Question

I have a function “HF” has type inside section S

Open S.
HF: forall f : dup_sig Sig, dup_ar f = ASignature.arity (F f)
End S.

Signature: Type
Sig: Signature    
dp_Sig : Signature
dup_sig : Signature -> Signature
F : dup_sig Sig -> Sig
dup_symb : Signature -> Type
dup_ar : forall Sig : Signature, dup_symb Sig -> nat
ASignature.arity : forall s : Signature, s -> nat

I want to write a lemma:

Lemma incl_fl : forall R R, Fl HF R [= R'.

Where

Fl: forall (S1 S2 : Signature) (F : S1 -> S2),
       (forall f : S1, ASignature.arity f = ASignature.arity (F f)) ->
       list (ATrs.rule S1) -> list (ATrs.rule S2)

It is Ok when I have this function inside the section S.

But I would like to write the function incl_fl outside the section S.
Here is the type of HF outside section S.

HF: forall (arity : symbol -> nat) (f : dup_sig (Sig arity)),
    dup_ar f = ASignature.arity (F f)

Lemma incl_fl : forall arity R R', Fl HF R [= R'.

I got an error at “HF” :

The term "HF" has type
 "forall (arity : symbol -> nat) (f : dup_sig (Sig arity)),
  dup_ar f = ASignature.arity (F f)" while it is expected to have type
 "forall f : ?35524, ASignature.arity f = ASignature.arity (?35526 f)".

I tried to find a place to add the arity in this function “HF“, but I don’t success.
Could you please help me write the lemma “incl_fl” outside the section S? Thank you very much.

Answer 1

I found the answer.

Lemma incl_Fl : forall arity R R', Fl (HF (arity:=arity)) R [= R'.