(* Paraphernalia: *)

Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.

Require Import Arith Bool.

(* ********** *)

Check Bool.eqb. (* : bool -> bool -> bool *)

Check eqb. (* : bool -> bool -> bool *)

Search (eqb _ _ = true -> _ = _).
(* eqb_prop: forall a b : bool, eqb a b = true -> a = b *)

Search (eqb _ _ = true).
(* eqb_reflx: forall b : bool, eqb b b = true *)

Search (_ * _).

(* Definition is_even (n: nat) := *)

(* Proposition product_of_n_and_even_is_even : *)
(*   forall n : nat, *)

Proposition product_of_2_consecutive_natural_numbers_is_even:
  forall n: nat,
  exists a: nat,
    n * (S n) = 2 * a.
Proof.
  intro n.
  induction n as [ | n' IHn'].
  - exists 0.
    rewrite -> (Nat.mul_0_l 1).
    rewrite -> (Nat.mul_0_r 2).
    reflexivity.
  - rewrite -> (Nat.mul_succ_r (S n') (S n')).
    rewrite -> (Nat.mul_succ_r (S n') n').
    rewrite -> (Nat.mul_comm (S n') n').
    destruct IHn' as [k IHn'].
    rewrite -> IHn'.
    rewrite <- (Nat.add_assoc (2 * k) (S n') (S n')).
    remember (S n' + S n') as x eqn:H_x.
    rewrite <- (Nat.mul_1_l (S n')) in H_x at 1.
    rewrite -> H_x.
    rewrite <- (Nat.mul_succ_l 1 (S n')).
    Search (_ * _ + _ * _).
    rewrite <- (Nat.mul_add_distr_l 2 k (S n')).
    exists (k + S n').
    reflexivity.
Qed.

Proposition product_of_3_consecutive_natural_numbers_is_divisible_by_2:
  forall n : nat,
  exists a : nat,
    n * (S n) * (S (S n)) = 2 * a.
Proof.
  intro n.
  induction n as [ | n' IHn' ].
  - exists 0.
    rewrite -> (Nat.mul_0_l 1).
    rewrite -> (Nat.mul_0_l 2).
    rewrite -> (Nat.mul_0_r 2).
    reflexivity.
  - destruct IHn' as [k IHn'].
    destruct (product_of_2_consecutive_natural_numbers_is_even (S n')) as [x H_product_of_2_consecutive_natural_numbers_is_even].
    rewrite -> H_product_of_2_consecutive_natural_numbers_is_even.
    exists (x * S (S (S n'))).
    Check (Nat.mul_assoc 2 x (S (S (S n')))).
    rewrite -> (Nat.mul_assoc 2 x (S (S (S n')))).
    reflexivity.
Qed.