Library Stdlib.Numbers.Natural.Abstract.NMulOrder


From Stdlib Require Export NAddOrder.

Module NMulOrderProp (Import N : NAxiomsMiniSig').
Include NAddOrderProp N.

Theorems that are either not valid on Z or have different proofs on N and Z

Theorem square_lt_mono : forall n m, n < m <-> n * n < m * m.

Theorem square_le_mono : forall n m, n <= m <-> n * n <= m * m.

Theorem mul_le_mono_l : forall n m p, n <= m -> p * n <= p * m.

Theorem mul_le_mono_r : forall n m p, n <= m -> n * p <= m * p.

Theorem le_mul_l : forall n m, m ~= 0 -> n <= m * n.

Theorem le_mul_r : forall n m, m ~= 0 -> n <= n * m.

Theorem mul_lt_mono : forall n m p q, n < m -> p < q -> n * p < m * q.

Theorem mul_le_mono : forall n m p q, n <= m -> p <= q -> n * p <= m * q.

Theorem lt_0_mul' : forall n m, n * m > 0 <-> n > 0 /\ m > 0.

Notation mul_pos := lt_0_mul' (only parsing).

Theorem eq_mul_1 : forall n m, n * m == 1 <-> n == 1 /\ m == 1.

Alternative name :

Definition mul_eq_1 := eq_mul_1.

End NMulOrderProp.