# Library Coq.Vectors.VectorSpec

Proofs of specification for functions defined over Vector
Author: Pierre Boutillier Institution: PPS, INRIA 12/2010
Require Fin.
Require Import VectorDef.
Import VectorNotations.
Definition cons_inj A a1 a2 n (v1 v2 : t A n)
(eq : a1 :: v1 = a2 :: v2) : a1 = a2 /\ v1 = v2 :=
match eq in _ = x return caseS _ (fun a2' _ v2' => fun v1' => a1 = a2' /\ v1' = v2') x v1
with | eq_refl => conj eq_refl eq_refl
end.
Lemmas are done for functions that use Fin.t but thanks to Peano_dec.le_unique, all is true for the one that use lt
Lemma eq_nth_iff A n (v1 v2: t A n):
( forall p1 p2, p1 = p2 -> v1 [@ p1 ] = v2 [@ p2 ] ) <-> v1 = v2.
Lemma nth_order_last A: forall n (v: t A (S n)) (H: n < S n),
nth_order v H = last v.
Lemma shiftin_nth A a n (v: t A n) k1 k2 (eq: k1 = k2):
nth (shiftin a v) (Fin.L_R 1 k1) = nth v k2.
Lemma shiftin_last A a n (v: t A n): last (shiftin a v) = a.
Lemma shiftrepeat_nth A: forall n k (v: t A (S n)),
nth (shiftrepeat v) (Fin.L_R 1 k) = nth v k.
Lemma shiftrepeat_last A: forall n (v: t A (S n)), last (shiftrepeat v) = last v.
Lemma const_nth A (a: A) n (p: Fin.t n): ( const a n )[@ p ] = a.
Lemma nth_map {A B} (f: A -> B) {n} v (p1 p2: Fin.t n) (eq: p1 = p2):
( map f v ) [@ p1 ] = f (v [@ p2 ]).
Lemma nth_map2 {A B C} (f: A -> B -> C) {n} v w (p1 p2 p3: Fin.t n):
p1 = p2 -> p2 = p3 -> ( map2 f v w ) [@ p1 ] = f (v [@ p2 ]) (w [@ p3 ]).
Lemma fold_left_right_assoc_eq {A B} {f: A -> B -> A}
(assoc: forall a b c, f (f a b) c = f (f a c) b)
{n} (v: t B n): forall a, fold_left f a v = fold_right (fun x y => f y x) v a.
Lemma to_list_of_list_opp {A} (l: list A): to_list (of_list l) = l.