author  chaieb 
Thu, 14 Jun 2007 22:59:39 +0200  
changeset 23390  01ef1135de73 
parent 23389  aaca6a8e5414 
child 23405  8993b3144358 
permissions  rwrr 
23148  1 
(* Title: HOL/Presburger.thy 
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ID: $Id$ 
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Author: Amine Chaieb, TU Muenchen 
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*) 
5 

15131  6 
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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15131  12 
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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90 
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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92 
next 
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93 
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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95 
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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98 
with nob tB have "False" by simp} 
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99 
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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100 
next 
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101 
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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103 
hence "x  (t  1) \<le> D" and "1 \<le> x  (t  1)" by simp+ 
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104 
hence "\<exists>j \<in> {1 .. D}. x  (t  1) = j" by auto 
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105 
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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108 
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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112 
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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113 
next 
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114 
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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136 
proof (blast, blast) 
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assume dp: "D > 0" and tA: "t + 1 \<in> A" 
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138 
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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140 
using dp tA by simp_all 
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141 
next 
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142 
assume dp: "D > 0" and tA: "t \<in> A" 
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143 
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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145 
using dp tA by simp_all 
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146 
next 
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147 
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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148 
next 
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149 
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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150 
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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153 
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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156 
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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158 
next 
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159 
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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162 
hence "\<exists>j \<in> {1 .. D}. (t + 1)  x = j" by auto 
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163 
hence "\<exists>j \<in> {1 .. D}. x = (t + 1)  j" by (auto simp add: ring_eq_simps) 
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164 
with nob tA have "False" by simp} 
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165 
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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166 
next 
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167 
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>A. x \<noteq> b  j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp 
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171 
next 
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172 
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>A. x \<noteq> b  j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto 
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176 
qed blast 
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section{* Cooper's Theorem @{text "\<infinity>"} and @{text "+\<infinity>"} Version *} 
13876  179 

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subsection{* First some trivial facts about periodic sets or predicates *} 
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lemma periodic_finite_ex: 
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assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x  k*d)" 
183 
shows "(EX x. P x) = (EX j : {1..d}. P j)" 

184 
(is "?LHS = ?RHS") 

185 
proof 

186 
assume ?LHS 

187 
then obtain x where P: "P x" .. 

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have "x mod d = x  (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) 
13876  189 
hence Pmod: "P x = P(x mod d)" using modd by simp 
190 
show ?RHS 

191 
proof (cases) 

192 
assume "x mod d = 0" 

193 
hence "P 0" using P Pmod by simp 

194 
moreover have "P 0 = P(0  (1)*d)" using modd by blast 

195 
ultimately have "P d" by simp 

196 
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) 

197 
ultimately show ?RHS .. 

198 
next 

199 
assume not0: "x mod d \<noteq> 0" 

200 
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) 

201 
moreover have "x mod d : {1..d}" 

202 
proof  

23389  203 
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) 
204 
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) 

13876  205 
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) 
206 
qed 

207 
ultimately show ?RHS .. 

208 
qed 

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qed auto 
13876  210 

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subsection{* The @{text "\<infinity>"} Version*} 
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212 

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lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x  (abs(xz)+1) * d < z" 
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by(induct rule: int_gr_induct,simp_all add:int_distrib) 
14577  215 

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lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(xz)+1) * d" 
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by(induct rule: int_gr_induct, simp_all add:int_distrib) 
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218 

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theorem int_induct[case_names base step1 step2]: 
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assumes 
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base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and 
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step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i  1)" 
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223 
shows "P i" 
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224 
proof  
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have "i \<le> k \<or> i\<ge> k" by arith 
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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 
13876  227 
qed 
228 

229 
lemma decr_mult_lemma: 

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assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x  d)" and knneg: "0 <= k" 
13876  231 
shows "ALL x. P x \<longrightarrow> P(x  k*d)" 
232 
using knneg 

233 
proof (induct rule:int_ge_induct) 

234 
case base thus ?case by simp 

235 
next 

236 
case (step i) 

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{fix x 
13876  238 
have "P x \<longrightarrow> P (x  i * d)" using step.hyps by blast 
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239 
also have "\<dots> \<longrightarrow> P(x  (i + 1) * d)" using minus[THEN spec, of "x  i * d"] 
14738  240 
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) 
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ultimately have "P x \<longrightarrow> P(x  (i + 1) * d)" by blast} 
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thus ?case .. 
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243 
qed 
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244 

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lemma minusinfinity: 
23389  246 
assumes dpos: "0 < d" and 
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P1eqP1: "ALL x k. P1 x = P1(x  k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" 
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shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" 
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249 
proof 
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250 
assume eP1: "EX x. P1 x" 
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251 
then obtain x where P1: "P1 x" .. 
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252 
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. 
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253 
let ?w = "x  (abs(xz)+1) * d" 
23389  254 
from dpos have w: "?w < z" by(rule decr_lemma) 
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255 
have "P1 x = P1 ?w" using P1eqP1 by blast 
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256 
also have "\<dots> = P(?w)" using w P1eqP by blast 
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257 
finally have "P ?w" using P1 by blast 
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258 
thus "EX x. P x" .. 
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259 
qed 
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260 

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261 
lemma cpmi: 
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assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" 
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263 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) > P (x) > P (x  D)" 
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264 
and pd: "\<forall> x k. P' x = P' (xk*D)" 
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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)))" 
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(is "?L = (?R1 \<or> ?R2)") 
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267 
proof 
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{assume "?R2" hence "?L" by blast} 
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moreover 
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{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 
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271 
moreover 
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272 
{ fix x 
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assume P: "P x" and H: "\<not> ?R2" 
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{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" 
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hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto 
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with nb P have "P (y  D)" by auto } 
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hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) > P (x) > P (x  D)" by blast 
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278 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x  D)" by auto 
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279 
from p1 obtain z where z: "ALL x. x < z > (P x = P' x)" by blast 
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280 
let ?y = "x  (\<bar>x  z\<bar> + 1)*D" 
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281 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 
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282 
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp 
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from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 
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284 
with periodic_finite_ex[OF dp pd] 
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285 
have "?R1" by blast} 
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286 
ultimately show ?thesis by blast 
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287 
qed 
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288 

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289 
subsection {* The @{text "+\<infinity>"} Version*} 
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290 

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291 
lemma plusinfinity: 
23389  292 
assumes dpos: "(0::int) < d" and 
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293 
P1eqP1: "\<forall>x k. P' x = P'(x  k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" 
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294 
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" 
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295 
proof 
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296 
assume eP1: "EX x. P' x" 
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297 
then obtain x where P1: "P' x" .. 
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298 
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. 
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299 
let ?w' = "x + (abs(xz)+1) * d" 
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300 
let ?w = "x  ((abs(xz) + 1))*d" 
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301 
have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps) 
23389  302 
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) 
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303 
hence "P' x = P' ?w" using P1eqP1 by blast 
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304 
also have "\<dots> = P(?w)" using w P1eqP by blast 
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305 
finally have "P ?w" using P1 by blast 
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306 
thus "EX x. P x" .. 
13876  307 
qed 
308 

309 
lemma incr_mult_lemma: 

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310 
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" 
13876  311 
shows "ALL x. P x \<longrightarrow> P(x + k*d)" 
312 
using knneg 

313 
proof (induct rule:int_ge_induct) 

314 
case base thus ?case by simp 

315 
next 

316 
case (step i) 

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317 
{fix x 
13876  318 
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast 
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319 
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] 
13876  320 
by (simp add:int_distrib zadd_ac) 
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321 
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} 
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322 
thus ?case .. 
13876  323 
qed 
324 

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325 
lemma cppi: 
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326 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" 
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327 
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b  j) > P (x) > P (x + D)" 
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328 
and pd: "\<forall> x k. P' x= P' (xk*D)" 
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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)") 
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330 
proof 
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331 
{assume "?R2" hence "?L" by blast} 
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332 
moreover 
6894137e854a
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333 
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 
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334 
moreover 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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335 
{ fix x 
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336 
assume P: "P x" and H: "\<not> ?R2" 
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337 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b  j))" and P: "P y" 
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338 
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b  j)" by auto 
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339 
with nb P have "P (y + D)" by auto } 
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340 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(bj)) > P (x) > P (x + D)" by blast 
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341 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto 
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342 
from p1 obtain z where z: "ALL x. x > z > (P x = P' x)" by blast 
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343 
let ?y = "x + (\<bar>x  z\<bar> + 1)*D" 
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344 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 
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345 
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp 
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346 
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 
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347 
with periodic_finite_ex[OF dp pd] 
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348 
have "?R1" by blast} 
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349 
ultimately show ?thesis by blast 
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350 
qed 
13876  351 

352 
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" 

353 
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) 

354 
apply(fastsimp) 

355 
done 

356 

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357 
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" 
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358 
apply (rule eq_reflection[symmetric]) 
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359 
apply (rule iffI) 
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360 
defer 
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361 
apply (erule exE) 
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362 
apply (rule_tac x = "l * x" in exI) 
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363 
apply (simp add: dvd_def) 
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364 
apply (rule_tac x="x" in exI, simp) 
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365 
apply (erule exE) 
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366 
apply (erule conjE) 
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367 
apply (erule dvdE) 
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368 
apply (rule_tac x = k in exI) 
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369 
apply simp 
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370 
done 
13876  371 

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372 
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" 
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373 
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
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374 
using not0 by (simp add: dvd_def) 
13876  375 

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376 
lemma all_not_ex: "(ALL x. P x) = (~ (EX x. ~ P x ))" 
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377 
by blast 
13876  378 

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379 
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd t)" 
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380 
by simp_all 
14577  381 
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} 
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382 
lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))" 
13876  383 
by (simp split add: split_nat) 
384 

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385 
lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))" 
23365  386 
apply (auto split add: split_nat) 
387 
apply (rule_tac x="int x" in exI, simp) 

388 
apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp) 

389 
done 

13876  390 

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391 
lemma zdiff_int_split: "P (int (x  y)) = 
13876  392 
((y \<le> x \<longrightarrow> P (int x  int y)) \<and> (x < y \<longrightarrow> P 0))" 
23365  393 
by (case_tac "y \<le> x", simp_all) 
13876  394 

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395 
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp 
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396 
lemma number_of2: "(0::int) <= Numeral0" by simp 
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397 
lemma Suc_plus1: "Suc n = n + 1" by simp 
13876  398 

14577  399 
text {* 
400 
\medskip Specific instances of congruence rules, to prevent 

401 
simplifier from looping. *} 

13876  402 

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403 
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp 
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404 

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405 
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
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406 
by (simp cong: conj_cong) 
20485  407 
lemma int_eq_number_of_eq: 
408 
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" 

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409 
by simp 
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410 

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411 
lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m" 
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412 
unfolding dvd_eq_mod_eq_0[symmetric] .. 
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413 

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414 
lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m" 
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415 
unfolding zdvd_iff_zmod_eq_0[symmetric] .. 
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416 
declare mod_1[presburger] 
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417 
declare mod_0[presburger] 
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418 
declare zmod_1[presburger] 
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419 
declare zmod_zero[presburger] 
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420 
declare zmod_self[presburger] 
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421 
declare mod_self[presburger] 
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422 
declare DIVISION_BY_ZERO_MOD[presburger] 
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423 
declare nat_mod_div_trivial[presburger] 
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424 
declare div_mod_equality2[presburger] 
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425 
declare div_mod_equality[presburger] 
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426 
declare mod_div_equality2[presburger] 
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427 
declare mod_div_equality[presburger] 
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428 
declare mod_mult_self1[presburger] 
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429 
declare mod_mult_self2[presburger] 
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430 
declare zdiv_zmod_equality2[presburger] 
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431 
declare zdiv_zmod_equality[presburger] 
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432 
declare mod2_Suc_Suc[presburger] 
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433 
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" 
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434 
using IntDiv.DIVISION_BY_ZERO by blast+ 
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435 

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436 
use "Tools/Presburger/cooper.ML" 
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437 
oracle linzqe_oracle ("term") = Coopereif.cooper_oracle 
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438 

23146  439 
use "Tools/Presburger/presburger.ML" 
13876  440 

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441 
setup {* 
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442 
arith_tactic_add 
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443 
(mk_arith_tactic "presburger" (fn i => fn st => 
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444 
(warning "Trying Presburger arithmetic ..."; 
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445 
Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st))) 
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446 
(* FIXME!!!!!!! get the right context!!*) 
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447 
*} 
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448 

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449 
method_setup presburger = {* 
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450 
let 
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fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 
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452 
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () 
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453 
val addN = "add" 
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454 
val delN = "del" 
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455 
val elimN = "elim" 
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456 
val any_keyword = keyword addN  keyword delN  simple_keyword elimN 
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457 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 
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458 
in 
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459 
fn src => Method.syntax 
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460 
((Scan.optional (simple_keyword elimN >> K false) true)  
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461 
(Scan.optional (keyword addN  thms) [])  
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462 
(Scan.optional (keyword delN  thms) [])) src 
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463 
#> (fn (((elim, add_ths), del_ths),ctxt) => 
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464 
Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) 
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465 
end 
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466 
*} "" 
22801  467 

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468 
lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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469 
lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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470 
lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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471 
lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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472 
lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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473 

22801  474 
subsection {* Code generator setup *} 
20595  475 
text {* 
22801  476 
Presburger arithmetic is convenient to prove some 
477 
of the following code lemmas on integer numerals: 

20595  478 
*} 
479 

480 
lemma eq_Pls_Pls: 

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481 
"Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger 
20595  482 

483 
lemma eq_Pls_Min: 

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484 
"Numeral.Pls = Numeral.Min \<longleftrightarrow> False" 
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485 
unfolding Pls_def Min_def by presburger 
20595  486 

487 
lemma eq_Pls_Bit0: 

21454  488 
"Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k" 
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489 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  490 

491 
lemma eq_Pls_Bit1: 

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492 
"Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False" 
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493 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  494 

495 
lemma eq_Min_Pls: 

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496 
"Numeral.Min = Numeral.Pls \<longleftrightarrow> False" 
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497 
unfolding Pls_def Min_def by presburger 
20595  498 

499 
lemma eq_Min_Min: 

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500 
"Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger 
20595  501 

502 
lemma eq_Min_Bit0: 

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503 
"Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False" 
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504 
unfolding Min_def Bit_def bit.cases by presburger 
20595  505 

506 
lemma eq_Min_Bit1: 

21454  507 
"Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k" 
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508 
unfolding Min_def Bit_def bit.cases by presburger 
20595  509 

510 
lemma eq_Bit0_Pls: 

21454  511 
"Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k" 
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512 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  513 

514 
lemma eq_Bit1_Pls: 

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515 
"Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False" 
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516 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  517 

518 
lemma eq_Bit0_Min: 

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519 
"Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False" 
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520 
unfolding Min_def Bit_def bit.cases by presburger 
20595  521 

522 
lemma eq_Bit1_Min: 

21454  523 
"(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k" 
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524 
unfolding Min_def Bit_def bit.cases by presburger 
20595  525 

526 
lemma eq_Bit_Bit: 

21454  527 
"Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow> 
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528 
v1 = v2 \<and> k1 = k2" 
21454  529 
unfolding Bit_def 
20595  530 
apply (cases v1) 
531 
apply (cases v2) 

532 
apply auto 

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533 
apply presburger 
20595  534 
apply (cases v2) 
535 
apply auto 

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536 
apply presburger 
20595  537 
apply (cases v2) 
538 
apply auto 

539 
done 

540 

22801  541 
lemma eq_number_of: 
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542 
"(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l" 
22801  543 
unfolding number_of_is_id .. 
20595  544 

22394  545 

20595  546 
lemma less_eq_Pls_Pls: 
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547 
"Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+ 
20595  548 

549 
lemma less_eq_Pls_Min: 

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changeset

550 
"Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

551 
unfolding Pls_def Min_def by presburger 
20595  552 

553 
lemma less_eq_Pls_Bit: 

554 
"Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k" 

555 
unfolding Pls_def Bit_def by (cases v) auto 

556 

557 
lemma less_eq_Min_Pls: 

22744
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Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset

558 
"Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

559 
unfolding Pls_def Min_def by presburger 
20595  560 

561 
lemma less_eq_Min_Min: 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset

562 
"Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+ 
20595  563 

564 
lemma less_eq_Min_Bit0: 

565 
"Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k" 

566 
unfolding Min_def Bit_def by auto 

567 

568 
lemma less_eq_Min_Bit1: 

569 
"Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k" 

570 
unfolding Min_def Bit_def by auto 

571 

572 
lemma less_eq_Bit0_Pls: 

573 
"Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls" 

574 
unfolding Pls_def Bit_def by simp 

575 

576 
lemma less_eq_Bit1_Pls: 

577 
"Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" 

578 
unfolding Pls_def Bit_def by auto 

579 

580 
lemma less_eq_Bit_Min: 

581 
"Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" 

582 
unfolding Min_def Bit_def by (cases v) auto 

583 

584 
lemma less_eq_Bit0_Bit: 

585 
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2" 

22394  586 
unfolding Bit_def bit.cases by (cases v) auto 
20595  587 

588 
lemma less_eq_Bit_Bit1: 

589 
"Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2" 

22394  590 
unfolding Bit_def bit.cases by (cases v) auto 
591 

592 
lemma less_eq_Bit1_Bit0: 

593 
"Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" 

594 
unfolding Bit_def by (auto split: bit.split) 

20595  595 

22801  596 
lemma less_eq_number_of: 
597 
"(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l" 

598 
unfolding number_of_is_id .. 

22394  599 

600 

601 
lemma less_Pls_Pls: 

23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

602 
"Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by presburger 
22394  603 

604 
lemma less_Pls_Min: 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset

605 
"Numeral.Pls < Numeral.Min \<longleftrightarrow> False" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

606 
unfolding Pls_def Min_def by presburger 
22394  607 

608 
lemma less_Pls_Bit0: 

609 
"Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k" 

610 
unfolding Pls_def Bit_def by auto 

611 

612 
lemma less_Pls_Bit1: 

613 
"Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k" 

614 
unfolding Pls_def Bit_def by auto 

615 

616 
lemma less_Min_Pls: 

22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset

617 
"Numeral.Min < Numeral.Pls \<longleftrightarrow> True" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

618 
unfolding Pls_def Min_def by presburger 
22394  619 

620 
lemma less_Min_Min: 

23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

621 
"Numeral.Min < Numeral.Min \<longleftrightarrow> False" by presburger 
22394  622 

623 
lemma less_Min_Bit: 

624 
"Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k" 

625 
unfolding Min_def Bit_def by (auto split: bit.split) 

626 

627 
lemma less_Bit_Pls: 

628 
"Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" 

629 
unfolding Pls_def Bit_def by (auto split: bit.split) 

630 

631 
lemma less_Bit0_Min: 

632 
"Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" 

633 
unfolding Min_def Bit_def by auto 

634 

635 
lemma less_Bit1_Min: 

636 
"Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min" 

637 
unfolding Min_def Bit_def by auto 

638 

639 
lemma less_Bit_Bit0: 

640 
"Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" 

641 
unfolding Bit_def by (auto split: bit.split) 

642 

643 
lemma less_Bit1_Bit: 

644 
"Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2" 

645 
unfolding Bit_def by (auto split: bit.split) 

646 

647 
lemma less_Bit0_Bit1: 

648 
"Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2" 

23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

649 
unfolding Bit_def bit.cases by arith 
22394  650 

22801  651 
lemma less_number_of: 
652 
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l" 

653 
unfolding number_of_is_id .. 

654 

655 
lemmas pred_succ_numeral_code [code func] = 

656 
arith_simps(512) 

657 

658 
lemmas plus_numeral_code [code func] = 

659 
arith_simps(1317) 

660 
arith_simps(2627) 

661 
arith_extra_simps(1) [where 'a = int] 

662 

663 
lemmas minus_numeral_code [code func] = 

664 
arith_simps(1821) 

665 
arith_extra_simps(2) [where 'a = int] 

666 
arith_extra_simps(5) [where 'a = int] 

667 

668 
lemmas times_numeral_code [code func] = 

669 
arith_simps(2225) 

670 
arith_extra_simps(4) [where 'a = int] 

671 

672 
lemmas eq_numeral_code [code func] = 

673 
eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1 

674 
eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1 

675 
eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit 

676 
eq_number_of 

677 

678 
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit 

679 
less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1 

680 
less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0 

681 
less_eq_number_of 

682 

22394  683 
lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0 
684 
less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls 

685 
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1 

22801  686 
less_number_of 
20595  687 

23365  688 
end 