Module VectorNotations2.
Notation "[||]" := (nil _) : vector_scope.
Notation "h ':::' t" := (cons _ h _ t) (at level 60, right associativity) : vector_scope.
Notation " [| x |] " := (x ::: [||]) : vector_scope.
Notation " [| x ; y ; .. ; z |] " := (cons _ x _ (cons _ y _ .. (cons _ z _ (nil _)) ..)) : vector_scope.
Notation "v [@ p ]" := (nth v p) (at level 1, format "v [@ p ]") : vector_scope.
End VectorNotations2.
Ltac existT_eq :=
match goal with
| [ H: existT ?X1 ?Y1 ?Z1 = existT ?X2 ?Y2 ?Z2 |- _] =>
apply EqdepFacts.eq_sigT_iff_eq_dep in H; inv H
end.
Ltac existT_eq' :=
match goal with
| [ H: existT ?X1 ?Y1 ?Z1 = existT ?X2 ?Y2 ?Z2 |- _] =>
apply EqdepFacts.eq_sigT_iff_eq_dep in H; induction H
end.

Lemma replace_nth2 X n (v : Vector.t X n) i j (x : X) :
i <> j -> (Vector.replace v i x) [@j] = v[@j].