✔ Stuck with a seemingly simple proof. · Coq users

Stream: Coq users

Topic: ✔ Stuck with a seemingly simple proof.

Lessness (Aug 07 2022 at 09:49):

Inductive vector A: nat -> Type :=
  | vnil: vector A 0
  | vcons: forall (n: nat), A -> vector A n -> vector A (S n).

Theorem test A (v: vector A 0): v = vnil A.
Proof.
Admitted.

Thanks in advance.

Gaëtan Gilbert (Aug 07 2022 at 10:03):

  set (n := 0) in v.
  change (match n return vector A n -> Prop with
          | 0 => fun v => v = vnil A
          | S _ => fun _ => True
          end v).
  destruct v;trivial.

Lessness (Aug 07 2022 at 10:08):

Thank you very much!

Yann Leray (Aug 07 2022 at 10:20):

You can even let Coq fill in the S _ case with something like exact (match v in vector _ 0 => vnil _ => eq_refl end).

Notification Bot (Aug 07 2022 at 12:15):

Lessness has marked this topic as resolved.

Malcolm Sharpe (Aug 08 2022 at 07:07):

Turns out even the in is not needed, so it comes out to the remarkably concise:

Inductive vector A: nat -> Type :=
  | vnil: vector A 0
  | vcons: forall (n: nat), A -> vector A n -> vector A (S n).

Theorem test A (v: vector A 0): v = vnil A.
Proof.
  exact (match v with vnil _ => eq_refl end).
Qed.

(I'm just a beginner, so I'm still working on understanding why this works...)

Last updated: Oct 13 2024 at 01:02 UTC