How to specialize nested hypotheses in Coq?

Question

I've already got the idea of specialize in Coq.

It works okay with specialize (H1 trm_int).

H1 : forall x, value x -> term x.

But what about this case

H1 : forall x, value x -> forall y, value y -> sub x y.

It doesn't work with specialize (H1 trm_int _trm_int).

Edit:

specialize (H1 trm_int _trm_int) works fine with

H1 : forall x y, value x -> value y -> sub x y.

Notice the forall y appears in the second premise .

I'm sorry but what doesn't work? – Théo Winterhalter Jan 16 '21 at 07:55 — Théo Winterhalter, Jan 16 '21 at 07:55
@ThéoWinterhalter Hi I updated description, plz check – DoubleX Jan 16 '21 at 08:58 — DoubleX, Jan 16 '21 at 08:58

score 1 · Answer 1 · answered Jan 16 '21 at 12:34

I am not sure I understand your problem. specialize will apply your hypothesis to the arguments you give it. It is not limited to forall quantifiers.

Say you have

h : nat -> bool

you can also use specialize (h 0) and you will get

h : bool

So when you have

H1 : forall x, value x -> forall y, value y -> sub x y.

and you specialise, you have to use specialize (H1 some_x some_value_x some_y) where some_x is an instance for x, some_value_x is a proof of value some_x and some_y is an instance for y.

Note that there is an alternative using with that allows you to specialise using names. Please consider the following example

Lemma foo :
  forall (h : forall x, x = 0 -> forall y, y = x -> x = y),
    True.
Proof.
  intro h.
  specialize h with (x := 0) (y := 0).

Here you will end up with

h: 0 = 0 -> 0 = 0 -> 0 = 0

But you could also give the proof directly with

specialize (h 0 eq_refl 0).

How to specialize nested hypotheses in Coq?

1 Answers1