src/HOL/Presburger.thy
author wenzelm
Thu Jun 14 18:33:31 2007 +0200 (2007-06-14)
changeset 23389 aaca6a8e5414
parent 23365 f31794033ae1
child 23390 01ef1135de73
permissions -rw-r--r--
tuned proofs: avoid implicit prems;
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(*  Title:      HOL/Presburger.thy
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    ID:         $Id$
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    Author:     Amine Chaieb, TU Muenchen
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*)
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theory Presburger
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imports NatSimprocs SetInterval
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  uses "Tools/Presburger/cooper_data" "Tools/Presburger/qelim" 
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       "Tools/Presburger/generated_cooper.ML"
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       ("Tools/Presburger/cooper.ML") ("Tools/Presburger/presburger.ML") 
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begin
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setup {* Cooper_Data.setup*}
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section{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
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lemma minf:
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  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
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     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)"
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  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x<z. P x = P' x; \<exists>z.\<forall>x<z. Q x = Q' x\<rbrakk> 
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     \<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False"
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  "\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False"
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  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (d dvd x + s) = (d dvd x + s)"
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  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
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  "\<exists>z.\<forall>x<z. F = F"
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  by ((erule exE, erule exE,rule_tac x="min z za" in exI,simp)+, (rule_tac x="t" in exI,fastsimp)+) simp_all
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lemma pinf:
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  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
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     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)"
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  "\<lbrakk>\<exists>(z ::'a::linorder).\<forall>x>z. P x = P' x; \<exists>z.\<forall>x>z. Q x = Q' x\<rbrakk> 
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     \<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True"
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  "\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True"
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  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (d dvd x + s) = (d dvd x + s)"
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  "\<exists>z.\<forall>(x::'a::{linorder,plus,times})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)"
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  "\<exists>z.\<forall>x>z. F = F"
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  by ((erule exE, erule exE,rule_tac x="max z za" in exI,simp)+,(rule_tac x="t" in exI,fastsimp)+) simp_all
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lemma inf_period:
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  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
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    \<Longrightarrow> \<forall>x k. (P x \<and> Q x) = (P (x - k*D) \<and> Q (x - k*D))"
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  "\<lbrakk>\<forall>x k. P x = P (x - k*D); \<forall>x k. Q x = Q (x - k*D)\<rbrakk> 
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    \<Longrightarrow> \<forall>x k. (P x \<or> Q x) = (P (x - k*D) \<or> Q (x - k*D))"
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  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
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  "(d::'a::{comm_ring}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
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  "\<forall>x k. F = F"
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by simp_all
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  (clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI,
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    simp add: ring_eq_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_eq_simps)+
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section{* The A and B sets *}
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lemma bset:
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  "\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
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     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
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  \<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))"
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  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ;
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     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> 
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  \<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))"
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  "\<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))"
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  "\<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))"
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  "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))"
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  "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))"
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  "\<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))"
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  "\<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))"
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  "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))"
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  "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))"
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  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F"
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proof (blast, blast)
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  assume dp: "D > 0" and tB: "t - 1\<in> B"
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  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" 
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    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"])
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    using dp tB by simp_all
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next
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  assume dp: "D > 0" and tB: "t \<in> B"
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  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))" 
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    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
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    using dp tB by simp_all
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next
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  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
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next
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  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
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next
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  assume dp: "D > 0" and tB:"t \<in> B"
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  {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"
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    hence "x -t \<le> D" and "1 \<le> x - t" by simp+
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      hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
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      hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_eq_simps)
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      with nob tB have "False" by simp}
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  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
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next
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  assume dp: "D > 0" and tB:"t - 1\<in> B"
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  {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"
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    hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
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      hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
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      hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps)
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      with nob tB have "False" by simp}
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  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
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next
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  assume d: "d dvd D"
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  {fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t"
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      by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)}
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  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
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next
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  assume d: "d dvd D"
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  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t"
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      by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)}
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  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
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qed blast
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lemma aset:
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  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
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     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
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  \<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))"
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  "\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ;
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     \<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> 
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  \<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))"
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  "\<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))"
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  "\<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))"
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  "\<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))"
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  "\<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))"
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  "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))"
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  "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))"
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  "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))"
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  "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))"
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  "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F"
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proof (blast, blast)
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  assume dp: "D > 0" and tA: "t + 1 \<in> A"
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  show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" 
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    apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"])
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    using dp tA by simp_all
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next
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  assume dp: "D > 0" and tA: "t \<in> A"
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  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))" 
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    apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"])
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    using dp tA by simp_all
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next
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  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
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next
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  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
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next
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  assume dp: "D > 0" and tA:"t \<in> A"
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  {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"
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    hence "t - x \<le> D" and "1 \<le> t - x" by simp+
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      hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
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      hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps) 
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      with nob tA have "False" by simp}
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  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
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next
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  assume dp: "D > 0" and tA:"t + 1\<in> A"
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  {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"
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    hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps)
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      hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
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      hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps)
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      with nob tA have "False" by simp}
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  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
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next
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  assume d: "d dvd D"
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   168
  {fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
chaieb@23314
   169
      by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)}
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   170
  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
chaieb@23314
   171
next
chaieb@23314
   172
  assume d: "d dvd D"
chaieb@23314
   173
  {fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
chaieb@23314
   174
      by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)}
chaieb@23314
   175
  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
chaieb@23314
   176
qed blast
wenzelm@14577
   177
chaieb@23314
   178
section{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *}
berghofe@13876
   179
chaieb@23314
   180
subsection{* First some trivial facts about periodic sets or predicates *}
chaieb@23314
   181
lemma periodic_finite_ex:
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   182
  assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)"
berghofe@13876
   183
  shows "(EX x. P x) = (EX j : {1..d}. P j)"
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   184
  (is "?LHS = ?RHS")
berghofe@13876
   185
proof
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   186
  assume ?LHS
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   187
  then obtain x where P: "P x" ..
chaieb@23314
   188
  have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
berghofe@13876
   189
  hence Pmod: "P x = P(x mod d)" using modd by simp
berghofe@13876
   190
  show ?RHS
berghofe@13876
   191
  proof (cases)
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   192
    assume "x mod d = 0"
berghofe@13876
   193
    hence "P 0" using P Pmod by simp
berghofe@13876
   194
    moreover have "P 0 = P(0 - (-1)*d)" using modd by blast
berghofe@13876
   195
    ultimately have "P d" by simp
berghofe@13876
   196
    moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff)
berghofe@13876
   197
    ultimately show ?RHS ..
berghofe@13876
   198
  next
berghofe@13876
   199
    assume not0: "x mod d \<noteq> 0"
berghofe@13876
   200
    have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
berghofe@13876
   201
    moreover have "x mod d : {1..d}"
berghofe@13876
   202
    proof -
wenzelm@23389
   203
      from dpos have "0 \<le> x mod d" by(rule pos_mod_sign)
wenzelm@23389
   204
      moreover from dpos have "x mod d < d" by(rule pos_mod_bound)
berghofe@13876
   205
      ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
berghofe@13876
   206
    qed
berghofe@13876
   207
    ultimately show ?RHS ..
berghofe@13876
   208
  qed
chaieb@23314
   209
qed auto
berghofe@13876
   210
chaieb@23314
   211
subsection{* The @{text "-\<infinity>"} Version*}
chaieb@23314
   212
chaieb@23314
   213
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z"
chaieb@23314
   214
by(induct rule: int_gr_induct,simp_all add:int_distrib)
wenzelm@14577
   215
chaieb@23314
   216
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d"
chaieb@23314
   217
by(induct rule: int_gr_induct, simp_all add:int_distrib)
chaieb@23314
   218
chaieb@23314
   219
theorem int_induct[case_names base step1 step2]:
chaieb@23314
   220
  assumes 
chaieb@23314
   221
  base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and
chaieb@23314
   222
  step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
chaieb@23314
   223
  shows "P i"
chaieb@23314
   224
proof -
chaieb@23314
   225
  have "i \<le> k \<or> i\<ge> k" by arith
chaieb@23314
   226
  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
berghofe@13876
   227
qed
berghofe@13876
   228
berghofe@13876
   229
lemma decr_mult_lemma:
chaieb@23314
   230
  assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k"
berghofe@13876
   231
  shows "ALL x. P x \<longrightarrow> P(x - k*d)"
berghofe@13876
   232
using knneg
berghofe@13876
   233
proof (induct rule:int_ge_induct)
berghofe@13876
   234
  case base thus ?case by simp
berghofe@13876
   235
next
berghofe@13876
   236
  case (step i)
chaieb@23314
   237
  {fix x
berghofe@13876
   238
    have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast
chaieb@23314
   239
    also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"]
obua@14738
   240
      by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric])
chaieb@23314
   241
    ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast}
chaieb@23314
   242
  thus ?case ..
chaieb@23314
   243
qed
chaieb@23314
   244
chaieb@23314
   245
lemma  minusinfinity:
wenzelm@23389
   246
  assumes dpos: "0 < d" and
chaieb@23314
   247
    P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)"
chaieb@23314
   248
  shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)"
chaieb@23314
   249
proof
chaieb@23314
   250
  assume eP1: "EX x. P1 x"
chaieb@23314
   251
  then obtain x where P1: "P1 x" ..
chaieb@23314
   252
  from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" ..
chaieb@23314
   253
  let ?w = "x - (abs(x-z)+1) * d"
wenzelm@23389
   254
  from dpos have w: "?w < z" by(rule decr_lemma)
chaieb@23314
   255
  have "P1 x = P1 ?w" using P1eqP1 by blast
chaieb@23314
   256
  also have "\<dots> = P(?w)" using w P1eqP by blast
chaieb@23314
   257
  finally have "P ?w" using P1 by blast
chaieb@23314
   258
  thus "EX x. P x" ..
chaieb@23314
   259
qed
chaieb@23314
   260
chaieb@23314
   261
lemma cpmi: 
chaieb@23314
   262
  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x"
chaieb@23314
   263
  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)"
chaieb@23314
   264
  and pd: "\<forall> x k. P' x = P' (x-k*D)"
chaieb@23314
   265
  shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" 
chaieb@23314
   266
         (is "?L = (?R1 \<or> ?R2)")
chaieb@23314
   267
proof-
chaieb@23314
   268
 {assume "?R2" hence "?L"  by blast}
chaieb@23314
   269
 moreover
chaieb@23314
   270
 {assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
chaieb@23314
   271
 moreover 
chaieb@23314
   272
 { fix x
chaieb@23314
   273
   assume P: "P x" and H: "\<not> ?R2"
chaieb@23314
   274
   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y"
chaieb@23314
   275
     hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto
chaieb@23314
   276
     with nb P  have "P (y - D)" by auto }
chaieb@23314
   277
   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast
chaieb@23314
   278
   with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto
chaieb@23314
   279
   from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast
chaieb@23314
   280
   let ?y = "x - (\<bar>x - z\<bar> + 1)*D"
chaieb@23314
   281
   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
chaieb@23314
   282
   from dp have yz: "?y < z" using decr_lemma[OF dp] by simp   
chaieb@23314
   283
   from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
chaieb@23314
   284
   with periodic_finite_ex[OF dp pd]
chaieb@23314
   285
   have "?R1" by blast}
chaieb@23314
   286
 ultimately show ?thesis by blast
chaieb@23314
   287
qed
chaieb@23314
   288
chaieb@23314
   289
subsection {* The @{text "+\<infinity>"} Version*}
chaieb@23314
   290
chaieb@23314
   291
lemma  plusinfinity:
wenzelm@23389
   292
  assumes dpos: "(0::int) < d" and
chaieb@23314
   293
    P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x"
chaieb@23314
   294
  shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)"
chaieb@23314
   295
proof
chaieb@23314
   296
  assume eP1: "EX x. P' x"
chaieb@23314
   297
  then obtain x where P1: "P' x" ..
chaieb@23314
   298
  from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
chaieb@23314
   299
  let ?w' = "x + (abs(x-z)+1) * d"
chaieb@23314
   300
  let ?w = "x - (-(abs(x-z) + 1))*d"
chaieb@23314
   301
  have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps)
wenzelm@23389
   302
  from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
chaieb@23314
   303
  hence "P' x = P' ?w" using P1eqP1 by blast
chaieb@23314
   304
  also have "\<dots> = P(?w)" using w P1eqP by blast
chaieb@23314
   305
  finally have "P ?w" using P1 by blast
chaieb@23314
   306
  thus "EX x. P x" ..
berghofe@13876
   307
qed
berghofe@13876
   308
berghofe@13876
   309
lemma incr_mult_lemma:
chaieb@23314
   310
  assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k"
berghofe@13876
   311
  shows "ALL x. P x \<longrightarrow> P(x + k*d)"
berghofe@13876
   312
using knneg
berghofe@13876
   313
proof (induct rule:int_ge_induct)
berghofe@13876
   314
  case base thus ?case by simp
berghofe@13876
   315
next
berghofe@13876
   316
  case (step i)
chaieb@23314
   317
  {fix x
berghofe@13876
   318
    have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast
chaieb@23314
   319
    also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"]
berghofe@13876
   320
      by (simp add:int_distrib zadd_ac)
chaieb@23314
   321
    ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast}
chaieb@23314
   322
  thus ?case ..
berghofe@13876
   323
qed
berghofe@13876
   324
chaieb@23314
   325
lemma cppi: 
chaieb@23314
   326
  assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x"
chaieb@23314
   327
  and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)"
chaieb@23314
   328
  and pd: "\<forall> x k. P' x= P' (x-k*D)"
chaieb@23314
   329
  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)")
chaieb@23314
   330
proof-
chaieb@23314
   331
 {assume "?R2" hence "?L"  by blast}
chaieb@23314
   332
 moreover
chaieb@23314
   333
 {assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp}
chaieb@23314
   334
 moreover 
chaieb@23314
   335
 { fix x
chaieb@23314
   336
   assume P: "P x" and H: "\<not> ?R2"
chaieb@23314
   337
   {fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y"
chaieb@23314
   338
     hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto
chaieb@23314
   339
     with nb P  have "P (y + D)" by auto }
chaieb@23314
   340
   hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast
chaieb@23314
   341
   with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto
chaieb@23314
   342
   from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast
chaieb@23314
   343
   let ?y = "x + (\<bar>x - z\<bar> + 1)*D"
chaieb@23314
   344
   have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith
chaieb@23314
   345
   from dp have yz: "?y > z" using incr_lemma[OF dp] by simp
chaieb@23314
   346
   from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto
chaieb@23314
   347
   with periodic_finite_ex[OF dp pd]
chaieb@23314
   348
   have "?R1" by blast}
chaieb@23314
   349
 ultimately show ?thesis by blast
chaieb@23314
   350
qed
berghofe@13876
   351
berghofe@13876
   352
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})"
berghofe@13876
   353
apply(simp add:atLeastAtMost_def atLeast_def atMost_def)
berghofe@13876
   354
apply(fastsimp)
berghofe@13876
   355
done
berghofe@13876
   356
chaieb@23314
   357
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)"
chaieb@23314
   358
  apply (rule eq_reflection[symmetric])
chaieb@23314
   359
  apply (rule iffI)
chaieb@23314
   360
  defer
chaieb@23314
   361
  apply (erule exE)
chaieb@23314
   362
  apply (rule_tac x = "l * x" in exI)
chaieb@23314
   363
  apply (simp add: dvd_def)
chaieb@23314
   364
  apply (rule_tac x="x" in exI, simp)
chaieb@23314
   365
  apply (erule exE)
chaieb@23314
   366
  apply (erule conjE)
chaieb@23314
   367
  apply (erule dvdE)
chaieb@23314
   368
  apply (rule_tac x = k in exI)
chaieb@23314
   369
  apply simp
chaieb@23314
   370
  done
berghofe@13876
   371
chaieb@23314
   372
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0"
chaieb@23314
   373
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
chaieb@23314
   374
  using not0 by (simp add: dvd_def)
berghofe@13876
   375
chaieb@23314
   376
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))"
chaieb@23314
   377
by blast
berghofe@13876
   378
chaieb@23314
   379
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)"
chaieb@23314
   380
  by simp_all
wenzelm@14577
   381
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*}
chaieb@23314
   382
lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))"
berghofe@13876
   383
  by (simp split add: split_nat)
berghofe@13876
   384
chaieb@23314
   385
lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))"
huffman@23365
   386
  apply (auto split add: split_nat)
huffman@23365
   387
  apply (rule_tac x="int x" in exI, simp)
huffman@23365
   388
  apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp)
huffman@23365
   389
  done
berghofe@13876
   390
chaieb@23314
   391
lemma zdiff_int_split: "P (int (x - y)) =
berghofe@13876
   392
  ((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))"
huffman@23365
   393
  by (case_tac "y \<le> x", simp_all)
berghofe@13876
   394
chaieb@23314
   395
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp
chaieb@23314
   396
lemma number_of2: "(0::int) <= Numeral0" by simp
chaieb@23314
   397
lemma Suc_plus1: "Suc n = n + 1" by simp
berghofe@13876
   398
wenzelm@14577
   399
text {*
wenzelm@14577
   400
  \medskip Specific instances of congruence rules, to prevent
wenzelm@14577
   401
  simplifier from looping. *}
berghofe@13876
   402
chaieb@23314
   403
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp
chaieb@18202
   404
chaieb@23314
   405
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
chaieb@23314
   406
  by (simp cong: conj_cong)
haftmann@20485
   407
lemma int_eq_number_of_eq:
haftmann@20485
   408
  "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
chaieb@18202
   409
  by simp
chaieb@18202
   410
chaieb@18202
   411
chaieb@23314
   412
use "Tools/Presburger/cooper.ML"
chaieb@23314
   413
oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
chaieb@18202
   414
wenzelm@23146
   415
use "Tools/Presburger/presburger.ML"
berghofe@13876
   416
chaieb@23314
   417
setup {* 
chaieb@23314
   418
  arith_tactic_add 
chaieb@23314
   419
    (mk_arith_tactic "presburger" (fn i => fn st =>
chaieb@23314
   420
       (warning "Trying Presburger arithmetic ...";   
chaieb@23333
   421
    Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st)))
chaieb@23314
   422
  (* FIXME!!!!!!! get the right context!!*)	
chaieb@23314
   423
*}
chaieb@23333
   424
chaieb@23314
   425
method_setup presburger = {*
chaieb@23333
   426
let
chaieb@23333
   427
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
chaieb@23333
   428
 fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
chaieb@23333
   429
 val addN = "add"
chaieb@23333
   430
 val delN = "del"
chaieb@23333
   431
 val elimN = "elim"
chaieb@23333
   432
 val any_keyword = keyword addN || keyword delN || simple_keyword elimN
chaieb@23333
   433
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
chaieb@23333
   434
in
chaieb@23333
   435
  fn src => Method.syntax 
chaieb@23333
   436
   ((Scan.optional (simple_keyword elimN >> K false) true) -- 
chaieb@23333
   437
    (Scan.optional (keyword addN |-- thms) []) -- 
chaieb@23333
   438
    (Scan.optional (keyword delN |-- thms) [])) src 
chaieb@23333
   439
  #> (fn (((elim, add_ths), del_ths),ctxt) => 
chaieb@23333
   440
         Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt))
chaieb@23333
   441
end
chaieb@23314
   442
*} ""
haftmann@22801
   443
haftmann@22801
   444
subsection {* Code generator setup *}
haftmann@20595
   445
text {*
haftmann@22801
   446
  Presburger arithmetic is convenient to prove some
haftmann@22801
   447
  of the following code lemmas on integer numerals:
haftmann@20595
   448
*}
haftmann@20595
   449
haftmann@20595
   450
lemma eq_Pls_Pls:
haftmann@22744
   451
  "Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by rule+
haftmann@20595
   452
haftmann@20595
   453
lemma eq_Pls_Min:
haftmann@22744
   454
  "Numeral.Pls = Numeral.Min \<longleftrightarrow> False"
haftmann@21454
   455
  unfolding Pls_def Min_def by auto
haftmann@20595
   456
haftmann@20595
   457
lemma eq_Pls_Bit0:
haftmann@21454
   458
  "Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k"
haftmann@21454
   459
  unfolding Pls_def Bit_def bit.cases by auto
haftmann@20595
   460
haftmann@20595
   461
lemma eq_Pls_Bit1:
haftmann@22744
   462
  "Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False"
haftmann@21454
   463
  unfolding Pls_def Bit_def bit.cases by arith
haftmann@20595
   464
haftmann@20595
   465
lemma eq_Min_Pls:
haftmann@22744
   466
  "Numeral.Min = Numeral.Pls \<longleftrightarrow> False"
haftmann@21454
   467
  unfolding Pls_def Min_def by auto
haftmann@20595
   468
haftmann@20595
   469
lemma eq_Min_Min:
haftmann@22744
   470
  "Numeral.Min = Numeral.Min \<longleftrightarrow> True" by rule+
haftmann@20595
   471
haftmann@20595
   472
lemma eq_Min_Bit0:
haftmann@22744
   473
  "Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False"
haftmann@21454
   474
  unfolding Min_def Bit_def bit.cases by arith
haftmann@20595
   475
haftmann@20595
   476
lemma eq_Min_Bit1:
haftmann@21454
   477
  "Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k"
haftmann@21454
   478
  unfolding Min_def Bit_def bit.cases by auto
haftmann@20595
   479
haftmann@20595
   480
lemma eq_Bit0_Pls:
haftmann@21454
   481
  "Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k"
haftmann@21454
   482
  unfolding Pls_def Bit_def bit.cases by auto
haftmann@20595
   483
haftmann@20595
   484
lemma eq_Bit1_Pls:
haftmann@22744
   485
  "Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False"
haftmann@21454
   486
  unfolding Pls_def Bit_def bit.cases by arith
haftmann@20595
   487
haftmann@20595
   488
lemma eq_Bit0_Min:
haftmann@22744
   489
  "Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False"
haftmann@21454
   490
  unfolding Min_def Bit_def bit.cases by arith
haftmann@20595
   491
haftmann@20595
   492
lemma eq_Bit1_Min:
haftmann@21454
   493
  "(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k"
haftmann@21454
   494
  unfolding Min_def Bit_def bit.cases by auto
haftmann@20595
   495
haftmann@20595
   496
lemma eq_Bit_Bit:
haftmann@21454
   497
  "Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow>
haftmann@21454
   498
    v1 = v2 \<and> k1 = k2"
haftmann@21454
   499
  unfolding Bit_def
haftmann@20595
   500
  apply (cases v1)
haftmann@20595
   501
  apply (cases v2)
haftmann@20595
   502
  apply auto
haftmann@20595
   503
  apply arith
haftmann@20595
   504
  apply (cases v2)
haftmann@20595
   505
  apply auto
haftmann@20595
   506
  apply arith
haftmann@20595
   507
  apply (cases v2)
haftmann@20595
   508
  apply auto
haftmann@20595
   509
done
haftmann@20595
   510
haftmann@22801
   511
lemma eq_number_of:
haftmann@22801
   512
  "(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l"
haftmann@22801
   513
  unfolding number_of_is_id ..
haftmann@20595
   514
haftmann@22394
   515
haftmann@20595
   516
lemma less_eq_Pls_Pls:
haftmann@22744
   517
  "Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+
haftmann@20595
   518
haftmann@20595
   519
lemma less_eq_Pls_Min:
haftmann@22744
   520
  "Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False"
haftmann@20595
   521
  unfolding Pls_def Min_def by auto
haftmann@20595
   522
haftmann@20595
   523
lemma less_eq_Pls_Bit:
haftmann@20595
   524
  "Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k"
haftmann@20595
   525
  unfolding Pls_def Bit_def by (cases v) auto
haftmann@20595
   526
haftmann@20595
   527
lemma less_eq_Min_Pls:
haftmann@22744
   528
  "Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True"
haftmann@20595
   529
  unfolding Pls_def Min_def by auto
haftmann@20595
   530
haftmann@20595
   531
lemma less_eq_Min_Min:
haftmann@22744
   532
  "Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+
haftmann@20595
   533
haftmann@20595
   534
lemma less_eq_Min_Bit0:
haftmann@20595
   535
  "Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k"
haftmann@20595
   536
  unfolding Min_def Bit_def by auto
haftmann@20595
   537
haftmann@20595
   538
lemma less_eq_Min_Bit1:
haftmann@20595
   539
  "Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k"
haftmann@20595
   540
  unfolding Min_def Bit_def by auto
haftmann@20595
   541
haftmann@20595
   542
lemma less_eq_Bit0_Pls:
haftmann@20595
   543
  "Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls"
haftmann@20595
   544
  unfolding Pls_def Bit_def by simp
haftmann@20595
   545
haftmann@20595
   546
lemma less_eq_Bit1_Pls:
haftmann@20595
   547
  "Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
haftmann@20595
   548
  unfolding Pls_def Bit_def by auto
haftmann@20595
   549
haftmann@20595
   550
lemma less_eq_Bit_Min:
haftmann@20595
   551
  "Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
haftmann@20595
   552
  unfolding Min_def Bit_def by (cases v) auto
haftmann@20595
   553
haftmann@20595
   554
lemma less_eq_Bit0_Bit:
haftmann@20595
   555
  "Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2"
haftmann@22394
   556
  unfolding Bit_def bit.cases by (cases v) auto
haftmann@20595
   557
haftmann@20595
   558
lemma less_eq_Bit_Bit1:
haftmann@20595
   559
  "Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
haftmann@22394
   560
  unfolding Bit_def bit.cases by (cases v) auto
haftmann@22394
   561
haftmann@22394
   562
lemma less_eq_Bit1_Bit0:
haftmann@22394
   563
  "Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
haftmann@22394
   564
  unfolding Bit_def by (auto split: bit.split)
haftmann@20595
   565
haftmann@22801
   566
lemma less_eq_number_of:
haftmann@22801
   567
  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
haftmann@22801
   568
  unfolding number_of_is_id ..
haftmann@22394
   569
haftmann@22394
   570
haftmann@22394
   571
lemma less_Pls_Pls:
haftmann@22744
   572
  "Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by auto
haftmann@22394
   573
haftmann@22394
   574
lemma less_Pls_Min:
haftmann@22744
   575
  "Numeral.Pls < Numeral.Min \<longleftrightarrow> False"
haftmann@22394
   576
  unfolding Pls_def Min_def by auto
haftmann@22394
   577
haftmann@22394
   578
lemma less_Pls_Bit0:
haftmann@22394
   579
  "Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k"
haftmann@22394
   580
  unfolding Pls_def Bit_def by auto
haftmann@22394
   581
haftmann@22394
   582
lemma less_Pls_Bit1:
haftmann@22394
   583
  "Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k"
haftmann@22394
   584
  unfolding Pls_def Bit_def by auto
haftmann@22394
   585
haftmann@22394
   586
lemma less_Min_Pls:
haftmann@22744
   587
  "Numeral.Min < Numeral.Pls \<longleftrightarrow> True"
haftmann@22394
   588
  unfolding Pls_def Min_def by auto
haftmann@22394
   589
haftmann@22394
   590
lemma less_Min_Min:
haftmann@22744
   591
  "Numeral.Min < Numeral.Min \<longleftrightarrow> False" by auto
haftmann@22394
   592
haftmann@22394
   593
lemma less_Min_Bit:
haftmann@22394
   594
  "Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k"
haftmann@22394
   595
  unfolding Min_def Bit_def by (auto split: bit.split)
haftmann@22394
   596
haftmann@22394
   597
lemma less_Bit_Pls:
haftmann@22394
   598
  "Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls"
haftmann@22394
   599
  unfolding Pls_def Bit_def by (auto split: bit.split)
haftmann@22394
   600
haftmann@22394
   601
lemma less_Bit0_Min:
haftmann@22394
   602
  "Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min"
haftmann@22394
   603
  unfolding Min_def Bit_def by auto
haftmann@22394
   604
haftmann@22394
   605
lemma less_Bit1_Min:
haftmann@22394
   606
  "Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min"
haftmann@22394
   607
  unfolding Min_def Bit_def by auto
haftmann@22394
   608
haftmann@22394
   609
lemma less_Bit_Bit0:
haftmann@22394
   610
  "Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2"
haftmann@22394
   611
  unfolding Bit_def by (auto split: bit.split)
haftmann@22394
   612
haftmann@22394
   613
lemma less_Bit1_Bit:
haftmann@22394
   614
  "Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2"
haftmann@22394
   615
  unfolding Bit_def by (auto split: bit.split)
haftmann@22394
   616
haftmann@22394
   617
lemma less_Bit0_Bit1:
haftmann@22394
   618
  "Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2"
haftmann@22394
   619
  unfolding Bit_def bit.cases by auto
haftmann@22394
   620
haftmann@22801
   621
lemma less_number_of:
haftmann@22801
   622
  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
haftmann@22801
   623
  unfolding number_of_is_id ..
haftmann@22801
   624
haftmann@22801
   625
lemmas pred_succ_numeral_code [code func] =
haftmann@22801
   626
  arith_simps(5-12)
haftmann@22801
   627
haftmann@22801
   628
lemmas plus_numeral_code [code func] =
haftmann@22801
   629
  arith_simps(13-17)
haftmann@22801
   630
  arith_simps(26-27)
haftmann@22801
   631
  arith_extra_simps(1) [where 'a = int]
haftmann@22801
   632
haftmann@22801
   633
lemmas minus_numeral_code [code func] =
haftmann@22801
   634
  arith_simps(18-21)
haftmann@22801
   635
  arith_extra_simps(2) [where 'a = int]
haftmann@22801
   636
  arith_extra_simps(5) [where 'a = int]
haftmann@22801
   637
haftmann@22801
   638
lemmas times_numeral_code [code func] =
haftmann@22801
   639
  arith_simps(22-25)
haftmann@22801
   640
  arith_extra_simps(4) [where 'a = int]
haftmann@22801
   641
haftmann@22801
   642
lemmas eq_numeral_code [code func] =
haftmann@22801
   643
  eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1
haftmann@22801
   644
  eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1
haftmann@22801
   645
  eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit
haftmann@22801
   646
  eq_number_of
haftmann@22801
   647
haftmann@22801
   648
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit
haftmann@22801
   649
  less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1
haftmann@22801
   650
  less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0
haftmann@22801
   651
  less_eq_number_of
haftmann@22801
   652
haftmann@22394
   653
lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0
haftmann@22394
   654
  less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls
haftmann@22394
   655
  less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1
haftmann@22801
   656
  less_number_of
haftmann@20595
   657
huffman@23365
   658
end