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".
15 inductive list (A:Type[0]) : Type[0] :=
17 | cons: A -> list A -> list A.
19 notation "hvbox(hd break :: tl)"
20 right associative with precedence 47
23 notation "[ list0 x sep ; ]"
24 non associative with precedence 90
25 for ${fold right @'nil rec acc @{'cons $x $acc}}.
27 notation "hvbox(l1 break @ l2)"
28 right associative with precedence 47
29 for @{'append $l1 $l2 }.
31 interpretation "nil" 'nil = (nil ?).
32 interpretation "cons" 'cons hd tl = (cons ? hd tl).
34 definition not_nil: ∀A:Type[0].list A → Prop ≝
35 λA.λl.match l with [ nil ⇒ True | cons hd tl ⇒ False ].
38 ∀A:Type[0].∀l:list A.∀a:A. a::l ≠ [].
39 #A #l #a @nmk #Heq (change with (not_nil ? (a::l))) >Heq //
43 let rec id_list A (l: list A) on l :=
46 | (cons hd tl) => hd :: id_list A tl ]. *)
48 let rec append A (l1: list A) l2 on l1 ≝
51 | cons hd tl ⇒ hd :: append A tl l2 ].
53 definition hd ≝ λA.λl: list A.λd:A.
54 match l with [ nil ⇒ d | cons a _ ⇒ a].
56 definition tail ≝ λA.λl: list A.
57 match l with [ nil ⇒ [] | cons hd tl ⇒ tl].
59 interpretation "append" 'append l1 l2 = (append ? l1 l2).
61 theorem append_nil: ∀A.∀l:list A.l @ [] = l.
62 #A #l (elim l) normalize // qed.
64 theorem associative_append:
65 ∀A.associative (list A) (append A).
66 #A #l1 #l2 #l3 (elim l1) normalize // qed.
68 theorem append_cons:∀A.∀a:A.∀l,l1.l@(a::l1)=(l@[a])@l1.
69 #A #a #l #l1 >associative_append // qed.
71 theorem nil_append_elim: ∀A.∀l1,l2: list A.∀P:?→?→Prop.
72 l1@l2=[] → P (nil A) (nil A) → P l1 l2.
73 #A #l1 #l2 #P (cases l1) normalize //
77 theorem nil_to_nil: ∀A.∀l1,l2:list A.
78 l1@l2 = [] → l1 = [] ∧ l2 = [].
79 #A #l1 #l2 #isnil @(nil_append_elim A l1 l2) /2/
82 (**************************** iterators ******************************)
84 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
85 match l with [ nil ⇒ nil ? | cons x tl ⇒ f x :: (map A B f tl)].
87 lemma map_append : ∀A,B,f,l1,l2.
88 (map A B f l1) @ (map A B f l2) = map A B f (l1@l2).
91 | #h #t #IH #l2 normalize //
94 let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:list A) on l :B ≝
95 match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
99 foldr T (list T) (λx,l0.if p x then x::l0 else l0) (nil T).
101 (* compose f [a1;...;an] [b1;...;bm] =
102 [f a1 b1; ... ;f an b1; ... ;f a1 bm; f an bm] *)
104 definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
105 foldr ?? (λi,acc.(map ?? (f i) l2)@acc) [ ] l1.
107 lemma filter_true : ∀A,l,a,p. p a = true →
108 filter A p (a::l) = a :: filter A p l.
109 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
111 lemma filter_false : ∀A,l,a,p. p a = false →
112 filter A p (a::l) = filter A p l.
113 #A #l #a #p #pa (elim l) normalize >pa normalize // qed.
115 theorem eq_map : ∀A,B,f,g,l. (∀x.f x = g x) → map A B f l = map A B g l.
116 #A #B #f #g #l #eqfg (elim l) normalize // qed.
118 (**************************** reverse *****************************)
119 let rec rev_append S (l1,l2:list S) on l1 ≝
122 | cons a tl ⇒ rev_append S tl (a::l2)
126 definition reverse ≝λS.λl.rev_append S l [].
128 lemma reverse_single : ∀S,a. reverse S [a] = [a].
131 lemma rev_append_def : ∀S,l1,l2.
132 rev_append S l1 l2 = (reverse S l1) @ l2 .
133 #S #l1 elim l1 normalize //
136 lemma reverse_cons : ∀S,a,l. reverse S (a::l) = (reverse S l)@[a].
137 #S #a #l whd in ⊢ (??%?); //
140 lemma reverse_append: ∀S,l1,l2.
141 reverse S (l1 @ l2) = (reverse S l2)@(reverse S l1).
142 #S #l1 elim l1 [normalize // | #a #tl #Hind #l2 >reverse_cons
143 >reverse_cons // qed.
145 lemma reverse_reverse : ∀S,l. reverse S (reverse S l) = l.
146 #S #l elim l // #a #tl #Hind >reverse_cons >reverse_append
149 (* an elimination principle for lists working on the tail;
150 useful for strings *)
151 lemma list_elim_left: ∀S.∀P:list S → Prop. P (nil S) →
152 (∀a.∀tl.P tl → P (tl@[a])) → ∀l. P l.
153 #S #P #Pnil #Pstep #l <(reverse_reverse … l)
154 generalize in match (reverse S l); #l elim l //
155 #a #tl #H >reverse_cons @Pstep //
158 (**************************** length ******************************)
160 let rec length (A:Type[0]) (l:list A) on l ≝
163 | cons a tl ⇒ S (length A tl)].
165 notation "|M|" non associative with precedence 60 for @{'norm $M}.
166 interpretation "norm" 'norm l = (length ? l).
168 lemma length_append: ∀A.∀l1,l2:list A.
170 #A #l1 elim l1 // normalize /2/
173 (****************************** nth ********************************)
174 let rec nth n (A:Type[0]) (l:list A) (d:A) ≝
177 |S m ⇒ nth m A (tail A l) d].
179 lemma nth_nil: ∀A,a,i. nth i A ([]) a = a.
180 #A #a #i elim i normalize //
183 (****************************** nth_opt ********************************)
184 let rec nth_opt (A:Type[0]) (n:nat) (l:list A) on l : option A ≝
187 | cons h t ⇒ match n with [ O ⇒ Some ? h | S m ⇒ nth_opt A m t ]
190 (**************************** All *******************************)
192 let rec All (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
195 | cons h t ⇒ P h ∧ All A P t
198 lemma All_mp : ∀A,P,Q. (∀a.P a → Q a) → ∀l. All A P l → All A Q l.
199 #A #P #Q #H #l elim l normalize //
203 lemma All_nth : ∀A,P,n,l.
206 nth_opt A n l = Some A a →
209 [ * [ #_ #a #E whd in E:(??%?); destruct
210 | #hd #tl * #H #_ #a #E whd in E:(??%?); destruct @H
213 [ #_ #a #E whd in E:(??%?); destruct
214 | #hd #tl * #_ whd in ⊢ (? → ∀_.??%? → ?); @IH
218 (**************************** Exists *******************************)
220 let rec Exists (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝
223 | cons h t ⇒ (P h) ∨ (Exists A P t)
226 lemma Exists_append : ∀A,P,l1,l2.
227 Exists A P (l1 @ l2) → Exists A P l1 ∨ Exists A P l2.
232 | #H cases (IH l2 H) /3/
236 lemma Exists_append_l : ∀A,P,l1,l2.
237 Exists A P l1 → Exists A P (l1@l2).
238 #A #P #l1 #l2 elim l1
246 lemma Exists_append_r : ∀A,P,l1,l2.
247 Exists A P l2 → Exists A P (l1@l2).
248 #A #P #l1 #l2 elim l1
250 | #h #t #IH #H %2 @IH @H
253 lemma Exists_add : ∀A,P,l1,x,l2. Exists A P (l1@l2) → Exists A P (l1@x::l2).
254 #A #P #l1 #x #l2 elim l1
256 | #h #t #IH normalize * [ #H %1 @H | #H %2 @IH @H ]
259 lemma Exists_mid : ∀A,P,l1,x,l2. P x → Exists A P (l1@x::l2).
260 #A #P #l1 #x #l2 #H elim l1
265 lemma Exists_map : ∀A,B,P,Q,f,l.
268 Exists B Q (map A B f l).
269 #A #B #P #Q #f #l elim l //
270 #h #t #IH * [ #H #F %1 @F @H | #H #F %2 @IH [ @H | @F ] ] qed.
272 lemma Exists_All : ∀A,P,Q,l.
276 #A #P #Q #l elim l [ * | #hd #tl #IH * [ #H1 * #H2 #_ %{hd} /2/ | #H1 * #_ #H2 @IH // ]
279 (**************************** fold *******************************)
281 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 ≝
285 if p a then op (f a) (fold A B op b p f l)
286 else fold A B op b p f l].
288 notation "\fold [ op , nil ]_{ ident i ∈ l | p} f"
290 for @{'fold $op $nil (λ${ident i}. $p) (λ${ident i}. $f) $l}.
292 notation "\fold [ op , nil ]_{ident i ∈ l } f"
294 for @{'fold $op $nil (λ${ident i}.true) (λ${ident i}. $f) $l}.
296 interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
299 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a = true →
300 \fold[op,nil]_{i ∈ a::l| p i} (f i) =
301 op (f a) \fold[op,nil]_{i ∈ l| p i} (f i).
302 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
305 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
306 p a = false → \fold[op,nil]_{i ∈ a::l| p i} (f i) =
307 \fold[op,nil]_{i ∈ l| p i} (f i).
308 #A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
311 ∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
312 \fold[op,nil]_{i ∈ l| p i} (f i) =
313 \fold[op,nil]_{i ∈ (filter A p l)} (f i).
314 #A #B #a #l #p #op #nil #f elim l //
315 #a #tl #Hind cases(true_or_false (p a)) #pa
316 [ >filter_true // > fold_true // >fold_true //
317 | >filter_false // >fold_false // ]
320 record Aop (A:Type[0]) (nil:A) : Type[0] ≝
322 nill:∀a. op nil a = a;
323 nilr:∀a. op a nil = a;
324 assoc: ∀a,b,c.op a (op b c) = op (op a b) c
327 theorem fold_sum: ∀A,B. ∀I,J:list A.∀nil.∀op:Aop B nil.∀f.
328 op (\fold[op,nil]_{i∈I} (f i)) (\fold[op,nil]_{i∈J} (f i)) =
329 \fold[op,nil]_{i∈(I@J)} (f i).
330 #A #B #I #J #nil #op #f (elim I) normalize
331 [>nill //|#a #tl #Hind <assoc //]
334 (********************** lhd and ltl ******************************)
336 let rec lhd (A:Type[0]) (l:list A) n on n ≝ match n with
338 | S n ⇒ match l with [ nil ⇒ nil … | cons a l ⇒ a :: lhd A l n ]
341 let rec ltl (A:Type[0]) (l:list A) n on n ≝ match n with
343 | S n ⇒ ltl A (tail … l) n
346 lemma lhd_nil: ∀A,n. lhd A ([]) n = [].
350 lemma ltl_nil: ∀A,n. ltl A ([]) n = [].
351 #A #n elim n normalize //
354 lemma lhd_cons_ltl: ∀A,n,l. lhd A l n @ ltl A l n = l.
356 #n #IHn #l elim l normalize //
359 lemma length_ltl: ∀A,n,l. |ltl A l n| = |l| - n.
361 #n #IHn *; normalize /2/
364 (********************** find ******************************)
365 let rec find (A,B:Type[0]) (f:A → option B) (l:list A) on l : option B ≝
370 [ None ⇒ find A B f t
375 (********************** position_of ******************************)
376 let rec position_of_aux (A:Type[0]) (found: A → bool) (l:list A) (acc:nat) on l : option nat ≝
380 match found h with [true ⇒ Some … acc | false ⇒ position_of_aux … found t (S acc)]].
382 definition position_of: ∀A:Type[0]. (A → bool) → list A → option nat ≝
383 λA,found,l. position_of_aux A found l 0.
386 (********************** make_list ******************************)
387 let rec make_list (A:Type[0]) (a:A) (n:nat) on n : list A ≝
390 | S m ⇒ a::(make_list A a m)