Module

Type.Data.Peano.Nat.Definition

data Nat :: Type

Represents a non-negative whole Number ℕ₀

data Z :: Nat

Represents 0

data Succ :: Nat -> Nat

Represents Successor of a Nat: (Succ a) ^= 1 + a

(IsNat a) => IsNat (Succ a)
(SumNat a b c) => SumNat (Succ a) b (Succ c)
ProductNat (Succ Z) a a
1 * a = a
(ProductNat a b ab, SumNat ab b result) => ProductNat (Succ a) b result
(1 + a) * b = b + a * b
ExponentiationNat a Z (Succ Z)
(ExponentiationNat a b ab, ProductNat ab a result) => ExponentiationNat a (Succ b) result
(CompareNat a b ord) => CompareNat (Succ a) (Succ b) ord
IsZeroNat (Succ a) False

data NProxy (n :: Nat)

Proxy from kind Nat to kind Type

class IsNat (a :: Nat)  where

reflectNat :: NProxy a -> Int
reflect typelevel Nat to a valuelevel Int
```
reflectNat (undefined :: NProxy D10) = 10
```

class SumNat (a :: Nat) (b :: Nat) (c :: Nat) | a b -> c

a + b = c

plusNat :: forall c b a. SumNat a b c => NProxy a -> NProxy b -> NProxy c

class ProductNat (a :: Nat) (b :: Nat) (c :: Nat) | a b -> c

a * b = c

ProductNat Z a Z
0 * a = 0
ProductNat (Succ Z) a a
1 * a = a
(ProductNat a b ab, SumNat ab b result) => ProductNat (Succ a) b result
(1 + a) * b = b + a * b

mulNat :: forall c b a. ProductNat a b c => NProxy a -> NProxy b -> NProxy c

class ExponentiationNat (a :: Nat) (b :: Nat) (c :: Nat) | a b -> c

powNat :: forall c b a. ExponentiationNat a b c => NProxy a -> NProxy b -> NProxy c

> powNat d2 d3
8 -- : NProxy D8

a raised to the power of b a^b = c

class CompareNat (a :: Nat) (b :: Nat) (ord :: Ordering) | a b -> ord

class IsZeroNat (a :: Nat) (isZero :: Boolean) | a -> isZero

Modules: Type.Data.Peano; Type.Data.Peano.Int; Type.Data.Peano.Int.Definition; Type.Data.Peano.Int.Parse; Type.Data.Peano.Int.TypeAliases; Type.Data.Peano.Nat; Type.Data.Peano.Nat.Definition; Type.Data.Peano.Nat.Parse; Type.Data.Peano.Nat.TypeAliases