Library Coq.Numbers.NatInt.NZMulOrder


Require Import NZAxioms.
Require Import NZAddOrder.

Module Type NZMulOrderProp (Import NZ : NZOrdAxiomsSig´).
Include NZAddOrderProp NZ.

Theorem mul_lt_pred :
  forall p q n m, S p == q -> (p * n < p * m <-> q * n + m < q * m + n).

Theorem mul_lt_mono_pos_l : forall p n m, 0 < p -> (n < m <-> p * n < p * m).

Theorem mul_lt_mono_pos_r : forall p n m, 0 < p -> (n < m <-> n * p < m * p).

Theorem mul_lt_mono_neg_l : forall p n m, p < 0 -> (n < m <-> p * m < p * n).

Theorem mul_lt_mono_neg_r : forall p n m, p < 0 -> (n < m <-> m * p < n * p).

Theorem mul_le_mono_nonneg_l : forall n m p, 0 <= p -> n <= m -> p * n <= p * m.

Theorem mul_le_mono_nonpos_l : forall n m p, p <= 0 -> n <= m -> p * m <= p * n.

Theorem mul_le_mono_nonneg_r : forall n m p, 0 <= p -> n <= m -> n * p <= m * p.

Theorem mul_le_mono_nonpos_r : forall n m p, p <= 0 -> n <= m -> m * p <= n * p.

Theorem mul_cancel_l : forall n m p, p ~= 0 -> (p * n == p * m <-> n == m).

Theorem mul_cancel_r : forall n m p, p ~= 0 -> (n * p == m * p <-> n == m).

Theorem mul_id_l : forall n m, m ~= 0 -> (n * m == m <-> n == 1).

Theorem mul_id_r : forall n m, n ~= 0 -> (n * m == n <-> m == 1).

Theorem mul_le_mono_pos_l : forall n m p, 0 < p -> (n <= m <-> p * n <= p * m).

Theorem mul_le_mono_pos_r : forall n m p, 0 < p -> (n <= m <-> n * p <= m * p).

Theorem mul_le_mono_neg_l : forall n m p, p < 0 -> (n <= m <-> p * m <= p * n).

Theorem mul_le_mono_neg_r : forall n m p, p < 0 -> (n <= m <-> m * p <= n * p).

Theorem mul_lt_mono_nonneg :
  forall n m p q, 0 <= n -> n < m -> 0 <= p -> p < q -> n * p < m * q.


Theorem mul_le_mono_nonneg :
  forall n m p q, 0 <= n -> n <= m -> 0 <= p -> p <= q -> n * p <= m * q.

Theorem mul_pos_pos : forall n m, 0 < n -> 0 < m -> 0 < n * m.

Theorem mul_neg_neg : forall n m, n < 0 -> m < 0 -> 0 < n * m.

Theorem mul_pos_neg : forall n m, 0 < n -> m < 0 -> n * m < 0.

Theorem mul_neg_pos : forall n m, n < 0 -> 0 < m -> n * m < 0.

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

Theorem mul_pos_cancel_l : forall n m, 0 < n -> (0 < n*m <-> 0 < m).

Theorem mul_pos_cancel_r : forall n m, 0 < m -> (0 < n*m <-> 0 < n).

Theorem mul_nonneg_cancel_l : forall n m, 0 < n -> (0 <= n*m <-> 0 <= m).

Theorem mul_nonneg_cancel_r : forall n m, 0 < m -> (0 <= n*m <-> 0 <= n).

Theorem lt_1_mul_pos : forall n m, 1 < n -> 0 < m -> 1 < n * m.

Theorem eq_mul_0 : forall n m, n * m == 0 <-> n == 0 \/ m == 0.

Theorem neq_mul_0 : forall n m, n ~= 0 /\ m ~= 0 <-> n * m ~= 0.

Theorem eq_square_0 : forall n, n * n == 0 <-> n == 0.

Theorem eq_mul_0_l : forall n m, n * m == 0 -> m ~= 0 -> n == 0.

Theorem eq_mul_0_r : forall n m, n * m == 0 -> n ~= 0 -> m == 0.

Some alternative names:

Definition mul_eq_0 := eq_mul_0.
Definition mul_eq_0_l := eq_mul_0_l.
Definition mul_eq_0_r := eq_mul_0_r.

Theorem lt_0_mul n m : 0 < n * m <-> (0 < n /\ 0 < m) \/ (m < 0 /\ n < 0).

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

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


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

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

Theorem mul_2_mono_l : forall n m, n < m -> 1 + 2 * n < 2 * m.

Lemma add_le_mul : forall a b, 1<a -> 1<b -> a+b <= a*b.

A few results about squares

Lemma square_nonneg : forall a, 0 <= a * a.

Lemma crossmul_le_addsquare : forall a b, 0<=a -> 0<=b -> b*a+a*b <= a*a+b*b.

Lemma add_square_le : forall a b, 0<=a -> 0<=b ->
 a*a + b*b <= (a+b)*(a+b).

Lemma square_add_le : forall a b, 0<=a -> 0<=b ->
 (a+b)*(a+b) <= 2*(a*a + b*b).

Lemma quadmul_le_squareadd : forall a b, 0<=a -> 0<=b ->
 2*2*a*b <= (a+b)*(a+b).

End NZMulOrderProp.