author | haftmann |
Thu, 19 Jul 2007 21:47:43 +0200 | |
changeset 23856 | ebec38420a85 |
parent 23685 | 1b0f4071946c |
child 24075 | 366d4d234814 |
permissions | -rw-r--r-- |
23465 | 1 |
(* 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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23472 | 6 |
header {* Decision Procedure for Presburger Arithmetic *} |
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theory Presburger |
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imports Arith_Tools SetInterval |
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uses |
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"Tools/Qelim/cooper_data.ML" |
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"Tools/Qelim/generated_cooper.ML" |
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("Tools/Qelim/cooper.ML") |
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("Tools/Qelim/presburger.ML") |
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begin |
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setup CooperData.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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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
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simp add: ring_simps, clarsimp,rule_tac x = "kb + ka*k" in exI,simp add: ring_simps)+ |
23465 | 64 |
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23472 | 65 |
subsection{* The A and B sets *} |
23465 | 66 |
lemma bset: |
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"\<lbrakk>\<forall>x.(\<forall>j \<in> {1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ; |
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\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> |
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\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> (P x \<and> Q x) \<longrightarrow> (P(x - D) \<and> Q (x - D))" |
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"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> P x \<longrightarrow> P(x - D) ; |
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\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> Q x \<longrightarrow> Q(x - D)\<rbrakk> \<Longrightarrow> |
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\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (P x \<or> Q x) \<longrightarrow> (P(x - D) \<or> Q (x - D))" |
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"\<lbrakk>D>0; t - 1\<in> B\<rbrakk> \<Longrightarrow> (\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" |
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"\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))" |
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"D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" |
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"D>0 \<Longrightarrow> (\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t))" |
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"\<lbrakk>D>0 ; t \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t))" |
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"\<lbrakk>D>0 ; t - 1 \<in> B\<rbrakk> \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t))" |
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"d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t))" |
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"d dvd D \<Longrightarrow>(\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not> d dvd (x - D) + t))" |
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"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j) \<longrightarrow> F \<longrightarrow> F" |
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proof (blast, blast) |
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assume dp: "D > 0" and tB: "t - 1\<in> B" |
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show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x = t) \<longrightarrow> (x - D = t))" |
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apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) |
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using dp tB by simp_all |
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next |
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assume dp: "D > 0" and tB: "t \<in> B" |
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show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<noteq> t) \<longrightarrow> (x - D \<noteq> t))" |
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apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) |
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using dp tB by simp_all |
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next |
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assume dp: "D > 0" thus "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x < t) \<longrightarrow> (x - D < t))" by arith |
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next |
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assume dp: "D > 0" thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x - D \<le> t)" by arith |
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next |
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assume dp: "D > 0" and tB:"t \<in> B" |
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{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t" |
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hence "x -t \<le> D" and "1 \<le> x - t" by simp+ |
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hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto |
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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
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hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps) |
23465 | 102 |
with nob tB have "False" by simp} |
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thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast |
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next |
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assume dp: "D > 0" and tB:"t - 1\<in> B" |
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{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t" |
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hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+ |
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hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto |
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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
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hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps) |
23465 | 110 |
with nob tB have "False" by simp} |
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thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast |
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next |
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assume d: "d dvd D" |
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{fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" |
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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
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by (clarsimp simp add: dvd_def,rule_tac x= "ka - k" in exI,simp add: ring_simps)} |
23465 | 116 |
thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x - D) + t)" by simp |
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next |
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assume d: "d dvd D" |
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{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x - D) + t" |
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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
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by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)} |
23465 | 121 |
thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto |
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qed blast |
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||
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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))" |
|
132 |
"\<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))" |
|
135 |
"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))" |
|
136 |
"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))" |
|
137 |
"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))" |
|
138 |
"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))" |
|
139 |
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F" |
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140 |
proof (blast, blast) |
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141 |
assume dp: "D > 0" and tA: "t + 1 \<in> A" |
|
142 |
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" |
|
143 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) |
|
144 |
using dp tA by simp_all |
|
145 |
next |
|
146 |
assume dp: "D > 0" and tA: "t \<in> A" |
|
147 |
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))" |
|
148 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) |
|
149 |
using dp tA by simp_all |
|
150 |
next |
|
151 |
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 |
|
152 |
next |
|
153 |
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 |
|
154 |
next |
|
155 |
assume dp: "D > 0" and tA:"t \<in> A" |
|
156 |
{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" |
|
157 |
hence "t - x \<le> D" and "1 \<le> t - x" by simp+ |
|
158 |
hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
159 |
hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps) |
23465 | 160 |
with nob tA have "False" by simp} |
161 |
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 |
|
162 |
next |
|
163 |
assume dp: "D > 0" and tA:"t + 1\<in> A" |
|
164 |
{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" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
165 |
hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps) |
23465 | 166 |
hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto |
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
167 |
hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps) |
23465 | 168 |
with nob tA have "False" by simp} |
169 |
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 |
|
170 |
next |
|
171 |
assume d: "d dvd D" |
|
172 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" |
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23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
173 |
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)} |
23465 | 174 |
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 |
175 |
next |
|
176 |
assume d: "d dvd D" |
|
177 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
178 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)} |
23465 | 179 |
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 |
180 |
qed blast |
|
181 |
||
182 |
subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *} |
|
183 |
||
184 |
subsubsection{* First some trivial facts about periodic sets or predicates *} |
|
185 |
lemma periodic_finite_ex: |
|
186 |
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)" |
|
187 |
shows "(EX x. P x) = (EX j : {1..d}. P j)" |
|
188 |
(is "?LHS = ?RHS") |
|
189 |
proof |
|
190 |
assume ?LHS |
|
191 |
then obtain x where P: "P x" .. |
|
192 |
have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) |
|
193 |
hence Pmod: "P x = P(x mod d)" using modd by simp |
|
194 |
show ?RHS |
|
195 |
proof (cases) |
|
196 |
assume "x mod d = 0" |
|
197 |
hence "P 0" using P Pmod by simp |
|
198 |
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast |
|
199 |
ultimately have "P d" by simp |
|
200 |
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) |
|
201 |
ultimately show ?RHS .. |
|
202 |
next |
|
203 |
assume not0: "x mod d \<noteq> 0" |
|
204 |
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) |
|
205 |
moreover have "x mod d : {1..d}" |
|
206 |
proof - |
|
207 |
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) |
|
208 |
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) |
|
209 |
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) |
|
210 |
qed |
|
211 |
ultimately show ?RHS .. |
|
212 |
qed |
|
213 |
qed auto |
|
214 |
||
215 |
subsubsection{* The @{text "-\<infinity>"} Version*} |
|
216 |
||
217 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z" |
|
218 |
by(induct rule: int_gr_induct,simp_all add:int_distrib) |
|
219 |
||
220 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d" |
|
221 |
by(induct rule: int_gr_induct, simp_all add:int_distrib) |
|
222 |
||
223 |
theorem int_induct[case_names base step1 step2]: |
|
224 |
assumes |
|
225 |
base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and |
|
226 |
step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
|
227 |
shows "P i" |
|
228 |
proof - |
|
229 |
have "i \<le> k \<or> i\<ge> k" by arith |
|
230 |
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 |
|
231 |
qed |
|
232 |
||
233 |
lemma decr_mult_lemma: |
|
234 |
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k" |
|
235 |
shows "ALL x. P x \<longrightarrow> P(x - k*d)" |
|
236 |
using knneg |
|
237 |
proof (induct rule:int_ge_induct) |
|
238 |
case base thus ?case by simp |
|
239 |
next |
|
240 |
case (step i) |
|
241 |
{fix x |
|
242 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast |
|
243 |
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"] |
|
244 |
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) |
|
245 |
ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast} |
|
246 |
thus ?case .. |
|
247 |
qed |
|
248 |
||
249 |
lemma minusinfinity: |
|
250 |
assumes dpos: "0 < d" and |
|
251 |
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" |
|
252 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" |
|
253 |
proof |
|
254 |
assume eP1: "EX x. P1 x" |
|
255 |
then obtain x where P1: "P1 x" .. |
|
256 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. |
|
257 |
let ?w = "x - (abs(x-z)+1) * d" |
|
258 |
from dpos have w: "?w < z" by(rule decr_lemma) |
|
259 |
have "P1 x = P1 ?w" using P1eqP1 by blast |
|
260 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
261 |
finally have "P ?w" using P1 by blast |
|
262 |
thus "EX x. P x" .. |
|
263 |
qed |
|
264 |
||
265 |
lemma cpmi: |
|
266 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" |
|
267 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)" |
|
268 |
and pd: "\<forall> x k. P' x = P' (x-k*D)" |
|
269 |
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" |
|
270 |
(is "?L = (?R1 \<or> ?R2)") |
|
271 |
proof- |
|
272 |
{assume "?R2" hence "?L" by blast} |
|
273 |
moreover |
|
274 |
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} |
|
275 |
moreover |
|
276 |
{ fix x |
|
277 |
assume P: "P x" and H: "\<not> ?R2" |
|
278 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" |
|
279 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto |
|
280 |
with nb P have "P (y - D)" by auto } |
|
281 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast |
|
282 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto |
|
283 |
from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast |
|
284 |
let ?y = "x - (\<bar>x - z\<bar> + 1)*D" |
|
285 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
|
286 |
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp |
|
287 |
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto |
|
288 |
with periodic_finite_ex[OF dp pd] |
|
289 |
have "?R1" by blast} |
|
290 |
ultimately show ?thesis by blast |
|
291 |
qed |
|
292 |
||
293 |
subsubsection {* The @{text "+\<infinity>"} Version*} |
|
294 |
||
295 |
lemma plusinfinity: |
|
296 |
assumes dpos: "(0::int) < d" and |
|
297 |
P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" |
|
298 |
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" |
|
299 |
proof |
|
300 |
assume eP1: "EX x. P' x" |
|
301 |
then obtain x where P1: "P' x" .. |
|
302 |
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. |
|
303 |
let ?w' = "x + (abs(x-z)+1) * d" |
|
304 |
let ?w = "x - (-(abs(x-z) + 1))*d" |
|
23477
f4b83f03cac9
tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents:
23472
diff
changeset
|
305 |
have ww'[simp]: "?w = ?w'" by (simp add: ring_simps) |
23465 | 306 |
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) |
307 |
hence "P' x = P' ?w" using P1eqP1 by blast |
|
308 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
309 |
finally have "P ?w" using P1 by blast |
|
310 |
thus "EX x. P x" .. |
|
311 |
qed |
|
312 |
||
313 |
lemma incr_mult_lemma: |
|
314 |
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" |
|
315 |
shows "ALL x. P x \<longrightarrow> P(x + k*d)" |
|
316 |
using knneg |
|
317 |
proof (induct rule:int_ge_induct) |
|
318 |
case base thus ?case by simp |
|
319 |
next |
|
320 |
case (step i) |
|
321 |
{fix x |
|
322 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast |
|
323 |
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] |
|
324 |
by (simp add:int_distrib zadd_ac) |
|
325 |
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} |
|
326 |
thus ?case .. |
|
327 |
qed |
|
328 |
||
329 |
lemma cppi: |
|
330 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" |
|
331 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)" |
|
332 |
and pd: "\<forall> x k. P' x= P' (x-k*D)" |
|
333 |
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)") |
|
334 |
proof- |
|
335 |
{assume "?R2" hence "?L" by blast} |
|
336 |
moreover |
|
337 |
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} |
|
338 |
moreover |
|
339 |
{ fix x |
|
340 |
assume P: "P x" and H: "\<not> ?R2" |
|
341 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y" |
|
342 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto |
|
343 |
with nb P have "P (y + D)" by auto } |
|
344 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast |
|
345 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto |
|
346 |
from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast |
|
347 |
let ?y = "x + (\<bar>x - z\<bar> + 1)*D" |
|
348 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
|
349 |
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp |
|
350 |
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto |
|
351 |
with periodic_finite_ex[OF dp pd] |
|
352 |
have "?R1" by blast} |
|
353 |
ultimately show ?thesis by blast |
|
354 |
qed |
|
355 |
||
356 |
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" |
|
357 |
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) |
|
358 |
apply(fastsimp) |
|
359 |
done |
|
360 |
||
361 |
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" |
|
362 |
apply (rule eq_reflection[symmetric]) |
|
363 |
apply (rule iffI) |
|
364 |
defer |
|
365 |
apply (erule exE) |
|
366 |
apply (rule_tac x = "l * x" in exI) |
|
367 |
apply (simp add: dvd_def) |
|
368 |
apply (rule_tac x="x" in exI, simp) |
|
369 |
apply (erule exE) |
|
370 |
apply (erule conjE) |
|
371 |
apply (erule dvdE) |
|
372 |
apply (rule_tac x = k in exI) |
|
373 |
apply simp |
|
374 |
done |
|
375 |
||
376 |
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" |
|
377 |
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" |
|
378 |
using not0 by (simp add: dvd_def) |
|
379 |
||
380 |
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)" |
|
381 |
by simp_all |
|
382 |
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} |
|
383 |
lemma all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))" |
|
384 |
by (simp split add: split_nat) |
|
385 |
||
386 |
lemma ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))" |
|
387 |
apply (auto split add: split_nat) |
|
388 |
apply (rule_tac x="int x" in exI, simp) |
|
389 |
apply (rule_tac x = "nat x" in exI,erule_tac x = "nat x" in allE, simp) |
|
390 |
done |
|
391 |
||
392 |
lemma zdiff_int_split: "P (int (x - y)) = |
|
393 |
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))" |
|
394 |
by (case_tac "y \<le> x", simp_all add: zdiff_int) |
|
395 |
||
396 |
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" by simp |
|
397 |
lemma number_of2: "(0::int) <= Numeral0" by simp |
|
398 |
lemma Suc_plus1: "Suc n = n + 1" by simp |
|
399 |
||
400 |
text {* |
|
401 |
\medskip Specific instances of congruence rules, to prevent |
|
402 |
simplifier from looping. *} |
|
403 |
||
404 |
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp |
|
405 |
||
406 |
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" |
|
407 |
by (simp cong: conj_cong) |
|
408 |
lemma int_eq_number_of_eq: |
|
409 |
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" |
|
410 |
by simp |
|
411 |
||
412 |
lemma mod_eq0_dvd_iff[presburger]: "(m::nat) mod n = 0 \<longleftrightarrow> n dvd m" |
|
413 |
unfolding dvd_eq_mod_eq_0[symmetric] .. |
|
414 |
||
415 |
lemma zmod_eq0_zdvd_iff[presburger]: "(m::int) mod n = 0 \<longleftrightarrow> n dvd m" |
|
416 |
unfolding zdvd_iff_zmod_eq_0[symmetric] .. |
|
417 |
declare mod_1[presburger] |
|
418 |
declare mod_0[presburger] |
|
419 |
declare zmod_1[presburger] |
|
420 |
declare zmod_zero[presburger] |
|
421 |
declare zmod_self[presburger] |
|
422 |
declare mod_self[presburger] |
|
423 |
declare DIVISION_BY_ZERO_MOD[presburger] |
|
424 |
declare nat_mod_div_trivial[presburger] |
|
425 |
declare div_mod_equality2[presburger] |
|
426 |
declare div_mod_equality[presburger] |
|
427 |
declare mod_div_equality2[presburger] |
|
428 |
declare mod_div_equality[presburger] |
|
429 |
declare mod_mult_self1[presburger] |
|
430 |
declare mod_mult_self2[presburger] |
|
431 |
declare zdiv_zmod_equality2[presburger] |
|
432 |
declare zdiv_zmod_equality[presburger] |
|
433 |
declare mod2_Suc_Suc[presburger] |
|
434 |
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" |
|
435 |
using IntDiv.DIVISION_BY_ZERO by blast+ |
|
436 |
||
437 |
use "Tools/Qelim/cooper.ML" |
|
438 |
oracle linzqe_oracle ("term") = Coopereif.cooper_oracle |
|
439 |
||
440 |
use "Tools/Qelim/presburger.ML" |
|
441 |
||
442 |
setup {* |
|
443 |
arith_tactic_add |
|
444 |
(mk_arith_tactic "presburger" (fn i => fn st => |
|
445 |
(warning "Trying Presburger arithmetic ..."; |
|
446 |
Presburger.cooper_tac true [] [] ((ProofContext.init o theory_of_thm) st) i st))) |
|
447 |
(* FIXME!!!!!!! get the right context!!*) |
|
448 |
*} |
|
449 |
||
450 |
method_setup presburger = {* |
|
451 |
let |
|
452 |
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () |
|
453 |
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () |
|
454 |
val addN = "add" |
|
455 |
val delN = "del" |
|
456 |
val elimN = "elim" |
|
457 |
val any_keyword = keyword addN || keyword delN || simple_keyword elimN |
|
458 |
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; |
|
459 |
in |
|
460 |
fn src => Method.syntax |
|
461 |
((Scan.optional (simple_keyword elimN >> K false) true) -- |
|
462 |
(Scan.optional (keyword addN |-- thms) []) -- |
|
463 |
(Scan.optional (keyword delN |-- thms) [])) src |
|
464 |
#> (fn (((elim, add_ths), del_ths),ctxt) => |
|
465 |
Method.SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) |
|
466 |
end |
|
467 |
*} "Cooper's algorithm for Presburger arithmetic" |
|
468 |
||
469 |
lemma [presburger]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
470 |
lemma [presburger]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
471 |
lemma [presburger]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
472 |
lemma [presburger]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
473 |
lemma [presburger]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
474 |
||
475 |
||
23685 | 476 |
lemma zdvd_period: |
477 |
fixes a d :: int |
|
478 |
assumes advdd: "a dvd d" |
|
479 |
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)" |
|
480 |
proof- |
|
481 |
{ |
|
482 |
fix x k |
|
483 |
from inf_period(3) [OF advdd, rule_format, where x=x and k="-k"] |
|
484 |
have "a dvd (x + t) \<longleftrightarrow> a dvd (x + k * d + t)" by simp |
|
485 |
} |
|
486 |
hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp |
|
487 |
then show ?thesis by simp |
|
488 |
qed |
|
489 |
||
490 |
||
23465 | 491 |
subsection {* Code generator setup *} |
492 |
||
493 |
text {* |
|
494 |
Presburger arithmetic is convenient to prove some |
|
495 |
of the following code lemmas on integer numerals: |
|
496 |
*} |
|
497 |
||
498 |
lemma eq_Pls_Pls: |
|
499 |
"Numeral.Pls = Numeral.Pls \<longleftrightarrow> True" by presburger |
|
500 |
||
501 |
lemma eq_Pls_Min: |
|
502 |
"Numeral.Pls = Numeral.Min \<longleftrightarrow> False" |
|
503 |
unfolding Pls_def Numeral.Min_def by presburger |
|
504 |
||
505 |
lemma eq_Pls_Bit0: |
|
506 |
"Numeral.Pls = Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls = k" |
|
507 |
unfolding Pls_def Bit_def bit.cases by presburger |
|
508 |
||
509 |
lemma eq_Pls_Bit1: |
|
510 |
"Numeral.Pls = Numeral.Bit k bit.B1 \<longleftrightarrow> False" |
|
511 |
unfolding Pls_def Bit_def bit.cases by presburger |
|
512 |
||
513 |
lemma eq_Min_Pls: |
|
514 |
"Numeral.Min = Numeral.Pls \<longleftrightarrow> False" |
|
515 |
unfolding Pls_def Numeral.Min_def by presburger |
|
516 |
||
517 |
lemma eq_Min_Min: |
|
518 |
"Numeral.Min = Numeral.Min \<longleftrightarrow> True" by presburger |
|
519 |
||
520 |
lemma eq_Min_Bit0: |
|
521 |
"Numeral.Min = Numeral.Bit k bit.B0 \<longleftrightarrow> False" |
|
522 |
unfolding Numeral.Min_def Bit_def bit.cases by presburger |
|
523 |
||
524 |
lemma eq_Min_Bit1: |
|
525 |
"Numeral.Min = Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min = k" |
|
526 |
unfolding Numeral.Min_def Bit_def bit.cases by presburger |
|
527 |
||
528 |
lemma eq_Bit0_Pls: |
|
529 |
"Numeral.Bit k bit.B0 = Numeral.Pls \<longleftrightarrow> Numeral.Pls = k" |
|
530 |
unfolding Pls_def Bit_def bit.cases by presburger |
|
531 |
||
532 |
lemma eq_Bit1_Pls: |
|
533 |
"Numeral.Bit k bit.B1 = Numeral.Pls \<longleftrightarrow> False" |
|
534 |
unfolding Pls_def Bit_def bit.cases by presburger |
|
535 |
||
536 |
lemma eq_Bit0_Min: |
|
537 |
"Numeral.Bit k bit.B0 = Numeral.Min \<longleftrightarrow> False" |
|
538 |
unfolding Numeral.Min_def Bit_def bit.cases by presburger |
|
539 |
||
540 |
lemma eq_Bit1_Min: |
|
541 |
"(Numeral.Bit k bit.B1) = Numeral.Min \<longleftrightarrow> Numeral.Min = k" |
|
542 |
unfolding Numeral.Min_def Bit_def bit.cases by presburger |
|
543 |
||
544 |
lemma eq_Bit_Bit: |
|
545 |
"Numeral.Bit k1 v1 = Numeral.Bit k2 v2 \<longleftrightarrow> |
|
546 |
v1 = v2 \<and> k1 = k2" |
|
547 |
unfolding Bit_def |
|
548 |
apply (cases v1) |
|
549 |
apply (cases v2) |
|
550 |
apply auto |
|
551 |
apply presburger |
|
552 |
apply (cases v2) |
|
553 |
apply auto |
|
554 |
apply presburger |
|
555 |
apply (cases v2) |
|
556 |
apply auto |
|
557 |
done |
|
558 |
||
559 |
lemma eq_number_of: |
|
560 |
"(number_of k \<Colon> int) = number_of l \<longleftrightarrow> k = l" |
|
561 |
unfolding number_of_is_id .. |
|
562 |
||
563 |
||
564 |
lemma less_eq_Pls_Pls: |
|
565 |
"Numeral.Pls \<le> Numeral.Pls \<longleftrightarrow> True" by rule+ |
|
566 |
||
567 |
lemma less_eq_Pls_Min: |
|
568 |
"Numeral.Pls \<le> Numeral.Min \<longleftrightarrow> False" |
|
569 |
unfolding Pls_def Numeral.Min_def by presburger |
|
570 |
||
571 |
lemma less_eq_Pls_Bit: |
|
572 |
"Numeral.Pls \<le> Numeral.Bit k v \<longleftrightarrow> Numeral.Pls \<le> k" |
|
573 |
unfolding Pls_def Bit_def by (cases v) auto |
|
574 |
||
575 |
lemma less_eq_Min_Pls: |
|
576 |
"Numeral.Min \<le> Numeral.Pls \<longleftrightarrow> True" |
|
577 |
unfolding Pls_def Numeral.Min_def by presburger |
|
578 |
||
579 |
lemma less_eq_Min_Min: |
|
580 |
"Numeral.Min \<le> Numeral.Min \<longleftrightarrow> True" by rule+ |
|
581 |
||
582 |
lemma less_eq_Min_Bit0: |
|
583 |
"Numeral.Min \<le> Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Min < k" |
|
584 |
unfolding Numeral.Min_def Bit_def by auto |
|
585 |
||
586 |
lemma less_eq_Min_Bit1: |
|
587 |
"Numeral.Min \<le> Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Min \<le> k" |
|
588 |
unfolding Numeral.Min_def Bit_def by auto |
|
589 |
||
590 |
lemma less_eq_Bit0_Pls: |
|
591 |
"Numeral.Bit k bit.B0 \<le> Numeral.Pls \<longleftrightarrow> k \<le> Numeral.Pls" |
|
592 |
unfolding Pls_def Bit_def by simp |
|
593 |
||
594 |
lemma less_eq_Bit1_Pls: |
|
595 |
"Numeral.Bit k bit.B1 \<le> Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" |
|
596 |
unfolding Pls_def Bit_def by auto |
|
597 |
||
598 |
lemma less_eq_Bit_Min: |
|
599 |
"Numeral.Bit k v \<le> Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" |
|
600 |
unfolding Numeral.Min_def Bit_def by (cases v) auto |
|
601 |
||
602 |
lemma less_eq_Bit0_Bit: |
|
603 |
"Numeral.Bit k1 bit.B0 \<le> Numeral.Bit k2 v \<longleftrightarrow> k1 \<le> k2" |
|
604 |
unfolding Bit_def bit.cases by (cases v) auto |
|
605 |
||
606 |
lemma less_eq_Bit_Bit1: |
|
607 |
"Numeral.Bit k1 v \<le> Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2" |
|
608 |
unfolding Bit_def bit.cases by (cases v) auto |
|
609 |
||
610 |
lemma less_eq_Bit1_Bit0: |
|
611 |
"Numeral.Bit k1 bit.B1 \<le> Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" |
|
612 |
unfolding Bit_def by (auto split: bit.split) |
|
613 |
||
614 |
lemma less_eq_number_of: |
|
615 |
"(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l" |
|
616 |
unfolding number_of_is_id .. |
|
617 |
||
618 |
||
619 |
lemma less_Pls_Pls: |
|
620 |
"Numeral.Pls < Numeral.Pls \<longleftrightarrow> False" by simp |
|
621 |
||
622 |
lemma less_Pls_Min: |
|
623 |
"Numeral.Pls < Numeral.Min \<longleftrightarrow> False" |
|
624 |
unfolding Pls_def Numeral.Min_def by presburger |
|
625 |
||
626 |
lemma less_Pls_Bit0: |
|
627 |
"Numeral.Pls < Numeral.Bit k bit.B0 \<longleftrightarrow> Numeral.Pls < k" |
|
628 |
unfolding Pls_def Bit_def by auto |
|
629 |
||
630 |
lemma less_Pls_Bit1: |
|
631 |
"Numeral.Pls < Numeral.Bit k bit.B1 \<longleftrightarrow> Numeral.Pls \<le> k" |
|
632 |
unfolding Pls_def Bit_def by auto |
|
633 |
||
634 |
lemma less_Min_Pls: |
|
635 |
"Numeral.Min < Numeral.Pls \<longleftrightarrow> True" |
|
636 |
unfolding Pls_def Numeral.Min_def by presburger |
|
637 |
||
638 |
lemma less_Min_Min: |
|
639 |
"Numeral.Min < Numeral.Min \<longleftrightarrow> False" by simp |
|
640 |
||
641 |
lemma less_Min_Bit: |
|
642 |
"Numeral.Min < Numeral.Bit k v \<longleftrightarrow> Numeral.Min < k" |
|
643 |
unfolding Numeral.Min_def Bit_def by (auto split: bit.split) |
|
644 |
||
645 |
lemma less_Bit_Pls: |
|
646 |
"Numeral.Bit k v < Numeral.Pls \<longleftrightarrow> k < Numeral.Pls" |
|
647 |
unfolding Pls_def Bit_def by (auto split: bit.split) |
|
648 |
||
649 |
lemma less_Bit0_Min: |
|
650 |
"Numeral.Bit k bit.B0 < Numeral.Min \<longleftrightarrow> k \<le> Numeral.Min" |
|
651 |
unfolding Numeral.Min_def Bit_def by auto |
|
652 |
||
653 |
lemma less_Bit1_Min: |
|
654 |
"Numeral.Bit k bit.B1 < Numeral.Min \<longleftrightarrow> k < Numeral.Min" |
|
655 |
unfolding Numeral.Min_def Bit_def by auto |
|
656 |
||
657 |
lemma less_Bit_Bit0: |
|
658 |
"Numeral.Bit k1 v < Numeral.Bit k2 bit.B0 \<longleftrightarrow> k1 < k2" |
|
659 |
unfolding Bit_def by (auto split: bit.split) |
|
660 |
||
661 |
lemma less_Bit1_Bit: |
|
662 |
"Numeral.Bit k1 bit.B1 < Numeral.Bit k2 v \<longleftrightarrow> k1 < k2" |
|
663 |
unfolding Bit_def by (auto split: bit.split) |
|
664 |
||
665 |
lemma less_Bit0_Bit1: |
|
666 |
"Numeral.Bit k1 bit.B0 < Numeral.Bit k2 bit.B1 \<longleftrightarrow> k1 \<le> k2" |
|
667 |
unfolding Bit_def bit.cases by arith |
|
668 |
||
669 |
lemma less_number_of: |
|
670 |
"(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l" |
|
671 |
unfolding number_of_is_id .. |
|
672 |
||
673 |
lemmas pred_succ_numeral_code [code func] = |
|
674 |
arith_simps(5-12) |
|
675 |
||
676 |
lemmas plus_numeral_code [code func] = |
|
677 |
arith_simps(13-17) |
|
678 |
arith_simps(26-27) |
|
679 |
arith_extra_simps(1) [where 'a = int] |
|
680 |
||
681 |
lemmas minus_numeral_code [code func] = |
|
682 |
arith_simps(18-21) |
|
683 |
arith_extra_simps(2) [where 'a = int] |
|
684 |
arith_extra_simps(5) [where 'a = int] |
|
685 |
||
686 |
lemmas times_numeral_code [code func] = |
|
687 |
arith_simps(22-25) |
|
688 |
arith_extra_simps(4) [where 'a = int] |
|
689 |
||
690 |
lemmas eq_numeral_code [code func] = |
|
691 |
eq_Pls_Pls eq_Pls_Min eq_Pls_Bit0 eq_Pls_Bit1 |
|
692 |
eq_Min_Pls eq_Min_Min eq_Min_Bit0 eq_Min_Bit1 |
|
693 |
eq_Bit0_Pls eq_Bit1_Pls eq_Bit0_Min eq_Bit1_Min eq_Bit_Bit |
|
694 |
eq_number_of |
|
695 |
||
696 |
lemmas less_eq_numeral_code [code func] = less_eq_Pls_Pls less_eq_Pls_Min less_eq_Pls_Bit |
|
697 |
less_eq_Min_Pls less_eq_Min_Min less_eq_Min_Bit0 less_eq_Min_Bit1 |
|
698 |
less_eq_Bit0_Pls less_eq_Bit1_Pls less_eq_Bit_Min less_eq_Bit0_Bit less_eq_Bit_Bit1 less_eq_Bit1_Bit0 |
|
699 |
less_eq_number_of |
|
700 |
||
701 |
lemmas less_numeral_code [code func] = less_Pls_Pls less_Pls_Min less_Pls_Bit0 |
|
702 |
less_Pls_Bit1 less_Min_Pls less_Min_Min less_Min_Bit less_Bit_Pls |
|
703 |
less_Bit0_Min less_Bit1_Min less_Bit_Bit0 less_Bit1_Bit less_Bit0_Bit1 |
|
704 |
less_number_of |
|
705 |
||
23856 | 706 |
|
707 |
lemma of_int_num [code func]: |
|
708 |
"of_int k = (if k = 0 then 0 else if k < 0 then |
|
709 |
- of_int (- k) else let |
|
710 |
(l, m) = divAlg (k, 2); |
|
711 |
l' = of_int l |
|
712 |
in if m = 0 then l' + l' else l' + l' + 1)" |
|
713 |
proof - |
|
714 |
have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> |
|
715 |
of_int k = of_int (k div 2 * 2 + 1)" |
|
716 |
proof - |
|
717 |
assume "k mod 2 \<noteq> 0" |
|
718 |
then have "k mod 2 = 1" by arith |
|
719 |
moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp |
|
720 |
ultimately show ?thesis by auto |
|
721 |
qed |
|
722 |
have aux2: "\<And>x. of_int 2 * x = x + x" |
|
723 |
proof - |
|
724 |
fix x |
|
725 |
have int2: "(2::int) = 1 + 1" by arith |
|
726 |
show "of_int 2 * x = x + x" |
|
727 |
unfolding int2 of_int_add left_distrib by simp |
|
728 |
qed |
|
729 |
have aux3: "\<And>x. x * of_int 2 = x + x" |
|
730 |
proof - |
|
731 |
fix x |
|
732 |
have int2: "(2::int) = 1 + 1" by arith |
|
733 |
show "x * of_int 2 = x + x" |
|
734 |
unfolding int2 of_int_add right_distrib by simp |
|
735 |
qed |
|
736 |
from aux1 show ?thesis by (auto simp add: divAlg_mod_div Let_def aux2 aux3) |
|
737 |
qed |
|
738 |
||
23465 | 739 |
end |