The Coq Standard Library

# Library Coq.Arith.Mult

# Properties of multiplication.

This file is mostly OBSOLETE now, see module PeanoNat.Nat instead.
Nat.mul is defined in Init/Nat.v.
Require Import PeanoNat.
For Compatibility:
Require Export Plus Minus Le Lt.
Local Open Scope nat_scope.

# nat is a semi-ring

## Zero property

Notation mult_0_l := Nat.mul_0_l (compat "8.4"). Notation mult_0_r := Nat.mul_0_r (compat "8.4").

## 1 is neutral

Notation mult_1_l := Nat.mul_1_l (compat "8.4"). Notation mult_1_r := Nat.mul_1_r (compat "8.4").
Hint Resolve mult_1_l mult_1_r: arith.

## Commutativity

Notation mult_comm := Nat.mul_comm (compat "8.4").
Hint Resolve mult_comm: arith.

## Distributivity

Notation mult_plus_distr_r :=
Nat.mul_add_distr_r (compat "8.4").
Notation mult_plus_distr_l :=
Nat.mul_add_distr_l (compat "8.4").
Notation mult_minus_distr_r :=
Nat.mul_sub_distr_r (compat "8.4").
Notation mult_minus_distr_l :=
Nat.mul_sub_distr_l (compat "8.4").
Hint Resolve mult_plus_distr_r: arith.
Hint Resolve mult_minus_distr_r: arith.
Hint Resolve mult_minus_distr_l: arith.

## Associativity

Notation mult_assoc := Nat.mul_assoc (compat "8.4").
Lemma mult_assoc_reverse n m p : n * m * p = n * (m * p).
Hint Resolve mult_assoc_reverse: arith.
Hint Resolve mult_assoc: arith.

## Inversion lemmas

Lemma mult_is_O n m : n * m = 0 -> n = 0 \/ m = 0.
Lemma mult_is_one n m : n * m = 1 -> n = 1 /\ m = 1.

## Multiplication and successor

Notation mult_succ_l := Nat.mul_succ_l (compat "8.4"). Notation mult_succ_r := Nat.mul_succ_r (compat "8.4").

# Compatibility with orders

Lemma mult_O_le n m : m = 0 \/ n <= m * n.
Hint Resolve mult_O_le: arith.
Lemma mult_le_compat_l n m p : n <= m -> p * n <= p * m.
Hint Resolve mult_le_compat_l: arith.
Lemma mult_le_compat_r n m p : n <= m -> n * p <= m * p.
Lemma mult_le_compat n m p q : n <= m -> p <= q -> n * p <= m * q.
Lemma mult_S_lt_compat_l n m p : m < p -> S n * m < S n * p.
Hint Resolve mult_S_lt_compat_l: arith.
Lemma mult_lt_compat_l n m p : n < m -> 0 < p -> p * n < p * m.
Lemma mult_lt_compat_r n m p : n < m -> 0 < p -> n * p < m * p.
Lemma mult_S_le_reg_l n m p : S n * m <= S n * p -> m <= p.

# n|->2*n and n|->2n+1 have disjoint image

Theorem odd_even_lem p q : 2 * p + 1 <> 2 * q.

# Tail-recursive mult

tail_mult is an alternative definition for mult which is tail-recursive, whereas mult is not. This can be useful when extracting programs.
Fixpoint mult_acc (s:nat) m n : nat :=
match n with
| O => s
| S p => mult_acc (tail_plus m s) m p
end.
Lemma mult_acc_aux : forall n m p, m + n * p = mult_acc m p n.
Definition tail_mult n m := mult_acc 0 m n.
Lemma mult_tail_mult : forall n m, n * m = tail_mult n m.
TailSimpl transforms any tail_plus and tail_mult into plus and mult and simplify
Ltac tail_simpl :=
repeat rewrite <- plus_tail_plus; repeat rewrite <- mult_tail_mult;
simpl.