Question

Testing out an "easy" example of identity types, mod equality, but transitivity proof wont type check, even from the template. More than a fix, I want to know why?

Here's a snippet of the minimal problem

data ModEq : (n:Nat) -> (x:Nat) -> (y:Nat) -> Type where
    {- 3 constructors
        reflexive:      x == x (mod n), 
        left-induct:    x == y (mod n) => (x+n) == y    (mod n)
        right-induct:   x == y (mod n) => x == (y+n)    (mod n)
    -}
  Reflex : (x:Nat) -> ModEq n x x  --- Needs syntatic sugar, for now prefix
  LInd : (ModEq n x y) -> ModEq n (x+n) y
  RInd : (ModEq n x y) -> ModEq n x (y+n)

{----- Proof of transitive property. -----}
total
isTrans : (ModEq n x y) -> (ModEq n y z) -> (ModEq n x z)
{- x=x & x=y => x=y -}
isTrans (Reflex x) v = v
isTrans u (Reflex y) = u
{- ((x=y=>(x+n)=y) & y=z) => x=y & y=z => x=z (induct) => (x+n)=z -}
isTrans (LInd u) v = LInd (isTrans u v)
isTrans u (RInd v) = RInd (isTrans u v)
{- (x=y=>x=(y+n)) & (y=z=>(y+n)=z) => x=y & y=z => x=z (induct) -}
isTrans (RInd u) (LInd v) = isTrans u v

The type mismatch is in the last line, even though from the comment line I really cannot tell why logically its wrong. Here's the error:

48 | isTrans (RInd u) (LInd v) = isTrans u v
   | ~~~~~~~
When checking left hand side of isTrans:
When checking an application of Main.isTrans:
        Type mismatch between
                ModEq n (x + n) z (Type of LInd v)
        and
                ModEq n (y + n) z (Expected type)

        Specifically:
                Type mismatch between
                        plus x n
                and
                        plus y n

Not only am I confused by how LInd v got assigned the (wrong seeming) type ModEq n (x+n) z, but I point out that when I simply try the "type-define-refine" approach with the built in template I get:

isTrans : (ModEq n x y) -> (ModEq n y z) -> (ModEq n x z)
isTrans (RInd _) (LInd _) = ?isTrans_rhs_1

And even this wont type-check, it complains:

40 | isTrans (RInd _) (LInd _) = ?isTrans_rhs_1
   | ~~~~~~~
When checking left hand side of isTrans:
When checking an application of Main.isTrans:
        Type mismatch between
                ModEq n (x + n) z (Type of LInd _)
        and
                ModEq n (y + n) z (Expected type)

        Specifically:
                Type mismatch between
                        plus x n
                and
                        plus y n

Answer 1

问题在于，在最后一种情况下，编译器无法推断出y = {x + n}。不过，您可以给它提示：

isTrans : (ModEq n x y) -> (ModEq n y z) -> (ModEq n x z)
isTrans (Reflex _) v = v
isTrans u (Reflex _) = u
isTrans (LInd u) v = LInd $ isTrans u v
isTrans u (RInd v) = RInd $ isTrans u v
isTrans (RInd u) (LInd v) {n} {x} {y = x + n} = ?isTrans_rhs

您为isTrans_rhs设定了以下目标：

  x : Nat
  n : Nat
  u : ModEq n x x
  z : Nat
  v : ModEq n x z
--------------------------------------
isTrans_rhs : ModEq n x z

因此，您可以得出isTrans (RInd u) (LInd v) {n} {x} {y = x + n} = v

Idris Type mismatch that occurs even from template

1 个答案: