add
   449     (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st =>
   450        (warning "Trying Presburger arithmetic ...";   
   451     Presburger.cooper_tac true [] [] ctxt i st)))
   452 *}
   453 
   454 method_setup presburger = {*
   455 let
   456  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
   457  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
   458  val addN = "add"
   459  val delN = "del"
   460  val elimN = "elim"
   461  val any_keyword = keyword addN || keyword delN || simple_keyword elimN
   462  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
   463 in
   464   fn src => Method.syntax 
   465    ((Scan.optional (simple_keyword elimN >> K false) true) -- 
   466     (Scan.optional (keyword addN |-- thms) []) -- 
   467     (Scan.optional (keyword delN |-- thms) [])) src 
   468   #> (fn (((elim, add_ths), del_ths),ctxt) => 
   469          Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
   470 end
   471 *} "Cooper's algorithm for Presburger arithmetic"
   472 
   473 lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   474 lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
   475 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   476 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
   477 lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
   478 
   479 
   480 lemma zdvd_period:
   481   fixes a d :: int
   482   assumes advdd: "a dvd d"
   483   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
   484   using advdd
   485   apply -
   486   apply (rule iffI)
   487   by algebra+
   488 
   489 end