是否有可能证明oddb 0 = true？ 库克

Question

I have define an odd like this:

 Inductive odd : nat -> Prop :=
  | odd_1 : odd 1
  | odd_S : forall n:nat, odd n -> odd (n + 1).

And below is testing if a number is odd or not.

Fixpoint oddb (n : nat) { struct n } : bool :=
  match n with
  | 0 => false
  | 1 => true
  | S (S n) => oddb n
  end.

I am trying to prove this:

Theorem odd_is_odd:
  forall n, (oddb n = true) -> odd n.
Proof.
  intros n. induction n.
  - simpl. intro H. exfalso. inversion H.
  - intro H. rewrite s_is_plus_one. constructor. apply IHn. destruct n.
    * 

But i got stuck here... it needs me prove oddb 0 = true which is false = true.. is it possible???

1 subgoal
IHn : oddb 0 = true -> odd 0
H : oddb 1 = true
______________________________________(1/1)
oddb 0 = true

Below is the lemma I used

Lemma s_is_plus_one : forall n:nat, S n = n + 1.
Proof.
intros. induction n.
 - reflexivity.
 - simpl. rewrite <- IHn. reflexivity.
Qed.

Answer 1

Your definition of odd seems wrong. Is it not n+2 . Anyway our problem is related to How to prove a odd number is the successor of double of nat in coq?