src/HOL/Presburger.thy
 author wenzelm Thu Oct 04 20:29:42 2007 +0200 (2007-10-04) changeset 24850 0cfd722ab579 parent 24404 317b207bc1ab child 24993 92dfacb32053 permissions -rw-r--r--
Name.uu, Name.aT;
```     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/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,times})<z. (d dvd x + s) = (d dvd x + s)"
```
```    34   "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<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,times})>z. (d dvd x + s) = (d dvd x + s)"
```
```    50   "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>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}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
```
```    60   "(d::'a::{comm_ring}) 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 by simp_all
```
```    63   (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
```
```    64     simp add: ring_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_simps)+
```
```    65
```
```    66 subsection{* The A and B sets *}
```
```    67 lemma bset:
```
```    68   "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
```
```    69      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
```
```    70   \<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))"
```
```    71   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
```
```    72      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow>
```
```    73   \<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))"
```
```    74   "\<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))"
```
```    75   "\<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))"
```
```    76   "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))"
```
```    77   "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))"
```
```    78   "\<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))"
```
```    79   "\<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))"
```
```    80   "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))"
```
```    81   "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))"
```
```    82   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
```
```    83 proof (blast, blast)
```
```    84   assume dp: "D > 0" and tB: "t - 1\<in> B"
```
```    85   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))"
```
```    86     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
```
```    87     using dp tB by simp_all
```
```    88 next
```
```    89   assume dp: "D > 0" and tB: "t \<in> B"
```
```    90   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))"
```
```    91     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
```
```    92     using dp tB by simp_all
```
```    93 next
```
```    94   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
```
```    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 \<le> t) \<longrightarrow> (x - D \<le> t)" by arith
```
```    97 next
```
```    98   assume dp: "D > 0" and tB:"t \<in> B"
```
```    99   {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"
```
```   100     hence "x -t \<le> D" and "1 \<le> x - t" by simp+
```
```   101       hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
```
```   102       hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps)
```
```   103       with nob tB have "False" by simp}
```
```   104   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
```
```   105 next
```
```   106   assume dp: "D > 0" and tB:"t - 1\<in> B"
```
```   107   {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"
```
```   108     hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
```
```   109       hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
```
```   110       hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps)
```
```   111       with nob tB have "False" by simp}
```
```   112   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
```
```   113 next
```
```   114   assume d: "d dvd D"
```
```   115   {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
```
```   116       by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_simps)}
```
```   117   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
```
```   118 next
```
```   119   assume d: "d dvd D"
```
```   120   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
```
```   121       by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)}
```
```   122   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
```
```   123 qed blast
```
```   124
```
```   125 lemma aset:
```
```   126   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
```
```   127      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
```
```   128   \<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))"
```
```   129   "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
```
```   130      \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow>
```
```   131   \<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))"
```
```   132   "\<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))"
```
```   133   "\<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))"
```
```   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 < t) \<longrightarrow> (x + D < t))"
```
```   135   "\<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))"
```
```   136   "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))"
```
```   137   "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))"
```
```   138   "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))"
```
```   139   "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))"
```
```   140   "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
```
```   141 proof (blast, blast)
```
```   142   assume dp: "D > 0" and tA: "t + 1 \<in> A"
```
```   143   show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))"
```
```   144     apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
```
```   145     using dp tA by simp_all
```
```   146 next
```
```   147   assume dp: "D > 0" and tA: "t \<in> A"
```
```   148   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))"
```
```   149     apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
```
```   150     using dp tA by simp_all
```
```   151 next
```
```   152   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
```
```   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 \<ge> t) \<longrightarrow> (x + D \<ge> t)" by arith
```
```   155 next
```
```   156   assume dp: "D > 0" and tA:"t \<in> A"
```
```   157   {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"
```
```   158     hence "t - x \<le> D" and "1 \<le> t - x" by simp+
```
```   159       hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
```
```   160       hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps)
```
```   161       with nob tA have "False" by simp}
```
```   162   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
```
```   163 next
```
```   164   assume dp: "D > 0" and tA:"t + 1\<in> A"
```
```   165   {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"
```
```   166     hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps)
```
```   167       hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
```
```   168       hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps)
```
```   169       with nob tA have "False" by simp}
```
```   170   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
```
```   171 next
```
```   172   assume d: "d dvd D"
```
```   173   {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
```
```   174       by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)}
```
```   175   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
```
```   176 next
```
```   177   assume d: "d dvd D"
```
```   178   {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
```
```   179       by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)}
```
```   180   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
```
```   181 qed blast
```
```   182
```
```   183 subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
```
```   184
```
```   185 subsubsection{* First some trivial facts about periodic sets or predicates *}
```
```   186 lemma periodic_finite_ex:
```
```   187   assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
```
```   188   shows "(EX x. P x) = (EX j : {1..d}. P j)"
```
```   189   (is "?LHS = ?RHS")
```
```   190 proof
```
```   191   assume ?LHS
```
```   192   then obtain x where P: "P x" ..
```
```   193   have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
```
```   194   hence Pmod: "P x = P(x mod d)" using modd by simp
```
```   195   show ?RHS
```
```   196   proof (cases)
```
```   197     assume "x mod d = 0"
```
```   198     hence "P 0" using P Pmod by simp
```
```   199     moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
```
```   200     ultimately have "P d" by simp
```
```   201     moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
```
```   202     ultimately show ?RHS ..
```
```   203   next
```
```   204     assume not0: "x mod d \<noteq> 0"
```
```   205     have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
```
```   206     moreover have "x mod d : {1..d}"
```
```   207     proof -
```
```   208       from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
```
```   209       moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
```
```   210       ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
```
```   211     qed
```
```   212     ultimately show ?RHS ..
```
```   213   qed
```
```   214 qed auto
```
```   215
```
```   216 subsubsection{* The @{text "-\<infinity>"} Version*}
```
```   217
```
```   218 lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
```
```   219 by(induct rule: int_gr_induct,simp_all add:int_distrib)
```
```   220
```
```   221 lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
```
```   222 by(induct rule: int_gr_induct, simp_all add:int_distrib)
```
```   223
```
```   224 theorem int_induct[case_names base step1 step2]:
```
```   225   assumes
```
```   226   base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
```
```   227   step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
```
```   228   shows "P i"
```
```   229 proof -
```
```   230   have "i \<le> k \<or> i\<ge> k" by arith
```
```   231   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
```
```   232 qed
```
```   233
```
```   234 lemma decr_mult_lemma:
```
```   235   assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
```
```   236   shows "ALL x. P x \<longrightarrow> P(x - k*d)"
```
```   237 using knneg
```
```   238 proof (induct rule:int_ge_induct)
```
```   239   case base thus ?case by simp
```
```   240 next
```
```   241   case (step i)
```
```   242   {fix x
```
```   243     have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
```
```   244     also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
```
```   245       by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
```
```   246     ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
```
```   247   thus ?case ..
```
```   248 qed
```
```   249
```
```   250 lemma  minusinfinity:
```
```   251   assumes dpos: "0 < d" and
```
```   252     P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
```
```   253   shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
```
```   254 proof
```
```   255   assume eP1: "EX x. P1 x"
```
```   256   then obtain x where P1: "P1 x" ..
```
```   257   from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
```
```   258   let ?w = "x - (abs(x-z)+1) * d"
```
```   259   from dpos have w: "?w < z" by(rule decr_lemma)
```
```   260   have "P1 x = P1 ?w" using P1eqP1 by blast
```
```   261   also have "\<dots> = P(?w)" using w P1eqP by blast
```
```   262   finally have "P ?w" using P1 by blast
```
```   263   thus "EX x. P x" ..
```
```   264 qed
```
```   265
```
```   266 lemma cpmi:
```
```   267   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
```
```   268   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
```
```   269   and pd: "\<forall> x k. P' x = P' (x-k*D)"
```
```   270   shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))"
```
```   271          (is "?L = (?R1 \<or> ?R2)")
```
```   272 proof-
```
```   273  {assume "?R2" hence "?L"  by blast}
```
```   274  moreover
```
```   275  {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
```
```   276  moreover
```
```   277  { fix x
```
```   278    assume P: "P x" and H: "\<not> ?R2"
```
```   279    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
```
```   280      hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
```
```   281      with nb P  have "P (y - D)" by auto }
```
```   282    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
```
```   283    with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
```
```   284    from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
```
```   285    let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
```
```   286    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
```
```   287    from dp have yz: "?y < z" using decr_lemma[OF dp] by simp
```
```   288    from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
```
```   289    with periodic_finite_ex[OF dp pd]
```
```   290    have "?R1" by blast}
```
```   291  ultimately show ?thesis by blast
```
```   292 qed
```
```   293
```
```   294 subsubsection {* The @{text "+\<infinity>"} Version*}
```
```   295
```
```   296 lemma  plusinfinity:
```
```   297   assumes dpos: "(0::int) < d" and
```
```   298     P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
```
```   299   shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
```
```   300 proof
```
```   301   assume eP1: "EX x. P' x"
```
```   302   then obtain x where P1: "P' x" ..
```
```   303   from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
```
```   304   let ?w' = "x + (abs(x-z)+1) * d"
```
```   305   let ?w = "x - (-(abs(x-z) + 1))*d"
```
```   306   have ww'[simp]: "?w = ?w'" by (simp add: ring_simps)
```
```   307   from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
```
```   308   hence "P' x = P' ?w" using P1eqP1 by blast
```
```   309   also have "\<dots> = P(?w)" using w P1eqP by blast
```
```   310   finally have "P ?w" using P1 by blast
```
```   311   thus "EX x. P x" ..
```
```   312 qed
```
```   313
```
```   314 lemma incr_mult_lemma:
```
```   315   assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
```
```   316   shows "ALL x. P x \<longrightarrow> P(x + k*d)"
```
```   317 using knneg
```
```   318 proof (induct rule:int_ge_induct)
```
```   319   case base thus ?case by simp
```
```   320 next
```
```   321   case (step i)
```
```   322   {fix x
```
```   323     have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
```
```   324     also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
```
```   325       by (simp add:int_distrib zadd_ac)
```
```   326     ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
```
```   327   thus ?case ..
```
```   328 qed
```
```   329
```
```   330 lemma cppi:
```
```   331   assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
```
```   332   and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
```
```   333   and pd: "\<forall> x k. P' x= P' (x-k*D)"
```
```   334   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)")
```
```   335 proof-
```
```   336  {assume "?R2" hence "?L"  by blast}
```
```   337  moreover
```
```   338  {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
```
```   339  moreover
```
```   340  { fix x
```
```   341    assume P: "P x" and H: "\<not> ?R2"
```
```   342    {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
```
```   343      hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
```
```   344      with nb P  have "P (y + D)" by auto }
```
```   345    hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
```
```   346    with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
```
```   347    from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
```
```   348    let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
```
```   349    have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
```
```   350    from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
```
```   351    from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
```
```   352    with periodic_finite_ex[OF dp pd]
```
```   353    have "?R1" by blast}
```
```   354  ultimately show ?thesis by blast
```
```   355 qed
```
```   356
```
```   357 lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
```
```   358 apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
```
```   359 apply(fastsimp)
```
```   360 done
```
```   361
```
```   362 theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
```
```   363   apply (rule eq_reflection[symmetric])
```
```   364   apply (rule iffI)
```
```   365   defer
```
```   366   apply (erule exE)
```
```   367   apply (rule_tac x = "l * x" in exI)
```
```   368   apply (simp add: dvd_def)
```
```   369   apply (rule_tac x="x" in exI, simp)
```
```   370   apply (erule exE)
```
```   371   apply (erule conjE)
```
```   372   apply (erule dvdE)
```
```   373   apply (rule_tac x = k in exI)
```
```   374   apply simp
```
```   375   done
```
```   376
```
```   377 lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
```
```   378 shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)"
```
```   379   using not0 by (simp add: dvd_def)
```
```   380
```
```   381 lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
```
```   382   by simp_all
```
```   383 text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
```
```   384 lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
```
```   385   by (simp split add: split_nat)
```
```   386
```
```   387 lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
```
```   388   apply (auto split add: split_nat)
```
```   389   apply (rule_tac x="int x" in exI, simp)
```
```   390   apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
```
```   391   done
```
```   392
```
```   393 lemma zdiff_int_split: "P (int (x - y)) =
```
```   394   ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
```
```   395   by (case_tac "y \<le> x", simp_all add: zdiff_int)
```
```   396
```
```   397 lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
```
```   398 lemma number_of2: "(0::int) <= Numeral0" by simp
```
```   399 lemma Suc_plus1: "Suc n = n + 1" by simp
```
```   400
```
```   401 text {*
```
```   402   \medskip Specific instances of congruence rules, to prevent
```
```   403   simplifier from looping. *}
```
```   404
```
```   405 theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
```
```   406
```
```   407 theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')"
```
```   408   by (simp cong: conj_cong)
```
```   409 lemma int_eq_number_of_eq:
```
```   410   "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
```
```   411   by simp
```
```   412
```
```   413 lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m"
```
```   414 unfolding dvd_eq_mod_eq_0[symmetric] ..
```
```   415
```
```   416 lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m"
```
```   417 unfolding zdvd_iff_zmod_eq_0[symmetric] ..
```
```   418 declare mod_1[presburger]
```
```   419 declare mod_0[presburger]
```
```   420 declare zmod_1[presburger]
```
```   421 declare zmod_zero[presburger]
```
```   422 declare zmod_self[presburger]
```
```   423 declare mod_self[presburger]
```
```   424 declare DIVISION_BY_ZERO_MOD[presburger]
```
```   425 declare nat_mod_div_trivial[presburger]
```
```   426 declare div_mod_equality2[presburger]
```
```   427 declare div_mod_equality[presburger]
```
```   428 declare mod_div_equality2[presburger]
```
```   429 declare mod_div_equality[presburger]
```
```   430 declare mod_mult_self1[presburger]
```
```   431 declare mod_mult_self2[presburger]
```
```   432 declare zdiv_zmod_equality2[presburger]
```
```   433 declare zdiv_zmod_equality[presburger]
```
```   434 declare mod2_Suc_Suc[presburger]
```
```   435 lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
```
```   436 using IntDiv.DIVISION_BY_ZERO by blast+
```
```   437
```
```   438 use "Tools/Qelim/cooper.ML"
```
```   439 oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
```
```   440
```
```   441 use "Tools/Qelim/presburger.ML"
```
```   442
```
```   443 declaration {* fn _ =>
```
```   444   arith_tactic_add
```
```   445     (mk_arith_tactic "presburger" (fn ctxt => fn i => fn st =>
```
```   446        (warning "Trying Presburger arithmetic ...";
```
```   447     Presburger.cooper_tac true [] [] ctxt i st)))
```
```   448 *}
```
```   449
```
```   450 method_setup presburger = {*
```
```   451 let
```
```   452  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```   453  fun simple_keyword k = Scan.lift (Args.\$\$\$ k) >> K ()
```
```   454  val addN = "add"
```
```   455  val delN = "del"
```
```   456  val elimN = "elim"
```
```   457  val any_keyword = keyword addN || keyword delN || simple_keyword elimN
```
```   458  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
```
```   459 in
```
```   460   fn src => Method.syntax
```
```   461    ((Scan.optional (simple_keyword elimN >> K false) true) --
```
```   462     (Scan.optional (keyword addN |-- thms) []) --
```
```   463     (Scan.optional (keyword delN |-- thms) [])) src
```
```   464   #> (fn (((elim, add_ths), del_ths),ctxt) =>
```
```   465          Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
```
```   466 end
```
```   467 *} "Cooper's algorithm for Presburger arithmetic"
```
```   468
```
```   469 lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   470 lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   471 lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   472 lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   473 lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger
```
```   474
```
```   475
```
```   476 lemma zdvd_period:
```
```   477   fixes a d :: int
```
```   478   assumes advdd: "a dvd d"
```
```   479   shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
```
```   480 proof-
```
```   481   {
```
```   482     fix x k
```
```   483     from inf_period(3) [OF advdd, rule_format, where x=x and k="-k"]
```
```   484     have "a dvd (x + t) \<longleftrightarrow> a dvd (x + k * d + t)" by simp
```
```   485   }
```
```   486   hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp
```
```   487   then show ?thesis by simp
```
```   488 qed
```
```   489
```
```   490
```
```   491 subsection {* Code generator setup *}
```
```   492
```
```   493 text {*
```
```   494   Presburger arithmetic is convenient to prove some
```
```   495   of the following code lemmas on integer numerals:
```
```   496 *}
```
```   497
```
```   498 lemma eq_Pls_Pls:
```
```   499   "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger
```
```   500
```
```   501 lemma eq_Pls_Min:
```
```   502   "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
```
```   503   unfolding Pls_def Numeral.Min_def by presburger
```
```   504
```
```   505 lemma eq_Pls_Bit0:
```
```   506   "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
```
```   507   unfolding Pls_def Bit_def bit.cases by presburger
```
```   508
```
```   509 lemma eq_Pls_Bit1:
```
```   510   "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
```
```   511   unfolding Pls_def Bit_def bit.cases by presburger
```
```   512
```
```   513 lemma eq_Min_Pls:
```
```   514   "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
```
```   515   unfolding Pls_def Numeral.Min_def by presburger
```
```   516
```
```   517 lemma eq_Min_Min:
```
```   518   "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger
```
```   519
```
```   520 lemma eq_Min_Bit0:
```
```   521   "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
```
```   522   unfolding Numeral.Min_def Bit_def bit.cases by presburger
```
```   523
```
```   524 lemma eq_Min_Bit1:
```
```   525   "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
```
```   526   unfolding Numeral.Min_def Bit_def bit.cases by presburger
```
```   527
```
```   528 lemma eq_Bit0_Pls:
```
```   529   "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
```
```   530   unfolding Pls_def Bit_def bit.cases by presburger
```
```   531
```
```   532 lemma eq_Bit1_Pls:
```
```   533   "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
```
```   534   unfolding Pls_def Bit_def bit.cases  by presburger
```
```   535
```
```   536 lemma eq_Bit0_Min:
```
```   537   "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
```
```   538   unfolding Numeral.Min_def Bit_def bit.cases  by presburger
```
```   539
```
```   540 lemma eq_Bit1_Min:
```
```   541   "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
```
```   542   unfolding Numeral.Min_def Bit_def bit.cases  by presburger
```
```   543
```
```   544 lemma eq_Bit_Bit:
```
```   545   "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
```
```   546     v1 = v2 \<and> k1 = k2"
```
```   547   unfolding Bit_def
```
```   548   apply (cases v1)
```
```   549   apply (cases v2)
```
```   550   apply auto
```
```   551   apply presburger
```
```   552   apply (cases v2)
```
```   553   apply auto
```
```   554   apply presburger
```
```   555   apply (cases v2)
```
```   556   apply auto
```
```   557   done
```
```   558
```
```   559 lemma eq_number_of:
```
```   560   "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l"
```
```   561   unfolding number_of_is_id ..
```
```   562
```
```   563
```
```   564 lemma less_eq_Pls_Pls:
```
```   565   "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
```
```   566
```
```   567 lemma less_eq_Pls_Min:
```
```   568   "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
```
```   569   unfolding Pls_def Numeral.Min_def by presburger
```
```   570
```
```   571 lemma less_eq_Pls_Bit:
```
```   572   "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
```
```   573   unfolding Pls_def Bit_def by (cases v) auto
```
```   574
```
```   575 lemma less_eq_Min_Pls:
```
```   576   "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
```
```   577   unfolding Pls_def Numeral.Min_def by presburger
```
```   578
```
```   579 lemma less_eq_Min_Min:
```
```   580   "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
```
```   581
```
```   582 lemma less_eq_Min_Bit0:
```
```   583   "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
```
```   584   unfolding Numeral.Min_def Bit_def by auto
```
```   585
```
```   586 lemma less_eq_Min_Bit1:
```
```   587   "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
```
```   588   unfolding Numeral.Min_def Bit_def by auto
```
```   589
```
```   590 lemma less_eq_Bit0_Pls:
```
```   591   "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
```
```   592   unfolding Pls_def Bit_def by simp
```
```   593
```
```   594 lemma less_eq_Bit1_Pls:
```
```   595   "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
```
```   596   unfolding Pls_def Bit_def by auto
```
```   597
```
```   598 lemma less_eq_Bit_Min:
```
```   599   "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
```
```   600   unfolding Numeral.Min_def Bit_def by (cases v) auto
```
```   601
```
```   602 lemma less_eq_Bit0_Bit:
```
```   603   "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
```
```   604   unfolding Bit_def bit.cases by (cases v) auto
```
```   605
```
```   606 lemma less_eq_Bit_Bit1:
```
```   607   "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
```
```   608   unfolding Bit_def bit.cases by (cases v) auto
```
```   609
```
```   610 lemma less_eq_Bit1_Bit0:
```
```   611   "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
```
```   612   unfolding Bit_def by (auto split: bit.split)
```
```   613
```
```   614 lemma less_eq_number_of:
```
```   615   "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
```
```   616   unfolding number_of_is_id ..
```
```   617
```
```   618
```
```   619 lemma less_Pls_Pls:
```
```   620   "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp
```
```   621
```
```   622 lemma less_Pls_Min:
```
```   623   "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
```
```   624   unfolding Pls_def Numeral.Min_def  by presburger
```
```   625
```
```   626 lemma less_Pls_Bit0:
```
```   627   "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
```
```   628   unfolding Pls_def Bit_def by auto
```
```   629
```
```   630 lemma less_Pls_Bit1:
```
```   631   "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
```
```   632   unfolding Pls_def Bit_def by auto
```
```   633
```
```   634 lemma less_Min_Pls:
```
```   635   "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
```
```   636   unfolding Pls_def Numeral.Min_def by presburger
```
```   637
```
```   638 lemma less_Min_Min:
```
```   639   "Numeral.Min < Numeral.Min \<longleftrightarrow> False"  by simp
```
```   640
```
```   641 lemma less_Min_Bit:
```
```   642   "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
```
```   643   unfolding Numeral.Min_def Bit_def by (auto split: bit.split)
```
```   644
```
```   645 lemma less_Bit_Pls:
```
```   646   "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
```
```   647   unfolding Pls_def Bit_def by (auto split: bit.split)
```
```   648
```
```   649 lemma less_Bit0_Min:
```
```   650   "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
```
```   651   unfolding Numeral.Min_def Bit_def by auto
```
```   652
```
```   653 lemma less_Bit1_Min:
```
```   654   "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
```
```   655   unfolding Numeral.Min_def Bit_def by auto
```
```   656
```
```   657 lemma less_Bit_Bit0:
```
```   658   "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
```
```   659   unfolding Bit_def by (auto split: bit.split)
```
```   660
```
```   661 lemma less_Bit1_Bit:
```
```   662   "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
```
```   663   unfolding Bit_def by (auto split: bit.split)
```
```   664
```
```   665 lemma less_Bit0_Bit1:
```
```   666   "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
```
```   667   unfolding Bit_def bit.cases  by arith
```
```   668
```
```   669 lemma less_number_of:
```
```   670   "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
```
```   671   unfolding number_of_is_id ..
```
```   672
```
```   673 lemmas pred_succ_numeral_code [code func] =
```
```   674   arith_simps(5-12)
```
```   675
```
```   676 lemmas plus_numeral_code [code func] =
```
```   677   arith_simps(13-17)
```
```   678   arith_simps(26-27)
```
```   679   arith_extra_simps(1) [where 'a = int]
```
```   680
```
```   681 lemmas minus_numeral_code [code func] =
```
```   682   arith_simps(18-21)
```
```   683   arith_extra_simps(2) [where 'a = int]
```
```   684   arith_extra_simps(5) [where 'a = int]
```
```   685
```
```   686 lemmas times_numeral_code [code func] =
```
```   687   arith_simps(22-25)
```
```   688   arith_extra_simps(4) [where 'a = int]
```
```   689
```
```   690 lemmas eq_numeral_code [code func] =
```
```   691   eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
```
```   692   eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
```
```   693   eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
```
```   694   eq_number_of
```
```   695
```
```   696 lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
```
```   697   less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
```
```   698   less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
```
```   699   less_eq_number_of
```
```   700
```
```   701 lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
```
```   702   less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
```
```   703   less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
```
```   704   less_number_of
```
```   705
```
```   706
```
```   707 lemma of_int_num [code func]:
```
```   708   "of_int k = (if k = 0 then 0 else if k < 0 then
```
```   709      - of_int (- k) else let
```
```   710        (l, m) = divAlg (k, 2);
```
```   711        l' = of_int l
```
```   712      in if m = 0 then l' + l' else l' + l' + 1)"
```
```   713 proof -
```
```   714   have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow>
```
```   715     of_int k = of_int (k div 2 * 2 + 1)"
```
```   716   proof -
```
```   717     assume "k mod 2 \<noteq> 0"
```
```   718     then have "k mod 2 = 1" by arith
```
```   719     moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
```
```   720     ultimately show ?thesis by auto
```
```   721   qed
```
```   722   have aux2: "\<And>x. of_int 2 * x = x + x"
```
```   723   proof -
```
```   724     fix x
```
```   725     have int2: "(2::int) = 1 + 1" by arith
```
```   726     show "of_int 2 * x = x + x"
```
```   727     unfolding int2 of_int_add left_distrib by simp
```
```   728   qed
```
```   729   have aux3: "\<And>x. x * of_int 2 = x + x"
```
```   730   proof -
```
```   731     fix x
```
```   732     have int2: "(2::int) = 1 + 1" by arith
```
```   733     show "x * of_int 2 = x + x"
```
```   734     unfolding int2 of_int_add right_distrib by simp
```
```   735   qed
```
```   736   from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3)
```
```   737 qed
```
```   738
```
```   739 end
```