10 - Scala. Co-product type (sum type)

1. Sum type (coproduct)
Dmytro Mitin
https://stepik.org/course/Introduction-to-programming-
with-dependent-types-in-Scala-2294/
March 2017
Dmytro Mitin Sum type (coproduct)

2. Sets
Union
A ∪ B = {x | x ∈ A ∨ x ∈ B}
Disjoint union
A B = (0, a) a ∈ A ∪ (1, b) b ∈ B
= {0} × A ∪ {1} × B
Dmytro Mitin Sum type (coproduct)

3. Sum type (coproduct)
1 Type Formation
Γ A : ∗ Γ B : ∗
Γ A + B : ∗
2 Constructors
Γ a : A Γ B : ∗
Γ inl a : A + B
Γ A : ∗ Γ b : B
Γ inr b : A + B
3 Eliminator
Γ C : ∗ Γ, a : A c1 : C Γ, b : B c2 : C
Γ, s : A + B case+(s)(λa.c1, λb.c2) : C
s match { case Inl(a) => c1(a); case Inr(b) => c2(b) }
Dmytro Mitin Sum type (coproduct)

4. Sum type (coproduct)
4 Computation rules (”β-reduction“)
Γ C : ∗ Γ, a : A c1 : C Γ, b : B c2 : C Γ a : A
Γ case+(inl a )(λa.c1, λb.c2) ≡ c1[a ← a ] : C
Γ C : ∗ Γ, a : A c1 : C Γ, b : B c2 : C Γ b : B
Γ case+(inr b )(λa.c1, λb.c2) ≡ c2[b ← b ] : C
5 Uniqueness principle (”η-conversion“)
Dmytro Mitin Sum type (coproduct)

5. Haskell, Scala and Java
Haskell
data Sum a b = Inl a | Inr b
Scala
sealed trait Sum[A, B]
case class Inl[A, B](a: A) extends Sum[A, B]
case class Inr[A, B](b: B) extends Sum[A, B]
Java
public interface Sum<A, B> {}
public class Inl<A, B> implements Sum<A, B> {
private final A a;
}
public class Inr<A, B> implements Sum<A, B> {
private final B b;
}
Dmytro Mitin Sum type (coproduct)

6. ProvingGround. Built-in type
git clone https://github.com/siddhartha-gadgil/ProvingGround.git
cd ProvingGround
sbt mantle/test:console
val A = "A" :: Type
val B = "B" :: Type
val a = "a" :: A
val b = "b" :: B
val a1 = PlusTyp(A, B).incl1(a) !: PlusTyp(A, B)
val b1 = PlusTyp(A, B).incl2(b) !: PlusTyp(A, B)
val recAorBA = PlusTyp(A, B).rec(A)
val recAorBB = PlusTyp(A, B).rec(B)
val a0 = "a0" :: A
val b0 = "b0" :: B
val idAOrDefault = recAorBA(a :-> a)(b :-> a0)
val idBOrDefault = recAorBB(a :-> b0)(b :-> b)
Dmytro Mitin Sum type (coproduct)

7. ProvingGround. Built-in type
idAOrDefault(a1) == a
idAOrDefault(b1) == a0
idBOrDefault(a1) == b0
idBOrDefault(b1) == b
val recAorBAorB = PlusTyp(A, B).rec(PlusTyp(A, B))
val id = recAorBAorB(a :-> PlusTyp(A, B).incl1(a))(b :->
PlusTyp(A, B).incl2(b))
id(a1) == a1
id(b1) == b1
Dmytro Mitin Sum type (coproduct)

8. ProvingGround. Custom type
import TLImplicits.
import shapeless.
val A = "A" :: Type
val B = "B" :: Type
val a = "a" :: A
val b = "b" :: B
val AorB = "A + B" :: Type
val SumInd = ("inl" ::: A ->>: AorB) |: ("inr" ::: B ->>: AorB)
=: AorB
val inl :: inr :: HNil = SumInd.intros
val a1 = inl(a) !: AorB
val b1 = inr(b) !: AorB
val recAorBA = SumInd.rec(A)
val recAorBB = SumInd.rec(B)
Dmytro Mitin Sum type (coproduct)

9. ProvingGround. Custom type
val a0 = "a0" :: A
val b0 = "b0" :: B
val idAOrDefault = recAorBA(a :-> a)(b :-> a0)
val idBOrDefault = recAorBB(a :-> b0)(b :-> b)
idAOrDefault(a1) == a
idAOrDefault(b1) == a0
idBOrDefault(a1) == b0
idBOrDefault(b1) == b
val recAorBAorB = SumInd.rec(AorB)
val id = recAorBAorB(a :-> inl(a))(b :-> inr(b))
id(a1) == a1
id(b1) == b1
Dmytro Mitin Sum type (coproduct)