NMulOrder.v

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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
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(* <O___,, *       (see CREDITS file for the list of authors)           *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
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(*                      Evgeny Makarov, INRIA, 2007                     *)
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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.forall n m : t, n < m <-> n * n < m * m
Proof.forall n m : t, n < m <-> n * n < m * m
intros n m; split; intro;
[apply square_lt_mono_nonneg | apply square_lt_simpl_nonneg];
try assumption; apply le_0_l.
Qed.

Theorem square_le_mono : forall n m, n <= m <-> n * n <= m * m.forall n m : t, n <= m <-> n * n <= m * m
Proof.forall n m : t, n <= m <-> n * n <= m * m
intros n m; split; intro;
[apply square_le_mono_nonneg | apply square_le_simpl_nonneg];
try assumption; apply le_0_l.
Qed.

Theorem mul_le_mono_l : forall n m p, n <= m -> p * n <= p * m.forall n m p : t, n <= m -> p * n <= p * m
Proof.forall n m p : t, n <= m -> p * n <= p * m
intros; apply mul_le_mono_nonneg_l.n, m, p:t
H:n <= m
0 <= p
n, m, p:t
H:n <= m
n <= m apply le_0_l.n, m, p:t
H:n <= m
n <= m assumption.
Qed.

Theorem mul_le_mono_r : forall n m p, n <= m -> n * p <= m * p.forall n m p : t, n <= m -> n * p <= m * p
Proof.forall n m p : t, n <= m -> n * p <= m * p
intros; apply mul_le_mono_nonneg_r.n, m, p:t
H:n <= m
0 <= p
n, m, p:t
H:n <= m
n <= m apply le_0_l.n, m, p:t
H:n <= m
n <= m assumption.
Qed.

Theorem mul_lt_mono : forall n m p q, n < m -> p < q -> n * p < m * q.forall n m p q : t, n < m -> p < q -> n * p < m * q
Proof.forall n m p q : t, n < m -> p < q -> n * p < m * q
intros; apply mul_lt_mono_nonneg; try assumption; apply le_0_l.
Qed.

Theorem mul_le_mono : forall n m p q, n <= m -> p <= q -> n * p <= m * q.forall n m p q : t, n <= m -> p <= q -> n * p <= m * q
Proof.forall n m p q : t, n <= m -> p <= q -> n * p <= m * q
intros; apply mul_le_mono_nonneg; try assumption; apply le_0_l.
Qed.

Theorem lt_0_mul' : forall n m, n * m > 0 <-> n > 0 /\ m > 0.forall n m : t, 0 < n * m <-> 0 < n /\ 0 < m
Proof.forall n m : t, 0 < n * m <-> 0 < n /\ 0 < m
intros n m; split; [intro H | intros [H1 H2]].n, m:t
H:0 < n * m
0 < n /\ 0 < m
n, m:t
H1:0 < n
H2:0 < m
0 < n * m
apply lt_0_mul in H.n, m:t
H:0 < n /\ 0 < m \/ m < 0 /\ n < 0
0 < n /\ 0 < m
n, m:t
H1:0 < n
H2:0 < m
0 < n * m destruct H as [[H1 H2] | [H1 H2]].n, m:t
H1:0 < n
H2:0 < m
0 < n /\ 0 < m
n, m:t
H1:m < 0
H2:n < 0
0 < n /\ 0 < m
n, m:t
H1:0 < n
H2:0 < m
0 < n * m now split.n, m:t
H1:m < 0
H2:n < 0
0 < n /\ 0 < m
n, m:t
H1:0 < n
H2:0 < m
0 < n * m
 false_hyp H1 nlt_0_r.n, m:t
H1:0 < n
H2:0 < m
0 < n * m
now apply mul_pos_pos.
Qed.

Notation mul_pos := lt_0_mul' (only parsing).

Theorem eq_mul_1 : forall n m, n * m == 1 <-> n == 1 /\ m == 1.forall n m : t, n * m == 1 <-> n == 1 /\ m == 1
Proof.forall n m : t, n * m == 1 <-> n == 1 /\ m == 1
intros n m.n, m:t
n * m == 1 <-> n == 1 /\ m == 1
split; [| intros [H1 H2]; now rewrite H1, H2, mul_1_l].n, m:t
n * m == 1 -> n == 1 /\ m == 1
intro H; destruct (lt_trichotomy n 1) as [H1 | [H1 | H1]].n, m:t
H:n * m == 1
H1:n < 1
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:n == 1
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
n == 1 /\ m == 1
apply lt_1_r in H1.n, m:t
H:n * m == 1
H1:n == 0
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:n == 1
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
n == 1 /\ m == 1 rewrite H1, mul_0_l in H.n, m:t
H:0 == 1
H1:n == 0
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:n == 1
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
n == 1 /\ m == 1 order'.n, m:t
H:n * m == 1
H1:n == 1
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
n == 1 /\ m == 1
rewrite H1, mul_1_l in H; now split.n, m:t
H:n * m == 1
H1:1 < n
n == 1 /\ m == 1
destruct (eq_0_gt_0_cases m) as [H2 | H2].n, m:t
H:n * m == 1
H1:1 < n
H2:m == 0
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
H2:0 < m
n == 1 /\ m == 1
rewrite H2, mul_0_r in H.n, m:t
H:0 == 1
H1:1 < n
H2:m == 0
n == 1 /\ m == 1
n, m:t
H:n * m == 1
H1:1 < n
H2:0 < m
n == 1 /\ m == 1 order'.n, m:t
H:n * m == 1
H1:1 < n
H2:0 < m
n == 1 /\ m == 1
apply (mul_lt_mono_pos_r m) in H1; [| assumption].n, m:t
H:n * m == 1
H1:1 * m < n * m
H2:0 < m
n == 1 /\ m == 1 rewrite mul_1_l in H1.n, m:t
H:n * m == 1
H1:m < n * m
H2:0 < m
n == 1 /\ m == 1
assert (H3 : 1 < n * m) by now apply (lt_1_l m).n, m:t
H:n * m == 1
H1:m < n * m
H2:0 < m
H3:1 < n * m
n == 1 /\ m == 1
rewrite H in H3; false_hyp H3 lt_irrefl.
Qed.

Alternative name :

Definition mul_eq_1 := eq_mul_1.

End NMulOrderProp.