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
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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
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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.
cut (forall n, P n /\ P (S n)).
intros. elim (H2 n). auto with arith.
induction n0. auto with arith.
intros. elim IHn0; auto with arith.
Qed.
0 <n => n/2 < n
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Lemma lt_div2 : forall n, 0 < n -> div2 n < n.
Proof.
intro n. pattern n in |- *. apply ind_0_1_SS.
intro. inversion H.
auto with arith.
intros. simpl in |- *.
case (zerop n0).
intro. rewrite e. 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.
split. split; auto with arith.
split. intro H. inversion H.
intro H. absurd (S (div2 0) = div2 1); auto with arith.
split. split. intro. inversion H. inversion H1.
intro H. absurd (div2 1 = div2 2).
simpl in |- *. discriminate. assumption.
split; auto with arith.
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)
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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.
split; split; auto with arith.
intro H. inversion H.
split; split; auto with arith.
intro H. inversion H. inversion H1.
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:
(Immediate: it is n/2)
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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.