From Hammer Require Import Hammer.

From Stdlib Require Import Arith.

Hammer_version.
Hammer_objects.

Set Hammer SAutoLimit 0.

Lemma lem_1 : le 1 2.
  hammer.
Qed.

Lemma lem_2 : forall n : nat, Nat.Odd n \/ Nat.Odd (n + 1).
  hammer.
Qed.

Lemma lem_2_1 : forall n : nat, Nat.Even n \/ Nat.Even (n + 1).
  hammer.
Qed.

Lemma lem_3 : le 2 3.
  hammer.
Qed.

Lemma mult_1 : forall m n k : nat, m * n + k = k + n * m.
Proof.
  hammer.
Qed.

Lemma lem_rew : forall m n : nat, 1 + n + m + 1 = m + 2 + n.
Proof.
  hammer.
Qed.

Lemma lem_pow : forall n : nat, 3 * 3 ^ n = 3 ^ (n + 1).
Proof.
  hammer.
Qed.

Lemma minus_neq_O : forall n i:nat, (i < n) -> (n - i) <> 0.
Proof.
  hammer.
Qed.

Lemma le_minusni_n : forall n i:nat, i <= n -> n - i <= n.
Proof.
  hammer.
Qed.

Lemma le_double : forall m n:nat, 2 * m <= 2 * n -> m <= n.
Proof.
  hammer.
Qed.

Lemma le_plus : forall x, x + x >= x.
Proof.
  hammer.
Qed.

Lemma le_plus_2 : forall x, x > 0 -> x < x + x.
Proof.
  hammer.
Qed.
