How to compare two functions for equivalence, as in (λx.2*x) == (λx.x+x)?

Question

Is there a way to compare two functions for equality? For example, (λx.2*x) == (λx.x+x) should return true, because those are obviously equivalent.

Answer 1

This is undecidable in general, but for a suitable subset, you can indeed do it today effectively using SMT solvers:

$ ghci
GHCi, version 8.0.1: http://www.haskell.org/ghc/  :? for help
Prelude> :m Data.SBV
Prelude Data.SBV> (\x ->  2 * x) === (\x -> x + x :: SInteger)
Q.E.D.
Prelude Data.SBV> (\x ->  2 * x) === (\x -> 1 + x + x :: SInteger)
Falsifiable. Counter-example:
  s0 = 0 :: Integer

For details, see: https://hackage.haskell.org/package/sbv