The Coq Proof Assistant

# Library Coq.Numbers.Natural.Abstract.NMulOrder

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.