How do I simplify a hypothesis of the form True -> P in Coq?

Question

If I have a hypothesis in my proof context that looks like H: True -> P and I want to convert it to H: P , what's the easiest way to do this? I tried simpl in H but it does nothing, and the only way I've found is the extremely unsatisfactory pose proof (H Coq.Init.Logic.I) as H . Isn't there a simpler way?

Answer 1

There are two ways to work with this, besides using pose proof .

Using specialize .

This tactics allows you to provide arguments to you hypotheses. In you case you could do

specialize (H I).

or even

specialize H with (1 := I).

and you can use as if you want to create a copy rather than instantiate H directly.

Using forward .

I think this is what you want here. forward H. will ask you to prove the first hypothesis of H . So you will do something like this:

forward H.
- auto.
- (* Then resume with H : P *)

but you can also provide it with a (goal-closing) tactic:

forward H by auto.
(* Now you have one goal, and H has type P *)

forward is—as of yet—not part of the standard library. It can however be defined pretty easily (here is the definition from the MetaCoq library).

Ltac forward_gen H tac :=
  match type of H with
  | ?X -> _ => let H' := fresh in assert (H':X) ; [tac|specialize (H H'); clear H']
  end.

Tactic Notation "forward" constr(H) := forward_gen H ltac:(idtac).
Tactic Notation "forward" constr(H) "by" tactic(tac) := forward_gen H tac.

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Note that simpl here doesn't work because it's not really a tactic to simplify hypotheses in the usual sense, it's really just a tactic to apply some computation rules, it basically evaluates the goal or the hypothesis you apply it on. True -> P does not reduce to P because it would then take one fewer argument.