1 | include "basics/lists/list.ma". |
---|
2 | include "basics/types.ma". |
---|
3 | include "arithmetics/nat.ma". |
---|
4 | include "basics/russell.ma". |
---|
5 | |
---|
6 | (* let's implement a daemon not used by automation *) |
---|
7 | inductive DAEMONXXX: Type[0] ≝ K1DAEMONXXX: DAEMONXXX | K2DAEMONXXX: DAEMONXXX. |
---|
8 | axiom IMPOSSIBLE: K1DAEMONXXX = K2DAEMONXXX. |
---|
9 | example daemon: False. generalize in match IMPOSSIBLE; #IMPOSSIBLE destruct(IMPOSSIBLE) qed. |
---|
10 | example not_implemented: False. cases daemon qed. |
---|
11 | |
---|
12 | notation "⊥" with precedence 90 |
---|
13 | for @{ match ? in False with [ ] }. |
---|
14 | notation "Ⓧ" with precedence 90 |
---|
15 | for @{ λabs.match abs in False with [ ] }. |
---|
16 | |
---|
17 | |
---|
18 | definition ltb ≝ |
---|
19 | λm, n: nat. |
---|
20 | leb (S m) n. |
---|
21 | |
---|
22 | definition geb ≝ |
---|
23 | λm, n: nat. |
---|
24 | leb n m. |
---|
25 | |
---|
26 | definition gtb ≝ |
---|
27 | λm, n: nat. |
---|
28 | ltb n m. |
---|
29 | |
---|
30 | (* dpm: unless I'm being stupid, this isn't defined in the stdlib? *) |
---|
31 | let rec eq_nat (n: nat) (m: nat) on n: bool ≝ |
---|
32 | match n with |
---|
33 | [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ] |
---|
34 | | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ] |
---|
35 | ]. |
---|
36 | |
---|
37 | let rec forall |
---|
38 | (A: Type[0]) (f: A → bool) (l: list A) |
---|
39 | on l ≝ |
---|
40 | match l with |
---|
41 | [ nil ⇒ true |
---|
42 | | cons hd tl ⇒ f hd ∧ forall A f tl |
---|
43 | ]. |
---|
44 | |
---|
45 | let rec prefix |
---|
46 | (A: Type[0]) (k: nat) (l: list A) |
---|
47 | on l ≝ |
---|
48 | match l with |
---|
49 | [ nil ⇒ [ ] |
---|
50 | | cons hd tl ⇒ |
---|
51 | match k with |
---|
52 | [ O ⇒ [ ] |
---|
53 | | S k' ⇒ hd :: prefix A k' tl |
---|
54 | ] |
---|
55 | ]. |
---|
56 | |
---|
57 | let rec fold_left2 |
---|
58 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → A) (accu: A) |
---|
59 | (left: list B) (right: list C) (proof: |left| = |right|) |
---|
60 | on left: A ≝ |
---|
61 | match left return λx. |x| = |right| → A with |
---|
62 | [ nil ⇒ λnil_prf. |
---|
63 | match right return λx. |[ ]| = |x| → A with |
---|
64 | [ nil ⇒ λnil_nil_prf. accu |
---|
65 | | cons hd tl ⇒ λcons_nil_absrd. ? |
---|
66 | ] nil_prf |
---|
67 | | cons hd tl ⇒ λcons_prf. |
---|
68 | match right return λx. |hd::tl| = |x| → A with |
---|
69 | [ nil ⇒ λcons_nil_absrd. ? |
---|
70 | | cons hd' tl' ⇒ λcons_cons_prf. |
---|
71 | fold_left2 … f (f accu hd hd') tl tl' ? |
---|
72 | ] cons_prf |
---|
73 | ] proof. |
---|
74 | [ 1: normalize in cons_nil_absrd; |
---|
75 | destruct(cons_nil_absrd) |
---|
76 | | 2: normalize in cons_nil_absrd; |
---|
77 | destruct(cons_nil_absrd) |
---|
78 | | 3: normalize in cons_cons_prf; |
---|
79 | @injective_S |
---|
80 | assumption |
---|
81 | ] |
---|
82 | qed. |
---|
83 | |
---|
84 | let rec remove_n_first_internal |
---|
85 | (i: nat) (A: Type[0]) (l: list A) (n: nat) |
---|
86 | on l ≝ |
---|
87 | match l with |
---|
88 | [ nil ⇒ [ ] |
---|
89 | | cons hd tl ⇒ |
---|
90 | match eq_nat i n with |
---|
91 | [ true ⇒ l |
---|
92 | | _ ⇒ remove_n_first_internal (S i) A tl n |
---|
93 | ] |
---|
94 | ]. |
---|
95 | |
---|
96 | definition remove_n_first ≝ |
---|
97 | λA: Type[0]. |
---|
98 | λn: nat. |
---|
99 | λl: list A. |
---|
100 | remove_n_first_internal 0 A l n. |
---|
101 | |
---|
102 | let rec foldi_from_until_internal |
---|
103 | (A: Type[0]) (i: nat) (res: ?) (rem: list A) (m: nat) (f: nat → list A → A → list A) |
---|
104 | on rem ≝ |
---|
105 | match rem with |
---|
106 | [ nil ⇒ res |
---|
107 | | cons e tl ⇒ |
---|
108 | match geb i m with |
---|
109 | [ true ⇒ res |
---|
110 | | _ ⇒ foldi_from_until_internal A (S i) (f i res e) tl m f |
---|
111 | ] |
---|
112 | ]. |
---|
113 | |
---|
114 | definition foldi_from_until ≝ |
---|
115 | λA: Type[0]. |
---|
116 | λn: nat. |
---|
117 | λm: nat. |
---|
118 | λf: ?. |
---|
119 | λa: ?. |
---|
120 | λl: ?. |
---|
121 | foldi_from_until_internal A 0 a (remove_n_first A n l) m f. |
---|
122 | |
---|
123 | definition foldi_from ≝ |
---|
124 | λA: Type[0]. |
---|
125 | λn. |
---|
126 | λf. |
---|
127 | λa. |
---|
128 | λl. |
---|
129 | foldi_from_until A n (|l|) f a l. |
---|
130 | |
---|
131 | definition foldi_until ≝ |
---|
132 | λA: Type[0]. |
---|
133 | λm. |
---|
134 | λf. |
---|
135 | λa. |
---|
136 | λl. |
---|
137 | foldi_from_until A 0 m f a l. |
---|
138 | |
---|
139 | definition foldi ≝ |
---|
140 | λA: Type[0]. |
---|
141 | λf. |
---|
142 | λa. |
---|
143 | λl. |
---|
144 | foldi_from_until A 0 (|l|) f a l. |
---|
145 | |
---|
146 | definition hd_safe ≝ |
---|
147 | λA: Type[0]. |
---|
148 | λl: list A. |
---|
149 | λproof: 0 < |l|. |
---|
150 | match l return λx. 0 < |x| → A with |
---|
151 | [ nil ⇒ λnil_absrd. ? |
---|
152 | | cons hd tl ⇒ λcons_prf. hd |
---|
153 | ] proof. |
---|
154 | normalize in nil_absrd; |
---|
155 | cases(not_le_Sn_O 0) |
---|
156 | #HYP |
---|
157 | cases(HYP nil_absrd) |
---|
158 | qed. |
---|
159 | |
---|
160 | definition tail_safe ≝ |
---|
161 | λA: Type[0]. |
---|
162 | λl: list A. |
---|
163 | λproof: 0 < |l|. |
---|
164 | match l return λx. 0 < |x| → list A with |
---|
165 | [ nil ⇒ λnil_absrd. ? |
---|
166 | | cons hd tl ⇒ λcons_prf. tl |
---|
167 | ] proof. |
---|
168 | normalize in nil_absrd; |
---|
169 | cases(not_le_Sn_O 0) |
---|
170 | #HYP |
---|
171 | cases(HYP nil_absrd) |
---|
172 | qed. |
---|
173 | |
---|
174 | let rec split |
---|
175 | (A: Type[0]) (l: list A) (index: nat) (proof: index ≤ |l|) |
---|
176 | on index ≝ |
---|
177 | match index return λx. x ≤ |l| → (list A) × (list A) with |
---|
178 | [ O ⇒ λzero_prf. 〈[], l〉 |
---|
179 | | S index' ⇒ λsucc_prf. |
---|
180 | match l return λx. S index' ≤ |x| → (list A) × (list A) with |
---|
181 | [ nil ⇒ λnil_absrd. ? |
---|
182 | | cons hd tl ⇒ λcons_prf. |
---|
183 | let 〈l1, l2〉 ≝ split A tl index' ? in |
---|
184 | 〈hd :: l1, l2〉 |
---|
185 | ] succ_prf |
---|
186 | ] proof. |
---|
187 | [1: normalize in nil_absrd; |
---|
188 | cases(not_le_Sn_O index') |
---|
189 | #HYP |
---|
190 | cases(HYP nil_absrd) |
---|
191 | |2: normalize in cons_prf; |
---|
192 | @le_S_S_to_le |
---|
193 | assumption |
---|
194 | ] |
---|
195 | qed. |
---|
196 | |
---|
197 | let rec nth_safe |
---|
198 | (elt_type: Type[0]) (index: nat) (the_list: list elt_type) |
---|
199 | (proof: index < | the_list |) |
---|
200 | on index ≝ |
---|
201 | match index return λs. s < | the_list | → elt_type with |
---|
202 | [ O ⇒ |
---|
203 | match the_list return λt. 0 < | t | → elt_type with |
---|
204 | [ nil ⇒ λnil_absurd. ? |
---|
205 | | cons hd tl ⇒ λcons_proof. hd |
---|
206 | ] |
---|
207 | | S index' ⇒ |
---|
208 | match the_list return λt. S index' < | t | → elt_type with |
---|
209 | [ nil ⇒ λnil_absurd. ? |
---|
210 | | cons hd tl ⇒ |
---|
211 | λcons_proof. nth_safe elt_type index' tl ? |
---|
212 | ] |
---|
213 | ] proof. |
---|
214 | [ normalize in nil_absurd; |
---|
215 | cases (not_le_Sn_O 0) |
---|
216 | #ABSURD |
---|
217 | elim (ABSURD nil_absurd) |
---|
218 | | normalize in nil_absurd; |
---|
219 | cases (not_le_Sn_O (S index')) |
---|
220 | #ABSURD |
---|
221 | elim (ABSURD nil_absurd) |
---|
222 | | normalize in cons_proof; |
---|
223 | @le_S_S_to_le |
---|
224 | assumption |
---|
225 | ] |
---|
226 | qed. |
---|
227 | |
---|
228 | definition last_safe ≝ |
---|
229 | λelt_type: Type[0]. |
---|
230 | λthe_list: list elt_type. |
---|
231 | λproof : 0 < | the_list |. |
---|
232 | nth_safe elt_type (|the_list| - 1) the_list ?. |
---|
233 | normalize /2 by lt_plus_to_minus/ |
---|
234 | qed. |
---|
235 | |
---|
236 | let rec reduce |
---|
237 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) on left ≝ |
---|
238 | match left with |
---|
239 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
---|
240 | | cons hd tl ⇒ |
---|
241 | match right with |
---|
242 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
---|
243 | | cons hd' tl' ⇒ |
---|
244 | let 〈cleft, cright〉 ≝ reduce A B tl tl' in |
---|
245 | let 〈commonl, restl〉 ≝ cleft in |
---|
246 | let 〈commonr, restr〉 ≝ cright in |
---|
247 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
---|
248 | ] |
---|
249 | ]. |
---|
250 | |
---|
251 | (* |
---|
252 | axiom reduce_strong: |
---|
253 | ∀A: Type[0]. |
---|
254 | ∀left: list A. |
---|
255 | ∀right: list A. |
---|
256 | Σret: ((list A) × (list A)) × ((list A) × (list A)). | \fst (\fst ret) | = | \fst (\snd ret) |. |
---|
257 | *) |
---|
258 | |
---|
259 | let rec reduce_strong |
---|
260 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
---|
261 | on left : Σret: ((list A) × (list A)) × ((list B) × (list B)). |\fst (\fst ret)| = |\fst (\snd ret)| ≝ |
---|
262 | match left with |
---|
263 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
---|
264 | | cons hd tl ⇒ |
---|
265 | match right with |
---|
266 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
---|
267 | | cons hd' tl' ⇒ |
---|
268 | let 〈cleft, cright〉 ≝ reduce_strong A B tl tl' in |
---|
269 | let 〈commonl, restl〉 ≝ cleft in |
---|
270 | let 〈commonr, restr〉 ≝ cright in |
---|
271 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
---|
272 | ] |
---|
273 | ]. |
---|
274 | [ 1: normalize % |
---|
275 | | 2: normalize % |
---|
276 | | 3: normalize >p3 in p2; >p4 cases (reduce_strong … tl tl1) normalize |
---|
277 | #X #H #EQ destruct // ] |
---|
278 | qed. |
---|
279 | |
---|
280 | let rec map2_opt |
---|
281 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
---|
282 | (left: list A) (right: list B) on left ≝ |
---|
283 | match left with |
---|
284 | [ nil ⇒ |
---|
285 | match right with |
---|
286 | [ nil ⇒ Some ? (nil C) |
---|
287 | | _ ⇒ None ? |
---|
288 | ] |
---|
289 | | cons hd tl ⇒ |
---|
290 | match right with |
---|
291 | [ nil ⇒ None ? |
---|
292 | | cons hd' tl' ⇒ |
---|
293 | match map2_opt A B C f tl tl' with |
---|
294 | [ None ⇒ None ? |
---|
295 | | Some tail ⇒ Some ? (f hd hd' :: tail) |
---|
296 | ] |
---|
297 | ] |
---|
298 | ]. |
---|
299 | |
---|
300 | let rec map2 |
---|
301 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
---|
302 | (left: list A) (right: list B) (proof: | left | = | right |) on left ≝ |
---|
303 | match left return λx. | x | = | right | → list C with |
---|
304 | [ nil ⇒ |
---|
305 | match right return λy. | [] | = | y | → list C with |
---|
306 | [ nil ⇒ λnil_prf. nil C |
---|
307 | | _ ⇒ λcons_absrd. ? |
---|
308 | ] |
---|
309 | | cons hd tl ⇒ |
---|
310 | match right return λy. | hd::tl | = | y | → list C with |
---|
311 | [ nil ⇒ λnil_absrd. ? |
---|
312 | | cons hd' tl' ⇒ λcons_prf. (f hd hd') :: map2 A B C f tl tl' ? |
---|
313 | ] |
---|
314 | ] proof. |
---|
315 | [1: normalize in cons_absrd; |
---|
316 | destruct(cons_absrd) |
---|
317 | |2: normalize in nil_absrd; |
---|
318 | destruct(nil_absrd) |
---|
319 | |3: normalize in cons_prf; |
---|
320 | destruct(cons_prf) |
---|
321 | assumption |
---|
322 | ] |
---|
323 | qed. |
---|
324 | |
---|
325 | let rec map3 |
---|
326 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (D: Type[0]) (f: A → B → C → D) |
---|
327 | (left: list A) (centre: list B) (right: list C) |
---|
328 | (prflc: |left| = |centre|) (prfcr: |centre| = |right|) on left ≝ |
---|
329 | match left return λx. |x| = |centre| → list D with |
---|
330 | [ nil ⇒ λnil_prf. |
---|
331 | match centre return λx. |x| = |right| → list D with |
---|
332 | [ nil ⇒ λnil_nil_prf. |
---|
333 | match right return λx. |nil ?| = |x| → list D with |
---|
334 | [ nil ⇒ λnil_nil_nil_prf. nil D |
---|
335 | | cons hd tl ⇒ λcons_nil_nil_absrd. ? |
---|
336 | ] nil_nil_prf |
---|
337 | | cons hd tl ⇒ λnil_cons_absrd. ? |
---|
338 | ] prfcr |
---|
339 | | cons hd tl ⇒ λcons_prf. |
---|
340 | match centre return λx. |x| = |right| → list D with |
---|
341 | [ nil ⇒ λcons_nil_absrd. ? |
---|
342 | | cons hd' tl' ⇒ λcons_cons_prf. |
---|
343 | match right return λx. |right| = |x| → |cons ? hd' tl'| = |x| → list D with |
---|
344 | [ nil ⇒ λrefl_prf. λcons_cons_nil_absrd. ? |
---|
345 | | cons hd'' tl'' ⇒ λrefl_prf. λcons_cons_cons_prf. |
---|
346 | (f hd hd' hd'') :: (map3 A B C D f tl tl' tl'' ? ?) |
---|
347 | ] (refl ? (|right|)) cons_cons_prf |
---|
348 | ] prfcr |
---|
349 | ] prflc. |
---|
350 | [ 1: normalize in cons_nil_nil_absrd; |
---|
351 | destruct(cons_nil_nil_absrd) |
---|
352 | | 2: generalize in match nil_cons_absrd; |
---|
353 | <prfcr <nil_prf #HYP |
---|
354 | normalize in HYP; |
---|
355 | destruct(HYP) |
---|
356 | | 3: generalize in match cons_nil_absrd; |
---|
357 | <prfcr <cons_prf #HYP |
---|
358 | normalize in HYP; |
---|
359 | destruct(HYP) |
---|
360 | | 4: normalize in cons_cons_nil_absrd; |
---|
361 | destruct(cons_cons_nil_absrd) |
---|
362 | | 5: normalize in cons_cons_cons_prf; |
---|
363 | destruct(cons_cons_cons_prf) |
---|
364 | assumption |
---|
365 | | 6: generalize in match cons_cons_cons_prf; |
---|
366 | <refl_prf <prfcr <cons_prf #HYP |
---|
367 | normalize in HYP; |
---|
368 | destruct(HYP) |
---|
369 | @sym_eq assumption |
---|
370 | ] |
---|
371 | qed. |
---|
372 | |
---|
373 | lemma eq_rect_Type0_r : |
---|
374 | ∀A: Type[0]. |
---|
375 | ∀a:A. |
---|
376 | ∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
377 | #A #a #P #H #x #p lapply H lapply P cases p // |
---|
378 | qed. |
---|
379 | |
---|
380 | let rec safe_nth (A: Type[0]) (n: nat) (l: list A) (p: n < length A l) on n: A ≝ |
---|
381 | match n return λo. o < length A l → A with |
---|
382 | [ O ⇒ |
---|
383 | match l return λm. 0 < length A m → A with |
---|
384 | [ nil ⇒ λabsd1. ? |
---|
385 | | cons hd tl ⇒ λprf1. hd |
---|
386 | ] |
---|
387 | | S n' ⇒ |
---|
388 | match l return λm. S n' < length A m → A with |
---|
389 | [ nil ⇒ λabsd2. ? |
---|
390 | | cons hd tl ⇒ λprf2. safe_nth A n' tl ? |
---|
391 | ] |
---|
392 | ] ?. |
---|
393 | [ 1: |
---|
394 | @ p |
---|
395 | | 4: |
---|
396 | normalize in prf2; |
---|
397 | normalize |
---|
398 | @ le_S_S_to_le |
---|
399 | assumption |
---|
400 | | 2: |
---|
401 | normalize in absd1; |
---|
402 | cases (not_le_Sn_O O) |
---|
403 | # H |
---|
404 | elim (H absd1) |
---|
405 | | 3: |
---|
406 | normalize in absd2; |
---|
407 | cases (not_le_Sn_O (S n')) |
---|
408 | # H |
---|
409 | elim (H absd2) |
---|
410 | ] |
---|
411 | qed. |
---|
412 | |
---|
413 | let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝ |
---|
414 | match n with |
---|
415 | [ O ⇒ |
---|
416 | match l with |
---|
417 | [ nil ⇒ [ ] |
---|
418 | | cons hd tl ⇒ l |
---|
419 | ] |
---|
420 | | S n ⇒ |
---|
421 | match l with |
---|
422 | [ nil ⇒ [ ] |
---|
423 | | cons hd tl ⇒ |
---|
424 | hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n |
---|
425 | ] |
---|
426 | ]. |
---|
427 | |
---|
428 | definition nub_by ≝ |
---|
429 | λA: Type[0]. |
---|
430 | λf: A → A → bool. |
---|
431 | λl: list A. |
---|
432 | nub_by_internal A f l (length ? l). |
---|
433 | |
---|
434 | let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝ |
---|
435 | match l with |
---|
436 | [ nil ⇒ false |
---|
437 | | cons hd tl ⇒ orb (eq a hd) (member A eq a tl) |
---|
438 | ]. |
---|
439 | |
---|
440 | let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝ |
---|
441 | match n with |
---|
442 | [ O ⇒ [ ] |
---|
443 | | S n ⇒ |
---|
444 | match l with |
---|
445 | [ nil ⇒ [ ] |
---|
446 | | cons hd tl ⇒ hd :: take A n tl |
---|
447 | ] |
---|
448 | ]. |
---|
449 | |
---|
450 | let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝ |
---|
451 | match n with |
---|
452 | [ O ⇒ l |
---|
453 | | S n ⇒ |
---|
454 | match l with |
---|
455 | [ nil ⇒ [ ] |
---|
456 | | cons hd tl ⇒ drop A n tl |
---|
457 | ] |
---|
458 | ]. |
---|
459 | |
---|
460 | definition list_split ≝ |
---|
461 | λA: Type[0]. |
---|
462 | λn: nat. |
---|
463 | λl: list A. |
---|
464 | 〈take A n l, drop A n l〉. |
---|
465 | |
---|
466 | let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B) |
---|
467 | (l: list A) on l: list B ≝ |
---|
468 | match l with |
---|
469 | [ nil ⇒ nil ? |
---|
470 | | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl) |
---|
471 | ]. |
---|
472 | |
---|
473 | definition mapi ≝ |
---|
474 | λA, B: Type[0]. |
---|
475 | λf: nat → A → B. |
---|
476 | λl: list A. |
---|
477 | mapi_internal A B 0 f l. |
---|
478 | |
---|
479 | let rec zip_pottier |
---|
480 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
---|
481 | on left ≝ |
---|
482 | match left with |
---|
483 | [ nil ⇒ [ ] |
---|
484 | | cons hd tl ⇒ |
---|
485 | match right with |
---|
486 | [ nil ⇒ [ ] |
---|
487 | | cons hd' tl' ⇒ 〈hd, hd'〉 :: zip_pottier A B tl tl' |
---|
488 | ] |
---|
489 | ]. |
---|
490 | |
---|
491 | let rec zip_safe |
---|
492 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) (prf: |left| = |right|) |
---|
493 | on left ≝ |
---|
494 | match left return λx. |x| = |right| → list (A × B) with |
---|
495 | [ nil ⇒ λnil_prf. |
---|
496 | match right return λx. |[ ]| = |x| → list (A × B) with |
---|
497 | [ nil ⇒ λnil_nil_prf. [ ] |
---|
498 | | cons hd tl ⇒ λnil_cons_absrd. ? |
---|
499 | ] nil_prf |
---|
500 | | cons hd tl ⇒ λcons_prf. |
---|
501 | match right return λx. |hd::tl| = |x| → list (A × B) with |
---|
502 | [ nil ⇒ λcons_nil_absrd. ? |
---|
503 | | cons hd' tl' ⇒ λcons_cons_prf. 〈hd, hd'〉 :: zip_safe A B tl tl' ? |
---|
504 | ] cons_prf |
---|
505 | ] prf. |
---|
506 | [ 1: normalize in nil_cons_absrd; |
---|
507 | destruct(nil_cons_absrd) |
---|
508 | | 2: normalize in cons_nil_absrd; |
---|
509 | destruct(cons_nil_absrd) |
---|
510 | | 3: normalize in cons_cons_prf; |
---|
511 | @injective_S |
---|
512 | assumption |
---|
513 | ] |
---|
514 | qed. |
---|
515 | |
---|
516 | let rec zip (A: Type[0]) (B: Type[0]) (l: list A) (r: list B) on l: option (list (A × B)) ≝ |
---|
517 | match l with |
---|
518 | [ nil ⇒ Some ? (nil (A × B)) |
---|
519 | | cons hd tl ⇒ |
---|
520 | match r with |
---|
521 | [ nil ⇒ None ? |
---|
522 | | cons hd' tl' ⇒ |
---|
523 | match zip ? ? tl tl' with |
---|
524 | [ None ⇒ None ? |
---|
525 | | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail) |
---|
526 | ] |
---|
527 | ] |
---|
528 | ]. |
---|
529 | |
---|
530 | let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝ |
---|
531 | match l with |
---|
532 | [ nil ⇒ a |
---|
533 | | cons hd tl ⇒ foldl A B f (f a hd) tl |
---|
534 | ]. |
---|
535 | |
---|
536 | lemma foldl_step: |
---|
537 | ∀A:Type[0]. |
---|
538 | ∀B: Type[0]. |
---|
539 | ∀H: A → B → A. |
---|
540 | ∀acc: A. |
---|
541 | ∀pre: list B. |
---|
542 | ∀hd:B. |
---|
543 | foldl A B H acc (pre@[hd]) = (H (foldl A B H acc pre) hd). |
---|
544 | #A #B #H #acc #pre generalize in match acc; -acc; elim pre |
---|
545 | [ normalize; // |
---|
546 | | #hd #tl #IH #acc #X normalize; @IH ] |
---|
547 | qed. |
---|
548 | |
---|
549 | lemma foldl_append: |
---|
550 | ∀A:Type[0]. |
---|
551 | ∀B: Type[0]. |
---|
552 | ∀H: A → B → A. |
---|
553 | ∀acc: A. |
---|
554 | ∀suff,pre: list B. |
---|
555 | foldl A B H acc (pre@suff) = (foldl A B H (foldl A B H acc pre) suff). |
---|
556 | #A #B #H #acc #suff elim suff |
---|
557 | [ #pre >append_nil % |
---|
558 | | #hd #tl #IH #pre whd in ⊢ (???%); <(foldl_step … H ??) applyS (IH (pre@[hd])) ] |
---|
559 | qed. |
---|
560 | |
---|
561 | definition flatten ≝ |
---|
562 | λA: Type[0]. |
---|
563 | λl: list (list A). |
---|
564 | foldr ? ? (append ?) [ ] l. |
---|
565 | |
---|
566 | (* redirecting to library reverse *) |
---|
567 | definition rev ≝ reverse. |
---|
568 | |
---|
569 | lemma append_length: |
---|
570 | ∀A: Type[0]. |
---|
571 | ∀l, r: list A. |
---|
572 | |(l @ r)| = |l| + |r|. |
---|
573 | #A #L #R |
---|
574 | elim L |
---|
575 | [ % |
---|
576 | | #HD #TL #IH |
---|
577 | normalize >IH % |
---|
578 | ] |
---|
579 | qed. |
---|
580 | |
---|
581 | lemma append_nil: |
---|
582 | ∀A: Type[0]. |
---|
583 | ∀l: list A. |
---|
584 | l @ [ ] = l. |
---|
585 | #A #L |
---|
586 | elim L // |
---|
587 | qed. |
---|
588 | |
---|
589 | lemma rev_append: |
---|
590 | ∀A: Type[0]. |
---|
591 | ∀l, r: list A. |
---|
592 | rev A (l @ r) = rev A r @ rev A l. |
---|
593 | #A #L #R |
---|
594 | elim L |
---|
595 | [ normalize >append_nil % |
---|
596 | | #HD #TL normalize #IH |
---|
597 | >rev_append_def |
---|
598 | >rev_append_def |
---|
599 | >rev_append_def |
---|
600 | >append_nil |
---|
601 | normalize |
---|
602 | >IH |
---|
603 | @associative_append |
---|
604 | ] |
---|
605 | qed. |
---|
606 | |
---|
607 | lemma rev_length: |
---|
608 | ∀A: Type[0]. |
---|
609 | ∀l: list A. |
---|
610 | |rev A l| = |l|. |
---|
611 | #A #L |
---|
612 | elim L |
---|
613 | [ % |
---|
614 | | #HD #TL normalize #IH |
---|
615 | >rev_append_def |
---|
616 | >(append_length A (rev A TL) [HD]) |
---|
617 | normalize /2 by / |
---|
618 | ] |
---|
619 | qed. |
---|
620 | |
---|
621 | lemma nth_append_first: |
---|
622 | ∀A:Type[0]. |
---|
623 | ∀n:nat.∀l1,l2:list A.∀d:A. |
---|
624 | n < |l1| → nth n A (l1@l2) d = nth n A l1 d. |
---|
625 | #A #n #l1 #l2 #d |
---|
626 | generalize in match n; -n; elim l1 |
---|
627 | [ normalize #k #Hk @⊥ @(absurd … Hk) @not_le_Sn_O |
---|
628 | | #h #t #Hind #k normalize |
---|
629 | cases k -k |
---|
630 | [ #Hk normalize @refl |
---|
631 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
632 | ] |
---|
633 | ] |
---|
634 | qed. |
---|
635 | |
---|
636 | lemma nth_append_second: |
---|
637 | ∀A: Type[0].∀n.∀l1,l2:list A.∀d.n ≥ length A l1 -> |
---|
638 | nth n A (l1@l2) d = nth (n - length A l1) A l2 d. |
---|
639 | #A #n #l1 #l2 #d |
---|
640 | generalize in match n; -n; elim l1 |
---|
641 | [ normalize #k #Hk <(minus_n_O) @refl |
---|
642 | | #h #t #Hind #k normalize |
---|
643 | cases k -k; |
---|
644 | [ #Hk @⊥ @(absurd (S (|t|) ≤ 0)) [ @Hk | @not_le_Sn_O ] |
---|
645 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
646 | ] |
---|
647 | ] |
---|
648 | qed. |
---|
649 | |
---|
650 | |
---|
651 | let rec fold_left_i_aux (A: Type[0]) (B: Type[0]) |
---|
652 | (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝ |
---|
653 | match l with |
---|
654 | [ nil ⇒ x |
---|
655 | | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl |
---|
656 | ]. |
---|
657 | |
---|
658 | definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O. |
---|
659 | |
---|
660 | notation "hvbox(t⌈o ↦ h⌉)" |
---|
661 | with precedence 45 |
---|
662 | for @{ match (? : $o=$h) with [ refl ⇒ $t ] }. |
---|
663 | |
---|
664 | definition function_apply ≝ |
---|
665 | λA, B: Type[0]. |
---|
666 | λf: A → B. |
---|
667 | λa: A. |
---|
668 | f a. |
---|
669 | |
---|
670 | notation "f break $ x" |
---|
671 | left associative with precedence 99 |
---|
672 | for @{ 'function_apply $f $x }. |
---|
673 | |
---|
674 | interpretation "Function application" 'function_apply f x = (function_apply ? ? f x). |
---|
675 | |
---|
676 | let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝ |
---|
677 | match n with |
---|
678 | [ O ⇒ a |
---|
679 | | S o ⇒ f (iterate A f a o) |
---|
680 | ]. |
---|
681 | |
---|
682 | let rec division_aux (m: nat) (n : nat) (p: nat) ≝ |
---|
683 | match ltb n (S p) with |
---|
684 | [ true ⇒ O |
---|
685 | | false ⇒ |
---|
686 | match m with |
---|
687 | [ O ⇒ O |
---|
688 | | (S q) ⇒ S (division_aux q (n - (S p)) p) |
---|
689 | ] |
---|
690 | ]. |
---|
691 | |
---|
692 | definition division ≝ |
---|
693 | λm, n: nat. |
---|
694 | match n with |
---|
695 | [ O ⇒ S m |
---|
696 | | S o ⇒ division_aux m m o |
---|
697 | ]. |
---|
698 | |
---|
699 | notation "hvbox(n break ÷ m)" |
---|
700 | right associative with precedence 47 |
---|
701 | for @{ 'division $n $m }. |
---|
702 | |
---|
703 | interpretation "Nat division" 'division n m = (division n m). |
---|
704 | |
---|
705 | let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝ |
---|
706 | match leb n p with |
---|
707 | [ true ⇒ n |
---|
708 | | false ⇒ |
---|
709 | match m with |
---|
710 | [ O ⇒ n |
---|
711 | | S o ⇒ modulus_aux o (n - (S p)) p |
---|
712 | ] |
---|
713 | ]. |
---|
714 | |
---|
715 | definition modulus ≝ |
---|
716 | λm, n: nat. |
---|
717 | match n with |
---|
718 | [ O ⇒ m |
---|
719 | | S o ⇒ modulus_aux m m o |
---|
720 | ]. |
---|
721 | |
---|
722 | notation "hvbox(n break 'mod' m)" |
---|
723 | right associative with precedence 47 |
---|
724 | for @{ 'modulus $n $m }. |
---|
725 | |
---|
726 | interpretation "Nat modulus" 'modulus m n = (modulus m n). |
---|
727 | |
---|
728 | definition divide_with_remainder ≝ |
---|
729 | λm, n: nat. |
---|
730 | mk_Prod … (m ÷ n) (modulus m n). |
---|
731 | |
---|
732 | let rec exponential (m: nat) (n: nat) on n ≝ |
---|
733 | match n with |
---|
734 | [ O ⇒ S O |
---|
735 | | S o ⇒ m * exponential m o |
---|
736 | ]. |
---|
737 | |
---|
738 | interpretation "Nat exponential" 'exp n m = (exponential n m). |
---|
739 | |
---|
740 | notation "hvbox(a break ⊎ b)" |
---|
741 | left associative with precedence 55 |
---|
742 | for @{ 'disjoint_union $a $b }. |
---|
743 | interpretation "sum" 'disjoint_union A B = (Sum A B). |
---|
744 | |
---|
745 | theorem less_than_or_equal_monotone: |
---|
746 | ∀m, n: nat. |
---|
747 | m ≤ n → (S m) ≤ (S n). |
---|
748 | #m #n #H |
---|
749 | elim H |
---|
750 | /2 by le_n, le_S/ |
---|
751 | qed. |
---|
752 | |
---|
753 | theorem less_than_or_equal_b_complete: |
---|
754 | ∀m, n: nat. |
---|
755 | leb m n = false → ¬(m ≤ n). |
---|
756 | #m; |
---|
757 | elim m; |
---|
758 | normalize |
---|
759 | [ #n #H |
---|
760 | destruct |
---|
761 | | #y #H1 #z |
---|
762 | cases z |
---|
763 | normalize |
---|
764 | [ #H |
---|
765 | /2 by / |
---|
766 | | /3 by not_le_to_not_le_S_S/ |
---|
767 | ] |
---|
768 | ] |
---|
769 | qed. |
---|
770 | |
---|
771 | theorem less_than_or_equal_b_correct: |
---|
772 | ∀m, n: nat. |
---|
773 | leb m n = true → m ≤ n. |
---|
774 | #m |
---|
775 | elim m |
---|
776 | // |
---|
777 | #y #H1 #z |
---|
778 | cases z |
---|
779 | normalize |
---|
780 | [ #H |
---|
781 | destruct |
---|
782 | | #n #H lapply (H1 … H) /2 by le_S_S/ |
---|
783 | ] |
---|
784 | qed. |
---|
785 | |
---|
786 | definition less_than_or_equal_b_elim: |
---|
787 | ∀m, n: nat. |
---|
788 | ∀P: bool → Type[0]. |
---|
789 | (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n). |
---|
790 | #m #n #P #H1 #H2; |
---|
791 | lapply (less_than_or_equal_b_correct m n) |
---|
792 | lapply (less_than_or_equal_b_complete m n) |
---|
793 | cases (leb m n) |
---|
794 | /3 by / |
---|
795 | qed. |
---|
796 | |
---|
797 | lemma inclusive_disjunction_true: |
---|
798 | ∀b, c: bool. |
---|
799 | (orb b c) = true → b = true ∨ c = true. |
---|
800 | # b |
---|
801 | # c |
---|
802 | elim b |
---|
803 | [ normalize |
---|
804 | # H |
---|
805 | @ or_introl |
---|
806 | % |
---|
807 | | normalize |
---|
808 | /3 by trans_eq, orb_true_l/ |
---|
809 | ] |
---|
810 | qed. |
---|
811 | |
---|
812 | lemma conjunction_true: |
---|
813 | ∀b, c: bool. |
---|
814 | andb b c = true → b = true ∧ c = true. |
---|
815 | # b |
---|
816 | # c |
---|
817 | elim b |
---|
818 | normalize |
---|
819 | [ /2 by conj/ |
---|
820 | | # K |
---|
821 | destruct |
---|
822 | ] |
---|
823 | qed. |
---|
824 | |
---|
825 | lemma eq_true_false: false=true → False. |
---|
826 | # K |
---|
827 | destruct |
---|
828 | qed. |
---|
829 | |
---|
830 | lemma inclusive_disjunction_b_true: ∀b. orb b true = true. |
---|
831 | # b |
---|
832 | cases b |
---|
833 | % |
---|
834 | qed. |
---|
835 | |
---|
836 | definition bool_to_Prop ≝ |
---|
837 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
---|
838 | |
---|
839 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
---|
840 | |
---|
841 | lemma bool_as_Prop_to_eq : ∀b : bool. b → b = true. |
---|
842 | **% |
---|
843 | qed. |
---|
844 | |
---|
845 | (* with this you can use prf : b with b : bool with rewriting |
---|
846 | >prf rewrites b as true *) |
---|
847 | coercion bool_to_Prop_to_eq : ∀b : bool.∀prf : b.b = true |
---|
848 | ≝ bool_as_Prop_to_eq on _prf : bool_to_Prop ? to (? = true). |
---|
849 | |
---|
850 | lemma andb_Prop : ∀b,d : bool.b → d → b∧d. |
---|
851 | #b #d #btrue #dtrue >btrue >dtrue % |
---|
852 | qed. |
---|
853 | |
---|
854 | lemma andb_Prop_true : ∀b,d : bool. (b∧d) → And (bool_to_Prop b) (bool_to_Prop d). |
---|
855 | #b #d #bdtrue elim (andb_true … bdtrue) #btrue #dtrue >btrue >dtrue % % |
---|
856 | qed. |
---|
857 | |
---|
858 | lemma orb_Prop_l : ∀b,d : bool.b → b∨d. |
---|
859 | #b #d #btrue >btrue % |
---|
860 | qed. |
---|
861 | |
---|
862 | lemma orb_Prop_r : ∀b,d : bool.d → b∨d. |
---|
863 | #b #d #dtrue >dtrue elim b % |
---|
864 | qed. |
---|
865 | |
---|
866 | lemma orb_Prop_true : ∀b,d : bool. (b∨d) → Or (bool_to_Prop b) (bool_to_Prop d). |
---|
867 | #b #d #bdtrue elim (orb_true_l … bdtrue) #xtrue >xtrue [%1 | %2] % |
---|
868 | qed. |
---|
869 | |
---|
870 | lemma notb_Prop : ∀b : bool. Not (bool_to_Prop b) → notb b. |
---|
871 | * * #H [@H % | %] |
---|
872 | qed. |
---|
873 | |
---|
874 | lemma eq_false_to_notb: ∀b. b = false → ¬ b. |
---|
875 | *; /2 by eq_true_false, I/ |
---|
876 | qed. |
---|
877 | |
---|
878 | lemma not_b_to_eq_false : ∀b : bool. Not (bool_to_Prop b) → b = false. |
---|
879 | ** #H [elim (H ?) % | %] |
---|
880 | qed. |
---|
881 | |
---|
882 | (* with this you can use prf : ¬b with b : bool with rewriting |
---|
883 | >prf rewrites b as false *) |
---|
884 | coercion not_bool_to_Prop_to_eq : ∀b : bool.∀prf : Not (bool_to_Prop b).b = false |
---|
885 | ≝ not_b_to_eq_false on _prf : Not (bool_to_Prop ?) to (? = false). |
---|
886 | |
---|
887 | |
---|
888 | lemma true_or_false_Prop : ∀b : bool.Or (bool_to_Prop b) (¬(bool_to_Prop b)). |
---|
889 | * [%1 % | %2 % *] |
---|
890 | qed. |
---|
891 | |
---|
892 | lemma eq_true_to_b : ∀b. b = true → b. |
---|
893 | #b #btrue >btrue % |
---|
894 | qed. |
---|
895 | |
---|
896 | definition if_then_else_safe : ∀A : Type[0].∀b : bool.(b → A) → (¬(bool_to_Prop b) → A) → A ≝ |
---|
897 | λA,b,f,g. |
---|
898 | match b return λx.match x with [true ⇒ bool_to_Prop b | false ⇒ ¬bool_to_Prop b] → A with |
---|
899 | [ true ⇒ f |
---|
900 | | false ⇒ g |
---|
901 | ] ?. elim b % * |
---|
902 | qed. |
---|
903 | |
---|
904 | notation > "'If' b 'then' 'with' ident prf1 'do' f 'else' 'with' ident prf2 'do' g" with precedence 46 for |
---|
905 | @{'if_then_else_safe $b (λ${ident prf1}.$f) (λ${ident prf2}.$g)}. |
---|
906 | notation > "'If' b 'then' 'with' ident prf1 : ty1 'do' f 'else' 'with' ident prf2 : ty2 'do' g" with precedence 46 for |
---|
907 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
908 | notation > "'If' b 'then' f 'else' 'with' ident prf2 'do' g" with precedence 46 for |
---|
909 | @{'if_then_else_safe $b (λ_.$f) (λ${ident prf2}.$g)}. |
---|
910 | notation > "'If' b 'then' f 'else' 'with' ident prf2 : ty2 'do' g" with precedence 46 for |
---|
911 | @{'if_then_else_safe $b (λ_.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
912 | notation > "'If' b 'then' 'with' ident prf1 'do' f 'else' g" with precedence 46 for |
---|
913 | @{'if_then_else_safe $b (λ${ident prf1}.$f) (λ_.$g)}. |
---|
914 | notation > "'If' b 'then' 'with' ident prf1 : ty1 'do' f 'else' g" with precedence 46 for |
---|
915 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ_.$g)}. |
---|
916 | |
---|
917 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break 'with' \nbsp ident prf1 : ty1 \nbsp 'do' \nbsp break f \nbsp break 'else' \nbsp break 'with' \nbsp ident prf2 : ty2 \nbsp 'do' \nbsp break g)" with precedence 46 for |
---|
918 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
919 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break f \nbsp break 'else' \nbsp break 'with' \nbsp ident prf2 : ty2 \nbsp 'do' \nbsp break g)" with precedence 46 for |
---|
920 | @{'if_then_else_safe $b (λ_:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
921 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break 'with' \nbsp ident prf1 : ty1 \nbsp 'do' \nbsp break f \nbsp break 'else' \nbsp break g)" with precedence 46 for |
---|
922 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ_:$ty2.$g)}. |
---|
923 | |
---|
924 | interpretation "dependent if then else" 'if_then_else_safe b f g = (if_then_else_safe ? b f g). |
---|
925 | |
---|
926 | lemma length_append: |
---|
927 | ∀A.∀l1,l2:list A. |
---|
928 | |l1 @ l2| = |l1| + |l2|. |
---|
929 | #A #l1 elim l1 |
---|
930 | [ // |
---|
931 | | #hd #tl #IH #l2 normalize <IH //] |
---|
932 | qed. |
---|