In Coq, is there a way to see the tactics applied by tauto?

Link:: https://stackoverflow.com/q/66812480

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Question

Is there a way to see the tactics applied by tauto? I.e., run tauto and get a list of tactics to apply (not including tauto)?

Answer

tauto is a tactic directly written in OCaml, so it does not apply other tactics - it constructs a proof term. But you can have a look at the proof term it constructs.

E.g.

Goal forall P Q : Prop, P /\ Q -> P.forall P Q : Prop, P /\ Q -> P
  tauto.
  Show Proof.(fun (P Q : Prop) (H : P /\ Q) =>
 and_ind (fun (H0 : P) (_ : Q) => H0) H)

The fun (P Q : Prop) (H : P /\ Q) corresponds to intros P Q H. Then it uses and_ind with a function as argument. This corresponds to exact (and_ind (fun P' Q' => P') H). As you can see the trick is in the construction of the function argument of and_ind.

It is instructive to look at these proof terms, but if you would do the proof manually, you would usually do it in a different way than tauto does.

And if you look at proof terms of tauto, isolate the goal you proof with tauto - otherwise the proof term will be hard to digest.