Either and Dec

In brief: A ⊎ B is data Either. Dec P is P ⊎ (P → Empty): a decision for a particular P, not LEM. There is no inhabitant of Π (P : Type) → Dec P.

Either

Declare it in the book. The infix ⊎ desugars to Either A B. ASCII is the name Either.

data Either (A : Type) (B : Type) : Type where
  left  : A → Either A B
  right : B → Either A B

left and right are checked against an expected Either. Match has an explicit motive, same shape as other data.

A run sum (examples/either_run.muro):

def fromLeft : run Π (e : Either Nat Unit) → Nat :=
  λ (e : Either Nat Unit) →
    match e motive (λ (_ : Either Nat Unit) → Nat)
      | left n => n
      | right _ => 0

Emit: {:left, n} / {:right, t}.

Either is not Data. Closures P → Empty are not Data. No + on a refutation.

Dec is a spec

def Dec : spec Π (P : Type) → Type :=
  λ (P : Type) → Either P (Π (_ : P) → Empty)

A value of Dec P is either a witness of P or a function that turns a witness into Empty. That is a decision for this P. It is not an axiom that every type is decidable.

There is no inhabitant of Π (P : Type) → Dec P.

evenDec

examples/even_dec.muro decides IsEven n by matching on n.

• 0 — left tt. Even.
• suc 0 — right of a function that matchEmptys the evidence. Odd.
• suc (suc p) — reuse evenDec p and keep the same injection.

def evenDec : evidence Π (n : Nat) → Dec (IsEven n) :=
  λ (n : Nat) →
    match n motive (λ (x : Nat) → Dec (IsEven x))
      | 0 => left tt
      | suc n1 =>
        match n1 motive (λ (y : Nat) → Dec (IsEven suc(y)))
          | 0 => right (λ (e : IsEven suc(0)) → matchEmpty e motive (λ _ → Empty))
          | suc p =>
            match evenDec p motive (λ (d : Dec (IsEven p)) → Dec (IsEven suc(suc(p))))
              | left e  => left e
              | right c => right c

evenDec is evidence. It is not emitted. You can still check it:

mix muro.check examples/even_dec.muro

Next: Machine numbers.