author | haftmann |
Thu, 22 Nov 2018 10:06:31 +0000 | |
changeset 69325 | 4b6ddc5989fc |
parent 69313 | b021008c5397 |
child 69597 | ff784d5a5bfb |
permissions | -rw-r--r-- |
30439 | 1 |
(* Title: HOL/Decision_Procs/MIR.thy |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2 |
Author: Amine Chaieb |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
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3 |
*) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
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4 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
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|
5 |
theory MIR |
41849 | 6 |
imports Complex_Main Dense_Linear_Order DP_Library |
66453
cc19f7ca2ed6
session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
wenzelm
parents:
66124
diff
changeset
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7 |
"HOL-Library.Code_Target_Numeral" "HOL-Library.Old_Recdef" |
27368 | 8 |
begin |
23264
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Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
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9 |
|
69325 | 10 |
section \<open>Prelude\<close> |
11 |
||
12 |
abbreviation (input) UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" |
|
13 |
where "UNION A f \<equiv> \<Union> (f ` A)" \<comment> \<open>legacy\<close> |
|
14 |
||
15 |
||
61586 | 16 |
section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, floor, <)\<close>\<close> |
27456 | 17 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
18 |
declare of_int_floor_cancel [simp del] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
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19 |
|
51369 | 20 |
lemma myle: |
21 |
fixes a b :: "'a::{ordered_ab_group_add}" |
|
41849 | 22 |
shows "(a \<le> b) = (0 \<le> b - a)" |
51369 | 23 |
by (metis add_0_left add_le_cancel_right diff_add_cancel) |
24 |
||
25 |
lemma myless: |
|
26 |
fixes a b :: "'a::{ordered_ab_group_add}" |
|
41849 | 27 |
shows "(a < b) = (0 < b - a)" |
51369 | 28 |
by (metis le_iff_diff_le_0 less_le_not_le myle) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
29 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
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30 |
(* Periodicity of dvd *) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
31 |
lemmas dvd_period = zdvd_period |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
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parents:
diff
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32 |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
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33 |
(* The Divisibility relation between reals *) |
51369 | 34 |
definition rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
35 |
where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real_of_int k)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
36 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
37 |
lemma int_rdvd_real: |
61942 | 38 |
"real_of_int (i::int) rdvd x = (i dvd \<lfloor>x\<rfloor> \<and> real_of_int \<lfloor>x\<rfloor> = x)" (is "?l = ?r") |
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324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
39 |
proof |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
40 |
assume "?l" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
41 |
hence th: "\<exists> k. x=real_of_int (i*k)" by (simp add: rdvd_def) |
61942 | 42 |
hence th': "real_of_int \<lfloor>x\<rfloor> = x" by (auto simp del: of_int_mult) |
43 |
with th have "\<exists> k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)" by simp |
|
44 |
hence "\<exists>k. \<lfloor>x\<rfloor> = i*k" by presburger |
|
45 |
thus ?r using th' by (simp add: dvd_def) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
46 |
next |
61076 | 47 |
assume "?r" hence "(i::int) dvd \<lfloor>x::real\<rfloor>" .. |
61942 | 48 |
hence "\<exists>k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
49 |
by (metis (no_types) dvd_def) |
60533 | 50 |
thus ?l using \<open>?r\<close> by (simp add: rdvd_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
51 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
52 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
53 |
lemma int_rdvd_iff: "(real_of_int (i::int) rdvd real_of_int t) = (i dvd t)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
54 |
by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only: of_int_mult[symmetric]) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
55 |
|
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
56 |
|
61945 | 57 |
lemma rdvd_abs1: "(\<bar>real_of_int d\<bar> rdvd t) = (real_of_int (d ::int) rdvd t)" |
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324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
58 |
proof |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
59 |
assume d: "real_of_int d rdvd t" |
61942 | 60 |
from d int_rdvd_real have d2: "d dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t" |
51369 | 61 |
by auto |
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324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
62 |
|
61945 | 63 |
from iffD2[OF abs_dvd_iff] d2 have "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" by blast |
64 |
with ti int_rdvd_real[symmetric] have "real_of_int \<bar>d\<bar> rdvd t" by blast |
|
65 |
thus "\<bar>real_of_int d\<bar> rdvd t" by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
66 |
next |
61945 | 67 |
assume "\<bar>real_of_int d\<bar> rdvd t" hence "real_of_int \<bar>d\<bar> rdvd t" by simp |
68 |
with int_rdvd_real[where i="\<bar>d\<bar>" and x="t"] |
|
69 |
have d2: "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t" |
|
51369 | 70 |
by auto |
61942 | 71 |
from iffD1[OF abs_dvd_iff] d2 have "d dvd \<lfloor>t\<rfloor>" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
72 |
with ti int_rdvd_real[symmetric] show "real_of_int d rdvd t" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
73 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
74 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
75 |
lemma rdvd_minus: "(real_of_int (d::int) rdvd t) = (real_of_int d rdvd -t)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
76 |
apply (auto simp add: rdvd_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
77 |
apply (rule_tac x="-k" in exI, simp) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
78 |
apply (rule_tac x="-k" in exI, simp) |
51369 | 79 |
done |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
80 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
81 |
lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)" |
51369 | 82 |
by (auto simp add: rdvd_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
83 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
84 |
lemma rdvd_mult: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
85 |
assumes knz: "k\<noteq>0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
86 |
shows "(real_of_int (n::int) * real_of_int (k::int) rdvd x * real_of_int k) = (real_of_int n rdvd x)" |
51369 | 87 |
using knz by (simp add: rdvd_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
88 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
89 |
(*********************************************************************************) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
90 |
(**** SHADOW SYNTAX AND SEMANTICS ****) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
91 |
(*********************************************************************************) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
92 |
|
66809 | 93 |
datatype (plugins del: size) num = C int | Bound nat | CN nat int num |
94 |
| Neg num | Add num num | Sub num num |
|
95 |
| Mul int num | Floor num | CF int num num |
|
96 |
||
97 |
instantiation num :: size |
|
98 |
begin |
|
99 |
||
100 |
primrec size_num :: "num \<Rightarrow> nat" |
|
101 |
where |
|
102 |
"size_num (C c) = 1" |
|
103 |
| "size_num (Bound n) = 1" |
|
104 |
| "size_num (Neg a) = 1 + size_num a" |
|
105 |
| "size_num (Add a b) = 1 + size_num a + size_num b" |
|
106 |
| "size_num (Sub a b) = 3 + size_num a + size_num b" |
|
107 |
| "size_num (CN n c a) = 4 + size_num a " |
|
108 |
| "size_num (CF c a b) = 4 + size_num a + size_num b" |
|
109 |
| "size_num (Mul c a) = 1 + size_num a" |
|
110 |
| "size_num (Floor a) = 1 + size_num a" |
|
111 |
||
112 |
instance .. |
|
113 |
||
114 |
end |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
115 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
116 |
(* Semantics of numeral terms (num) *) |
66809 | 117 |
primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" |
118 |
where |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
119 |
"Inum bs (C c) = (real_of_int c)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
120 |
| "Inum bs (Bound n) = bs!n" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
121 |
| "Inum bs (CN n c a) = (real_of_int c) * (bs!n) + (Inum bs a)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
122 |
| "Inum bs (Neg a) = -(Inum bs a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
123 |
| "Inum bs (Add a b) = Inum bs a + Inum bs b" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
124 |
| "Inum bs (Sub a b) = Inum bs a - Inum bs b" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
125 |
| "Inum bs (Mul c a) = (real_of_int c) * Inum bs a" |
61942 | 126 |
| "Inum bs (Floor a) = real_of_int \<lfloor>Inum bs a\<rfloor>" |
127 |
| "Inum bs (CF c a b) = real_of_int c * real_of_int \<lfloor>Inum bs a\<rfloor> + Inum bs b" |
|
128 |
definition "isint t bs \<equiv> real_of_int \<lfloor>Inum bs t\<rfloor> = Inum bs t" |
|
129 |
||
130 |
lemma isint_iff: "isint n bs = (real_of_int \<lfloor>Inum bs n\<rfloor> = Inum bs n)" |
|
51369 | 131 |
by (simp add: isint_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
132 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
133 |
lemma isint_Floor: "isint (Floor n) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
134 |
by (simp add: isint_iff) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
135 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
136 |
lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
137 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
138 |
let ?e = "Inum bs e" |
61942 | 139 |
assume be: "isint e bs" hence efe:"real_of_int \<lfloor>?e\<rfloor> = ?e" by (simp add: isint_iff) |
140 |
have "real_of_int \<lfloor>Inum bs (Mul c e)\<rfloor> = real_of_int \<lfloor>real_of_int (c * \<lfloor>?e\<rfloor>)\<rfloor>" |
|
141 |
using efe by simp |
|
142 |
also have "\<dots> = real_of_int (c* \<lfloor>?e\<rfloor>)" by (metis floor_of_int) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
143 |
also have "\<dots> = real_of_int c * ?e" using efe by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
144 |
finally show ?thesis using isint_iff by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
145 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
146 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
147 |
lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
148 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
149 |
let ?I = "\<lambda> t. Inum bs t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
150 |
assume ie: "isint e bs" |
61942 | 151 |
hence th: "real_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def) |
152 |
have "real_of_int \<lfloor>?I (Neg e)\<rfloor> = real_of_int \<lfloor>- (real_of_int \<lfloor>?I e\<rfloor>)\<rfloor>" |
|
153 |
by (simp add: th) |
|
154 |
also have "\<dots> = - real_of_int \<lfloor>?I e\<rfloor>" by simp |
|
23264
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chaieb
parents:
diff
changeset
|
155 |
finally show "isint (Neg e) bs" by (simp add: isint_def th) |
324622260d29
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chaieb
parents:
diff
changeset
|
156 |
qed |
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chaieb
parents:
diff
changeset
|
157 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
158 |
lemma isint_sub: |
23264
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chaieb
parents:
diff
changeset
|
159 |
assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs" |
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chaieb
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diff
changeset
|
160 |
proof- |
324622260d29
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chaieb
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|
161 |
let ?I = "\<lambda> t. Inum bs t" |
61942 | 162 |
from ie have th: "real_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def) |
163 |
have "real_of_int \<lfloor>?I (Sub (C c) e)\<rfloor> = real_of_int \<lfloor>real_of_int (c - \<lfloor>?I e\<rfloor>)\<rfloor>" |
|
164 |
by (simp add: th) |
|
165 |
also have "\<dots> = real_of_int (c - \<lfloor>?I e\<rfloor>)" by simp |
|
23264
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chaieb
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diff
changeset
|
166 |
finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th) |
324622260d29
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chaieb
parents:
diff
changeset
|
167 |
qed |
324622260d29
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chaieb
parents:
diff
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|
168 |
|
51369 | 169 |
lemma isint_add: |
170 |
assumes ai: "isint a bs" and bi: "isint b bs" |
|
171 |
shows "isint (Add a b) bs" |
|
23264
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diff
changeset
|
172 |
proof- |
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parents:
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|
173 |
let ?a = "Inum bs a" |
324622260d29
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chaieb
parents:
diff
changeset
|
174 |
let ?b = "Inum bs b" |
61942 | 175 |
from ai bi isint_iff have "real_of_int \<lfloor>?a + ?b\<rfloor> = real_of_int \<lfloor>real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>\<rfloor>" |
51369 | 176 |
by simp |
61942 | 177 |
also have "\<dots> = real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>" by simp |
23264
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chaieb
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|
178 |
also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp |
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chaieb
parents:
diff
changeset
|
179 |
finally show "isint (Add a b) bs" by (simp add: isint_iff) |
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chaieb
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diff
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|
180 |
qed |
324622260d29
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chaieb
parents:
diff
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|
181 |
|
324622260d29
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|
182 |
lemma isint_c: "isint (C j) bs" |
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parents:
diff
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|
183 |
by (simp add: isint_iff) |
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chaieb
parents:
diff
changeset
|
184 |
|
324622260d29
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chaieb
parents:
diff
changeset
|
185 |
|
324622260d29
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chaieb
parents:
diff
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|
186 |
(* FORMULAE *) |
66809 | 187 |
datatype (plugins del: size) fm = |
188 |
T | F | Lt num | Le num | Gt num | Ge num | Eq num | NEq num | |
|
189 |
Dvd int num | NDvd int num | |
|
190 |
NOT fm | And fm fm | Or fm fm | Imp fm fm | Iff fm fm | E fm | A fm |
|
191 |
||
192 |
instantiation fm :: size |
|
193 |
begin |
|
194 |
||
195 |
primrec size_fm :: "fm \<Rightarrow> nat" |
|
196 |
where |
|
197 |
"size_fm (NOT p) = 1 + size_fm p" |
|
198 |
| "size_fm (And p q) = 1 + size_fm p + size_fm q" |
|
199 |
| "size_fm (Or p q) = 1 + size_fm p + size_fm q" |
|
200 |
| "size_fm (Imp p q) = 3 + size_fm p + size_fm q" |
|
201 |
| "size_fm (Iff p q) = 3 + 2 * (size_fm p + size_fm q)" |
|
202 |
| "size_fm (E p) = 1 + size_fm p" |
|
203 |
| "size_fm (A p) = 4 + size_fm p" |
|
204 |
| "size_fm (Dvd i t) = 2" |
|
205 |
| "size_fm (NDvd i t) = 2" |
|
206 |
| "size_fm T = 1" |
|
207 |
| "size_fm F = 1" |
|
208 |
| "size_fm (Lt _) = 1" |
|
209 |
| "size_fm (Le _) = 1" |
|
210 |
| "size_fm (Gt _) = 1" |
|
211 |
| "size_fm (Ge _) = 1" |
|
212 |
| "size_fm (Eq _) = 1" |
|
213 |
| "size_fm (NEq _) = 1" |
|
214 |
||
215 |
instance .. |
|
216 |
||
217 |
end |
|
218 |
||
219 |
lemma size_fm_pos [simp]: "size p > 0" for p :: fm |
|
220 |
by (induct p) simp_all |
|
23264
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chaieb
parents:
diff
changeset
|
221 |
|
324622260d29
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chaieb
parents:
diff
changeset
|
222 |
(* Semantics of formulae (fm) *) |
66809 | 223 |
primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" |
224 |
where |
|
225 |
"Ifm bs T \<longleftrightarrow> True" |
|
226 |
| "Ifm bs F \<longleftrightarrow> False" |
|
227 |
| "Ifm bs (Lt a) \<longleftrightarrow> Inum bs a < 0" |
|
228 |
| "Ifm bs (Gt a) \<longleftrightarrow> Inum bs a > 0" |
|
229 |
| "Ifm bs (Le a) \<longleftrightarrow> Inum bs a \<le> 0" |
|
230 |
| "Ifm bs (Ge a) \<longleftrightarrow> Inum bs a \<ge> 0" |
|
231 |
| "Ifm bs (Eq a) \<longleftrightarrow> Inum bs a = 0" |
|
232 |
| "Ifm bs (NEq a) \<longleftrightarrow> Inum bs a \<noteq> 0" |
|
233 |
| "Ifm bs (Dvd i b) \<longleftrightarrow> real_of_int i rdvd Inum bs b" |
|
234 |
| "Ifm bs (NDvd i b) \<longleftrightarrow> \<not> (real_of_int i rdvd Inum bs b)" |
|
235 |
| "Ifm bs (NOT p) \<longleftrightarrow> \<not> (Ifm bs p)" |
|
236 |
| "Ifm bs (And p q) \<longleftrightarrow> Ifm bs p \<and> Ifm bs q" |
|
237 |
| "Ifm bs (Or p q) \<longleftrightarrow> Ifm bs p \<or> Ifm bs q" |
|
238 |
| "Ifm bs (Imp p q) \<longleftrightarrow> (Ifm bs p \<longrightarrow> Ifm bs q)" |
|
239 |
| "Ifm bs (Iff p q) \<longleftrightarrow> (Ifm bs p \<longleftrightarrow> Ifm bs q)" |
|
240 |
| "Ifm bs (E p) \<longleftrightarrow> (\<exists>x. Ifm (x # bs) p)" |
|
241 |
| "Ifm bs (A p) \<longleftrightarrow> (\<forall>x. Ifm (x # bs) p)" |
|
242 |
||
243 |
fun prep :: "fm \<Rightarrow> fm" |
|
244 |
where |
|
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chaieb
parents:
diff
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|
245 |
"prep (E T) = T" |
66124 | 246 |
| "prep (E F) = F" |
247 |
| "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))" |
|
248 |
| "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))" |
|
249 |
| "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" |
|
250 |
| "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))" |
|
251 |
| "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" |
|
252 |
| "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" |
|
253 |
| "prep (E p) = E (prep p)" |
|
254 |
| "prep (A (And p q)) = And (prep (A p)) (prep (A q))" |
|
255 |
| "prep (A p) = prep (NOT (E (NOT p)))" |
|
256 |
| "prep (NOT (NOT p)) = prep p" |
|
257 |
| "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))" |
|
258 |
| "prep (NOT (A p)) = prep (E (NOT p))" |
|
259 |
| "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))" |
|
260 |
| "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))" |
|
261 |
| "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))" |
|
262 |
| "prep (NOT p) = NOT (prep p)" |
|
263 |
| "prep (Or p q) = Or (prep p) (prep q)" |
|
264 |
| "prep (And p q) = And (prep p) (prep q)" |
|
265 |
| "prep (Imp p q) = prep (Or (NOT p) q)" |
|
266 |
| "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))" |
|
267 |
| "prep p = p" |
|
268 |
||
23264
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chaieb
parents:
diff
changeset
|
269 |
lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p" |
51369 | 270 |
by (induct p rule: prep.induct) auto |
23264
324622260d29
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chaieb
parents:
diff
changeset
|
271 |
|
324622260d29
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chaieb
parents:
diff
changeset
|
272 |
|
324622260d29
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chaieb
parents:
diff
changeset
|
273 |
(* Quantifier freeness *) |
66809 | 274 |
fun qfree:: "fm \<Rightarrow> bool" |
275 |
where |
|
23264
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chaieb
parents:
diff
changeset
|
276 |
"qfree (E p) = False" |
66809 | 277 |
| "qfree (A p) = False" |
278 |
| "qfree (NOT p) = qfree p" |
|
279 |
| "qfree (And p q) = (qfree p \<and> qfree q)" |
|
280 |
| "qfree (Or p q) = (qfree p \<and> qfree q)" |
|
281 |
| "qfree (Imp p q) = (qfree p \<and> qfree q)" |
|
282 |
| "qfree (Iff p q) = (qfree p \<and> qfree q)" |
|
283 |
| "qfree p = True" |
|
23264
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chaieb
parents:
diff
changeset
|
284 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
285 |
(* Boundedness and substitution *) |
66809 | 286 |
primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) |
287 |
where |
|
23264
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chaieb
parents:
diff
changeset
|
288 |
"numbound0 (C c) = True" |
66809 | 289 |
| "numbound0 (Bound n) = (n>0)" |
290 |
| "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)" |
|
291 |
| "numbound0 (Neg a) = numbound0 a" |
|
292 |
| "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)" |
|
293 |
| "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" |
|
294 |
| "numbound0 (Mul i a) = numbound0 a" |
|
295 |
| "numbound0 (Floor a) = numbound0 a" |
|
296 |
| "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)" |
|
25765 | 297 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
298 |
lemma numbound0_I: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
299 |
assumes nb: "numbound0 a" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
300 |
shows "Inum (b#bs) a = Inum (b'#bs) a" |
41849 | 301 |
using nb by (induct a) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
302 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
303 |
lemma numbound0_gen: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
304 |
assumes nb: "numbound0 t" and ti: "isint t (x#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
305 |
shows "\<forall> y. isint t (y#bs)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
306 |
using nb ti |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
307 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
308 |
fix y |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
309 |
from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
310 |
show "isint t (y#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
311 |
by (simp add: isint_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
312 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
313 |
|
66809 | 314 |
primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) |
315 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
316 |
"bound0 T = True" |
66809 | 317 |
| "bound0 F = True" |
318 |
| "bound0 (Lt a) = numbound0 a" |
|
319 |
| "bound0 (Le a) = numbound0 a" |
|
320 |
| "bound0 (Gt a) = numbound0 a" |
|
321 |
| "bound0 (Ge a) = numbound0 a" |
|
322 |
| "bound0 (Eq a) = numbound0 a" |
|
323 |
| "bound0 (NEq a) = numbound0 a" |
|
324 |
| "bound0 (Dvd i a) = numbound0 a" |
|
325 |
| "bound0 (NDvd i a) = numbound0 a" |
|
326 |
| "bound0 (NOT p) = bound0 p" |
|
327 |
| "bound0 (And p q) = (bound0 p \<and> bound0 q)" |
|
328 |
| "bound0 (Or p q) = (bound0 p \<and> bound0 q)" |
|
329 |
| "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))" |
|
330 |
| "bound0 (Iff p q) = (bound0 p \<and> bound0 q)" |
|
331 |
| "bound0 (E p) = False" |
|
332 |
| "bound0 (A p) = False" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
333 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
334 |
lemma bound0_I: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
335 |
assumes bp: "bound0 p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
336 |
shows "Ifm (b#bs) p = Ifm (b'#bs) p" |
51369 | 337 |
using bp numbound0_I [where b="b" and bs="bs" and b'="b'"] |
41849 | 338 |
by (induct p) auto |
25765 | 339 |
|
66809 | 340 |
primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) |
341 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
342 |
"numsubst0 t (C c) = (C c)" |
66809 | 343 |
| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)" |
344 |
| "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))" |
|
345 |
| "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)" |
|
346 |
| "numsubst0 t (Neg a) = Neg (numsubst0 t a)" |
|
347 |
| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)" |
|
348 |
| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" |
|
349 |
| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)" |
|
350 |
| "numsubst0 t (Floor a) = Floor (numsubst0 t a)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
351 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
352 |
lemma numsubst0_I: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
353 |
shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" |
41849 | 354 |
by (induct t) simp_all |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
355 |
|
66809 | 356 |
primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) |
357 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
358 |
"subst0 t T = T" |
66809 | 359 |
| "subst0 t F = F" |
360 |
| "subst0 t (Lt a) = Lt (numsubst0 t a)" |
|
361 |
| "subst0 t (Le a) = Le (numsubst0 t a)" |
|
362 |
| "subst0 t (Gt a) = Gt (numsubst0 t a)" |
|
363 |
| "subst0 t (Ge a) = Ge (numsubst0 t a)" |
|
364 |
| "subst0 t (Eq a) = Eq (numsubst0 t a)" |
|
365 |
| "subst0 t (NEq a) = NEq (numsubst0 t a)" |
|
366 |
| "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)" |
|
367 |
| "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)" |
|
368 |
| "subst0 t (NOT p) = NOT (subst0 t p)" |
|
369 |
| "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" |
|
370 |
| "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" |
|
371 |
| "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" |
|
372 |
| "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
373 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
374 |
lemma subst0_I: assumes qfp: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
375 |
shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
376 |
using qfp numsubst0_I[where b="b" and bs="bs" and a="a"] |
41849 | 377 |
by (induct p) simp_all |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
378 |
|
66809 | 379 |
fun decrnum:: "num \<Rightarrow> num" |
380 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
381 |
"decrnum (Bound n) = Bound (n - 1)" |
41839 | 382 |
| "decrnum (Neg a) = Neg (decrnum a)" |
383 |
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)" |
|
384 |
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" |
|
385 |
| "decrnum (Mul c a) = Mul c (decrnum a)" |
|
386 |
| "decrnum (Floor a) = Floor (decrnum a)" |
|
387 |
| "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" |
|
388 |
| "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)" |
|
389 |
| "decrnum a = a" |
|
390 |
||
66809 | 391 |
fun decr :: "fm \<Rightarrow> fm" |
392 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
393 |
"decr (Lt a) = Lt (decrnum a)" |
41839 | 394 |
| "decr (Le a) = Le (decrnum a)" |
395 |
| "decr (Gt a) = Gt (decrnum a)" |
|
396 |
| "decr (Ge a) = Ge (decrnum a)" |
|
397 |
| "decr (Eq a) = Eq (decrnum a)" |
|
398 |
| "decr (NEq a) = NEq (decrnum a)" |
|
399 |
| "decr (Dvd i a) = Dvd i (decrnum a)" |
|
400 |
| "decr (NDvd i a) = NDvd i (decrnum a)" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
401 |
| "decr (NOT p) = NOT (decr p)" |
41839 | 402 |
| "decr (And p q) = And (decr p) (decr q)" |
403 |
| "decr (Or p q) = Or (decr p) (decr q)" |
|
404 |
| "decr (Imp p q) = Imp (decr p) (decr q)" |
|
405 |
| "decr (Iff p q) = Iff (decr p) (decr q)" |
|
406 |
| "decr p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
407 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
408 |
lemma decrnum: assumes nb: "numbound0 t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
409 |
shows "Inum (x#bs) t = Inum bs (decrnum t)" |
51369 | 410 |
using nb by (induct t rule: decrnum.induct) simp_all |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
411 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
412 |
lemma decr: assumes nb: "bound0 p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
413 |
shows "Ifm (x#bs) p = Ifm bs (decr p)" |
51369 | 414 |
using nb by (induct p rule: decr.induct) (simp_all add: decrnum) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
415 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
416 |
lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" |
51369 | 417 |
by (induct p) simp_all |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
418 |
|
66809 | 419 |
fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) |
420 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
421 |
"isatom T = True" |
41839 | 422 |
| "isatom F = True" |
423 |
| "isatom (Lt a) = True" |
|
424 |
| "isatom (Le a) = True" |
|
425 |
| "isatom (Gt a) = True" |
|
426 |
| "isatom (Ge a) = True" |
|
427 |
| "isatom (Eq a) = True" |
|
428 |
| "isatom (NEq a) = True" |
|
429 |
| "isatom (Dvd i b) = True" |
|
430 |
| "isatom (NDvd i b) = True" |
|
431 |
| "isatom p = False" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
432 |
|
51369 | 433 |
lemma numsubst0_numbound0: |
434 |
assumes nb: "numbound0 t" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
435 |
shows "numbound0 (numsubst0 t a)" |
51369 | 436 |
using nb by (induct a) auto |
437 |
||
438 |
lemma subst0_bound0: |
|
439 |
assumes qf: "qfree p" and nb: "numbound0 t" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
440 |
shows "bound0 (subst0 t p)" |
51369 | 441 |
using qf numsubst0_numbound0[OF nb] by (induct p) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
442 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
443 |
lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" |
51369 | 444 |
by (induct p) simp_all |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
445 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
446 |
|
25765 | 447 |
definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
448 |
"djf f p q = (if q=T then T else if q=F then f p else |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
449 |
(let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))" |
25765 | 450 |
|
451 |
definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where |
|
452 |
"evaldjf f ps = foldr (djf f) ps F" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
453 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
454 |
lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
455 |
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
456 |
(cases "f p", simp_all add: Let_def djf_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
457 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
458 |
lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))" |
51369 | 459 |
by (induct ps) (simp_all add: evaldjf_def djf_Or) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
460 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
461 |
lemma evaldjf_bound0: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
462 |
assumes nb: "\<forall> x\<in> set xs. bound0 (f x)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
463 |
shows "bound0 (evaldjf f xs)" |
51369 | 464 |
using nb |
465 |
apply (induct xs) |
|
466 |
apply (auto simp add: evaldjf_def djf_def Let_def) |
|
467 |
apply (case_tac "f a") |
|
468 |
apply auto |
|
469 |
done |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
470 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
471 |
lemma evaldjf_qf: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
472 |
assumes nb: "\<forall> x\<in> set xs. qfree (f x)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
473 |
shows "qfree (evaldjf f xs)" |
51369 | 474 |
using nb |
475 |
apply (induct xs) |
|
476 |
apply (auto simp add: evaldjf_def djf_def Let_def) |
|
477 |
apply (case_tac "f a") |
|
478 |
apply auto |
|
479 |
done |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
480 |
|
66809 | 481 |
fun disjuncts :: "fm \<Rightarrow> fm list" |
482 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
483 |
"disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" |
41839 | 484 |
| "disjuncts F = []" |
485 |
| "disjuncts p = [p]" |
|
486 |
||
66809 | 487 |
fun conjuncts :: "fm \<Rightarrow> fm list" |
488 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
489 |
"conjuncts (And p q) = (conjuncts p) @ (conjuncts q)" |
41839 | 490 |
| "conjuncts T = []" |
491 |
| "conjuncts p = [p]" |
|
492 |
||
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
493 |
lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p" |
51369 | 494 |
by (induct p rule: conjuncts.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
495 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
496 |
lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q" |
51369 | 497 |
proof - |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
498 |
assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
499 |
hence "list_all qfree (disjuncts p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
500 |
by (induct p rule: disjuncts.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
501 |
thus ?thesis by (simp only: list_all_iff) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
502 |
qed |
51369 | 503 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
504 |
lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
505 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
506 |
assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
507 |
hence "list_all qfree (conjuncts p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
508 |
by (induct p rule: conjuncts.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
509 |
thus ?thesis by (simp only: list_all_iff) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
510 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
511 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
512 |
definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
513 |
"DJ f p \<equiv> evaldjf f (disjuncts p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
514 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
515 |
lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
516 |
and fF: "f F = F" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
517 |
shows "Ifm bs (DJ f p) = Ifm bs (f p)" |
51369 | 518 |
proof - |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
519 |
have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
520 |
by (simp add: DJ_def evaldjf_ex) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
521 |
also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
522 |
finally show ?thesis . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
523 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
524 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
525 |
lemma DJ_qf: assumes |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
526 |
fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
527 |
shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) " |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
528 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
529 |
fix p assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
530 |
have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
531 |
from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
532 |
with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
533 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
534 |
from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
535 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
536 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
537 |
lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
538 |
shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
539 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
540 |
fix p::fm and bs |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
541 |
assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
542 |
from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
543 |
from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
544 |
have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
545 |
by (simp add: DJ_def evaldjf_ex) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
546 |
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
547 |
also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
548 |
finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
549 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
550 |
(* Simplification *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
551 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
552 |
(* Algebraic simplifications for nums *) |
66809 | 553 |
fun bnds:: "num \<Rightarrow> nat list" |
554 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
555 |
"bnds (Bound n) = [n]" |
41839 | 556 |
| "bnds (CN n c a) = n#(bnds a)" |
557 |
| "bnds (Neg a) = bnds a" |
|
558 |
| "bnds (Add a b) = (bnds a)@(bnds b)" |
|
559 |
| "bnds (Sub a b) = (bnds a)@(bnds b)" |
|
560 |
| "bnds (Mul i a) = bnds a" |
|
561 |
| "bnds (Floor a) = bnds a" |
|
562 |
| "bnds (CF c a b) = (bnds a)@(bnds b)" |
|
563 |
| "bnds a = []" |
|
66809 | 564 |
|
565 |
fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool" |
|
566 |
where |
|
41839 | 567 |
"lex_ns [] ms = True" |
568 |
| "lex_ns ns [] = False" |
|
569 |
| "lex_ns (n#ns) (m#ms) = (n<m \<or> ((n = m) \<and> lex_ns ns ms)) " |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
570 |
definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" where |
41839 | 571 |
"lex_bnd t s \<equiv> lex_ns (bnds t) (bnds s)" |
572 |
||
66809 | 573 |
fun maxcoeff:: "num \<Rightarrow> int" |
574 |
where |
|
61945 | 575 |
"maxcoeff (C i) = \<bar>i\<bar>" |
576 |
| "maxcoeff (CN n c t) = max \<bar>c\<bar> (maxcoeff t)" |
|
577 |
| "maxcoeff (CF c t s) = max \<bar>c\<bar> (maxcoeff s)" |
|
41839 | 578 |
| "maxcoeff t = 1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
579 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
580 |
lemma maxcoeff_pos: "maxcoeff t \<ge> 0" |
51369 | 581 |
by (induct t rule: maxcoeff.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
582 |
|
66809 | 583 |
fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" |
584 |
where |
|
31706 | 585 |
"numgcdh (C i) = (\<lambda>g. gcd i g)" |
41839 | 586 |
| "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))" |
587 |
| "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))" |
|
588 |
| "numgcdh t = (\<lambda>g. 1)" |
|
23858 | 589 |
|
51369 | 590 |
definition numgcd :: "num \<Rightarrow> int" |
591 |
where "numgcd t = numgcdh t (maxcoeff t)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
592 |
|
66809 | 593 |
fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" |
594 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
595 |
"reducecoeffh (C i) = (\<lambda> g. C (i div g))" |
41839 | 596 |
| "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))" |
597 |
| "reducecoeffh (CF c s t) = (\<lambda> g. CF (c div g) s (reducecoeffh t g))" |
|
598 |
| "reducecoeffh t = (\<lambda>g. t)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
599 |
|
51369 | 600 |
definition reducecoeff :: "num \<Rightarrow> num" |
23858 | 601 |
where |
51369 | 602 |
"reducecoeff t = |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
603 |
(let g = numgcd t in |
51369 | 604 |
if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
605 |
|
66809 | 606 |
fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" |
607 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
608 |
"dvdnumcoeff (C i) = (\<lambda> g. g dvd i)" |
41839 | 609 |
| "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" |
610 |
| "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" |
|
611 |
| "dvdnumcoeff t = (\<lambda>g. False)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
612 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
613 |
lemma dvdnumcoeff_trans: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
614 |
assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
615 |
shows "dvdnumcoeff t g" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
616 |
using dgt' gdg |
51369 | 617 |
by (induct t rule: dvdnumcoeff.induct) (simp_all add: gdg dvd_trans[OF gdg]) |
30042 | 618 |
|
619 |
declare dvd_trans [trans add] |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
620 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
621 |
lemma numgcd0: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
622 |
assumes g0: "numgcd t = 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
623 |
shows "Inum bs t = 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
624 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
625 |
have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0" |
31706 | 626 |
by (induct t rule: numgcdh.induct, auto) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
627 |
thus ?thesis using g0[simplified numgcd_def] by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
628 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
629 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
630 |
lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0" |
51369 | 631 |
using gp by (induct t rule: numgcdh.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
632 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
633 |
lemma numgcd_pos: "numgcd t \<ge>0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
634 |
by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
635 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
636 |
lemma reducecoeffh: |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
637 |
assumes gt: "dvdnumcoeff t g" and gp: "g > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
638 |
shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
639 |
using gt |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
640 |
proof(induct t rule: reducecoeffh.induct) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
641 |
case (1 i) hence gd: "g dvd i" by simp |
46670 | 642 |
from assms 1 show ?case by (simp add: real_of_int_div[OF gd]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
643 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
644 |
case (2 n c t) hence gd: "g dvd c" by simp |
46670 | 645 |
from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
646 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
647 |
case (3 c s t) hence gd: "g dvd c" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
648 |
from assms 3 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
649 |
qed (auto simp add: numgcd_def gp) |
41807 | 650 |
|
66809 | 651 |
fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" |
652 |
where |
|
61945 | 653 |
"ismaxcoeff (C i) = (\<lambda> x. \<bar>i\<bar> \<le> x)" |
654 |
| "ismaxcoeff (CN n c t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> (ismaxcoeff t x))" |
|
655 |
| "ismaxcoeff (CF c s t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> (ismaxcoeff t x))" |
|
41839 | 656 |
| "ismaxcoeff t = (\<lambda>x. True)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
657 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
658 |
lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'" |
51369 | 659 |
by (induct t rule: ismaxcoeff.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
660 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
661 |
lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
662 |
proof (induct t rule: maxcoeff.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
663 |
case (2 n c t) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
664 |
hence H:"ismaxcoeff t (maxcoeff t)" . |
61945 | 665 |
have thh: "maxcoeff t \<le> max \<bar>c\<bar> (maxcoeff t)" by simp |
51369 | 666 |
from ismaxcoeff_mono[OF H thh] show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
667 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
668 |
case (3 c t s) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
669 |
hence H1:"ismaxcoeff s (maxcoeff s)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
670 |
have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def) |
51369 | 671 |
from ismaxcoeff_mono[OF H1 thh1] show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
672 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
673 |
|
67118 | 674 |
lemma zgcd_gt1: |
675 |
"\<bar>i\<bar> > 1 \<and> \<bar>j\<bar> > 1 \<or> \<bar>i\<bar> = 0 \<and> \<bar>j\<bar> > 1 \<or> \<bar>i\<bar> > 1 \<and> \<bar>j\<bar> = 0" |
|
676 |
if "gcd i j > 1" for i j :: int |
|
677 |
proof - |
|
678 |
have "\<bar>k\<bar> \<le> 1 \<longleftrightarrow> k = - 1 \<or> k = 0 \<or> k = 1" for k :: int |
|
679 |
by auto |
|
680 |
with that show ?thesis |
|
681 |
by (auto simp add: not_less) |
|
682 |
qed |
|
51369 | 683 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
684 |
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0" |
41807 | 685 |
by (induct t rule: numgcdh.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
686 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
687 |
lemma dvdnumcoeff_aux: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
688 |
assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
689 |
shows "dvdnumcoeff t (numgcdh t m)" |
41807 | 690 |
using assms |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
691 |
proof(induct t rule: numgcdh.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
692 |
case (2 n c t) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
693 |
let ?g = "numgcdh t m" |
41807 | 694 |
from 2 have th:"gcd c ?g > 1" by simp |
27556 | 695 |
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] |
61945 | 696 |
have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp |
697 |
moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 2 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
698 |
have th: "dvdnumcoeff t ?g" by simp |
31706 | 699 |
have th': "gcd c ?g dvd ?g" by simp |
700 |
from dvdnumcoeff_trans[OF th' th] have ?case by simp } |
|
61945 | 701 |
moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1" |
41807 | 702 |
with 2 have th: "dvdnumcoeff t ?g" by simp |
31706 | 703 |
have th': "gcd c ?g dvd ?g" by simp |
704 |
from dvdnumcoeff_trans[OF th' th] have ?case by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
705 |
hence ?case by simp } |
61945 | 706 |
moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0" |
41807 | 707 |
from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
708 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
709 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
710 |
case (3 c s t) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
711 |
let ?g = "numgcdh t m" |
41807 | 712 |
from 3 have th:"gcd c ?g > 1" by simp |
27556 | 713 |
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] |
61945 | 714 |
have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp |
715 |
moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 3 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
716 |
have th: "dvdnumcoeff t ?g" by simp |
31706 | 717 |
have th': "gcd c ?g dvd ?g" by simp |
718 |
from dvdnumcoeff_trans[OF th' th] have ?case by simp } |
|
61945 | 719 |
moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1" |
41807 | 720 |
with 3 have th: "dvdnumcoeff t ?g" by simp |
31706 | 721 |
have th': "gcd c ?g dvd ?g" by simp |
722 |
from dvdnumcoeff_trans[OF th' th] have ?case by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
723 |
hence ?case by simp } |
61945 | 724 |
moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0" |
41807 | 725 |
from numgcdh0[OF g0] have "m=0". with 3 g0 have ?case by simp } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
726 |
ultimately show ?case by blast |
31706 | 727 |
qed auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
728 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
729 |
lemma dvdnumcoeff_aux2: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
730 |
assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
731 |
using assms |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
732 |
proof (simp add: numgcd_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
733 |
let ?mc = "maxcoeff t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
734 |
let ?g = "numgcdh t ?mc" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
735 |
have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
736 |
have th2: "?mc \<ge> 0" by (rule maxcoeff_pos) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
737 |
assume H: "numgcdh t ?mc > 1" |
41807 | 738 |
from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
739 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
740 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
741 |
lemma reducecoeff: "real_of_int (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
742 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
743 |
let ?g = "numgcd t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
744 |
have "?g \<ge> 0" by (simp add: numgcd_pos) |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
745 |
hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
746 |
moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
747 |
moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
748 |
moreover { assume g1:"?g > 1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
749 |
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
750 |
from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
751 |
by (simp add: reducecoeff_def Let_def)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
752 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
753 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
754 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
755 |
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)" |
51369 | 756 |
by (induct t rule: reducecoeffh.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
757 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
758 |
lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)" |
51369 | 759 |
using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) |
760 |
||
761 |
consts numadd:: "num \<times> num \<Rightarrow> num" |
|
66809 | 762 |
recdef numadd "measure (\<lambda>(t, s). size t + size s)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
763 |
"numadd (CN n1 c1 r1,CN n2 c2 r2) = |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
764 |
(if n1=n2 then |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
765 |
(let c = c1 + c2 |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
766 |
in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
767 |
else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2)) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
768 |
else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
769 |
"numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
770 |
"numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
771 |
"numadd (CF c1 t1 r1,CF c2 t2 r2) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
772 |
(if t1 = t2 then |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
773 |
(let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s)) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
774 |
else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2)) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
775 |
else CF c2 t2 (numadd(CF c1 t1 r1,r2)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
776 |
"numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
777 |
"numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
778 |
"numadd (C b1, C b2) = C (b1+b2)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
779 |
"numadd (a,b) = Add a b" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
780 |
|
66809 | 781 |
lemma numadd [simp]: "Inum bs (numadd (t, s)) = Inum bs (Add t s)" |
782 |
by (induct t s rule: numadd.induct) (simp_all add: Let_def algebra_simps add_eq_0_iff) |
|
783 |
||
784 |
lemma numadd_nb [simp]: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numadd (t, s))" |
|
785 |
by (induct t s rule: numadd.induct) (simp_all add: Let_def) |
|
786 |
||
787 |
fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" |
|
788 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
789 |
"nummul (C j) = (\<lambda> i. C (i*j))" |
41839 | 790 |
| "nummul (CN n c t) = (\<lambda> i. CN n (c*i) (nummul t i))" |
791 |
| "nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))" |
|
792 |
| "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))" |
|
793 |
| "nummul t = (\<lambda> i. Mul i t)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
794 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
795 |
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)" |
51369 | 796 |
by (induct t rule: nummul.induct) (auto simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
797 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
798 |
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)" |
51369 | 799 |
by (induct t rule: nummul.induct) auto |
800 |
||
801 |
definition numneg :: "num \<Rightarrow> num" |
|
802 |
where "numneg t \<equiv> nummul t (- 1)" |
|
803 |
||
804 |
definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" |
|
805 |
where "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
806 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
807 |
lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" |
51369 | 808 |
using numneg_def nummul by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
809 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
810 |
lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)" |
51369 | 811 |
using numneg_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
812 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
813 |
lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" |
51369 | 814 |
using numsub_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
815 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
816 |
lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)" |
51369 | 817 |
using numsub_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
818 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
819 |
lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
820 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
821 |
have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
822 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
823 |
have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
824 |
also have "\<dots>" by (simp add: isint_add cti si) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
825 |
finally show ?thesis . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
826 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
827 |
|
66809 | 828 |
fun split_int:: "num \<Rightarrow> num \<times> num" |
829 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
830 |
"split_int (C c) = (C 0, C c)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
831 |
| "split_int (CN n c b) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
832 |
(let (bv,bi) = split_int b |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
833 |
in (CN n c bv, bi))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
834 |
| "split_int (CF c a b) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
835 |
(let (bv,bi) = split_int b |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
836 |
in (bv, CF c a bi))" |
41839 | 837 |
| "split_int a = (a,C 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
838 |
|
41807 | 839 |
lemma split_int: "\<And>tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
840 |
proof (induct t rule: split_int.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
841 |
case (2 c n b tv ti) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
842 |
let ?bv = "fst (split_int b)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
843 |
let ?bi = "snd (split_int b)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
844 |
have "split_int b = (?bv,?bi)" by simp |
41807 | 845 |
with 2(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ |
846 |
from 2(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def) |
|
847 |
from 2(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
848 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
849 |
case (3 c a b tv ti) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
850 |
let ?bv = "fst (split_int b)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
851 |
let ?bi = "snd (split_int b)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
852 |
have "split_int b = (?bv,?bi)" by simp |
41807 | 853 |
with 3(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ |
854 |
from 3(2) have tibi: "ti = CF c a ?bi" |
|
855 |
by (simp add: Let_def split_def) |
|
856 |
from 3(2) b[symmetric] bii show ?case |
|
857 |
by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF) |
|
29667 | 858 |
qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
859 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
860 |
lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) " |
41807 | 861 |
by (induct t rule: split_int.induct) (auto simp add: Let_def split_def) |
862 |
||
863 |
definition numfloor:: "num \<Rightarrow> num" |
|
23858 | 864 |
where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
865 |
"numfloor t = (let (tv,ti) = split_int t in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
866 |
(case tv of C i \<Rightarrow> numadd (tv,ti) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
867 |
| _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
868 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
869 |
lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
870 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
871 |
let ?tv = "fst (split_int t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
872 |
let ?ti = "snd (split_int t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
873 |
have tvti:"split_int t = (?tv,?ti)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
874 |
{assume H: "\<forall> v. ?tv \<noteq> C v" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
875 |
hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" |
51369 | 876 |
by (cases ?tv) (auto simp add: numfloor_def Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
877 |
from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ |
61942 | 878 |
hence "?N (Floor t) = real_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp |
879 |
also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
880 |
by (simp,subst tii[simplified isint_iff, symmetric]) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
881 |
also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
882 |
finally have ?thesis using th1 by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
883 |
moreover {fix v assume H:"?tv = C v" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
884 |
from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ |
61942 | 885 |
hence "?N (Floor t) = real_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp |
886 |
also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
887 |
by (simp,subst tii[simplified isint_iff, symmetric]) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
888 |
also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) |
51369 | 889 |
finally have ?thesis by (simp add: H numfloor_def Let_def split_def) } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
890 |
ultimately show ?thesis by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
891 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
892 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
893 |
lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
894 |
using split_int_nb[where t="t"] |
51369 | 895 |
by (cases "fst (split_int t)") (auto simp add: numfloor_def Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
896 |
|
66809 | 897 |
fun simpnum:: "num \<Rightarrow> num" |
898 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
899 |
"simpnum (C j) = C j" |
41839 | 900 |
| "simpnum (Bound n) = CN n 1 (C 0)" |
901 |
| "simpnum (Neg t) = numneg (simpnum t)" |
|
902 |
| "simpnum (Add t s) = numadd (simpnum t,simpnum s)" |
|
903 |
| "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" |
|
904 |
| "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" |
|
905 |
| "simpnum (Floor t) = numfloor (simpnum t)" |
|
906 |
| "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))" |
|
907 |
| "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
908 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
909 |
lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" |
51369 | 910 |
by (induct t rule: simpnum.induct) auto |
911 |
||
912 |
lemma simpnum_numbound0[simp]: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)" |
|
913 |
by (induct t rule: simpnum.induct) auto |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
914 |
|
66809 | 915 |
fun nozerocoeff:: "num \<Rightarrow> bool" |
916 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
917 |
"nozerocoeff (C c) = True" |
41839 | 918 |
| "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)" |
919 |
| "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)" |
|
920 |
| "nozerocoeff (Mul c t) = (c\<noteq>0 \<and> nozerocoeff t)" |
|
921 |
| "nozerocoeff t = True" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
922 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
923 |
lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))" |
51369 | 924 |
by (induct a b rule: numadd.induct) (auto simp add: Let_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
925 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
926 |
lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)" |
51369 | 927 |
by (induct a rule: nummul.induct) (auto simp add: Let_def numadd_nz) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
928 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
929 |
lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)" |
51369 | 930 |
by (simp add: numneg_def nummul_nz) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
931 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
932 |
lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)" |
51369 | 933 |
by (simp add: numsub_def numneg_nz numadd_nz) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
934 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
935 |
lemma split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))" |
51369 | 936 |
by (induct t rule: split_int.induct) (auto simp add: Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
937 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
938 |
lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)" |
51369 | 939 |
by (simp add: numfloor_def Let_def split_def) |
940 |
(cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
941 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
942 |
lemma simpnum_nz: "nozerocoeff (simpnum t)" |
51369 | 943 |
by (induct t rule: simpnum.induct) |
944 |
(auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
945 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
946 |
lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
947 |
proof (induct t rule: maxcoeff.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
948 |
case (2 n c t) |
61945 | 949 |
hence cnz: "c \<noteq>0" and mx: "max \<bar>c\<bar> (maxcoeff t) = 0" by simp+ |
950 |
have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp |
|
951 |
with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith |
|
41807 | 952 |
with 2 show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
953 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
954 |
case (3 c s t) |
61945 | 955 |
hence cnz: "c \<noteq>0" and mx: "max \<bar>c\<bar> (maxcoeff t) = 0" by simp+ |
956 |
have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp |
|
957 |
with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith |
|
41807 | 958 |
with 3 show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
959 |
qed auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
960 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
961 |
lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
962 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
963 |
from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
964 |
from numgcdh0[OF th] have th:"maxcoeff t = 0" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
965 |
from maxcoeff_nz[OF nz th] show ?thesis . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
966 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
967 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
968 |
definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
969 |
"simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
970 |
(let t' = simpnum t ; g = numgcd t' in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
971 |
if g > 1 then (let g' = gcd n g in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
972 |
if g' = 1 then (t',n) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
973 |
else (reducecoeffh t' g', n div g')) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
974 |
else (t',n))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
975 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
976 |
lemma simp_num_pair_ci: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
977 |
shows "((\<lambda> (t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real_of_int n) (t,n))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
978 |
(is "?lhs = ?rhs") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
979 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
980 |
let ?t' = "simpnum t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
981 |
let ?g = "numgcd ?t'" |
31706 | 982 |
let ?g' = "gcd n ?g" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
983 |
{assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
984 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
985 |
{ assume nnz: "n \<noteq> 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
986 |
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
987 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
988 |
{assume g1:"?g>1" hence g0: "?g > 0" by simp |
31706 | 989 |
from g1 nnz have gp0: "?g' \<noteq> 0" by simp |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
990 |
hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
991 |
hence "?g'= 1 \<or> ?g' > 1" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
992 |
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
993 |
moreover {assume g'1:"?g'>1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
994 |
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
995 |
let ?tt = "reducecoeffh ?t' ?g'" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
996 |
let ?t = "Inum bs ?tt" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
997 |
have gpdg: "?g' dvd ?g" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
998 |
have gpdd: "?g' dvd n" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
999 |
have gpdgp: "?g' dvd ?g'" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1000 |
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1001 |
have th2:"real_of_int ?g' * ?t = Inum bs ?t'" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1002 |
from nnz g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')" by (simp add: simp_num_pair_def Let_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1003 |
also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1004 |
also have "\<dots> = (Inum bs ?t' / real_of_int n)" |
46670 | 1005 |
using real_of_int_div[OF gpdd] th2 gp0 by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1006 |
finally have "?lhs = Inum bs t / real_of_int n" by simp |
41807 | 1007 |
then have ?thesis using nnz g1 g'1 by (simp add: simp_num_pair_def) } |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
1008 |
ultimately have ?thesis by auto } |
41807 | 1009 |
ultimately have ?thesis by blast } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1010 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1011 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1012 |
|
41807 | 1013 |
lemma simp_num_pair_l: |
1014 |
assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1015 |
shows "numbound0 t' \<and> n' >0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1016 |
proof- |
41807 | 1017 |
let ?t' = "simpnum t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1018 |
let ?g = "numgcd ?t'" |
31706 | 1019 |
let ?g' = "gcd n ?g" |
41807 | 1020 |
{ assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1021 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1022 |
{ assume nnz: "n \<noteq> 0" |
41807 | 1023 |
{assume "\<not> ?g > 1" hence ?thesis using assms by (auto simp add: Let_def simp_num_pair_def) } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1024 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1025 |
{assume g1:"?g>1" hence g0: "?g > 0" by simp |
31706 | 1026 |
from g1 nnz have gp0: "?g' \<noteq> 0" by simp |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
1027 |
hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1028 |
hence "?g'= 1 \<or> ?g' > 1" by arith |
41807 | 1029 |
moreover {assume "?g'=1" hence ?thesis using assms g1 g0 |
1030 |
by (auto simp add: Let_def simp_num_pair_def) } |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1031 |
moreover {assume g'1:"?g'>1" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1032 |
have gpdg: "?g' dvd ?g" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1033 |
have gpdd: "?g' dvd n" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1034 |
have gpdgp: "?g' dvd ?g'" by simp |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1035 |
from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" . |
47142 | 1036 |
from zdiv_mono1[OF g'n g'p, simplified div_self[OF gp0]] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1037 |
have "n div ?g' >0" by simp |
41807 | 1038 |
hence ?thesis using assms g1 g'1 |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1039 |
by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)} |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
1040 |
ultimately have ?thesis by auto } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1041 |
ultimately have ?thesis by blast } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1042 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1043 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1044 |
|
66809 | 1045 |
fun not:: "fm \<Rightarrow> fm" |
1046 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1047 |
"not (NOT p) = p" |
41839 | 1048 |
| "not T = F" |
1049 |
| "not F = T" |
|
1050 |
| "not (Lt t) = Ge t" |
|
1051 |
| "not (Le t) = Gt t" |
|
1052 |
| "not (Gt t) = Le t" |
|
1053 |
| "not (Ge t) = Lt t" |
|
1054 |
| "not (Eq t) = NEq t" |
|
1055 |
| "not (NEq t) = Eq t" |
|
1056 |
| "not (Dvd i t) = NDvd i t" |
|
1057 |
| "not (NDvd i t) = Dvd i t" |
|
1058 |
| "not (And p q) = Or (not p) (not q)" |
|
1059 |
| "not (Or p q) = And (not p) (not q)" |
|
1060 |
| "not p = NOT p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1061 |
lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" |
41807 | 1062 |
by (induct p) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1063 |
lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)" |
41807 | 1064 |
by (induct p) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1065 |
lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)" |
41807 | 1066 |
by (induct p) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1067 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1068 |
definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1069 |
"conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1070 |
if p = q then p else And p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1071 |
lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" |
41807 | 1072 |
by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1073 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1074 |
lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1075 |
using conj_def by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1076 |
lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1077 |
using conj_def by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1078 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1079 |
definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1080 |
"disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1081 |
else if p=q then p else Or p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1082 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1083 |
lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" |
41807 | 1084 |
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1085 |
lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1086 |
using disj_def by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1087 |
lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1088 |
using disj_def by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1089 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1090 |
definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1091 |
"imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1092 |
else Imp p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1093 |
lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" |
41807 | 1094 |
by (cases "p=F \<or> q=T",simp_all add: imp_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1095 |
lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)" |
41807 | 1096 |
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1097 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1098 |
definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1099 |
"iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1100 |
if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1101 |
Iff p q)" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
1102 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1103 |
lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" |
66809 | 1104 |
by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp) (cases "not p= q", auto) |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61610
diff
changeset
|
1105 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1106 |
lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1107 |
by (unfold iff_def,cases "p=q", auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1108 |
|
66809 | 1109 |
fun check_int:: "num \<Rightarrow> bool" |
1110 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1111 |
"check_int (C i) = True" |
41839 | 1112 |
| "check_int (Floor t) = True" |
1113 |
| "check_int (Mul i t) = check_int t" |
|
1114 |
| "check_int (Add t s) = (check_int t \<and> check_int s)" |
|
1115 |
| "check_int (Neg t) = check_int t" |
|
1116 |
| "check_int (CF c t s) = check_int s" |
|
1117 |
| "check_int t = False" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1118 |
lemma check_int: "check_int t \<Longrightarrow> isint t bs" |
51369 | 1119 |
by (induct t) (auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1120 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1121 |
lemma rdvd_left1_int: "real_of_int \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1122 |
by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1123 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1124 |
lemma rdvd_reduce: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1125 |
assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1126 |
shows "real_of_int (d::int) rdvd real_of_int (c::int)*t = (real_of_int (d div g) rdvd real_of_int (c div g)*t)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1127 |
proof |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1128 |
assume d: "real_of_int d rdvd real_of_int c * t" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1129 |
from d rdvd_def obtain k where k_def: "real_of_int c * t = real_of_int d* real_of_int (k::int)" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1130 |
from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1131 |
from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1132 |
from k_def kd_def kc_def have "real_of_int g * real_of_int kc * t = real_of_int g * real_of_int kd * real_of_int k" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1133 |
hence "real_of_int kc * t = real_of_int kd * real_of_int k" using gp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1134 |
hence th:"real_of_int kd rdvd real_of_int kc * t" using rdvd_def by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1135 |
from kd_def gp have th':"kd = d div g" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1136 |
from kc_def gp have "kc = c div g" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1137 |
with th th' show "real_of_int (d div g) rdvd real_of_int (c div g) * t" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1138 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1139 |
assume d: "real_of_int (d div g) rdvd real_of_int (c div g) * t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1140 |
from gp have gnz: "g \<noteq> 0" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1141 |
thus "real_of_int d rdvd real_of_int c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real_of_int (c div g) * t"] real_of_int_div[OF gd] real_of_int_div[OF gc] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1142 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1143 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1144 |
definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1145 |
"simpdvd d t \<equiv> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1146 |
(let g = numgcd t in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1147 |
if g > 1 then (let g' = gcd d g in |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1148 |
if g' = 1 then (d, t) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1149 |
else (d div g',reducecoeffh t g')) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1150 |
else (d, t))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1151 |
lemma simpdvd: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1152 |
assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1153 |
shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1154 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1155 |
let ?g = "numgcd t" |
31706 | 1156 |
let ?g' = "gcd d ?g" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1157 |
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1158 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1159 |
{assume g1:"?g>1" hence g0: "?g > 0" by simp |
31706 | 1160 |
from g1 dnz have gp0: "?g' \<noteq> 0" by simp |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
1161 |
hence g'p: "?g' > 0" using gcd_ge_0_int[where x="d" and y="numgcd t"] by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1162 |
hence "?g'= 1 \<or> ?g' > 1" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1163 |
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1164 |
moreover {assume g'1:"?g'>1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1165 |
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" .. |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1166 |
let ?tt = "reducecoeffh t ?g'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1167 |
let ?t = "Inum bs ?tt" |
31706 | 1168 |
have gpdg: "?g' dvd ?g" by simp |
1169 |
have gpdd: "?g' dvd d" by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1170 |
have gpdgp: "?g' dvd ?g'" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1171 |
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1172 |
have th2:"real_of_int ?g' * ?t = Inum bs t" by simp |
41807 | 1173 |
from assms g1 g0 g'1 |
1174 |
have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)" |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1175 |
by (simp add: simpdvd_def Let_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1176 |
also have "\<dots> = (real_of_int d rdvd (Inum bs t))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1177 |
using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1178 |
th2[symmetric] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1179 |
finally have ?thesis by simp } |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
1180 |
ultimately have ?thesis by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1181 |
} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1182 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1183 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1184 |
|
66809 | 1185 |
fun simpfm :: "fm \<Rightarrow> fm" |
1186 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1187 |
"simpfm (And p q) = conj (simpfm p) (simpfm q)" |
41839 | 1188 |
| "simpfm (Or p q) = disj (simpfm p) (simpfm q)" |
1189 |
| "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" |
|
1190 |
| "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" |
|
1191 |
| "simpfm (NOT p) = not (simpfm p)" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1192 |
| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F |
66809 | 1193 |
| _ \<Rightarrow> Lt (reducecoeff a'))" |
41839 | 1194 |
| "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0) then T else F | _ \<Rightarrow> Le (reducecoeff a'))" |
1195 |
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt (reducecoeff a'))" |
|
1196 |
| "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0) then T else F | _ \<Rightarrow> Ge (reducecoeff a'))" |
|
1197 |
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq (reducecoeff a'))" |
|
1198 |
| "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0) then T else F | _ \<Rightarrow> NEq (reducecoeff a'))" |
|
1199 |
| "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) |
|
61945 | 1200 |
else if (\<bar>i\<bar> = 1) \<and> check_int a then T |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1201 |
else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1202 |
| "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) |
61945 | 1203 |
else if (\<bar>i\<bar> = 1) \<and> check_int a then F |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1204 |
else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))" |
41839 | 1205 |
| "simpfm p = p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1206 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1207 |
lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1208 |
proof(induct p rule: simpfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1209 |
case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1210 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1211 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1212 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1213 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1214 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1215 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1216 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1217 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1218 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1219 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1220 |
with sa have "Inum bs a < 0 = (real_of_int ?g * ?r < real_of_int ?g * 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1221 |
also have "\<dots> = (?r < 0)" using gp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1222 |
by (simp only: mult_less_cancel_left) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1223 |
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1224 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1225 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1226 |
case (7 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1227 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1228 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1229 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1230 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1231 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1232 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1233 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1234 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1235 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1236 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1237 |
with sa have "Inum bs a \<le> 0 = (real_of_int ?g * ?r \<le> real_of_int ?g * 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1238 |
also have "\<dots> = (?r \<le> 0)" using gp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1239 |
by (simp only: mult_le_cancel_left) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1240 |
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1241 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1242 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1243 |
case (8 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1244 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1245 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1246 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1247 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1248 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1249 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1250 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1251 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1252 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1253 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1254 |
with sa have "Inum bs a > 0 = (real_of_int ?g * ?r > real_of_int ?g * 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1255 |
also have "\<dots> = (?r > 0)" using gp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1256 |
by (simp only: mult_less_cancel_left) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1257 |
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1258 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1259 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1260 |
case (9 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1261 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1262 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1263 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1264 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1265 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1266 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1267 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1268 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1269 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1270 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1271 |
with sa have "Inum bs a \<ge> 0 = (real_of_int ?g * ?r \<ge> real_of_int ?g * 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1272 |
also have "\<dots> = (?r \<ge> 0)" using gp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1273 |
by (simp only: mult_le_cancel_left) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1274 |
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1275 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1276 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1277 |
case (10 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1278 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1279 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1280 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1281 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1282 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1283 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1284 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1285 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1286 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1287 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1288 |
with sa have "Inum bs a = 0 = (real_of_int ?g * ?r = 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1289 |
also have "\<dots> = (?r = 0)" using gp |
51369 | 1290 |
by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1291 |
finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1292 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1293 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1294 |
case (11 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1295 |
{fix v assume "?sa = C v" hence ?case using sa by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1296 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1297 |
let ?g = "numgcd ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1298 |
let ?rsa = "reducecoeff ?sa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1299 |
let ?r = "Inum bs ?rsa" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1300 |
have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1301 |
{assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1302 |
with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1303 |
hence gp: "real_of_int ?g > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1304 |
have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1305 |
with sa have "Inum bs a \<noteq> 0 = (real_of_int ?g * ?r \<noteq> 0)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1306 |
also have "\<dots> = (?r \<noteq> 0)" using gp |
51369 | 1307 |
by simp |
1308 |
finally have ?case using H by (cases "?sa") (simp_all add: Let_def) } |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1309 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1310 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1311 |
case (12 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
61945 | 1312 |
have "i=0 \<or> (\<bar>i\<bar> = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)))" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1313 |
{assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1314 |
moreover |
61945 | 1315 |
{assume ai1: "\<bar>i\<bar> = 1" and ai: "check_int a" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1316 |
hence "i=1 \<or> i= - 1" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1317 |
moreover {assume i1: "i = 1" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1318 |
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1319 |
have ?case using i1 ai by simp } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1320 |
moreover {assume i1: "i = - 1" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1321 |
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1322 |
rdvd_abs1[where d="- 1" and t="Inum bs a"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1323 |
have ?case using i1 ai by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1324 |
ultimately have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1325 |
moreover |
61945 | 1326 |
{assume inz: "i\<noteq>0" and cond: "(\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1327 |
{fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond |
61945 | 1328 |
by (cases "\<bar>i\<bar> = 1", auto simp add: int_rdvd_iff) } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1329 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1330 |
hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1331 |
from simpnum_nz have nz:"nozerocoeff ?sa" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1332 |
from simpdvd [OF nz inz] th have ?case using sa by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1333 |
ultimately have ?case by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1334 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1335 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1336 |
case (13 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp |
61945 | 1337 |
have "i=0 \<or> (\<bar>i\<bar> = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)))" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1338 |
{assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1339 |
moreover |
61945 | 1340 |
{assume ai1: "\<bar>i\<bar> = 1" and ai: "check_int a" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1341 |
hence "i=1 \<or> i= - 1" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1342 |
moreover {assume i1: "i = 1" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1343 |
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1344 |
have ?case using i1 ai by simp } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1345 |
moreover {assume i1: "i = - 1" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1346 |
from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1347 |
rdvd_abs1[where d="- 1" and t="Inum bs a"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1348 |
have ?case using i1 ai by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1349 |
ultimately have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1350 |
moreover |
61945 | 1351 |
{assume inz: "i\<noteq>0" and cond: "(\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1352 |
{fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond |
61945 | 1353 |
by (cases "\<bar>i\<bar> = 1", auto simp add: int_rdvd_iff) } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1354 |
moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1355 |
hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
1356 |
by (cases ?sa, auto simp add: Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1357 |
from simpnum_nz have nz:"nozerocoeff ?sa" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1358 |
from simpdvd [OF nz inz] th have ?case using sa by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1359 |
ultimately have ?case by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1360 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1361 |
qed (induct p rule: simpfm.induct, simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1362 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1363 |
lemma simpdvd_numbound0: "numbound0 t \<Longrightarrow> numbound0 (snd (simpdvd d t))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1364 |
by (simp add: simpdvd_def Let_def split_def reducecoeffh_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1365 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1366 |
lemma simpfm_bound0[simp]: "bound0 p \<Longrightarrow> bound0 (simpfm p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1367 |
proof(induct p rule: simpfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1368 |
case (6 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1369 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1370 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1371 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1372 |
case (7 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1373 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1374 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1375 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1376 |
case (8 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1377 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1378 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1379 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1380 |
case (9 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1381 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1382 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1383 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1384 |
case (10 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1385 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1386 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1387 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1388 |
case (11 a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1389 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1390 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1391 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1392 |
case (12 i a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1393 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1394 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1395 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1396 |
case (13 i a) hence nb: "numbound0 a" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1397 |
hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1398 |
thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1399 |
qed(auto simp add: disj_def imp_def iff_def conj_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1400 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1401 |
lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1402 |
by (induct p rule: simpfm.induct, auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1403 |
(case_tac "simpnum a",auto simp add: split_def Let_def)+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1404 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1405 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1406 |
(* Generic quantifier elimination *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1407 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1408 |
definition list_conj :: "fm list \<Rightarrow> fm" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1409 |
"list_conj ps \<equiv> foldr conj ps T" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1410 |
lemma list_conj: "Ifm bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm bs p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1411 |
by (induct ps, auto simp add: list_conj_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1412 |
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1413 |
by (induct ps, auto simp add: list_conj_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1414 |
lemma list_conj_nb: " \<forall>p\<in> set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1415 |
by (induct ps, auto simp add: list_conj_def) |
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
1416 |
definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where |
29788 | 1417 |
"CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = List.partition bound0 cjs |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1418 |
in conj (decr (list_conj yes)) (f (list_conj no)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1419 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1420 |
lemma CJNB_qe: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1421 |
assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1422 |
shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm bs ((CJNB qe p)) = Ifm bs (E p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1423 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1424 |
fix bs p |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1425 |
assume qfp: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1426 |
let ?cjs = "conjuncts p" |
29788 | 1427 |
let ?yes = "fst (List.partition bound0 ?cjs)" |
1428 |
let ?no = "snd (List.partition bound0 ?cjs)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1429 |
let ?cno = "list_conj ?no" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1430 |
let ?cyes = "list_conj ?yes" |
29788 | 1431 |
have part: "List.partition bound0 ?cjs = (?yes,?no)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1432 |
from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1433 |
hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1434 |
hence yes_qf: "qfree (decr ?cyes )" by (simp add: decr_qf) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1435 |
from conjuncts_qf[OF qfp] partition_set[OF part] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1436 |
have " \<forall>q\<in> set ?no. qfree q" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1437 |
hence no_qf: "qfree ?cno"by (simp add: list_conj_qf) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1438 |
with qe have cno_qf:"qfree (qe ?cno )" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1439 |
and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+ |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1440 |
from cno_qf yes_qf have qf: "qfree (CJNB qe p)" |
51369 | 1441 |
by (simp add: CJNB_def Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1442 |
{fix bs |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1443 |
from conjuncts have "Ifm bs p = (\<forall>q\<in> set ?cjs. Ifm bs q)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1444 |
also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm bs q) \<and> (\<forall>q\<in> set ?no. Ifm bs q))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1445 |
using partition_set[OF part] by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1446 |
finally have "Ifm bs p = ((Ifm bs ?cyes) \<and> (Ifm bs ?cno))" using list_conj by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1447 |
hence "Ifm bs (E p) = (\<exists>x. (Ifm (x#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" by simp |
26932 | 1448 |
also fix y have "\<dots> = (\<exists>x. (Ifm (y#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1449 |
using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1450 |
also have "\<dots> = (Ifm bs (decr ?cyes) \<and> Ifm bs (E ?cno))" |
33639
603320b93668
New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents:
33063
diff
changeset
|
1451 |
by (auto simp add: decr[OF yes_nb] simp del: partition_filter_conv) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1452 |
also have "\<dots> = (Ifm bs (conj (decr ?cyes) (qe ?cno)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1453 |
using qe[rule_format, OF no_qf] by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1454 |
finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1455 |
by (simp add: Let_def CJNB_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1456 |
with qf show "qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1457 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1458 |
|
66809 | 1459 |
fun qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" |
1460 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1461 |
"qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))" |
41839 | 1462 |
| "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))" |
1463 |
| "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1464 |
| "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1465 |
| "qelim (Or p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" |
41839 | 1466 |
| "qelim (Imp p q) = (\<lambda> qe. disj (qelim (NOT p) qe) (qelim q qe))" |
1467 |
| "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))" |
|
1468 |
| "qelim p = (\<lambda> y. simpfm p)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1469 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1470 |
lemma qelim_ci: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1471 |
assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1472 |
shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1473 |
using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] |
41807 | 1474 |
by (induct p rule: qelim.induct) (auto simp del: simpfm.simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1475 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1476 |
|
61586 | 1477 |
text \<open>The \<open>\<int>\<close> Part\<close> |
1478 |
text\<open>Linearity for fm where Bound 0 ranges over \<open>\<int>\<close>\<close> |
|
41839 | 1479 |
|
66809 | 1480 |
fun zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*) |
1481 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1482 |
"zsplit0 (C c) = (0,C c)" |
41839 | 1483 |
| "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" |
1484 |
| "zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)" |
|
1485 |
| "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)" |
|
1486 |
| "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1487 |
| "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1488 |
(ib,b') = zsplit0 b |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1489 |
in (ia+ib, Add a' b'))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1490 |
| "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1491 |
(ib,b') = zsplit0 b |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1492 |
in (ia-ib, Sub a' b'))" |
41839 | 1493 |
| "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" |
1494 |
| "zsplit0 (Floor a) = (let (i',a') = zsplit0 a in (i',Floor a'))" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1495 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1496 |
lemma zsplit0_I: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1497 |
shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real_of_int (x::int)) #bs) (CN 0 n a) = Inum (real_of_int x #bs) t) \<and> numbound0 a" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1498 |
(is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1499 |
proof(induct t rule: zsplit0.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1500 |
case (1 c n a) thus ?case by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1501 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1502 |
case (2 m n a) thus ?case by (cases "m=0") auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1503 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1504 |
case (3 n i a n a') thus ?case by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1505 |
next |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1506 |
case (4 c a b n a') thus ?case by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1507 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1508 |
case (5 t n a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1509 |
let ?nt = "fst (zsplit0 t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1510 |
let ?at = "snd (zsplit0 t)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1511 |
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using 5 |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1512 |
by (simp add: Let_def split_def) |
41891 | 1513 |
from abj 5 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1514 |
from th2[simplified] th[simplified] show ?case by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1515 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1516 |
case (6 s t n a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1517 |
let ?ns = "fst (zsplit0 s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1518 |
let ?as = "snd (zsplit0 s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1519 |
let ?nt = "fst (zsplit0 t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1520 |
let ?at = "snd (zsplit0 t)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1521 |
have abjs: "zsplit0 s = (?ns,?as)" by simp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1522 |
moreover have abjt: "zsplit0 t = (?nt,?at)" by simp |
41891 | 1523 |
ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using 6 |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1524 |
by (simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1525 |
from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1526 |
from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1527 |
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast |
41891 | 1528 |
from abjs 6 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1529 |
from th3[simplified] th2[simplified] th[simplified] show ?case |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49069
diff
changeset
|
1530 |
by (simp add: distrib_right) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1531 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1532 |
case (7 s t n a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1533 |
let ?ns = "fst (zsplit0 s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1534 |
let ?as = "snd (zsplit0 s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1535 |
let ?nt = "fst (zsplit0 t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1536 |
let ?at = "snd (zsplit0 t)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1537 |
have abjs: "zsplit0 s = (?ns,?as)" by simp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1538 |
moreover have abjt: "zsplit0 t = (?nt,?at)" by simp |
41891 | 1539 |
ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using 7 |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1540 |
by (simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1541 |
from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1542 |
from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1543 |
with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast |
41891 | 1544 |
from abjs 7 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1545 |
from th3[simplified] th2[simplified] th[simplified] show ?case |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1546 |
by (simp add: left_diff_distrib) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1547 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1548 |
case (8 i t n a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1549 |
let ?nt = "fst (zsplit0 t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1550 |
let ?at = "snd (zsplit0 t)" |
41891 | 1551 |
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using 8 |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1552 |
by (simp add: Let_def split_def) |
41891 | 1553 |
from abj 8 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1554 |
hence "?I x (Mul i t) = (real_of_int i) * ?I x (CN 0 ?nt ?at)" by simp |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49069
diff
changeset
|
1555 |
also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: distrib_left) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1556 |
finally show ?case using th th2 by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1557 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1558 |
case (9 t n a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1559 |
let ?nt = "fst (zsplit0 t)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1560 |
let ?at = "snd (zsplit0 t)" |
41891 | 1561 |
have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using 9 |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1562 |
by (simp add: Let_def split_def) |
41891 | 1563 |
from abj 9 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1564 |
hence na: "?N a" using th by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1565 |
have th': "(real_of_int ?nt)*(real_of_int x) = real_of_int (?nt * x)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1566 |
have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp |
61942 | 1567 |
also have "\<dots> = real_of_int \<lfloor>real_of_int ?nt * real_of_int x + ?I x ?at\<rfloor>" by simp |
1568 |
also have "\<dots> = real_of_int \<lfloor>?I x ?at + real_of_int (?nt * x)\<rfloor>" by (simp add: ac_simps) |
|
1569 |
also have "\<dots> = real_of_int (\<lfloor>?I x ?at\<rfloor> + (?nt * x))" |
|
63600 | 1570 |
by (simp add: of_int_mult[symmetric] del: of_int_mult) |
61942 | 1571 |
also have "\<dots> = real_of_int (?nt)*(real_of_int x) + real_of_int \<lfloor>?I x ?at\<rfloor>" by (simp add: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1572 |
finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1573 |
with na show ?case by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1574 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1575 |
|
66809 | 1576 |
fun iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool" (* Linearity test for fm *) |
1577 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1578 |
"iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" |
66809 | 1579 |
| "iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" |
1580 |
| "iszlfm (Eq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1581 |
| "iszlfm (NEq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1582 |
| "iszlfm (Lt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1583 |
| "iszlfm (Le (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1584 |
| "iszlfm (Gt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1585 |
| "iszlfm (Ge (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" |
|
1586 |
| "iszlfm (Dvd i (CN 0 c e)) = |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1587 |
(\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)" |
66809 | 1588 |
| "iszlfm (NDvd i (CN 0 c e))= |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1589 |
(\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)" |
66809 | 1590 |
| "iszlfm p = (\<lambda> bs. isatom p \<and> (bound0 p))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1591 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1592 |
lemma zlin_qfree: "iszlfm p bs \<Longrightarrow> qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1593 |
by (induct p rule: iszlfm.induct) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1594 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1595 |
lemma iszlfm_gen: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1596 |
assumes lp: "iszlfm p (x#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1597 |
shows "\<forall> y. iszlfm p (y#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1598 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1599 |
fix y |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1600 |
show "iszlfm p (y#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1601 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1602 |
by(induct p rule: iszlfm.induct, simp_all add: numbound0_gen[rule_format, where x="x" and y="y"]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1603 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1604 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1605 |
lemma conj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (conj p q) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1606 |
using conj_def by (cases p,auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1607 |
lemma disj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (disj p q) bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1608 |
using disj_def by (cases p,auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1609 |
|
66809 | 1610 |
fun zlfm :: "fm \<Rightarrow> fm" (* Linearity transformation for fm *) |
1611 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1612 |
"zlfm (And p q) = conj (zlfm p) (zlfm q)" |
66809 | 1613 |
| "zlfm (Or p q) = disj (zlfm p) (zlfm q)" |
1614 |
| "zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)" |
|
1615 |
| "zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))" |
|
1616 |
| "zlfm (Lt a) = (let (c,r) = zsplit0 a in |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1617 |
if c=0 then Lt r else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1618 |
if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1619 |
else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" |
66809 | 1620 |
| "zlfm (Le a) = (let (c,r) = zsplit0 a in |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1621 |
if c=0 then Le r else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1622 |
if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1623 |
else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" |
66809 | 1624 |
| "zlfm (Gt a) = (let (c,r) = zsplit0 a in |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1625 |
if c=0 then Gt r else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1626 |
if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1627 |
else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" |
66809 | 1628 |
| "zlfm (Ge a) = (let (c,r) = zsplit0 a in |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1629 |
if c=0 then Ge r else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1630 |
if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1631 |
else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" |
66809 | 1632 |
| "zlfm (Eq a) = (let (c,r) = zsplit0 a in |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1633 |
if c=0 then Eq r else |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1634 |
if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1635 |
else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))" |
66809 | 1636 |
| "zlfm (NEq a) = (let (c,r) = zsplit0 a in |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1637 |
if c=0 then NEq r else |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1638 |
if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1639 |
else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))" |
66809 | 1640 |
| "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1641 |
else (let (c,r) = zsplit0 a in |
61945 | 1642 |
if c=0 then Dvd \<bar>i\<bar> r else |
1643 |
if c>0 then And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (CN 0 c (Floor r))) |
|
1644 |
else And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (CN 0 (-c) (Neg (Floor r))))))" |
|
66809 | 1645 |
| "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1646 |
else (let (c,r) = zsplit0 a in |
61945 | 1647 |
if c=0 then NDvd \<bar>i\<bar> r else |
1648 |
if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (CN 0 c (Floor r))) |
|
1649 |
else Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (CN 0 (-c) (Neg (Floor r))))))" |
|
66809 | 1650 |
| "zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))" |
1651 |
| "zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))" |
|
1652 |
| "zlfm (NOT (Imp p q)) = conj (zlfm p) (zlfm (NOT q))" |
|
1653 |
| "zlfm (NOT (Iff p q)) = disj (conj(zlfm p) (zlfm(NOT q))) (conj (zlfm(NOT p)) (zlfm q))" |
|
1654 |
| "zlfm (NOT (NOT p)) = zlfm p" |
|
1655 |
| "zlfm (NOT T) = F" |
|
1656 |
| "zlfm (NOT F) = T" |
|
1657 |
| "zlfm (NOT (Lt a)) = zlfm (Ge a)" |
|
1658 |
| "zlfm (NOT (Le a)) = zlfm (Gt a)" |
|
1659 |
| "zlfm (NOT (Gt a)) = zlfm (Le a)" |
|
1660 |
| "zlfm (NOT (Ge a)) = zlfm (Lt a)" |
|
1661 |
| "zlfm (NOT (Eq a)) = zlfm (NEq a)" |
|
1662 |
| "zlfm (NOT (NEq a)) = zlfm (Eq a)" |
|
1663 |
| "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)" |
|
1664 |
| "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)" |
|
1665 |
| "zlfm p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1666 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1667 |
lemma split_int_less_real: |
61942 | 1668 |
"(real_of_int (a::int) < b) = (a < \<lfloor>b\<rfloor> \<or> (a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1669 |
proof( auto) |
61942 | 1670 |
assume alb: "real_of_int a < b" and agb: "\<not> a < \<lfloor>b\<rfloor>" |
1671 |
from agb have "\<lfloor>b\<rfloor> \<le> a" by simp |
|
1672 |
hence th: "b < real_of_int a + 1" by (simp only: floor_le_iff) |
|
1673 |
from floor_eq[OF alb th] show "a = \<lfloor>b\<rfloor>" by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1674 |
next |
61942 | 1675 |
assume alb: "a < \<lfloor>b\<rfloor>" |
1676 |
hence "real_of_int a < real_of_int \<lfloor>b\<rfloor>" by simp |
|
1677 |
moreover have "real_of_int \<lfloor>b\<rfloor> \<le> b" by simp |
|
1678 |
ultimately show "real_of_int a < b" by arith |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1679 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1680 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1681 |
lemma split_int_less_real': |
61942 | 1682 |
"(real_of_int (a::int) + b < 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> < 0 \<or> (real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1683 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1684 |
have "(real_of_int a + b <0) = (real_of_int a < -b)" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1685 |
with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1686 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1687 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1688 |
lemma split_int_gt_real': |
61942 | 1689 |
"(real_of_int (a::int) + b > 0) = (real_of_int a + real_of_int \<lfloor>b\<rfloor> > 0 \<or> (real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1690 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1691 |
have th: "(real_of_int a + b >0) = (real_of_int (-a) + (-b)< 0)" by arith |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1692 |
show ?thesis |
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
1693 |
by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) (auto simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1694 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1695 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1696 |
lemma split_int_le_real: |
61942 | 1697 |
"(real_of_int (a::int) \<le> b) = (a \<le> \<lfloor>b\<rfloor> \<or> (a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1698 |
proof( auto) |
61942 | 1699 |
assume alb: "real_of_int a \<le> b" and agb: "\<not> a \<le> \<lfloor>b\<rfloor>" |
1700 |
from alb have "\<lfloor>real_of_int a\<rfloor> \<le> \<lfloor>b\<rfloor>" by (simp only: floor_mono) |
|
1701 |
hence "a \<le> \<lfloor>b\<rfloor>" by simp with agb show "False" by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1702 |
next |
61942 | 1703 |
assume alb: "a \<le> \<lfloor>b\<rfloor>" |
1704 |
hence "real_of_int a \<le> real_of_int \<lfloor>b\<rfloor>" by (simp only: floor_mono) |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1705 |
also have "\<dots>\<le> b" by simp finally show "real_of_int a \<le> b" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1706 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1707 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1708 |
lemma split_int_le_real': |
61942 | 1709 |
"(real_of_int (a::int) + b \<le> 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> \<le> 0 \<or> (real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1710 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1711 |
have "(real_of_int a + b \<le>0) = (real_of_int a \<le> -b)" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1712 |
with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1713 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1714 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1715 |
lemma split_int_ge_real': |
61942 | 1716 |
"(real_of_int (a::int) + b \<ge> 0) = (real_of_int a + real_of_int \<lfloor>b\<rfloor> \<ge> 0 \<or> (real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1717 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1718 |
have th: "(real_of_int a + b \<ge>0) = (real_of_int (-a) + (-b) \<le> 0)" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1719 |
show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"]) |
51369 | 1720 |
(simp add: algebra_simps ,arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1721 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1722 |
|
61942 | 1723 |
lemma split_int_eq_real: "(real_of_int (a::int) = b) = ( a = \<lfloor>b\<rfloor> \<and> b = real_of_int \<lfloor>b\<rfloor>)" (is "?l = ?r") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1724 |
by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1725 |
|
61942 | 1726 |
lemma split_int_eq_real': "(real_of_int (a::int) + b = 0) = ( a - \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b = 0)" (is "?l = ?r") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1727 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1728 |
have "?l = (real_of_int a = -b)" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1729 |
with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1730 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1731 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1732 |
lemma zlfm_I: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1733 |
assumes qfp: "qfree p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1734 |
shows "(Ifm (real_of_int i #bs) (zlfm p) = Ifm (real_of_int i# bs) p) \<and> iszlfm (zlfm p) (real_of_int (i::int) #bs)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1735 |
(is "(?I (?l p) = ?I p) \<and> ?L (?l p)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1736 |
using qfp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1737 |
proof(induct p rule: zlfm.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1738 |
case (5 a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1739 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1740 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1741 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1742 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1743 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1744 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1745 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1746 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1747 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1748 |
by (cases "?r", simp_all add: Let_def split_def,rename_tac nat a b,case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1749 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1750 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1751 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1752 |
have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1753 |
also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1754 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1755 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1756 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1757 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1758 |
have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1759 |
also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1760 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1761 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1762 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1763 |
case (6 a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1764 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1765 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1766 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1767 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1768 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1769 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1770 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1771 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1772 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1773 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat",simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1774 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1775 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1776 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1777 |
have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1778 |
also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1779 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1780 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1781 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1782 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1783 |
have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1784 |
also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1785 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1786 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1787 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1788 |
case (7 a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1789 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1790 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1791 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1792 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1793 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1794 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1795 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1796 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1797 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1798 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1799 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1800 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1801 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1802 |
have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1803 |
also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1804 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1805 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1806 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1807 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1808 |
have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1809 |
also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1810 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1811 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1812 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1813 |
case (8 a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1814 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1815 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1816 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1817 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1818 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1819 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1820 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1821 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1822 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1823 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1824 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1825 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1826 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1827 |
have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
1828 |
also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1829 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1830 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1831 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1832 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1833 |
have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1834 |
also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1835 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1836 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1837 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1838 |
case (9 a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1839 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1840 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1841 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1842 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1843 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1844 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1845 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1846 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1847 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1848 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1849 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1850 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1851 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1852 |
have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1853 |
also have "\<dots> = (?I (?l (Eq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1854 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1855 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1856 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1857 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1858 |
have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1859 |
also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1860 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1861 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1862 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1863 |
case (10 a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1864 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1865 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1866 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1867 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1868 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1869 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1870 |
have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1871 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1872 |
{assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] |
58259 | 1873 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1874 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1875 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1876 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1877 |
have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1878 |
also have "\<dots> = (?I (?l (NEq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1879 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1880 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1881 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1882 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1883 |
have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1884 |
also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1885 |
finally have ?case using l by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1886 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1887 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1888 |
case (11 j a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1889 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1890 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1891 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1892 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1893 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1894 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1895 |
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1896 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1897 |
{ assume j: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) |
41891 | 1898 |
hence ?case using 11 j by (simp del: zlfm.simps add: rdvd_left_0_eq)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1899 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1900 |
{assume "?c=0" and "j\<noteq>0" hence ?case |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1901 |
using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] |
58259 | 1902 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1903 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1904 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1905 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1906 |
have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1907 |
using Ia by (simp add: Let_def split_def) |
61945 | 1908 |
also have "\<dots> = (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1909 |
by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp |
61945 | 1910 |
also have "\<dots> = (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> |
61942 | 1911 |
(real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r))))" |
61945 | 1912 |
by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1913 |
also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1914 |
by (simp add: Let_def split_def int_rdvd_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1915 |
del: of_int_mult) (auto simp add: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1916 |
finally have ?case using l jnz by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1917 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1918 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1919 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1920 |
have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1921 |
using Ia by (simp add: Let_def split_def) |
61945 | 1922 |
also have "\<dots> = (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1923 |
by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp |
61945 | 1924 |
also have "\<dots> = (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> |
61942 | 1925 |
(real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r))))" |
61945 | 1926 |
by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1927 |
also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz |
61945 | 1928 |
using rdvd_minus [where d="\<bar>j\<bar>" and t="real_of_int (?c*i + \<lfloor>?N ?r\<rfloor>)", simplified, symmetric] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1929 |
by (simp add: Let_def split_def int_rdvd_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1930 |
del: of_int_mult) (auto simp add: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1931 |
finally have ?case using l jnz by blast } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1932 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1933 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1934 |
case (12 j a) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1935 |
let ?c = "fst (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1936 |
let ?r = "snd (zsplit0 a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1937 |
have spl: "zsplit0 a = (?c,?r)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1938 |
from zsplit0_I[OF spl, where x="i" and bs="bs"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1939 |
have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1940 |
let ?N = "\<lambda> t. Inum (real_of_int i#bs) t" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1941 |
have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1942 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1943 |
{assume j: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) |
41891 | 1944 |
hence ?case using 12 j by (simp del: zlfm.simps add: rdvd_left_0_eq)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1945 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1946 |
{assume "?c=0" and "j\<noteq>0" hence ?case |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1947 |
using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] |
58259 | 1948 |
by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1949 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1950 |
{assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1951 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1952 |
have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1953 |
using Ia by (simp add: Let_def split_def) |
61945 | 1954 |
also have "\<dots> = (\<not> (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r)))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1955 |
by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp |
61945 | 1956 |
also have "\<dots> = (\<not> (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> |
61942 | 1957 |
(real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r)))))" |
61945 | 1958 |
by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1959 |
also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1960 |
by (simp add: Let_def split_def int_rdvd_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1961 |
del: of_int_mult) (auto simp add: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1962 |
finally have ?case using l jnz by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1963 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1964 |
{assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1965 |
by (simp add: nb Let_def split_def isint_Floor isint_neg) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1966 |
have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1967 |
using Ia by (simp add: Let_def split_def) |
61945 | 1968 |
also have "\<dots> = (\<not> (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r)))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1969 |
by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp |
61945 | 1970 |
also have "\<dots> = (\<not> (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> |
61942 | 1971 |
(real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r)))))" |
61945 | 1972 |
by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1973 |
also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz |
61945 | 1974 |
using rdvd_minus [where d="\<bar>j\<bar>" and t="real_of_int (?c*i + \<lfloor>?N ?r\<rfloor>)", simplified, symmetric] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1975 |
by (simp add: Let_def split_def int_rdvd_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1976 |
del: of_int_mult) (auto simp add: ac_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1977 |
finally have ?case using l jnz by blast } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1978 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1979 |
qed auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1980 |
|
61586 | 1981 |
text\<open>plusinf : Virtual substitution of \<open>+\<infinity>\<close> |
1982 |
minusinf: Virtual substitution of \<open>-\<infinity>\<close> |
|
1983 |
\<open>\<delta>\<close> Compute lcm \<open>d| Dvd d c*x+t \<in> p\<close> |
|
1984 |
\<open>d_\<delta>\<close> checks if a given l divides all the ds above\<close> |
|
23316 | 1985 |
|
66809 | 1986 |
fun minusinf:: "fm \<Rightarrow> fm" |
1987 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1988 |
"minusinf (And p q) = conj (minusinf p) (minusinf q)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
1989 |
| "minusinf (Or p q) = disj (minusinf p) (minusinf q)" |
41839 | 1990 |
| "minusinf (Eq (CN 0 c e)) = F" |
1991 |
| "minusinf (NEq (CN 0 c e)) = T" |
|
1992 |
| "minusinf (Lt (CN 0 c e)) = T" |
|
1993 |
| "minusinf (Le (CN 0 c e)) = T" |
|
1994 |
| "minusinf (Gt (CN 0 c e)) = F" |
|
1995 |
| "minusinf (Ge (CN 0 c e)) = F" |
|
1996 |
| "minusinf p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1997 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1998 |
lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
1999 |
by (induct p rule: minusinf.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2000 |
|
66809 | 2001 |
fun plusinf:: "fm \<Rightarrow> fm" |
2002 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2003 |
"plusinf (And p q) = conj (plusinf p) (plusinf q)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2004 |
| "plusinf (Or p q) = disj (plusinf p) (plusinf q)" |
41839 | 2005 |
| "plusinf (Eq (CN 0 c e)) = F" |
2006 |
| "plusinf (NEq (CN 0 c e)) = T" |
|
2007 |
| "plusinf (Lt (CN 0 c e)) = F" |
|
2008 |
| "plusinf (Le (CN 0 c e)) = F" |
|
2009 |
| "plusinf (Gt (CN 0 c e)) = T" |
|
2010 |
| "plusinf (Ge (CN 0 c e)) = T" |
|
2011 |
| "plusinf p = p" |
|
2012 |
||
66809 | 2013 |
fun \<delta> :: "fm \<Rightarrow> int" |
2014 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2015 |
"\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2016 |
| "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)" |
41839 | 2017 |
| "\<delta> (Dvd i (CN 0 c e)) = i" |
2018 |
| "\<delta> (NDvd i (CN 0 c e)) = i" |
|
2019 |
| "\<delta> p = 1" |
|
2020 |
||
66809 | 2021 |
fun d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" |
2022 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2023 |
"d_\<delta> (And p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2024 |
| "d_\<delta> (Or p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)" |
50252 | 2025 |
| "d_\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)" |
2026 |
| "d_\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)" |
|
2027 |
| "d_\<delta> p = (\<lambda> d. True)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2028 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2029 |
lemma delta_mono: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2030 |
assumes lin: "iszlfm p bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2031 |
and d: "d dvd d'" |
50252 | 2032 |
and ad: "d_\<delta> p d" |
2033 |
shows "d_\<delta> p d'" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2034 |
using lin ad d |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2035 |
proof(induct p rule: iszlfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2036 |
case (9 i c e) thus ?case using d |
30042 | 2037 |
by (simp add: dvd_trans[of "i" "d" "d'"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2038 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2039 |
case (10 i c e) thus ?case using d |
30042 | 2040 |
by (simp add: dvd_trans[of "i" "d" "d'"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2041 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2042 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2043 |
lemma \<delta> : assumes lin:"iszlfm p bs" |
50252 | 2044 |
shows "d_\<delta> p (\<delta> p) \<and> \<delta> p >0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2045 |
using lin |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2046 |
proof (induct p rule: iszlfm.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2047 |
case (1 p q) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2048 |
let ?d = "\<delta> (And p q)" |
41891 | 2049 |
from 1 lcm_pos_int have dp: "?d >0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2050 |
have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2051 |
hence th: "d_\<delta> p ?d" |
41891 | 2052 |
using delta_mono 1 by (simp only: iszlfm.simps) blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2053 |
have "\<delta> q dvd \<delta> (And p q)" using 1 by simp |
50252 | 2054 |
hence th': "d_\<delta> q ?d" using delta_mono 1 by (simp only: iszlfm.simps) blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2055 |
from th th' dp show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2056 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2057 |
case (2 p q) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2058 |
let ?d = "\<delta> (And p q)" |
41891 | 2059 |
from 2 lcm_pos_int have dp: "?d >0" by simp |
2060 |
have "\<delta> p dvd \<delta> (And p q)" using 2 by simp |
|
50252 | 2061 |
hence th: "d_\<delta> p ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast |
41891 | 2062 |
have "\<delta> q dvd \<delta> (And p q)" using 2 by simp |
50252 | 2063 |
hence th': "d_\<delta> q ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast |
31730 | 2064 |
from th th' dp show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2065 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2066 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2067 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2068 |
lemma minusinf_inf: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2069 |
assumes linp: "iszlfm p (a # bs)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2070 |
shows "\<exists> (z::int). \<forall> x < z. Ifm ((real_of_int x)#bs) (minusinf p) = Ifm ((real_of_int x)#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2071 |
(is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2072 |
using linp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2073 |
proof (induct p rule: minusinf.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2074 |
case (1 f g) |
41891 | 2075 |
then have "?P f" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2076 |
then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast |
41891 | 2077 |
with 1 have "?P g" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2078 |
then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2079 |
let ?z = "min z1 z2" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2080 |
from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2081 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2082 |
next |
41891 | 2083 |
case (2 f g) |
2084 |
then have "?P f" by simp |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2085 |
then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast |
41891 | 2086 |
with 2 have "?P g" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2087 |
then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2088 |
let ?z = "min z1 z2" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2089 |
from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2090 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2091 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2092 |
case (3 c e) |
41891 | 2093 |
then have "c > 0" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2094 |
hence rcpos: "real_of_int c > 0" by simp |
41891 | 2095 |
from 3 have nbe: "numbound0 e" by simp |
26932 | 2096 |
fix y |
61942 | 2097 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2098 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2099 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2100 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2101 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2102 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2103 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2104 |
hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2105 |
thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2106 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2107 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2108 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2109 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2110 |
case (4 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2111 |
then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp |
41891 | 2112 |
from 4 have nbe: "numbound0 e" by simp |
26932 | 2113 |
fix y |
61942 | 2114 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2115 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2116 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2117 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2118 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2119 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2120 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2121 |
hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2122 |
thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2123 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2124 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2125 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2126 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2127 |
case (5 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2128 |
then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp |
41891 | 2129 |
from 5 have nbe: "numbound0 e" by simp |
26932 | 2130 |
fix y |
61942 | 2131 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2132 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2133 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2134 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2135 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2136 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2137 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2138 |
thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2139 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2140 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2141 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2142 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2143 |
case (6 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2144 |
then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp |
41891 | 2145 |
from 6 have nbe: "numbound0 e" by simp |
26932 | 2146 |
fix y |
61942 | 2147 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2148 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2149 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2150 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2151 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2152 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2153 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2154 |
thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2155 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2156 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2157 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2158 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2159 |
case (7 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2160 |
then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp |
41891 | 2161 |
from 7 have nbe: "numbound0 e" by simp |
26932 | 2162 |
fix y |
61942 | 2163 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2164 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2165 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2166 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2167 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2168 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2169 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2170 |
thus "\<not> (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e>0)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2171 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2172 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2173 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2174 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2175 |
case (8 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2176 |
then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp |
41891 | 2177 |
from 8 have nbe: "numbound0 e" by simp |
26932 | 2178 |
fix y |
61942 | 2179 |
have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2180 |
proof (simp add: less_floor_iff , rule allI, rule impI) |
51369 | 2181 |
fix x :: int |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2182 |
assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2183 |
hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2184 |
with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36531
diff
changeset
|
2185 |
by (simp only: mult_strict_left_mono [OF th1 rcpos]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2186 |
thus "\<not> real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2187 |
using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2188 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2189 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2190 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2191 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2192 |
lemma minusinf_repeats: |
50252 | 2193 |
assumes d: "d_\<delta> p d" and linp: "iszlfm p (a # bs)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2194 |
shows "Ifm ((real_of_int(x - k*d))#bs) (minusinf p) = Ifm (real_of_int x #bs) (minusinf p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2195 |
using linp d |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2196 |
proof(induct p rule: iszlfm.induct) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2197 |
case (9 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2198 |
hence "\<exists> k. d=i*k" by (simp add: dvd_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2199 |
then obtain "di" where di_def: "d=i*di" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2200 |
show ?case |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2201 |
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2202 |
assume |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2203 |
"real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2204 |
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2205 |
hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2206 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2207 |
by (simp add: algebra_simps di_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2208 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2209 |
by (simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2210 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2211 |
thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2212 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2213 |
assume |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2214 |
"real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e") |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2215 |
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" by (simp add: rdvd_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2216 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2217 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)" by (simp add: di_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2218 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2219 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2220 |
by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2221 |
thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" using rdvd_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2222 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2223 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2224 |
case (10 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2225 |
hence "\<exists> k. d=i*k" by (simp add: dvd_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2226 |
then obtain "di" where di_def: "d=i*di" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2227 |
show ?case |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2228 |
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2229 |
assume |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2230 |
"real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2231 |
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2232 |
hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2233 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2234 |
by (simp add: algebra_simps di_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2235 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2236 |
by (simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2237 |
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2238 |
thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2239 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2240 |
assume |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2241 |
"real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e") |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2242 |
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" |
51369 | 2243 |
by (simp add: rdvd_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2244 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" |
51369 | 2245 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2246 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)" |
51369 | 2247 |
by (simp add: di_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2248 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))" |
51369 | 2249 |
by (simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2250 |
hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2251 |
by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2252 |
thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" |
51369 | 2253 |
using rdvd_def by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2254 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2255 |
qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int(x - k*d)" and b'="real_of_int x"] simp del: of_int_mult of_int_diff) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2256 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2257 |
lemma minusinf_ex: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2258 |
assumes lin: "iszlfm p (real_of_int (a::int) #bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2259 |
and exmi: "\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)" (is "\<exists> x. ?P1 x") |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2260 |
shows "\<exists> (x::int). Ifm (real_of_int x#bs) p" (is "\<exists> x. ?P x") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2261 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2262 |
let ?d = "\<delta> p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2263 |
from \<delta> [OF lin] have dpos: "?d >0" by simp |
50252 | 2264 |
from \<delta> [OF lin] have alld: "d_\<delta> p ?d" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2265 |
from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P1 x = ?P1 (x - (k * ?d))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2266 |
from minusinf_inf[OF lin] have th2:"\<exists> z. \<forall> x. x<z \<longrightarrow> (?P x = ?P1 x)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2267 |
from minusinfinity [OF dpos th1 th2] exmi show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2268 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2269 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2270 |
lemma minusinf_bex: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2271 |
assumes lin: "iszlfm p (real_of_int (a::int) #bs)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2272 |
shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)) = |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2273 |
(\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real_of_int x#bs) (minusinf p))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2274 |
(is "(\<exists> x. ?P x) = _") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2275 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2276 |
let ?d = "\<delta> p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2277 |
from \<delta> [OF lin] have dpos: "?d >0" by simp |
50252 | 2278 |
from \<delta> [OF lin] have alld: "d_\<delta> p ?d" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2279 |
from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P x = ?P (x - (k * ?d))" by simp |
23316 | 2280 |
from periodic_finite_ex[OF dpos th1] show ?thesis by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2281 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2282 |
|
66809 | 2283 |
lemma dvd1_eq1: "x > 0 \<Longrightarrow> is_unit x \<longleftrightarrow> x = 1" for x :: int |
2284 |
by simp |
|
2285 |
||
2286 |
fun a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coefficients of a formula *) |
|
2287 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2288 |
"a_\<beta> (And p q) = (\<lambda> k. And (a_\<beta> p k) (a_\<beta> q k))" |
66809 | 2289 |
| "a_\<beta> (Or p q) = (\<lambda> k. Or (a_\<beta> p k) (a_\<beta> q k))" |
2290 |
| "a_\<beta> (Eq (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))" |
|
2291 |
| "a_\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))" |
|
2292 |
| "a_\<beta> (Lt (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))" |
|
2293 |
| "a_\<beta> (Le (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))" |
|
2294 |
| "a_\<beta> (Gt (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))" |
|
2295 |
| "a_\<beta> (Ge (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))" |
|
2296 |
| "a_\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" |
|
2297 |
| "a_\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" |
|
2298 |
| "a_\<beta> p = (\<lambda> k. p)" |
|
2299 |
||
2300 |
fun d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*) |
|
2301 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2302 |
"d_\<beta> (And p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))" |
66809 | 2303 |
| "d_\<beta> (Or p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))" |
2304 |
| "d_\<beta> (Eq (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2305 |
| "d_\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2306 |
| "d_\<beta> (Lt (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2307 |
| "d_\<beta> (Le (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2308 |
| "d_\<beta> (Gt (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2309 |
| "d_\<beta> (Ge (CN 0 c e)) = (\<lambda> k. c dvd k)" |
|
2310 |
| "d_\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)" |
|
2311 |
| "d_\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)" |
|
2312 |
| "d_\<beta> p = (\<lambda> k. True)" |
|
2313 |
||
2314 |
fun \<zeta> :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*) |
|
2315 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2316 |
"\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)" |
66809 | 2317 |
| "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)" |
2318 |
| "\<zeta> (Eq (CN 0 c e)) = c" |
|
2319 |
| "\<zeta> (NEq (CN 0 c e)) = c" |
|
2320 |
| "\<zeta> (Lt (CN 0 c e)) = c" |
|
2321 |
| "\<zeta> (Le (CN 0 c e)) = c" |
|
2322 |
| "\<zeta> (Gt (CN 0 c e)) = c" |
|
2323 |
| "\<zeta> (Ge (CN 0 c e)) = c" |
|
2324 |
| "\<zeta> (Dvd i (CN 0 c e)) = c" |
|
2325 |
| "\<zeta> (NDvd i (CN 0 c e))= c" |
|
2326 |
| "\<zeta> p = 1" |
|
2327 |
||
2328 |
fun \<beta> :: "fm \<Rightarrow> num list" |
|
2329 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2330 |
"\<beta> (And p q) = (\<beta> p @ \<beta> q)" |
66809 | 2331 |
| "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" |
2332 |
| "\<beta> (Eq (CN 0 c e)) = [Sub (C (- 1)) e]" |
|
2333 |
| "\<beta> (NEq (CN 0 c e)) = [Neg e]" |
|
2334 |
| "\<beta> (Lt (CN 0 c e)) = []" |
|
2335 |
| "\<beta> (Le (CN 0 c e)) = []" |
|
2336 |
| "\<beta> (Gt (CN 0 c e)) = [Neg e]" |
|
2337 |
| "\<beta> (Ge (CN 0 c e)) = [Sub (C (- 1)) e]" |
|
2338 |
| "\<beta> p = []" |
|
2339 |
||
2340 |
fun \<alpha> :: "fm \<Rightarrow> num list" |
|
2341 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2342 |
"\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" |
66809 | 2343 |
| "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" |
2344 |
| "\<alpha> (Eq (CN 0 c e)) = [Add (C (- 1)) e]" |
|
2345 |
| "\<alpha> (NEq (CN 0 c e)) = [e]" |
|
2346 |
| "\<alpha> (Lt (CN 0 c e)) = [e]" |
|
2347 |
| "\<alpha> (Le (CN 0 c e)) = [Add (C (- 1)) e]" |
|
2348 |
| "\<alpha> (Gt (CN 0 c e)) = []" |
|
2349 |
| "\<alpha> (Ge (CN 0 c e)) = []" |
|
2350 |
| "\<alpha> p = []" |
|
2351 |
||
2352 |
fun mirror :: "fm \<Rightarrow> fm" |
|
2353 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2354 |
"mirror (And p q) = And (mirror p) (mirror q)" |
66809 | 2355 |
| "mirror (Or p q) = Or (mirror p) (mirror q)" |
2356 |
| "mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))" |
|
2357 |
| "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))" |
|
2358 |
| "mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))" |
|
2359 |
| "mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))" |
|
2360 |
| "mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))" |
|
2361 |
| "mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))" |
|
2362 |
| "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))" |
|
2363 |
| "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))" |
|
2364 |
| "mirror p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2365 |
|
50252 | 2366 |
lemma mirror_\<alpha>_\<beta>: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2367 |
assumes lp: "iszlfm p (a#bs)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2368 |
shows "(Inum (real_of_int (i::int)#bs)) ` set (\<alpha> p) = (Inum (real_of_int i#bs)) ` set (\<beta> (mirror p))" |
51369 | 2369 |
using lp by (induct p rule: mirror.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2370 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2371 |
lemma mirror: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2372 |
assumes lp: "iszlfm p (a#bs)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2373 |
shows "Ifm (real_of_int (x::int)#bs) (mirror p) = Ifm (real_of_int (- x)#bs) p" |
51369 | 2374 |
using lp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2375 |
proof(induct p rule: iszlfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2376 |
case (9 j c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2377 |
have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2378 |
(real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2379 |
by (simp only: rdvd_minus[symmetric]) |
41891 | 2380 |
from 9 th show ?case |
29667 | 2381 |
by (simp add: algebra_simps |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2382 |
numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2383 |
next |
41891 | 2384 |
case (10 j c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2385 |
have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2386 |
(real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2387 |
by (simp only: rdvd_minus[symmetric]) |
41891 | 2388 |
from 10 th show ?case |
29667 | 2389 |
by (simp add: algebra_simps |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2390 |
numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"]) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2391 |
qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int x" and b'="- real_of_int x"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2392 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2393 |
lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)" |
51369 | 2394 |
by (induct p rule: mirror.induct) (auto simp add: isint_neg) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2395 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2396 |
lemma mirror_d_\<beta>: "iszlfm p (a#bs) \<and> d_\<beta> p 1 |
50252 | 2397 |
\<Longrightarrow> iszlfm (mirror p) (a#bs) \<and> d_\<beta> (mirror p) 1" |
51369 | 2398 |
by (induct p rule: mirror.induct) (auto simp add: isint_neg) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2399 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2400 |
lemma mirror_\<delta>: "iszlfm p (a#bs) \<Longrightarrow> \<delta> (mirror p) = \<delta> p" |
51369 | 2401 |
by (induct p rule: mirror.induct) auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2402 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2403 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2404 |
lemma mirror_ex: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2405 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2406 |
shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2407 |
(is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2408 |
proof(auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2409 |
fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2410 |
thus "\<exists> x. ?I x p" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2411 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2412 |
fix x assume "?I x p" hence "?I (- x) ?mp" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2413 |
using mirror[OF lp, where x="- x", symmetric] by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2414 |
thus "\<exists> x. ?I x ?mp" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2415 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2416 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2417 |
lemma \<beta>_numbound0: assumes lp: "iszlfm p bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2418 |
shows "\<forall> b\<in> set (\<beta> p). numbound0 b" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2419 |
using lp by (induct p rule: \<beta>.induct,auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2420 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2421 |
lemma d_\<beta>_mono: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2422 |
assumes linp: "iszlfm p (a #bs)" |
50252 | 2423 |
and dr: "d_\<beta> p l" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2424 |
and d: "l dvd l'" |
50252 | 2425 |
shows "d_\<beta> p l'" |
30042 | 2426 |
using dr linp dvd_trans[of _ "l" "l'", simplified d] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2427 |
by (induct p rule: iszlfm.induct) simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2428 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2429 |
lemma \<alpha>_l: assumes lp: "iszlfm p (a#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2430 |
shows "\<forall> b\<in> set (\<alpha> p). numbound0 b \<and> isint b (a#bs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2431 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2432 |
by(induct p rule: \<alpha>.induct, auto simp add: isint_add isint_c) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2433 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2434 |
lemma \<zeta>: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2435 |
assumes linp: "iszlfm p (a #bs)" |
50252 | 2436 |
shows "\<zeta> p > 0 \<and> d_\<beta> p (\<zeta> p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2437 |
using linp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2438 |
proof(induct p rule: iszlfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2439 |
case (1 p q) |
41891 | 2440 |
then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp |
2441 |
from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2442 |
from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2443 |
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
2444 |
dl1 dl2 show ?case by (auto simp add: lcm_pos_int) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2445 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2446 |
case (2 p q) |
41891 | 2447 |
then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp |
2448 |
from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2449 |
from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2450 |
d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] |
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
2451 |
dl1 dl2 show ?case by (auto simp add: lcm_pos_int) |
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31730
diff
changeset
|
2452 |
qed (auto simp add: lcm_pos_int) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2453 |
|
50252 | 2454 |
lemma a_\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d_\<beta> p l" and lp: "l > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2455 |
shows "iszlfm (a_\<beta> p l) (a #bs) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> (Ifm (real_of_int (l * x) #bs) (a_\<beta> p l) = Ifm ((real_of_int x)#bs) p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2456 |
using linp d |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2457 |
proof (induct p rule: iszlfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2458 |
case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2459 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2460 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2461 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2462 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2463 |
then have ldcp:"0 < l div c" |
47142 | 2464 |
by (simp add: div_self[OF cnz]) |
30042 | 2465 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2466 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2467 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2468 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2469 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2470 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2471 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) < (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2472 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2473 |
using mult_less_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2474 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2475 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2476 |
case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2477 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2478 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2479 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2480 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2481 |
then have ldcp:"0 < l div c" |
47142 | 2482 |
by (simp add: div_self[OF cnz]) |
30042 | 2483 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2484 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2485 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2486 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2487 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2488 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2489 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<le> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2490 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2491 |
using mult_le_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2492 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2493 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2494 |
case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2495 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2496 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2497 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2498 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2499 |
then have ldcp:"0 < l div c" |
47142 | 2500 |
by (simp add: div_self[OF cnz]) |
30042 | 2501 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2502 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2503 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2504 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2505 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2506 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2507 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) > (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2508 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e > 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2509 |
using zero_less_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2510 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2511 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2512 |
case (8 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2513 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2514 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2515 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2516 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2517 |
then have ldcp:"0 < l div c" |
47142 | 2518 |
by (simp add: div_self[OF cnz]) |
30042 | 2519 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2520 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2521 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2522 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2523 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2524 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2525 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<ge> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2526 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2527 |
using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2528 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2529 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2530 |
case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2531 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2532 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2533 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2534 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2535 |
then have ldcp:"0 < l div c" |
47142 | 2536 |
by (simp add: div_self[OF cnz]) |
30042 | 2537 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2538 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2539 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2540 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2541 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2542 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2543 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) = (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2544 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2545 |
using mult_eq_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2546 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2547 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2548 |
case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2549 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2550 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2551 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2552 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2553 |
then have ldcp:"0 < l div c" |
47142 | 2554 |
by (simp add: div_self[OF cnz]) |
30042 | 2555 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2556 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2557 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2558 |
hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> (0::real)) = |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2559 |
(real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2560 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2561 |
also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<noteq> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2562 |
also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2563 |
using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2564 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] be isint_Mul[OF ei] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2565 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2566 |
case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2567 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2568 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2569 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2570 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2571 |
then have ldcp:"0 < l div c" |
47142 | 2572 |
by (simp add: div_self[OF cnz]) |
30042 | 2573 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2574 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2575 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2576 |
hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2577 |
also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2578 |
also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2579 |
using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2580 |
also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2581 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2582 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2583 |
case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2584 |
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2585 |
from cp have cnz: "c \<noteq> 0" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2586 |
have "c div c\<le> l div c" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2587 |
by (simp add: zdiv_mono1[OF clel cp]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2588 |
then have ldcp:"0 < l div c" |
47142 | 2589 |
by (simp add: div_self[OF cnz]) |
30042 | 2590 |
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp |
64246 | 2591 |
hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2592 |
by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2593 |
hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2594 |
also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2595 |
also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2596 |
using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2597 |
also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2598 |
finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2599 |
qed (simp_all add: numbound0_I[where bs="bs" and b="real_of_int (l * x)" and b'="real_of_int x"] isint_Mul del: of_int_mult) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2600 |
|
50252 | 2601 |
lemma a_\<beta>_ex: assumes linp: "iszlfm p (a#bs)" and d: "d_\<beta> p l" and lp: "l>0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2602 |
shows "(\<exists> x. l dvd x \<and> Ifm (real_of_int x #bs) (a_\<beta> p l)) = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2603 |
(is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2604 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2605 |
have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2606 |
using unity_coeff_ex[where l="l" and P="?P", simplified] by simp |
50252 | 2607 |
also have "\<dots> = (\<exists> (x::int). ?P' x)" using a_\<beta>[OF linp d lp] by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2608 |
finally show ?thesis . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2609 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2610 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2611 |
lemma \<beta>: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2612 |
assumes lp: "iszlfm p (a#bs)" |
50252 | 2613 |
and u: "d_\<beta> p 1" |
2614 |
and d: "d_\<delta> p d" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2615 |
and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2616 |
and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real_of_int x = b + real_of_int j)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2617 |
and p: "Ifm (real_of_int x#bs) p" (is "?P x") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2618 |
shows "?P (x - d)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2619 |
using lp u d dp nob p |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2620 |
proof(induct p rule: iszlfm.induct) |
41891 | 2621 |
case (5 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2622 |
with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 5 |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2623 |
show ?case by (simp del: of_int_minus) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2624 |
next |
41891 | 2625 |
case (6 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2626 |
with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 6 |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2627 |
show ?case by (simp del: of_int_minus) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2628 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2629 |
case (7 c e) hence p: "Ifm (real_of_int x #bs) (Gt (CN 0 c e))" and c1: "c=1" |
41891 | 2630 |
and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp_all |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2631 |
let ?e = "Inum (real_of_int x # bs) e" |
61942 | 2632 |
from ie1 have ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="a#bs"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2633 |
numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"] |
41891 | 2634 |
by (simp add: isint_iff) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2635 |
{assume "real_of_int (x-d) +?e > 0" hence ?case using c1 |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2636 |
numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2637 |
by (simp del: of_int_minus)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2638 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2639 |
{assume H: "\<not> real_of_int (x-d) + ?e > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2640 |
let ?v="Neg e" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2641 |
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2642 |
from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2643 |
have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x = - ?e + real_of_int j)" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2644 |
from H p have "real_of_int x + ?e > 0 \<and> real_of_int x + ?e \<le> real_of_int d" by (simp add: c1) |
61942 | 2645 |
hence "real_of_int (x + \<lfloor>?e\<rfloor>) > real_of_int (0::int) \<and> real_of_int (x + \<lfloor>?e\<rfloor>) \<le> real_of_int d" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2646 |
using ie by simp |
61942 | 2647 |
hence "x + \<lfloor>?e\<rfloor> \<ge> 1 \<and> x + \<lfloor>?e\<rfloor> \<le> d" by simp |
2648 |
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + \<lfloor>?e\<rfloor>" by simp |
|
2649 |
hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = real_of_int (- \<lfloor>?e\<rfloor> + j)" by force |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2650 |
hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = - ?e + real_of_int j" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2651 |
by (simp add: ie[simplified isint_iff]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2652 |
with nob have ?case by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2653 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2654 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2655 |
case (8 c e) hence p: "Ifm (real_of_int x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2656 |
and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2657 |
let ?e = "Inum (real_of_int x # bs) e" |
61942 | 2658 |
from ie1 have ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2659 |
by (simp add: isint_iff) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2660 |
{assume "real_of_int (x-d) +?e \<ge> 0" hence ?case using c1 |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2661 |
numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2662 |
by (simp del: of_int_minus)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2663 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2664 |
{assume H: "\<not> real_of_int (x-d) + ?e \<ge> 0" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2665 |
let ?v="Sub (C (- 1)) e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2666 |
have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2667 |
from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2668 |
have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x = - ?e - 1 + real_of_int j)" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2669 |
from H p have "real_of_int x + ?e \<ge> 0 \<and> real_of_int x + ?e < real_of_int d" by (simp add: c1) |
61942 | 2670 |
hence "real_of_int (x + \<lfloor>?e\<rfloor>) \<ge> real_of_int (0::int) \<and> real_of_int (x + \<lfloor>?e\<rfloor>) < real_of_int d" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2671 |
using ie by simp |
61942 | 2672 |
hence "x + \<lfloor>?e\<rfloor> + 1 \<ge> 1 \<and> x + \<lfloor>?e\<rfloor> + 1 \<le> d" by simp |
2673 |
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + \<lfloor>?e\<rfloor> + 1" by simp |
|
2674 |
hence "\<exists> (j::int) \<in> {1 .. d}. x= - \<lfloor>?e\<rfloor> - 1 + j" by (simp add: algebra_simps) |
|
2675 |
hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= real_of_int (- \<lfloor>?e\<rfloor> - 1 + j)" by presburger |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2676 |
hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= - ?e - 1 + real_of_int j" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2677 |
by (simp add: ie[simplified isint_iff]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2678 |
with nob have ?case by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2679 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2680 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2681 |
case (3 c e) hence p: "Ifm (real_of_int x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2682 |
and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2683 |
let ?e = "Inum (real_of_int x # bs) e" |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
2684 |
let ?v="(Sub (C (- 1)) e)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2685 |
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2686 |
from p have "real_of_int x= - ?e" by (simp add: c1) with 3(5) show ?case using dp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2687 |
by simp (erule ballE[where x="1"], |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2688 |
simp_all add:algebra_simps numbound0_I[OF bn,where b="real_of_int x"and b'="a"and bs="bs"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2689 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2690 |
case (4 c e)hence p: "Ifm (real_of_int x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2691 |
and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2692 |
let ?e = "Inum (real_of_int x # bs) e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2693 |
let ?v="Neg e" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2694 |
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2695 |
{assume "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e \<noteq> 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2696 |
hence ?case by (simp add: c1)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2697 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2698 |
{assume H: "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e = 0" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2699 |
hence "real_of_int x = - Inum ((real_of_int (x -d)) # bs) e + real_of_int d" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2700 |
hence "real_of_int x = - Inum (a # bs) e + real_of_int d" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2701 |
by (simp add: numbound0_I[OF bn,where b="real_of_int x - real_of_int d"and b'="a"and bs="bs"]) |
41891 | 2702 |
with 4(5) have ?case using dp by simp} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2703 |
ultimately show ?case by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2704 |
next |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2705 |
case (9 j c e) hence p: "Ifm (real_of_int x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2706 |
and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2707 |
let ?e = "Inum (real_of_int x # bs) e" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2708 |
from 9 have "isint e (a #bs)" by simp |
61942 | 2709 |
hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="(real_of_int x)#bs"] numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] |
41891 | 2710 |
by (simp add: isint_iff) |
2711 |
from 9 have id: "j dvd d" by simp |
|
61942 | 2712 |
from c1 ie[symmetric] have "?p x = (real_of_int j rdvd real_of_int (x + \<lfloor>?e\<rfloor>))" by simp |
2713 |
also have "\<dots> = (j dvd x + \<lfloor>?e\<rfloor>)" |
|
2714 |
using int_rdvd_real[where i="j" and x="real_of_int (x + \<lfloor>?e\<rfloor>)"] by simp |
|
2715 |
also have "\<dots> = (j dvd x - d + \<lfloor>?e\<rfloor>)" |
|
2716 |
using dvd_period[OF id, where x="x" and c="-1" and t="\<lfloor>?e\<rfloor>"] by simp |
|
2717 |
also have "\<dots> = (real_of_int j rdvd real_of_int (x - d + \<lfloor>?e\<rfloor>))" |
|
2718 |
using int_rdvd_real[where i="j" and x="real_of_int (x - d + \<lfloor>?e\<rfloor>)",symmetric, simplified] |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2719 |
ie by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2720 |
also have "\<dots> = (real_of_int j rdvd real_of_int x - real_of_int d + ?e)" |
41891 | 2721 |
using ie by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2722 |
finally show ?case |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2723 |
using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2724 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2725 |
case (10 j c e) hence p: "Ifm (real_of_int x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2726 |
let ?e = "Inum (real_of_int x # bs) e" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2727 |
from 10 have "isint e (a#bs)" by simp |
61942 | 2728 |
hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"] |
41891 | 2729 |
by (simp add: isint_iff) |
2730 |
from 10 have id: "j dvd d" by simp |
|
61942 | 2731 |
from c1 ie[symmetric] have "?p x = (\<not> real_of_int j rdvd real_of_int (x + \<lfloor>?e\<rfloor>))" by simp |
2732 |
also have "\<dots> = (\<not> j dvd x + \<lfloor>?e\<rfloor>)" |
|
2733 |
using int_rdvd_real[where i="j" and x="real_of_int (x + \<lfloor>?e\<rfloor>)"] by simp |
|
2734 |
also have "\<dots> = (\<not> j dvd x - d + \<lfloor>?e\<rfloor>)" |
|
2735 |
using dvd_period[OF id, where x="x" and c="-1" and t="\<lfloor>?e\<rfloor>"] by simp |
|
2736 |
also have "\<dots> = (\<not> real_of_int j rdvd real_of_int (x - d + \<lfloor>?e\<rfloor>))" |
|
2737 |
using int_rdvd_real[where i="j" and x="real_of_int (x - d + \<lfloor>?e\<rfloor>)",symmetric, simplified] |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2738 |
ie by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2739 |
also have "\<dots> = (\<not> real_of_int j rdvd real_of_int x - real_of_int d + ?e)" |
41891 | 2740 |
using ie by simp |
2741 |
finally show ?case |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2742 |
using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2743 |
qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int (x - d)" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2744 |
simp del: of_int_diff) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2745 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2746 |
lemma \<beta>': |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2747 |
assumes lp: "iszlfm p (a #bs)" |
50252 | 2748 |
and u: "d_\<beta> p 1" |
2749 |
and d: "d_\<delta> p d" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2750 |
and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2751 |
shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2752 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2753 |
fix x |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2754 |
assume nb:"?b" and px: "?P x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2755 |
hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real_of_int x = b + real_of_int j)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2756 |
by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2757 |
from \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2758 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2759 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2760 |
lemma \<beta>_int: assumes lp: "iszlfm p bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2761 |
shows "\<forall> b\<in> set (\<beta> p). isint b bs" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2762 |
using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2763 |
|
67613 | 2764 |
lemma cpmi_eq: "0 < D \<Longrightarrow> (\<exists>z::int. \<forall>x. x < z \<longrightarrow> (P x = P1 x)) |
2765 |
\<Longrightarrow> \<forall>x. \<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. P(b+j)) \<longrightarrow> P (x) \<longrightarrow> P (x - D) |
|
2766 |
\<Longrightarrow> (\<forall>(x::int). \<forall>(k::int). ((P1 x)= (P1 (x-k*D)))) |
|
2767 |
\<Longrightarrow> (\<exists>(x::int). P(x)) = ((\<exists>(j::int) \<in> {1..D} . (P1(j))) | (\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. P (b+j)))" |
|
23316 | 2768 |
apply(rule iffI) |
2769 |
prefer 2 |
|
2770 |
apply(drule minusinfinity) |
|
2771 |
apply assumption+ |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44121
diff
changeset
|
2772 |
apply(fastforce) |
23316 | 2773 |
apply clarsimp |
67613 | 2774 |
apply(subgoal_tac "\<And>k. 0<=k \<Longrightarrow> \<forall>x. P x \<longrightarrow> P (x - k*D)") |
23316 | 2775 |
apply(frule_tac x = x and z=z in decr_lemma) |
2776 |
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)") |
|
2777 |
prefer 2 |
|
2778 |
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)") |
|
2779 |
prefer 2 apply arith |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44121
diff
changeset
|
2780 |
apply fastforce |
23316 | 2781 |
apply(drule (1) periodic_finite_ex) |
2782 |
apply blast |
|
2783 |
apply(blast dest:decr_mult_lemma) |
|
2784 |
done |
|
2785 |
||
2786 |
||
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2787 |
theorem cp_thm: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2788 |
assumes lp: "iszlfm p (a #bs)" |
50252 | 2789 |
and u: "d_\<beta> p 1" |
2790 |
and d: "d_\<delta> p d" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2791 |
and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2792 |
shows "(\<exists> (x::int). Ifm (real_of_int x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real_of_int j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p))" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2793 |
(is "(\<exists> (x::int). ?P (real_of_int x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real_of_int j)))") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2794 |
proof- |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2795 |
from minusinf_inf[OF lp] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2796 |
have th: "\<exists>(z::int). \<forall>x<z. ?P (real_of_int x) = ?M x" by blast |
61942 | 2797 |
let ?B' = "{\<lfloor>?I b\<rfloor> | b. b\<in> ?B}" |
2798 |
from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real_of_int \<lfloor>?I b\<rfloor> = ?I b" by simp |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2799 |
from B[rule_format] |
61942 | 2800 |
have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int \<lfloor>?I b\<rfloor> + real_of_int j))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2801 |
by simp |
61942 | 2802 |
also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (\<lfloor>?I b\<rfloor> + j)))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2803 |
also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2804 |
finally have BB': |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2805 |
"(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2806 |
by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2807 |
hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j))) \<longrightarrow> ?P (real_of_int x) \<longrightarrow> ?P (real_of_int (x - d))" using \<beta>'[OF lp u d dp] by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2808 |
from minusinf_repeats[OF d lp] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2809 |
have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2810 |
from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2811 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2812 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2813 |
(* Reddy and Loveland *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2814 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2815 |
|
66809 | 2816 |
fun \<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy and Loveland Bset*) |
2817 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2818 |
"\<rho> (And p q) = (\<rho> p @ \<rho> q)" |
66809 | 2819 |
| "\<rho> (Or p q) = (\<rho> p @ \<rho> q)" |
2820 |
| "\<rho> (Eq (CN 0 c e)) = [(Sub (C (- 1)) e,c)]" |
|
2821 |
| "\<rho> (NEq (CN 0 c e)) = [(Neg e,c)]" |
|
2822 |
| "\<rho> (Lt (CN 0 c e)) = []" |
|
2823 |
| "\<rho> (Le (CN 0 c e)) = []" |
|
2824 |
| "\<rho> (Gt (CN 0 c e)) = [(Neg e, c)]" |
|
2825 |
| "\<rho> (Ge (CN 0 c e)) = [(Sub (C (-1)) e, c)]" |
|
2826 |
| "\<rho> p = []" |
|
2827 |
||
2828 |
fun \<sigma>_\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy and Loveland*) |
|
2829 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2830 |
"\<sigma>_\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))" |
66809 | 2831 |
| "\<sigma>_\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))" |
2832 |
| "\<sigma>_\<rho> (Eq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e)) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2833 |
else (Eq (Add (Mul c t) (Mul k e))))" |
66809 | 2834 |
| "\<sigma>_\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2835 |
else (NEq (Add (Mul c t) (Mul k e))))" |
66809 | 2836 |
| "\<sigma>_\<rho> (Lt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2837 |
else (Lt (Add (Mul c t) (Mul k e))))" |
66809 | 2838 |
| "\<sigma>_\<rho> (Le (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2839 |
else (Le (Add (Mul c t) (Mul k e))))" |
66809 | 2840 |
| "\<sigma>_\<rho> (Gt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2841 |
else (Gt (Add (Mul c t) (Mul k e))))" |
66809 | 2842 |
| "\<sigma>_\<rho> (Ge (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2843 |
else (Ge (Add (Mul c t) (Mul k e))))" |
66809 | 2844 |
| "\<sigma>_\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2845 |
else (Dvd (i*k) (Add (Mul c t) (Mul k e))))" |
66809 | 2846 |
| "\<sigma>_\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2847 |
else (NDvd (i*k) (Add (Mul c t) (Mul k e))))" |
66809 | 2848 |
| "\<sigma>_\<rho> p = (\<lambda> (t,k). p)" |
2849 |
||
2850 |
fun \<alpha>_\<rho> :: "fm \<Rightarrow> (num \<times> int) list" |
|
2851 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2852 |
"\<alpha>_\<rho> (And p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)" |
66809 | 2853 |
| "\<alpha>_\<rho> (Or p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)" |
2854 |
| "\<alpha>_\<rho> (Eq (CN 0 c e)) = [(Add (C (- 1)) e,c)]" |
|
2855 |
| "\<alpha>_\<rho> (NEq (CN 0 c e)) = [(e,c)]" |
|
2856 |
| "\<alpha>_\<rho> (Lt (CN 0 c e)) = [(e,c)]" |
|
2857 |
| "\<alpha>_\<rho> (Le (CN 0 c e)) = [(Add (C (- 1)) e,c)]" |
|
2858 |
| "\<alpha>_\<rho> p = []" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2859 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2860 |
(* Simulates normal substituion by modifying the formula see correctness theorem *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2861 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
2862 |
definition \<sigma> :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where |
50252 | 2863 |
"\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>_\<rho> p (t,k))" |
2864 |
||
2865 |
lemma \<sigma>_\<rho>: |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2866 |
assumes linp: "iszlfm p (real_of_int (x::int)#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2867 |
and kpos: "real_of_int k > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2868 |
and tnb: "numbound0 t" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2869 |
and tint: "isint t (real_of_int x#bs)" |
61942 | 2870 |
and kdt: "k dvd \<lfloor>Inum (b'#bs) t\<rfloor>" |
2871 |
shows "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (t,k)) = (Ifm ((real_of_int (\<lfloor>Inum (b'#bs) t\<rfloor> div k))#bs) p)" |
|
2872 |
(is "?I (real_of_int x) (?s p) = (?I (real_of_int (\<lfloor>?N b' t\<rfloor> div k)) p)" is "_ = (?I ?tk p)") |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2873 |
using linp kpos tnb |
50252 | 2874 |
proof(induct p rule: \<sigma>_\<rho>.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2875 |
case (3 c e) |
41891 | 2876 |
from 3 have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2877 |
{ assume kdc: "k dvd c" |
61942 | 2878 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 2879 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2880 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2881 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2882 |
moreover |
41891 | 2883 |
{ assume *: "\<not> k dvd c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2884 |
from kpos have knz': "real_of_int k \<noteq> 0" by simp |
61942 | 2885 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" |
41891 | 2886 |
using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2887 |
from assms * have "?I (real_of_int x) (?s (Eq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k = 0)" |
46670 | 2888 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2889 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2890 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2891 |
by (simp add: ti algebra_simps) |
2892 |
also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" |
|
2893 |
using nonzero_eq_divide_eq[OF knz', |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2894 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2895 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2896 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
2897 |
by (simp add: ti) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2898 |
finally have ?case . } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2899 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2900 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2901 |
case (4 c e) |
41891 | 2902 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2903 |
{ assume kdc: "k dvd c" |
61942 | 2904 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 2905 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2906 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2907 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2908 |
moreover |
41891 | 2909 |
{ assume *: "\<not> k dvd c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2910 |
from kpos have knz': "real_of_int k \<noteq> 0" by simp |
61942 | 2911 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2912 |
from assms * have "?I (real_of_int x) (?s (NEq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<noteq> 0)" |
46670 | 2913 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2914 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2915 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2916 |
by (simp add: ti algebra_simps) |
2917 |
also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" |
|
2918 |
using nonzero_eq_divide_eq[OF knz', |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2919 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2920 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2921 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2922 |
by (simp add: ti) |
2923 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2924 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2925 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2926 |
case (5 c e) |
41891 | 2927 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2928 |
{ assume kdc: "k dvd c" |
61942 | 2929 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 2930 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2931 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2932 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2933 |
moreover |
41891 | 2934 |
{ assume *: "\<not> k dvd c" |
61942 | 2935 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2936 |
from assms * have "?I (real_of_int x) (?s (Lt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k < 0)" |
46670 | 2937 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2938 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2939 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2940 |
by (simp add: ti algebra_simps) |
2941 |
also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" |
|
2942 |
using pos_less_divide_eq[OF kpos, |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2943 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2944 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2945 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2946 |
by (simp add: ti) |
2947 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2948 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2949 |
next |
46670 | 2950 |
case (6 c e) |
41891 | 2951 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2952 |
{ assume kdc: "k dvd c" |
61942 | 2953 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 2954 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2955 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2956 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2957 |
moreover |
41891 | 2958 |
{ assume *: "\<not> k dvd c" |
61942 | 2959 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2960 |
from assms * have "?I (real_of_int x) (?s (Le (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<le> 0)" |
46670 | 2961 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2962 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2963 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2964 |
by (simp add: ti algebra_simps) |
2965 |
also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" |
|
2966 |
using pos_le_divide_eq[OF kpos, |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2967 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2968 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2969 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2970 |
by (simp add: ti) |
2971 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2972 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2973 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2974 |
case (7 c e) |
41891 | 2975 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2976 |
{ assume kdc: "k dvd c" |
61942 | 2977 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 2978 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2979 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2980 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2981 |
moreover |
41891 | 2982 |
{ assume *: "\<not> k dvd c" |
61942 | 2983 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2984 |
from assms * have "?I (real_of_int x) (?s (Gt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k > 0)" |
46670 | 2985 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2986 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2987 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2988 |
by (simp add: ti algebra_simps) |
2989 |
also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" |
|
2990 |
using pos_divide_less_eq[OF kpos, |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2991 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2992 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2993 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 2994 |
by (simp add: ti) |
2995 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2996 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
2997 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
2998 |
case (8 c e) |
41891 | 2999 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3000 |
{ assume kdc: "k dvd c" |
61942 | 3001 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 3002 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3003 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3004 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3005 |
moreover |
41891 | 3006 |
{ assume *: "\<not> k dvd c" |
61942 | 3007 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3008 |
from assms * have "?I (real_of_int x) (?s (Ge (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<ge> 0)" |
46670 | 3009 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3010 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3011 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3012 |
by (simp add: ti algebra_simps) |
3013 |
also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" |
|
3014 |
using pos_divide_le_eq[OF kpos, |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3015 |
where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3016 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3017 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3018 |
by (simp add: ti) |
3019 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3020 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3021 |
next |
41891 | 3022 |
case (9 i c e) |
3023 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3024 |
{ assume kdc: "k dvd c" |
61942 | 3025 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 3026 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3027 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3028 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3029 |
moreover |
41891 | 3030 |
{ assume *: "\<not> k dvd c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3031 |
from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp |
61942 | 3032 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3033 |
from assms * have "?I (real_of_int x) (?s (Dvd i (CN 0 c e))) = (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k)" |
46670 | 3034 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3035 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3036 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3037 |
by (simp add: ti algebra_simps) |
3038 |
also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" |
|
3039 |
using rdvd_mult[OF knz, where n="i"] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3040 |
real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3041 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3042 |
by (simp add: ti) |
3043 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3044 |
ultimately show ?case by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3045 |
next |
41891 | 3046 |
case (10 i c e) |
3047 |
then have cp: "c > 0" and nb: "numbound0 e" by auto |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3048 |
{ assume kdc: "k dvd c" |
61942 | 3049 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
46670 | 3050 |
from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3051 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3052 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3053 |
moreover |
41891 | 3054 |
{ assume *: "\<not> k dvd c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3055 |
from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp |
61942 | 3056 |
from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3057 |
from assms * have "?I (real_of_int x) (?s (NDvd i (CN 0 c e))) = (\<not> (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k))" |
46670 | 3058 |
using real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3059 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3060 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3061 |
by (simp add: ti algebra_simps) |
3062 |
also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" |
|
46670 | 3063 |
using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF kdt] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3064 |
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3065 |
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] |
41891 | 3066 |
by (simp add: ti) |
3067 |
finally have ?case . } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3068 |
ultimately show ?case by blast |
61942 | 3069 |
qed (simp_all add: bound0_I[where bs="bs" and b="real_of_int (\<lfloor>?N b' t\<rfloor> div k)" and b'="real_of_int x"] |
3070 |
numbound0_I[where bs="bs" and b="real_of_int (\<lfloor>?N b' t\<rfloor> div k)" and b'="real_of_int x"]) |
|
41849 | 3071 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3072 |
|
50252 | 3073 |
lemma \<sigma>_\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t" |
3074 |
shows "bound0 (\<sigma>_\<rho> p (t,k))" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3075 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3076 |
by (induct p rule: iszlfm.induct, auto simp add: nb) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3077 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3078 |
lemma \<rho>_l: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3079 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3080 |
shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int i#bs)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3081 |
using lp by (induct p rule: \<rho>.induct, auto simp add: isint_sub isint_neg) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3082 |
|
50252 | 3083 |
lemma \<alpha>_\<rho>_l: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3084 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3085 |
shows "\<forall> (b,k) \<in> set (\<alpha>_\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int i#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3086 |
using lp isint_add [OF isint_c[where j="- 1"],where bs="real_of_int i#bs"] |
50252 | 3087 |
by (induct p rule: \<alpha>_\<rho>.induct, auto) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3088 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3089 |
lemma \<rho>: assumes lp: "iszlfm p (real_of_int (i::int) #bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3090 |
and pi: "Ifm (real_of_int i#bs) p" |
50252 | 3091 |
and d: "d_\<delta> p d" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3092 |
and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3093 |
and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> Inum (real_of_int i#bs) e + real_of_int j" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3094 |
(is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3095 |
shows "Ifm (real_of_int(i - d)#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3096 |
using lp pi d nob |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3097 |
proof(induct p rule: iszlfm.induct) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3098 |
case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3099 |
and pi: "real_of_int (c*i) = - 1 - ?N i e + real_of_int (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> -1 - ?N i e + real_of_int j" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3100 |
by simp+ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3101 |
from mult_strict_left_mono[OF dp cp] have one:"1 \<in> {1 .. c*d}" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3102 |
from nob[rule_format, where j="1", OF one] pi show ?case by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3103 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3104 |
case (4 c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3105 |
hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3106 |
and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3107 |
by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3108 |
{assume "real_of_int (c*i) \<noteq> - ?N i e + real_of_int (c*d)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3109 |
with numbound0_I[OF nb, where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] |
29667 | 3110 |
have ?case by (simp add: algebra_simps)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3111 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3112 |
{assume pi: "real_of_int (c*i) = - ?N i e + real_of_int (c*d)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3113 |
from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3114 |
from nob[rule_format, where j="c*d", OF d] pi have ?case by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3115 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3116 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3117 |
case (5 c e) hence cp: "c > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3118 |
from 5 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3119 |
of_int_mult] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3120 |
show ?case using 5 dp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3121 |
apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] |
56544 | 3122 |
algebra_simps del: mult_pos_pos) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3123 |
by (metis add.right_neutral of_int_0_less_iff of_int_mult pos_add_strict) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3124 |
next |
41891 | 3125 |
case (6 c e) hence cp: "c > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3126 |
from 6 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3127 |
of_int_mult] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3128 |
show ?case using 6 dp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3129 |
apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] |
56544 | 3130 |
algebra_simps del: mult_pos_pos) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3131 |
using order_trans by fastforce |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3132 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3133 |
case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3134 |
and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3135 |
and pi: "real_of_int (c*i) + ?N i e > 0" and cp': "real_of_int c >0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3136 |
by simp+ |
61942 | 3137 |
let ?fe = "\<lfloor>?N i e\<rfloor>" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3138 |
from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c > 0" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3139 |
from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) > real_of_int (0::int)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3140 |
hence pi': "c*i + ?fe > 0" by (simp only: of_int_less_iff[symmetric]) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3141 |
have "real_of_int (c*i) + ?N i e > real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3142 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3143 |
{assume "real_of_int (c*i) + ?N i e > real_of_int (c*d)" hence ?case |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3144 |
by (simp add: algebra_simps |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3145 |
numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3146 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3147 |
{assume H:"real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3148 |
with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<le> real_of_int (c*d)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3149 |
hence pid: "c*i + ?fe \<le> c*d" by (simp only: of_int_le_iff) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3150 |
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3151 |
hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = - ?N i e + real_of_int j1" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3152 |
unfolding Bex_def using ei[simplified isint_iff] by fastforce |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3153 |
with nob have ?case by blast } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3154 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3155 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3156 |
case (8 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3157 |
and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - 1 - ?N i e + real_of_int j" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3158 |
and pi: "real_of_int (c*i) + ?N i e \<ge> 0" and cp': "real_of_int c >0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3159 |
by simp+ |
61942 | 3160 |
let ?fe = "\<lfloor>?N i e\<rfloor>" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3161 |
from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c \<ge> 0" by (simp add: algebra_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3162 |
from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<ge> real_of_int (0::int)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3163 |
hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: of_int_le_iff[symmetric]) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3164 |
have "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e < real_of_int (c*d)" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3165 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3166 |
{assume "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d)" hence ?case |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3167 |
by (simp add: algebra_simps |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3168 |
numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3169 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3170 |
{assume H:"real_of_int (c*i) + ?N i e < real_of_int (c*d)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3171 |
with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) < real_of_int (c*d)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3172 |
hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: of_int_le_iff) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3173 |
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3174 |
hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) + 1= - ?N i e + real_of_int j1" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3175 |
unfolding Bex_def using ei[simplified isint_iff] by fastforce |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3176 |
hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = (- ?N i e + real_of_int j1) - 1" |
51369 | 3177 |
by (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3178 |
hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = - 1 - ?N i e + real_of_int j1" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
3179 |
by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3180 |
with nob have ?case by blast } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3181 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3182 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3183 |
case (9 j c e) hence p: "real_of_int j rdvd real_of_int (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3184 |
let ?e = "Inum (real_of_int i # bs) e" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3185 |
from 9 have "isint e (real_of_int i #bs)" by simp |
61942 | 3186 |
hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"] |
41891 | 3187 |
by (simp add: isint_iff) |
3188 |
from 9 have id: "j dvd d" by simp |
|
61942 | 3189 |
from ie[symmetric] have "?p i = (real_of_int j rdvd real_of_int (c*i + \<lfloor>?e\<rfloor>))" by simp |
3190 |
also have "\<dots> = (j dvd c*i + \<lfloor>?e\<rfloor>)" |
|
3191 |
using int_rdvd_iff [where i="j" and t="c*i + \<lfloor>?e\<rfloor>"] by simp |
|
3192 |
also have "\<dots> = (j dvd c*i - c*d + \<lfloor>?e\<rfloor>)" |
|
3193 |
using dvd_period[OF id, where x="c*i" and c="-c" and t="\<lfloor>?e\<rfloor>"] by simp |
|
3194 |
also have "\<dots> = (real_of_int j rdvd real_of_int (c*i - c*d + \<lfloor>?e\<rfloor>))" |
|
3195 |
using int_rdvd_iff[where i="j" and t="(c*i - c*d + \<lfloor>?e\<rfloor>)",symmetric, simplified] |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3196 |
ie by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3197 |
also have "\<dots> = (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" |
41891 | 3198 |
using ie by (simp add:algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3199 |
finally show ?case |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3200 |
using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p |
41891 | 3201 |
by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3202 |
next |
41891 | 3203 |
case (10 j c e) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3204 |
hence p: "\<not> (real_of_int j rdvd real_of_int (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" |
41891 | 3205 |
by simp+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3206 |
let ?e = "Inum (real_of_int i # bs) e" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3207 |
from 10 have "isint e (real_of_int i #bs)" by simp |
61942 | 3208 |
hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3209 |
using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"] |
41891 | 3210 |
by (simp add: isint_iff) |
3211 |
from 10 have id: "j dvd d" by simp |
|
61942 | 3212 |
from ie[symmetric] have "?p i = (\<not> (real_of_int j rdvd real_of_int (c*i + \<lfloor>?e\<rfloor>)))" by simp |
3213 |
also have "\<dots> = Not (j dvd c*i + \<lfloor>?e\<rfloor>)" |
|
3214 |
using int_rdvd_iff [where i="j" and t="c*i + \<lfloor>?e\<rfloor>"] by simp |
|
3215 |
also have "\<dots> = Not (j dvd c*i - c*d + \<lfloor>?e\<rfloor>)" |
|
3216 |
using dvd_period[OF id, where x="c*i" and c="-c" and t="\<lfloor>?e\<rfloor>"] by simp |
|
3217 |
also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*i - c*d + \<lfloor>?e\<rfloor>))" |
|
3218 |
using int_rdvd_iff[where i="j" and t="(c*i - c*d + \<lfloor>?e\<rfloor>)",symmetric, simplified] |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3219 |
ie by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3220 |
also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" |
41891 | 3221 |
using ie by (simp add:algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3222 |
finally show ?case |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3223 |
using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p |
41891 | 3224 |
by (simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3225 |
qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3226 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3227 |
lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3228 |
shows "bound0 (\<sigma> p k t)" |
50252 | 3229 |
using \<sigma>_\<rho>_nb[OF lp nb] nb by (simp add: \<sigma>_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3230 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3231 |
lemma \<rho>': assumes lp: "iszlfm p (a #bs)" |
50252 | 3232 |
and d: "d_\<delta> p d" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3233 |
and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3234 |
shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3235 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3236 |
fix x |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3237 |
assume nob1:"?b x" and px: "?P x" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3238 |
from iszlfm_gen[OF lp, rule_format, where y="real_of_int x"] have lp': "iszlfm p (real_of_int x#bs)". |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3239 |
have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * x) \<noteq> Inum (real_of_int x # bs) e + real_of_int j" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3240 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3241 |
fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3242 |
and cx: "real_of_int (c*x) = Inum (real_of_int x#bs) e + real_of_int j" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3243 |
let ?e = "Inum (real_of_int x#bs) e" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3244 |
from \<rho>_l[OF lp'] ecR have ei:"isint e (real_of_int x#bs)" and cp:"c>0" and nb:"numbound0 e" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3245 |
by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3246 |
from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" . |
61942 | 3247 |
from cx ei[simplified isint_iff] have "real_of_int (c*x) = real_of_int (\<lfloor>?e\<rfloor> + j)" by simp |
3248 |
hence cx: "c*x = \<lfloor>?e\<rfloor> + j" by (simp only: of_int_eq_iff) |
|
3249 |
hence cdej:"c dvd \<lfloor>?e\<rfloor> + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp) |
|
3250 |
hence "real_of_int c rdvd real_of_int (\<lfloor>?e\<rfloor> + j)" by (simp only: int_rdvd_iff) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3251 |
hence rcdej: "real_of_int c rdvd ?e + real_of_int j" by (simp add: ei[simplified isint_iff]) |
61942 | 3252 |
from cx have "(c*x) div c = (\<lfloor>?e\<rfloor> + j) div c" by simp |
3253 |
with cp have "x = (\<lfloor>?e\<rfloor> + j) div c" by simp |
|
3254 |
with px have th: "?P ((\<lfloor>?e\<rfloor> + j) div c)" by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3255 |
from cp have cp': "real_of_int c > 0" by simp |
61942 | 3256 |
from cdej have cdej': "c dvd \<lfloor>Inum (real_of_int x#bs) (Add e (C j))\<rfloor>" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3257 |
from nb have nb': "numbound0 (Add e (C j))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3258 |
have ji: "isint (C j) (real_of_int x#bs)" by (simp add: isint_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3259 |
from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real_of_int x#bs)" . |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3260 |
from th \<sigma>_\<rho>[where b'="real_of_int x", OF lp' cp' nb' ei' cdej',symmetric] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3261 |
have "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (Add e (C j), c))" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3262 |
with rcdej have th: "Ifm (real_of_int x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3263 |
from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real_of_int x" and b'="a"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3264 |
have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3265 |
with ecR jD nob1 show "False" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3266 |
qed |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3267 |
from \<rho>[OF lp' px d dp nob] show "?P (x -d )" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3268 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3269 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3270 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3271 |
lemma rl_thm: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3272 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3273 |
shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3274 |
(is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3275 |
is "?lhs = (?MD \<or> ?RD)" is "?lhs = ?rhs") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3276 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3277 |
let ?d= "\<delta> p" |
50252 | 3278 |
from \<delta>[OF lp] have d:"d_\<delta> p ?d" and dp: "?d > 0" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3279 |
{ assume H:"?MD" hence th:"\<exists> (x::int). ?MP x" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3280 |
from H minusinf_ex[OF lp th] have ?thesis by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3281 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3282 |
{ fix e c j assume exR:"(e,c) \<in> ?R" and jD:"j\<in> {1 .. c*?d}" and spx:"?SP c e j" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3283 |
from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real_of_int i#bs)" and cp: "c > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3284 |
by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3285 |
have "isint (C j) (real_of_int i#bs)" by (simp add: isint_iff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3286 |
with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real_of_int i"]] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3287 |
have eji:"isint (Add e (C j)) (real_of_int i#bs)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3288 |
from nb have nb': "numbound0 (Add e (C j))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3289 |
from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real_of_int i"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3290 |
have spx': "Ifm (real_of_int i # bs) (\<sigma> p c (Add e (C j)))" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3291 |
from spx' have rcdej:"real_of_int c rdvd (Inum (real_of_int i#bs) (Add e (C j)))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3292 |
and sr:"Ifm (real_of_int i#bs) (\<sigma>_\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+ |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3293 |
from rcdej eji[simplified isint_iff] |
61942 | 3294 |
have "real_of_int c rdvd real_of_int \<lfloor>Inum (real_of_int i#bs) (Add e (C j))\<rfloor>" by simp |
3295 |
hence cdej:"c dvd \<lfloor>Inum (real_of_int i#bs) (Add e (C j))\<rfloor>" by (simp only: int_rdvd_iff) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3296 |
from cp have cp': "real_of_int c > 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3297 |
from \<sigma>_\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real_of_int i # bs) (Add e (C j))\<rfloor> div c)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3298 |
by (simp add: \<sigma>_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3299 |
hence ?lhs by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3300 |
with exR jD spx have ?thesis by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3301 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3302 |
{ fix x assume px: "?P x" and nob: "\<not> ?RD" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3303 |
from iszlfm_gen [OF lp,rule_format, where y="a"] have lp':"iszlfm p (a#bs)" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3304 |
from \<rho>'[OF lp' d dp, rule_format, OF nob] have th:"\<forall> x. ?P x \<longrightarrow> ?P (x - ?d)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3305 |
from minusinf_inf[OF lp] obtain z where z:"\<forall> x<z. ?MP x = ?P x" by blast |
61945 | 3306 |
have zp: "\<bar>x - z\<bar> + 1 \<ge> 0" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3307 |
from decr_lemma[OF dp,where x="x" and z="z"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3308 |
decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\<exists> x. ?MP x" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3309 |
with minusinf_bex[OF lp] px nob have ?thesis by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3310 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3311 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3312 |
|
50252 | 3313 |
lemma mirror_\<alpha>_\<rho>: assumes lp: "iszlfm p (a#bs)" |
3314 |
shows "(\<lambda> (t,k). (Inum (a#bs) t, k)) ` set (\<alpha>_\<rho> p) = (\<lambda> (t,k). (Inum (a#bs) t,k)) ` set (\<rho> (mirror p))" |
|
51369 | 3315 |
using lp |
3316 |
by (induct p rule: mirror.induct) (simp_all add: split_def image_Un) |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3317 |
|
61586 | 3318 |
text \<open>The \<open>\<real>\<close> part\<close> |
3319 |
||
3320 |
text\<open>Linearity for fm where Bound 0 ranges over \<open>\<real>\<close>\<close> |
|
66809 | 3321 |
fun isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *) |
3322 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3323 |
"isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" |
66809 | 3324 |
| "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" |
3325 |
| "isrlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3326 |
| "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3327 |
| "isrlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3328 |
| "isrlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3329 |
| "isrlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3330 |
| "isrlfm (Ge (CN 0 c e)) = (c>0 \<and> numbound0 e)" |
|
3331 |
| "isrlfm p = (isatom p \<and> (bound0 p))" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3332 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
3333 |
definition fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3334 |
"fp p n s j \<equiv> (if n > 0 then |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3335 |
(And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j))))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3336 |
(Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1)))))))) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3337 |
else |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3338 |
(And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j))))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3339 |
(Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3340 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3341 |
(* splits the bounded from the unbounded part*) |
66809 | 3342 |
fun rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list" |
3343 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3344 |
"rsplit0 (Bound 0) = [(T,1,C 0)]" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3345 |
| "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b |
24336 | 3346 |
in map (\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) [(a,b). a\<leftarrow>acs,b\<leftarrow>bcs])" |
41839 | 3347 |
| "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" |
3348 |
| "rsplit0 (Neg a) = map (\<lambda> (p,n,s). (p,-n,Neg s)) (rsplit0 a)" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3349 |
| "rsplit0 (Floor a) = concat (map |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3350 |
(\<lambda> (p,n,s). if n=0 then [(p,0,Floor s)] |
41836 | 3351 |
else (map (\<lambda> j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then [0 .. n] else [n .. 0]))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3352 |
(rsplit0 a))" |
41839 | 3353 |
| "rsplit0 (CN 0 c a) = map (\<lambda> (p,n,s). (p,n+c,s)) (rsplit0 a)" |
3354 |
| "rsplit0 (CN m c a) = map (\<lambda> (p,n,s). (p,n,CN m c s)) (rsplit0 a)" |
|
3355 |
| "rsplit0 (CF c t s) = rsplit0 (Add (Mul c (Floor t)) s)" |
|
3356 |
| "rsplit0 (Mul c a) = map (\<lambda> (p,n,s). (p,c*n,Mul c s)) (rsplit0 a)" |
|
3357 |
| "rsplit0 t = [(T,0,t)]" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3358 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3359 |
lemma conj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3360 |
using conj_def by (cases p, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3361 |
lemma disj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3362 |
using disj_def by (cases p, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3363 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3364 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3365 |
lemma rsplit0_cs: |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3366 |
shows "\<forall> (p,n,s) \<in> set (rsplit0 t). |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3367 |
(Ifm (x#bs) p \<longrightarrow> (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3368 |
(is "\<forall> (p,n,s) \<in> ?SS t. (?I p \<longrightarrow> ?N t = ?N (CN 0 n s)) \<and> _ \<and> _ ") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3369 |
proof(induct t rule: rsplit0.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3370 |
case (5 a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3371 |
let ?p = "\<lambda> (p,n,s) j. fp p n s j" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3372 |
let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))" |
41836 | 3373 |
let ?J = "\<lambda> n. if n>0 then [0..n] else [n..0]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3374 |
let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3375 |
have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3376 |
have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3377 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3378 |
have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. |
41836 | 3379 |
?ff (p,n,s) = map (?f(p,n,s)) [0..n]" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3380 |
hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3381 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). |
41836 | 3382 |
set (map (?f(p,n,s)) [0..n])))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3383 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3384 |
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3385 |
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" |
69313 | 3386 |
thus "(\<Union>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3387 |
by (auto simp add: split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3388 |
qed |
41836 | 3389 |
have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) [n..0]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3390 |
by auto |
69313 | 3391 |
hence U3: "(\<Union> ((\<lambda>(p,n,s). set (?ff (p,n,s))) ` {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0})) = |
3392 |
(\<Union> ((\<lambda>(p,n,s). set (map (?f(p,n,s)) [n..0])) ` {(p,n,s). (p,n,s)\<in> ?SS a\<and>n<0}))" |
|
3393 |
proof - |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3394 |
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3395 |
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" |
69313 | 3396 |
thus "(\<Union>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3397 |
by (auto simp add: split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3398 |
qed |
69313 | 3399 |
have "?SS (Floor a) = \<Union> ((\<lambda>x. set (?ff x)) ` ?SS a)" |
46130 | 3400 |
by auto |
69313 | 3401 |
also have "\<dots> = \<Union> ((\<lambda> (p,n,s). set (?ff (p,n,s))) ` ?SS a)" |
3402 |
by blast |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3403 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3404 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3405 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3406 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3407 |
by (auto split: if_splits) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3408 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3409 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3410 |
(UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) [0..n]))) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3411 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). |
41836 | 3412 |
set (map (?f(p,n,s)) [n..0]))))" by (simp only: U1 U2 U3) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3413 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3414 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3415 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3416 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))" |
57816
d8bbb97689d3
no need for 'set_simps' now that 'datatype_new' generates the desired 'set' property
blanchet
parents:
57514
diff
changeset
|
3417 |
by (simp only: set_map set_upto list.set) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3418 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3419 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3420 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3421 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" |
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3422 |
by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3423 |
finally |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3424 |
have FS: "?SS (Floor a) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3425 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3426 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3427 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" |
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3428 |
by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3429 |
show ?case |
41891 | 3430 |
proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -) |
3431 |
fix p n s |
|
3432 |
let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" |
|
3433 |
assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or> |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3434 |
(\<exists>ab ac ba. |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3435 |
(ab, ac, ba) \<in> set (rsplit0 a) \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3436 |
0 < ac \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3437 |
(\<exists>j. p = fp ab ac ba j \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3438 |
n = 0 \<and> s = Add (Floor ba) (C j) \<and> 0 \<le> j \<and> j \<le> ac)) \<or> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3439 |
(\<exists>ab ac ba. |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3440 |
(ab, ac, ba) \<in> set (rsplit0 a) \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3441 |
ac < 0 \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3442 |
(\<exists>j. p = fp ab ac ba j \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3443 |
n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3444 |
moreover |
41891 | 3445 |
{ fix s' |
3446 |
assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'" |
|
3447 |
hence ?ths using 5(1) by auto } |
|
3448 |
moreover |
|
3449 |
{ fix p' n' s' j |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3450 |
assume pns: "(p', n', s') \<in> ?SS a" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3451 |
and np: "0 < n'" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3452 |
and p_def: "p = ?p (p',n',s') j" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3453 |
and n0: "n = 0" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3454 |
and s_def: "s = (Add (Floor s') (C j))" |
41891 | 3455 |
and jp: "0 \<le> j" and jn: "j \<le> n'" |
61076 | 3456 |
from 5 pns have H:"(Ifm ((x::real) # (bs::real list)) p' \<longrightarrow> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3457 |
Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3458 |
numbound0 s' \<and> isrlfm p'" by blast |
41891 | 3459 |
hence nb: "numbound0 s'" by simp |
51369 | 3460 |
from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by simp |
41891 | 3461 |
let ?nxs = "CN 0 n' s'" |
61942 | 3462 |
let ?l = "\<lfloor>?N s'\<rfloor> + j" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3463 |
from H |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3464 |
have "?I (?p (p',n',s') j) \<longrightarrow> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3465 |
(((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" |
51369 | 3466 |
by (simp add: fp_def np algebra_simps) |
61942 | 3467 |
also have "\<dots> \<longrightarrow> \<lfloor>?N ?nxs\<rfloor> = ?l \<and> ?N a = ?N ?nxs" |
66515 | 3468 |
using floor_eq_iff[where x="?N ?nxs" and a="?l"] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3469 |
moreover |
41891 | 3470 |
have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp |
3471 |
ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" |
|
3472 |
by blast |
|
3473 |
with s_def n0 p_def nb nf have ?ths by auto} |
|
3474 |
moreover |
|
3475 |
{ fix p' n' s' j |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3476 |
assume pns: "(p', n', s') \<in> ?SS a" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3477 |
and np: "n' < 0" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3478 |
and p_def: "p = ?p (p',n',s') j" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3479 |
and n0: "n = 0" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3480 |
and s_def: "s = (Add (Floor s') (C j))" |
41891 | 3481 |
and jp: "n' \<le> j" and jn: "j \<le> 0" |
61076 | 3482 |
from 5 pns have H:"(Ifm ((x::real) # (bs::real list)) p' \<longrightarrow> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3483 |
Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and> |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3484 |
numbound0 s' \<and> isrlfm p'" by blast |
41891 | 3485 |
hence nb: "numbound0 s'" by simp |
51369 | 3486 |
from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by simp |
41891 | 3487 |
let ?nxs = "CN 0 n' s'" |
61942 | 3488 |
let ?l = "\<lfloor>?N s'\<rfloor> + j" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3489 |
from H |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3490 |
have "?I (?p (p',n',s') j) \<longrightarrow> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3491 |
(((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" |
51369 | 3492 |
by (simp add: np fp_def algebra_simps) |
61942 | 3493 |
also have "\<dots> \<longrightarrow> \<lfloor>?N ?nxs\<rfloor> = ?l \<and> ?N a = ?N ?nxs" |
66515 | 3494 |
using floor_eq_iff[where x="?N ?nxs" and a="?l"] by simp |
41891 | 3495 |
moreover |
3496 |
have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp |
|
3497 |
ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" |
|
3498 |
by blast |
|
3499 |
with s_def n0 p_def nb nf have ?ths by auto} |
|
61652
90c65a811257
MIR decision procedure again working
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
3500 |
ultimately show ?ths by fastforce |
41891 | 3501 |
qed |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3502 |
next |
28741 | 3503 |
case (3 a b) then show ?case |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
3504 |
by auto |
51369 | 3505 |
qed (auto simp add: Let_def split_def algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3506 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3507 |
lemma real_in_int_intervals: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3508 |
assumes xb: "real_of_int m \<le> x \<and> x < real_of_int ((n::int) + 1)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3509 |
shows "\<exists> j\<in> {m.. n}. real_of_int j \<le> x \<and> x < real_of_int (j+1)" (is "\<exists> j\<in> ?N. ?P j") |
61942 | 3510 |
by (rule bexI[where P="?P" and x="\<lfloor>x\<rfloor>" and A="?N"]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3511 |
(auto simp add: floor_less_iff[where x="x" and z="n+1", simplified] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3512 |
xb[simplified] floor_mono[where x="real_of_int m" and y="x", OF conjunct1[OF xb], simplified floor_of_int[where z="m"]]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3513 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3514 |
lemma rsplit0_complete: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3515 |
assumes xp:"0 \<le> x" and x1:"x < 1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3516 |
shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3517 |
proof(induct t rule: rsplit0.induct) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3518 |
case (2 a b) |
41891 | 3519 |
then have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3520 |
then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast |
41891 | 3521 |
with 2 have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3522 |
then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\<in> ?SS b \<and> ?I pb" by blast |
24336 | 3523 |
from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \<in> set[(x,y). x\<leftarrow>rsplit0 a, y\<leftarrow>rsplit0 b]" |
3524 |
by (auto) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3525 |
let ?f="(\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3526 |
from imageI[OF th, where f="?f"] have "?f ((pa,na,sa),(pb,nb,sb)) \<in> ?SS (Add a b)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3527 |
by (simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3528 |
hence "(And pa pb, na +nb, Add sa sb) \<in> ?SS (Add a b)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3529 |
moreover from pa pb have "?I (And pa pb)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3530 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3531 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3532 |
case (5 a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3533 |
let ?p = "\<lambda> (p,n,s) j. fp p n s j" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3534 |
let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))" |
41836 | 3535 |
let ?J = "\<lambda> n. if n>0 then [0..n] else [n..0]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3536 |
let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3537 |
have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3538 |
have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto |
41836 | 3539 |
have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. ?ff (p,n,s) = map (?f(p,n,s)) [0..n]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3540 |
by auto |
41836 | 3541 |
hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [0..n])))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3542 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3543 |
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3544 |
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" |
69313 | 3545 |
thus "(\<Union>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3546 |
by (auto simp add: split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3547 |
qed |
41836 | 3548 |
have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) [n..0]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3549 |
by auto |
41836 | 3550 |
hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [n..0])))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3551 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3552 |
fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3553 |
assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" |
69313 | 3554 |
thus "(\<Union>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3555 |
by (auto simp add: split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3556 |
qed |
24473 | 3557 |
|
69313 | 3558 |
have "?SS (Floor a) = \<Union> ((\<lambda>x. set (?ff x)) ` ?SS a)" by auto |
3559 |
also have "\<dots> = \<Union> ((\<lambda> (p,n,s). set (?ff (p,n,s))) ` ?SS a)" by blast |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3560 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3561 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3562 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3563 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" |
68270
2bc921b2159b
treat gcd_eq_1_imp_coprime analogously to mod_0_imp_dvd
haftmann
parents:
67613
diff
changeset
|
3564 |
by (auto split: if_splits) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3565 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3566 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3567 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [0..n]))) Un |
51369 | 3568 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [n..0]))))" |
3569 |
by (simp only: U1 U2 U3) |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3570 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3571 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3572 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3573 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))" |
57816
d8bbb97689d3
no need for 'set_simps' now that 'datatype_new' generates the desired 'set' property
blanchet
parents:
57514
diff
changeset
|
3574 |
by (simp only: set_map set_upto list.set) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3575 |
also have "\<dots> = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3576 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3577 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un |
51369 | 3578 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" |
3579 |
by blast |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3580 |
finally |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3581 |
have FS: "?SS (Floor a) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3582 |
((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3583 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un |
41891 | 3584 |
(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" |
3585 |
by blast |
|
3586 |
from 5 have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3587 |
then obtain "p" "n" "s" where pns: "(p,n,s) \<in> ?SS a \<and> ?I p" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3588 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3589 |
from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3590 |
by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3591 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3592 |
have "n=0 \<or> n >0 \<or> n <0" by arith |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3593 |
moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3594 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3595 |
{ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3596 |
assume np: "n > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3597 |
from of_int_floor_le[of "?N s"] have "?N (Floor s) \<le> ?N s" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3598 |
also from mult_left_mono[OF xp] np have "?N s \<le> real_of_int n * x + ?N s" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3599 |
finally have "?N (Floor s) \<le> real_of_int n * x + ?N s" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3600 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3601 |
{from x1 np have "real_of_int n *x + ?N s < real_of_int n + ?N s" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3602 |
also from real_of_int_floor_add_one_gt[where r="?N s"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3603 |
have "\<dots> < real_of_int n + ?N (Floor s) + 1" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3604 |
finally have "real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp} |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3605 |
ultimately have "?N (Floor s) \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3606 |
hence th: "0 \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (n+1)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3607 |
from real_in_int_intervals th have "\<exists> j\<in> {0 .. n}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3608 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3609 |
hence "\<exists> j\<in> {0 .. n}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3610 |
by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3611 |
hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)" |
51369 | 3612 |
using pns by (simp add: fp_def np algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3613 |
then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3614 |
hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3615 |
hence ?case using pns |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3616 |
by (simp only: FS,simp add: bex_Un) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3617 |
(rule disjI2, rule disjI1,rule exI [where x="p"], |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3618 |
rule exI [where x="n"],rule exI [where x="s"],simp_all add: np) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3619 |
} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3620 |
moreover |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3621 |
{ assume nn: "n < 0" hence np: "-n >0" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3622 |
from of_int_floor_le[of "?N s"] have "?N (Floor s) + 1 > ?N s" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3623 |
moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real_of_int n * x + ?N s" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3624 |
ultimately have "?N (Floor s) + 1 > real_of_int n * x + ?N s" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3625 |
moreover |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3626 |
{from x1 nn have "real_of_int n *x + ?N s \<ge> real_of_int n + ?N s" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3627 |
moreover from of_int_floor_le[of "?N s"] have "real_of_int n + ?N s \<ge> real_of_int n + ?N (Floor s)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3628 |
ultimately have "real_of_int n *x + ?N s \<ge> ?N (Floor s) + real_of_int n" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
3629 |
by (simp only: algebra_simps)} |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3630 |
ultimately have "?N (Floor s) + real_of_int n \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (1::int)" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3631 |
hence th: "real_of_int n \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (1::int)" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3632 |
have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3633 |
have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3634 |
from real_in_int_intervals th have "\<exists> j\<in> {n .. 0}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3635 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3636 |
hence "\<exists> j\<in> {n .. 0}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3637 |
by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3638 |
hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j) \<and> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3639 |
hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
3640 |
using pns by (simp add: fp_def nn algebra_simps |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3641 |
del: diff_less_0_iff_less diff_le_0_iff_le) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3642 |
then obtain "j" where j_def: "j\<in> {n .. 0} \<and> ?I (?p (p,n,s) j)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3643 |
hence "\<exists>x \<in> {?p (p,n,s) j |j. n\<le> j \<and> j \<le> 0 }. ?I x" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3644 |
hence ?case using pns |
23464 | 3645 |
by (simp only: FS,simp add: bex_Un) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3646 |
(rule disjI2, rule disjI2,rule exI [where x="p"], |
23464 | 3647 |
rule exI [where x="n"],rule exI [where x="s"],simp_all add: nn) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3648 |
} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3649 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3650 |
qed (auto simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3651 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3652 |
(* Linearize a formula where Bound 0 ranges over [0,1) *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3653 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
3654 |
definition rsplit :: "(int \<Rightarrow> num \<Rightarrow> fm) \<Rightarrow> num \<Rightarrow> fm" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3655 |
"rsplit f a \<equiv> foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) (rsplit0 a)) F" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3656 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3657 |
lemma foldr_disj_map: "Ifm bs (foldr disj (map f xs) F) = (\<exists> x \<in> set xs. Ifm bs (f x))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3658 |
by(induct xs, simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3659 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3660 |
lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\<forall> x \<in> set xs. Ifm bs (f x))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3661 |
by(induct xs, simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3662 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3663 |
lemma foldr_disj_map_rlfm: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3664 |
assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3665 |
and \<phi>: "\<forall> (\<phi>,n,s) \<in> set xs. numbound0 s \<and> isrlfm \<phi>" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3666 |
shows "isrlfm (foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) xs) F)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3667 |
using lf \<phi> by (induct xs, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3668 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3669 |
lemma rsplit_ex: "Ifm bs (rsplit f a) = (\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). Ifm bs (conj \<phi> (f n s)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3670 |
using foldr_disj_map[where xs="rsplit0 a"] rsplit_def by (simp add: split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3671 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3672 |
lemma rsplit_l: assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3673 |
shows "isrlfm (rsplit f a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3674 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3675 |
from rsplit0_cs[where t="a"] have th: "\<forall> (\<phi>,n,s) \<in> set (rsplit0 a). numbound0 s \<and> isrlfm \<phi>" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3676 |
from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3677 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3678 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3679 |
lemma rsplit: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3680 |
assumes xp: "x \<ge> 0" and x1: "x < 1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3681 |
and f: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3682 |
shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3683 |
proof(auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3684 |
let ?I = "\<lambda>x p. Ifm (x#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3685 |
let ?N = "\<lambda> x t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3686 |
assume "?I x (rsplit f a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3687 |
hence "\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). ?I x (And \<phi> (f n s))" using rsplit_ex by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3688 |
then obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and "?I x (And \<phi> (f n s))" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3689 |
hence \<phi>: "?I x \<phi>" and fns: "?I x (f n s)" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3690 |
from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3691 |
have th: "(?N x a = ?N x (CN 0 n s)) \<and> numbound0 s" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3692 |
from f[rule_format, OF th] fns show "?I x (g a)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3693 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3694 |
let ?I = "\<lambda>x p. Ifm (x#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3695 |
let ?N = "\<lambda> x t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3696 |
assume ga: "?I x (g a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3697 |
from rsplit0_complete[OF xp x1, where bs="bs" and t="a"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3698 |
obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and fx: "?I x \<phi>" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3699 |
from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3700 |
have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3701 |
with ga f have "?I x (f n s)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3702 |
with rsplit_ex fnsS fx show "?I x (rsplit f a)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3703 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3704 |
|
23997 | 3705 |
definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3706 |
lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3707 |
else (Gt (CN 0 (-c) (Neg t))))" |
23858 | 3708 |
|
23997 | 3709 |
definition le :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3710 |
le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3711 |
else (Ge (CN 0 (-c) (Neg t))))" |
23858 | 3712 |
|
23997 | 3713 |
definition gt :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3714 |
gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3715 |
else (Lt (CN 0 (-c) (Neg t))))" |
23858 | 3716 |
|
23997 | 3717 |
definition ge :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3718 |
ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3719 |
else (Le (CN 0 (-c) (Neg t))))" |
23858 | 3720 |
|
23997 | 3721 |
definition eq :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3722 |
eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3723 |
else (Eq (CN 0 (-c) (Neg t))))" |
23858 | 3724 |
|
23997 | 3725 |
definition neq :: "int \<Rightarrow> num \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3726 |
neq_def: "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3727 |
else (NEq (CN 0 (-c) (Neg t))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3728 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3729 |
lemma lt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3730 |
(is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _\<longrightarrow> ?I (lt n s) = ?I (Lt a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3731 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3732 |
fix a n s |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3733 |
assume H: "?N a = ?N (CN 0 n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3734 |
show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def)) |
41849 | 3735 |
(cases "n > 0", simp_all add: lt_def algebra_simps myless[of _ "0"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3736 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3737 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3738 |
lemma lt_l: "isrlfm (rsplit lt a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3739 |
by (rule rsplit_l[where f="lt" and a="a"], auto simp add: lt_def, |
58259 | 3740 |
case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3741 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3742 |
lemma le_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (le n s) = Ifm (x#bs) (Le a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (le n s) = ?I (Le a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3743 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3744 |
fix a n s |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3745 |
assume H: "?N a = ?N (CN 0 n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3746 |
show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def)) |
41849 | 3747 |
(cases "n > 0", simp_all add: le_def algebra_simps myle[of _ "0"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3748 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3749 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3750 |
lemma le_l: "isrlfm (rsplit le a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3751 |
by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def) |
58259 | 3752 |
(case_tac s, simp_all, rename_tac nat a b, case_tac "nat",simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3753 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3754 |
lemma gt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (gt n s) = ?I (Gt a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3755 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3756 |
fix a n s |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3757 |
assume H: "?N a = ?N (CN 0 n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3758 |
show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def)) |
41849 | 3759 |
(cases "n > 0", simp_all add: gt_def algebra_simps myless[of _ "0"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3760 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3761 |
lemma gt_l: "isrlfm (rsplit gt a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3762 |
by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def) |
58259 | 3763 |
(case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3764 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3765 |
lemma ge_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (ge n s) = ?I (Ge a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3766 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3767 |
fix a n s |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3768 |
assume H: "?N a = ?N (CN 0 n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3769 |
show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def)) |
41849 | 3770 |
(cases "n > 0", simp_all add: ge_def algebra_simps myle[of _ "0"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3771 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3772 |
lemma ge_l: "isrlfm (rsplit ge a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3773 |
by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def) |
58259 | 3774 |
(case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3775 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3776 |
lemma eq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (eq n s) = ?I (Eq a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3777 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3778 |
fix a n s |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3779 |
assume H: "?N a = ?N (CN 0 n s)" |
29667 | 3780 |
show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3781 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3782 |
lemma eq_l: "isrlfm (rsplit eq a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3783 |
by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def) |
58259 | 3784 |
(case_tac s, simp_all, rename_tac nat a b, case_tac"nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3785 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3786 |
lemma neq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (neq n s) = ?I (NEq a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3787 |
proof(clarify) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3788 |
fix a n s bs |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3789 |
assume H: "?N a = ?N (CN 0 n s)" |
29667 | 3790 |
show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3791 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3792 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3793 |
lemma neq_l: "isrlfm (rsplit neq a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3794 |
by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def) |
58259 | 3795 |
(case_tac s, simp_all, rename_tac nat a b, case_tac"nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3796 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3797 |
lemma small_le: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3798 |
assumes u0:"0 \<le> u" and u1: "u < 1" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3799 |
shows "(-u \<le> real_of_int (n::int)) = (0 \<le> n)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3800 |
using u0 u1 by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3801 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3802 |
lemma small_lt: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3803 |
assumes u0:"0 \<le> u" and u1: "u < 1" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3804 |
shows "(real_of_int (n::int) < real_of_int (m::int) - u) = (n < m)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3805 |
using u0 u1 by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3806 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3807 |
lemma rdvd01_cs: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3808 |
assumes up: "u \<ge> 0" and u1: "u<1" and np: "real_of_int n > 0" |
61942 | 3809 |
shows "(real_of_int (i::int) rdvd real_of_int (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real_of_int n * u = s - real_of_int \<lfloor>s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>s\<rfloor>))" (is "?lhs = ?rhs") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3810 |
proof- |
61942 | 3811 |
let ?ss = "s - real_of_int \<lfloor>s\<rfloor>" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3812 |
from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]] |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
3813 |
of_int_floor_le have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" by (auto simp: floor_less_cancel) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3814 |
from np have n0: "real_of_int n \<ge> 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3815 |
from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3816 |
have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3817 |
from int_rdvd_real[where i="i" and x="real_of_int (n::int) * u - s"] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3818 |
have "real_of_int i rdvd real_of_int n * u - s = |
61942 | 3819 |
(i dvd \<lfloor>real_of_int n * u - s\<rfloor> \<and> (real_of_int \<lfloor>real_of_int n * u - s\<rfloor> = real_of_int n * u - s ))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3820 |
(is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp |
61942 | 3821 |
also have "\<dots> = (?DE \<and> real_of_int (\<lfloor>real_of_int n * u - s\<rfloor> + \<lfloor>s\<rfloor>) \<ge> -?ss |
3822 |
\<and> real_of_int (\<lfloor>real_of_int n * u - s\<rfloor> + \<lfloor>s\<rfloor>) < real_of_int n - ?ss)" (is "_=(?DE \<and>real_of_int ?a \<ge> _ \<and> real_of_int ?a < _)") |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3823 |
using nu0 nun by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3824 |
also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3825 |
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3826 |
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int (\<lfloor>real_of_int n * u - s\<rfloor>) = real_of_int j - real_of_int \<lfloor>s\<rfloor> ))" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3827 |
by (simp only: algebra_simps of_int_diff[symmetric] of_int_eq_iff) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3828 |
also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * u - s = real_of_int j - real_of_int \<lfloor>s\<rfloor> \<and> real_of_int i rdvd real_of_int n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real_of_int n * u - s\<rfloor>"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3829 |
by (auto cong: conj_cong) |
29667 | 3830 |
also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps ) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3831 |
finally show ?thesis . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3832 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3833 |
|
23858 | 3834 |
definition |
3835 |
DVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" |
|
3836 |
where |
|
41836 | 3837 |
DVDJ_def: "DVDJ i n s = (foldr disj (map (\<lambda> j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) F)" |
23858 | 3838 |
|
3839 |
definition |
|
3840 |
NDVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" |
|
3841 |
where |
|
41836 | 3842 |
NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) T)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3843 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3844 |
lemma DVDJ_DVD: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3845 |
assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3846 |
shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3847 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3848 |
let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3849 |
let ?s= "Inum (x#bs) s" |
41836 | 3850 |
from foldr_disj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3851 |
have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" |
41836 | 3852 |
by (simp add: np DVDJ_def) |
61942 | 3853 |
also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int \<lfloor>- ?s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>- ?s\<rfloor>))" |
51369 | 3854 |
by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3855 |
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3856 |
have "\<dots> = (real_of_int i rdvd real_of_int n * x - (-?s))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3857 |
finally show ?thesis by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3858 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3859 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3860 |
lemma NDVDJ_NDVD: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3861 |
assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3862 |
shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3863 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3864 |
let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3865 |
let ?s= "Inum (x#bs) s" |
41836 | 3866 |
from foldr_conj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3867 |
have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" |
41836 | 3868 |
by (simp add: np NDVDJ_def) |
61942 | 3869 |
also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int \<lfloor>- ?s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>- ?s\<rfloor>)))" |
51369 | 3870 |
by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3871 |
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
3872 |
have "\<dots> = (\<not> (real_of_int i rdvd real_of_int n * x - (-?s)))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3873 |
finally show ?thesis by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3874 |
qed |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3875 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3876 |
lemma foldr_disj_map_rlfm2: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3877 |
assumes lf: "\<forall> n . isrlfm (f n)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3878 |
shows "isrlfm (foldr disj (map f xs) F)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3879 |
using lf by (induct xs, auto) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3880 |
lemma foldr_And_map_rlfm2: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3881 |
assumes lf: "\<forall> n . isrlfm (f n)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3882 |
shows "isrlfm (foldr conj (map f xs) T)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3883 |
using lf by (induct xs, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3884 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3885 |
lemma DVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3886 |
shows "isrlfm (DVDJ i n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3887 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3888 |
let ?f="\<lambda>j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3889 |
(Dvd i (Sub (C j) (Floor (Neg s))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3890 |
have th: "\<forall> j. isrlfm (?f j)" using nb np by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3891 |
from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3892 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3893 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3894 |
lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3895 |
shows "isrlfm (NDVDJ i n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3896 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3897 |
let ?f="\<lambda>j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3898 |
(NDvd i (Sub (C j) (Floor (Neg s))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3899 |
have th: "\<forall> j. isrlfm (?f j)" using nb np by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3900 |
from NDVDJ_def foldr_And_map_rlfm2[OF th] show ?thesis by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3901 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3902 |
|
23997 | 3903 |
definition DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where |
23858 | 3904 |
DVD_def: "DVD i c t = |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3905 |
(if i=0 then eq c t else |
61945 | 3906 |
if c = 0 then (Dvd i t) else if c >0 then DVDJ \<bar>i\<bar> c t else DVDJ \<bar>i\<bar> (-c) (Neg t))" |
23858 | 3907 |
|
23997 | 3908 |
definition NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where |
23858 | 3909 |
"NDVD i c t = |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3910 |
(if i=0 then neq c t else |
61945 | 3911 |
if c = 0 then (NDvd i t) else if c >0 then NDVDJ \<bar>i\<bar> c t else NDVDJ \<bar>i\<bar> (-c) (Neg t))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3912 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3913 |
lemma DVD_mono: |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3914 |
assumes xp: "0\<le> x" and x1: "x < 1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3915 |
shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3916 |
(is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (DVD i n s) = ?I (Dvd i a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3917 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3918 |
fix a n s |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3919 |
assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3920 |
let ?th = "?I (DVD i n s) = ?I (Dvd i a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3921 |
have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3922 |
moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3923 |
by (simp add: DVD_def rdvd_left_0_eq)} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3924 |
moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3925 |
moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3926 |
by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3927 |
rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3928 |
moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3929 |
ultimately show ?th by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3930 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3931 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3932 |
lemma NDVD_mono: assumes xp: "0\<le> x" and x1: "x < 1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3933 |
shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3934 |
(is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (NDVD i n s) = ?I (NDvd i a)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3935 |
proof(clarify) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3936 |
fix a n s |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3937 |
assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3938 |
let ?th = "?I (NDVD i n s) = ?I (NDvd i a)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3939 |
have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3940 |
moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3941 |
by (simp add: NDVD_def rdvd_left_0_eq)} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3942 |
moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3943 |
moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3944 |
by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3945 |
rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3946 |
moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3947 |
by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3948 |
ultimately show ?th by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3949 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3950 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3951 |
lemma DVD_l: "isrlfm (rsplit (DVD i) a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3952 |
by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l) |
58259 | 3953 |
(case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3954 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3955 |
lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3956 |
by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l) |
58259 | 3957 |
(case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3958 |
|
66809 | 3959 |
fun rlfm :: "fm \<Rightarrow> fm" |
3960 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3961 |
"rlfm (And p q) = conj (rlfm p) (rlfm q)" |
66809 | 3962 |
| "rlfm (Or p q) = disj (rlfm p) (rlfm q)" |
3963 |
| "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" |
|
3964 |
| "rlfm (Iff p q) = disj (conj(rlfm p) (rlfm q)) (conj(rlfm (NOT p)) (rlfm (NOT q)))" |
|
3965 |
| "rlfm (Lt a) = rsplit lt a" |
|
3966 |
| "rlfm (Le a) = rsplit le a" |
|
3967 |
| "rlfm (Gt a) = rsplit gt a" |
|
3968 |
| "rlfm (Ge a) = rsplit ge a" |
|
3969 |
| "rlfm (Eq a) = rsplit eq a" |
|
3970 |
| "rlfm (NEq a) = rsplit neq a" |
|
3971 |
| "rlfm (Dvd i a) = rsplit (\<lambda> t. DVD i t) a" |
|
3972 |
| "rlfm (NDvd i a) = rsplit (\<lambda> t. NDVD i t) a" |
|
3973 |
| "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" |
|
3974 |
| "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" |
|
3975 |
| "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" |
|
3976 |
| "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" |
|
3977 |
| "rlfm (NOT (NOT p)) = rlfm p" |
|
3978 |
| "rlfm (NOT T) = F" |
|
3979 |
| "rlfm (NOT F) = T" |
|
3980 |
| "rlfm (NOT (Lt a)) = simpfm (rlfm (Ge a))" |
|
3981 |
| "rlfm (NOT (Le a)) = simpfm (rlfm (Gt a))" |
|
3982 |
| "rlfm (NOT (Gt a)) = simpfm (rlfm (Le a))" |
|
3983 |
| "rlfm (NOT (Ge a)) = simpfm (rlfm (Lt a))" |
|
3984 |
| "rlfm (NOT (Eq a)) = simpfm (rlfm (NEq a))" |
|
3985 |
| "rlfm (NOT (NEq a)) = simpfm (rlfm (Eq a))" |
|
3986 |
| "rlfm (NOT (Dvd i a)) = simpfm (rlfm (NDvd i a))" |
|
3987 |
| "rlfm (NOT (NDvd i a)) = simpfm (rlfm (Dvd i a))" |
|
3988 |
| "rlfm p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3989 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3990 |
lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3991 |
by (induct p rule: isrlfm.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3992 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3993 |
lemma simpfm_rl: "isrlfm p \<Longrightarrow> isrlfm (simpfm p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3994 |
proof (induct p) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3995 |
case (Lt a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3996 |
hence "bound0 (Lt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 3997 |
by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
3998 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
3999 |
{assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4000 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4001 |
have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4002 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4003 |
moreover |
41891 | 4004 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4005 |
{ assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4006 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4007 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4008 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4009 |
by (simp add: numgcd_def) |
60533 | 4010 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4011 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4012 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4013 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4014 |
} |
41891 | 4015 |
with Lt a have ?case |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4016 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4017 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4018 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4019 |
case (Le a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4020 |
hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 4021 |
by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4022 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4023 |
{ assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4024 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4025 |
have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4026 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4027 |
moreover |
41891 | 4028 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4029 |
{ assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4030 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4031 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4032 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4033 |
by (simp add: numgcd_def) |
60533 | 4034 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4035 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4036 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4037 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4038 |
} |
41891 | 4039 |
with Le a have ?case |
51369 | 4040 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4041 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4042 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4043 |
case (Gt a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4044 |
hence "bound0 (Gt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 4045 |
by (cases a, simp_all, rename_tac nat a b,case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4046 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4047 |
{assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4048 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4049 |
have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4050 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4051 |
moreover |
41891 | 4052 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4053 |
{ assume cn1: "numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4054 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4055 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4056 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4057 |
by (simp add: numgcd_def) |
60533 | 4058 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4059 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4060 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4061 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4062 |
} |
41891 | 4063 |
with Gt a have ?case |
51369 | 4064 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4065 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4066 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4067 |
case (Ge a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4068 |
hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 4069 |
by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4070 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4071 |
{ assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4072 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4073 |
have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4074 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4075 |
moreover |
41891 | 4076 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4077 |
{ assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4078 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4079 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4080 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4081 |
by (simp add: numgcd_def) |
60533 | 4082 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4083 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4084 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4085 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4086 |
} |
41891 | 4087 |
with Ge a have ?case |
51369 | 4088 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4089 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4090 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4091 |
case (Eq a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4092 |
hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 4093 |
by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4094 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4095 |
{ assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4096 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4097 |
have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4098 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4099 |
moreover |
41891 | 4100 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4101 |
{ assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4102 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4103 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4104 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4105 |
by (simp add: numgcd_def) |
60533 | 4106 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4107 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4108 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4109 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4110 |
} |
41891 | 4111 |
with Eq a have ?case |
51369 | 4112 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4113 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4114 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4115 |
case (NEq a) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4116 |
hence "bound0 (NEq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" |
58259 | 4117 |
by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4118 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4119 |
{assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4120 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4121 |
have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4122 |
with bn bound0at_l have ?case by blast} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4123 |
moreover |
41891 | 4124 |
{ fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e" |
4125 |
{ assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4126 |
with numgcd_pos[where t="CN 0 c (simpnum e)"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4127 |
have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4128 |
from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" |
41849 | 4129 |
by (simp add: numgcd_def) |
60533 | 4130 |
from \<open>c > 0\<close> have th': "c\<noteq>0" by auto |
4131 |
from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp |
|
47142 | 4132 |
from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] |
41891 | 4133 |
have "0 < c div numgcd (CN 0 c (simpnum e))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4134 |
} |
41891 | 4135 |
with NEq a have ?case |
51369 | 4136 |
by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4137 |
ultimately show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4138 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4139 |
case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4140 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4141 |
have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4142 |
with bn bound0at_l show ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4143 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4144 |
case (NDvd i a) hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4145 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4146 |
have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4147 |
with bn bound0at_l show ?case by blast |
51369 | 4148 |
qed(auto simp add: conj_def imp_def disj_def iff_def Let_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4149 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4150 |
lemma rlfm_I: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4151 |
assumes qfp: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4152 |
and xp: "0 \<le> x" and x1: "x < 1" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4153 |
shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \<and> isrlfm (rlfm p)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4154 |
using qfp |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4155 |
by (induct p rule: rlfm.induct) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4156 |
(auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4157 |
rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4158 |
rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4159 |
lemma rlfm_l: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4160 |
assumes qfp: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4161 |
shows "isrlfm (rlfm p)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4162 |
using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l |
51369 | 4163 |
by (induct p rule: rlfm.induct) (auto simp add: simpfm_rl) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4164 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4165 |
(* Operations needed for Ferrante and Rackoff *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4166 |
lemma rminusinf_inf: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4167 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4168 |
shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4169 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4170 |
proof (induct p rule: minusinf.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4171 |
case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4172 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4173 |
case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4174 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4175 |
case (3 c e) |
41891 | 4176 |
from 3 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4177 |
from 3 have cp: "real_of_int c > 0" by simp |
26932 | 4178 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4179 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4180 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4181 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4182 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4183 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4184 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4185 |
hence "real_of_int c * x + ?e < 0" by arith |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4186 |
hence "real_of_int c * x + ?e \<noteq> 0" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4187 |
with xz have "?P ?z x (Eq (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4188 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4189 |
hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4190 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4191 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4192 |
case (4 c e) |
41891 | 4193 |
from 4 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4194 |
from 4 have cp: "real_of_int c > 0" by simp |
26932 | 4195 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4196 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4197 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4198 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4199 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4200 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4201 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4202 |
hence "real_of_int c * x + ?e < 0" by arith |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4203 |
hence "real_of_int c * x + ?e \<noteq> 0" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4204 |
with xz have "?P ?z x (NEq (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4205 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4206 |
hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4207 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4208 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4209 |
case (5 c e) |
41891 | 4210 |
from 5 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4211 |
from 5 have cp: "real_of_int c > 0" by simp |
26932 | 4212 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4213 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4214 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4215 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4216 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4217 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4218 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4219 |
hence "real_of_int c * x + ?e < 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4220 |
with xz have "?P ?z x (Lt (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4221 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4222 |
hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4223 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4224 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4225 |
case (6 c e) |
41891 | 4226 |
from 6 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4227 |
from 6 have cp: "real_of_int c > 0" by simp |
26932 | 4228 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4229 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4230 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4231 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4232 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4233 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4234 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4235 |
hence "real_of_int c * x + ?e < 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4236 |
with xz have "?P ?z x (Le (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4237 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4238 |
hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4239 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4240 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4241 |
case (7 c e) |
41891 | 4242 |
from 7 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4243 |
from 7 have cp: "real_of_int c > 0" by simp |
26932 | 4244 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4245 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4246 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4247 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4248 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4249 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4250 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4251 |
hence "real_of_int c * x + ?e < 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4252 |
with xz have "?P ?z x (Gt (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4253 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4254 |
hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4255 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4256 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4257 |
case (8 c e) |
41891 | 4258 |
from 8 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4259 |
from 8 have cp: "real_of_int c > 0" by simp |
26932 | 4260 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4261 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4262 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4263 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4264 |
assume xz: "x < ?z" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4265 |
hence "(real_of_int c * x < - ?e)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4266 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4267 |
hence "real_of_int c * x + ?e < 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4268 |
with xz have "?P ?z x (Ge (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4269 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4270 |
hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4271 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4272 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4273 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4274 |
lemma rplusinf_inf: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4275 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4276 |
shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4277 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4278 |
proof (induct p rule: isrlfm.induct) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4279 |
case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4280 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4281 |
case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4282 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4283 |
case (3 c e) |
41891 | 4284 |
from 3 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4285 |
from 3 have cp: "real_of_int c > 0" by simp |
26932 | 4286 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4287 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4288 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4289 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4290 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4291 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4292 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4293 |
hence "real_of_int c * x + ?e > 0" by arith |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4294 |
hence "real_of_int c * x + ?e \<noteq> 0" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4295 |
with xz have "?P ?z x (Eq (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4296 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4297 |
hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4298 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4299 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4300 |
case (4 c e) |
41891 | 4301 |
from 4 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4302 |
from 4 have cp: "real_of_int c > 0" by simp |
26932 | 4303 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4304 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4305 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4306 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4307 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4308 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4309 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4310 |
hence "real_of_int c * x + ?e > 0" by arith |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4311 |
hence "real_of_int c * x + ?e \<noteq> 0" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4312 |
with xz have "?P ?z x (NEq (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4313 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4314 |
hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4315 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4316 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4317 |
case (5 c e) |
41891 | 4318 |
from 5 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4319 |
from 5 have cp: "real_of_int c > 0" by simp |
26932 | 4320 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4321 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4322 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4323 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4324 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4325 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4326 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4327 |
hence "real_of_int c * x + ?e > 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4328 |
with xz have "?P ?z x (Lt (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4329 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4330 |
hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4331 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4332 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4333 |
case (6 c e) |
41891 | 4334 |
from 6 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4335 |
from 6 have cp: "real_of_int c > 0" by simp |
26932 | 4336 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4337 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4338 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4339 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4340 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4341 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4342 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4343 |
hence "real_of_int c * x + ?e > 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4344 |
with xz have "?P ?z x (Le (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4345 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4346 |
hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4347 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4348 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4349 |
case (7 c e) |
41891 | 4350 |
from 7 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4351 |
from 7 have cp: "real_of_int c > 0" by simp |
26932 | 4352 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4353 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4354 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4355 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4356 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4357 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4358 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4359 |
hence "real_of_int c * x + ?e > 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4360 |
with xz have "?P ?z x (Gt (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4361 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4362 |
hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4363 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4364 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4365 |
case (8 c e) |
41891 | 4366 |
from 8 have nb: "numbound0 e" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4367 |
from 8 have cp: "real_of_int c > 0" by simp |
26932 | 4368 |
fix a |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4369 |
let ?e="Inum (a#bs) e" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4370 |
let ?z = "(- ?e) / real_of_int c" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4371 |
{fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4372 |
assume xz: "x > ?z" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4373 |
with mult_strict_right_mono [OF xz cp] cp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4374 |
have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4375 |
hence "real_of_int c * x + ?e > 0" by arith |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4376 |
with xz have "?P ?z x (Ge (CN 0 c e))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4377 |
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4378 |
hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4379 |
thus ?case by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4380 |
qed simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4381 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4382 |
lemma rminusinf_bound0: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4383 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4384 |
shows "bound0 (minusinf p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4385 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4386 |
by (induct p rule: minusinf.induct) simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4387 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4388 |
lemma rplusinf_bound0: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4389 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4390 |
shows "bound0 (plusinf p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4391 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4392 |
by (induct p rule: plusinf.induct) simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4393 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4394 |
lemma rminusinf_ex: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4395 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4396 |
and ex: "Ifm (a#bs) (minusinf p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4397 |
shows "\<exists> x. Ifm (x#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4398 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4399 |
from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4400 |
have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4401 |
from rminusinf_inf[OF lp, where bs="bs"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4402 |
obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4403 |
from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4404 |
moreover have "z - 1 < z" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4405 |
ultimately show ?thesis using z_def by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4406 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4407 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4408 |
lemma rplusinf_ex: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4409 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4410 |
and ex: "Ifm (a#bs) (plusinf p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4411 |
shows "\<exists> x. Ifm (x#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4412 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4413 |
from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4414 |
have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4415 |
from rplusinf_inf[OF lp, where bs="bs"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4416 |
obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4417 |
from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4418 |
moreover have "z + 1 > z" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4419 |
ultimately show ?thesis using z_def by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4420 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4421 |
|
66809 | 4422 |
fun \<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list" |
4423 |
where |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4424 |
"\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)" |
66809 | 4425 |
| "\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)" |
4426 |
| "\<Upsilon> (Eq (CN 0 c e)) = [(Neg e,c)]" |
|
4427 |
| "\<Upsilon> (NEq (CN 0 c e)) = [(Neg e,c)]" |
|
4428 |
| "\<Upsilon> (Lt (CN 0 c e)) = [(Neg e,c)]" |
|
4429 |
| "\<Upsilon> (Le (CN 0 c e)) = [(Neg e,c)]" |
|
4430 |
| "\<Upsilon> (Gt (CN 0 c e)) = [(Neg e,c)]" |
|
4431 |
| "\<Upsilon> (Ge (CN 0 c e)) = [(Neg e,c)]" |
|
4432 |
| "\<Upsilon> p = []" |
|
4433 |
||
4434 |
fun \<upsilon> :: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" |
|
4435 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4436 |
"\<upsilon> (And p q) = (\<lambda> (t,n). And (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))" |
66809 | 4437 |
| "\<upsilon> (Or p q) = (\<lambda> (t,n). Or (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))" |
4438 |
| "\<upsilon> (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))" |
|
4439 |
| "\<upsilon> (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))" |
|
4440 |
| "\<upsilon> (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))" |
|
4441 |
| "\<upsilon> (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))" |
|
4442 |
| "\<upsilon> (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))" |
|
4443 |
| "\<upsilon> (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))" |
|
4444 |
| "\<upsilon> p = (\<lambda> (t,n). p)" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4445 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4446 |
lemma \<upsilon>_I: assumes lp: "isrlfm p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4447 |
and np: "real_of_int n > 0" and nbt: "numbound0 t" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4448 |
shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real_of_int n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4449 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4450 |
proof(induct p rule: \<upsilon>.induct) |
41891 | 4451 |
case (5 c e) |
4452 |
from 5 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4453 |
have "?I ?u (Lt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) < 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4454 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4455 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) < 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4456 |
by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4457 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4458 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) < 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4459 |
using np by simp |
29667 | 4460 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4461 |
next |
41891 | 4462 |
case (6 c e) |
4463 |
from 6 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4464 |
have "?I ?u (Le (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<le> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4465 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4466 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<le> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4467 |
by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4468 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4469 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<le> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4470 |
using np by simp |
29667 | 4471 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4472 |
next |
41891 | 4473 |
case (7 c e) |
4474 |
from 7 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4475 |
have "?I ?u (Gt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) > 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4476 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4477 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) > 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4478 |
by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4479 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4480 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) > 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4481 |
using np by simp |
29667 | 4482 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4483 |
next |
41891 | 4484 |
case (8 c e) |
4485 |
from 8 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4486 |
have "?I ?u (Ge (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<ge> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4487 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4488 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<ge> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4489 |
by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4490 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4491 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<ge> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4492 |
using np by simp |
29667 | 4493 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4494 |
next |
41891 | 4495 |
case (3 c e) |
4496 |
from 3 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4497 |
from np have np: "real_of_int n \<noteq> 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4498 |
have "?I ?u (Eq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) = 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4499 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4500 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) = 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4501 |
by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4502 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4503 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) = 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4504 |
using np by simp |
29667 | 4505 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4506 |
next |
41891 | 4507 |
case (4 c e) |
4508 |
from 4 have cp: "c >0" and nb: "numbound0 e" by simp_all |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4509 |
from np have np: "real_of_int n \<noteq> 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4510 |
have "?I ?u (NEq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<noteq> 0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4511 |
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4512 |
also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4513 |
by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" |
64240 | 4514 |
and b="0", simplified div_0]) (simp only: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4515 |
also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<noteq> 0)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4516 |
using np by simp |
29667 | 4517 |
finally show ?case using nbt nb by (simp add: algebra_simps) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4518 |
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real_of_int n" and b'="x"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4519 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4520 |
lemma \<Upsilon>_l: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4521 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4522 |
shows "\<forall> (t,k) \<in> set (\<Upsilon> p). numbound0 t \<and> k >0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4523 |
using lp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4524 |
by(induct p rule: \<Upsilon>.induct) auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4525 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4526 |
lemma rminusinf_\<Upsilon>: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4527 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4528 |
and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4529 |
and ex: "Ifm (x#bs) p" (is "?I x p") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4530 |
shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<ge> Inum (a#bs) s / real_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real_of_int m") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4531 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4532 |
have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int m *x \<ge> ?N a s") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4533 |
using lp nmi ex |
41849 | 4534 |
by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4535 |
then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<ge> ?N a s" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4536 |
from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4537 |
from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real_of_int m" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
4538 |
by (auto simp add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4539 |
thus ?thesis using smU by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4540 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4541 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4542 |
lemma rplusinf_\<Upsilon>: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4543 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4544 |
and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4545 |
and ex: "Ifm (x#bs) p" (is "?I x p") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4546 |
shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<le> Inum (a#bs) s / real_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real_of_int m") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4547 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4548 |
have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int m *x \<le> ?N a s") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4549 |
using lp nmi ex |
41849 | 4550 |
by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4551 |
then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<le> ?N a s" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4552 |
from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4553 |
from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real_of_int m" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
4554 |
by (auto simp add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4555 |
thus ?thesis using smU by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4556 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4557 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4558 |
lemma lin_dense: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4559 |
assumes lp: "isrlfm p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4560 |
and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real_of_int n) ` set (\<Upsilon> p)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4561 |
(is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real_of_int n ) ` (?U p)") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4562 |
and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4563 |
and ly: "l < y" and yu: "y < u" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4564 |
shows "Ifm (y#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4565 |
using lp px noS |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4566 |
proof (induct p rule: isrlfm.induct) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4567 |
case (5 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4568 |
from 5 have "x * real_of_int c + ?N x e < 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4569 |
hence pxc: "x < (- ?N x e) / real_of_int c" |
41891 | 4570 |
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4571 |
from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4572 |
with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4573 |
hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4574 |
moreover {assume y: "y < (-?N x e)/ real_of_int c" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4575 |
hence "y * real_of_int c < - ?N x e" |
41891 | 4576 |
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4577 |
hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps) |
41891 | 4578 |
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4579 |
moreover {assume y: "y > (- ?N x e) / real_of_int c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4580 |
with yu have eu: "u > (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4581 |
with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto) |
41891 | 4582 |
with lx pxc have "False" by auto |
4583 |
hence ?case by simp } |
|
4584 |
ultimately show ?case by blast |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4585 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4586 |
case (6 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4587 |
from 6 have "x * real_of_int c + ?N x e \<le> 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4588 |
hence pxc: "x \<le> (- ?N x e) / real_of_int c" |
41891 | 4589 |
by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4590 |
from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4591 |
with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4592 |
hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4593 |
moreover {assume y: "y < (-?N x e)/ real_of_int c" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4594 |
hence "y * real_of_int c < - ?N x e" |
41891 | 4595 |
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4596 |
hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps) |
41891 | 4597 |
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4598 |
moreover {assume y: "y > (- ?N x e) / real_of_int c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4599 |
with yu have eu: "u > (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4600 |
with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto) |
41891 | 4601 |
with lx pxc have "False" by auto |
4602 |
hence ?case by simp } |
|
4603 |
ultimately show ?case by blast |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4604 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4605 |
case (7 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4606 |
from 7 have "x * real_of_int c + ?N x e > 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4607 |
hence pxc: "x > (- ?N x e) / real_of_int c" |
41891 | 4608 |
by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4609 |
from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4610 |
with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4611 |
hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4612 |
moreover {assume y: "y > (-?N x e)/ real_of_int c" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4613 |
hence "y * real_of_int c > - ?N x e" |
41891 | 4614 |
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4615 |
hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps) |
41891 | 4616 |
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4617 |
moreover {assume y: "y < (- ?N x e) / real_of_int c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4618 |
with ly have eu: "l < (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4619 |
with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto) |
41891 | 4620 |
with xu pxc have "False" by auto |
4621 |
hence ?case by simp } |
|
4622 |
ultimately show ?case by blast |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4623 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4624 |
case (8 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4625 |
from 8 have "x * real_of_int c + ?N x e \<ge> 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4626 |
hence pxc: "x \<ge> (- ?N x e) / real_of_int c" |
41891 | 4627 |
by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4628 |
from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4629 |
with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4630 |
hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4631 |
moreover {assume y: "y > (-?N x e)/ real_of_int c" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4632 |
hence "y * real_of_int c > - ?N x e" |
41891 | 4633 |
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4634 |
hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps) |
41891 | 4635 |
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4636 |
moreover {assume y: "y < (- ?N x e) / real_of_int c" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4637 |
with ly have eu: "l < (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4638 |
with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto) |
41891 | 4639 |
with xu pxc have "False" by auto |
4640 |
hence ?case by simp } |
|
4641 |
ultimately show ?case by blast |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4642 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4643 |
case (3 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4644 |
from cp have cnz: "real_of_int c \<noteq> 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4645 |
from 3 have "x * real_of_int c + ?N x e = 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4646 |
hence pxc: "x = (- ?N x e) / real_of_int c" |
41891 | 4647 |
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4648 |
from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4649 |
with lx xu have yne: "x \<noteq> - ?N x e / real_of_int c" by auto |
41891 | 4650 |
with pxc show ?case by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4651 |
next |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4652 |
case (4 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4653 |
from cp have cnz: "real_of_int c \<noteq> 0" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4654 |
from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4655 |
with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4656 |
hence "y* real_of_int c \<noteq> -?N x e" |
41891 | 4657 |
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4658 |
hence "y* real_of_int c + ?N x e \<noteq> 0" by (simp add: algebra_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4659 |
thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] |
41891 | 4660 |
by (simp add: algebra_simps) |
41849 | 4661 |
qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"]) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4662 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4663 |
lemma rinf_\<Upsilon>: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4664 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4665 |
and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4666 |
and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4667 |
and ex: "\<exists> x. Ifm (x#bs) p" (is "\<exists> x. ?I x p") |
41891 | 4668 |
shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4669 |
?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4670 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4671 |
let ?N = "\<lambda> x t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4672 |
let ?U = "set (\<Upsilon> p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4673 |
from ex obtain a where pa: "?I a p" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4674 |
from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4675 |
have nmi': "\<not> (?I a (?M p))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4676 |
from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4677 |
have npi': "\<not> (?I a (?P p))" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4678 |
have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((?N a l/real_of_int n + ?N a s /real_of_int m) / 2) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4679 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4680 |
let ?M = "(\<lambda> (t,c). ?N a t / real_of_int c) ` ?U" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4681 |
have fM: "finite ?M" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4682 |
from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4683 |
have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real_of_int n \<and> a \<ge> ?N x s / real_of_int m" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4684 |
then obtain "t" "n" "s" "m" where |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4685 |
tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4686 |
and xs1: "a \<le> ?N x s / real_of_int m" and tx1: "a \<ge> ?N x t / real_of_int n" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4687 |
from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real_of_int m" and tx: "a \<ge> ?N a t / real_of_int n" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4688 |
from tnU have Mne: "?M \<noteq> {}" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4689 |
hence Une: "?U \<noteq> {}" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4690 |
let ?l = "Min ?M" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4691 |
let ?u = "Max ?M" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4692 |
have linM: "?l \<in> ?M" using fM Mne by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4693 |
have uinM: "?u \<in> ?M" using fM Mne by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4694 |
have tnM: "?N a t / real_of_int n \<in> ?M" using tnU by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4695 |
have smM: "?N a s / real_of_int m \<in> ?M" using smU by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4696 |
have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4697 |
have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4698 |
have "?l \<le> ?N a t / real_of_int n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4699 |
have "?N a s / real_of_int m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4700 |
from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4701 |
have "(\<exists> s\<in> ?M. ?I s p) \<or> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4702 |
(\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4703 |
moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4704 |
hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real_of_int nu" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4705 |
then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real_of_int nu" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4706 |
have "(u + u) / 2 = u" by auto with pu tuu |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4707 |
have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int nu)) / 2) p" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4708 |
with tuU have ?thesis by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4709 |
moreover{ |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4710 |
assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4711 |
then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
4712 |
and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" |
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
4713 |
by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4714 |
from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real_of_int t1n" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4715 |
then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real_of_int t1n" by blast |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4716 |
from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real_of_int t2n" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4717 |
then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real_of_int t2n" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4718 |
from t1x xt2 have t1t2: "t1 < t2" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4719 |
let ?u = "(t1 + t2) / 2" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4720 |
from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4721 |
from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4722 |
with t1uU t2uU t1u t2u have ?thesis by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4723 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4724 |
qed |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4725 |
then obtain "l" "n" "s" "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4726 |
and pu: "?I ((?N a l / real_of_int n + ?N a s / real_of_int m) / 2) p" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4727 |
from lnU smU \<Upsilon>_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4728 |
from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4729 |
numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4730 |
have "?I ((?N x l / real_of_int n + ?N x s / real_of_int m) / 2) p" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4731 |
with lnU smU |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4732 |
show ?thesis by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4733 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4734 |
(* The Ferrante - Rackoff Theorem *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4735 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4736 |
theorem fr_eq: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4737 |
assumes lp: "isrlfm p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4738 |
shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/ real_of_int n + (Inum (x#bs) s) / real_of_int m) /2)#bs) p))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4739 |
(is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4740 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4741 |
assume px: "\<exists> x. ?I x p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4742 |
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4743 |
moreover {assume "?M \<or> ?P" hence "?D" by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4744 |
moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4745 |
from rinf_\<Upsilon>[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4746 |
ultimately show "?D" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4747 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4748 |
assume "?D" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4749 |
moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4750 |
moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4751 |
moreover {assume f:"?F" hence "?E" by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4752 |
ultimately show "?E" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4753 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4754 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4755 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4756 |
lemma fr_eq_\<upsilon>: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4757 |
assumes lp: "isrlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4758 |
shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (\<Upsilon> p). \<exists> (s,l) \<in> set (\<Upsilon> p). Ifm (x#bs) (\<upsilon> p (Add(Mul l t) (Mul k s) , 2*k*l))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4759 |
(is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4760 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4761 |
assume px: "\<exists> x. ?I x p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4762 |
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4763 |
moreover {assume "?M \<or> ?P" hence "?D" by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4764 |
moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4765 |
let ?f ="\<lambda> (t,n). Inum (x#bs) t / real_of_int n" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4766 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4767 |
{fix t n s m assume "(t,n)\<in> set (\<Upsilon> p)" and "(s,m) \<in> set (\<Upsilon> p)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4768 |
with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
4769 |
by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4770 |
let ?st = "Add (Mul m t) (Mul n s)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4771 |
from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4772 |
from tnb snb have st_nb: "numbound0 ?st" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4773 |
have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" |
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31952
diff
changeset
|
4774 |
using mnp mp np by (simp add: algebra_simps add_divide_distrib) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4775 |
from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"] |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4776 |
have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real_of_int n + ?N s / real_of_int m) /2) p" by (simp only: st[symmetric])} |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4777 |
with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4778 |
ultimately show "?D" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4779 |
next |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4780 |
assume "?D" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4781 |
moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4782 |
moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4783 |
moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4784 |
and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4785 |
with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real_of_int k > 0" and snb: "numbound0 s" and mp:"real_of_int l > 0" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4786 |
let ?st = "Add (Mul l t) (Mul k s)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4787 |
from np mp have mnp: "real_of_int (2*k*l) > 0" by (simp add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4788 |
from tnb snb have st_nb: "numbound0 ?st" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4789 |
from \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4790 |
ultimately show "?E" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4791 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4792 |
|
60533 | 4793 |
text\<open>The overall Part\<close> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4794 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4795 |
lemma real_ex_int_real01: |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4796 |
shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real_of_int i + u))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4797 |
proof(auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4798 |
fix x |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4799 |
assume Px: "P x" |
61942 | 4800 |
let ?i = "\<lfloor>x\<rfloor>" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4801 |
let ?u = "x - real_of_int ?i" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4802 |
have "x = real_of_int ?i + ?u" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4803 |
hence "P (real_of_int ?i + ?u)" using Px by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4804 |
moreover have "real_of_int ?i \<le> x" using of_int_floor_le by simp hence "0 \<le> ?u" by arith |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4805 |
moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4806 |
ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real_of_int i + u))" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4807 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4808 |
|
66809 | 4809 |
fun exsplitnum :: "num \<Rightarrow> num" |
4810 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4811 |
"exsplitnum (C c) = (C c)" |
41839 | 4812 |
| "exsplitnum (Bound 0) = Add (Bound 0) (Bound 1)" |
4813 |
| "exsplitnum (Bound n) = Bound (n+1)" |
|
4814 |
| "exsplitnum (Neg a) = Neg (exsplitnum a)" |
|
4815 |
| "exsplitnum (Add a b) = Add (exsplitnum a) (exsplitnum b) " |
|
4816 |
| "exsplitnum (Sub a b) = Sub (exsplitnum a) (exsplitnum b) " |
|
4817 |
| "exsplitnum (Mul c a) = Mul c (exsplitnum a)" |
|
4818 |
| "exsplitnum (Floor a) = Floor (exsplitnum a)" |
|
4819 |
| "exsplitnum (CN 0 c a) = CN 0 c (Add (Mul c (Bound 1)) (exsplitnum a))" |
|
4820 |
| "exsplitnum (CN n c a) = CN (n+1) c (exsplitnum a)" |
|
4821 |
| "exsplitnum (CF c s t) = CF c (exsplitnum s) (exsplitnum t)" |
|
4822 |
||
66809 | 4823 |
fun exsplit :: "fm \<Rightarrow> fm" |
4824 |
where |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4825 |
"exsplit (Lt a) = Lt (exsplitnum a)" |
41839 | 4826 |
| "exsplit (Le a) = Le (exsplitnum a)" |
4827 |
| "exsplit (Gt a) = Gt (exsplitnum a)" |
|
4828 |
| "exsplit (Ge a) = Ge (exsplitnum a)" |
|
4829 |
| "exsplit (Eq a) = Eq (exsplitnum a)" |
|
4830 |
| "exsplit (NEq a) = NEq (exsplitnum a)" |
|
4831 |
| "exsplit (Dvd i a) = Dvd i (exsplitnum a)" |
|
4832 |
| "exsplit (NDvd i a) = NDvd i (exsplitnum a)" |
|
4833 |
| "exsplit (And p q) = And (exsplit p) (exsplit q)" |
|
4834 |
| "exsplit (Or p q) = Or (exsplit p) (exsplit q)" |
|
4835 |
| "exsplit (Imp p q) = Imp (exsplit p) (exsplit q)" |
|
4836 |
| "exsplit (Iff p q) = Iff (exsplit p) (exsplit q)" |
|
4837 |
| "exsplit (NOT p) = NOT (exsplit p)" |
|
4838 |
| "exsplit p = p" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4839 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4840 |
lemma exsplitnum: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4841 |
"Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t" |
29667 | 4842 |
by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4843 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4844 |
lemma exsplit: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4845 |
assumes qfp: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4846 |
shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4847 |
using qfp exsplitnum[where x="x" and y="y" and bs="bs"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4848 |
by(induct p rule: exsplit.induct) simp_all |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4849 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4850 |
lemma splitex: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4851 |
assumes qf: "qfree p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4852 |
shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real_of_int i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4853 |
proof- |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4854 |
have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real_of_int i)#bs) (exsplit p))" |
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61694
diff
changeset
|
4855 |
by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4856 |
also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real_of_int i + x) #bs) p)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
4857 |
by (simp only: exsplit[OF qf] ac_simps) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4858 |
also have "\<dots> = (\<exists> x. Ifm (x#bs) p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4859 |
by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4860 |
finally show ?thesis by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4861 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4862 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4863 |
(* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4864 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
4865 |
definition ferrack01 :: "fm \<Rightarrow> fm" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4866 |
"ferrack01 p \<equiv> (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p); |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4867 |
U = remdups(map simp_num_pair |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4868 |
(map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4869 |
(alluopairs (\<Upsilon> p')))) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4870 |
in decr (evaldjf (\<upsilon> p') U ))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4871 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4872 |
lemma fr_eq_01: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4873 |
assumes qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4874 |
shows "(\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\<exists> (t,n) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \<exists> (s,m) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\<upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4875 |
(is "(\<exists> x. ?I x ?q) = ?F") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4876 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4877 |
let ?rq = "rlfm ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4878 |
let ?M = "?I x (minusinf ?rq)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4879 |
let ?P = "?I x (plusinf ?rq)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4880 |
have MF: "?M = False" |
31706 | 4881 |
apply (simp add: Let_def reducecoeff_def numgcd_def rsplit_def ge_def lt_def conj_def disj_def) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4882 |
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))", simp_all) |
31706 | 4883 |
have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def rsplit_def ge_def lt_def conj_def disj_def) |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58310
diff
changeset
|
4884 |
by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))", simp_all) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4885 |
have "(\<exists> x. ?I x ?q ) = |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4886 |
((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4887 |
(is "(\<exists> x. ?I x ?q) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4888 |
proof |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4889 |
assume "\<exists> x. ?I x ?q" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4890 |
then obtain x where qx: "?I x ?q" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4891 |
hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4892 |
by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4893 |
from qx have "?I x ?rq " |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4894 |
by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4895 |
hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4896 |
from qf have qfq:"isrlfm ?rq" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4897 |
by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) |
50252 | 4898 |
with lqx fr_eq_\<upsilon>[OF qfq] show "?M \<or> ?P \<or> ?F" by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4899 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4900 |
assume D: "?D" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4901 |
let ?U = "set (\<Upsilon> ?rq )" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4902 |
from MF PF D have "?F" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4903 |
then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4904 |
from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4905 |
by (auto simp add: rsplit_def lt_def ge_def) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4906 |
from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0" by (auto simp add: split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4907 |
let ?st = "Add (Mul m t) (Mul n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4908 |
from tnb snb have stnb: "numbound0 ?st" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4909 |
from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4910 |
from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4911 |
have "\<exists> x. ?I x ?rq" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4912 |
thus "?E" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4913 |
using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4914 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4915 |
with MF PF show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4916 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4917 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4918 |
lemma \<Upsilon>_cong_aux: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4919 |
assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4920 |
shows "((\<lambda> (t,n). Inum (x#bs) t /real_of_int n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real_of_int n + Inum (x#bs) s /real_of_int m)/2) ` (set U \<times> set U))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4921 |
(is "?lhs = ?rhs") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4922 |
proof(auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4923 |
fix t n s m |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4924 |
assume "((t,n),(s,m)) \<in> set (alluopairs U)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4925 |
hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4926 |
using alluopairs_set1[where xs="U"] by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4927 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4928 |
let ?st= "Add (Mul m t) (Mul n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4929 |
from Ul th have mnz: "m \<noteq> 0" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4930 |
from Ul th have nnz: "n \<noteq> 0" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4931 |
have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" |
29667 | 4932 |
using mnz nnz by (simp add: algebra_simps add_divide_distrib) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4933 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4934 |
thus "(real_of_int m * Inum (x # bs) t + real_of_int n * Inum (x # bs) s) / |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4935 |
(2 * real_of_int n * real_of_int m) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4936 |
\<in> (\<lambda>((t, n), s, m). |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4937 |
(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4938 |
(set U \<times> set U)"using mnz nnz th |
29667 | 4939 |
apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4940 |
by (rule_tac x="(s,m)" in bexI,simp_all) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
4941 |
(rule_tac x="(t,n)" in bexI,simp_all add: mult.commute) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4942 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4943 |
fix t n s m |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4944 |
assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4945 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4946 |
let ?st= "Add (Mul m t) (Mul n s)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4947 |
from Ul smU have mnz: "m \<noteq> 0" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4948 |
from Ul tnU have nnz: "n \<noteq> 0" by auto |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4949 |
have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" |
29667 | 4950 |
using mnz nnz by (simp add: algebra_simps add_divide_distrib) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4951 |
let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4952 |
have Pc:"\<forall> a b. ?P a b = ?P b a" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4953 |
by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4954 |
from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4955 |
from alluopairs_ex[OF Pc, where xs="U"] tnU smU |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4956 |
have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4957 |
by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4958 |
then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4959 |
and Pts': "?P (t',n') (s',m')" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4960 |
from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4961 |
let ?st' = "Add (Mul m' t') (Mul n' s')" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4962 |
have st': "(?N t' / real_of_int n' + ?N s' / real_of_int m')/2 = ?N ?st' / real_of_int (2*n'*m')" |
29667 | 4963 |
using mnz' nnz' by (simp add: algebra_simps add_divide_distrib) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4964 |
from Pts' have |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4965 |
"(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2" by simp |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4966 |
also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4967 |
finally show "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4968 |
\<in> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4969 |
(\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4970 |
set (alluopairs U)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4971 |
using ts'_U by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4972 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4973 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4974 |
lemma \<Upsilon>_cong: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4975 |
assumes lp: "isrlfm p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4976 |
and UU': "((\<lambda> (t,n). Inum (x#bs) t /real_of_int n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real_of_int n + Inum (x#bs) s /real_of_int m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)") |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4977 |
and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4978 |
and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4979 |
shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (\<upsilon> p (t,n)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4980 |
(is "?lhs = ?rhs") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4981 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4982 |
assume ?lhs |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4983 |
then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4984 |
Pst: "Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4985 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4986 |
from tnU smU U have tnb: "numbound0 t" and np: "n > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4987 |
and snb: "numbound0 s" and mp:"m > 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4988 |
let ?st= "Add (Mul m t) (Mul n s)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4989 |
from np mp have mnp: "real_of_int (2*n*m) > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4990 |
by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4991 |
from tnb snb have stnb: "numbound0 ?st" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4992 |
have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" |
29667 | 4993 |
using mp np by (simp add: algebra_simps add_divide_distrib) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4994 |
from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4995 |
hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4996 |
by auto (rule_tac x="(a,b)" in bexI, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
4997 |
then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4998 |
from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
4999 |
from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5000 |
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5001 |
from conjunct1[OF \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5002 |
have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5003 |
then show ?rhs using tnU' by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5004 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5005 |
assume ?rhs |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5006 |
then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (\<upsilon> p (t', n'))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5007 |
by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5008 |
from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5009 |
hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5010 |
by auto (rule_tac x="(a,b)" in bexI, auto) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5011 |
then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5012 |
th: "?f (t',n') = ?g((t,n),(s,m)) "by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5013 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5014 |
from tnU smU U have tnb: "numbound0 t" and np: "n > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5015 |
and snb: "numbound0 s" and mp:"m > 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5016 |
let ?st= "Add (Mul m t) (Mul n s)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5017 |
from np mp have mnp: "real_of_int (2*n*m) > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5018 |
by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5019 |
from tnb snb have stnb: "numbound0 ?st" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5020 |
have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" |
29667 | 5021 |
using mp np by (simp add: algebra_simps add_divide_distrib) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5022 |
from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5023 |
from \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5024 |
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5025 |
with \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5026 |
qed |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5027 |
|
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5028 |
lemma ferrack01: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5029 |
assumes qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5030 |
shows "((\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \<and> qfree (ferrack01 p)" (is "(?lhs = ?rhs) \<and> _") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5031 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5032 |
let ?I = "\<lambda> x p. Ifm (x#bs) p" |
26935 | 5033 |
fix x |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5034 |
let ?N = "\<lambda> t. Inum (x#bs) t" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5035 |
let ?q = "rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5036 |
let ?U = "\<Upsilon> ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5037 |
let ?Up = "alluopairs ?U" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5038 |
let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5039 |
let ?S = "map ?g ?Up" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5040 |
let ?SS = "map simp_num_pair ?S" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5041 |
let ?Y = "remdups ?SS" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5042 |
let ?f= "(\<lambda> (t,n). ?N t / real_of_int n)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5043 |
let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real_of_int n + ?N s/ real_of_int m) /2" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5044 |
let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5045 |
let ?ep = "evaldjf (\<upsilon> ?q) ?Y" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5046 |
from rlfm_l[OF qf] have lq: "isrlfm ?q" |
31706 | 5047 |
by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5048 |
from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5049 |
from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" . |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5050 |
from U_l UpU |
50241 | 5051 |
have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5052 |
hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 " |
56544 | 5053 |
by (auto) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5054 |
have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5055 |
proof- |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5056 |
{ fix t n assume tnY: "(t,n) \<in> set ?Y" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5057 |
hence "(t,n) \<in> set ?SS" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5058 |
hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)" |
33639
603320b93668
New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents:
33063
diff
changeset
|
5059 |
by (auto simp add: split_def simp del: map_map) |
603320b93668
New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents:
33063
diff
changeset
|
5060 |
(rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5061 |
then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5062 |
from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5063 |
from simp_num_pair_l[OF tnb np tns] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5064 |
have "numbound0 t \<and> n > 0" . } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5065 |
thus ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5066 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5067 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5068 |
have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5069 |
proof- |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5070 |
from simp_num_pair_ci[where bs="x#bs"] have |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5071 |
"\<forall>x. (?f o simp_num_pair) x = ?f x" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5072 |
hence th: "?f o simp_num_pair = ?f" using ext by blast |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55584
diff
changeset
|
5073 |
have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_comp comp_assoc) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5074 |
also have "\<dots> = (?f ` set ?S)" by (simp add: th) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5075 |
also have "\<dots> = ((?f o ?g) ` set ?Up)" |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55584
diff
changeset
|
5076 |
by (simp only: set_map o_def image_comp) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5077 |
also have "\<dots> = (?h ` (set ?U \<times> set ?U))" |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55584
diff
changeset
|
5078 |
using \<Upsilon>_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_comp] by blast |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5079 |
finally show ?thesis . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5080 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5081 |
have "\<forall> (t,n) \<in> set ?Y. bound0 (\<upsilon> ?q (t,n))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5082 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5083 |
{ fix t n assume tnY: "(t,n) \<in> set ?Y" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5084 |
with Y_l have tnb: "numbound0 t" and np: "real_of_int n > 0" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5085 |
from \<upsilon>_I[OF lq np tnb] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5086 |
have "bound0 (\<upsilon> ?q (t,n))" by simp} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5087 |
thus ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5088 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5089 |
hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="\<upsilon> ?q"] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5090 |
by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5091 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5092 |
from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5093 |
by (simp only: split_def fst_conv snd_conv) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5094 |
also have "\<dots> = (\<exists> (t,n) \<in> set ?Y. ?I x (\<upsilon> ?q (t,n)))" using \<Upsilon>_cong[OF lq YU U_l Y_l] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5095 |
by (simp only: split_def fst_conv snd_conv) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5096 |
also have "\<dots> = (Ifm (x#bs) ?ep)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5097 |
using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\<upsilon> ?q",symmetric] |
61424
c3658c18b7bc
prod_case as canonical name for product type eliminator
haftmann
parents:
61076
diff
changeset
|
5098 |
by (simp only: split_def prod.collapse) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5099 |
also have "\<dots> = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5100 |
finally have lr: "?lhs = ?rhs" by (simp only: ferrack01_def Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5101 |
from decr_qf[OF ep_nb] have "qfree (ferrack01 p)" by (simp only: Let_def ferrack01_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5102 |
with lr show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5103 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5104 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5105 |
lemma cp_thm': |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5106 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
50252 | 5107 |
and up: "d_\<beta> p 1" and dd: "d_\<delta> p d" and dp: "d > 0" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5108 |
shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real_of_int i#bs)) ` set (\<beta> p). Ifm ((b+real_of_int j)#bs) p))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5109 |
using cp_thm[OF lp up dd dp] by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5110 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
5111 |
definition unit :: "fm \<Rightarrow> fm \<times> num list \<times> int" where |
50252 | 5112 |
"unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a_\<beta> p' l); d = \<delta> q; |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5113 |
B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q)) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5114 |
in if length B \<le> length a then (q,B,d) else (mirror q, a,d))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5115 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5116 |
lemma unit: assumes qf: "qfree p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5117 |
shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5118 |
((\<exists> (x::int). Ifm (real_of_int x#bs) p) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5119 |
(\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5120 |
(Inum (real_of_int i#bs)) ` set B = (Inum (real_of_int i#bs)) ` set (\<beta> q) \<and> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5121 |
d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5122 |
proof- |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5123 |
fix q B d |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5124 |
assume qBd: "unit p = (q,B,d)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5125 |
let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5126 |
Inum (real_of_int i#bs) ` set B = Inum (real_of_int i#bs) ` set (\<beta> q) \<and> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5127 |
d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q (real_of_int i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5128 |
let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5129 |
let ?p' = "zlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5130 |
let ?l = "\<zeta> ?p'" |
50252 | 5131 |
let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a_\<beta> ?p' ?l)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5132 |
let ?d = "\<delta> ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5133 |
let ?B = "set (\<beta> ?q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5134 |
let ?B'= "remdups (map simpnum (\<beta> ?q))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5135 |
let ?A = "set (\<alpha> ?q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5136 |
let ?A'= "remdups (map simpnum (\<alpha> ?q))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5137 |
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5138 |
have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5139 |
from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5140 |
have lp': "\<forall> (i::int). iszlfm ?p' (real_of_int i#bs)" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5141 |
hence lp'': "iszlfm ?p' (real_of_int (i::int)#bs)" by simp |
50252 | 5142 |
from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d_\<beta> ?p' ?l" by auto |
5143 |
from a_\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp' |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5144 |
have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5145 |
from lp'' lp a_\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real_of_int i#bs)" and uq: "d_\<beta> ?q 1" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5146 |
by (auto simp add: isint_def) |
50252 | 5147 |
from \<delta>[OF lq] have dp:"?d >0" and dd: "d_\<delta> ?q ?d" by blast+ |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5148 |
let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5149 |
have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_comp) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5150 |
also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real_of_int i #bs"] by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5151 |
finally have BB': "?N ` set ?B' = ?N ` ?B" . |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5152 |
have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_comp) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5153 |
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"] by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5154 |
finally have AA': "?N ` set ?A' = ?N ` ?A" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5155 |
from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b" |
51369 | 5156 |
by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5157 |
from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b" |
51369 | 5158 |
by simp |
5159 |
{ assume "length ?B' \<le> length ?A'" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5160 |
hence q:"q=?q" and "B = ?B'" and d:"d = ?d" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5161 |
using qBd by (auto simp add: Let_def unit_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5162 |
with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)" |
51369 | 5163 |
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ |
5164 |
with pq_ex dp uq dd lq q d have ?thes by simp } |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5165 |
moreover |
51369 | 5166 |
{ assume "\<not> (length ?B' \<le> length ?A')" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5167 |
hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5168 |
using qBd by (auto simp add: Let_def unit_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5169 |
with AA' mirror_\<alpha>_\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5170 |
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5171 |
from mirror_ex[OF lq] pq_ex q |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5172 |
have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5173 |
from lq uq q mirror_d_\<beta> [where p="?q" and bs="bs" and a="real_of_int i"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5174 |
have lq': "iszlfm q (real_of_int i#bs)" and uq: "d_\<beta> q 1" by auto |
50252 | 5175 |
from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d_\<delta> q d " by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5176 |
from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5177 |
} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5178 |
ultimately show ?thes by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5179 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5180 |
(* Cooper's Algorithm *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5181 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
5182 |
definition cooper :: "fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5183 |
"cooper p \<equiv> |
41836 | 5184 |
(let (q,B,d) = unit p; js = [1..d]; |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5185 |
mq = simpfm (minusinf q); |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5186 |
md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5187 |
in if md = T then T else |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5188 |
(let qd = evaldjf (\<lambda> t. simpfm (subst0 t q)) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5189 |
(remdups (map (\<lambda> (b,j). simpnum (Add b (C j))) |
24336 | 5190 |
[(b,j). b\<leftarrow>B,j\<leftarrow>js])) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5191 |
in decr (disj md qd)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5192 |
lemma cooper: assumes qf: "qfree p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5193 |
shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5194 |
(is "(?lhs = ?rhs) \<and> _") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5195 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5196 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5197 |
let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5198 |
let ?q = "fst (unit p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5199 |
let ?B = "fst (snd(unit p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5200 |
let ?d = "snd (snd (unit p))" |
41836 | 5201 |
let ?js = "[1..?d]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5202 |
let ?mq = "minusinf ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5203 |
let ?smq = "simpfm ?mq" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5204 |
let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" |
26935 | 5205 |
fix i |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5206 |
let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t" |
24336 | 5207 |
let ?bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]" |
5208 |
let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5209 |
let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5210 |
have qbf:"unit p = (?q,?B,?d)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5211 |
from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5212 |
B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5213 |
uq:"d_\<beta> ?q 1" and dd: "d_\<delta> ?q ?d" and dp: "?d > 0" and |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5214 |
lq: "iszlfm ?q (real_of_int i#bs)" and |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5215 |
Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5216 |
from zlin_qfree[OF lq] have qfq: "qfree ?q" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5217 |
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5218 |
have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5219 |
hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5220 |
by (auto simp only: subst0_bound0[OF qfmq]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5221 |
hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53168
diff
changeset
|
5222 |
by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5223 |
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp |
24336 | 5224 |
from Bn jsnb have "\<forall> (b,j) \<in> set ?bjs. numbound0 (Add b (C j))" |
5225 |
by simp |
|
5226 |
hence "\<forall> (b,j) \<in> set ?bjs. numbound0 (simpnum (Add b (C j)))" |
|
5227 |
using simpnum_numbound0 by blast |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5228 |
hence "\<forall> t \<in> set ?sbjs. numbound0 t" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5229 |
hence "\<forall> t \<in> set (remdups ?sbjs). bound0 (subst0 t ?q)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5230 |
using subst0_bound0[OF qfq] by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5231 |
hence th': "\<forall> t \<in> set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5232 |
using simpfm_bound0 by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5233 |
from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5234 |
from mdb qdb |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5235 |
have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5236 |
from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5237 |
have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real_of_int j)#bs) ?q))" by auto |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5238 |
also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real_of_int j)#bs) ?q))" by auto |
24336 | 5239 |
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp |
5240 |
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci) |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5241 |
also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5242 |
by (auto simp add: split_def) |
51369 | 5243 |
also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> t \<in> set (remdups ?sbjs). (\<lambda> t. ?I i (simpfm (subst0 t ?q))) t))" |
5244 |
by (simp only: simpfm subst0_I[OF qfq] Inum.simps subst0_I[OF qfmq] set_remdups) |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5245 |
also have "\<dots> = ((?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js)) \<or> (?I i (evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex) |
51369 | 5246 |
finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5247 |
hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5248 |
{assume mdT: "?md = T" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5249 |
hence cT:"cooper p = T" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5250 |
by (simp only: cooper_def unit_def split_def Let_def if_True) simp |
51369 | 5251 |
from mdT mdqd have lhs:"?lhs" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5252 |
from mdT have "?rhs" by (simp add: cooper_def unit_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5253 |
with lhs cT have ?thesis by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5254 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5255 |
{assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5256 |
by (simp only: cooper_def unit_def split_def Let_def if_False) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5257 |
with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5258 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5259 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5260 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5261 |
lemma DJcooper: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5262 |
assumes qf: "qfree p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5263 |
shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5264 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5265 |
from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5266 |
from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5267 |
have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5268 |
by (simp add: DJ_def evaldjf_ex) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5269 |
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs) q)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5270 |
using cooper disjuncts_qf[OF qf] by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5271 |
also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5272 |
finally show ?thesis using thqf by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5273 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5274 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5275 |
(* Redy and Loveland *) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5276 |
|
50252 | 5277 |
lemma \<sigma>_\<rho>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" |
5278 |
shows "Ifm (a#bs) (\<sigma>_\<rho> p (t,c)) = Ifm (a#bs) (\<sigma>_\<rho> p (t',c))" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5279 |
using lp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5280 |
by (induct p rule: iszlfm.induct, auto simp add: tt') |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5281 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5282 |
lemma \<sigma>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5283 |
shows "Ifm (a#bs) (\<sigma> p c t) = Ifm (a#bs) (\<sigma> p c t')" |
50252 | 5284 |
by (simp add: \<sigma>_def tt' \<sigma>_\<rho>_cong[OF lp tt']) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5285 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5286 |
lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5287 |
and RR: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5288 |
shows "(\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))) = (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j))))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5289 |
(is "?lhs = ?rhs") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5290 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5291 |
let ?d = "\<delta> p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5292 |
assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5293 |
and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5294 |
from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5295 |
hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" using RR by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5296 |
hence "\<exists> (e',c') \<in> set (\<rho> p). Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5297 |
then obtain e' c' where ecRo:"(e',c') \<in> set (\<rho> p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5298 |
and cc':"c = c'" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5299 |
from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5300 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5301 |
from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
56544
diff
changeset
|
5302 |
from ecRo jD px' show ?rhs apply (auto simp: cc') |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5303 |
by (rule_tac x="(e', c')" in bexI,simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5304 |
(rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5305 |
next |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5306 |
let ?d = "\<delta> p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5307 |
assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5308 |
and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5309 |
from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5310 |
hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5311 |
hence "\<exists> (e',c') \<in> R. Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5312 |
then obtain e' c' where ecRo:"(e',c') \<in> R" and ee':"Inum (a#bs) e = Inum (a#bs) e'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5313 |
and cc':"c = c'" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5314 |
from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5315 |
from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp |
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
56544
diff
changeset
|
5316 |
from ecRo jD px' show ?lhs apply (auto simp: cc') |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5317 |
by (rule_tac x="(e', c')" in bexI,simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5318 |
(rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5319 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5320 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5321 |
lemma rl_thm': |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5322 |
assumes lp: "iszlfm p (real_of_int (i::int)#bs)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5323 |
and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5324 |
shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5325 |
using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5326 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
5327 |
definition chooset :: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int" where |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5328 |
"chooset p \<equiv> (let q = zlfm p ; d = \<delta> q; |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5329 |
B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ; |
50252 | 5330 |
a = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<alpha>_\<rho> q)) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5331 |
in if length B \<le> length a then (q,B,d) else (mirror q, a,d))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5332 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5333 |
lemma chooset: assumes qf: "qfree p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5334 |
shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5335 |
((\<exists> (x::int). Ifm (real_of_int x#bs) p) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5336 |
(\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5337 |
((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5338 |
(\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5339 |
proof- |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5340 |
fix q B d |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5341 |
assume qBd: "chooset p = (q,B,d)" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5342 |
let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5343 |
(\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5344 |
(\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5345 |
let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5346 |
let ?q = "zlfm p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5347 |
let ?d = "\<delta> ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5348 |
let ?B = "set (\<rho> ?q)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5349 |
let ?f = "\<lambda> (t,k). (simpnum t,k)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5350 |
let ?B'= "remdups (map ?f (\<rho> ?q))" |
50252 | 5351 |
let ?A = "set (\<alpha>_\<rho> ?q)" |
5352 |
let ?A'= "remdups (map ?f (\<alpha>_\<rho> ?q))" |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5353 |
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5354 |
have pp': "\<forall> i. ?I i ?q = ?I i p" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5355 |
hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5356 |
from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real_of_int i"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5357 |
have lq: "iszlfm ?q (real_of_int (i::int)#bs)" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5358 |
from \<delta>[OF lq] have dp:"?d >0" by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5359 |
let ?N = "\<lambda> (t,c). (Inum (real_of_int (i::int)#bs) t,c)" |
56154
f0a927235162
more complete set of lemmas wrt. image and composition
haftmann
parents:
55584
diff
changeset
|
5360 |
have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_comp) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5361 |
also have "\<dots> = ?N ` ?B" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5362 |
by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5363 |
finally have BB': "?N ` set ?B' = ?N ` ?B" . |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5364 |
have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_comp) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5365 |
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"] |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5366 |
by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5367 |
finally have AA': "?N ` set ?A' = ?N ` ?A" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5368 |
from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0" |
51369 | 5369 |
by (simp add: split_def) |
50252 | 5370 |
from \<alpha>_\<rho>_l[OF lq] have A_nb: "\<forall> (e,c)\<in> set ?A'. numbound0 e \<and> c > 0" |
51369 | 5371 |
by (simp add: split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5372 |
{assume "length ?B' \<le> length ?A'" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5373 |
hence q:"q=?q" and "B = ?B'" and d:"d = ?d" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5374 |
using qBd by (auto simp add: Let_def chooset_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5375 |
with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5376 |
and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5377 |
with pq_ex dp lq q d have ?thes by simp} |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5378 |
moreover |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5379 |
{assume "\<not> (length ?B' \<le> length ?A')" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5380 |
hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5381 |
using qBd by (auto simp add: Let_def chooset_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5382 |
with AA' mirror_\<alpha>_\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)" |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5383 |
and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5384 |
from mirror_ex[OF lq] pq_ex q |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5385 |
have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5386 |
from lq q mirror_l [where p="?q" and bs="bs" and a="real_of_int i"] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5387 |
have lq': "iszlfm q (real_of_int i#bs)" by auto |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5388 |
from mirror_\<delta>[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5389 |
} |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5390 |
ultimately show ?thes by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5391 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5392 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
5393 |
definition stage :: "fm \<Rightarrow> int \<Rightarrow> (num \<times> int) \<Rightarrow> fm" where |
41836 | 5394 |
"stage p d \<equiv> (\<lambda> (e,c). evaldjf (\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))) [1..c*d])" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5395 |
lemma stage: |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5396 |
shows "Ifm bs (stage p d (e,c)) = (\<exists> j\<in>{1 .. c*d}. Ifm bs (\<sigma> p c (Add e (C j))))" |
41836 | 5397 |
by (unfold stage_def split_def ,simp only: evaldjf_ex simpfm) simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5398 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5399 |
lemma stage_nb: assumes lp: "iszlfm p (a#bs)" and cp: "c >0" and nb:"numbound0 e" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5400 |
shows "bound0 (stage p d (e,c))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5401 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5402 |
let ?f = "\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))" |
41836 | 5403 |
have th: "\<forall> j\<in> set [1..c*d]. bound0 (?f j)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5404 |
proof |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5405 |
fix j |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5406 |
from nb have nb':"numbound0 (Add e (C j))" by simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5407 |
from simpfm_bound0[OF \<sigma>_nb[OF lp nb', where k="c"]] |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5408 |
show "bound0 (simpfm (\<sigma> p c (Add e (C j))))" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5409 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5410 |
from evaldjf_bound0[OF th] show ?thesis by (unfold stage_def split_def) simp |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5411 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5412 |
|
35416
d8d7d1b785af
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
haftmann
parents:
35028
diff
changeset
|
5413 |
definition redlove :: "fm \<Rightarrow> fm" where |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5414 |
"redlove p \<equiv> |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5415 |
(let (q,B,d) = chooset p; |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5416 |
mq = simpfm (minusinf q); |
41836 | 5417 |
md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) [1..d] |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5418 |
in if md = T then T else |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5419 |
(let qd = evaldjf (stage q d) B |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5420 |
in decr (disj md qd)))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5421 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5422 |
lemma redlove: assumes qf: "qfree p" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5423 |
shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5424 |
(is "(?lhs = ?rhs) \<and> _") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5425 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5426 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5427 |
let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5428 |
let ?q = "fst (chooset p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5429 |
let ?B = "fst (snd(chooset p))" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5430 |
let ?d = "snd (snd (chooset p))" |
41836 | 5431 |
let ?js = "[1..?d]" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5432 |
let ?mq = "minusinf ?q" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5433 |
let ?smq = "simpfm ?mq" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5434 |
let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" |
26935 | 5435 |
fix i |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5436 |
let ?N = "\<lambda> (t,k). (Inum (real_of_int (i::int)#bs) t,k)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5437 |
let ?qd = "evaldjf (stage ?q ?d) ?B" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5438 |
have qbf:"chooset p = (?q,?B,?d)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5439 |
from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5440 |
B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5441 |
lq: "iszlfm ?q (real_of_int i#bs)" and |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5442 |
Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5443 |
from zlin_qfree[OF lq] have qfq: "qfree ?q" . |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5444 |
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5445 |
have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5446 |
hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5447 |
by (auto simp only: subst0_bound0[OF qfmq]) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5448 |
hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))" |
51369 | 5449 |
by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5450 |
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5451 |
from Bn stage_nb[OF lq] have th:"\<forall> x \<in> set ?B. bound0 (stage ?q ?d x)" by auto |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5452 |
from evaldjf_bound0[OF th] have qdb: "bound0 ?qd" . |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5453 |
from mdb qdb |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5454 |
have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5455 |
from trans [OF pq_ex rl_thm'[OF lq B]] dd |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5456 |
have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real_of_int i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5457 |
also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))" |
51369 | 5458 |
by (simp add: stage split_def) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5459 |
also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq)) \<or> ?I i ?qd)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5460 |
by (simp add: evaldjf_ex subst0_I[OF qfmq]) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5461 |
finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex set_upto simpfm) |
51369 | 5462 |
also have "\<dots> = (?I i (disj ?md ?qd))" by simp |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5463 |
also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5464 |
finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" . |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5465 |
{assume mdT: "?md = T" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5466 |
hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5467 |
from mdT have lhs:"?lhs" using mdqd by simp |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5468 |
from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5469 |
with lhs cT have ?thesis by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5470 |
moreover |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5471 |
{assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5472 |
by (simp add: redlove_def chooset_def split_def Let_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5473 |
with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5474 |
ultimately show ?thesis by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5475 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5476 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5477 |
lemma DJredlove: |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5478 |
assumes qf: "qfree p" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5479 |
shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5480 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5481 |
from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5482 |
from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5483 |
have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5484 |
by (simp add: DJ_def evaldjf_ex) |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5485 |
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs) q)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5486 |
using redlove disjuncts_qf[OF qf] by blast |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5487 |
also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto) |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5488 |
finally show ?thesis using thqf by blast |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5489 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5490 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5491 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5492 |
lemma exsplit_qf: assumes qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5493 |
shows "qfree (exsplit p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5494 |
using qf by (induct p rule: exsplit.induct, auto) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5495 |
|
27456 | 5496 |
definition mircfr :: "fm \<Rightarrow> fm" where |
5497 |
"mircfr = DJ cooper o ferrack01 o simpfm o exsplit" |
|
5498 |
||
5499 |
definition mirlfr :: "fm \<Rightarrow> fm" where |
|
5500 |
"mirlfr = DJ redlove o ferrack01 o simpfm o exsplit" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5501 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5502 |
lemma mircfr: "\<forall> bs p. qfree p \<longrightarrow> qfree (mircfr p) \<and> Ifm bs (mircfr p) = Ifm bs (E p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5503 |
proof(clarsimp simp del: Ifm.simps) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5504 |
fix bs p |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5505 |
assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5506 |
show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5507 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5508 |
let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5509 |
have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5510 |
using splitex[OF qf] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5511 |
with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5512 |
with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5513 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5514 |
qed |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5515 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5516 |
lemma mirlfr: "\<forall> bs p. qfree p \<longrightarrow> qfree(mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5517 |
proof(clarsimp simp del: Ifm.simps) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5518 |
fix bs p |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5519 |
assume qf: "qfree p" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5520 |
show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)") |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5521 |
proof- |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5522 |
let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5523 |
have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5524 |
using splitex[OF qf] by simp |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5525 |
with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5526 |
with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5527 |
qed |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5528 |
qed |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5529 |
|
27456 | 5530 |
definition mircfrqe:: "fm \<Rightarrow> fm" where |
5531 |
"mircfrqe p = qelim (prep p) mircfr" |
|
5532 |
||
5533 |
definition mirlfrqe:: "fm \<Rightarrow> fm" where |
|
5534 |
"mirlfrqe p = qelim (prep p) mirlfr" |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5535 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5536 |
theorem mircfrqe: "(Ifm bs (mircfrqe p) = Ifm bs p) \<and> qfree (mircfrqe p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5537 |
using qelim_ci[OF mircfr] prep by (auto simp add: mircfrqe_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5538 |
|
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5539 |
theorem mirlfrqe: "(Ifm bs (mirlfrqe p) = Ifm bs p) \<and> qfree (mirlfrqe p)" |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5540 |
using qelim_ci[OF mirlfr] prep by (auto simp add: mirlfrqe_def) |
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5541 |
|
23858 | 5542 |
definition |
36870 | 5543 |
"problem1 = A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0))))))" |
23858 | 5544 |
|
5545 |
definition |
|
36870 | 5546 |
"problem2 = A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0))))" |
23858 | 5547 |
|
5548 |
definition |
|
36870 | 5549 |
"problem3 = A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0))))))" |
23858 | 5550 |
|
5551 |
definition |
|
36870 | 5552 |
"problem4 = E (And (Ge (Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg (Bound 0))))))" |
5553 |
||
60533 | 5554 |
ML_val \<open>@{code mircfrqe} @{code problem1}\<close> |
5555 |
ML_val \<open>@{code mirlfrqe} @{code problem1}\<close> |
|
5556 |
ML_val \<open>@{code mircfrqe} @{code problem2}\<close> |
|
5557 |
ML_val \<open>@{code mirlfrqe} @{code problem2}\<close> |
|
5558 |
ML_val \<open>@{code mircfrqe} @{code problem3}\<close> |
|
5559 |
ML_val \<open>@{code mirlfrqe} @{code problem3}\<close> |
|
5560 |
ML_val \<open>@{code mircfrqe} @{code problem4}\<close> |
|
5561 |
ML_val \<open>@{code mirlfrqe} @{code problem4}\<close> |
|
51272 | 5562 |
|
24249 | 5563 |
|
36531
19f6e3b0d9b6
code_reflect: specify module name directly after keyword
haftmann
parents:
36526
diff
changeset
|
5564 |
(*code_reflect Mir |
36526 | 5565 |
functions mircfrqe mirlfrqe |
5566 |
file "mir.ML"*) |
|
23858 | 5567 |
|
60533 | 5568 |
oracle mirfr_oracle = \<open> |
27456 | 5569 |
let |
5570 |
||
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5571 |
val mk_C = @{code C} o @{code int_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5572 |
val mk_Dvd = @{code Dvd} o apfst @{code int_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5573 |
val mk_Bound = @{code Bound} o @{code nat_of_integer}; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5574 |
|
67399 | 5575 |
fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (=) vs t |
27456 | 5576 |
of NONE => error "Variable not found in the list!" |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5577 |
| SOME n => mk_Bound n) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5578 |
| num_of_term vs @{term "of_int (0::int)"} = mk_C 0 |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5579 |
| num_of_term vs @{term "of_int (1::int)"} = mk_C 1 |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5580 |
| num_of_term vs @{term "0::real"} = mk_C 0 |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5581 |
| num_of_term vs @{term "1::real"} = mk_C 1 |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
5582 |
| num_of_term vs @{term "- 1::real"} = mk_C (~ 1) |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5583 |
| num_of_term vs (Bound i) = mk_Bound i |
27456 | 5584 |
| num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t') |
67399 | 5585 |
| num_of_term vs (@{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = |
27456 | 5586 |
@{code Add} (num_of_term vs t1, num_of_term vs t2) |
67399 | 5587 |
| num_of_term vs (@{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = |
27456 | 5588 |
@{code Sub} (num_of_term vs t1, num_of_term vs t2) |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68270
diff
changeset
|
5589 |
| num_of_term vs (@{term "(*) :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = |
27456 | 5590 |
(case (num_of_term vs t1) |
5591 |
of @{code C} i => @{code Mul} (i, num_of_term vs t2) |
|
5592 |
| _ => error "num_of_term: unsupported Multiplication") |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5593 |
| num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t')) = |
62342 | 5594 |
mk_C (HOLogic.dest_numeral t') |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5595 |
| num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t')) = |
62342 | 5596 |
mk_C (~ (HOLogic.dest_numeral t')) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5597 |
| num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ t')) = |
27456 | 5598 |
@{code Floor} (num_of_term vs t') |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5599 |
| num_of_term vs (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ t')) = |
27456 | 5600 |
@{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t'))) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46670
diff
changeset
|
5601 |
| num_of_term vs (@{term "numeral :: _ \<Rightarrow> real"} $ t') = |
62342 | 5602 |
mk_C (HOLogic.dest_numeral t') |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54230
diff
changeset
|
5603 |
| num_of_term vs (@{term "- numeral :: _ \<Rightarrow> real"} $ t') = |
62342 | 5604 |
mk_C (~ (HOLogic.dest_numeral t')) |
28264 | 5605 |
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t); |
27456 | 5606 |
|
5607 |
fun fm_of_term vs @{term True} = @{code T} |
|
5608 |
| fm_of_term vs @{term False} = @{code F} |
|
67399 | 5609 |
| fm_of_term vs (@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = |
27456 | 5610 |
@{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
67399 | 5611 |
| fm_of_term vs (@{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = |
27456 | 5612 |
@{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
67399 | 5613 |
| fm_of_term vs (@{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5614 |
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) |
67399 | 5615 |
| fm_of_term vs (@{term "(rdvd)"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) = |
62342 | 5616 |
mk_Dvd (HOLogic.dest_numeral t1, num_of_term vs t2) |
67399 | 5617 |
| fm_of_term vs (@{term "(rdvd)"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) = |
62342 | 5618 |
mk_Dvd (~ (HOLogic.dest_numeral t1), num_of_term vs t2) |
67399 | 5619 |
| fm_of_term vs (@{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = |
27456 | 5620 |
@{code Iff} (fm_of_term vs t1, fm_of_term vs t2) |
38795
848be46708dc
formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents:
38786
diff
changeset
|
5621 |
| fm_of_term vs (@{term HOL.conj} $ t1 $ t2) = |
27456 | 5622 |
@{code And} (fm_of_term vs t1, fm_of_term vs t2) |
38795
848be46708dc
formerly unnamed infix conjunction and disjunction now named HOL.conj and HOL.disj
haftmann
parents:
38786
diff
changeset
|
5623 |
| fm_of_term vs (@{term HOL.disj} $ t1 $ t2) = |
27456 | 5624 |
@{code Or} (fm_of_term vs t1, fm_of_term vs t2) |
38786
e46e7a9cb622
formerly unnamed infix impliciation now named HOL.implies
haftmann
parents:
38558
diff
changeset
|
5625 |
| fm_of_term vs (@{term HOL.implies} $ t1 $ t2) = |
27456 | 5626 |
@{code Imp} (fm_of_term vs t1, fm_of_term vs t2) |
5627 |
| fm_of_term vs (@{term "Not"} $ t') = |
|
5628 |
@{code NOT} (fm_of_term vs t') |
|
38558 | 5629 |
| fm_of_term vs (Const (@{const_name Ex}, _) $ Abs (xn, xT, p)) = |
27456 | 5630 |
@{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) |
38558 | 5631 |
| fm_of_term vs (Const (@{const_name All}, _) $ Abs (xn, xT, p)) = |
27456 | 5632 |
@{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) |
28264 | 5633 |
| fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t); |
27456 | 5634 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5635 |
fun term_of_num vs (@{code C} i) = @{term "of_int :: int \<Rightarrow> real"} $ |
51143
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5636 |
HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i) |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5637 |
| term_of_num vs (@{code Bound} n) = |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5638 |
let |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5639 |
val m = @{code integer_of_nat} n; |
0a2371e7ced3
two target language numeral types: integer and natural, as replacement for code_numeral;
haftmann
parents:
50252
diff
changeset
|
5640 |
in fst (the (find_first (fn (_, q) => m = q) vs)) end |
27456 | 5641 |
| term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) = |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5642 |
@{term "of_int :: int \<Rightarrow> real"} $ (@{term "ceiling :: real \<Rightarrow> int"} $ term_of_num vs t') |
27456 | 5643 |
| term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t' |
67399 | 5644 |
| term_of_num vs (@{code Add} (t1, t2)) = @{term "(+) :: real \<Rightarrow> real \<Rightarrow> real"} $ |
27456 | 5645 |
term_of_num vs t1 $ term_of_num vs t2 |
67399 | 5646 |
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "(-) :: real \<Rightarrow> real \<Rightarrow> real"} $ |
27456 | 5647 |
term_of_num vs t1 $ term_of_num vs t2 |
69064
5840724b1d71
Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents:
68270
diff
changeset
|
5648 |
| term_of_num vs (@{code Mul} (i, t2)) = @{term "(*) :: real \<Rightarrow> real \<Rightarrow> real"} $ |
27456 | 5649 |
term_of_num vs (@{code C} i) $ term_of_num vs t2 |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5650 |
| term_of_num vs (@{code Floor} t) = @{term "of_int :: int \<Rightarrow> real"} $ (@{term "floor :: real \<Rightarrow> int"} $ term_of_num vs t) |
27456 | 5651 |
| term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)) |
5652 |
| term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s)); |
|
5653 |
||
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5654 |
fun term_of_fm vs @{code T} = @{term True} |
45740 | 5655 |
| term_of_fm vs @{code F} = @{term False} |
27456 | 5656 |
| term_of_fm vs (@{code Lt} t) = |
67399 | 5657 |
@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} |
27456 | 5658 |
| term_of_fm vs (@{code Le} t) = |
67399 | 5659 |
@{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} |
27456 | 5660 |
| term_of_fm vs (@{code Gt} t) = |
67399 | 5661 |
@{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t |
27456 | 5662 |
| term_of_fm vs (@{code Ge} t) = |
67399 | 5663 |
@{term "(\<le>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t |
27456 | 5664 |
| term_of_fm vs (@{code Eq} t) = |
67399 | 5665 |
@{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} |
27456 | 5666 |
| term_of_fm vs (@{code NEq} t) = |
5667 |
term_of_fm vs (@{code NOT} (@{code Eq} t)) |
|
5668 |
| term_of_fm vs (@{code Dvd} (i, t)) = |
|
67399 | 5669 |
@{term "(rdvd)"} $ term_of_num vs (@{code C} i) $ term_of_num vs t |
27456 | 5670 |
| term_of_fm vs (@{code NDvd} (i, t)) = |
5671 |
term_of_fm vs (@{code NOT} (@{code Dvd} (i, t))) |
|
5672 |
| term_of_fm vs (@{code NOT} t') = |
|
5673 |
HOLogic.Not $ term_of_fm vs t' |
|
5674 |
| term_of_fm vs (@{code And} (t1, t2)) = |
|
5675 |
HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2 |
|
5676 |
| term_of_fm vs (@{code Or} (t1, t2)) = |
|
5677 |
HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2 |
|
5678 |
| term_of_fm vs (@{code Imp} (t1, t2)) = |
|
5679 |
HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2 |
|
5680 |
| term_of_fm vs (@{code Iff} (t1, t2)) = |
|
67399 | 5681 |
@{term "(=) :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2; |
27456 | 5682 |
|
28290 | 5683 |
in |
60325 | 5684 |
fn (ctxt, t) => |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5685 |
let |
44121 | 5686 |
val fs = Misc_Legacy.term_frees t; |
33063 | 5687 |
val vs = map_index swap fs; |
60325 | 5688 |
(*If quick_and_dirty then run without proof generation as oracle*) |
5689 |
val qe = if Config.get ctxt quick_and_dirty then @{code mircfrqe} else @{code mirlfrqe}; |
|
5690 |
val t' = term_of_fm vs (qe (fm_of_term vs t)); |
|
5691 |
in Thm.cterm_of ctxt (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t'))) end |
|
69266
7cc2d66a92a6
proper ML expressions, without trailing semicolons;
wenzelm
parents:
69064
diff
changeset
|
5692 |
end |
60533 | 5693 |
\<close> |
27456 | 5694 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5695 |
lemmas iff_real_of_int = of_int_eq_iff [where 'a = real, symmetric] |
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5696 |
of_int_less_iff [where 'a = real, symmetric] |
61652
90c65a811257
MIR decision procedure again working
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
5697 |
of_int_le_iff [where 'a = real, symmetric] |
90c65a811257
MIR decision procedure again working
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
5698 |
|
48891 | 5699 |
ML_file "mir_tac.ML" |
47432 | 5700 |
|
60533 | 5701 |
method_setup mir = \<open> |
53168 | 5702 |
Scan.lift (Args.mode "no_quantify") >> |
47432 | 5703 |
(fn q => fn ctxt => SIMPLE_METHOD' (Mir_Tac.mir_tac ctxt (not q))) |
60533 | 5704 |
\<close> "decision procedure for MIR arithmetic" |
61652
90c65a811257
MIR decision procedure again working
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
5705 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5706 |
lemma "\<forall>x::real. (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> \<longleftrightarrow> (x = real_of_int \<lfloor>x\<rfloor>))" |
41891 | 5707 |
by mir |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5708 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
5709 |
lemma "\<forall>x::real. real_of_int (2::int)*x - (real_of_int (1::int)) < real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil> \<and> real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil> \<le> real_of_int (2::int)*x + (real_of_int (1::int))" |
41891 | 5710 |
by mir |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5711 |
|
58909 | 5712 |
lemma "\<forall>x::real. 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61652
diff
changeset
|
5713 |
by mir |
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5714 |
|
58909 | 5715 |
lemma "\<forall>x::real. \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)" |
41891 | 5716 |
by mir |
23858 | 5717 |
|
61945 | 5718 |
lemma "\<forall>(x::real) (y::real). \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> \<bar>y - x\<bar> \<and> \<bar>y - x\<bar> \<le> 1" |
41891 | 5719 |
by mir |
61652
90c65a811257
MIR decision procedure again working
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
5720 |
|
23264
324622260d29
Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers
chaieb
parents:
diff
changeset
|
5721 |
end |