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Util.v
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Util.v
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Require Import List Omega SetoidClass Recdef.
Import ListNotations.
Require Import Class.
Fixpoint even n :=
match n with
| 0 => true
| S m => negb (even m)
end.
Lemma even_odd_half : forall n, (even n = true -> exists k, 2 * k = n) /\
(even n = false -> exists k, S (2 * k) = n).
Proof.
induction n; split; intros.
- exists 0; auto.
- discriminate.
- simpl in H.
destruct IHn.
destruct H1 as [k Hk].
+ destruct (even n); [discriminate | auto].
+ exists (S k).
omega.
- destruct IHn.
destruct H0 as [k Hk].
+ simpl in H.
destruct (even n); [auto | discriminate].
+ exists k; omega.
Qed.
Lemma even_half : forall n, even n = true -> exists k, 2 * k = n.
Proof.
intros; apply even_odd_half; auto.
Qed.
Lemma odd_half : forall n, even n = false -> exists k, S (2 * k) = n.
Proof.
intros; apply even_odd_half; auto.
Qed.
Lemma even_2k : forall k, even (2 * k) = true.
Proof.
induction k.
- auto.
- simpl; rewrite <- plus_n_Sm; simpl.
simpl in IHk; rewrite IHk; auto.
Qed.
Lemma odd_2k1 : forall k, even (S (2 * k)) = false.
Proof.
induction k.
- auto.
- simpl; rewrite <- plus_n_Sm.
simpl; simpl in IHk.
rewrite IHk; auto.
Qed.
Lemma half_2k : forall k, (2*k)/2 = k.
Proof.
intro.
rewrite Nat.mul_comm.
rewrite Nat.div_mul; omega.
Qed.
Lemma half_2k1 : forall k, (S (2 * k))/2 = k.
Proof.
intro.
pose (Nat.add_b2n_double_div2 true).
simpl Nat.b2n in e.
apply e.
Qed.
Section Fin.
Fixpoint Fin(n : nat) : Type :=
match n with
| 0 => Empty_set
| S m => unit + Fin m
end.
Fixpoint Fin_le{n} : Fin n -> Fin n -> Prop :=
match n with
| 0 => fun i _ => match i with end
| S m => fun i j => match i,j with
| inl _, inl _ => False
| inl _, inr _ => True
| inr _, inl _ => False
| inr i', inr j' => Fin_le i' j'
end
end.
Lemma Fin_le_irref : forall (n : nat)(i : Fin n), ~ Fin_le i i.
Proof.
induction n.
- intros [].
- destruct i as [[]|j].
+ tauto.
+ apply IHn.
Qed.
Lemma Fin_le_trans : forall (n : nat)(i j k : Fin n), Fin_le i j -> Fin_le j k -> Fin_le i k.
Proof.
induction n; intros.
- destruct i.
- destruct i as [|i']; destruct k as [|k'].
+ destruct j; auto.
+ exact I.
+ destruct j; auto.
+ destruct j as [|j'].
* destruct H.
* apply (IHn _ j' _); auto.
Qed.
Lemma Fin_trich : forall (n : nat)(i j : Fin n), {i = j} + {Fin_le i j} + {Fin_le j i}.
Proof.
induction n.
- intros [].
- intros [[]|i'] [[]|j'].
+ left; left; auto.
+ left; right; exact I.
+ right; exact I.
+ destruct (IHn i' j') as [[Heq|Hle]|Hge].
* left; left; congruence.
* left; right; exact Hle.
* right; exact Hge.
Qed.
Fixpoint list_index{X}(xs : list X){struct xs} : Fin (length xs) -> X :=
match xs return Fin (length xs) -> X with
| [] => fun i => match i with end
| y::ys => fun i => match i with
| inl _ => y
| inr j => list_index ys j
end
end.
Lemma in_index : forall {X}`{Eq X}(xs : list X)(x : X), setoidIn x xs -> exists (i : Fin (length xs)),
list_index xs i == x.
Proof.
induction xs; intros.
- destruct H1.
- destruct H1.
+ exists (inl tt); simpl; symmetry; auto.
+ destruct (IHxs x H1) as [i Hi].
exists (inr i); auto.
Qed.
End Fin.