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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: Div2.v 9245 2006-10-17 12:53:34Z notin $ i*)
Require Import Lt.
Require Import Plus.
Require Import Compare_dec.
Require Import Even.
Open Local Scope nat_scope.
Implicit Type n : nat.
(** Here we define [n/2] and prove some of its properties *)
Fixpoint div2 n : nat :=
match n with
| O => 0
| S O => 0
| S (S n') => S (div2 n')
end.
(** Since [div2] is recursively defined on [0], [1] and [(S (S n))], it is
useful to prove the corresponding induction principle *)
Lemma ind_0_1_SS :
forall P:nat -> Prop,
P 0 -> P 1 -> (forall n, P n -> P (S (S n))) -> forall n, P n.
Proof.
intros P H0 H1 Hn.
cut (forall n, P n /\ P (S n)).
intros H'n n. elim (H'n n). auto with arith.
induction n. auto with arith.
intros. elim IHn; auto with arith.
Qed.
(** [0 <n => n/2 < n] *)
Lemma lt_div2 : forall n, 0 < n -> div2 n < n.
Proof.
intro n. pattern n in |- *. apply ind_0_1_SS.
(* n = 0 *)
inversion 1.
(* n=1 *)
simpl; trivial.
(* n=S S n' *)
intro n'; case (zerop n').
intro n'_eq_0. rewrite n'_eq_0. auto with arith.
auto with arith.
Qed.
Hint Resolve lt_div2: arith.
(** Properties related to the parity *)
Lemma even_odd_div2 :
forall n,
(even n <-> div2 n = div2 (S n)) /\ (odd n <-> S (div2 n) = div2 (S n)).
Proof.
intro n. pattern n in |- *. apply ind_0_1_SS.
(* n = 0 *)
split. split; auto with arith.
split. intro H. inversion H.
intro H. absurd (S (div2 0) = div2 1); auto with arith.
(* n = 1 *)
split. split. intro. inversion H. inversion H1.
intro H. absurd (div2 1 = div2 2).
simpl in |- *. discriminate. assumption.
split; auto with arith.
(* n = (S (S n')) *)
intros. decompose [and] H. unfold iff in H0, H1.
decompose [and] H0. decompose [and] H1. clear H H0 H1.
split; split; auto with arith.
intro H. inversion H. inversion H1.
change (S (div2 n0) = S (div2 (S n0))) in |- *. auto with arith.
intro H. inversion H. inversion H1.
change (S (S (div2 n0)) = S (div2 (S n0))) in |- *. auto with arith.
Qed.
(** Specializations *)
Lemma even_div2 : forall n, even n -> div2 n = div2 (S n).
Proof fun n => proj1 (proj1 (even_odd_div2 n)).
Lemma div2_even : forall n, div2 n = div2 (S n) -> even n.
Proof fun n => proj2 (proj1 (even_odd_div2 n)).
Lemma odd_div2 : forall n, odd n -> S (div2 n) = div2 (S n).
Proof fun n => proj1 (proj2 (even_odd_div2 n)).
Lemma div2_odd : forall n, S (div2 n) = div2 (S n) -> odd n.
Proof fun n => proj2 (proj2 (even_odd_div2 n)).
Hint Resolve even_div2 div2_even odd_div2 div2_odd: arith.
(** Properties related to the double ([2n]) *)
Definition double n := n + n.
Hint Unfold double: arith.
Lemma double_S : forall n, double (S n) = S (S (double n)).
Proof.
intro. unfold double in |- *. simpl in |- *. auto with arith.
Qed.
Lemma double_plus : forall n (m:nat), double (n + m) = double n + double m.
Proof.
intros m n. unfold double in |- *.
do 2 rewrite plus_assoc_reverse. rewrite (plus_permute n).
reflexivity.
Qed.
Hint Resolve double_S: arith.
Lemma even_odd_double :
forall n,
(even n <-> n = double (div2 n)) /\ (odd n <-> n = S (double (div2 n))).
Proof.
intro n. pattern n in |- *. apply ind_0_1_SS.
(* n = 0 *)
split; split; auto with arith.
intro H. inversion H.
(* n = 1 *)
split; split; auto with arith.
intro H. inversion H. inversion H1.
(* n = (S (S n')) *)
intros. decompose [and] H. unfold iff in H0, H1.
decompose [and] H0. decompose [and] H1. clear H H0 H1.
split; split.
intro H. inversion H. inversion H1.
simpl in |- *. rewrite (double_S (div2 n0)). auto with arith.
simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith.
intro H. inversion H. inversion H1.
simpl in |- *. rewrite (double_S (div2 n0)). auto with arith.
simpl in |- *. rewrite (double_S (div2 n0)). intro H. injection H. auto with arith.
Qed.
(** Specializations *)
Lemma even_double : forall n, even n -> n = double (div2 n).
Proof fun n => proj1 (proj1 (even_odd_double n)).
Lemma double_even : forall n, n = double (div2 n) -> even n.
Proof fun n => proj2 (proj1 (even_odd_double n)).
Lemma odd_double : forall n, odd n -> n = S (double (div2 n)).
Proof fun n => proj1 (proj2 (even_odd_double n)).
Lemma double_odd : forall n, n = S (double (div2 n)) -> odd n.
Proof fun n => proj2 (proj2 (even_odd_double n)).
Hint Resolve even_double double_even odd_double double_odd: arith.
(** Application:
- if [n] is even then there is a [p] such that [n = 2p]
- if [n] is odd then there is a [p] such that [n = 2p+1]
(Immediate: it is [n/2]) *)
Lemma even_2n : forall n, even n -> {p : nat | n = double p}.
Proof.
intros n H. exists (div2 n). auto with arith.
Qed.
Lemma odd_S2n : forall n, odd n -> {p : nat | n = S (double p)}.
Proof.
intros n H. exists (div2 n). auto with arith.
Qed.
(** Doubling before dividing by two brings back to the initial number. *)
Lemma div2_double : forall n:nat, div2 (2*n) = n.
Proof.
induction n.
simpl; auto.
simpl.
replace (n+S(n+0)) with (S (2*n)).
f_equal; auto.
simpl; auto with arith.
Qed.
Lemma div2_double_plus_one : forall n:nat, div2 (S (2*n)) = n.
Proof.
induction n.
simpl; auto.
simpl.
replace (n+S(n+0)) with (S (2*n)).
f_equal; auto.
simpl; auto with arith.
Qed.
|
