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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
Require Export Plus.
Require Export Minus.
Require Export Lt.
Require Export Le.
Local Open Scope nat_scope.
Implicit Types m n p : nat.
(** Theorems about multiplication in [nat]. [mult] is defined in module [Init/Peano.v]. *)
(** * [nat] is a semi-ring *)
(** ** Zero property *)
Lemma mult_0_r : forall n, n * 0 = 0.
Proof.
intro; symmetry ; apply mult_n_O.
Qed.
Lemma mult_0_l : forall n, 0 * n = 0.
Proof.
reflexivity.
Qed.
(** ** 1 is neutral *)
Lemma mult_1_l : forall n, 1 * n = n.
Proof.
simpl; auto with arith.
Qed.
Hint Resolve mult_1_l: arith v62.
Lemma mult_1_r : forall n, n * 1 = n.
Proof.
induction n; [ trivial |
simpl; rewrite IHn; reflexivity].
Qed.
Hint Resolve mult_1_r: arith v62.
(** ** Commutativity *)
Lemma mult_comm : forall n m, n * m = m * n.
Proof.
intros; induction n; simpl; auto with arith.
rewrite <- mult_n_Sm.
rewrite IHn; apply plus_comm.
Qed.
Hint Resolve mult_comm: arith v62.
(** ** Distributivity *)
Lemma mult_plus_distr_r : forall n m p, (n + m) * p = n * p + m * p.
Proof.
intros; induction n; simpl; auto with arith.
rewrite <- plus_assoc, IHn; auto with arith.
Qed.
Hint Resolve mult_plus_distr_r: arith v62.
Lemma mult_plus_distr_l : forall n m p, n * (m + p) = n * m + n * p.
Proof.
induction n. trivial.
intros. simpl. rewrite IHn. symmetry. apply plus_permute_2_in_4.
Qed.
Lemma mult_minus_distr_r : forall n m p, (n - m) * p = n * p - m * p.
Proof.
intros; induction n, m using nat_double_ind; simpl; auto with arith.
rewrite <- minus_plus_simpl_l_reverse; auto with arith.
Qed.
Hint Resolve mult_minus_distr_r: arith v62.
Lemma mult_minus_distr_l : forall n m p, n * (m - p) = n * m - n * p.
Proof.
intros n m p.
rewrite mult_comm, mult_minus_distr_r, (mult_comm m n), (mult_comm p n); reflexivity.
Qed.
Hint Resolve mult_minus_distr_l: arith v62.
(** ** Associativity *)
Lemma mult_assoc_reverse : forall n m p, n * m * p = n * (m * p).
Proof.
intros; induction n; simpl; auto with arith.
rewrite mult_plus_distr_r.
induction IHn; auto with arith.
Qed.
Hint Resolve mult_assoc_reverse: arith v62.
Lemma mult_assoc : forall n m p, n * (m * p) = n * m * p.
Proof.
auto with arith.
Qed.
Hint Resolve mult_assoc: arith v62.
(** ** Inversion lemmas *)
Lemma mult_is_O : forall n m, n * m = 0 -> n = 0 \/ m = 0.
Proof.
destruct n as [| n]; simpl; intros m H.
left; trivial.
right; apply plus_is_O in H; destruct H; trivial.
Qed.
Lemma mult_is_one : forall n m, n * m = 1 -> n = 1 /\ m = 1.
Proof.
destruct n as [|n]; simpl; intros m H.
edestruct O_S; eauto.
destruct plus_is_one with (1:=H) as [[-> Hnm] | [-> Hnm]].
simpl in H; rewrite mult_0_r in H; elim (O_S _ H).
rewrite mult_1_r in Hnm; auto.
Qed.
(** ** Multiplication and successor *)
Lemma mult_succ_l : forall n m:nat, S n * m = n * m + m.
Proof.
intros; simpl. rewrite plus_comm. reflexivity.
Qed.
Lemma mult_succ_r : forall n m:nat, n * S m = n * m + n.
Proof.
induction n as [| p H]; intro m; simpl.
reflexivity.
rewrite H, <- plus_n_Sm; apply f_equal; rewrite plus_assoc; reflexivity.
Qed.
(** * Compatibility with orders *)
Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n.
Proof.
induction m; simpl; auto with arith.
Qed.
Hint Resolve mult_O_le: arith v62.
Lemma mult_le_compat_l : forall n m p, n <= m -> p * n <= p * m.
Proof.
induction p as [| p IHp]; intros; simpl.
apply le_n.
auto using plus_le_compat.
Qed.
Hint Resolve mult_le_compat_l: arith.
Lemma mult_le_compat_r : forall n m p, n <= m -> n * p <= m * p.
Proof.
intros m n p H; rewrite mult_comm, (mult_comm n); auto with arith.
Qed.
Lemma mult_le_compat :
forall n m p (q:nat), n <= m -> p <= q -> n * p <= m * q.
Proof.
intros m n p q Hmn Hpq; induction Hmn.
induction Hpq.
(* m*p<=m*p *)
apply le_n.
(* m*p<=m*m0 -> m*p<=m*(S m0) *)
rewrite <- mult_n_Sm; apply le_trans with (m * m0).
assumption.
apply le_plus_l.
(* m*p<=m0*q -> m*p<=(S m0)*q *)
simpl; apply le_trans with (m0 * q).
assumption.
apply le_plus_r.
Qed.
Lemma mult_S_lt_compat_l : forall n m p, m < p -> S n * m < S n * p.
Proof.
induction n; intros; simpl in *.
rewrite <- 2 plus_n_O; assumption.
auto using plus_lt_compat.
Qed.
Hint Resolve mult_S_lt_compat_l: arith.
Lemma mult_lt_compat_l : forall n m p, n < m -> 0 < p -> p * n < p * m.
Proof.
intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp).
now apply mult_S_lt_compat_l.
Qed.
Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p.
Proof.
intros m n p H Hp. destruct p. elim (lt_irrefl _ Hp).
rewrite (mult_comm m), (mult_comm n). now apply mult_S_lt_compat_l.
Qed.
Lemma mult_S_le_reg_l : forall n m p, S n * m <= S n * p -> m <= p.
Proof.
intros m n p H; destruct (le_or_lt n p). trivial.
assert (H1:S m * n < S m * n).
apply le_lt_trans with (m := S m * p). assumption.
apply mult_S_lt_compat_l. assumption.
elim (lt_irrefl _ H1).
Qed.
(** * n|->2*n and n|->2n+1 have disjoint image *)
Theorem odd_even_lem : forall p q, 2 * p + 1 <> 2 * q.
Proof.
induction p; destruct q.
discriminate.
simpl; rewrite plus_comm. discriminate.
discriminate.
intro H0; destruct (IHp q).
replace (2 * q) with (2 * S q - 2).
rewrite <- H0; simpl.
repeat rewrite (fun x y => plus_comm x (S y)); simpl; auto.
simpl; rewrite (fun y => plus_comm q (S y)); destruct q; simpl; auto.
Qed.
(** * 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.
Proof.
induction n as [| p IHp]; simpl; auto.
intros s m; rewrite <- plus_tail_plus; rewrite <- IHp.
rewrite <- plus_assoc_reverse; apply f_equal2; auto.
rewrite plus_comm; auto.
Qed.
Definition tail_mult n m := mult_acc 0 m n.
Lemma mult_tail_mult : forall n m, n * m = tail_mult n m.
Proof.
intros; unfold tail_mult; rewrite <- mult_acc_aux; auto.
Qed.
(** [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.
|
