Liquid Haskell: Error with Proof Combinators and Types Refined by Predicates

Question

As a minimal example of the problem I'm having, here's a definition of natural numbers, a doubling function, and a type refined by an even-ness predicate:

data Nat' = Z | S Nat' deriving Show

{-@ reflect double' @-}
double' :: Nat' -> Nat'
double' Z = Z
double' (S x) = (S (S (double' x)))

{-@ type Even' = {v:Nat' | even' v} @-}

{-@ reflect even' @-}
even' :: Nat' -> Bool
even' Z = True
even' (S Z) = False
even' (S (S x)) = even' x

I'd like to first declare {-@ double' :: Nat' -> Even' @-} and then prove this to be true, but I'm under the impression that I instead must first write the proof and then use castWithTheorem (which itself has worked for me) as such:

{-@ even_double :: x:Nat' -> {even' (double' x)} @-}
even_double Z =     even' (double' Z) 
              ==.   even' Z  
              ==.   True 
              ***   QED
even_double (S x) =     even' (double' (S x))
                  ==.   even' (S (S (double' x)))
                  ==.   even' (double' x)
                  ?     even_double x
                  ==.   True
                  ***   QED

{-@ double :: Nat' -> Even' @-}
double x = castWithTheorem (even_double x) (double' x)

However, this gives fairly illegible errors like:

:1:1-1:1: Error
  elaborate solver elabBE 177 "lq_anf$##7205759403792806976##d3tK" {lq_tmp$x##1556 : (GHC.Types.$126$$126$ (GHC.Prim.TYPE GHC.Types.LiftedRep) (GHC.Prim.TYPE GHC.Types.LiftedRep) bool bool) | [(lq_tmp$x##1556 = GHC.Types.Eq#)]} failed on:
      lq_tmp$x##1556 == GHC.Types.Eq#
  with error
      Cannot unify (GHC.Types.$126$$126$ (GHC.Prim.TYPE GHC.Types.LiftedRep) (GHC.Prim.TYPE GHC.Types.LiftedRep) bool bool) with func(0 , [(GHC.Prim.$126$$35$ @(42) @(43) @(44) @(45));
          (GHC.Types.$126$$126$ @(42) @(43) @(44) @(45))]) in expression: lq_tmp$x##1556 == GHC.Types.Eq# 
  because
Elaborate fails on lq_tmp$x##1556 == GHC.Types.Eq#
  in environment
      GHC.Types.Eq# := func(4 , [(GHC.Prim.$126$$35$ @(0) @(1) @(2) @(3));
                                 (GHC.Types.$126$$126$ @(0) @(1) @(2) @(3))])

      lq_tmp$x##1556 := (GHC.Types.$126$$126$ (GHC.Prim.TYPE GHC.Types.LiftedRep) (GHC.Prim.TYPE GHC.Types.LiftedRep) bool bool)

What am I doing wrong? From my experiments, it seems to be caused by trying to prove that some predicate function is true of some argument.

Answer 1

The issue was that I should have been using NewProofCombinators instead of ProofCombinators. Then replacing ==. with === and castWithTheorem (even_double x) (double' x) with (double' x) `withProof` (even_double x) fixes the problem: http://goto.ucsd.edu:8090/index.html#?demo=permalink%2F1543595949_5844.hs

All the online resources I've found use ProofCombinators so hopefully this saves someone some pain.

Source: https://github.com/ucsd-progsys/liquidhaskell/issues/1378#issuecomment-443262472