\ / GNU General Public License Version 2
V_____________________________________________________________*)
-include "basics/finset.ma".
+include "basics/lists/list.ma".
record Vector (A:Type[0]) (n:nat): Type[0] ≝
{ vec :> list A;
len: length ? vec = n
}.
+lemma Vector_of_list ≝ λA,l.
+ mk_Vector A (|l|) l (refl ??).
+
+lemma Vector_eq : ∀A,n,v1,v2.
+ vec A n v1 = vec A n v2 → v1 = v2.
+#A #n * #l1 #H1 * #l2 #H2 #eql1l2 generalize in match H1;
+-H1 >eql1l2 //
+qed.
+
definition vec_tail ≝ λA.λn.λv:Vector A n.
mk_Vector A (pred n) (tail A v) ?.
>length_tail >(len A n v) //
normalize >(len A n v) //
qed.
+definition vec_hd ≝ λA.λn.λv:Vector A (S n).
+hd A v ?. elim v * [normalize #H destruct | #a #H #eq @a]
+qed.
+
+lemma vec_expand: ∀A,n,v.
+ v = vec_cons A (vec_hd A n v) n (vec_tail A (S n) v).
+#A #n * #l cases l [normalize in ⊢ (%→?); #H destruct |//]
+qed.
+
+lemma vector_nil: ∀A.∀v:Vector A 0.
+ v = mk_Vector A 0 (nil A) (refl ??).
+#A * * // #a #tl normalize #H destruct
+qed.
+
definition vec_append ≝ λA.λn1,n2.λv1:Vector A n1.λv2: Vector A n2.
mk_Vector A (n1+n2) (v1@v2).
definition vec_map ≝ λA,B.λf:A→B.λn.λv:Vector A n.
mk_Vector B n (map ?? f v)
(trans_eq … (length_map …) (len A n v)).
-
+
+lemma nth_default : ∀A,i,n.∀v:Vector A n.∀d1,d2. i < n →
+ nth i ? v d1 = nth i ? v d2.
+#A #i elim i -i
+ [#n #v #d1 #d2 #ltOn lapply v @(lt_O_n_elim … ltOn)
+ -v #m #v >(vec_expand … v) //
+ |#i #Hind #n #v #d1 #d2 #ltn lapply ltn lapply v @(lt_O_n_elim … (ltn_to_ltO … ltn))
+ -v -ltn #m #v #ltim >(vec_expand … v) @(Hind m (vec_tail A (S m) v) d1 d2 ?)
+ @le_S_S_to_le //
+ ]
+qed.
+
+lemma eq_vec: ∀A,n.∀v1,v2:Vector A n.∀d.
+ (∀i. i < n → nth i A v1 d = nth i A v2 d) → v1 = v2.
+#A #n elim n -n
+ [#v1 #v2 #H >(vector_nil A v1) >(vector_nil A v2) //
+ |#n #Hind #v1 #v2 #d #H >(vec_expand … v1) >(vec_expand … v2)
+ >(Hind (vec_tail … v1) (vec_tail … v2) d)
+ [cut (vec_hd A n v1 = vec_hd A n v2) //
+ cut (∀i,d1,d2. i < S n → nth i A v1 d1 = nth i A v2 d2)
+ [#i #d1 #d2 #Hi >(nth_default ????? d) // >(nth_default ???? d2 d) // @H //]
+ -H #H @(H 0) //
+ |#i #ltin @(H (S i)) @le_S_S //
+ ]
+ ]
+qed.
+
+lemma nth_vec_map :
+ ∀A,B,f,i,n.∀v:Vector A n.∀d.
+ f (nth i ? v d) = nth i ? (vec_map A B f n v) (f d).
+#A #B #f #i elim i
+[ *
+ [ #v #d >(vector_nil … v) %
+ | #n0 #v #d >(vec_expand … v) % ]
+| #i0 #IH *
+ [ #v #d >(vector_nil … v) normalize cases i0 //
+ | #n #v #d >(vec_expand … v) whd in ⊢ (??(?%)%);
+ >(IH n (vec_tail A (S n) v) d) % ] ]
+qed.
+
+
+(* mapi: map with index to move in list.ma *)
+let rec change_vec (A:Type[0]) (n:nat) on n ≝
+match n return λn0.∀v:Vector A n0.A→nat→Vector A n0 with
+[ O ⇒ λv,a,i.v
+| S m ⇒ λv,a,i.
+ match i with
+ [ O ⇒ vec_cons A a m (vec_tail … v)
+ | S j ⇒ vec_cons A (vec_hd A m v) m (change_vec A m (vec_tail … v) a j)
+ ]
+].
+
+lemma nth_change_vec : ∀A,i,n,v,a,d. i < n →
+ nth i ? (change_vec A n v a i) d = a.
+#A #i elim i
+ [#n #v #a #d #ltOn lapply v -v @(lt_O_n_elim n ltOn ??) //
+ |#m #Hind #n #v #a #d #Hlt
+ lapply Hlt lapply v @(lt_O_n_elim … (ltn_to_ltO … Hlt))
+ #p #v #ltmp @Hind @le_S_S_to_le //
+ ]
+qed.
+
+lemma nth_change_vec_neq : ∀A,j,i,n,v,a,d. i ≠ j →
+ nth j ? (change_vec A n v a i) d = nth j ? v d.
+#A #j elim j
+ [#i * // #n #v #a #d cases i
+ [#H @False_ind @(absurd ?? H) //
+ |#i0 #_ >(vec_expand ?? v) in ⊢ (???%); //
+ ]
+ |#m #Hind #i * // cases i // #i0 #n #v #a #d #neqim
+ whd in ⊢(??%?); whd in match (tail ??); >Hind
+ [>(vec_expand ??v) in ⊢ (???%); // |@(not_to_not … neqim) // ]
+ ]
+qed.
+
+lemma change_vec_same : ∀sig,n,v,i,d.
+ change_vec sig n v (nth i ? v d) i = v.
+#sig #n #v #i #d @(eq_vec … d)
+#i0 #Hi0 cases (decidable_eq_nat i i0) #Hi0
+[ >Hi0 >nth_change_vec //
+| >nth_change_vec_neq //
+]
+qed.
+
+lemma change_vec_cons_tail :∀A,n,vA,a,b,i.
+ change_vec A (S n) (vec_cons ? a n vA) b (S i) =
+ vec_cons ? a n (change_vec A n vA b i).
+#A #n #vA cases vA //
+qed.
+
+lemma change_vec_commute : ∀A,n,v,a,b,i,j. i ≠ j →
+ change_vec A n (change_vec A n v a i) b j
+ = change_vec A n (change_vec A n v b j) a i.
+#A #n #v #a #b #i #j #Hij @(eq_vec … a)
+#k #Hk cases (decidable_eq_nat k i) #Hki
+[ >Hki >nth_change_vec // >(nth_change_vec_neq ??????? (sym_not_eq … Hij))
+ >nth_change_vec //
+| cases (decidable_eq_nat k j) #Hkj
+ [ >Hkj >nth_change_vec // >(nth_change_vec_neq ??????? Hij) >nth_change_vec //
+ | >(nth_change_vec_neq ??????? (sym_not_eq … Hki))
+ >(nth_change_vec_neq ??????? (sym_not_eq … Hkj))
+ >(nth_change_vec_neq ??????? (sym_not_eq … Hki))
+ >(nth_change_vec_neq ??????? (sym_not_eq … Hkj)) //
+ ]
+]
+qed.
+
+lemma change_vec_change_vec : ∀A,n,v,a,b,i.
+ change_vec A n (change_vec A n v a i) b i = change_vec A n v b i.
+#A #n #v #a #b #i @(eq_vec … a) #i0 #Hi0
+cases (decidable_eq_nat i i0) #Hii0
+[ >Hii0 >nth_change_vec // >nth_change_vec //
+| >nth_change_vec_neq // >nth_change_vec_neq //
+ >nth_change_vec_neq // ]
+qed.
+
+lemma eq_vec_change_vec : ∀sig,n.∀v1,v2:Vector sig n.∀i,t,d.
+ nth i ? v2 d = t →
+ (∀j.i ≠ j → nth j ? v1 d = nth j ? v2 d) →
+ v2 = change_vec ?? v1 t i.
+#sig #n #v1 #v2 #i #t #d #H1 #H2 @(eq_vec … d)
+#i0 #Hlt cases (decidable_eq_nat i0 i) #Hii0
+[ >Hii0 >nth_change_vec //
+| >nth_change_vec_neq [|@sym_not_eq //] @sym_eq @H2 @sym_not_eq // ]
+qed-.
+
+(* map *)
+
let rec pmap A B C (f:A→B→C) l1 l2 on l1 ≝
match l1 with
[ nil ⇒ nil C
>(le_to_leb_true … (le_n n)) //
qed.
+lemma pmap_vec_cons :∀A,B,C,f,n,vA,vB,a,b.
+ pmap_vec A B C f (S n) (vec_cons ? a n vA) (vec_cons ? b n vB) =
+ vec_cons ? (f a b) n (pmap_vec A B C f n vA vB).
+// qed.
+
+lemma pmap_change : ∀A,B,C,f,n,vA,vB,a,b,i.
+ pmap_vec A B C f n (change_vec ? n vA a i) (change_vec ? n vB b i) =
+ change_vec ? n (pmap_vec A B C f n vA vB) (f a b) i.
+#A #B #C #f #n elim n //
+#m #Hind #vA #vB #a #b >(vec_expand … vA) >(vec_expand … vB) * //
+#i >pmap_vec_cons >pmap_vec_cons >change_vec_cons_tail @eq_f @Hind
+qed.
\ No newline at end of file
projects / helm.git / blobdiff