even_Sn_not_even_n - apply 1 hypothesis in another

Question

Unfortunately I got stuck again:

Inductive even : nat > Prop :=
| ev_0 : even 0
| ev_SS (n : nat) (H : even n) : even (S (S n)).

Lemma even_Sn_not_even_n : forall n,
    even (S n) <-> not (even n).
Proof.
  intros n. split.
  + intros H. unfold not. intros H1. induction H1 as [|n' E' IHn].
    - inversion H.
    - inversion_clear H. apply IHn in H0. apply H0.
  + intros H. induction n as [|n' IHn].
    - exfalso. apply H. apply ev_0.
    - apply evSS_inv'.

Here is the result:

1 subgoal (ID 179)

n' : nat
H : ~ even (S n')
IHn : ~ even n' -> even (S n')
============================
even n'

As far I could prove it in words:

(n' + 1) is not even according to H. Therefore according to IHn, it is not true that n' is not even (double negation):

IHn : ~ ~ even n'

Unfolding double negation, we conclude that n' is even.

But how to write it in coq?

Answer 1

The usual way to strip double negation is to introduce the "excluded middle" axiom, which is defined under the name classic in Coq.Logic.Classical_Prop, and apply the lemma NNPP.

However, in this particular case, you can use the technique called reflection by showing that the Prop is consistent with a boolean function (you might remember the evenb function introduced earlier in the book).

(Assuming you're at the beginning of IndProp) You'll soon see the following definition later in that chapter:

Inductive reflect (P : Prop) : bool -> Prop :=
| ReflectT (H : P) : reflect P true
| ReflectF (H : ~ P) : reflect P false.

You can prove the statement

Lemma even_reflect : forall n : nat, reflect (even n) (evenb n).

and then use it to move between a Prop and a boolean (which contain the same information i.e. the (non-)evenness of n) at the same time. This also means that you can do classical reasoning on that particular property without using the classic axiom.

I suggest to complete the exercises under Reflection section in IndProp, and then try the following exercises. (Edit: I uploaded the full answer here.)

(* Since `evenb` has a nontrivial recursion structure, you need the following lemma: *)
Lemma nat_ind2 :
  forall P : nat -> Prop,
  P 0 -> P 1 -> (forall n : nat, P n -> P (S (S n))) -> forall n : nat, P n.
Proof. fix IH 5. intros. destruct n as [| [| ]]; auto.
  apply H1. apply IH; auto. Qed.

(* This is covered in an earlier chapter *)
Lemma negb_involutive : forall x : bool, negb (negb x) = x.
Proof. intros []; auto. Qed.

(* This one too. *)
Lemma evenb_S : forall n : nat, evenb (S n) = negb (evenb n).
Proof. induction n.
  - auto.
  - rewrite IHn. simpl. destruct (evenb n); auto. Qed.

(* Exercises. *)
Lemma evenb_even : forall n : nat, evenb n = true -> even n.
Proof. induction n using nat_ind2.
  (* Fill in here *) Admitted.

Lemma evenb_odd : forall n : nat, evenb n = false -> ~ (even n).
Proof. induction n using nat_ind2.
  (* Fill in here *) Admitted.

Lemma even_reflect : forall n : nat, reflect (even n) (evenb n).
Proof. (* Fill in here. Hint: You don't need induction. *) Admitted.

Lemma even_iff_evenb : forall n, even n <-> evenb n = true.
Proof. (* Fill in here. Hint: use `reflect_iff` from IndProp. *) Admitted.

Theorem reflect_iff_false : forall P b, reflect P b -> (~ P <-> b = false).
Proof. (* Fill in here. *) Admitted.

Lemma n_even_iff_evenb : forall n, ~ (even n) <-> evenb n = false.
Proof. (* Fill in here. *) Admitted.

Lemma even_Sn_not_even_n : forall n,
    even (S n) <-> not (even n).
Proof. (* Fill in here.
  Hint: Now you can convert all the (non-)evenness properties to booleans,
  and then work with boolean logic! *) Admitted.