author  huffman 
Wed, 20 Jun 2007 05:06:56 +0200  
changeset 23430  771117253634 
parent 23405  8993b3144358 
child 23438  dd824e86fa8a 
permissions  rwrr 
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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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header {* Decision Procedure for Presburger Arithmetic *} 
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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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15131  14 
begin 
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setup {* Cooper_Data.setup*} 
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subsection{* 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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94 
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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97 
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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100 
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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102 
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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105 
hence "x  (t  1) \<le> D" and "1 \<le> x  (t  1)" by simp+ 
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106 
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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110 
next 
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111 
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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114 
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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115 
next 
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116 
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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118 
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)} 
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119 
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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120 
qed blast 
13876  121 

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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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136 
"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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137 
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b  j) \<longrightarrow> F \<longrightarrow> F" 
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138 
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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142 
using dp tA by simp_all 
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143 
next 
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144 
assume dp: "D > 0" and tA: "t \<in> A" 
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145 
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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147 
using dp tA by simp_all 
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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 > t) \<longrightarrow> (x + D > t))" by arith 
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150 
next 
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151 
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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152 
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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155 
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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158 
with nob tA have "False" by simp} 
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159 
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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160 
next 
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161 
assume dp: "D > 0" and tA:"t + 1\<in> A" 
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162 
{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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163 
hence "(t + 1)  x \<le> D" and "1 \<le> (t + 1)  x" by (simp_all add: ring_eq_simps) 
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164 
hence "\<exists>j \<in> {1 .. D}. (t + 1)  x = j" by auto 
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165 
hence "\<exists>j \<in> {1 .. D}. x = (t + 1)  j" by (auto simp add: ring_eq_simps) 
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166 
with nob tA have "False" by simp} 
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167 
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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168 
next 
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169 
assume d: "d dvd D" 
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170 
{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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172 
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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173 
next 
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174 
assume d: "d dvd D" 
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175 
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" 
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176 
by (clarsimp simp add: dvd_def,erule_tac x= "ka  k" in allE,simp add: ring_eq_simps)} 
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177 
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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178 
qed blast 
14577  179 

23430  180 
subsection{* Cooper's Theorem @{text "\<infinity>"} and @{text "+\<infinity>"} Version *} 
13876  181 

23430  182 
subsubsection{* First some trivial facts about periodic sets or predicates *} 
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lemma periodic_finite_ex: 
13876  184 
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x  k*d)" 
185 
shows "(EX x. P x) = (EX j : {1..d}. P j)" 

186 
(is "?LHS = ?RHS") 

187 
proof 

188 
assume ?LHS 

189 
then obtain x where P: "P x" .. 

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190 
have "x mod d = x  (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) 
13876  191 
hence Pmod: "P x = P(x mod d)" using modd by simp 
192 
show ?RHS 

193 
proof (cases) 

194 
assume "x mod d = 0" 

195 
hence "P 0" using P Pmod by simp 

196 
moreover have "P 0 = P(0  (1)*d)" using modd by blast 

197 
ultimately have "P d" by simp 

198 
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) 

199 
ultimately show ?RHS .. 

200 
next 

201 
assume not0: "x mod d \<noteq> 0" 

202 
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) 

203 
moreover have "x mod d : {1..d}" 

204 
proof  

23389  205 
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) 
206 
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) 

13876  207 
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) 
208 
qed 

209 
ultimately show ?RHS .. 

210 
qed 

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qed auto 
13876  212 

23430  213 
subsubsection{* The @{text "\<infinity>"} Version*} 
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214 

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lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x  (abs(xz)+1) * d < z" 
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216 
by(induct rule: int_gr_induct,simp_all add:int_distrib) 
14577  217 

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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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220 

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theorem int_induct[case_names base step1 step2]: 
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222 
assumes 
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223 
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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225 
shows "P i" 
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226 
proof  
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have "i \<le> k \<or> i\<ge> k" by arith 
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228 
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  229 
qed 
230 

231 
lemma decr_mult_lemma: 

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232 
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x  d)" and knneg: "0 <= k" 
13876  233 
shows "ALL x. P x \<longrightarrow> P(x  k*d)" 
234 
using knneg 

235 
proof (induct rule:int_ge_induct) 

236 
case base thus ?case by simp 

237 
next 

238 
case (step i) 

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239 
{fix x 
13876  240 
have "P x \<longrightarrow> P (x  i * d)" using step.hyps by blast 
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241 
also have "\<dots> \<longrightarrow> P(x  (i + 1) * d)" using minus[THEN spec, of "x  i * d"] 
14738  242 
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) 
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243 
ultimately have "P x \<longrightarrow> P(x  (i + 1) * d)" by blast} 
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244 
thus ?case .. 
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245 
qed 
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246 

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247 
lemma minusinfinity: 
23389  248 
assumes dpos: "0 < d" and 
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249 
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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250 
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" 
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251 
proof 
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252 
assume eP1: "EX x. P1 x" 
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253 
then obtain x where P1: "P1 x" .. 
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254 
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. 
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255 
let ?w = "x  (abs(xz)+1) * d" 
23389  256 
from dpos have w: "?w < z" by(rule decr_lemma) 
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257 
have "P1 x = P1 ?w" using P1eqP1 by blast 
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258 
also have "\<dots> = P(?w)" using w P1eqP by blast 
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259 
finally have "P ?w" using P1 by blast 
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260 
thus "EX x. P x" .. 
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261 
qed 
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262 

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263 
lemma cpmi: 
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264 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" 
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265 
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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266 
and pd: "\<forall> x k. P' x = P' (xk*D)" 
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267 
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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268 
(is "?L = (?R1 \<or> ?R2)") 
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269 
proof 
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270 
{assume "?R2" hence "?L" by blast} 
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271 
moreover 
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272 
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 
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273 
moreover 
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274 
{ fix x 
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275 
assume P: "P x" and H: "\<not> ?R2" 
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276 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" 
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277 
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto 
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278 
with nb P have "P (y  D)" by auto } 
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279 
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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280 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x  D)" by auto 
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281 
from p1 obtain z where z: "ALL x. x < z > (P x = P' x)" by blast 
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282 
let ?y = "x  (\<bar>x  z\<bar> + 1)*D" 
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283 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 
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284 
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp 
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285 
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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286 
with periodic_finite_ex[OF dp pd] 
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287 
have "?R1" by blast} 
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288 
ultimately show ?thesis by blast 
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289 
qed 
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290 

23430  291 
subsubsection {* The @{text "+\<infinity>"} Version*} 
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292 

6894137e854a
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293 
lemma plusinfinity: 
23389  294 
assumes dpos: "(0::int) < d" and 
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295 
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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296 
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" 
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297 
proof 
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298 
assume eP1: "EX x. P' x" 
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299 
then obtain x where P1: "P' x" .. 
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300 
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. 
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301 
let ?w' = "x + (abs(xz)+1) * d" 
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302 
let ?w = "x  ((abs(xz) + 1))*d" 
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303 
have ww'[simp]: "?w = ?w'" by (simp add: ring_eq_simps) 
23389  304 
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) 
23314
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305 
hence "P' x = P' ?w" using P1eqP1 by blast 
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306 
also have "\<dots> = P(?w)" using w P1eqP by blast 
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307 
finally have "P ?w" using P1 by blast 
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308 
thus "EX x. P x" .. 
13876  309 
qed 
310 

311 
lemma incr_mult_lemma: 

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312 
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" 
13876  313 
shows "ALL x. P x \<longrightarrow> P(x + k*d)" 
314 
using knneg 

315 
proof (induct rule:int_ge_induct) 

316 
case base thus ?case by simp 

317 
next 

318 
case (step i) 

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319 
{fix x 
13876  320 
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast 
23314
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321 
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] 
13876  322 
by (simp add:int_distrib zadd_ac) 
23314
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323 
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} 
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324 
thus ?case .. 
13876  325 
qed 
326 

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327 
lemma cppi: 
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328 
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" 
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329 
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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330 
and pd: "\<forall> x k. P' x= P' (xk*D)" 
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changeset

331 
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)") 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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332 
proof 
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changeset

333 
{assume "?R2" hence "?L" by blast} 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

334 
moreover 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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diff
changeset

335 
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

336 
moreover 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

337 
{ fix x 
6894137e854a
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chaieb
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338 
assume P: "P x" and H: "\<not> ?R2" 
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A new and cleaned up Theory for QE. for Presburger arithmetic
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339 
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b  j))" and P: "P y" 
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340 
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b  j)" by auto 
6894137e854a
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341 
with nb P have "P (y + D)" by auto } 
6894137e854a
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changeset

342 
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(bj)) > P (x) > P (x + D)" by blast 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

343 
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto 
6894137e854a
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344 
from p1 obtain z where z: "ALL x. x > z > (P x = P' x)" by blast 
6894137e854a
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345 
let ?y = "x + (\<bar>x  z\<bar> + 1)*D" 
6894137e854a
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346 
have zp: "0 <= (\<bar>x  z\<bar> + 1)" by arith 
6894137e854a
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chaieb
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changeset

347 
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp 
6894137e854a
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chaieb
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348 
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto 
6894137e854a
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chaieb
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349 
with periodic_finite_ex[OF dp pd] 
6894137e854a
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chaieb
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350 
have "?R1" by blast} 
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chaieb
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changeset

351 
ultimately show ?thesis by blast 
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A new and cleaned up Theory for QE. for Presburger arithmetic
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352 
qed 
13876  353 

354 
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" 

355 
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) 

356 
apply(fastsimp) 

357 
done 

358 

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359 
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" 
6894137e854a
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360 
apply (rule eq_reflection[symmetric]) 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

361 
apply (rule iffI) 
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A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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changeset

362 
defer 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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363 
apply (erule exE) 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
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364 
apply (rule_tac x = "l * x" in exI) 
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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365 
apply (simp add: dvd_def) 
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366 
apply (rule_tac x="x" in exI, simp) 
6894137e854a
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367 
apply (erule exE) 
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368 
apply (erule conjE) 
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369 
apply (erule dvdE) 
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370 
apply (rule_tac x = k in exI) 
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371 
apply simp 
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372 
done 
13876  373 

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374 
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" 
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375 
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" 
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376 
using not0 by (simp add: dvd_def) 
13876  377 

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378 
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd t)" 
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379 
by simp_all 
14577  380 
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} 
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381 
lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))" 
13876  382 
by (simp split add: split_nat) 
383 

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384 
lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))" 
23365  385 
apply (auto split add: split_nat) 
386 
apply (rule_tac x="int x" in exI, simp) 

387 
apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp) 

388 
done 

13876  389 

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390 
lemma zdiff_int_split: "P (int (x  y)) = 
13876  391 
((y \<le> x \<longrightarrow> P (int x  int y)) \<and> (x < y \<longrightarrow> P 0))" 
23365  392 
by (case_tac "y \<le> x", simp_all) 
13876  393 

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394 
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp 
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395 
lemma number_of2: "(0::int) <= Numeral0" by simp 
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396 
lemma Suc_plus1: "Suc n = n + 1" by simp 
13876  397 

14577  398 
text {* 
399 
\medskip Specific instances of congruence rules, to prevent 

400 
simplifier from looping. *} 

13876  401 

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402 
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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403 

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404 
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" 
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405 
by (simp cong: conj_cong) 
20485  406 
lemma int_eq_number_of_eq: 
407 
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" 

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408 
by simp 
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409 

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410 
lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m" 
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411 
unfolding dvd_eq_mod_eq_0[symmetric] .. 
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412 

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413 
lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m" 
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414 
unfolding zdvd_iff_zmod_eq_0[symmetric] .. 
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415 
declare mod_1[presburger] 
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416 
declare mod_0[presburger] 
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417 
declare zmod_1[presburger] 
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418 
declare zmod_zero[presburger] 
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419 
declare zmod_self[presburger] 
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420 
declare mod_self[presburger] 
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declare DIVISION_BY_ZERO_MOD[presburger] 
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422 
declare nat_mod_div_trivial[presburger] 
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423 
declare div_mod_equality2[presburger] 
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424 
declare div_mod_equality[presburger] 
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425 
declare mod_div_equality2[presburger] 
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426 
declare mod_div_equality[presburger] 
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427 
declare mod_mult_self1[presburger] 
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428 
declare mod_mult_self2[presburger] 
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429 
declare zdiv_zmod_equality2[presburger] 
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430 
declare zdiv_zmod_equality[presburger] 
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431 
declare mod2_Suc_Suc[presburger] 
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432 
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" 
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433 
using IntDiv.DIVISION_BY_ZERO by blast+ 
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434 

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435 
use "Tools/Presburger/cooper.ML" 
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436 
oracle linzqe_oracle ("term") = Coopereif.cooper_oracle 
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437 

23146  438 
use "Tools/Presburger/presburger.ML" 
13876  439 

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440 
setup {* 
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441 
arith_tactic_add 
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442 
(mk_arith_tactic "presburger" (fn i => fn st => 
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443 
(warning "Trying Presburger arithmetic ..."; 
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444 
Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st))) 
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445 
(* FIXME!!!!!!! get the right context!!*) 
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446 
*} 
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447 

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448 
method_setup presburger = {* 
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449 
let 
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450 
fun keyword k = Scan.lift (Args.$$$ k  Args.colon) >> K () 
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451 
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () 
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452 
val addN = "add" 
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453 
val delN = "del" 
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454 
val elimN = "elim" 
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455 
val any_keyword = keyword addN  keyword delN  simple_keyword elimN 
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456 
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; 
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457 
in 
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458 
fn src => Method.syntax 
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459 
((Scan.optional (simple_keyword elimN >> K false) true)  
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460 
(Scan.optional (keyword addN  thms) [])  
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461 
(Scan.optional (keyword delN  thms) [])) src 
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462 
#> (fn (((elim, add_ths), del_ths),ctxt) => 
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463 
Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) 
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464 
end 
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465 
*} "" 
22801  466 

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467 
lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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468 
lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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469 
lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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470 
lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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471 
lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger 
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472 

22801  473 
subsection {* Code generator setup *} 
20595  474 
text {* 
22801  475 
Presburger arithmetic is convenient to prove some 
476 
of the following code lemmas on integer numerals: 

20595  477 
*} 
478 

479 
lemma eq_Pls_Pls: 

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480 
"Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger 
20595  481 

482 
lemma eq_Pls_Min: 

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483 
"Numeral.Pls = Numeral.Min \<longleftrightarrow> False" 
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484 
unfolding Pls_def Min_def by presburger 
20595  485 

486 
lemma eq_Pls_Bit0: 

21454  487 
"Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k" 
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488 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  489 

490 
lemma eq_Pls_Bit1: 

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491 
"Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False" 
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492 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  493 

494 
lemma eq_Min_Pls: 

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495 
"Numeral.Min = Numeral.Pls \<longleftrightarrow> False" 
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496 
unfolding Pls_def Min_def by presburger 
20595  497 

498 
lemma eq_Min_Min: 

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499 
"Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger 
20595  500 

501 
lemma eq_Min_Bit0: 

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502 
"Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False" 
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503 
unfolding Min_def Bit_def bit.cases by presburger 
20595  504 

505 
lemma eq_Min_Bit1: 

21454  506 
"Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k" 
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507 
unfolding Min_def Bit_def bit.cases by presburger 
20595  508 

509 
lemma eq_Bit0_Pls: 

21454  510 
"Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k" 
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511 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  512 

513 
lemma eq_Bit1_Pls: 

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514 
"Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False" 
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515 
unfolding Pls_def Bit_def bit.cases by presburger 
20595  516 

517 
lemma eq_Bit0_Min: 

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518 
"Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False" 
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519 
unfolding Min_def Bit_def bit.cases by presburger 
20595  520 

521 
lemma eq_Bit1_Min: 

21454  522 
"(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k" 
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523 
unfolding Min_def Bit_def bit.cases by presburger 
20595  524 

525 
lemma eq_Bit_Bit: 

21454  526 
"Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow> 
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527 
v1 = v2 \<and> k1 = k2" 
21454  528 
unfolding Bit_def 
20595  529 
apply (cases v1) 
530 
apply (cases v2) 

531 
apply auto 

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532 
apply presburger 
20595  533 
apply (cases v2) 
534 
apply auto 

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535 
apply presburger 
20595  536 
apply (cases v2) 
537 
apply auto 

538 
done 

539 

22801  540 
lemma eq_number_of: 
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541 
"(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l" 
22801  542 
unfolding number_of_is_id .. 
20595  543 

22394  544 

20595  545 
lemma less_eq_Pls_Pls: 
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546 
"Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+ 
20595  547 

548 
lemma less_eq_Pls_Min: 

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549 
"Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False" 
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550 
unfolding Pls_def Min_def by presburger 
20595  551 

552 
lemma less_eq_Pls_Bit: 

553 
"Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k" 

554 
unfolding Pls_def Bit_def by (cases v) auto 

555 

556 
lemma less_eq_Min_Pls: 

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557 
"Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True" 
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558 
unfolding Pls_def Min_def by presburger 
20595  559 

560 
lemma less_eq_Min_Min: 

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561 
"Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+ 
20595  562 

563 
lemma less_eq_Min_Bit0: 

564 
"Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k" 

565 
unfolding Min_def Bit_def by auto 

566 

567 
lemma less_eq_Min_Bit1: 

568 
"Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k" 

569 
unfolding Min_def Bit_def by auto 

570 

571 
lemma less_eq_Bit0_Pls: 

572 
"Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls" 

573 
unfolding Pls_def Bit_def by simp 

574 

575 
lemma less_eq_Bit1_Pls: 

576 
"Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" 

577 
unfolding Pls_def Bit_def by auto 

578 

579 
lemma less_eq_Bit_Min: 

580 
"Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" 

581 
unfolding Min_def Bit_def by (cases v) auto 

582 

583 
lemma less_eq_Bit0_Bit: 

584 
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2" 

22394  585 
unfolding Bit_def bit.cases by (cases v) auto 
20595  586 

587 
lemma less_eq_Bit_Bit1: 

588 
"Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2" 

22394  589 
unfolding Bit_def bit.cases by (cases v) auto 
590 

591 
lemma less_eq_Bit1_Bit0: 

592 
"Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" 

593 
unfolding Bit_def by (auto split: bit.split) 

20595  594 

22801  595 
lemma less_eq_number_of: 
596 
"(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l" 

597 
unfolding number_of_is_id .. 

22394  598 

599 

600 
lemma less_Pls_Pls: 

23405  601 
"Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp 
22394  602 

603 
lemma less_Pls_Min: 

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

604 
"Numeral.Pls < Numeral.Min \<longleftrightarrow> False" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

605 
unfolding Pls_def Min_def by presburger 
22394  606 

607 
lemma less_Pls_Bit0: 

608 
"Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k" 

609 
unfolding Pls_def Bit_def by auto 

610 

611 
lemma less_Pls_Bit1: 

612 
"Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k" 

613 
unfolding Pls_def Bit_def by auto 

614 

615 
lemma less_Min_Pls: 

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

616 
"Numeral.Min < Numeral.Pls \<longleftrightarrow> True" 
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset

617 
unfolding Pls_def Min_def by presburger 
22394  618 

619 
lemma less_Min_Min: 

23405  620 
"Numeral.Min < Numeral.Min \<longleftrightarrow> False" by simp 
22394  621 

622 
lemma less_Min_Bit: 

623 
"Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k" 

624 
unfolding Min_def Bit_def by (auto split: bit.split) 

625 

626 
lemma less_Bit_Pls: 

627 
"Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" 

628 
unfolding Pls_def Bit_def by (auto split: bit.split) 

629 

630 
lemma less_Bit0_Min: 

631 
"Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" 

632 
unfolding Min_def Bit_def by auto 

633 

634 
lemma less_Bit1_Min: 

635 
"Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min" 

636 
unfolding Min_def Bit_def by auto 

637 

638 
lemma less_Bit_Bit0: 

639 
"Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" 

640 
unfolding Bit_def by (auto split: bit.split) 

641 

642 
lemma less_Bit1_Bit: 

643 
"Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2" 

644 
unfolding Bit_def by (auto split: bit.split) 

645 

646 
lemma less_Bit0_Bit1: 

647 
"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

648 
unfolding Bit_def bit.cases by arith 
22394  649 

22801  650 
lemma less_number_of: 
651 
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l" 

652 
unfolding number_of_is_id .. 

653 

654 
lemmas pred_succ_numeral_code [code func] = 

655 
arith_simps(512) 

656 

657 
lemmas plus_numeral_code [code func] = 

658 
arith_simps(1317) 

659 
arith_simps(2627) 

660 
arith_extra_simps(1) [where 'a = int] 

661 

662 
lemmas minus_numeral_code [code func] = 

663 
arith_simps(1821) 

664 
arith_extra_simps(2) [where 'a = int] 

665 
arith_extra_simps(5) [where 'a = int] 

666 

667 
lemmas times_numeral_code [code func] = 

668 
arith_simps(2225) 

669 
arith_extra_simps(4) [where 'a = int] 

670 

671 
lemmas eq_numeral_code [code func] = 

672 
eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1 

673 
eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1 

674 
eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit 

675 
eq_number_of 

676 

677 
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit 

678 
less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1 

679 
less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0 

680 
less_eq_number_of 

681 

22394  682 
lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0 
683 
less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls 

684 
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1 

22801  685 
less_number_of 
20595  686 

23365  687 
end 