author | huffman |
Wed, 20 Jun 2007 17:28:55 +0200 | |
changeset 23438 | dd824e86fa8a |
parent 23430 | 771117253634 |
child 23460 | f9ae34d5f8da |
permissions | -rw-r--r-- |
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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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||
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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"Tools/Presburger/generated_cooper.ML" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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("Tools/Presburger/cooper.ML") ("Tools/Presburger/presburger.ML") |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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|
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begin |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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setup {* Cooper_Data.setup*} |
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|
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subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *} |
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|
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A new and cleaned up Theory for QE. for Presburger arithmetic
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lemma minf: |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<and> Q x) = (P' x \<and> Q' x)" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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\<Longrightarrow> \<exists>z.\<forall>x<z. (P x \<or> Q x) = (P' x \<or> Q' x)" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x = t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<noteq> t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x < t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<le> t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x > t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x<z.(x \<ge> t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>z.\<forall>x<z. F = F" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<and> Q x) = (P' x \<and> Q' x)" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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\<Longrightarrow> \<exists>z.\<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x = t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<noteq> t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x < t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<le> t) = False" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x > t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>(z ::'a::{linorder}).\<forall>x>z.(x \<ge> t) = True" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<exists>z.\<forall>x>z. F = F" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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"\<forall>x k. F = F" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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by simp_all |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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(clarsimp simp add: dvd_def, rule iffI, clarsimp,rule_tac x = "kb - ka*k" in exI, |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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75 |
"\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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77 |
"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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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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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A new and cleaned up Theory for QE. for Presburger arithmetic
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proof (blast, blast) |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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assume dp: "D > 0" and tB: "t - 1\<in> B" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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84 |
using dp tB by simp_all |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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85 |
next |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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86 |
assume dp: "D > 0" and tB: "t \<in> B" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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87 |
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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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88 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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89 |
using dp tB by simp_all |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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90 |
next |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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91 |
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
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92 |
next |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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93 |
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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A new and cleaned up Theory for QE. for Presburger arithmetic
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94 |
next |
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95 |
assume dp: "D > 0" and tB:"t \<in> B" |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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96 |
{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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A new and cleaned up Theory for QE. for Presburger arithmetic
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97 |
hence "x -t \<le> D" and "1 \<le> x - t" by simp+ |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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98 |
hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto |
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A new and cleaned up Theory for QE. for Presburger arithmetic
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99 |
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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A new and cleaned up Theory for QE. for Presburger arithmetic
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101 |
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
102 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
103 |
assume dp: "D > 0" and tB:"t - 1\<in> B" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
104 |
{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" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
105 |
hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+ |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
106 |
hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
107 |
hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_eq_simps) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
108 |
with nob tB have "False" by simp} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
109 |
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
110 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
111 |
assume d: "d dvd D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
112 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
113 |
by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_eq_simps)} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
115 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
116 |
assume d: "d dvd D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
117 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
118 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_eq_simps)} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
120 |
qed blast |
13876 | 121 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
122 |
lemma aset: |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
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diff
changeset
|
123 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
124 |
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
125 |
\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
126 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
127 |
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
128 |
\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
129 |
"\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
130 |
"\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
131 |
"\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
132 |
"\<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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
133 |
"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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
134 |
"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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
135 |
"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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
137 |
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
138 |
proof (blast, blast) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
139 |
assume dp: "D > 0" and tA: "t + 1 \<in> A" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
140 |
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
141 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
142 |
using dp tA by simp_all |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
143 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
144 |
assume dp: "D > 0" and tA: "t \<in> A" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
146 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
147 |
using dp tA by simp_all |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
148 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
150 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
152 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
153 |
assume dp: "D > 0" and tA:"t \<in> A" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
154 |
{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" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
155 |
hence "t - x \<le> D" and "1 \<le> t - x" by simp+ |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
156 |
hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
157 |
hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_eq_simps) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
158 |
with nob tA have "False" by simp} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
160 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
161 |
assume dp: "D > 0" and tA:"t + 1\<in> A" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
163 |
hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_eq_simps) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
164 |
hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
165 |
hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_eq_simps) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
166 |
with nob tA have "False" by simp} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
168 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
169 |
assume d: "d dvd D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
170 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
171 |
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_eq_simps)} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
173 |
next |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
174 |
assume d: "d dvd D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
175 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
176 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_eq_simps)} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 *} |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
183 |
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" .. |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
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diff
changeset
|
211 |
qed auto |
13876 | 212 |
|
23430 | 213 |
subsubsection{* The @{text "-\<infinity>"} Version*} |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
214 |
|
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
215 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
216 |
by(induct rule: int_gr_induct,simp_all add:int_distrib) |
14577 | 217 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
218 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
219 |
by(induct rule: int_gr_induct, simp_all add:int_distrib) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
220 |
|
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
221 |
theorem int_induct[case_names base step1 step2]: |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
222 |
assumes |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
223 |
base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
224 |
step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
225 |
shows "P i" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
226 |
proof - |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
227 |
have "i \<le> k \<or> i\<ge> k" by arith |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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: |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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) |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
239 |
{fix x |
13876 | 240 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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]) |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
243 |
ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
244 |
thus ?case .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
245 |
qed |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
246 |
|
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
247 |
lemma minusinfinity: |
23389 | 248 |
assumes dpos: "0 < d" and |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
250 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
251 |
proof |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
252 |
assume eP1: "EX x. P1 x" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
253 |
then obtain x where P1: "P1 x" .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
254 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
255 |
let ?w = "x - (abs(x-z)+1) * d" |
23389 | 256 |
from dpos have w: "?w < z" by(rule decr_lemma) |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
257 |
have "P1 x = P1 ?w" using P1eqP1 by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
258 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
259 |
finally have "P ?w" using P1 by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
260 |
thus "EX x. P x" .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
261 |
qed |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
262 |
|
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
263 |
lemma cpmi: |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
264 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
265 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
266 |
and pd: "\<forall> x k. P' x = P' (x-k*D)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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)))" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
268 |
(is "?L = (?R1 \<or> ?R2)") |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
269 |
proof- |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
270 |
{assume "?R2" hence "?L" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
271 |
moreover |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
272 |
{assume H:"?R1" hence "?L" using minusinfinity[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
parents:
23253
diff
changeset
|
273 |
moreover |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
274 |
{ fix x |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
275 |
assume P: "P x" and H: "\<not> ?R2" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
276 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
277 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
278 |
with nb P have "P (y - D)" by auto } |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
279 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
280 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
281 |
from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
282 |
let ?y = "x - (\<bar>x - z\<bar> + 1)*D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
283 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
284 |
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
285 |
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
286 |
with periodic_finite_ex[OF dp pd] |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
287 |
have "?R1" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
288 |
ultimately show ?thesis by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
289 |
qed |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
290 |
|
23430 | 291 |
subsubsection {* The @{text "+\<infinity>"} Version*} |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
292 |
|
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
293 |
lemma plusinfinity: |
23389 | 294 |
assumes dpos: "(0::int) < d" and |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
295 |
P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
296 |
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
297 |
proof |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
298 |
assume eP1: "EX x. P' x" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
299 |
then obtain x where P1: "P' x" .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
300 |
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
301 |
let ?w' = "x + (abs(x-z)+1) * d" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
302 |
let ?w = "x - (-(abs(x-z) + 1))*d" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
305 |
hence "P' x = P' ?w" using P1eqP1 by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
306 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
307 |
finally have "P ?w" using P1 by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
308 |
thus "EX x. P x" .. |
13876 | 309 |
qed |
310 |
||
311 |
lemma incr_mult_lemma: |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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) |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
319 |
{fix x |
13876 | 320 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
323 |
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
324 |
thus ?case .. |
13876 | 325 |
qed |
326 |
||
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
327 |
lemma cppi: |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
328 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
329 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
330 |
and pd: "\<forall> x k. P' x= P' (x-k*D)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
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
chaieb
parents:
23253
diff
changeset
|
332 |
proof- |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
333 |
{assume "?R2" hence "?L" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
334 |
moreover |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
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
parents:
23253
diff
changeset
|
336 |
moreover |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
337 |
{ fix x |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
338 |
assume P: "P x" and H: "\<not> ?R2" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
339 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
340 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
341 |
with nb P have "P (y + D)" by auto } |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
342 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
343 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
344 |
from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
345 |
let ?y = "x + (\<bar>x - z\<bar> + 1)*D" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
346 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
347 |
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
349 |
with periodic_finite_ex[OF dp pd] |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
350 |
have "?R1" by blast} |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
351 |
ultimately show ?thesis by blast |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
||
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
359 |
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
360 |
apply (rule eq_reflection[symmetric]) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
361 |
apply (rule iffI) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
362 |
defer |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
363 |
apply (erule exE) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
364 |
apply (rule_tac x = "l * x" in exI) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
365 |
apply (simp add: dvd_def) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
366 |
apply (rule_tac x="x" in exI, simp) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
367 |
apply (erule exE) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
368 |
apply (erule conjE) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
369 |
apply (erule dvdE) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
370 |
apply (rule_tac x = k in exI) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
371 |
apply simp |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
372 |
done |
13876 | 373 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
374 |
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
375 |
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
376 |
using not0 by (simp add: dvd_def) |
13876 | 377 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
378 |
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
379 |
by simp_all |
14577 | 380 |
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
||
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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))" |
23438
dd824e86fa8a
remove simp attribute from of_nat_diff, for backward compatibility with zdiff_int
huffman
parents:
23430
diff
changeset
|
392 |
by (case_tac "y \<le> x", simp_all add: zdiff_int) |
13876 | 393 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
394 |
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
395 |
lemma number_of2: "(0::int) <= Numeral0" by simp |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
402 |
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp |
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
17589
diff
changeset
|
403 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
404 |
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
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)" |
|
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
17589
diff
changeset
|
408 |
by simp |
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
17589
diff
changeset
|
409 |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
410 |
lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m" |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
411 |
unfolding dvd_eq_mod_eq_0[symmetric] .. |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
412 |
|
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
413 |
lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m" |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
414 |
unfolding zdvd_iff_zmod_eq_0[symmetric] .. |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
415 |
declare mod_1[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
416 |
declare mod_0[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
417 |
declare zmod_1[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
418 |
declare zmod_zero[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
419 |
declare zmod_self[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
420 |
declare mod_self[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
421 |
declare DIVISION_BY_ZERO_MOD[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
422 |
declare nat_mod_div_trivial[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
423 |
declare div_mod_equality2[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
424 |
declare div_mod_equality[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
425 |
declare mod_div_equality2[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
426 |
declare mod_div_equality[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
427 |
declare mod_mult_self1[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
428 |
declare mod_mult_self2[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
429 |
declare zdiv_zmod_equality2[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
430 |
declare zdiv_zmod_equality[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
431 |
declare mod2_Suc_Suc[presburger] |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
432 |
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
433 |
using IntDiv.DIVISION_BY_ZERO by blast+ |
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
17589
diff
changeset
|
434 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
435 |
use "Tools/Presburger/cooper.ML" |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
436 |
oracle linzqe_oracle ("term") = Coopereif.cooper_oracle |
18202
46af82efd311
presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents:
17589
diff
changeset
|
437 |
|
23146 | 438 |
use "Tools/Presburger/presburger.ML" |
13876 | 439 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
440 |
setup {* |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
441 |
arith_tactic_add |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
442 |
(mk_arith_tactic "presburger" (fn i => fn st => |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
443 |
(warning "Trying Presburger arithmetic ..."; |
23333
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
444 |
Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st))) |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
445 |
(* FIXME!!!!!!! get the right context!!*) |
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
446 |
*} |
23333
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
447 |
|
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
448 |
method_setup presburger = {* |
23333
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
449 |
let |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
450 |
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
451 |
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
452 |
val addN = "add" |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
453 |
val delN = "del" |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
454 |
val elimN = "elim" |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
455 |
val any_keyword = keyword addN || keyword delN || simple_keyword elimN |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
456 |
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
457 |
in |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
458 |
fn src => Method.syntax |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
459 |
((Scan.optional (simple_keyword elimN >> K false) true) -- |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
460 |
(Scan.optional (keyword addN |-- thms) []) -- |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
461 |
(Scan.optional (keyword delN |-- thms) [])) src |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
462 |
#> (fn (((elim, add_ths), del_ths),ctxt) => |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
463 |
Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) |
ec5b4ab52026
Method now takes theorems to be added or deleted from a simpset for simplificatio before the core method starts
chaieb
parents:
23314
diff
changeset
|
464 |
end |
23314
6894137e854a
A new and cleaned up Theory for QE. for Presburger arithmetic
chaieb
parents:
23253
diff
changeset
|
465 |
*} "" |
22801 | 466 |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
467 |
lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
468 |
lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
469 |
lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
470 |
lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
471 |
lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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: |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
480 |
"Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger |
20595 | 481 |
|
482 |
lemma eq_Pls_Min: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
483 |
"Numeral.Pls = Numeral.Min \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
488 |
unfolding Pls_def Bit_def bit.cases by presburger |
20595 | 489 |
|
490 |
lemma eq_Pls_Bit1: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
491 |
"Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
492 |
unfolding Pls_def Bit_def bit.cases by presburger |
20595 | 493 |
|
494 |
lemma eq_Min_Pls: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
495 |
"Numeral.Min = Numeral.Pls \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
496 |
unfolding Pls_def Min_def by presburger |
20595 | 497 |
|
498 |
lemma eq_Min_Min: |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
499 |
"Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger |
20595 | 500 |
|
501 |
lemma eq_Min_Bit0: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
502 |
"Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
511 |
unfolding Pls_def Bit_def bit.cases by presburger |
20595 | 512 |
|
513 |
lemma eq_Bit1_Pls: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
514 |
"Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
515 |
unfolding Pls_def Bit_def bit.cases by presburger |
20595 | 516 |
|
517 |
lemma eq_Bit0_Min: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
518 |
"Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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> |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
527 |
v1 = v2 \<and> k1 = k2" |
21454 | 528 |
unfolding Bit_def |
20595 | 529 |
apply (cases v1) |
530 |
apply (cases v2) |
|
531 |
apply auto |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
532 |
apply presburger |
20595 | 533 |
apply (cases v2) |
534 |
apply auto |
|
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
535 |
apply presburger |
20595 | 536 |
apply (cases v2) |
537 |
apply auto |
|
538 |
done |
|
539 |
||
22801 | 540 |
lemma eq_number_of: |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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: |
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
546 |
"Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+ |
20595 | 547 |
|
548 |
lemma less_eq_Pls_Min: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
549 |
"Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
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: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
557 |
"Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True" |
23390
01ef1135de73
Added some lemmas to default presburger simpset; tuned proofs
chaieb
parents:
23389
diff
changeset
|
558 |
unfolding Pls_def Min_def by presburger |
20595 | 559 |
|
560 |
lemma less_eq_Min_Min: |
|
22744
5cbe966d67a2
Isar definitions are now added explicitly to code theorem table
haftmann
parents:
22394
diff
changeset
|
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(5-12) |
|
656 |
||
657 |
lemmas plus_numeral_code [code func] = |
|
658 |
arith_simps(13-17) |
|
659 |
arith_simps(26-27) |
|
660 |
arith_extra_simps(1) [where 'a = int] |
|
661 |
||
662 |
lemmas minus_numeral_code [code func] = |
|
663 |
arith_simps(18-21) |
|
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(22-25) |
|
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 |