Introduction to programming with dependent types in Scala
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This document introduces programming with dependent types in Scala. It discusses values and types, functions that relate values to types or types to other types, dependent pairs and fibers, using types for calculations at compile-time, and theorem proving using dependent types.
![Value functions, type families and dependent types
Value-level function (value depending on another value)
(x: Int) => x * x, (x: String) => x + "ab", . . .
Type family (type depending on another type)
Seq[Int], Seq[Boolean], Seq[String], . . .
List[Int], List[Boolean], List[String], . . .
Dependent type (type depending on value of another type)
Sized[Seq[String], 0], Sized[Seq[String], 1], Sized[Seq[String], 2], . . .
val x: Seq[String] = Seq("a", "b", "c")
val x1: Seq[String] = Seq("a", "b", "c", "d")
import shapeless.Sized // Vector type
val x2 : Sized[Seq[String], 3] = Sized("a", "b", "c")
val x3 : Sized[Seq[String], 3] = Sized("a", "b", "c", "d")
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-3-320.jpg)
![Homotopy type theory (HoTT)
https://homotopytypetheory.org/book/
Fiber bundle
Dependent pair type
mkPair(n, vn) !: Sgma(n !: Nat, Vec(n))
( 3, Sized("a", "b", "c")) : Σ(n : Nat, Sized[Seq[String], n])
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-4-320.jpg)
![Type-level programming
Types
True, False, 0, 1, 2, . . . , Int, Boolean, String, . . .
Values
True, False, 0, 1, 2, . . .
Type-level calculations (compile time)
implicitly[(True# && [False])# || [True] =:= True]
implicitly[( 1# + [ 2])# * [ 3] =:= 9]
Value-level calculations (run time)
(true && false) || true == true
(True.&&(False)).||(True) == True
(1 + 2) * 3 == 9
( 1.+( 2)).*( 3) == 9
Dmytro Mitin Introduction to programming with dependent types in Scala](https://image.slidesharecdn.com/intro-170404145715/85/Introduction-to-programming-with-dependent-types-in-Scala-5-320.jpg)
Introduction to programming with dependent types in Scala
- 1. Introduction to programming with dependent types in Scala Dmytro Mitin https://stepik.org/2294 April 2017 Dmytro Mitin Introduction to programming with dependent types in Scala
- 2. Values and types Types Int, Boolean, String, . . . Values 10, true, "abc", . . . 10 : Int true : Boolean "abc" : String . . . Dmytro Mitin Introduction to programming with dependent types in Scala
- 3. Value functions, type families and dependent types Value-level function (value depending on another value) (x: Int) => x * x, (x: String) => x + "ab", . . . Type family (type depending on another type) Seq[Int], Seq[Boolean], Seq[String], . . . List[Int], List[Boolean], List[String], . . . Dependent type (type depending on value of another type) Sized[Seq[String], 0], Sized[Seq[String], 1], Sized[Seq[String], 2], . . . val x: Seq[String] = Seq("a", "b", "c") val x1: Seq[String] = Seq("a", "b", "c", "d") import shapeless.Sized // Vector type val x2 : Sized[Seq[String], 3] = Sized("a", "b", "c") val x3 : Sized[Seq[String], 3] = Sized("a", "b", "c", "d") Dmytro Mitin Introduction to programming with dependent types in Scala
- 4. Homotopy type theory (HoTT) https://homotopytypetheory.org/book/ Fiber bundle Dependent pair type mkPair(n, vn) !: Sgma(n !: Nat, Vec(n)) ( 3, Sized("a", "b", "c")) : Σ(n : Nat, Sized[Seq[String], n]) Dmytro Mitin Introduction to programming with dependent types in Scala
- 5. Type-level programming Types True, False, 0, 1, 2, . . . , Int, Boolean, String, . . . Values True, False, 0, 1, 2, . . . Type-level calculations (compile time) implicitly[(True# && [False])# || [True] =:= True] implicitly[( 1# + [ 2])# * [ 3] =:= 9] Value-level calculations (run time) (true && false) || true == true (True.&&(False)).||(True) == True (1 + 2) * 3 == 9 ( 1.+( 2)).*( 3) == 9 Dmytro Mitin Introduction to programming with dependent types in Scala
- 6. Theorem proving import provingground. import HoTT. import TLImplicits. import shapeless. val indN assoc = NatInd.induc(n :-> (m ∼>: (k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) )))) val hyp = "(n+m)+k=n+(m+k)" :: m ∼>: k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) ) val assoc = indN assoc(m :∼> (k :∼> add(m)(k).refl))( n :∼> (hyp :-> (m :∼> (k :∼> IdentityTyp.extnslty(succ)(add(add(n)(m))(k)) (add(n)(add(m)(k))) (hyp(m)(k)) )))) assoc !: n ∼>: m ∼>: k ∼>: ( add(add(n)(m))(k) =:= add(n)(add(m)(k)) ) Dmytro Mitin Introduction to programming with dependent types in Scala