11 - Scala. Function type

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

2. Sets
Relation
R ⊂ A × B
a R b ⇐⇒ (a, b) ∈ R
Function
f : A → B
y = f (x) ⇐⇒ x f y ⇐⇒ (x, y) ∈ f
∀x ∈ A ∃! y ∈ B y = f (x)
Dmytro Mitin Function type

3. Function type
1 Type Formation
Γ A : ∗ Γ B : ∗
Γ A → B : ∗
2 Constructor
Γ, x : A b : B Γ B : ∗
Γ λx.b : A → B
3 Eliminator
Γ f : A → B Γ a : A
Γ f a : B
Dmytro Mitin Function type

4. Function type
4 Computation rule (β-reduction)
Γ, x : A b : B Γ a : A Γ B : ∗
Γ (λx.b) a ≡ b[x ← a] : B
5 Uniqueness principle (η-conversion)
Γ f : A → B
Γ λx. f x ≡ f : A → B
Dmytro Mitin Function type

5. Haskell, Scala and Java
Haskell
f :: a -> b
f = undefined
Scala
def f[A, B](x: A): B = ???
def g[A, B]: Function1[A, B] = ???
def h[A, B]: A => B = ???
Java
public <A, B> B f(A x) {
return null;
}
public <A, B> Function<A, B> g() {
return null;
}
import java.util.function.Function;
Dmytro Mitin Function type

6. ProvingGround
git clone https://github.com/siddhartha-gadgil/ProvingGround.git
cd ProvingGround
sbt mantle/test:console
val A = "A" :: Type
val B = "B" :: Type
val f = "f" :: A ->: B
val a = "a" :: A
val b = f (a) !: B
val g = a :-> a !: A ->: A
import ScalaRep.
import NatInt.rep
val h = ((x: Int) => x * x).term !: NatInt ->: NatInt
h(2.term) == 4.term
Dmytro Mitin Function type