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import Mathlib.LinearAlgebra.Basis.VectorSpace
-- import data.nat.basic

inductive Vect (α : Type u) : Nat → Type u where
   | nil : Vect α 0
   | cons : α → Vect α n → Vect α (n + 1)

def Vect.zip : Vect α n → Vect β n → Vect (α × β) n
  | .nil, .nil => .nil
  | .cons x xs, .cons y ys => .cons (x, y) (zip xs ys)

-- #eval (Vect.cons "Hello" (Vect.cons "world" Vect.nil))
-- .zip (Vect.cons "Hello" (Vect.cons "world" Vect.nil))

def hi : Vect String 2 := Vect.cons "Hello" (Vect.cons "world" Vect.nil)

-- #eval hi.zip hi

-- def main : IO Unit := IO.println "Hello, world!"

-- #eval main

-- structure Pos where
--   succ ::
--   pred : Nat



def merge [Ord α] (xs : List α) (ys : List α) : List α :=
  match xs, ys with
  | [], _ => ys
  | _, [] => xs
  | x'::xs', y'::ys' =>
    match Ord.compare x' y' with
    | .lt | .eq => x' :: merge xs' (y' :: ys')
    | .gt => y' :: merge (x'::xs') ys'


theorem blah (i : ℕ) : i ≠ 0 → i / 2 < i := by
  exact fun a => Nat.bitwise_rec_lemma a

def lsb (i : Nat) : Nat :=
  if i = 0 then 0 else if i % 2 == 1 then 1 else 2 * lsb (i / 2)
termination_by i
decreasing_by
  cases i
  case zero =>
    simp
    contradiction
  case succ =>
    apply blah
    simp

theorem lsb_le_i (i : Nat) : lsb i ≤ i := by
  if h₁ : i = 0 then
    simp [h₁, lsb]
  else if h₂ : i % 2 == 1 then
    simp [h₁, h₂, lsb]
    exact Nat.one_le_iff_ne_zero.mpr h₁
  else
    calc
      lsb i = 2 * lsb (i / 2) := by rw [lsb]; simp [h₁, h₂]
      _ ≤ 2 * (i / 2) := by simp [lsb_le_i (i / 2)]
      _ ≤ i := by exact Nat.mul_div_le i 2;