Portability	non-portable (MPTC, non-standard instances)
Stability	experimental
Maintainer	[email protected]

Data.TypeLevel.Num.Ops

Contents

Successor/Predecessor
Addition/Subtraction
Multiplication/Division
- Special efficiency cases
Exponientiation/Logarithm
- Special efficiency cases
Comparison assertions
- General comparison assertion
  - Type-level values denoting comparison results
- Abbreviated comparison assertions
Maximum/Minimum
Greatest Common Divisor

Description

Type-level numerical operations and its value-level reflection functions.

Synopsis

class (Nat x, Pos y) => Succ x y | x -> y, y -> x
succ :: Succ x y => x -> y
class (Pos x, Nat y) => Pred x y | x -> y, y -> x
pred :: Pred x y => x -> y
class (Add' x y z, Add' y x z) => Add x y z | x y -> z, z x -> y, z y -> x
(+) :: Add x y z => x -> y -> z
class Sub x y z | x y -> z, z x -> y, z y -> x
(-) :: Sub x y z => x -> y -> z
class (Nat x, Nat y, Nat z) => Mul x y z | x y -> z
(*) :: Mul x y z => x -> y -> z
class Div x y z | x y -> z, x z -> y, y z -> x
div :: Div x y z => x -> y -> z
class Mod x y r | x y -> r
mod :: Mod x y r => x -> y -> r
class (Nat x, Pos y) => DivMod x y q r | x y -> q r
divMod :: DivMod x y q r => x -> y -> (q, r)
class (Pos d, Nat x) => IsDivBy d x
isDivBy :: IsDivBy d x => d -> x
class (Nat x, Nat q) => Mul10 x q | x -> q, q -> x
mul10 :: Mul10 x q => x -> q
class (Nat x, Nat q) => Div10 x q | x -> q, q -> x
div10 :: Div10 x q => x -> q
class (Nat i, Nat x) => DivMod10 x i l | i l -> x, x -> i l
divMod10 :: DivMod10 x q r => x -> (q, r)
class (Nat b, Nat e, Nat r) => ExpBase b e r | b e -> r
(^) :: ExpBase b e r => b -> e -> r
class (Pos b, b :>=: D2, Pos x, Nat e) => LogBase b x e | b x -> e
logBase :: LogBaseF b x e f => b -> x -> e
class (Pos b, b :>=: D2, Pos x, Nat e, Bool f) => LogBaseF b x e f | b x -> e f
logBaseF :: LogBaseF b x e f => b -> x -> (e, f)
class (Pos b, b :>=: D2, Pos x) => IsPowOf b x
isPowOf :: IsPowOf b x => b -> x -> ()
class (Nat x, Pos y) => Exp10 x y | x -> y, y -> x
exp10 :: Exp10 x y => x -> y
class (Pos x, Nat y) => Log10 x y | x -> y
log10 :: Log10 x y => x -> y
class (Nat x, Nat y) => Trich x y r | x y -> r
trich :: Trich x y r => z -> x -> r
data LT
data EQ
data GT
class x :==: y
class x :>: y
class x :<: y
class x :>=: y
class x :<=: y
(==) :: x :==: y => x -> y -> ()
(>) :: x :>: y => x -> y -> ()
(<) :: x :<: y => x -> y -> ()
(>=) :: x :>=: y => x -> y -> ()
(<=) :: x :<=: y => x -> y -> ()
class Max x y z | x y -> z
max :: Max x y z => x -> y -> z
class Min x y z | x y -> z
min :: Min x y z => x -> y -> z
class (Nat x, Nat y, Nat gcd) => GCD x y gcd | x y -> gcd
gcd :: GCD x y z => x -> y -> z

Successor/Predecessor

class (Nat x, Pos y) => Succ x y | x -> y, y -> xSource

Successor type-level relation. Succ x y establishes that succ x = y.

Instances

(Pos y, IsZero y yz, DivMod10 x xi xl, Succ' xi xl yi yl yz, DivMod10 y yi yl) => Succ x y

succ :: Succ x y => x -> ySource

value-level reflection function for the Succ type-level relation

class (Pos x, Nat y) => Pred x y | x -> y, y -> xSource

Predecessor type-level relation. Pred x y establishes that pred x = y.

Instances

Succ x y => Pred y x

pred :: Pred x y => x -> ySource

value-level reflection function for the Pred type-level relation

Addition/Subtraction

class (Add' x y z, Add' y x z) => Add x y z | x y -> z, z x -> y, z y -> xSource

Addition type-level relation. Add x y z establishes that x + y = z.

Instances

(Add' x y z, Add' y x z) => Add x y z

(+) :: Add x y z => x -> y -> zSource

value-level reflection function for the Add type-level relation

class Sub x y z | x y -> z, z x -> y, z y -> xSource

Subtraction type-level relation. Sub x y z establishes that x - y = z

Instances

Add x y z => Sub z y x

(-) :: Sub x y z => x -> y -> zSource

value-level reflection function for the Sub type-level relation

Multiplication/Division

class (Nat x, Nat y, Nat z) => Mul x y z | x y -> zSource

Multiplication type-level relation. Mul x y z establishes that x * y = z. Note it isn't relational (i.e. its inverse cannot be used for division, however, even if it could, the resulting division would only work for zero-remainder divisions)

Instances

(Add z y z', Mul D8 y z) => Mul D9 y z'
(Add z y z', Mul D7 y z) => Mul D8 y z'
(Add z y z', Mul D6 y z) => Mul D7 y z'
(Add z y z', Mul D5 y z) => Mul D6 y z'
(Add z y z', Mul D4 y z) => Mul D5 y z'
(Add z y z', Mul D3 y z) => Mul D4 y z'
(Add z y z', Mul D2 y z) => Mul D3 y z'
Add y y z => Mul D2 y z
Nat y => Mul D1 y y
Nat y => Mul D0 y D0
(Pos (:* xi xl), Nat y, Mul xi y z, Mul10 z z10, Mul xl y dy, Add dy z10 z') => Mul (:* xi xl) y z'

(*) :: Mul x y z => x -> y -> zSource

value-level reflection function for the multiplication type-level relation

class Div x y z | x y -> z, x z -> y, y z -> xSource

Division type-level relation. Remainder-discarding version of DivMod. Note it is not relational (due to DivMod not being relational)

Instances

DivMod x y q r => Div x y q

div :: Div x y z => x -> y -> zSource

value-level reflection function for the Div type-level relation

class Mod x y r | x y -> rSource

Remainder of division, type-level relation. Mod x y r establishes that r is the reminder of dividing x by y.

Instances

DivMod x y q r => Mod x y r

mod :: Mod x y r => x -> y -> rSource

value-level reflection function for the Mod type-level relation

class (Nat x, Pos y) => DivMod x y q r | x y -> q rSource

Division and Remainder type-level relation. DivMod x y q r establishes that xy = q + ry Note it is not relational (i.e. its inverse cannot be used for multiplication).

Instances

(Pos y, Trich x y cmp, DivMod' x y q r cmp) => DivMod x y q r

divMod :: DivMod x y q r => x -> y -> (q, r)Source

value-level reflection function for the DivMod type-level relation

class (Pos d, Nat x) => IsDivBy d x Source

Is-divisible-by type-level assertion. e.g IsDivBy d x establishes that x is divisible by d.

Instances

DivMod x d q D0 => IsDivBy d x

isDivBy :: IsDivBy d x => d -> xSource

value-level reflection function for IsDivBy

Special efficiency cases

class (Nat x, Nat q) => Mul10 x q | x -> q, q -> xSource

Multiplication by 10 type-level relation (based on DivMod10). Mul10 x y establishes that 10 * x = y.

Instances

DivMod10 x q D0 => Mul10 q x

mul10 :: Mul10 x q => x -> qSource

value-level reflection function for Mul10

class (Nat x, Nat q) => Div10 x q | x -> q, q -> xSource

Division by 10 type-level relation (based on DivMod10)

Instances

DivMod10 x q r => Div10 x q

div10 :: Div10 x q => x -> qSource

value-level reflection function for Mul10

class (Nat i, Nat x) => DivMod10 x i l | i l -> x, x -> i lSource

Division by 10 and Remainer type-level relation (similar to DivMod).

This operation is much faster than DivMod. Furthermore, it is the general, non-structural, constructor/deconstructor since it splits a decimal numeral into its initial digits and last digit. Thus, it allows to inspect the structure of a number and is normally used to create type-level operations.

Note that contrary to DivMod, DivMod10 is relational (it can be used to multiply by 10)

Instances

DivMod10 D9 D0 D9
DivMod10 D8 D0 D8
DivMod10 D7 D0 D7
DivMod10 D6 D0 D6
DivMod10 D5 D0 D5
DivMod10 D4 D0 D4
DivMod10 D3 D0 D3
DivMod10 D2 D0 D2
DivMod10 D1 D0 D1
DivMod10 D0 D0 D0
Nat (:* D9 l) => DivMod10 (:* D9 l) D9 l
Nat (:* D8 l) => DivMod10 (:* D8 l) D8 l
Nat (:* D7 l) => DivMod10 (:* D7 l) D7 l
Nat (:* D6 l) => DivMod10 (:* D6 l) D6 l
Nat (:* D5 l) => DivMod10 (:* D5 l) D5 l
Nat (:* D4 l) => DivMod10 (:* D4 l) D4 l
Nat (:* D3 l) => DivMod10 (:* D3 l) D3 l
Nat (:* D2 l) => DivMod10 (:* D2 l) D2 l
Nat (:* D1 l) => DivMod10 (:* D1 l) D1 l
(Nat (:* x l), Nat (:* (:* x l) l')) => DivMod10 (:* (:* x l) l') (:* x l) l'

divMod10 :: DivMod10 x q r => x -> (q, r)Source

value-level reflection function for DivMod10

Exponientiation/Logarithm

class (Nat b, Nat e, Nat r) => ExpBase b e r | b e -> rSource

Exponentation type-level relation. ExpBase b e r establishes that b^e = r Note it is not relational (i.e. it cannot be used to express logarithms)

Instances

(Mul r b r', ExpBase b D8 r) => ExpBase b D9 r'
(Mul r b r', ExpBase b D7 r) => ExpBase b D8 r'
(Mul r b r', ExpBase b D6 r) => ExpBase b D7 r'
(Mul r b r', ExpBase b D5 r) => ExpBase b D6 r'
(Mul r b r', ExpBase b D4 r) => ExpBase b D5 r'
(Mul r b r', ExpBase b D3 r) => ExpBase b D4 r'
(Mul r b r', ExpBase b D2 r) => ExpBase b D3 r'
Mul b b r => ExpBase b D2 r
Nat b => ExpBase b D1 b
Nat b => ExpBase b D0 D1
(Nat b, Pos (:* ei el), Nat r, Mul b r r', Pred (:* ei el) e', ExpBase b e' r) => ExpBase b (:* ei el) r'

(^) :: ExpBase b e r => b -> e -> rSource

value-level reflection function for the ExpBase type-level relation

class (Pos b, b :>=: D2, Pos x, Nat e) => LogBase b x e | b x -> eSource

Instances

LogBaseF b x e f => LogBase b x e

logBase :: LogBaseF b x e f => b -> x -> eSource

value-level reflection function for LogBase

class (Pos b, b :>=: D2, Pos x, Nat e, Bool f) => LogBaseF b x e f | b x -> e fSource

Version of LogBase which also outputs if the logarithm calculated was exact. f indicates if the resulting logarithm has no fractional part (i.e. tells if the result provided is exact)

Instances

(Trich x b cmp, LogBaseF' b x e f cmp) => LogBaseF b x e f

logBaseF :: LogBaseF b x e f => b -> x -> (e, f)Source

value-level reflection function for LogBaseF

class (Pos b, b :>=: D2, Pos x) => IsPowOf b x Source

Assert that a number (x) can be expressed as the power of another one (b) (i.e. the fractional part of log_base_b x = 0, or, in a different way, exists y . b^y = x).

Instances

(Trich x b cmp, IsPowOf' b x cmp) => IsPowOf b x

isPowOf :: IsPowOf b x => b -> x -> ()Source

Special efficiency cases

class (Nat x, Pos y) => Exp10 x y | x -> y, y -> xSource

Base-10 Exponentiation type-level relation

Instances

Exp10 D0 D1
Exp10 D9 (:* (:* (:* (:* (:* (:* (:* (:* (:* D1 D0) D0) D0) D0) D0) D0) D0) D0) D0)
Exp10 D8 (:* (:* (:* (:* (:* (:* (:* (:* D1 D0) D0) D0) D0) D0) D0) D0) D0)
Exp10 D7 (:* (:* (:* (:* (:* (:* (:* D1 D0) D0) D0) D0) D0) D0) D0)
Exp10 D6 (:* (:* (:* (:* (:* (:* D1 D0) D0) D0) D0) D0) D0)
Exp10 D5 (:* (:* (:* (:* (:* D1 D0) D0) D0) D0) D0)
Exp10 D4 (:* (:* (:* (:* D1 D0) D0) D0) D0)
Exp10 D3 (:* (:* (:* D1 D0) D0) D0)
Exp10 D2 (:* (:* D1 D0) D0)
Exp10 D1 (:* D1 D0)
(Pred (:* xi xl) x', Exp10 x' (:* (:* (:* (:* (:* (:* (:* (:* (:* y D0) D0) D0) D0) D0) D0) D0) D0) D0)) => Exp10 (:* xi xl) (:* (:* (:* (:* (:* (:* (:* (:* (:* (:* y D0) D0) D0) D0) D0) D0) D0) D0) D0) D0)

exp10 :: Exp10 x y => x -> ySource

value-level reflection function for Exp10

class (Pos x, Nat y) => Log10 x y | x -> ySource

Base-10 logarithm type-level relation Note it is not relational (cannot be used to express Exponentation to 10) However, it works with any positive numeral (not just powers of 10)

Instances

Log10 D9 D0
Log10 D8 D0
Log10 D7 D0
Log10 D6 D0
Log10 D5 D0
Log10 D4 D0
Log10 D3 D0
Log10 D2 D0
Log10 D1 D0
(Pos (:* xi xl), Pred y y', Log10 xi y') => Log10 (:* xi xl) y

log10 :: Log10 x y => x -> ySource

value-level reflection function for Log10

Comparison assertions

General comparison assertion

class (Nat x, Nat y) => Trich x y r | x y -> rSource

Trichotomy type-level relation. 'Trich x y r' establishes the relation (r) between x and y. The obtained relation (r) Can be LT (if x is lower than y), EQ (if x equals y) or GT (if x is greater than y)

Instances

Trich D9 D9 EQ
Trich D9 D8 GT
Trich D9 D7 GT
Trich D9 D6 GT
Trich D9 D5 GT
Trich D9 D4 GT
Trich D9 D3 GT
Trich D9 D2 GT
Trich D9 D1 GT
Trich D9 D0 GT
Trich D8 D9 LT
Trich D8 D8 EQ
Trich D8 D7 GT
Trich D8 D6 GT
Trich D8 D5 GT
Trich D8 D4 GT
Trich D8 D3 GT
Trich D8 D2 GT
Trich D8 D1 GT
Trich D8 D0 GT
Trich D7 D9 LT
Trich D7 D8 LT
Trich D7 D7 EQ
Trich D7 D6 GT
Trich D7 D5 GT
Trich D7 D4 GT
Trich D7 D3 GT
Trich D7 D2 GT
Trich D7 D1 GT
Trich D7 D0 GT
Trich D6 D9 LT
Trich D6 D8 LT
Trich D6 D7 LT
Trich D6 D6 EQ
Trich D6 D5 GT
Trich D6 D4 GT
Trich D6 D3 GT
Trich D6 D2 GT
Trich D6 D1 GT
Trich D6 D0 GT
Trich D5 D9 LT
Trich D5 D8 LT
Trich D5 D7 LT
Trich D5 D6 LT
Trich D5 D5 EQ
Trich D5 D4 GT
Trich D5 D3 GT
Trich D5 D2 GT
Trich D5 D1 GT
Trich D5 D0 GT
Trich D4 D9 LT
Trich D4 D8 LT
Trich D4 D7 LT
Trich D4 D6 LT
Trich D4 D5 LT
Trich D4 D4 EQ
Trich D4 D3 GT
Trich D4 D2 GT
Trich D4 D1 GT
Trich D4 D0 GT
Trich D3 D9 LT
Trich D3 D8 LT
Trich D3 D7 LT
Trich D3 D6 LT
Trich D3 D5 LT
Trich D3 D4 LT
Trich D3 D3 EQ
Trich D3 D2 GT
Trich D3 D1 GT
Trich D3 D0 GT
Trich D2 D9 LT
Trich D2 D8 LT
Trich D2 D7 LT
Trich D2 D6 LT
Trich D2 D5 LT
Trich D2 D4 LT
Trich D2 D3 LT
Trich D2 D2 EQ
Trich D2 D1 GT
Trich D2 D0 GT
Trich D1 D9 LT
Trich D1 D8 LT
Trich D1 D7 LT
Trich D1 D6 LT
Trich D1 D5 LT
Trich D1 D4 LT
Trich D1 D3 LT
Trich D1 D2 LT
Trich D1 D1 EQ
Trich D1 D0 GT
Trich D0 D9 LT
Trich D0 D8 LT
Trich D0 D7 LT
Trich D0 D6 LT
Trich D0 D5 LT
Trich D0 D4 LT
Trich D0 D3 LT
Trich D0 D2 LT
Trich D0 D1 LT
Trich D0 D0 EQ
Pos (:* yi yl) => Trich D9 (:* yi yl) LT
Pos (:* yi yl) => Trich D8 (:* yi yl) LT
Pos (:* yi yl) => Trich D7 (:* yi yl) LT
Pos (:* yi yl) => Trich D6 (:* yi yl) LT
Pos (:* yi yl) => Trich D5 (:* yi yl) LT
Pos (:* yi yl) => Trich D4 (:* yi yl) LT
Pos (:* yi yl) => Trich D3 (:* yi yl) LT
Pos (:* yi yl) => Trich D2 (:* yi yl) LT
Pos (:* yi yl) => Trich D1 (:* yi yl) LT
Pos (:* yi yl) => Trich D0 (:* yi yl) LT
Pos (:* yi yl) => Trich (:* yi yl) D9 GT
Pos (:* yi yl) => Trich (:* yi yl) D8 GT
Pos (:* yi yl) => Trich (:* yi yl) D7 GT
Pos (:* yi yl) => Trich (:* yi yl) D6 GT
Pos (:* yi yl) => Trich (:* yi yl) D5 GT
Pos (:* yi yl) => Trich (:* yi yl) D4 GT
Pos (:* yi yl) => Trich (:* yi yl) D3 GT
Pos (:* yi yl) => Trich (:* yi yl) D2 GT
Pos (:* yi yl) => Trich (:* yi yl) D1 GT
Pos (:* yi yl) => Trich (:* yi yl) D0 GT
(Pos (:* xi xl), Pos (:* yi yl), Trich xl yl rl, Trich xi yi ri, CS ri rl r) => Trich (:* xi xl) (:* yi yl) r

trich :: Trich x y r => z -> x -> rSource

value-level reflection function for the comparison type-level assertion

Type-level values denoting comparison results

data LT Source

Lower than

Instances

CS LT r LT
Trich D8 D9 LT
Trich D7 D9 LT
Trich D7 D8 LT
Trich D6 D9 LT
Trich D6 D8 LT
Trich D6 D7 LT
Trich D5 D9 LT
Trich D5 D8 LT
Trich D5 D7 LT
Trich D5 D6 LT
Trich D4 D9 LT
Trich D4 D8 LT
Trich D4 D7 LT
Trich D4 D6 LT
Trich D4 D5 LT
Trich D3 D9 LT
Trich D3 D8 LT
Trich D3 D7 LT
Trich D3 D6 LT
Trich D3 D5 LT
Trich D3 D4 LT
Trich D2 D9 LT
Trich D2 D8 LT
Trich D2 D7 LT
Trich D2 D6 LT
Trich D2 D5 LT
Trich D2 D4 LT
Trich D2 D3 LT
Trich D1 D9 LT
Trich D1 D8 LT
Trich D1 D7 LT
Trich D1 D6 LT
Trich D1 D5 LT
Trich D1 D4 LT
Trich D1 D3 LT
Trich D1 D2 LT
Trich D0 D9 LT
Trich D0 D8 LT
Trich D0 D7 LT
Trich D0 D6 LT
Trich D0 D5 LT
Trich D0 D4 LT
Trich D0 D3 LT
Trich D0 D2 LT
Trich D0 D1 LT
Max' x y LT y
(Nat x, Nat y, GCD y x gcd) => GCD' x y False LT gcd
(Pos b, b :>=: D2, Pos x) => LogBaseF' b x D0 False LT
(Nat x, Pos y) => DivMod' x y D0 x LT
Pos (:* yi yl) => Trich D9 (:* yi yl) LT
Pos (:* yi yl) => Trich D8 (:* yi yl) LT
Pos (:* yi yl) => Trich D7 (:* yi yl) LT
Pos (:* yi yl) => Trich D6 (:* yi yl) LT
Pos (:* yi yl) => Trich D5 (:* yi yl) LT
Pos (:* yi yl) => Trich D4 (:* yi yl) LT
Pos (:* yi yl) => Trich D3 (:* yi yl) LT
Pos (:* yi yl) => Trich D2 (:* yi yl) LT
Pos (:* yi yl) => Trich D1 (:* yi yl) LT
Pos (:* yi yl) => Trich D0 (:* yi yl) LT

data EQ Source

Equal

Instances

CS EQ r r
Trich D9 D9 EQ
Trich D8 D8 EQ
Trich D7 D7 EQ
Trich D6 D6 EQ
Trich D5 D5 EQ
Trich D4 D4 EQ
Trich D3 D3 EQ
Trich D2 D2 EQ
Trich D1 D1 EQ
Trich D0 D0 EQ
(Pos b, b :>=: D2) => IsPowOf' b b EQ
Max' x y EQ y
Nat x => GCD' x x False EQ x
(Pos b, b :>=: D2) => LogBaseF' b b D1 True EQ
(Nat x, Pos y) => DivMod' x y D1 D0 EQ

data GT Source

Greater than

Instances

CS GT r GT
Trich D9 D8 GT
Trich D9 D7 GT
Trich D9 D6 GT
Trich D9 D5 GT
Trich D9 D4 GT
Trich D9 D3 GT
Trich D9 D2 GT
Trich D9 D1 GT
Trich D9 D0 GT
Trich D8 D7 GT
Trich D8 D6 GT
Trich D8 D5 GT
Trich D8 D4 GT
Trich D8 D3 GT
Trich D8 D2 GT
Trich D8 D1 GT
Trich D8 D0 GT
Trich D7 D6 GT
Trich D7 D5 GT
Trich D7 D4 GT
Trich D7 D3 GT
Trich D7 D2 GT
Trich D7 D1 GT
Trich D7 D0 GT
Trich D6 D5 GT
Trich D6 D4 GT
Trich D6 D3 GT
Trich D6 D2 GT
Trich D6 D1 GT
Trich D6 D0 GT
Trich D5 D4 GT
Trich D5 D3 GT
Trich D5 D2 GT
Trich D5 D1 GT
Trich D5 D0 GT
Trich D4 D3 GT
Trich D4 D2 GT
Trich D4 D1 GT
Trich D4 D0 GT
Trich D3 D2 GT
Trich D3 D1 GT
Trich D3 D0 GT
Trich D2 D1 GT
Trich D2 D0 GT
Trich D1 D0 GT
(Pos b, b :>=: D2, Pos x, DivMod x b q D0, IsPowOf b q) => IsPowOf' b x GT
Max' x y GT x
(Nat x, Nat y, Sub x y x', GCD x' y gcd) => GCD' x y False GT gcd
(Pos b, b :>=: D2, Pos x, DivMod x b q r, IsZero r rz, And rz f' f, Pred e e', LogBaseF b q e' f') => LogBaseF' b x e f GT
(Nat x, Pos y, Sub x y x', Pred q q', DivMod x' y q' r) => DivMod' x y q r GT
Pos (:* yi yl) => Trich (:* yi yl) D9 GT
Pos (:* yi yl) => Trich (:* yi yl) D8 GT
Pos (:* yi yl) => Trich (:* yi yl) D7 GT
Pos (:* yi yl) => Trich (:* yi yl) D6 GT
Pos (:* yi yl) => Trich (:* yi yl) D5 GT
Pos (:* yi yl) => Trich (:* yi yl) D4 GT
Pos (:* yi yl) => Trich (:* yi yl) D3 GT
Pos (:* yi yl) => Trich (:* yi yl) D2 GT
Pos (:* yi yl) => Trich (:* yi yl) D1 GT
Pos (:* yi yl) => Trich (:* yi yl) D0 GT

Abbreviated comparison assertions

class x :==: y Source

Equality abbreviated type-level assertion

Instances

Trich x y EQ => x :==: y

class x :>: y Source

Greater-than abbreviated type-level assertion

Instances

Trich x y GT => x :>: y

class x :<: y Source

Lower-than abbreviated type-level assertion

Instances

Trich x y LT => x :<: y

class x :>=: y Source

Greater-than or equal abbreviated type-level assertion

Instances

(Succ x x', Trich x' y GT) => x :>=: y

class x :<=: y Source

Lower-than or equal abbreviated type-level assertion

Instances

(Succ x' x, Trich x' y LT) => x :<=: y

(==) :: x :==: y => x -> y -> ()Source

value-level reflection function for the equality abbreviated type-level assertion

(>) :: x :>: y => x -> y -> ()Source

value-level reflection function for the equality abbreviated type-level assertion

(<) :: x :<: y => x -> y -> ()Source

value-level reflection function for the lower-than abbreviated type-level assertion

(>=) :: x :>=: y => x -> y -> ()Source

value-level reflection function for the greater-than or equal abbreviated type-level assertion

(<=) :: x :<=: y => x -> y -> ()Source

value-level reflection function for the lower-than or equal abbreviated type-level assertion

Maximum/Minimum

class Max x y z | x y -> zSource

Maximum type-level relation

Instances

(Max' x y b z, Trich x y b) => Max x y z

max :: Max x y z => x -> y -> zSource

value-level reflection function for the maximum type-level relation

class Min x y z | x y -> zSource

Minimum type-level relation

Instances

(Max' y x b z, Trich x y b) => Min x y z

min :: Min x y z => x -> y -> zSource

value-level reflection function for the minimum type-level relation

Greatest Common Divisor

class (Nat x, Nat y, Nat gcd) => GCD x y gcd | x y -> gcdSource

Greatest Common Divisor type-level relation

Instances

(Nat x, Nat y, Trich x y cmp, IsZero y yz, GCD' x y yz cmp gcd) => GCD x y gcd

gcd :: GCD x y z => x -> y -> zSource

value-level reflection function for the GCD type-level relation