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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id: Mult.v 9245 2006-10-17 12:53:34Z notin $ i*)
Require Export Plus.
Require Export Minus.
Require Export Lt.
Require Export Le.
Open Local 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 in |- *; 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 in |- *; 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; elim n; intros; simpl in |- *; auto with arith.
elim mult_n_Sm.
elim H; 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; elim n; simpl in |- *; intros; auto with arith.
elim plus_assoc; elim H; 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 in |- *. rewrite (IHn m p). apply sym_eq. 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; pattern n, m in |- *; apply nat_double_ind; simpl in |- *; intros;
auto with arith.
elim 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. rewrite mult_minus_distr_r.
rewrite (mult_comm m n); rewrite (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; elim n; intros; simpl in |- *; auto with arith.
rewrite mult_plus_distr_r.
elim H; 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.
(** * Compatibility with orders *)
Lemma mult_O_le : forall n m, m = 0 \/ n <= m * n.
Proof.
induction m; simpl in |- *; 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 in |- *. apply le_n.
intros. simpl in |- *. apply plus_le_compat. assumption.
apply IHp. assumption.
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. rewrite (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 in |- *; 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.
intro m; induction m. intros. simpl in |- *. rewrite <- plus_n_O. rewrite <- plus_n_O. assumption.
intros. exact (plus_lt_compat _ _ _ _ H (IHm _ _ H)).
Qed.
Hint Resolve mult_S_lt_compat_l: arith.
Lemma mult_lt_compat_r : forall n m p, n < m -> 0 < p -> n * p < m * p.
Proof.
intros m n p H H0.
induction p.
elim (lt_irrefl _ H0).
rewrite mult_comm.
replace (n * S p) with (S p * n); auto with arith.
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. elim (le_or_lt n p). trivial.
intro H0. cut (S m * n < S m * n). intro. elim (lt_irrefl _ H1).
apply le_lt_trans with (m := S m * p). assumption.
apply mult_S_lt_compat_l. assumption.
Qed.
(** * n|->2*n and n|->2n+1 have disjoint image *)
Theorem odd_even_lem : forall p q, 2 * p + 1 <> 2 * q.
Proof.
intros p; elim p; auto.
intros q; case q; simpl in |- *.
red in |- *; intros; discriminate.
intros q'; rewrite (fun x y => plus_comm x (S y)); simpl in |- *; red in |- *;
intros; discriminate.
intros p' H q; case q.
simpl in |- *; red in |- *; intros; discriminate.
intros q'; red in |- *; intros H0; case (H q').
replace (2 * q') with (2 * S q' - 2).
rewrite <- H0; simpl in |- *; auto.
repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *; auto.
simpl in |- *; repeat rewrite (fun x y => plus_comm x (S y)); simpl in |- *;
auto.
case q'; simpl in |- *; 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 {struct 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 in |- *; auto.
intros s m; rewrite <- plus_tail_plus; rewrite <- IHp.
rewrite <- plus_assoc_reverse; apply (f_equal2 (A1:=nat) (A2:=nat)); 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 in |- *; 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 in |- *.
|
