An assumption in the goal is completely the same with a function I defined beforehand. How can I tell Coq they are indeed the same?

Question

My current goal is like this:

   1 subgoal
    fun1 : forall W : Type, list W -> list W -> list W
    H_fun1 : S_fun fun1
    V : Type
    h : V
    t : list V
    ______________________________________(1/1)
    fun2 V (fun3 fun2 V t) (h :: nil) =
    fun1 V (fun3 fun2 V t) (h :: nil)

fun2 is a function I defined before which has type forall W : Type, list W -> list W -> list W and also satisfy S_fun fun2. How can I replace fun2 by fun1 or replace fun1 by fun2 or tell Coq fun2 and fun1 are actually the same to use reflexivity to finish the proof?

score 1 · Answer 1 · answered Sep 13 '21 at 01:53

In general, it is not possible to substitute fun2 this way, because fun1 could actually be different. However, it might be the case that the S_fun hypothesis implies that fun1 and fun2 are equal for all inputs, in which case we could conclude. Could you give us some more details about your example?

An assumption in the goal is completely the same with a function I defined beforehand. How can I tell Coq they are indeed the same?

1 Answers1