2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/types.ma".
13 include "arithmetics/nat.ma".
14 include "basics/core_notation/card_1.ma".
16 inductive list (A:Type[0]) : Type[0] :=
18 | cons: A -> list A -> list A.
20 notation "hvbox(hd break :: tl)"
21 right associative with precedence 47
24 notation "[ list0 term 19 x sep ; ]"
25 non associative with precedence 90
26 for ${fold right @'nil rec acc @{'cons $x $acc}}.
28 notation "hvbox(l1 break @ l2)"
29 right associative with precedence 47
30 for @{'append $l1 $l2 }.
32 interpretation "nil" 'nil = (nil ?).
33 interpretation "cons" 'cons hd tl = (cons ? hd tl).
35 definition is_nil: ∀A:Type[0].list A → Prop ≝
36 λA.λl.match l with [ nil ⇒ True | cons hd tl ⇒ False ].
39 ∀A:Type[0].∀l:list A.∀a:A. a::l ≠ [].
40 #A #l #a @nmk #Heq (change with (is_nil ? (a::l))) >Heq //
44 let rec id_list A (l: list A) on l :=
47 | (cons hd tl) => hd :: id_list A tl ]. *)
49 let rec append A (l1: list A) l2 on l1 ≝
52 | cons hd tl ⇒ hd :: append A tl l2 ].
54 definition hd ≝ λA.λl: list A.λd:A.
55 match l with [ nil ⇒ d | cons a _ ⇒ a].
57 definition tail ≝ λA.λl: list A.
58 match l with [ nil ⇒ [] | cons hd tl ⇒ tl].
60 definition option_hd ≝
61 λA.λl:list A. match l with
63 | cons a _ ⇒ Some ? a ].
65 interpretation "append" 'append l1 l2 = (append ? l1 l2).
67 theorem append_nil: ∀A.∀l:list A.l @ [] = l.
68 #A #l (elim l) normalize // qed.
70 theorem associative_append:
71 ∀A.associative (list A) (append A).
72 #A #l1 #l2 #l3 (elim l1) normalize // qed.
74 theorem append_cons:∀A.∀a:A.∀l,l1.l@(a::l1)=(l@[a])@l1.
75 #A #a #l #l1 >associative_append // qed.
77 theorem nil_append_elim: ∀A.∀l1,l2: list A.∀P:?→?→Prop.
78 l1@l2=[] → P (nil A) (nil A) → P l1 l2.
79 #A #l1 #l2 #P (cases l1) normalize //
83 theorem nil_to_nil: ∀A.∀l1,l2:list A.
84 l1@l2 = [] → l1 = [] ∧ l2 = [].
85 #A #l1 #l2 #isnil @(nil_append_elim A l1 l2) /2/
88 lemma cons_injective_l : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → a1 = a2.
89 #A #a1 #a2 #l1 #l2 #Heq destruct //
92 lemma cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
93 #A #a1 #a2 #l1 #l2 #Heq destruct //
98 definition option_cons ≝ λsig.λc:option sig.λl.
99 match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
101 lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l).
105 (* comparing lists *)
107 lemma compare_append : ∀A,l1,l2,l3,l4. l1@l2 = l3@l4 →
108 ∃l:list A.(l1 = l3@l ∧ l4=l@l2) ∨ (l3 = l1@l ∧ l2=l@l4).
110 [#l2 #l3 #l4 #Heq %{l3} %2 % // @Heq
111 |#a1 #tl1 #Hind #l2 #l3 cases l3
112 [#l4 #Heq %{(a1::tl1)} %1 % // @sym_eq @Heq
113 |#a3 #tl3 #l4 normalize in ⊢ (%→?); #Heq cases (Hind l2 tl3 l4 ?)
114 [#l * * #Heq1 #Heq2 %{l}
115 [%1 % // >Heq1 >(cons_injective_l ????? Heq) //
116 |%2 % // >Heq1 >(cons_injective_l ????? Heq) //
118 |@(cons_injective_r ????? Heq)
124 (**************************** iterators ******************************)
126 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
127 match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
129 lemma map_append : ∀A,B,f,l1,l2.
130 (map A B f l1) @ (map A B f l2) = map A B f (l1@l2).
133 | #h #t #IH #l2 normalize //
136 let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:list A) on l :B ≝
137 match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
141 foldr T (list T) (λx,l0.if p x then x::l0 else l0) (nil T).
143 (* compose f [a1;...;an] [b1;...;bm] =
144 [f a1 b1; ... ;f an b1; ... ;f a1 bm; f an bm] *)
146 definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
147 foldr ?? (λi,acc.(map ?? (f i) l2)@acc) [ ] l1.
149 lemma filter_true : ∀A,l,a,p. p a = true →
150 filter A p (a::l) = a :: filter A p l.
151 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
153 lemma filter_false : ∀A,l,a,p. p a = false →
154 filter A p (a::l) = filter A p l.
155 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
157 theorem eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
158 #A #B #f #g #l #eqfg (elim l) normalize // qed.
160 (**************************** reverse *****************************)
161 let rec rev_append S (l1,l2:list S) on l1 ≝
164 | cons a tl ⇒ rev_append S tl (a::l2)
168 definition reverse ≝λS.λl.rev_append S l [].
170 lemma reverse_single : ∀S,a. reverse S [a] = [a].
173 lemma rev_append_def : ∀S,l1,l2.
174 rev_append S l1 l2 = (reverse S l1) @ l2 .
175 #S #l1 elim l1 normalize //
178 lemma reverse_cons : ∀S,a,l. reverse S (a::l) = (reverse S l)@[a].
179 #S #a #l whd in ⊢ (??%?); //
182 lemma reverse_append: ∀S,l1,l2.
183 reverse S (l1 @ l2) = (reverse S l2)@(reverse S l1).
184 #S #l1 elim l1 [normalize // | #a #tl #Hind #l2 >reverse_cons
185 >reverse_cons // qed.
187 lemma reverse_reverse : ∀S,l. reverse S (reverse S l) = l.
188 #S #l elim l // #a #tl #Hind >reverse_cons >reverse_append
191 (* an elimination principle for lists working on the tail;
192 useful for strings *)
193 lemma list_elim_left: ∀S.∀P:list S → Prop. P (nil S) →
194 (∀a.∀tl.P tl → P (tl@[a])) → ∀l. P l.
195 #S #P #Pnil #Pstep #l <(reverse_reverse … l)
196 generalize in match (reverse S l); #l elim l //
197 #a #tl #H >reverse_cons @Pstep //
200 (**************************** length ******************************)
202 let rec length (A:Type[0]) (l:list A) on l ≝
205 | cons a tl ⇒ S (length A tl)].
207 interpretation "list length" 'card l = (length ? l).
209 lemma length_tail: ∀A,l. length ? (tail A l) = pred (length ? l).
213 lemma length_tail1 : ∀A,l.0 < |l| → |tail A l| < |l|.
217 lemma length_append: ∀A.∀l1,l2:list A.
219 #A #l1 elim l1 // normalize /2/
222 lemma length_map: ∀A,B,l.∀f:A→B. length ? (map ?? f l) = length ? l.
223 #A #B #l #f elim l // #a #tl #Hind normalize //
226 lemma length_reverse: ∀A.∀l:list A.
228 #A #l elim l // #a #l0 #IH >reverse_cons >length_append normalize //
231 lemma lenght_to_nil: ∀A.∀l:list A.
233 #A * // #a #tl normalize #H destruct
236 lemma lists_length_split :
237 ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
239 [ #l2 %{[ ]} %{l2} % % %
241 [ %{[ ]} %{(hd1::tl1)} %2 % %
242 | #hd2 #tl2 cases (IH tl2) #x * #y *
243 [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % //
244 | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ]
249 (****************** traversing two lists in parallel *****************)
251 ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
252 length ? l1 = length ? l2 →
254 (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) →
256 #T1 #T2 #l1 #l2 #P #Hl #Pnil #Pcons
257 generalize in match Hl; generalize in match l2;
259 [#l2 cases l2 // normalize #t2 #tl2 #H destruct
260 |#t1 #tl1 #IH #l2 cases l2
261 [normalize #H destruct
262 |#t2 #tl2 #H @Pcons @IH normalize in H; destruct // ]
267 ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:Prop.
268 length ? l1 = length ? l2 →
269 (l1 = [] → l2 = [] → P) →
270 (∀hd1,hd2,tl1,tl2.l1 = hd1::tl1 → l2 = hd2::tl2 → P) → P.
271 #T1 #T2 #l1 #l2 #P #Hl @(list_ind2 … Hl)
272 [ #Pnil #Pcons @Pnil //
273 | #tl1 #tl2 #hd1 #hd2 #IH1 #IH2 #Hp @Hp // ]
276 (*********************** properties of append ***********************)
277 lemma append_l1_injective :
278 ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l1 = l2.
279 #a #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen) //
280 #tl1 #tl2 #hd1 #hd2 #IH normalize #Heq destruct @eq_f /2/
283 lemma append_l2_injective :
284 ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l3 = l4.
285 #a #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen) normalize //
286 #tl1 #tl2 #hd1 #hd2 #IH normalize #Heq destruct /2/
289 lemma append_l1_injective_r :
290 ∀A.∀l1,l2,l3,l4:list A. |l3| = |l4| → l1@l3 = l2@l4 → l1 = l2.
291 #a #l1 #l2 #l3 #l4 #Hlen #Heq lapply (eq_f … (reverse ?) … Heq)
292 >reverse_append >reverse_append #Heq1
293 lapply (append_l2_injective … Heq1) [ // ] #Heq2
294 lapply (eq_f … (reverse ?) … Heq2) //
297 lemma append_l2_injective_r :
298 ∀A.∀l1,l2,l3,l4:list A. |l3| = |l4| → l1@l3 = l2@l4 → l3 = l4.
299 #a #l1 #l2 #l3 #l4 #Hlen #Heq lapply (eq_f … (reverse ?) … Heq)
300 >reverse_append >reverse_append #Heq1
301 lapply (append_l1_injective … Heq1) [ // ] #Heq2
302 lapply (eq_f … (reverse ?) … Heq2) //
305 lemma length_rev_append: ∀A.∀l,acc:list A.
306 |rev_append ? l acc| = |l|+|acc|.
307 #A #l elim l // #a #tl #Hind normalize
308 #acc >Hind normalize //
311 (****************************** mem ********************************)
312 let rec mem A (a:A) (l:list A) on l ≝
315 | cons hd tl ⇒ a=hd ∨ mem A a tl
318 lemma mem_append: ∀A,a,l1,l2.mem A a (l1@l2) →
319 mem ? a l1 ∨ mem ? a l2.
324 |#Hmema cases (Hind ? Hmema) -Hmema #Hmema [%1 %2 //|%2 //]
329 lemma mem_append_l1: ∀A,a,l1,l2.mem A a l1 → mem A a (l1@l2).
330 #A #a #l1 #l2 elim l1
331 [whd in ⊢ (%→?); @False_ind
332 |#b #tl #Hind * [#eqab %1 @eqab |#Hmema %2 @Hind //]
336 lemma mem_append_l2: ∀A,a,l1,l2.mem A a l2 → mem A a (l1@l2).
337 #A #a #l1 #l2 elim l1 [//|#b #tl #Hind #Hmema %2 @Hind //]
340 lemma mem_single: ∀A,a,b. mem A a [b] → a=b.
341 #A #a #b * // @False_ind
344 lemma mem_map: ∀A,B.∀f:A→B.∀l,b.
345 mem ? b (map … f l) → ∃a. mem ? a l ∧ f a = b.
347 [#b normalize @False_ind
348 |#a #tl #Hind #b normalize *
349 [#eqb @(ex_intro … a) /3/
350 |#memb cases (Hind … memb) #a * #mema #eqb
356 lemma mem_map_forward: ∀A,B.∀f:A→B.∀a,l.
357 mem A a l → mem B (f a) (map ?? f l).
358 #A #B #f #a #l elim l
359 [normalize @False_ind
361 [#eqab <eqab normalize %1 % |#memtl normalize %2 @Hind @memtl]
365 (****************************** mem filter ***************************)
366 lemma mem_filter: ∀S,f,a,l.
367 mem S a (filter S f l) → mem S a l.
368 #S #f #a #l elim l [normalize //]
369 #b #tl #Hind normalize (cases (f b)) normalize
370 [* [#eqab %1 @eqab | #H %2 @Hind @H]
374 lemma mem_filter_true: ∀S,f,a,l.
375 mem S a (filter S f l) → f a = true.
376 #S #f #a #l elim l [normalize @False_ind ]
377 #b #tl #Hind cases (true_or_false (f b)) #H
378 normalize >H normalize [2:@Hind]
382 lemma mem_filter_l: ∀S,f,x,l. (f x = true) → mem S x l →
383 mem S x (filter ? f l).
384 #S #f #x #l #fx elim l [@False_ind]
386 [#eqxb <eqxb >(filter_true ???? fx) %1 %
387 |#Htl cases (true_or_false (f b)) #fb
388 [>(filter_true ???? fb) %2 @Hind @Htl
389 |>(filter_false ???? fb) @Hind @Htl
394 lemma filter_case: ∀A,p,l,x. mem ? x l →
395 mem ? x (filter A p l) ∨ mem ? x (filter A (λx.¬ p x) l).
399 [#eqxa >eqxa cases (true_or_false (p a)) #Hcase
400 [%1 >(filter_true A tl a p Hcase) %1 %
401 |%2 >(filter_true A tl a ??) [%1 % | >Hcase %]
403 |#memx cases (Hind … memx) -memx #memx
404 [%1 cases (true_or_false (p a)) #Hpa
405 [>(filter_true A tl a p Hpa) %2 @memx
406 |>(filter_false A tl a p Hpa) @memx
408 |cases (true_or_false (p a)) #Hcase
409 [%2 >(filter_false A tl a) [@memx |>Hcase %]
410 |%2 >(filter_true A tl a) [%2 @memx|>Hcase %]
417 lemma filter_length2: ∀A,p,l. |filter A p l|+|filter A (λx.¬ p x) l| = |l|.
419 #a #tl #Hind cases (true_or_false (p a)) #Hcase
420 [>(filter_true A tl a p Hcase) >(filter_false A tl a ??)
421 [@(eq_f ?? S) @Hind | >Hcase %]
422 |>(filter_false A tl a p Hcase) >(filter_true A tl a ??)
423 [<plus_n_Sm @(eq_f ?? S) @Hind | >Hcase %]
427 (***************************** unique *******************************)
428 let rec unique A (l:list A) on l ≝
431 |cons a tl ⇒ ¬ mem A a tl ∧ unique A tl].
433 lemma unique_filter : ∀S,l,f.
434 unique S l → unique S (filter S f l).
437 #memba #uniquetl cases (true_or_false … (f a)) #Hfa
438 [>(filter_true ???? Hfa) %
439 [@(not_to_not … memba) @mem_filter |/2/ ]
444 lemma filter_eqb : ∀m,l. unique ? l →
445 (mem ? m l ∧ filter ? (eqb m) l = [m])∨(¬mem ? m l ∧ filter ? (eqb m) l = []).
447 [#_ %2 % [% @False_ind | //]
448 |#a #tl #Hind * #Hmema #Hunique
450 [* #Hmemm #Hind %1 % [%2 //]
451 >filter_false // @not_eq_to_eqb_false % #eqma @(absurd ? Hmemm) //
452 |* #Hmemm #Hind cases (decidable_eq_nat m a) #eqma
453 [%1 <eqma % [%1 //] >filter_true [2: @eq_to_eqb_true //] >Hind //
455 [@(not_to_not … Hmemm) * // #H @False_ind @(absurd … H) //
456 |>filter_false // @not_eq_to_eqb_false @eqma
463 lemma length_filter_eqb: ∀m,l. unique ? l →
464 |filter ? (eqb m) l| ≤ 1.
465 #m #l #Huni cases (filter_eqb m l Huni) * #_ #H >H //
468 (***************************** split *******************************)
469 let rec split_rev A (l:list A) acc n on n ≝
474 |cons a tl ⇒ split_rev A tl (a::acc) m
478 definition split ≝ λA,l,n.
479 let 〈l1,l2〉 ≝ split_rev A l [] n in 〈reverse ? l1,l2〉.
481 lemma split_rev_len: ∀A,n,l,acc. n ≤ |l| →
482 |\fst (split_rev A l acc n)| = n+|acc|.
483 #A #n elim n // #m #Hind *
484 [normalize #acc #Hfalse @False_ind /2/
485 |#a #tl #acc #Hlen normalize >Hind
486 [normalize // |@le_S_S_to_le //]
490 lemma split_len: ∀A,n,l. n ≤ |l| →
491 |\fst (split A l n)| = n.
492 #A #n #l #Hlen normalize >(eq_pair_fst_snd ?? (split_rev …))
493 normalize >length_reverse >(split_rev_len … [ ] Hlen) normalize //
496 lemma split_rev_eq: ∀A,n,l,acc. n ≤ |l| →
498 reverse ? (\fst (split_rev A l acc n))@(\snd (split_rev A l acc n)).
501 [#acc whd in ⊢ ((??%)→?); #False_ind /2/
502 |#a #tl #acc #Hlen >append_cons <reverse_single <reverse_append
503 @(Hind tl) @le_S_S_to_le @Hlen
507 lemma split_eq: ∀A,n,l. n ≤ |l| →
508 l = (\fst (split A l n))@(\snd (split A l n)).
509 #A #n #l #Hlen change with ((reverse ? [ ])@l) in ⊢ (??%?);
510 >(split_rev_eq … Hlen) normalize
511 >(eq_pair_fst_snd ?? (split_rev A l [] n)) %
514 lemma split_exists: ∀A,n.∀l:list A. n ≤ |l| →
515 ∃l1,l2. l = l1@l2 ∧ |l1| = n.
516 #A #n #l #Hlen @(ex_intro … (\fst (split A l n)))
517 @(ex_intro … (\snd (split A l n))) % /2/
520 (****************************** flatten ******************************)
521 definition flatten ≝ λA.foldr (list A) (list A) (append A) [].
523 lemma flatten_to_mem: ∀A,n,l,l1,l2.∀a:list A. 0 < n →
524 (∀x. mem ? x l → |x| = n) → |a| = n → flatten ? l = l1@a@l2 →
525 (∃q.|l1| = n*q) → mem ? a l.
527 [normalize #l1 #l2 #a #posn #Hlen #Ha #Hnil @False_ind
528 cut (|a|=0) [@sym_eq @le_n_O_to_eq
529 @(transitive_le ? (|nil A|)) // >Hnil >length_append >length_append //] /2/
530 |#hd #tl #Hind #l1 #l2 #a #posn #Hlen #Ha
531 whd in match (flatten ??); #Hflat * #q cases q
533 cut (a = hd) [>(lenght_to_nil… Hl1) in Hflat;
534 whd in ⊢ ((???%)→?); #Hflat @sym_eq @(append_l1_injective … Hflat)
537 |#q1 #Hl1 lapply (split_exists … n l1 ?) //
538 * #l11 * #l12 * #Heql1 #Hlenl11 %2
539 @(Hind l12 l2 … posn ? Ha)
540 [#x #memx @Hlen %2 //
541 |@(append_l2_injective ? hd l11)
543 |>Hflat >Heql1 >associative_append %
545 |@(ex_intro …q1) @(injective_plus_r n)
546 <Hlenl11 in ⊢ (??%?); <length_append <Heql1 >Hl1 //
552 (****************************** nth ********************************)
553 let rec nth n (A:Type[0]) (l:list A) (d:A) ≝
556 |S m ⇒ nth m A (tail A l) d].
558 lemma nth_nil: ∀A,a,i. nth i A ([]) a = a.
559 #A #a #i elim i normalize //
562 (****************************** nth_opt ********************************)
563 let rec nth_opt (A:Type[0]) (n:nat) (l:list A) on l : option A ≝
566 | cons h t ⇒ match n with [ O ⇒ Some ? h | S m ⇒ nth_opt A m t ]
569 (**************************** All *******************************)
571 let rec All (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
574 | cons h t ⇒ P h ∧ All A P t
577 lemma All_mp : ∀A,P,Q. (∀a.P a → Q a) → ∀l. All A P l → All A Q l.
578 #A #P #Q #H #l elim l normalize //
582 lemma All_nth : ∀A,P,n,l.
585 nth_opt A n l = Some A a →
588 [ * [ #_ #a #E whd in E:(??%?); destruct
589 | #hd #tl * #H #_ #a #E whd in E:(??%?); destruct @H
592 [ #_ #a #E whd in E:(??%?); destruct
593 | #hd #tl * #_ whd in ⊢ (? → ∀_.??%? → ?); @IH
597 lemma All_append: ∀A,P,l1,l2. All A P l1 → All A P l2 → All A P (l1@l2).
598 #A #P #l1 elim l1 -l1 //
599 #a #l1 #IHl1 #l2 * /3 width=1/
602 lemma All_inv_append: ∀A,P,l1,l2. All A P (l1@l2) → All A P l1 ∧ All A P l2.
603 #A #P #l1 elim l1 -l1 /2 width=1/
604 #a #l1 #IHl1 #l2 * #Ha #Hl12
605 elim (IHl1 … Hl12) -IHl1 -Hl12 /3 width=1/
608 (**************************** Allr ******************************)
610 let rec Allr (A:Type[0]) (R:relation A) (l:list A) on l : Prop ≝
613 | cons a1 l ⇒ match l with [ nil ⇒ True | cons a2 _ ⇒ R a1 a2 ∧ Allr A R l ]
616 lemma Allr_fwd_append_sn: ∀A,R,l1,l2. Allr A R (l1@l2) → Allr A R l1.
617 #A #R #l1 elim l1 -l1 // #a1 * // #a2 #l1 #IHl1 #l2 * /3 width=2/
620 lemma Allr_fwd_cons: ∀A,R,a,l. Allr A R (a::l) → Allr A R l.
621 #A #R #a * // #a0 #l * //
624 lemma Allr_fwd_append_dx: ∀A,R,l1,l2. Allr A R (l1@l2) → Allr A R l2.
625 #A #R #l1 elim l1 -l1 // #a1 #l1 #IHl1 #l2 #H
626 lapply (Allr_fwd_cons … (l1@l2) H) -H /2 width=1/
629 (**************************** Exists *******************************)
631 let rec Exists (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
634 | cons h t ⇒ (P h) ∨ (Exists A P t)
637 lemma Exists_append : ∀A,P,l1,l2.
638 Exists A P (l1 @ l2) → Exists A P l1 ∨ Exists A P l2.
643 | #H cases (IH l2 H) /3/
647 lemma Exists_append_l : ∀A,P,l1,l2.
648 Exists A P l1 → Exists A P (l1@l2).
649 #A #P #l1 #l2 elim l1
657 lemma Exists_append_r : ∀A,P,l1,l2.
658 Exists A P l2 → Exists A P (l1@l2).
659 #A #P #l1 #l2 elim l1
661 | #h #t #IH #H %2 @IH @H
664 lemma Exists_add : ∀A,P,l1,x,l2. Exists A P (l1@l2) → Exists A P (l1@x::l2).
665 #A #P #l1 #x #l2 elim l1
667 | #h #t #IH normalize * [ #H %1 @H | #H %2 @IH @H ]
670 lemma Exists_mid : ∀A,P,l1,x,l2. P x → Exists A P (l1@x::l2).
671 #A #P #l1 #x #l2 #H elim l1
676 lemma Exists_map : ∀A,B,P,Q,f,l.
679 Exists B Q (map A B f l).
680 #A #B #P #Q #f #l elim l //
681 #h #t #IH * [ #H #F %1 @F @H | #H #F %2 @IH [ @H | @F ] ] qed.
683 lemma Exists_All : ∀A,P,Q,l.
687 #A #P #Q #l elim l [ * | #hd #tl #IH * [ #H1 * #H2 #_ %{hd} /2/ | #H1 * #_ #H2 @IH // ]
690 (**************************** fold *******************************)
692 let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→bool) (f:A→B) (l:list A) on l :B ≝
696 if p a then op (f a) (fold A B op b p f l)
697 else fold A B op b p f l].
699 notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
701 for @{'fold $op $nil (λ${ident i}. $p) (λ${ident i}. $f) $l}.
703 notation "\fold [ op , nil ]_{ident i ∈ l } f"
705 for @{'fold $op $nil (λ${ident i}.true) (λ${ident i}. $f) $l}.
707 interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
710 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a = true →
711 \fold[op,nil]_{i ∈ a::l| p i} (f i) =
712 op (f a) \fold[op,nil]_{i ∈ l| p i} (f i).
713 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
716 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
717 p a = false → \fold[op,nil]_{i ∈ a::l| p i} (f i) =
718 \fold[op,nil]_{i ∈ l| p i} (f i).
719 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
722 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
723 \fold[op,nil]_{i ∈ l| p i} (f i) =
724 \fold[op,nil]_{i ∈ (filter A p l)} (f i).
725 #A #B #a #l #p #op #nil #f elim l //
726 #a #tl #Hind cases(true_or_false (p a)) #pa
727 [ >filter_true // > fold_true // >fold_true //
728 | >filter_false // >fold_false // ]
731 record Aop (A:Type[0]) (nil:A) : Type[0] ≝
733 nill:∀a. op nil a = a;
734 nilr:∀a. op a nil = a;
735 assoc: ∀a,b,c.op a (op b c) = op (op a b) c
738 theorem fold_sum: ∀A,B. ∀I,J:list A.∀nil.∀op:Aop B nil.∀f.
739 op (\fold[op,nil]_{i∈I} (f i)) (\fold[op,nil]_{i∈J} (f i)) =
740 \fold[op,nil]_{i∈(I@J)} (f i).
741 #A #B #I #J #nil #op #f (elim I) normalize
742 [>nill //|#a #tl #Hind <assoc //]
745 (********************** lhd and ltl ******************************)
747 let rec lhd (A:Type[0]) (l:list A) n on n ≝ match n with
749 | S n ⇒ match l with [ nil ⇒ nil … | cons a l ⇒ a :: lhd A l n ]
752 let rec ltl (A:Type[0]) (l:list A) n on n ≝ match n with
754 | S n ⇒ ltl A (tail … l) n
757 lemma lhd_nil: ∀A,n. lhd A ([]) n = [].
761 lemma ltl_nil: ∀A,n. ltl A ([]) n = [].
762 #A #n elim n normalize //
765 lemma lhd_cons_ltl: ∀A,n,l. lhd A l n @ ltl A l n = l.
767 #n #IHn #l elim l normalize //
770 lemma length_ltl: ∀A,n,l. |ltl A l n| = |l| - n.
772 #n #IHn *; normalize /2/
775 (********************** find ******************************)
776 let rec find (A,B:Type[0]) (f:A → option B) (l:list A) on l : option B ≝
781 [ None ⇒ find A B f t
786 (********************** position_of ******************************)
787 let rec position_of_aux (A:Type[0]) (found: A → bool) (l:list A) (acc:nat) on l : option nat ≝
791 match found h with [true ⇒ Some … acc | false ⇒ position_of_aux … found t (S acc)]].
793 definition position_of: ∀A:Type[0]. (A → bool) → list A → option nat ≝
794 λA,found,l. position_of_aux A found l 0.
797 (********************** make_list ******************************)
798 let rec make_list (A:Type[0]) (a:A) (n:nat) on n : list A ≝
801 | S m ⇒ a::(make_list A a m)