author | haftmann |
Fri, 07 May 2010 09:51:55 +0200 | |
changeset 36749 | a8dc19a352e6 |
parent 35216 | 7641e8d831d2 |
child 36798 | 3981db162131 |
permissions | -rw-r--r-- |
23465 | 1 |
(* Title: HOL/Presburger.thy |
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Author: Amine Chaieb, TU Muenchen |
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*) |
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23472 | 5 |
header {* Decision Procedure for Presburger Arithmetic *} |
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theory Presburger |
28402 | 8 |
imports Groebner_Basis SetInterval |
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uses |
30656
ddb1fafa2dcb
moved import of module qelim to theory Presburger
haftmann
parents:
30549
diff
changeset
|
10 |
"Tools/Qelim/qelim.ML" |
23465 | 11 |
"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" |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
33 |
"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (d dvd x + s) = (d dvd x + s)" |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
34 |
"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})<z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" |
23465 | 35 |
"\<exists>z.\<forall>x<z. F = F" |
36 |
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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||
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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" |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
49 |
"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (d dvd x + s) = (d dvd x + s)" |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
50 |
"\<exists>z.\<forall>(x::'a::{linorder,plus,Rings.dvd})>z. (\<not> d dvd x + s) = (\<not> d dvd x + s)" |
23465 | 51 |
"\<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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||
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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))" |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
59 |
"(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)" |
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
60 |
"(d::'a::{comm_ring,Rings.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)" |
23465 | 61 |
"\<forall>x k. F = F" |
29667 | 62 |
apply (auto elim!: dvdE simp add: algebra_simps) |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
63 |
unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric] |
27668 | 64 |
unfolding dvd_def mult_commute [of d] |
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by auto |
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23465 | 66 |
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23472 | 67 |
subsection{* The A and B sets *} |
23465 | 68 |
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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27668 | 87 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t - 1"]) |
88 |
apply algebra using dp tB by simp_all |
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23465 | 89 |
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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27668 | 93 |
apply algebra |
23465 | 94 |
using dp tB by simp_all |
95 |
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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97 |
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 |
|
99 |
next |
|
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assume dp: "D > 0" and tB:"t \<in> B" |
|
101 |
{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" |
|
102 |
hence "x -t \<le> D" and "1 \<le> x - t" by simp+ |
|
103 |
hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto |
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29667 | 104 |
hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps) |
23465 | 105 |
with nob tB have "False" by simp} |
106 |
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 |
|
107 |
next |
|
108 |
assume dp: "D > 0" and tB:"t - 1\<in> B" |
|
109 |
{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" |
|
110 |
hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+ |
|
111 |
hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto |
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29667 | 112 |
hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps) |
23465 | 113 |
with nob tB have "False" by simp} |
114 |
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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115 |
next |
|
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assume d: "d dvd D" |
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27668 | 117 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x - D) + t" by algebra} |
23465 | 118 |
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 |
119 |
next |
|
120 |
assume d: "d dvd D" |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
121 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t" |
29667 | 122 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)} |
23465 | 123 |
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 |
124 |
qed blast |
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125 |
||
126 |
lemma aset: |
|
127 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; |
|
128 |
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> |
|
129 |
\<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))" |
|
130 |
"\<lbrakk>\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> P x \<longrightarrow> P(x + D) ; |
|
131 |
\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> Q x \<longrightarrow> Q(x + D)\<rbrakk> \<Longrightarrow> |
|
132 |
\<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))" |
|
133 |
"\<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))" |
|
134 |
"\<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))" |
|
135 |
"\<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))" |
|
136 |
"\<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))" |
|
137 |
"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))" |
|
138 |
"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))" |
|
139 |
"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))" |
|
140 |
"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))" |
|
141 |
"\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j) \<longrightarrow> F \<longrightarrow> F" |
|
142 |
proof (blast, blast) |
|
143 |
assume dp: "D > 0" and tA: "t + 1 \<in> A" |
|
144 |
show "(\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x = t) \<longrightarrow> (x + D = t))" |
|
145 |
apply (rule allI, rule impI,erule ballE[where x="1"],erule ballE[where x="t + 1"]) |
|
146 |
using dp tA by simp_all |
|
147 |
next |
|
148 |
assume dp: "D > 0" and tA: "t \<in> A" |
|
149 |
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))" |
|
150 |
apply (rule allI, rule impI,erule ballE[where x="D"],erule ballE[where x="t"]) |
|
151 |
using dp tA by simp_all |
|
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 > t) \<longrightarrow> (x + D > t))" by arith |
|
154 |
next |
|
155 |
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 |
|
156 |
next |
|
157 |
assume dp: "D > 0" and tA:"t \<in> A" |
|
158 |
{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" |
|
159 |
hence "t - x \<le> D" and "1 \<le> t - x" by simp+ |
|
160 |
hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto |
|
29667 | 161 |
hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: algebra_simps) |
23465 | 162 |
with nob tA have "False" by simp} |
163 |
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 |
|
164 |
next |
|
165 |
assume dp: "D > 0" and tA:"t + 1\<in> A" |
|
166 |
{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" |
|
29667 | 167 |
hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: algebra_simps) |
23465 | 168 |
hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto |
29667 | 169 |
hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps) |
23465 | 170 |
with nob tA have "False" by simp} |
171 |
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 |
|
172 |
next |
|
173 |
assume d: "d dvd D" |
|
174 |
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t" |
|
29667 | 175 |
by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)} |
23465 | 176 |
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 |
177 |
next |
|
178 |
assume d: "d dvd D" |
|
179 |
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t" |
|
29667 | 180 |
by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: algebra_simps)} |
23465 | 181 |
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 |
182 |
qed blast |
|
183 |
||
184 |
subsection{* Cooper's Theorem @{text "-\<infinity>"} and @{text "+\<infinity>"} Version *} |
|
185 |
||
186 |
subsubsection{* First some trivial facts about periodic sets or predicates *} |
|
187 |
lemma periodic_finite_ex: |
|
188 |
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)" |
|
189 |
shows "(EX x. P x) = (EX j : {1..d}. P j)" |
|
190 |
(is "?LHS = ?RHS") |
|
191 |
proof |
|
192 |
assume ?LHS |
|
193 |
then obtain x where P: "P x" .. |
|
194 |
have "x mod d = x - (x div d)*d" by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) |
|
195 |
hence Pmod: "P x = P(x mod d)" using modd by simp |
|
196 |
show ?RHS |
|
197 |
proof (cases) |
|
198 |
assume "x mod d = 0" |
|
199 |
hence "P 0" using P Pmod by simp |
|
200 |
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast |
|
201 |
ultimately have "P d" by simp |
|
35216 | 202 |
moreover have "d : {1..d}" using dpos by simp |
23465 | 203 |
ultimately show ?RHS .. |
204 |
next |
|
205 |
assume not0: "x mod d \<noteq> 0" |
|
35216 | 206 |
have "P(x mod d)" using dpos P Pmod by simp |
23465 | 207 |
moreover have "x mod d : {1..d}" |
208 |
proof - |
|
209 |
from dpos have "0 \<le> x mod d" by(rule pos_mod_sign) |
|
210 |
moreover from dpos have "x mod d < d" by(rule pos_mod_bound) |
|
35216 | 211 |
ultimately show ?thesis using not0 by simp |
23465 | 212 |
qed |
213 |
ultimately show ?RHS .. |
|
214 |
qed |
|
215 |
qed auto |
|
216 |
||
217 |
subsubsection{* The @{text "-\<infinity>"} Version*} |
|
218 |
||
219 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z" |
|
220 |
by(induct rule: int_gr_induct,simp_all add:int_distrib) |
|
221 |
||
222 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d" |
|
223 |
by(induct rule: int_gr_induct, simp_all add:int_distrib) |
|
224 |
||
225 |
theorem int_induct[case_names base step1 step2]: |
|
226 |
assumes |
|
227 |
base: "P(k::int)" and step1: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" and |
|
228 |
step2: "\<And>i. \<lbrakk>k \<ge> i; P i\<rbrakk> \<Longrightarrow> P(i - 1)" |
|
229 |
shows "P i" |
|
230 |
proof - |
|
231 |
have "i \<le> k \<or> i\<ge> k" by arith |
|
232 |
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 |
|
233 |
qed |
|
234 |
||
235 |
lemma decr_mult_lemma: |
|
236 |
assumes dpos: "(0::int) < d" and minus: "\<forall>x. P x \<longrightarrow> P(x - d)" and knneg: "0 <= k" |
|
237 |
shows "ALL x. P x \<longrightarrow> P(x - k*d)" |
|
238 |
using knneg |
|
239 |
proof (induct rule:int_ge_induct) |
|
240 |
case base thus ?case by simp |
|
241 |
next |
|
242 |
case (step i) |
|
243 |
{fix x |
|
244 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast |
|
245 |
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" using minus[THEN spec, of "x - i * d"] |
|
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
246 |
by (simp add: algebra_simps) |
23465 | 247 |
ultimately have "P x \<longrightarrow> P(x - (i + 1) * d)" by blast} |
248 |
thus ?case .. |
|
249 |
qed |
|
250 |
||
251 |
lemma minusinfinity: |
|
252 |
assumes dpos: "0 < d" and |
|
253 |
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" |
|
254 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" |
|
255 |
proof |
|
256 |
assume eP1: "EX x. P1 x" |
|
257 |
then obtain x where P1: "P1 x" .. |
|
258 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. |
|
259 |
let ?w = "x - (abs(x-z)+1) * d" |
|
260 |
from dpos have w: "?w < z" by(rule decr_lemma) |
|
261 |
have "P1 x = P1 ?w" using P1eqP1 by blast |
|
262 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
263 |
finally have "P ?w" using P1 by blast |
|
264 |
thus "EX x. P x" .. |
|
265 |
qed |
|
266 |
||
267 |
lemma cpmi: |
|
268 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x< z. P x = P' x" |
|
269 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> B. x \<noteq> b+j) --> P (x) --> P (x - D)" |
|
270 |
and pd: "\<forall> x k. P' x = P' (x-k*D)" |
|
271 |
shows "(\<exists>x. P x) = ((\<exists> j\<in> {1..D} . P' j) | (\<exists> j \<in> {1..D}.\<exists> b\<in> B. P (b+j)))" |
|
272 |
(is "?L = (?R1 \<or> ?R2)") |
|
273 |
proof- |
|
274 |
{assume "?R2" hence "?L" by blast} |
|
275 |
moreover |
|
276 |
{assume H:"?R1" hence "?L" using minusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} |
|
277 |
moreover |
|
278 |
{ fix x |
|
279 |
assume P: "P x" and H: "\<not> ?R2" |
|
280 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>B. P (b + j))" and P: "P y" |
|
281 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : B. y = b+j)" by auto |
|
282 |
with nb P have "P (y - D)" by auto } |
|
283 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)" by blast |
|
284 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x - D)" by auto |
|
285 |
from p1 obtain z where z: "ALL x. x < z --> (P x = P' x)" by blast |
|
286 |
let ?y = "x - (\<bar>x - z\<bar> + 1)*D" |
|
287 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
|
288 |
from dp have yz: "?y < z" using decr_lemma[OF dp] by simp |
|
289 |
from z[rule_format, OF yz] decr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto |
|
290 |
with periodic_finite_ex[OF dp pd] |
|
291 |
have "?R1" by blast} |
|
292 |
ultimately show ?thesis by blast |
|
293 |
qed |
|
294 |
||
295 |
subsubsection {* The @{text "+\<infinity>"} Version*} |
|
296 |
||
297 |
lemma plusinfinity: |
|
298 |
assumes dpos: "(0::int) < d" and |
|
299 |
P1eqP1: "\<forall>x k. P' x = P'(x - k*d)" and ePeqP1: "\<exists> z. \<forall> x>z. P x = P' x" |
|
300 |
shows "(\<exists> x. P' x) \<longrightarrow> (\<exists> x. P x)" |
|
301 |
proof |
|
302 |
assume eP1: "EX x. P' x" |
|
303 |
then obtain x where P1: "P' x" .. |
|
304 |
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" .. |
|
305 |
let ?w' = "x + (abs(x-z)+1) * d" |
|
306 |
let ?w = "x - (-(abs(x-z) + 1))*d" |
|
29667 | 307 |
have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps) |
23465 | 308 |
from dpos have w: "?w > z" by(simp only: ww' incr_lemma) |
309 |
hence "P' x = P' ?w" using P1eqP1 by blast |
|
310 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
311 |
finally have "P ?w" using P1 by blast |
|
312 |
thus "EX x. P x" .. |
|
313 |
qed |
|
314 |
||
315 |
lemma incr_mult_lemma: |
|
316 |
assumes dpos: "(0::int) < d" and plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and knneg: "0 <= k" |
|
317 |
shows "ALL x. P x \<longrightarrow> P(x + k*d)" |
|
318 |
using knneg |
|
319 |
proof (induct rule:int_ge_induct) |
|
320 |
case base thus ?case by simp |
|
321 |
next |
|
322 |
case (step i) |
|
323 |
{fix x |
|
324 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast |
|
325 |
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" using plus[THEN spec, of "x + i * d"] |
|
326 |
by (simp add:int_distrib zadd_ac) |
|
327 |
ultimately have "P x \<longrightarrow> P(x + (i + 1) * d)" by blast} |
|
328 |
thus ?case .. |
|
329 |
qed |
|
330 |
||
331 |
lemma cppi: |
|
332 |
assumes dp: "0 < D" and p1:"\<exists>z. \<forall> x> z. P x = P' x" |
|
333 |
and nb:"\<forall>x.(\<forall> j\<in> {1..D}. \<forall>(b::int) \<in> A. x \<noteq> b - j) --> P (x) --> P (x + D)" |
|
334 |
and pd: "\<forall> x k. P' x= P' (x-k*D)" |
|
335 |
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)") |
|
336 |
proof- |
|
337 |
{assume "?R2" hence "?L" by blast} |
|
338 |
moreover |
|
339 |
{assume H:"?R1" hence "?L" using plusinfinity[OF dp pd p1] periodic_finite_ex[OF dp pd] by simp} |
|
340 |
moreover |
|
341 |
{ fix x |
|
342 |
assume P: "P x" and H: "\<not> ?R2" |
|
343 |
{fix y assume "\<not> (\<exists>j\<in>{1..D}. \<exists>b\<in>A. P (b - j))" and P: "P y" |
|
344 |
hence "~(EX (j::int) : {1..D}. EX (b::int) : A. y = b - j)" by auto |
|
345 |
with nb P have "P (y + D)" by auto } |
|
346 |
hence "ALL x.~(EX (j::int) : {1..D}. EX (b::int) : A. P(b-j)) --> P (x) --> P (x + D)" by blast |
|
347 |
with H P have th: " \<forall>x. P x \<longrightarrow> P (x + D)" by auto |
|
348 |
from p1 obtain z where z: "ALL x. x > z --> (P x = P' x)" by blast |
|
349 |
let ?y = "x + (\<bar>x - z\<bar> + 1)*D" |
|
350 |
have zp: "0 <= (\<bar>x - z\<bar> + 1)" by arith |
|
351 |
from dp have yz: "?y > z" using incr_lemma[OF dp] by simp |
|
352 |
from z[rule_format, OF yz] incr_mult_lemma[OF dp th zp, rule_format, OF P] have th2: " P' ?y" by auto |
|
353 |
with periodic_finite_ex[OF dp pd] |
|
354 |
have "?R1" by blast} |
|
355 |
ultimately show ?thesis by blast |
|
356 |
qed |
|
357 |
||
358 |
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" |
|
359 |
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) |
|
360 |
apply(fastsimp) |
|
361 |
done |
|
362 |
||
35050
9f841f20dca6
renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents:
33318
diff
changeset
|
363 |
theorem unity_coeff_ex: "(\<exists>(x::'a::{semiring_0,Rings.dvd}). P (l * x)) \<equiv> (\<exists>x. l dvd (x + 0) \<and> P x)" |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
364 |
apply (rule eq_reflection [symmetric]) |
23465 | 365 |
apply (rule iffI) |
366 |
defer |
|
367 |
apply (erule exE) |
|
368 |
apply (rule_tac x = "l * x" in exI) |
|
369 |
apply (simp add: dvd_def) |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
370 |
apply (rule_tac x = x in exI, simp) |
23465 | 371 |
apply (erule exE) |
372 |
apply (erule conjE) |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
373 |
apply simp |
23465 | 374 |
apply (erule dvdE) |
375 |
apply (rule_tac x = k in exI) |
|
376 |
apply simp |
|
377 |
done |
|
378 |
||
379 |
lemma zdvd_mono: assumes not0: "(k::int) \<noteq> 0" |
|
380 |
shows "((m::int) dvd t) \<equiv> (k*m dvd k*t)" |
|
381 |
using not0 by (simp add: dvd_def) |
|
382 |
||
383 |
lemma uminus_dvd_conv: "(d dvd (t::int)) \<equiv> (-d dvd t)" "(d dvd (t::int)) \<equiv> (d dvd -t)" |
|
384 |
by simp_all |
|
32553 | 385 |
|
23465 | 386 |
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} |
32553 | 387 |
|
23465 | 388 |
lemma zdiff_int_split: "P (int (x - y)) = |
389 |
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))" |
|
390 |
by (case_tac "y \<le> x", simp_all add: zdiff_int) |
|
391 |
||
26086
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
392 |
lemma number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (Int.Bit0 n) \<and> (0::int) <= number_of (Int.Bit1 n)" |
3c243098b64a
New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
huffman
parents:
26075
diff
changeset
|
393 |
by simp |
23465 | 394 |
lemma number_of2: "(0::int) <= Numeral0" by simp |
395 |
||
396 |
text {* |
|
397 |
\medskip Specific instances of congruence rules, to prevent |
|
398 |
simplifier from looping. *} |
|
399 |
||
400 |
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" by simp |
|
401 |
||
402 |
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" |
|
403 |
by (simp cong: conj_cong) |
|
404 |
lemma int_eq_number_of_eq: |
|
405 |
"(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)" |
|
28967
3bdb1eae352c
enable eq_bin_simps for simplifying equalities on numerals
huffman
parents:
28402
diff
changeset
|
406 |
by (rule eq_number_of_eq) |
23465 | 407 |
|
30031 | 408 |
declare dvd_eq_mod_eq_0[symmetric, presburger] |
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
409 |
declare mod_1[presburger] |
23465 | 410 |
declare mod_0[presburger] |
30031 | 411 |
declare mod_by_1[presburger] |
23465 | 412 |
declare zmod_zero[presburger] |
413 |
declare zmod_self[presburger] |
|
414 |
declare mod_self[presburger] |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
415 |
declare mod_by_0[presburger] |
30027 | 416 |
declare mod_div_trivial[presburger] |
23465 | 417 |
declare div_mod_equality2[presburger] |
418 |
declare div_mod_equality[presburger] |
|
419 |
declare mod_div_equality2[presburger] |
|
420 |
declare mod_div_equality[presburger] |
|
421 |
declare mod_mult_self1[presburger] |
|
422 |
declare mod_mult_self2[presburger] |
|
423 |
declare zdiv_zmod_equality2[presburger] |
|
424 |
declare zdiv_zmod_equality[presburger] |
|
425 |
declare mod2_Suc_Suc[presburger] |
|
426 |
lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a" |
|
27651
16a26996c30e
moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents:
27540
diff
changeset
|
427 |
by simp_all |
23465 | 428 |
|
429 |
use "Tools/Qelim/cooper.ML" |
|
28290 | 430 |
oracle linzqe_oracle = Coopereif.cooper_oracle |
23465 | 431 |
|
432 |
use "Tools/Qelim/presburger.ML" |
|
433 |
||
30686
47a32dd1b86e
moved generic arith_tac (formerly silent_arith_tac), verbose_arith_tac (formerly arith_tac) to Arith_Data; simple_arith-tac now named linear_arith_tac
haftmann
parents:
30656
diff
changeset
|
434 |
setup {* Arith_Data.add_tactic "Presburger arithmetic" (K (Presburger.cooper_tac true [] [])) *} |
23465 | 435 |
|
436 |
method_setup presburger = {* |
|
437 |
let |
|
438 |
fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K () |
|
439 |
fun simple_keyword k = Scan.lift (Args.$$$ k) >> K () |
|
440 |
val addN = "add" |
|
441 |
val delN = "del" |
|
442 |
val elimN = "elim" |
|
443 |
val any_keyword = keyword addN || keyword delN || simple_keyword elimN |
|
444 |
val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat; |
|
445 |
in |
|
30549 | 446 |
Scan.optional (simple_keyword elimN >> K false) true -- |
447 |
Scan.optional (keyword addN |-- thms) [] -- |
|
448 |
Scan.optional (keyword delN |-- thms) [] >> |
|
449 |
(fn ((elim, add_ths), del_ths) => fn ctxt => |
|
450 |
SIMPLE_METHOD' (Presburger.cooper_tac elim add_ths del_ths ctxt)) |
|
23465 | 451 |
end |
452 |
*} "Cooper's algorithm for Presburger arithmetic" |
|
453 |
||
27668 | 454 |
lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
455 |
lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
456 |
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
457 |
lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
458 |
lemma [presburger, algebra]: "m mod 2 = (1::int) \<longleftrightarrow> \<not> 2 dvd m " by presburger |
|
23465 | 459 |
|
460 |
end |