0

When I was trying to prove if two functions are equivalent and come to the step:

I : f(S a') b = S (f a' b)

f (S a') (S b) = S (f a' (S b))

I am wondering whether it's possible to use exact(I) to prove it, namely, to replace (S b) by b, since that's the only difference.

Serene
  • 1
  • 1

1 Answers1

1

The inference from the premise to the conclusion is generally false for an arbitrary function f: consider the function f a b such that f a 0 := a and f a (S b) := S b, you can prove the premise and contradict the conclusion.

You could substitute b in I only if it was quantified universally in that hypotesis: I : forall b, f (S a') b = S (f a' b) ; in that case substituting would amount to application of I to S b.

If it's not possible to strenghten your hypothesis, you need to use something specific to the function f to conclude.

kyo dralliam
  • 1,197
  • 5
  • 6