author | kleing |
Mon, 21 Jun 2004 10:25:57 +0200 | |
changeset 14981 | e73f8140af78 |
parent 14941 | 1edb674e0c33 |
child 15013 | 34264f5e4691 |
permissions | -rw-r--r-- |
13876 | 1 |
(* Title: HOL/Integ/Presburger.thy |
2 |
ID: $Id$ |
|
3 |
Author: Amine Chaieb, Tobias Nipkow and Stefan Berghofer, TU Muenchen |
|
4 |
||
5 |
File containing necessary theorems for the proof |
|
6 |
generation for Cooper Algorithm |
|
7 |
*) |
|
8 |
||
14577 | 9 |
header {* Presburger Arithmetic: Cooper Algorithm *} |
10 |
||
14485 | 11 |
theory Presburger = NatSimprocs + SetInterval |
13876 | 12 |
files |
13 |
("cooper_dec.ML") |
|
14 |
("cooper_proof.ML") |
|
15 |
("qelim.ML") |
|
16 |
("presburger.ML"): |
|
17 |
||
14577 | 18 |
text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*} |
13876 | 19 |
|
20 |
theorem unity_coeff_ex: "(\<exists>x::int. P (l * x)) = (\<exists>x. l dvd (1*x+0) \<and> P x)" |
|
21 |
apply (rule iffI) |
|
22 |
apply (erule exE) |
|
23 |
apply (rule_tac x = "l * x" in exI) |
|
24 |
apply simp |
|
25 |
apply (erule exE) |
|
26 |
apply (erule conjE) |
|
27 |
apply (erule dvdE) |
|
28 |
apply (rule_tac x = k in exI) |
|
29 |
apply simp |
|
30 |
done |
|
31 |
||
32 |
lemma uminus_dvd_conv: "(d dvd (t::int)) = (-d dvd t)" |
|
33 |
apply(unfold dvd_def) |
|
34 |
apply(rule iffI) |
|
35 |
apply(clarsimp) |
|
36 |
apply(rename_tac k) |
|
37 |
apply(rule_tac x = "-k" in exI) |
|
38 |
apply simp |
|
39 |
apply(clarsimp) |
|
40 |
apply(rename_tac k) |
|
41 |
apply(rule_tac x = "-k" in exI) |
|
42 |
apply simp |
|
43 |
done |
|
44 |
||
45 |
lemma uminus_dvd_conv': "(d dvd (t::int)) = (d dvd -t)" |
|
46 |
apply(unfold dvd_def) |
|
47 |
apply(rule iffI) |
|
48 |
apply(clarsimp) |
|
49 |
apply(rule_tac x = "-k" in exI) |
|
50 |
apply simp |
|
51 |
apply(clarsimp) |
|
52 |
apply(rule_tac x = "-k" in exI) |
|
53 |
apply simp |
|
54 |
done |
|
55 |
||
56 |
||
57 |
||
14577 | 58 |
text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} less than some integer @{text z}.*} |
13876 | 59 |
|
60 |
theorem eq_minf_conjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow> |
|
61 |
\<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow> |
|
62 |
\<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))" |
|
63 |
apply (erule exE)+ |
|
64 |
apply (rule_tac x = "min z1 z2" in exI) |
|
65 |
apply simp |
|
66 |
done |
|
67 |
||
68 |
||
69 |
theorem eq_minf_disjI: "\<exists>z1::int. \<forall>x. x < z1 \<longrightarrow> (A1 x = A2 x) \<Longrightarrow> |
|
70 |
\<exists>z2::int. \<forall>x. x < z2 \<longrightarrow> (B1 x = B2 x) \<Longrightarrow> |
|
71 |
\<exists>z::int. \<forall>x. x < z \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))" |
|
72 |
||
73 |
apply (erule exE)+ |
|
74 |
apply (rule_tac x = "min z1 z2" in exI) |
|
75 |
apply simp |
|
76 |
done |
|
77 |
||
78 |
||
14577 | 79 |
text {*Theorems for the combination of proofs of the equality of @{text P} and @{text P_m} for integers @{text x} greather than some integer @{text z}.*} |
13876 | 80 |
|
81 |
theorem eq_pinf_conjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow> |
|
82 |
\<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow> |
|
83 |
\<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<and> B1 x) = (A2 x \<and> B2 x))" |
|
84 |
apply (erule exE)+ |
|
85 |
apply (rule_tac x = "max z1 z2" in exI) |
|
86 |
apply simp |
|
87 |
done |
|
88 |
||
89 |
||
90 |
theorem eq_pinf_disjI: "\<exists>z1::int. \<forall>x. z1 < x \<longrightarrow> (A1 x = A2 x) \<Longrightarrow> |
|
91 |
\<exists>z2::int. \<forall>x. z2 < x \<longrightarrow> (B1 x = B2 x) \<Longrightarrow> |
|
92 |
\<exists>z::int. \<forall>x. z < x \<longrightarrow> ((A1 x \<or> B1 x) = (A2 x \<or> B2 x))" |
|
93 |
apply (erule exE)+ |
|
94 |
apply (rule_tac x = "max z1 z2" in exI) |
|
95 |
apply simp |
|
96 |
done |
|
14577 | 97 |
|
98 |
text {* |
|
99 |
\medskip Theorems for the combination of proofs of the modulo @{text |
|
100 |
D} property for @{text "P plusinfinity"} |
|
101 |
||
102 |
FIXME: This is THE SAME theorem as for the @{text minusinf} version, |
|
103 |
but with @{text "+k.."} instead of @{text "-k.."} In the future |
|
104 |
replace these both with only one. *} |
|
13876 | 105 |
|
106 |
theorem modd_pinf_conjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow> |
|
107 |
\<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow> |
|
108 |
\<forall>(x::int) (k::int). (A x \<and> B x) = (A (x+k*d) \<and> B (x+k*d))" |
|
109 |
by simp |
|
110 |
||
111 |
theorem modd_pinf_disjI: "\<forall>(x::int) k. A x = A (x+k*d) \<Longrightarrow> |
|
112 |
\<forall>(x::int) k. B x = B (x+k*d) \<Longrightarrow> |
|
113 |
\<forall>(x::int) (k::int). (A x \<or> B x) = (A (x+k*d) \<or> B (x+k*d))" |
|
114 |
by simp |
|
115 |
||
14577 | 116 |
text {* |
117 |
This is one of the cases where the simplifed formula is prooved to |
|
118 |
habe some property (in relation to @{text P_m}) but we need to prove |
|
119 |
the property for the original formula (@{text P_m}) |
|
120 |
||
121 |
FIXME: This is exaclty the same thm as for @{text minusinf}. *} |
|
122 |
||
13876 | 123 |
lemma pinf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x)) ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) " |
14577 | 124 |
by blast |
13876 | 125 |
|
126 |
||
14577 | 127 |
text {* |
128 |
\medskip Theorems for the combination of proofs of the modulo @{text D} |
|
129 |
property for @{text "P minusinfinity"} *} |
|
13876 | 130 |
|
131 |
theorem modd_minf_conjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow> |
|
132 |
\<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow> |
|
133 |
\<forall>(x::int) (k::int). (A x \<and> B x) = (A (x-k*d) \<and> B (x-k*d))" |
|
134 |
by simp |
|
135 |
||
136 |
theorem modd_minf_disjI: "\<forall>(x::int) k. A x = A (x-k*d) \<Longrightarrow> |
|
137 |
\<forall>(x::int) k. B x = B (x-k*d) \<Longrightarrow> |
|
138 |
\<forall>(x::int) (k::int). (A x \<or> B x) = (A (x-k*d) \<or> B (x-k*d))" |
|
139 |
by simp |
|
140 |
||
14577 | 141 |
text {* |
142 |
This is one of the cases where the simplifed formula is prooved to |
|
143 |
have some property (in relation to @{text P_m}) but we need to |
|
144 |
prove the property for the original formula (@{text P_m}). *} |
|
13876 | 145 |
|
146 |
lemma minf_simp_eq: "ALL x. P(x) = Q(x) ==> (EX (x::int). P(x)) --> (EX (x::int). F(x)) ==> (EX (x::int). Q(x)) --> (EX (x::int). F(x)) " |
|
14577 | 147 |
by blast |
13876 | 148 |
|
14577 | 149 |
text {* |
150 |
Theorem needed for proving at runtime divide properties using the |
|
151 |
arithmetic tactic (which knows only about modulo = 0). *} |
|
13876 | 152 |
|
153 |
lemma zdvd_iff_zmod_eq_0: "(m dvd n) = (n mod m = (0::int))" |
|
14577 | 154 |
by(simp add:dvd_def zmod_eq_0_iff) |
13876 | 155 |
|
14577 | 156 |
text {* |
157 |
\medskip Theorems used for the combination of proof for the |
|
158 |
backwards direction of Cooper's Theorem. They rely exclusively on |
|
159 |
Predicate calculus.*} |
|
13876 | 160 |
|
161 |
lemma not_ast_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P1(x) --> P1(x + d)) |
|
162 |
==> |
|
163 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d)) |
|
164 |
==> |
|
165 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<or> P2(x)) --> (P1(x + d) \<or> P2(x + d))) " |
|
14577 | 166 |
by blast |
13876 | 167 |
|
168 |
||
169 |
lemma not_ast_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a- j)) --> P1(x) --> P1(x + d)) |
|
170 |
==> |
|
171 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> P2(x) --> P2(x + d)) |
|
172 |
==> |
|
173 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->(P1(x) \<and> P2(x)) --> (P1(x + d) |
|
174 |
\<and> P2(x + d))) " |
|
14577 | 175 |
by blast |
13876 | 176 |
|
177 |
lemma not_ast_p_Q_elim: " |
|
178 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) -->P(x) --> P(x + d)) |
|
179 |
==> ( P = Q ) |
|
180 |
==> (ALL x. ~(EX (j::int) : {1..d}. EX (a::int) : A. P(a - j)) -->P(x) --> P(x + d))" |
|
14577 | 181 |
by blast |
13876 | 182 |
|
14577 | 183 |
text {* |
184 |
\medskip Theorems used for the combination of proof for the |
|
185 |
backwards direction of Cooper's Theorem. They rely exclusively on |
|
186 |
Predicate calculus.*} |
|
13876 | 187 |
|
188 |
lemma not_bst_p_disjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d)) |
|
189 |
==> |
|
190 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d)) |
|
191 |
==> |
|
192 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<or> P2(x)) --> (P1(x - d) |
|
193 |
\<or> P2(x-d))) " |
|
14577 | 194 |
by blast |
13876 | 195 |
|
196 |
lemma not_bst_p_conjI: "(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P1(x) --> P1(x - d)) |
|
197 |
==> |
|
198 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> P2(x) --> P2(x - d)) |
|
199 |
==> |
|
200 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->(P1(x) \<and> P2(x)) --> (P1(x - d) |
|
201 |
\<and> P2(x-d))) " |
|
14577 | 202 |
by blast |
13876 | 203 |
|
204 |
lemma not_bst_p_Q_elim: " |
|
205 |
(ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) -->P(x) --> P(x - d)) |
|
206 |
==> ( P = Q ) |
|
207 |
==> (ALL x. ~(EX (j::int) : {1..d}. EX (b::int) : B. P(b+j)) -->P(x) --> P(x - d))" |
|
14577 | 208 |
by blast |
13876 | 209 |
|
14577 | 210 |
text {* \medskip This is the first direction of Cooper's Theorem. *} |
13876 | 211 |
lemma cooper_thm: "(R --> (EX x::int. P x)) ==> (Q -->(EX x::int. P x )) ==> ((R|Q) --> (EX x::int. P x )) " |
14577 | 212 |
by blast |
13876 | 213 |
|
14577 | 214 |
text {* |
215 |
\medskip The full Cooper's Theorem in its equivalence Form. Given |
|
216 |
the premises it is trivial too, it relies exclusively on prediacte calculus.*} |
|
13876 | 217 |
lemma cooper_eq_thm: "(R --> (EX x::int. P x)) ==> (Q -->(EX x::int. P x )) ==> ((~Q) |
218 |
--> (EX x::int. P x ) --> R) ==> (EX x::int. P x) = R|Q " |
|
14577 | 219 |
by blast |
13876 | 220 |
|
14577 | 221 |
text {* |
222 |
\medskip Some of the atomic theorems generated each time the atom |
|
223 |
does not depend on @{text x}, they are trivial.*} |
|
13876 | 224 |
|
225 |
lemma fm_eq_minf: "EX z::int. ALL x. x < z --> (P = P) " |
|
14577 | 226 |
by blast |
13876 | 227 |
|
228 |
lemma fm_modd_minf: "ALL (x::int). ALL (k::int). (P = P)" |
|
14577 | 229 |
by blast |
13876 | 230 |
|
231 |
lemma not_bst_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> fm --> fm" |
|
14577 | 232 |
by blast |
13876 | 233 |
|
234 |
lemma fm_eq_pinf: "EX z::int. ALL x. z < x --> (P = P) " |
|
14577 | 235 |
by blast |
13876 | 236 |
|
14577 | 237 |
text {* The next two thms are the same as the @{text minusinf} version. *} |
238 |
||
13876 | 239 |
lemma fm_modd_pinf: "ALL (x::int). ALL (k::int). (P = P)" |
14577 | 240 |
by blast |
13876 | 241 |
|
242 |
lemma not_ast_p_fm: "ALL (x::int). Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> fm --> fm" |
|
14577 | 243 |
by blast |
13876 | 244 |
|
14577 | 245 |
text {* Theorems to be deleted from simpset when proving simplified formulaes. *} |
13876 | 246 |
|
247 |
lemma P_eqtrue: "(P=True) = P" |
|
248 |
by rules |
|
249 |
||
250 |
lemma P_eqfalse: "(P=False) = (~P)" |
|
251 |
by rules |
|
252 |
||
14577 | 253 |
text {* |
254 |
\medskip Theorems for the generation of the bachwards direction of |
|
255 |
Cooper's Theorem. |
|
13876 | 256 |
|
14577 | 257 |
These are the 6 interesting atomic cases which have to be proved relying on the |
258 |
properties of B-set and the arithmetic and contradiction proofs. *} |
|
13876 | 259 |
|
260 |
lemma not_bst_p_lt: "0 < (d::int) ==> |
|
261 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ( 0 < -x + a) --> (0 < -(x - d) + a )" |
|
14577 | 262 |
by arith |
13876 | 263 |
|
264 |
lemma not_bst_p_gt: "\<lbrakk> (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow> |
|
265 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 < (x) + a) --> ( 0 < (x - d) + a)" |
|
266 |
apply clarsimp |
|
267 |
apply(rule ccontr) |
|
268 |
apply(drule_tac x = "x+a" in bspec) |
|
269 |
apply(simp add:atLeastAtMost_iff) |
|
270 |
apply(drule_tac x = "-a" in bspec) |
|
271 |
apply assumption |
|
272 |
apply(simp) |
|
273 |
done |
|
274 |
||
275 |
lemma not_bst_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a - 1 \<rbrakk> \<Longrightarrow> |
|
276 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> (0 = x + a) --> (0 = (x - d) + a )" |
|
277 |
apply clarsimp |
|
278 |
apply(subgoal_tac "x = -a") |
|
279 |
prefer 2 apply arith |
|
280 |
apply(drule_tac x = "1" in bspec) |
|
281 |
apply(simp add:atLeastAtMost_iff) |
|
282 |
apply(drule_tac x = "-a- 1" in bspec) |
|
283 |
apply assumption |
|
284 |
apply(simp) |
|
285 |
done |
|
286 |
||
287 |
||
288 |
lemma not_bst_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> B; g = -a \<rbrakk> \<Longrightarrow> |
|
289 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(0 = x + a) --> ~(0 = (x - d) + a)" |
|
290 |
apply clarsimp |
|
291 |
apply(subgoal_tac "x = -a+d") |
|
292 |
prefer 2 apply arith |
|
293 |
apply(drule_tac x = "d" in bspec) |
|
294 |
apply(simp add:atLeastAtMost_iff) |
|
295 |
apply(drule_tac x = "-a" in bspec) |
|
296 |
apply assumption |
|
297 |
apply(simp) |
|
298 |
done |
|
299 |
||
300 |
||
301 |
lemma not_bst_p_dvd: "(d1::int) dvd d ==> |
|
302 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> d1 dvd (x + a) --> d1 dvd ((x - d) + a )" |
|
303 |
apply(clarsimp simp add:dvd_def) |
|
304 |
apply(rename_tac m) |
|
305 |
apply(rule_tac x = "m - k" in exI) |
|
306 |
apply(simp add:int_distrib) |
|
307 |
done |
|
308 |
||
309 |
lemma not_bst_p_ndvd: "(d1::int) dvd d ==> |
|
310 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (b::int) : B. Q(b+j)) --> ~(d1 dvd (x + a)) --> ~(d1 dvd ((x - d) + a ))" |
|
311 |
apply(clarsimp simp add:dvd_def) |
|
312 |
apply(rename_tac m) |
|
313 |
apply(erule_tac x = "m + k" in allE) |
|
314 |
apply(simp add:int_distrib) |
|
315 |
done |
|
316 |
||
14577 | 317 |
text {* |
318 |
\medskip Theorems for the generation of the bachwards direction of |
|
319 |
Cooper's Theorem. |
|
13876 | 320 |
|
14577 | 321 |
These are the 6 interesting atomic cases which have to be proved |
322 |
relying on the properties of A-set ant the arithmetic and |
|
323 |
contradiction proofs. *} |
|
13876 | 324 |
|
325 |
lemma not_ast_p_gt: "0 < (d::int) ==> |
|
326 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ( 0 < x + t) --> (0 < (x + d) + t )" |
|
14577 | 327 |
by arith |
13876 | 328 |
|
329 |
lemma not_ast_p_lt: "\<lbrakk>0 < d ;(t::int) \<in> A \<rbrakk> \<Longrightarrow> |
|
330 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 < -x + t) --> ( 0 < -(x + d) + t)" |
|
331 |
apply clarsimp |
|
332 |
apply (rule ccontr) |
|
333 |
apply (drule_tac x = "t-x" in bspec) |
|
334 |
apply simp |
|
335 |
apply (drule_tac x = "t" in bspec) |
|
336 |
apply assumption |
|
337 |
apply simp |
|
338 |
done |
|
339 |
||
340 |
lemma not_ast_p_eq: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t + 1 \<rbrakk> \<Longrightarrow> |
|
341 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> (0 = x + t) --> (0 = (x + d) + t )" |
|
342 |
apply clarsimp |
|
343 |
apply (drule_tac x="1" in bspec) |
|
344 |
apply simp |
|
345 |
apply (drule_tac x="- t + 1" in bspec) |
|
346 |
apply assumption |
|
347 |
apply(subgoal_tac "x = -t") |
|
348 |
prefer 2 apply arith |
|
349 |
apply simp |
|
350 |
done |
|
351 |
||
352 |
lemma not_ast_p_ne: "\<lbrakk> 0 < d; (g::int) \<in> A; g = -t \<rbrakk> \<Longrightarrow> |
|
353 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(0 = x + t) --> ~(0 = (x + d) + t)" |
|
354 |
apply clarsimp |
|
355 |
apply (subgoal_tac "x = -t-d") |
|
356 |
prefer 2 apply arith |
|
357 |
apply (drule_tac x = "d" in bspec) |
|
358 |
apply simp |
|
359 |
apply (drule_tac x = "-t" in bspec) |
|
360 |
apply assumption |
|
361 |
apply simp |
|
362 |
done |
|
363 |
||
364 |
lemma not_ast_p_dvd: "(d1::int) dvd d ==> |
|
365 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> d1 dvd (x + t) --> d1 dvd ((x + d) + t )" |
|
366 |
apply(clarsimp simp add:dvd_def) |
|
367 |
apply(rename_tac m) |
|
368 |
apply(rule_tac x = "m + k" in exI) |
|
369 |
apply(simp add:int_distrib) |
|
370 |
done |
|
371 |
||
372 |
lemma not_ast_p_ndvd: "(d1::int) dvd d ==> |
|
373 |
ALL x. Q(x::int) \<and> ~(EX (j::int) : {1..d}. EX (a::int) : A. Q(a - j)) --> ~(d1 dvd (x + t)) --> ~(d1 dvd ((x + d) + t ))" |
|
374 |
apply(clarsimp simp add:dvd_def) |
|
375 |
apply(rename_tac m) |
|
376 |
apply(erule_tac x = "m - k" in allE) |
|
377 |
apply(simp add:int_distrib) |
|
378 |
done |
|
379 |
||
14577 | 380 |
text {* |
381 |
\medskip These are the atomic cases for the proof generation for the |
|
382 |
modulo @{text D} property for @{text "P plusinfinity"} |
|
13876 | 383 |
|
14577 | 384 |
They are fully based on arithmetics. *} |
13876 | 385 |
|
386 |
lemma dvd_modd_pinf: "((d::int) dvd d1) ==> |
|
387 |
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x+k*d1 + t))))" |
|
388 |
apply(clarsimp simp add:dvd_def) |
|
389 |
apply(rule iffI) |
|
390 |
apply(clarsimp) |
|
391 |
apply(rename_tac n m) |
|
392 |
apply(rule_tac x = "m + n*k" in exI) |
|
393 |
apply(simp add:int_distrib) |
|
394 |
apply(clarsimp) |
|
395 |
apply(rename_tac n m) |
|
396 |
apply(rule_tac x = "m - n*k" in exI) |
|
14271 | 397 |
apply(simp add:int_distrib mult_ac) |
13876 | 398 |
done |
399 |
||
400 |
lemma not_dvd_modd_pinf: "((d::int) dvd d1) ==> |
|
401 |
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x+k*d1 + t))))" |
|
402 |
apply(clarsimp simp add:dvd_def) |
|
403 |
apply(rule iffI) |
|
404 |
apply(clarsimp) |
|
405 |
apply(rename_tac n m) |
|
406 |
apply(erule_tac x = "m - n*k" in allE) |
|
14271 | 407 |
apply(simp add:int_distrib mult_ac) |
13876 | 408 |
apply(clarsimp) |
409 |
apply(rename_tac n m) |
|
410 |
apply(erule_tac x = "m + n*k" in allE) |
|
14271 | 411 |
apply(simp add:int_distrib mult_ac) |
13876 | 412 |
done |
413 |
||
14577 | 414 |
text {* |
415 |
\medskip These are the atomic cases for the proof generation for the |
|
416 |
equivalence of @{text P} and @{text "P plusinfinity"} for integers |
|
417 |
@{text x} greater than some integer @{text z}. |
|
418 |
||
419 |
They are fully based on arithmetics. *} |
|
13876 | 420 |
|
421 |
lemma eq_eq_pinf: "EX z::int. ALL x. z < x --> (( 0 = x +t ) = False )" |
|
422 |
apply(rule_tac x = "-t" in exI) |
|
423 |
apply simp |
|
424 |
done |
|
425 |
||
426 |
lemma neq_eq_pinf: "EX z::int. ALL x. z < x --> ((~( 0 = x +t )) = True )" |
|
427 |
apply(rule_tac x = "-t" in exI) |
|
428 |
apply simp |
|
429 |
done |
|
430 |
||
431 |
lemma le_eq_pinf: "EX z::int. ALL x. z < x --> ( 0 < x +t = True )" |
|
432 |
apply(rule_tac x = "-t" in exI) |
|
433 |
apply simp |
|
434 |
done |
|
435 |
||
436 |
lemma len_eq_pinf: "EX z::int. ALL x. z < x --> (0 < -x +t = False )" |
|
437 |
apply(rule_tac x = "t" in exI) |
|
438 |
apply simp |
|
439 |
done |
|
440 |
||
441 |
lemma dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((d dvd (x + t)) = (d dvd (x + t))) " |
|
14577 | 442 |
by simp |
13876 | 443 |
|
444 |
lemma not_dvd_eq_pinf: "EX z::int. ALL x. z < x --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) " |
|
14577 | 445 |
by simp |
13876 | 446 |
|
14577 | 447 |
text {* |
448 |
\medskip These are the atomic cases for the proof generation for the |
|
449 |
modulo @{text D} property for @{text "P minusinfinity"}. |
|
450 |
||
451 |
They are fully based on arithmetics. *} |
|
13876 | 452 |
|
453 |
lemma dvd_modd_minf: "((d::int) dvd d1) ==> |
|
454 |
(ALL (x::int). ALL (k::int). (((d::int) dvd (x + t)) = (d dvd (x-k*d1 + t))))" |
|
455 |
apply(clarsimp simp add:dvd_def) |
|
456 |
apply(rule iffI) |
|
457 |
apply(clarsimp) |
|
458 |
apply(rename_tac n m) |
|
459 |
apply(rule_tac x = "m - n*k" in exI) |
|
460 |
apply(simp add:int_distrib) |
|
461 |
apply(clarsimp) |
|
462 |
apply(rename_tac n m) |
|
463 |
apply(rule_tac x = "m + n*k" in exI) |
|
14271 | 464 |
apply(simp add:int_distrib mult_ac) |
13876 | 465 |
done |
466 |
||
467 |
||
468 |
lemma not_dvd_modd_minf: "((d::int) dvd d1) ==> |
|
469 |
(ALL (x::int). ALL k. (~((d::int) dvd (x + t))) = (~(d dvd (x-k*d1 + t))))" |
|
470 |
apply(clarsimp simp add:dvd_def) |
|
471 |
apply(rule iffI) |
|
472 |
apply(clarsimp) |
|
473 |
apply(rename_tac n m) |
|
474 |
apply(erule_tac x = "m + n*k" in allE) |
|
14271 | 475 |
apply(simp add:int_distrib mult_ac) |
13876 | 476 |
apply(clarsimp) |
477 |
apply(rename_tac n m) |
|
478 |
apply(erule_tac x = "m - n*k" in allE) |
|
14271 | 479 |
apply(simp add:int_distrib mult_ac) |
13876 | 480 |
done |
481 |
||
14577 | 482 |
text {* |
483 |
\medskip These are the atomic cases for the proof generation for the |
|
484 |
equivalence of @{text P} and @{text "P minusinfinity"} for integers |
|
485 |
@{text x} less than some integer @{text z}. |
|
13876 | 486 |
|
14577 | 487 |
They are fully based on arithmetics. *} |
13876 | 488 |
|
489 |
lemma eq_eq_minf: "EX z::int. ALL x. x < z --> (( 0 = x +t ) = False )" |
|
490 |
apply(rule_tac x = "-t" in exI) |
|
491 |
apply simp |
|
492 |
done |
|
493 |
||
494 |
lemma neq_eq_minf: "EX z::int. ALL x. x < z --> ((~( 0 = x +t )) = True )" |
|
495 |
apply(rule_tac x = "-t" in exI) |
|
496 |
apply simp |
|
497 |
done |
|
498 |
||
499 |
lemma le_eq_minf: "EX z::int. ALL x. x < z --> ( 0 < x +t = False )" |
|
500 |
apply(rule_tac x = "-t" in exI) |
|
501 |
apply simp |
|
502 |
done |
|
503 |
||
504 |
||
505 |
lemma len_eq_minf: "EX z::int. ALL x. x < z --> (0 < -x +t = True )" |
|
506 |
apply(rule_tac x = "t" in exI) |
|
507 |
apply simp |
|
508 |
done |
|
509 |
||
510 |
lemma dvd_eq_minf: "EX z::int. ALL x. x < z --> ((d dvd (x + t)) = (d dvd (x + t))) " |
|
14577 | 511 |
by simp |
13876 | 512 |
|
513 |
lemma not_dvd_eq_minf: "EX z::int. ALL x. x < z --> ((~(d dvd (x + t))) = (~(d dvd (x + t)))) " |
|
14577 | 514 |
by simp |
13876 | 515 |
|
14577 | 516 |
text {* |
517 |
\medskip This Theorem combines whithnesses about @{text "P |
|
518 |
minusinfinity"} to show one component of the equivalence proof for |
|
519 |
Cooper's Theorem. |
|
13876 | 520 |
|
14577 | 521 |
FIXME: remove once they are part of the distribution. *} |
522 |
||
13876 | 523 |
theorem int_ge_induct[consumes 1,case_names base step]: |
524 |
assumes ge: "k \<le> (i::int)" and |
|
525 |
base: "P(k)" and |
|
526 |
step: "\<And>i. \<lbrakk>k \<le> i; P i\<rbrakk> \<Longrightarrow> P(i+1)" |
|
527 |
shows "P i" |
|
528 |
proof - |
|
529 |
{ fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k <= i \<Longrightarrow> P i" |
|
530 |
proof (induct n) |
|
531 |
case 0 |
|
532 |
hence "i = k" by arith |
|
533 |
thus "P i" using base by simp |
|
534 |
next |
|
535 |
case (Suc n) |
|
536 |
hence "n = nat((i - 1) - k)" by arith |
|
537 |
moreover |
|
538 |
have ki1: "k \<le> i - 1" using Suc.prems by arith |
|
539 |
ultimately |
|
540 |
have "P(i - 1)" by(rule Suc.hyps) |
|
541 |
from step[OF ki1 this] show ?case by simp |
|
542 |
qed |
|
543 |
} |
|
544 |
from this ge show ?thesis by fast |
|
545 |
qed |
|
546 |
||
547 |
theorem int_gr_induct[consumes 1,case_names base step]: |
|
548 |
assumes gr: "k < (i::int)" and |
|
549 |
base: "P(k+1)" and |
|
550 |
step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)" |
|
551 |
shows "P i" |
|
552 |
apply(rule int_ge_induct[of "k + 1"]) |
|
553 |
using gr apply arith |
|
554 |
apply(rule base) |
|
555 |
apply(rule step) |
|
556 |
apply simp+ |
|
557 |
done |
|
558 |
||
559 |
lemma decr_lemma: "0 < (d::int) \<Longrightarrow> x - (abs(x-z)+1) * d < z" |
|
560 |
apply(induct rule: int_gr_induct) |
|
561 |
apply simp |
|
562 |
apply arith |
|
563 |
apply (simp add:int_distrib) |
|
564 |
apply arith |
|
565 |
done |
|
566 |
||
567 |
lemma incr_lemma: "0 < (d::int) \<Longrightarrow> z < x + (abs(x-z)+1) * d" |
|
568 |
apply(induct rule: int_gr_induct) |
|
569 |
apply simp |
|
570 |
apply arith |
|
571 |
apply (simp add:int_distrib) |
|
572 |
apply arith |
|
573 |
done |
|
574 |
||
575 |
lemma minusinfinity: |
|
576 |
assumes "0 < d" and |
|
577 |
P1eqP1: "ALL x k. P1 x = P1(x - k*d)" and |
|
578 |
ePeqP1: "EX z::int. ALL x. x < z \<longrightarrow> (P x = P1 x)" |
|
579 |
shows "(EX x. P1 x) \<longrightarrow> (EX x. P x)" |
|
580 |
proof |
|
581 |
assume eP1: "EX x. P1 x" |
|
582 |
then obtain x where P1: "P1 x" .. |
|
583 |
from ePeqP1 obtain z where P1eqP: "ALL x. x < z \<longrightarrow> (P x = P1 x)" .. |
|
584 |
let ?w = "x - (abs(x-z)+1) * d" |
|
585 |
show "EX x. P x" |
|
586 |
proof |
|
587 |
have w: "?w < z" by(rule decr_lemma) |
|
588 |
have "P1 x = P1 ?w" using P1eqP1 by blast |
|
589 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
590 |
finally show "P ?w" using P1 by blast |
|
591 |
qed |
|
592 |
qed |
|
593 |
||
14577 | 594 |
text {* |
595 |
\medskip This Theorem combines whithnesses about @{text "P |
|
596 |
minusinfinity"} to show one component of the equivalence proof for |
|
597 |
Cooper's Theorem. *} |
|
13876 | 598 |
|
599 |
lemma plusinfinity: |
|
600 |
assumes "0 < d" and |
|
601 |
P1eqP1: "ALL (x::int) (k::int). P1 x = P1 (x + k * d)" and |
|
602 |
ePeqP1: "EX z::int. ALL x. z < x --> (P x = P1 x)" |
|
603 |
shows "(EX x::int. P1 x) --> (EX x::int. P x)" |
|
604 |
proof |
|
605 |
assume eP1: "EX x. P1 x" |
|
606 |
then obtain x where P1: "P1 x" .. |
|
607 |
from ePeqP1 obtain z where P1eqP: "ALL x. z < x \<longrightarrow> (P x = P1 x)" .. |
|
608 |
let ?w = "x + (abs(x-z)+1) * d" |
|
609 |
show "EX x. P x" |
|
610 |
proof |
|
611 |
have w: "z < ?w" by(rule incr_lemma) |
|
612 |
have "P1 x = P1 ?w" using P1eqP1 by blast |
|
613 |
also have "\<dots> = P(?w)" using w P1eqP by blast |
|
614 |
finally show "P ?w" using P1 by blast |
|
615 |
qed |
|
616 |
qed |
|
617 |
||
14577 | 618 |
text {* |
619 |
\medskip Theorem for periodic function on discrete sets. *} |
|
13876 | 620 |
|
621 |
lemma minf_vee: |
|
622 |
assumes dpos: "(0::int) < d" and modd: "ALL x k. P x = P(x - k*d)" |
|
623 |
shows "(EX x. P x) = (EX j : {1..d}. P j)" |
|
624 |
(is "?LHS = ?RHS") |
|
625 |
proof |
|
626 |
assume ?LHS |
|
627 |
then obtain x where P: "P x" .. |
|
628 |
have "x mod d = x - (x div d)*d" |
|
14271 | 629 |
by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) |
13876 | 630 |
hence Pmod: "P x = P(x mod d)" using modd by simp |
631 |
show ?RHS |
|
632 |
proof (cases) |
|
633 |
assume "x mod d = 0" |
|
634 |
hence "P 0" using P Pmod by simp |
|
635 |
moreover have "P 0 = P(0 - (-1)*d)" using modd by blast |
|
636 |
ultimately have "P d" by simp |
|
637 |
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) |
|
638 |
ultimately show ?RHS .. |
|
639 |
next |
|
640 |
assume not0: "x mod d \<noteq> 0" |
|
641 |
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) |
|
642 |
moreover have "x mod d : {1..d}" |
|
643 |
proof - |
|
644 |
have "0 \<le> x mod d" by(rule pos_mod_sign) |
|
645 |
moreover have "x mod d < d" by(rule pos_mod_bound) |
|
646 |
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) |
|
647 |
qed |
|
648 |
ultimately show ?RHS .. |
|
649 |
qed |
|
650 |
next |
|
651 |
assume ?RHS thus ?LHS by blast |
|
652 |
qed |
|
653 |
||
14577 | 654 |
text {* |
655 |
\medskip Theorem for periodic function on discrete sets. *} |
|
656 |
||
13876 | 657 |
lemma pinf_vee: |
658 |
assumes dpos: "0 < (d::int)" and modd: "ALL (x::int) (k::int). P x = P (x+k*d)" |
|
659 |
shows "(EX x::int. P x) = (EX (j::int) : {1..d} . P j)" |
|
660 |
(is "?LHS = ?RHS") |
|
661 |
proof |
|
662 |
assume ?LHS |
|
663 |
then obtain x where P: "P x" .. |
|
664 |
have "x mod d = x + (-(x div d))*d" |
|
14271 | 665 |
by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq) |
13876 | 666 |
hence Pmod: "P x = P(x mod d)" using modd by (simp only:) |
667 |
show ?RHS |
|
668 |
proof (cases) |
|
669 |
assume "x mod d = 0" |
|
670 |
hence "P 0" using P Pmod by simp |
|
671 |
moreover have "P 0 = P(0 + 1*d)" using modd by blast |
|
672 |
ultimately have "P d" by simp |
|
673 |
moreover have "d : {1..d}" using dpos by(simp add:atLeastAtMost_iff) |
|
674 |
ultimately show ?RHS .. |
|
675 |
next |
|
676 |
assume not0: "x mod d \<noteq> 0" |
|
677 |
have "P(x mod d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound) |
|
678 |
moreover have "x mod d : {1..d}" |
|
679 |
proof - |
|
680 |
have "0 \<le> x mod d" by(rule pos_mod_sign) |
|
681 |
moreover have "x mod d < d" by(rule pos_mod_bound) |
|
682 |
ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff) |
|
683 |
qed |
|
684 |
ultimately show ?RHS .. |
|
685 |
qed |
|
686 |
next |
|
687 |
assume ?RHS thus ?LHS by blast |
|
688 |
qed |
|
689 |
||
690 |
lemma decr_mult_lemma: |
|
691 |
assumes dpos: "(0::int) < d" and |
|
692 |
minus: "ALL x::int. P x \<longrightarrow> P(x - d)" and |
|
693 |
knneg: "0 <= k" |
|
694 |
shows "ALL x. P x \<longrightarrow> P(x - k*d)" |
|
695 |
using knneg |
|
696 |
proof (induct rule:int_ge_induct) |
|
697 |
case base thus ?case by simp |
|
698 |
next |
|
699 |
case (step i) |
|
700 |
show ?case |
|
701 |
proof |
|
702 |
fix x |
|
703 |
have "P x \<longrightarrow> P (x - i * d)" using step.hyps by blast |
|
704 |
also have "\<dots> \<longrightarrow> P(x - (i + 1) * d)" |
|
705 |
using minus[THEN spec, of "x - i * d"] |
|
14738 | 706 |
by (simp add:int_distrib OrderedGroup.diff_diff_eq[symmetric]) |
13876 | 707 |
ultimately show "P x \<longrightarrow> P(x - (i + 1) * d)" by blast |
708 |
qed |
|
709 |
qed |
|
710 |
||
711 |
lemma incr_mult_lemma: |
|
712 |
assumes dpos: "(0::int) < d" and |
|
713 |
plus: "ALL x::int. P x \<longrightarrow> P(x + d)" and |
|
714 |
knneg: "0 <= k" |
|
715 |
shows "ALL x. P x \<longrightarrow> P(x + k*d)" |
|
716 |
using knneg |
|
717 |
proof (induct rule:int_ge_induct) |
|
718 |
case base thus ?case by simp |
|
719 |
next |
|
720 |
case (step i) |
|
721 |
show ?case |
|
722 |
proof |
|
723 |
fix x |
|
724 |
have "P x \<longrightarrow> P (x + i * d)" using step.hyps by blast |
|
725 |
also have "\<dots> \<longrightarrow> P(x + (i + 1) * d)" |
|
726 |
using plus[THEN spec, of "x + i * d"] |
|
727 |
by (simp add:int_distrib zadd_ac) |
|
728 |
ultimately show "P x \<longrightarrow> P(x + (i + 1) * d)" by blast |
|
729 |
qed |
|
730 |
qed |
|
731 |
||
732 |
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x)) |
|
733 |
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) |
|
734 |
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D)))) |
|
735 |
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))" |
|
736 |
apply(rule iffI) |
|
737 |
prefer 2 |
|
738 |
apply(drule minusinfinity) |
|
739 |
apply assumption+ |
|
740 |
apply(fastsimp) |
|
741 |
apply clarsimp |
|
742 |
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)") |
|
743 |
apply(frule_tac x = x and z=z in decr_lemma) |
|
744 |
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)") |
|
745 |
prefer 2 |
|
746 |
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)") |
|
747 |
prefer 2 apply arith |
|
748 |
apply fastsimp |
|
749 |
apply(drule (1) minf_vee) |
|
750 |
apply blast |
|
751 |
apply(blast dest:decr_mult_lemma) |
|
752 |
done |
|
753 |
||
14577 | 754 |
text {* Cooper Theorem, plus infinity version. *} |
13876 | 755 |
lemma cppi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. z < x --> (P x = P1 x)) |
756 |
==> ALL x.~(EX (j::int) : {1..D}. EX (a::int) : A. P(a - j)) --> P (x) --> P (x + D) |
|
757 |
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x+k*D)))) |
|
758 |
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (a::int) : A. P (a - j)))" |
|
759 |
apply(rule iffI) |
|
760 |
prefer 2 |
|
761 |
apply(drule plusinfinity) |
|
762 |
apply assumption+ |
|
763 |
apply(fastsimp) |
|
764 |
apply clarsimp |
|
765 |
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x + k*D)") |
|
766 |
apply(frule_tac x = x and z=z in incr_lemma) |
|
767 |
apply(subgoal_tac "P1(x + (\<bar>x - z\<bar> + 1) * D)") |
|
768 |
prefer 2 |
|
769 |
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)") |
|
770 |
prefer 2 apply arith |
|
771 |
apply fastsimp |
|
772 |
apply(drule (1) pinf_vee) |
|
773 |
apply blast |
|
774 |
apply(blast dest:incr_mult_lemma) |
|
775 |
done |
|
776 |
||
777 |
||
14577 | 778 |
text {* |
779 |
\bigskip Theorems for the quantifier elminination Functions. *} |
|
13876 | 780 |
|
781 |
lemma qe_ex_conj: "(EX (x::int). A x) = R |
|
782 |
==> (EX (x::int). P x) = (Q & (EX x::int. A x)) |
|
783 |
==> (EX (x::int). P x) = (Q & R)" |
|
784 |
by blast |
|
785 |
||
786 |
lemma qe_ex_nconj: "(EX (x::int). P x) = (True & Q) |
|
787 |
==> (EX (x::int). P x) = Q" |
|
788 |
by blast |
|
789 |
||
790 |
lemma qe_conjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 & Q1) = (P2 & Q2)" |
|
791 |
by blast |
|
792 |
||
793 |
lemma qe_disjI: "P1 = P2 ==> Q1 = Q2 ==> (P1 | Q1) = (P2 | Q2)" |
|
794 |
by blast |
|
795 |
||
796 |
lemma qe_impI: "P1 = P2 ==> Q1 = Q2 ==> (P1 --> Q1) = (P2 --> Q2)" |
|
797 |
by blast |
|
798 |
||
799 |
lemma qe_eqI: "P1 = P2 ==> Q1 = Q2 ==> (P1 = Q1) = (P2 = Q2)" |
|
800 |
by blast |
|
801 |
||
802 |
lemma qe_Not: "P = Q ==> (~P) = (~Q)" |
|
803 |
by blast |
|
804 |
||
805 |
lemma qe_ALL: "(EX x. ~P x) = R ==> (ALL x. P x) = (~R)" |
|
806 |
by blast |
|
807 |
||
14577 | 808 |
text {* \bigskip Theorems for proving NNF *} |
13876 | 809 |
|
810 |
lemma nnf_im: "((~P) = P1) ==> (Q=Q1) ==> ((P --> Q) = (P1 | Q1))" |
|
811 |
by blast |
|
812 |
||
813 |
lemma nnf_eq: "((P & Q) = (P1 & Q1)) ==> (((~P) & (~Q)) = (P2 & Q2)) ==> ((P = Q) = ((P1 & Q1)|(P2 & Q2)))" |
|
814 |
by blast |
|
815 |
||
816 |
lemma nnf_nn: "(P = Q) ==> ((~~P) = Q)" |
|
817 |
by blast |
|
818 |
lemma nnf_ncj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P & Q)) = (P1 | Q1))" |
|
819 |
by blast |
|
820 |
||
821 |
lemma nnf_ndj: "((~P) = P1) ==> ((~Q) = Q1) ==> ((~(P | Q)) = (P1 & Q1))" |
|
822 |
by blast |
|
823 |
lemma nnf_nim: "(P = P1) ==> ((~Q) = Q1) ==> ((~(P --> Q)) = (P1 & Q1))" |
|
824 |
by blast |
|
825 |
lemma nnf_neq: "((P & (~Q)) = (P1 & Q1)) ==> (((~P) & Q) = (P2 & Q2)) ==> ((~(P = Q)) = ((P1 & Q1)|(P2 & Q2)))" |
|
826 |
by blast |
|
827 |
lemma nnf_sdj: "((A & (~B)) = (A1 & B1)) ==> ((C & (~D)) = (C1 & D1)) ==> (A = (~C)) ==> ((~((A & B) | (C & D))) = ((A1 & B1) | (C1 & D1)))" |
|
828 |
by blast |
|
829 |
||
830 |
||
831 |
lemma qe_exI2: "A = B ==> (EX (x::int). A(x)) = (EX (x::int). B(x))" |
|
832 |
by simp |
|
833 |
||
834 |
lemma qe_exI: "(!!x::int. A x = B x) ==> (EX (x::int). A(x)) = (EX (x::int). B(x))" |
|
835 |
by rules |
|
836 |
||
837 |
lemma qe_ALLI: "(!!x::int. A x = B x) ==> (ALL (x::int). A(x)) = (ALL (x::int). B(x))" |
|
838 |
by rules |
|
839 |
||
840 |
lemma cp_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) |
|
841 |
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (b::int) : B. (P1 (j) | P(b+j))) " |
|
842 |
by blast |
|
843 |
||
844 |
lemma cppi_expand: "(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) |
|
845 |
==>(EX (x::int). P (x)) = (EX (j::int) : {1..d}. EX (a::int) : A. (P1 (j) | P(a - j))) " |
|
846 |
by blast |
|
847 |
||
848 |
||
849 |
lemma simp_from_to: "{i..j::int} = (if j < i then {} else insert i {i+1..j})" |
|
850 |
apply(simp add:atLeastAtMost_def atLeast_def atMost_def) |
|
851 |
apply(fastsimp) |
|
852 |
done |
|
853 |
||
14577 | 854 |
text {* \bigskip Theorems required for the @{text adjustcoeffitienteq} *} |
13876 | 855 |
|
856 |
lemma ac_dvd_eq: assumes not0: "0 ~= (k::int)" |
|
857 |
shows "((m::int) dvd (c*n+t)) = (k*m dvd ((k*c)*n+(k*t)))" (is "?P = ?Q") |
|
858 |
proof |
|
859 |
assume ?P |
|
860 |
thus ?Q |
|
861 |
apply(simp add:dvd_def) |
|
862 |
apply clarify |
|
863 |
apply(rename_tac d) |
|
864 |
apply(drule_tac f = "op * k" in arg_cong) |
|
865 |
apply(simp only:int_distrib) |
|
866 |
apply(rule_tac x = "d" in exI) |
|
14271 | 867 |
apply(simp only:mult_ac) |
13876 | 868 |
done |
869 |
next |
|
870 |
assume ?Q |
|
871 |
then obtain d where "k * c * n + k * t = (k*m)*d" by(fastsimp simp:dvd_def) |
|
14271 | 872 |
hence "(c * n + t) * k = (m*d) * k" by(simp add:int_distrib mult_ac) |
13876 | 873 |
hence "((c * n + t) * k) div k = ((m*d) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"]) |
874 |
hence "c*n+t = m*d" by(simp add: zdiv_zmult_self1[OF not0[symmetric]]) |
|
875 |
thus ?P by(simp add:dvd_def) |
|
876 |
qed |
|
877 |
||
878 |
lemma ac_lt_eq: assumes gr0: "0 < (k::int)" |
|
879 |
shows "((m::int) < (c*n+t)) = (k*m <((k*c)*n+(k*t)))" (is "?P = ?Q") |
|
880 |
proof |
|
881 |
assume P: ?P |
|
14271 | 882 |
show ?Q using zmult_zless_mono2[OF P gr0] by(simp add: int_distrib mult_ac) |
13876 | 883 |
next |
884 |
assume ?Q |
|
14271 | 885 |
hence "0 < k*(c*n + t - m)" by(simp add: int_distrib mult_ac) |
14353
79f9fbef9106
Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents:
14271
diff
changeset
|
886 |
with gr0 have "0 < (c*n + t - m)" by(simp add: zero_less_mult_iff) |
13876 | 887 |
thus ?P by(simp) |
888 |
qed |
|
889 |
||
890 |
lemma ac_eq_eq : assumes not0: "0 ~= (k::int)" shows "((m::int) = (c*n+t)) = (k*m =((k*c)*n+(k*t)) )" (is "?P = ?Q") |
|
891 |
proof |
|
892 |
assume ?P |
|
893 |
thus ?Q |
|
894 |
apply(drule_tac f = "op * k" in arg_cong) |
|
895 |
apply(simp only:int_distrib) |
|
896 |
done |
|
897 |
next |
|
898 |
assume ?Q |
|
14271 | 899 |
hence "m * k = (c*n + t) * k" by(simp add:int_distrib mult_ac) |
13876 | 900 |
hence "((m) * k) div k = ((c*n + t) * k) div k" by(rule arg_cong[of _ _ "%t. t div k"]) |
901 |
thus ?P by(simp add: zdiv_zmult_self1[OF not0[symmetric]]) |
|
902 |
qed |
|
903 |
||
904 |
lemma ac_pi_eq: assumes gr0: "0 < (k::int)" shows "(~((0::int) < (c*n + t))) = (0 < ((-k)*c)*n + ((-k)*t + k))" |
|
905 |
proof - |
|
906 |
have "(~ (0::int) < (c*n + t)) = (0<1-(c*n + t))" by arith |
|
14271 | 907 |
also have "(1-(c*n + t)) = (-1*c)*n + (-t+1)" by(simp add: int_distrib mult_ac) |
13876 | 908 |
also have "0<(-1*c)*n + (-t+1) = (0 < (k*(-1*c)*n) + (k*(-t+1)))" by(rule ac_lt_eq[of _ 0,OF gr0,simplified]) |
14271 | 909 |
also have "(k*(-1*c)*n) + (k*(-t+1)) = ((-k)*c)*n + ((-k)*t + k)" by(simp add: int_distrib mult_ac) |
13876 | 910 |
finally show ?thesis . |
911 |
qed |
|
912 |
||
913 |
lemma binminus_uminus_conv: "(a::int) - b = a + (-b)" |
|
914 |
by arith |
|
915 |
||
916 |
lemma linearize_dvd: "(t::int) = t1 ==> (d dvd t) = (d dvd t1)" |
|
917 |
by simp |
|
918 |
||
919 |
lemma lf_lt: "(l::int) = ll ==> (r::int) = lr ==> (l < r) =(ll < lr)" |
|
920 |
by simp |
|
921 |
||
922 |
lemma lf_eq: "(l::int) = ll ==> (r::int) = lr ==> (l = r) =(ll = lr)" |
|
923 |
by simp |
|
924 |
||
925 |
lemma lf_dvd: "(l::int) = ll ==> (r::int) = lr ==> (l dvd r) =(ll dvd lr)" |
|
926 |
by simp |
|
927 |
||
14577 | 928 |
text {* \bigskip Theorems for transforming predicates on nat to predicates on @{text int}*} |
13876 | 929 |
|
930 |
theorem all_nat: "(\<forall>x::nat. P x) = (\<forall>x::int. 0 <= x \<longrightarrow> P (nat x))" |
|
931 |
by (simp split add: split_nat) |
|
932 |
||
933 |
theorem ex_nat: "(\<exists>x::nat. P x) = (\<exists>x::int. 0 <= x \<and> P (nat x))" |
|
934 |
apply (simp split add: split_nat) |
|
935 |
apply (rule iffI) |
|
936 |
apply (erule exE) |
|
937 |
apply (rule_tac x = "int x" in exI) |
|
938 |
apply simp |
|
939 |
apply (erule exE) |
|
940 |
apply (rule_tac x = "nat x" in exI) |
|
941 |
apply (erule conjE) |
|
942 |
apply (erule_tac x = "nat x" in allE) |
|
943 |
apply simp |
|
944 |
done |
|
945 |
||
946 |
theorem zdiff_int_split: "P (int (x - y)) = |
|
947 |
((y \<le> x \<longrightarrow> P (int x - int y)) \<and> (x < y \<longrightarrow> P 0))" |
|
948 |
apply (case_tac "y \<le> x") |
|
949 |
apply (simp_all add: zdiff_int) |
|
950 |
done |
|
951 |
||
952 |
theorem zdvd_int: "(x dvd y) = (int x dvd int y)" |
|
953 |
apply (simp only: dvd_def ex_nat int_int_eq [symmetric] zmult_int [symmetric] |
|
954 |
nat_0_le cong add: conj_cong) |
|
955 |
apply (rule iffI) |
|
956 |
apply rules |
|
957 |
apply (erule exE) |
|
958 |
apply (case_tac "x=0") |
|
959 |
apply (rule_tac x=0 in exI) |
|
960 |
apply simp |
|
961 |
apply (case_tac "0 \<le> k") |
|
962 |
apply rules |
|
963 |
apply (simp add: linorder_not_le) |
|
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14353
diff
changeset
|
964 |
apply (drule mult_strict_left_mono_neg [OF iffD2 [OF zero_less_int_conv]]) |
13876 | 965 |
apply assumption |
14271 | 966 |
apply (simp add: mult_ac) |
13876 | 967 |
done |
968 |
||
969 |
theorem number_of1: "(0::int) <= number_of n \<Longrightarrow> (0::int) <= number_of (n BIT b)" |
|
970 |
by simp |
|
971 |
||
972 |
theorem number_of2: "(0::int) <= number_of bin.Pls" by simp |
|
973 |
||
974 |
theorem Suc_plus1: "Suc n = n + 1" by simp |
|
975 |
||
14577 | 976 |
text {* |
977 |
\medskip Specific instances of congruence rules, to prevent |
|
978 |
simplifier from looping. *} |
|
13876 | 979 |
|
14758
af3b71a46a1c
A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
chaieb
parents:
14738
diff
changeset
|
980 |
theorem imp_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<longrightarrow> P) = (0 <= x \<longrightarrow> P')" |
13876 | 981 |
by simp |
982 |
||
14758
af3b71a46a1c
A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
chaieb
parents:
14738
diff
changeset
|
983 |
theorem conj_le_cong: "(0 <= x \<Longrightarrow> P = P') \<Longrightarrow> (0 <= (x::int) \<and> P) = (0 <= x \<and> P')" |
af3b71a46a1c
A new implementation for presburger arithmetic following the one suggested in technical report Chaieb Amine and Tobias Nipkow. It is generic an smaller.
chaieb
parents:
14738
diff
changeset
|
984 |
by (simp cong: conj_cong) |
13876 | 985 |
|
986 |
use "cooper_dec.ML" |
|
14941 | 987 |
oracle |
988 |
presburger_oracle = CooperDec.mk_presburger_oracle |
|
989 |
||
13876 | 990 |
use "cooper_proof.ML" |
991 |
use "qelim.ML" |
|
992 |
use "presburger.ML" |
|
993 |
||
994 |
setup "Presburger.setup" |
|
995 |
||
996 |
end |