29 lines
898 B
Text
29 lines
898 B
Text
inductive Vector (α : Type u): Nat → Type u where
|
||
| nil : Vector α 0
|
||
| cons (head : α) (tail : Vector α n) : Vector α (n+1)
|
||
|
||
namespace Vector
|
||
def nth : ∀{n}, Vector α n → Fin n → α
|
||
| n+1, cons x xs, ⟨ 0, _⟩ => x
|
||
| n+1, cons x xs, ⟨k+1, h⟩ => xs.nth ⟨k, sorry⟩
|
||
|
||
def snoc : ∀{n : Nat} (xs : Vector α n) (x : α), Vector α (n+1)
|
||
| _, nil, x' => cons x' nil
|
||
| _, cons x xs, x' => cons x (snoc xs x')
|
||
|
||
theorem nth_snoc_eq (k: Fin (n+1))(v : Vector α n)
|
||
(h: k.val = n):
|
||
(v.snoc x).nth k = x := by
|
||
cases k; rename_i k hk
|
||
induction v generalizing k <;> subst h
|
||
· simp only [nth]
|
||
· simp_all[nth]
|
||
|
||
theorem nth_snoc_eq_works (k: Fin (n+1))(v : Vector α n)
|
||
(h: k.val = n):
|
||
(v.snoc x).nth k = x := by
|
||
cases k; rename_i k hk
|
||
induction v generalizing k <;> subst h
|
||
· simp only [nth]
|
||
· simp[*,nth]
|
||
end Vector
|