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shamansir/purs-typeclasses.cue

Last active May 15, 2021 10:46
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PureScript typeclasses in CUE -> JSON, for https://imgur.com/a/ESOR7
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purs-typeclasses.cue
plus: {
name: "Plus"
info: "Left and Right identity for Alt"
parent: "alt"
"package": "Control*"
members:
[
{ name: "empty"
, def: "f a"
, laws : // FIXME: laws are incorrect
[
{ law: "associativity"
, example:
{ left: "(x <|> y) <|> z"
, right: "x <|> (y <|> z)"
}
}
,
{ law: "distributivity"
, example:
{ left: "f <$> (x <|> y)"
, right: "(f <$> x) <|> (f <$> y)"
}
}
]
}
,
]
instances: [ "Array", ]
}
comonad: {
name: "Comonad"
info: "Left and Right identity for Alt"
parent: "extend"
"package": "Control*"
members:
[
{ name: "extract"
, def: "w a -> a"
, laws :
[
{ law: "left identity"
, example:
{ left: "extract <<== xs"
, right: "xs"
}
}
,
{ law: "right identity"
, example:
{ left: "extract (f <<== xs)"
, right: "f xs"
}
}
]
}
,
]
instances: [ ]
}
semigroupoid: {
name: "Semigroupoid"
info: "Category without identity"
"package": "Control"
members:
[
{ name: "compose"
, def: "a c d -> a b c -> a b d"
, op: "<<<"
, opEmoji: "🔙"
, laws :
[
{ law: "associativity"
, example:
{ left: "p <<< (q <<< r)"
, right: "(p <<< q) <<< r"
}
}
]
}
,
{ name: "composeFlipped"
, def: "Semigroupoid a => a b c -> a c d -> a b d"
, op: ">>>"
, opEmoji: "🔜"
}
]
instances: [ ]
}
category: {
name: "Category"
info: "Objects and composable morphisms with their identity"
parent: "semigroupoid"
"package": "Control"
members:
[
{ name: "id"
, def: "a t t"
, laws :
[
{ law: "identity"
, example:
{ left: "id <<< p"
, middle: "p <<< id"
, right: "p"
}
}
]
}
]
instances: [ "(->)" ]
}
lazy: {
name: "Lazy"
info: "Deferred operations"
"package": "Control*"
members:
[
{ name: "defer"
, def: "(Unit -> l) -> l"
}
, { name: "fix"
, def: "Lazy l => (l -> l) -> l"
}
]
instances: [ "(a -> b)", "Unit" ]
}
alt: {
name: "Alt"
info: "Associative operation on a constructor"
parent: "functor"
"package": "Control*"
members:
[
{ name: "alt"
, def: "f a -> f a -> f a"
, op: "<|>"
, opEmoji: "🔗"
, laws :
[
{ law: "associativity"
, example:
{ left: "(x <|> y) <|> z"
, right: "x <|> (y <|> z)"
}
}
,
{ law: "distributivity"
, example:
{ left: "f <$> (x <|> y)"
, right: "(f <$> x) <|> (f <$> y)"
}
}
]
}
,
]
instances: [ "Array", ]
}
extend : {
name: "Extend"
info: "Extend local computation to a global one"
parent: "functor"
"package": "Control*"
members:
[
{ name: "extend"
, def: "m a -> (a -> m b) -> m b"
, op: "<<=="
, laws :
[
{ law: "associativity"
, example:
{ left: "extend f <<< extend g"
, right: "extend (f <<< extend g)"
}
}
]
}
,
{ name: "extendFlipped"
, def: "Extend w => w a -> (w a -> b) -> w b"
, op: "==>>"
}
,
{ name: "composeCoKleisli"
, def: "Extend w => (w a -> b) -> (w b -> c) -> w a -> c"
, op: "=>="
}
,
{ name: "composeCoKleisliFlipped"
, def: "Extend w => (w b -> c) -> (w a -> b) -> w a -> c"
, op: "=<="
}
,
{ name: "duplicate"
, def: "Extend w => w a -> w (w a)"
}
]
instances: [ "Semigroup w => Extend ((->) w)", "Array", ]
}
bind : {
name: "Bind"
info: "Compose computations"
parent: "apply"
members:
[
{ name: "bind"
, def: "m a -> (a -> m b) -> m b"
, op: ">>="
, opEmoji: "🎉"
, laws :
[
{ law: "associativity"
, example:
{ left: "(x >>= f) >>= g"
, right: "x >>= (\\k -> f k >>= g)"
}
}
]
}
,
{ name: "bindFlipped"
, def: "Bind m => (a -> m b) -> m a -> m b"
, op: "==<<"
}
,
{ name: "join"
, def: "Bind m => m (m a) -> m a"
}
,
{ name: "composeKleisli"
, def: "Bind m => (a -> m b) -> (b -> m c) -> a -> m c"
, op: ">==>"
}
,
{ name: "composeKleisliFlipped"
, def: "Bind m => (b -> m c) -> (a -> m b) -> a -> m c"
, op: "<==<"
}
,
{ name: "ifM"
, def: "Bind m => m Boolean -> m a -> m a -> m a"
}
]
instances: [ "((->) r)", "Array" ]
}
monad : {
name: "Monad"
info: "Compose computations"
laws:
[
{ law: "left identity"
, example:
{ left: "pure x >>= f"
, right: "f x"
}
}
,
{ law: "right identity"
, example:
{ left: "x >>= pure"
, right: "x"
}
}
]
members:
[
{ name: "liftM1"
, def: "Monad m => (a -> b) -> m a -> m b"
}
,
{ name: "ap"
, def: "Monad m => m (a -> b) -> m a -> m b"
}
,
{ name: "whenM"
, def: "Monad m => m Boolean -> m Unit -> m Unit"
}
,
{ name: "unlessM"
, def: "Monad m => m Boolean -> m Unit -> m Unit"
}
]
parents: [ "bind", "applicative" ]
instances: [ "((->) r)", "Array" ]
}
monadZero : {
name: "MonadZero"
info: "Compose computations"
parents: [ "monad", "alternative" ]
laws:
[
{ law: "annihilation"
, example:
{ left: "empty >>= f"
, right: "empty"
}
}
]
members:
{ name: "guard"
, def: "MonadZero m => Boolean -> m Unit"
}
instances: [ "Array" ]
}
monadPlus : {
name: "MonadPluse"
info: "Distributivity for Monads"
parent: "monadZero"
laws:
[
{ law: "distributivity"
, example:
{ left: "(x <|> y) >>= f"
, right: "(x >>= f) <|> (y >>= f)"
}
}
]
members:
{ name: "guard"
, def: "MonadZero m => Boolean -> m Unit"
}
instances: [ "Array" ]
}
functor : {
name: "Functor"
info: "Convert and forget"
members:
[
{ name: "map"
, def: "(a -> b) -> f a -> f b"
, op: "<$>"
, opEmoji: "🚂"
, laws :
[
{ law: "identity"
, example:
{ left: "(<$>) id"
, right: "id"
}
}
,
{ law: "composition"
, example:
{ left: "(<$>) (f <<< g)"
, right: "(f <$>) <<< (g <$>)"
}
}
]
}
,
{ name: "mapFlipped"
, def: "Functor f => f a -> (a -> b) -> f b"
, op: "<#>"
}
,
{ name: "void"
, def: "Functor f => f a -> f Unit"
}
,
{ name: "voidRight"
, op: "<$"
, def: "Functor f => a -> f b -> f a"
}
,
{ name: "voidLeft"
, op: "$>"
, def: "Functor f => f a -> b -> f b"
}
,
{ name: "flap"
, op: "<@>"
, def: "Functor f => f (a -> b) -> a -> f b"
}
]
instances: [ "Array", "((->) r)" ]
}
apply : {
name: "Apply"
info: "Unwrap, convert, and wrap again"
parent: "functor"
members:
[
{ name: "apply"
, def: "f (a -> b) -> f a -> f b"
, op: "<*>"
, opEmoji: "🚋"
, laws :
[
{ law: "ap. composition"
, example:
{ left: "(<<<) <$> f <*> g <*> h"
, right: "f <*> (g <*> h)"
}
}
]
}
,
{ name: "applyFirst"
, def: "Apply f => f a -> f b -> f a"
, op: "<*"
}
,
{ name: "applySecond"
, def: "Apply f => f a -> f b -> f b"
, op: "*>"
, opEmoji: "👉"
}
,
{ name: "lift2"
, def: "Apply f => (a -> b -> c) -> f a -> f b -> f c"
}
,
{ name: "lift3"
, def: "Apply f => (a -> b -> c -> d) -> f a -> f b -> f c -> f d"
}
,
{ name: "lift4"
, def: "Apply f => (a -> b -> c -> d -> e) -> f a -> f b -> f c -> f d -> f e"
}
,
{ name: "lift5"
, def: "Apply f => (a -> b -> c -> d -> e -> g) -> f a -> f b -> f c -> f d -> f e -> f g"
}
]
instances: [ "Array", "((->) r)" ]
}
applicative : {
name: "Applicative"
info: "Lift with zero arguments, wrap values"
parent: "apply"
members:
[
{ name: "pure"
, def: "a -> f a"
, laws :
[
{ law: "identity"
, example:
{ left: "(pure id) <*> v"
, right: "v"
}
}
,
{ law: "composition"
, example:
{ left: "(pure <<<) <*> f <*> g <*> h"
, right: "f <*> (g <*> h)"
}
}
,
{ law: "homomorphism"
, example:
{ left: "(pure f) <*> (pure x)"
, right: "pure (f x)"
}
}
,
{ law: "interchange"
, example:
{ left: "u <*> (pure y)"
, right: "(pure ($ y)) <*> u"
}
}
]
}
,
{ name: "liftA1"
, def: "Applicative f => (a -> b) -> f a -> f b"
}
,
{ name: "when"
, def: "Applicative m => Boolean -> m Unit -> m Unit"
}
,
{ name: "unless"
, def: "Applicative m => Boolean -> m Unit -> m Unit"
}
]
instances: [ "Array", "((->) r)" ]
}
alternative : {
name: "Alternative"
info: "To have both Plus and Applicative"
parents: [ "plus", "applicative" ]
laws:
[
{ law: "distributivity"
, example:
{ left: "(f <|> g) <*> x"
, right: "(f <*> x) <|> (g <*> x)"
}
}
,
{ law: "annihilation"
, example:
{ left: "empty <*> f"
, right: "empty"
}
}
]
}
unit : {
name: "Unit"
info: "No computation needed"
package: "Data"
members:
[
{ name: "unit"
, def: "Unit"
}
]
instances: [ "Show Unit" ]
}
void : {
name: "Void"
info: "Uninhabited data type"
package: "Data"
members:
[
{ name: "absurd"
, def: "Void -> a"
, opEmoji: "💣"
}
]
instances: [ "Show Void" ]
}
function : {
name: "Function"
info: "Helpers for the core functions"
package: "Data"
members: [
{ name: "flip"
, def: "(a -> b -> c) -> b -> a -> c"
}
,
{ name: "const"
, def: "a -> b -> a"
}
,
{ name: "apply"
, def: "(a -> b) -> a -> b"
, op: "$"
, opEmoji: "💨"
}
,
{ name: "applyFlipped"
, def: "a -> (a -> b) -> b"
, op: "#"
}
,
{ name: "on"
, def: "(b -> b -> c) -> (a -> b) -> a -> a -> c"
}
]
}
naturalTransformation : {
name: "NaturalTransformation"
info: "Mapping b/w type constructors with no manipulation on inner value"
package: "Data"
members: [
{ name: "NaturalTransformation"
, def: "f a -> g a"
, op: "~>"
, opEmoji: "🐛"
}
]
}
semigroup : {
name: "Semigroup"
info: "Associativity"
package: "Data"
members: [
{ name: "append"
, def: "a -> a -> a"
, op: "<>"
, opEmoji: "🙏"
, laws:
[
{ law: "associativity"
, example:
{ left: "(x <> y) <> z"
, right: "x <> (y <> z)"
}
}
]
}
]
instances: [ "String", "Unit", "Void", "Array a", "Semigroup s' => Semigroup (s -> s')" ]
}
monoid : {
name: "Monoid"
parent: "semigroup"
info: "Folding empty collection"
members:
[
{ name: "mempty"
, def: "m"
, laws:
[
{ law: "associativity"
, example:
{ left: "mempty <> x"
, middle: "x <> mempty"
, right: "x forall x"
}
}
]
}
,
{ name: "power"
, def: "Monoid m => m -> Int -> m"
}
,
{ name: "guard"
, def: "Monoid m => Boolean -> m -> m"
}
]
instances: [ "Unit", "Ordering", "Number", "String", "Array a", "Monoid b => Monoid (a -> b)" ]
}
eq : {
name: "Eq"
package: "Data"
info: "Equality"
members:
[
{ name: "eq"
, def: "a -> a -> Boolean"
, op: "=="
, laws:
[
{ law: "reflexivity"
, example:
{ left: "x == x"
, right: "true"
}
}
,
{ law: "symmetry"
, example:
{ left: "x == y"
, right: "y == x"
}
}
,
{ law: "transitivity"
, example:
{ fact: "(x == y) and (y == z)"
, conclusion: "y == x"
}
}
]
}
,
{ name: "notEq"
, op: "/="
, def: "Eq a => a -> a -> Boolean"
}
]
instances:
[ "Boolean", "Int", "Number", "Char", "String", "Unit", "Void", "Eq a => Eq (Array a)" ]
}
ord : {
name: "Ord"
info: "Ordering"
parent: "eq"
members:
[
{ name: "compart"
, def: "a -> a -> Ordering"
, op: "=="
, laws:
[
{ law: "reflexivity"
, example:
{ fact: "a <= a"
}
}
,
{ law: "antisymmetry"
, example:
{ fact: "(a <= b) and (b <= a)"
, conclusion: "a = b"
}
}
,
{ law: "transitivity"
, example:
{ fact: "(a <= b) and (b <= c)"
, conclusion: "a <= c"
}
}
]
}
,
{ name: "lessThan"
, op: "<"
, def: "Ord a => a -> a -> Boolean"
}
,
{ name: "greaterThan"
, op: ">"
, def: "Ord a => a -> a -> Boolean"
}
,
{ name: "lessThanOrEq"
, op: "<="
, def: "Ord a => a -> a -> Boolean"
}
,
{ name: "greaterThanOrEq"
, op: ">="
, def: "Ord a => a -> a -> Boolean"
}
,
{ name: "min"
, def: "Ord a => a -> a -> a"
}
,
{ name: "max"
, def: "Ord a => a -> a -> a"
}
,
{ name: "clamp"
, def: "Ord a => a -> a -> a -> a"
}
,
{ name: "between"
, def: "Ord a => a -> a -> a -> Boolean"
}
,
{ name: "abs"
, def: "(Ord a, Ring a) => a -> a"
}
,
{ name: "signum"
, def: "(Ord a, Ring a) => a -> a"
}
,
{ name: "comparing"
, def: "Ord b => (a -> b) -> (a -> a -> Ordering)"
}
]
}
bounded : {
name: "Bounded"
parent: "ord"
info: "Have upper and lower boundary"
package: "Data"
laws:
[
{ name: "bounded"
, fact: "bottom <= a <= top"
}
]
members:
[
{ name: "top"
, def: "a"
}
,
{ name: "bottom"
, def: "a"
}
]
instances: [ "Boolean", "Int", "Char", "Ordering", "Unit" ]
}
ordering : {
name: "Ordering"
info: "Just to know the order"
values: [ "LT", "GT", "EQ" ]
members:
[
{ name: "invert"
, def: "Ordering -> Ordering"
}
]
instances: [ "Eq Ordering", "Semigroup Ordering", "Show Ordering" ]
}
show : {
name: "Show"
info: "Displaying values"
package: "Data"
members:
[
{ name: "show"
, def: "a -> String"
}
]
instances: [ "Show Boolean", "Show Int", "Show Number", "Show Char", "Show String", "Show a => Show (Array a)" ]
}
heytingAlgebra : {
name: "HeytingAlgebra"
info: "Bounded lattices, boolean starters"
members:
[
{ name: "ff"
, def: "a"
}
,
{ name: "tt"
, def: "a"
}
,
{ name: "implies"
, def: "a -> a -> a"
}
,
{ name: "conj"
, def: "a -> a -> a"
, op: "&&"
}
,
{ name: "disj"
, def: "a -> a -> a"
, op: "||"
}
,
{ name: "not"
, def: "a"
}
]
laws:
[
{
law: "associativity"
examples:
[ { left: "a || (b || c)", right: "(a || b) || c" }
, { left: "a && (b && c)", right: "(a && b) && c" }
]
}
,
{
law: "commutativity"
examples:
[ { left: "a || b", right: "b || a" }
, { left: "a && b", right: "b && a" }
]
}
,
{
law: "absorption"
examples:
[ { left: "a || (a && b)", right: "a" }
, { left: "a && a", right: "a" }
]
}
,
{
law: "idempotent"
examples:
[ { left: "a || a", right: "a" }
, { left: "a && (a || b)", right: "a" }
]
}
,
{
law: "identity"
examples:
[ { left: "a || ff", right: "a" }
, { left: "a && tt", right: "a" }
]
}
,
{
law: "implication"
examples:
[ { left: "a `implies` a", right: "tt" }
, { left: "a && (a `implies` b)", right: "a && b" }
, { left: "b && (a `implies` b)", right: "b" }
, { left: "a `implies` (b && c)", right: "(a `implies` b) && (a `implies` c)" }
]
}
,
{
law: "complemented"
example: { left: "not a", right: "a `implies` ff" }
}
]
instances: [ "Boolean", "Unit", "(a -> b)" ]
}
booleanAlgebra : {
name: "BooleanAlgebra"
info: "Behave like boolean"
parent: "heytingAlgebra"
laws:
[
{ name: "excluded middle"
, def: "a || not a == tt"
}
]
instances: [ "Boolean", "Unit", "BooleanAlgebra b => BooleanAlgebra (a -> b)" ]
}
boolean : {
name: "Boolean"
info: "Helpers for boolean types"
members:
[
{ name: "otherwise"
, def: "Boolean"
}
]
}
divisionRing : {
name: "DivisionRing"
parent: "ring"
info: "Non-zero rings in which every non-zero element has a multiplicative inverse / skew fields"
members:
[
{ name: "recip"
, def: "a -> a"
, laws:
[
{ law: "Non-zero Ring"
, fact: "one /= zero"
}
,
{ law: "Non-zero multplicative inverse"
, left: "recip a * a"
, middle: "a * recip a"
, right: "one forall a /= 0"
}
]
}
,
{ name: "leftDiv"
, def: "DivisionRing a => a -> a -> a"
}
,
{ name: "rightDiv"
, def: "DivisionRing a => a -> a -> a"
}
]
instances: [ "Number" ]
}
semiring : {
name: "Semiring"
info: "Sum and Multiply (a.k.a. extended Monoid)"
members:
[
{ name: "add"
, def: "a -> a -> a"
, op: "+"
}
,
{ name: "zero"
, def: "a"
}
,
{ name: "mul"
, def: "a -> a -> a"
, op: "*"
}
,
{ name: "one"
, def: "a"
}
]
laws:
[
{ law: "associativity"
, left: "(a + b) + c"
, right: "a + (b + c)"
}
,
{ law: "identity"
, left: "zero + a"
, middle: "a + zero"
, right: "a"
}
,
{ law: "commutative"
, left: "a + b"
, right: "b + a"
}
,
{ law: "associativity"
, left: "(a + b) * c"
, right: "a * (b * c)"
}
,
{ law: "identity"
, left: "one * a"
, middle: "a * one"
, right: "a"
}
,
{ law: "left distributivity"
, left: "a * (b + c)"
, right: "(a * b) + (a * c)"
}
,
{ law: "right distributivity"
, left: "a + (b * c)"
, right: "(a * c) + (b * c)"
}
,
{ law: "annihilation"
, left: "zero * a"
, middle: "a * zero"
, right: "zero"
}
]
instances: [ "Int", "Number", "Semiring b => Semiring (a -> b)" ]
}
ring : {
parent: "semiring"
name: "Ring"
info: "Subtraction"
members:
[
{ name: "sub"
, def: "a -> a -> a"
, op: "-"
, laws:
[
{ law: "Additive inverse"
, left: "a - a"
, middle: "zero - a"
, right: "zero"
}
]
}
,
{ name: "negate"
, def: "Ring a => a -> a"
}
]
instances: [ "Int", "Number", "Unit", "Ring b => Ring (a -> b)" ]
}
commutativeRing : {
name: "CommutativeRing"
info: "Multiplication behaves commutatively"
parent: "ring"
laws:
[
{ law: "commutative"
, left: "a * b"
, right: "b * a"
}
]
instances: [ "Int", "Number", "Unit", "CommutativeRing b => CommutativeRing (a -> b)" ]
}
euclidianRing : {
name: "EuclidianRing"
info: "Divide and Conquer"
parent: "commutativeRing"
members:
[
{ name: "degree"
, def: "a -> Int"
}
,
{ name: "div"
, def: "a -> a -> a"
, op: "/"
}
,
{ name: "mode"
, def: "a -> a -> a"
}
,
{ name: "gcd"
, def: "Eq a => EuclideanRing a => a -> a -> a"
}
,
{ name: "lcm"
, def: "Eq a => EuclideanRing a => a -> a -> a"
}
]
laws:
[
{ law: "integral domain"
, fact: "(a /= 0) and (b /= 0)"
, conclusion: "a * b /= 0"
}
,
{ law: "multiplicative Euclidean function"
, right: "a"
, left: "(a / b) * b + (a `mod` b)"
, conditions:
[ "degree a > 0"
, "degree a <= degree (a * b)"
]
}
]
instances: [ "Int", "Number" ]
}
field : {
info: "Commutative fields"
parent: "euclidianRing"
laws:
[
{ law: "non-zero multiplicative inverse"
, right: "a `mod` b"
, left: "0"
}
]
instances: [ "Int", "Number" ]
}
endo : {
name: "Endo"
info: "By itself"
package: "Data.Monoid"
statements:
[
{ left: "Endo f <> Endo g"
, right: "Endo (f <<< g)"
}
,
{ left: "mempty :: Endo _"
, right: "Endo id"
}
]
instances: [ "Newtype (Endo a) _", "Invariant Endo", "Semigroup (Endo a)", "Monoid (Endo a)" ]
}
dual : {
name: "Dual"
info: "Monoid under dualism"
package: "Data.Monoid"
statements:
[
{ left: "Dual x <> Dual y"
, right: "Dual (y <> x)"
}
,
{ left: "mempty :: Dual _"
, right: "Dual mempty"
}
]
instances: [ "Newtype (Dual a) _", "Eq a => Eq (Dual a)", "Ord a => Ord (Dual a)", "Bounded a => Bounded (Dual a)", "Functor Dual", "Invariant Dual", "Apply Dual", "Applicative Dual", "Bind Dual", "Monad Dual", "Extend Dual", "Comonad Dual", "Show a => Show (Dual a)", "Semigroup a => Semigroup (Dual a)", "Monoid a => Monoid (Dual a)" ]
}
additive : {
name: "Additive"
info: "May be added to something"
package: "Data.Monoid"
statements:
[
{ left: "Additive x <> Additive y"
, right: "Additive (x + y)"
}
,
{ left: "mempty :: Additive _"
, right: "Additive zero"
}
]
instances: [ "Newtype (Additive a) _", "Eq a => Eq (Additive a)", "Ord a => Ord (Additive a)", "Bounded a => Bounded (Additive a)", "Functor Additive", "Invariant Additive", "Apply Additive", "Applicative Additive", "Bind Additive", "Monad Additive", "Extend Additive", "Comonad Additive", "Show a => Show (Additive a)", "Semiring a => Semigroup (Additive a)", "Semiring a => Monoid (Additive a)" ]
}
multiplicative : {
name: "Multiplicative"
info: "May be multiplied on something"
package: "Data.Monoid"
statements:
[
{ left: "Multiplicative x <> Multiplicative y"
, right: "Multiplicative (x * y)"
}
,
{ left: "mempty :: Multiplicative _"
, right: "Multiplicative one"
}
]
instances: [ "Newtype (Multiplicative a) _", "Eq a => Eq (Multiplicative a)", "Ord a => Ord (Multiplicative a)", "Bounded a => Bounded (Multiplicative a)", "Functor Multiplicative", "Invariant Multiplicative", "Apply Multiplicative", "Applicative Multiplicative", "Bind Multiplicative", "Monad Multiplicative", "Extend Multiplicative", "Comonad Multiplicative", "Show a => Show (Multiplicative a)", "Semiring a => Semigroup (Multiplicative a)", "Semiring a => Monoid (Multiplicative a)" ]
}
conj : {
name: "Multiplicative"
info: "Monoid under conjunction"
package: "Data.Monoid"
statements:
[
{ left: "Conj x <> Conj y"
, right: "Conj (x && y)"
}
,
{ left: "mempty :: Conj _"
, right: "Conj top"
}
]
instances: [ "Newtype (Conj a) _", "Eq a => Eq (Conj a)", "Ord a => Ord (Conj a)", "Bounded a => Bounded (Conj a)", "Functor Conj", "Invariant Conj", "Apply Conj", "Applicative Conj", "Bind Conj", "Monad Conj", "Extend Conj", "Comonad Conj", "Show a => Show (Conj a)", "HeytingAlgebra a => Semigroup (Conj a)", "HeytingAlgebra a => Monoid (Conj a)", "HeytingAlgebra a => Semiring (Conj a)" ]
}
disj : {
name: "Disj"
info: "Monoid under disjunction"
package: "Data.Monoid"
statements:
[
{ left: "Conj x <> Conj y"
, right: "Disj (x || y)"
}
,
{ left: "mempty :: Disj _"
, right: "Disj bottom"
}
]
instances: [ "Newtype (Disj a) _", "Eq a => Eq (Disj a)", "Ord a => Ord (Disj a)", "Bounded a => Bounded (Disj a)", "Functor Disj", "Invariant Disj", "Apply Disj", "Applicative Disj", "Bind Disj", "Monad Disj", "Extend Disj", "Comonad Disj", "Show a => Show (Disj a)", "HeytingAlgebra a => Semigroup (Disj a)", "HeytingAlgebra a => Monoid (Disj a)", "HeytingAlgebra a => Semiring (Disj a)" ]
}
alternate : {
name: "Alternate"
info: "Monoid and Semigroup instances corresponding to Plus and Alt for f"
package: "Data.Monoid"
statements:
[
{ left: "Alternate fx <> Alternate fy"
, right: "Alternate (fx <|> fy)"
}
,
{ left: "mempty :: Alternate _"
, right: "Alternate empty"
}
]
instances: [ "Newtype (Alternate f a) _", "Eq a => Eq (Alternate f)", "Ord a => Ord (Alternate a)", "Bounded a => Bounded (Alternate a)", "Functor f => Functor Alternate", "Invariant f => Invariant (Alternate f)", "Apply f => Apply Alternate", "Applicative f => Applicative Alternate", "Bind f => Bind Alternate", "Monad Additive", "Extend Additive", "Comonad Additive", "Show (Alternate f a)", "Semigroup (Alternate f a)", "Monoid (Alternate f a)" ]
}
Raw
purs-typeclasses.json
{
"plus": {
"name": "Plus",
"info": "Left and Right identity for Alt",
"parent": "alt",
"package": "Control*",
"members": [
{
"name": "empty",
"def": "f a",
"laws": [
{
"law": "associativity",
"example": {
"left": "(x \u003c|\u003e y) \u003c|\u003e z",
"right": "x \u003c|\u003e (y \u003c|\u003e z)"
}
},
{
"law": "distributivity",
"example": {
"left": "f \u003c$\u003e (x \u003c|\u003e y)",
"right": "(f \u003c$\u003e x) \u003c|\u003e (f \u003c$\u003e y)"
}
}
]
}
],
"instances": [
"Array"
]
},
"comonad": {
"name": "Comonad",
"info": "Left and Right identity for Alt",
"parent": "extend",
"package": "Control*",
"members": [
{
"name": "extract",
"def": "w a -\u003e a",
"laws": [
{
"law": "left identity",
"example": {
"left": "extract \u003c\u003c== xs",
"right": "xs"
}
},
{
"law": "right identity",
"example": {
"left": "extract (f \u003c\u003c== xs)",
"right": "f xs"
}
}
]
}
],
"instances": []
},
"semigroupoid": {
"name": "Semigroupoid",
"info": "Category without identity",
"package": "Control",
"members": [
{
"name": "compose",
"def": "a c d -\u003e a b c -\u003e a b d",
"op": "\u003c\u003c\u003c",
"opEmoji": "🔙",
"laws": [
{
"law": "associativity",
"example": {
"left": "p \u003c\u003c\u003c (q \u003c\u003c\u003c r)",
"right": "(p \u003c\u003c\u003c q) \u003c\u003c\u003c r"
}
}
]
},
{
"name": "composeFlipped",
"def": "Semigroupoid a =\u003e a b c -\u003e a c d -\u003e a b d",
"op": "\u003e\u003e\u003e",
"opEmoji": "🔜"
}
],
"instances": []
},
"category": {
"name": "Category",
"info": "Objects and composable morphisms with their identity",
"parent": "semigroupoid",
"package": "Control",
"members": [
{
"name": "id",
"def": "a t t",
"laws": [
{
"law": "identity",
"example": {
"left": "id \u003c\u003c\u003c p",
"middle": "p \u003c\u003c\u003c id",
"right": "p"
}
}
]
}
],
"instances": [
"(-\u003e)"
]
},
"lazy": {
"name": "Lazy",
"info": "Deferred operations",
"package": "Control*",
"members": [
{
"name": "defer",
"def": "(Unit -\u003e l) -\u003e l"
},
{
"name": "fix",
"def": "Lazy l =\u003e (l -\u003e l) -\u003e l"
}
],
"instances": [
"(a -\u003e b)",
"Unit"
]
},
"alt": {
"name": "Alt",
"info": "Associative operation on a constructor",
"parent": "functor",
"package": "Control*",
"members": [
{
"name": "alt",
"def": "f a -\u003e f a -\u003e f a",
"op": "\u003c|\u003e",
"opEmoji": "🔗",
"laws": [
{
"law": "associativity",
"example": {
"left": "(x \u003c|\u003e y) \u003c|\u003e z",
"right": "x \u003c|\u003e (y \u003c|\u003e z)"
}
},
{
"law": "distributivity",
"example": {
"left": "f \u003c$\u003e (x \u003c|\u003e y)",
"right": "(f \u003c$\u003e x) \u003c|\u003e (f \u003c$\u003e y)"
}
}
]
}
],
"instances": [
"Array"
]
},
"extend": {
"name": "Extend",
"info": "Extend local computation to a global one",
"parent": "functor",
"package": "Control*",
"members": [
{
"name": "extend",
"def": "m a -\u003e (a -\u003e m b) -\u003e m b",
"op": "\u003c\u003c==",
"laws": [
{
"law": "associativity",
"example": {
"left": "extend f \u003c\u003c\u003c extend g",
"right": "extend (f \u003c\u003c\u003c extend g)"
}
}
]
},
{
"name": "extendFlipped",
"def": "Extend w =\u003e w a -\u003e (w a -\u003e b) -\u003e w b",
"op": "==\u003e\u003e"
},
{
"name": "composeCoKleisli",
"def": "Extend w =\u003e (w a -\u003e b) -\u003e (w b -\u003e c) -\u003e w a -\u003e c",
"op": "=\u003e="
},
{
"name": "composeCoKleisliFlipped",
"def": "Extend w =\u003e (w b -\u003e c) -\u003e (w a -\u003e b) -\u003e w a -\u003e c",
"op": "=\u003c="
},
{
"name": "duplicate",
"def": "Extend w =\u003e w a -\u003e w (w a)"
}
],
"instances": [
"Semigroup w =\u003e Extend ((-\u003e) w)",
"Array"
]
},
"bind": {
"name": "Bind",
"info": "Compose computations",
"parent": "apply",
"members": [
{
"name": "bind",
"def": "m a -\u003e (a -\u003e m b) -\u003e m b",
"op": "\u003e\u003e=",
"opEmoji": "🎉",
"laws": [
{
"law": "associativity",
"example": {
"left": "(x \u003e\u003e= f) \u003e\u003e= g",
"right": "x \u003e\u003e= (\\k -\u003e f k \u003e\u003e= g)"
}
}
]
},
{
"name": "bindFlipped",
"def": "Bind m =\u003e (a -\u003e m b) -\u003e m a -\u003e m b",
"op": "==\u003c\u003c"
},
{
"name": "join",
"def": "Bind m =\u003e m (m a) -\u003e m a"
},
{
"name": "composeKleisli",
"def": "Bind m =\u003e (a -\u003e m b) -\u003e (b -\u003e m c) -\u003e a -\u003e m c",
"op": "\u003e==\u003e"
},
{
"name": "composeKleisliFlipped",
"def": "Bind m =\u003e (b -\u003e m c) -\u003e (a -\u003e m b) -\u003e a -\u003e m c",
"op": "\u003c==\u003c"
},
{
"name": "ifM",
"def": "Bind m =\u003e m Boolean -\u003e m a -\u003e m a -\u003e m a"
}
],
"instances": [
"((-\u003e) r)",
"Array"
]
},
"monad": {
"name": "Monad",
"info": "Compose computations",
"laws": [
{
"law": "left identity",
"example": {
"left": "pure x \u003e\u003e= f",
"right": "f x"
}
},
{
"law": "right identity",
"example": {
"left": "x \u003e\u003e= pure",
"right": "x"
}
}
],
"members": [
{
"name": "liftM1",
"def": "Monad m =\u003e (a -\u003e b) -\u003e m a -\u003e m b"
},
{
"name": "ap",
"def": "Monad m =\u003e m (a -\u003e b) -\u003e m a -\u003e m b"
},
{
"name": "whenM",
"def": "Monad m =\u003e m Boolean -\u003e m Unit -\u003e m Unit"
},
{
"name": "unlessM",
"def": "Monad m =\u003e m Boolean -\u003e m Unit -\u003e m Unit"
}
],
"parents": [
"bind",
"applicative"
],
"instances": [
"((-\u003e) r)",
"Array"
]
},
"monadZero": {
"name": "MonadZero",
"info": "Compose computations",
"parents": [
"monad",
"alternative"
],
"laws": [
{
"law": "annihilation",
"example": {
"left": "empty \u003e\u003e= f",
"right": "empty"
}
}
],
"members": {
"name": "guard",
"def": "MonadZero m =\u003e Boolean -\u003e m Unit"
},
"instances": [
"Array"
]
},
"monadPlus": {
"name": "MonadPluse",
"info": "Distributivity for Monads",
"parent": "monadZero",
"laws": [
{
"law": "distributivity",
"example": {
"left": "(x \u003c|\u003e y) \u003e\u003e= f",
"right": "(x \u003e\u003e= f) \u003c|\u003e (y \u003e\u003e= f)"
}
}
],
"members": {
"name": "guard",
"def": "MonadZero m =\u003e Boolean -\u003e m Unit"
},
"instances": [
"Array"
]
},
"functor": {
"name": "Functor",
"info": "Convert and forget",
"members": [
{
"name": "map",
"def": "(a -\u003e b) -\u003e f a -\u003e f b",
"op": "\u003c$\u003e",
"opEmoji": "🚂",
"laws": [
{
"law": "identity",
"example": {
"left": "(\u003c$\u003e) id",
"right": "id"
}
},
{
"law": "composition",
"example": {
"left": "(\u003c$\u003e) (f \u003c\u003c\u003c g)",
"right": "(f \u003c$\u003e) \u003c\u003c\u003c (g \u003c$\u003e)"
}
}
]
},
{
"name": "mapFlipped",
"def": "Functor f =\u003e f a -\u003e (a -\u003e b) -\u003e f b",
"op": "\u003c#\u003e"
},
{
"name": "void",
"def": "Functor f =\u003e f a -\u003e f Unit"
},
{
"name": "voidRight",
"op": "\u003c$",
"def": "Functor f =\u003e a -\u003e f b -\u003e f a"
},
{
"name": "voidLeft",
"op": "$\u003e",
"def": "Functor f =\u003e f a -\u003e b -\u003e f b"
},
{
"name": "flap",
"op": "\u003c@\u003e",
"def": "Functor f =\u003e f (a -\u003e b) -\u003e a -\u003e f b"
}
],
"instances": [
"Array",
"((-\u003e) r)"
]
},
"apply": {
"name": "Apply",
"info": "Unwrap, convert, and wrap again",
"parent": "functor",
"members": [
{
"name": "apply",
"def": "f (a -\u003e b) -\u003e f a -\u003e f b",
"op": "\u003c*\u003e",
"opEmoji": "🚋",
"laws": [
{
"law": "ap. composition",
"example": {
"left": "(\u003c\u003c\u003c) \u003c$\u003e f \u003c*\u003e g \u003c*\u003e h",
"right": "f \u003c*\u003e (g \u003c*\u003e h)"
}
}
]
},
{
"name": "applyFirst",
"def": "Apply f =\u003e f a -\u003e f b -\u003e f a",
"op": "\u003c*"
},
{
"name": "applySecond",
"def": "Apply f =\u003e f a -\u003e f b -\u003e f b",
"op": "*\u003e",
"opEmoji": "👉"
},
{
"name": "lift2",
"def": "Apply f =\u003e (a -\u003e b -\u003e c) -\u003e f a -\u003e f b -\u003e f c"
},
{
"name": "lift3",
"def": "Apply f =\u003e (a -\u003e b -\u003e c -\u003e d) -\u003e f a -\u003e f b -\u003e f c -\u003e f d"
},
{
"name": "lift4",
"def": "Apply f =\u003e (a -\u003e b -\u003e c -\u003e d -\u003e e) -\u003e f a -\u003e f b -\u003e f c -\u003e f d -\u003e f e"
},
{
"name": "lift5",
"def": "Apply f =\u003e (a -\u003e b -\u003e c -\u003e d -\u003e e -\u003e g) -\u003e f a -\u003e f b -\u003e f c -\u003e f d -\u003e f e -\u003e f g"
}
],
"instances": [
"Array",
"((-\u003e) r)"
]
},
"applicative": {
"name": "Applicative",
"info": "Lift with zero arguments, wrap values",
"parent": "apply",
"members": [
{
"name": "pure",
"def": "a -\u003e f a",
"laws": [
{
"law": "identity",
"example": {
"left": "(pure id) \u003c*\u003e v",
"right": "v"
}
},
{
"law": "composition",
"example": {
"left": "(pure \u003c\u003c\u003c) \u003c*\u003e f \u003c*\u003e g \u003c*\u003e h",
"right": "f \u003c*\u003e (g \u003c*\u003e h)"
}
},
{
"law": "homomorphism",
"example": {
"left": "(pure f) \u003c*\u003e (pure x)",
"right": "pure (f x)"
}
},
{
"law": "interchange",
"example": {
"left": "u \u003c*\u003e (pure y)",
"right": "(pure ($ y)) \u003c*\u003e u"
}
}
]
},
{
"name": "liftA1",
"def": "Applicative f =\u003e (a -\u003e b) -\u003e f a -\u003e f b"
},
{
"name": "when",
"def": "Applicative m =\u003e Boolean -\u003e m Unit -\u003e m Unit"
},
{
"name": "unless",
"def": "Applicative m =\u003e Boolean -\u003e m Unit -\u003e m Unit"
}
],
"instances": [
"Array",
"((-\u003e) r)"
]
},
"alternative": {
"name": "Alternative",
"info": "To have both Plus and Applicative",
"parents": [
"plus",
"applicative"
],
"laws": [
{
"law": "distributivity",
"example": {
"left": "(f \u003c|\u003e g) \u003c*\u003e x",
"right": "(f \u003c*\u003e x) \u003c|\u003e (g \u003c*\u003e x)"
}
},
{
"law": "annihilation",
"example": {
"left": "empty \u003c*\u003e f",
"right": "empty"
}
}
]
},
"unit": {
"name": "Unit",
"info": "No computation needed",
"package": "Data",
"members": [
{
"name": "unit",
"def": "Unit"
}
],
"instances": [
"Show Unit"
]
},
"void": {
"name": "Void",
"info": "Uninhabited data type",
"package": "Data",
"members": [
{
"name": "absurd",
"def": "Void -\u003e a",
"opEmoji": "💣"
}
],
"instances": [
"Show Void"
]
},
"function": {
"name": "Function",
"info": "Helpers for the core functions",
"package": "Data",
"members": [
{
"name": "flip",
"def": "(a -\u003e b -\u003e c) -\u003e b -\u003e a -\u003e c"
},
{
"name": "const",
"def": "a -\u003e b -\u003e a"
},
{
"name": "apply",
"def": "(a -\u003e b) -\u003e a -\u003e b",
"op": "$",
"opEmoji": "💨"
},
{
"name": "applyFlipped",
"def": "a -\u003e (a -\u003e b) -\u003e b",
"op": "#"
},
{
"name": "on",
"def": "(b -\u003e b -\u003e c) -\u003e (a -\u003e b) -\u003e a -\u003e a -\u003e c"
}
]
},
"naturalTransformation": {
"name": "NaturalTransformation",
"info": "Mapping b/w type constructors with no manipulation on inner value",
"package": "Data",
"members": [
{
"name": "NaturalTransformation",
"def": "f a -\u003e g a",
"op": "~\u003e",
"opEmoji": "🐛"
}
]
},
"semigroup": {
"name": "Semigroup",
"info": "Associativity",
"package": "Data",
"members": [
{
"name": "append",
"def": "a -\u003e a -\u003e a",
"op": "\u003c\u003e",
"opEmoji": "🙏",
"laws": [
{
"law": "associativity",
"example": {
"left": "(x \u003c\u003e y) \u003c\u003e z",
"right": "x \u003c\u003e (y \u003c\u003e z)"
}
}
]
}
],
"instances": [
"String",
"Unit",
"Void",
"Array a",
"Semigroup s' =\u003e Semigroup (s -\u003e s')"
]
},
"monoid": {
"name": "Monoid",
"parent": "semigroup",
"info": "Folding empty collection",
"members": [
{
"name": "mempty",
"def": "m",
"laws": [
{
"law": "associativity",
"example": {
"left": "mempty \u003c\u003e x",
"middle": "x \u003c\u003e mempty",
"right": "x forall x"
}
}
]
},
{
"name": "power",
"def": "Monoid m =\u003e m -\u003e Int -\u003e m"
},
{
"name": "guard",
"def": "Monoid m =\u003e Boolean -\u003e m -\u003e m"
}
],
"instances": [
"Unit",
"Ordering",
"Number",
"String",
"Array a",
"Monoid b =\u003e Monoid (a -\u003e b)"
]
},
"eq": {
"name": "Eq",
"package": "Data",
"info": "Equality",
"members": [
{
"name": "eq",
"def": "a -\u003e a -\u003e Boolean",
"op": "==",
"laws": [
{
"law": "reflexivity",
"example": {
"left": "x == x",
"right": "true"
}
},
{
"law": "symmetry",
"example": {
"left": "x == y",
"right": "y == x"
}
},
{
"law": "transitivity",
"example": {
"fact": "(x == y) and (y == z)",
"conclusion": "y == x"
}
}
]
},
{
"name": "notEq",
"op": "/=",
"def": "Eq a =\u003e a -\u003e a -\u003e Boolean"
}
],
"instances": [
"Boolean",
"Int",
"Number",
"Char",
"String",
"Unit",
"Void",
"Eq a =\u003e Eq (Array a)"
]
},
"ord": {
"name": "Ord",
"info": "Ordering",
"parent": "eq",
"members": [
{
"name": "compart",
"def": "a -\u003e a -\u003e Ordering",
"op": "==",
"laws": [
{
"law": "reflexivity",
"example": {
"fact": "a \u003c= a"
}
},
{
"law": "antisymmetry",
"example": {
"fact": "(a \u003c= b) and (b \u003c= a)",
"conclusion": "a = b"
}
},
{
"law": "transitivity",
"example": {
"fact": "(a \u003c= b) and (b \u003c= c)",
"conclusion": "a \u003c= c"
}
}
]
},
{
"name": "lessThan",
"op": "\u003c",
"def": "Ord a =\u003e a -\u003e a -\u003e Boolean"
},
{
"name": "greaterThan",
"op": "\u003e",
"def": "Ord a =\u003e a -\u003e a -\u003e Boolean"
},
{
"name": "lessThanOrEq",
"op": "\u003c=",
"def": "Ord a =\u003e a -\u003e a -\u003e Boolean"
},
{
"name": "greaterThanOrEq",
"op": "\u003e=",
"def": "Ord a =\u003e a -\u003e a -\u003e Boolean"
},
{
"name": "min",
"def": "Ord a =\u003e a -\u003e a -\u003e a"
},
{
"name": "max",
"def": "Ord a =\u003e a -\u003e a -\u003e a"
},
{
"name": "clamp",
"def": "Ord a =\u003e a -\u003e a -\u003e a -\u003e a"
},
{
"name": "between",
"def": "Ord a =\u003e a -\u003e a -\u003e a -\u003e Boolean"
},
{
"name": "abs",
"def": "(Ord a, Ring a) =\u003e a -\u003e a"
},
{
"name": "signum",
"def": "(Ord a, Ring a) =\u003e a -\u003e a"
},
{
"name": "comparing",
"def": "Ord b =\u003e (a -\u003e b) -\u003e (a -\u003e a -\u003e Ordering)"
}
]
},
"bounded": {
"name": "Bounded",
"parent": "ord",
"info": "Have upper and lower boundary",
"package": "Data",
"laws": [
{
"name": "bounded",
"fact": "bottom \u003c= a \u003c= top"
}
],
"members": [
{
"name": "top",
"def": "a"
},
{
"name": "bottom",
"def": "a"
}
],
"instances": [
"Boolean",
"Int",
"Char",
"Ordering",
"Unit"
]
},
"ordering": {
"name": "Ordering",
"info": "Just to know the order",
"values": [
"LT",
"GT",
"EQ"
],
"members": [
{
"name": "invert",
"def": "Ordering -\u003e Ordering"
}
],
"instances": [
"Eq Ordering",
"Semigroup Ordering",
"Show Ordering"
]
},
"show": {
"name": "Show",
"info": "Displaying values",
"package": "Data",
"members": [
{
"name": "show",
"def": "a -\u003e String"
}
],
"instances": [
"Show Boolean",
"Show Int",
"Show Number",
"Show Char",
"Show String",
"Show a =\u003e Show (Array a)"
]
},
"heytingAlgebra": {
"name": "HeytingAlgebra",
"info": "Bounded lattices, boolean starters",
"members": [
{
"name": "ff",
"def": "a"
},
{
"name": "tt",
"def": "a"
},
{
"name": "implies",
"def": "a -\u003e a -\u003e a"
},
{
"name": "conj",
"def": "a -\u003e a -\u003e a",
"op": "\u0026\u0026"
},
{
"name": "disj",
"def": "a -\u003e a -\u003e a",
"op": "||"
},
{
"name": "not",
"def": "a"
}
],
"laws": [
{
"law": "associativity",
"examples": [
{
"left": "a || (b || c)",
"right": "(a || b) || c"
},
{
"left": "a \u0026\u0026 (b \u0026\u0026 c)",
"right": "(a \u0026\u0026 b) \u0026\u0026 c"
}
]
},
{
"law": "commutativity",
"examples": [
{
"left": "a || b",
"right": "b || a"
},
{
"left": "a \u0026\u0026 b",
"right": "b \u0026\u0026 a"
}
]
},
{
"law": "absorption",
"examples": [
{
"left": "a || (a \u0026\u0026 b)",
"right": "a"
},
{
"left": "a \u0026\u0026 a",
"right": "a"
}
]
},
{
"law": "idempotent",
"examples": [
{
"left": "a || a",
"right": "a"
},
{
"left": "a \u0026\u0026 (a || b)",
"right": "a"
}
]
},
{
"law": "identity",
"examples": [
{
"left": "a || ff",
"right": "a"
},
{
"left": "a \u0026\u0026 tt",
"right": "a"
}
]
},
{
"law": "implication",
"examples": [
{
"left": "a `implies` a",
"right": "tt"
},
{
"left": "a \u0026\u0026 (a `implies` b)",
"right": "a \u0026\u0026 b"
},
{
"left": "b \u0026\u0026 (a `implies` b)",
"right": "b"
},
{
"left": "a `implies` (b \u0026\u0026 c)",
"right": "(a `implies` b) \u0026\u0026 (a `implies` c)"
}
]
},
{
"law": "complemented",
"example": {
"left": "not a",
"right": "a `implies` ff"
}
}
],
"instances": [
"Boolean",
"Unit",
"(a -\u003e b)"
]
},
"booleanAlgebra": {
"name": "BooleanAlgebra",
"info": "Behave like boolean",
"parent": "heytingAlgebra",
"laws": [
{
"name": "excluded middle",
"def": "a || not a == tt"
}
],
"instances": [
"Boolean",
"Unit",
"BooleanAlgebra b =\u003e BooleanAlgebra (a -\u003e b)"
]
},
"boolean": {
"name": "Boolean",
"info": "Helpers for boolean types",
"members": [
{
"name": "otherwise",
"def": "Boolean"
}
]
},
"divisionRing": {
"name": "DivisionRing",
"parent": "ring",
"info": "Non-zero rings in which every non-zero element has a multiplicative inverse / skew fields",
"members": [
{
"name": "recip",
"def": "a -\u003e a",
"laws": [
{
"law": "Non-zero Ring",
"fact": "one /= zero"
},
{
"law": "Non-zero multplicative inverse",
"left": "recip a * a",
"middle": "a * recip a",
"right": "one forall a /= 0"
}
]
},
{
"name": "leftDiv",
"def": "DivisionRing a =\u003e a -\u003e a -\u003e a"
},
{
"name": "rightDiv",
"def": "DivisionRing a =\u003e a -\u003e a -\u003e a"
}
],
"instances": [
"Number"
]
},
"semiring": {
"name": "Semiring",
"info": "Sum and Multiply (a.k.a. extended Monoid)",
"members": [
{
"name": "add",
"def": "a -\u003e a -\u003e a",
"op": "+"
},
{
"name": "zero",
"def": "a"
},
{
"name": "mul",
"def": "a -\u003e a -\u003e a",
"op": "*"
},
{
"name": "one",
"def": "a"
}
],
"laws": [
{
"law": "associativity",
"left": "(a + b) + c",
"right": "a + (b + c)"
},
{
"law": "identity",
"left": "zero + a",
"middle": "a + zero",
"right": "a"
},
{
"law": "commutative",
"left": "a + b",
"right": "b + a"
},
{
"law": "associativity",
"left": "(a + b) * c",
"right": "a * (b * c)"
},
{
"law": "identity",
"left": "one * a",
"middle": "a * one",
"right": "a"
},
{
"law": "left distributivity",
"left": "a * (b + c)",
"right": "(a * b) + (a * c)"
},
{
"law": "right distributivity",
"left": "a + (b * c)",
"right": "(a * c) + (b * c)"
},
{
"law": "annihilation",
"left": "zero * a",
"middle": "a * zero",
"right": "zero"
}
],
"instances": [
"Int",
"Number",
"Semiring b =\u003e Semiring (a -\u003e b)"
]
},
"ring": {
"parent": "semiring",
"name": "Ring",
"info": "Subtraction",
"members": [
{
"name": "sub",
"def": "a -\u003e a -\u003e a",
"op": "-",
"laws": [
{
"law": "Additive inverse",
"left": "a - a",
"middle": "zero - a",
"right": "zero"
}
]
},
{
"name": "negate",
"def": "Ring a =\u003e a -\u003e a"
}
],
"instances": [
"Int",
"Number",
"Unit",
"Ring b =\u003e Ring (a -\u003e b)"
]
},
"commutativeRing": {
"name": "CommutativeRing",
"info": "Multiplication behaves commutatively",
"parent": "ring",
"laws": [
{
"law": "commutative",
"left": "a * b",
"right": "b * a"
}
],
"instances": [
"Int",
"Number",
"Unit",
"CommutativeRing b =\u003e CommutativeRing (a -\u003e b)"
]
},
"euclidianRing": {
"name": "EuclidianRing",
"info": "Divide and Conquer",
"parent": "commutativeRing",
"members": [
{
"name": "degree",
"def": "a -\u003e Int"
},
{
"name": "div",
"def": "a -\u003e a -\u003e a",
"op": "/"
},
{
"name": "mode",
"def": "a -\u003e a -\u003e a"
},
{
"name": "gcd",
"def": "Eq a =\u003e EuclideanRing a =\u003e a -\u003e a -\u003e a"
},
{
"name": "lcm",
"def": "Eq a =\u003e EuclideanRing a =\u003e a -\u003e a -\u003e a"
}
],
"laws": [
{
"law": "integral domain",
"fact": "(a /= 0) and (b /= 0)",
"conclusion": "a * b /= 0"
},
{
"law": "multiplicative Euclidean function",
"right": "a",
"left": "(a / b) * b + (a `mod` b)",
"conditions": [
"degree a \u003e 0",
"degree a \u003c= degree (a * b)"
]
}
],
"instances": [
"Int",
"Number"
]
},
"field": {
"info": "Commutative fields",
"parent": "euclidianRing",
"laws": [
{
"law": "non-zero multiplicative inverse",
"right": "a `mod` b",
"left": "0"
}
],
"instances": [
"Int",
"Number"
]
},
"endo": {
"name": "Endo",
"info": "By itself",
"package": "Data.Monoid",
"statements": [
{
"left": "Endo f \u003c\u003e Endo g",
"right": "Endo (f \u003c\u003c\u003c g)"
},
{
"left": "mempty :: Endo _",
"right": "Endo id"
}
],
"instances": [
"Newtype (Endo a) _",
"Invariant Endo",
"Semigroup (Endo a)",
"Monoid (Endo a)"
]
},
"dual": {
"name": "Dual",
"info": "Monoid under dualism",
"package": "Data.Monoid",
"statements": [
{
"left": "Dual x \u003c\u003e Dual y",
"right": "Dual (y \u003c\u003e x)"
},
{
"left": "mempty :: Dual _",
"right": "Dual mempty"
}
],
"instances": [
"Newtype (Dual a) _",
"Eq a =\u003e Eq (Dual a)",
"Ord a =\u003e Ord (Dual a)",
"Bounded a =\u003e Bounded (Dual a)",
"Functor Dual",
"Invariant Dual",
"Apply Dual",
"Applicative Dual",
"Bind Dual",
"Monad Dual",
"Extend Dual",
"Comonad Dual",
"Show a =\u003e Show (Dual a)",
"Semigroup a =\u003e Semigroup (Dual a)",
"Monoid a =\u003e Monoid (Dual a)"
]
},
"additive": {
"name": "Additive",
"info": "May be added to something",
"package": "Data.Monoid",
"statements": [
{
"left": "Additive x \u003c\u003e Additive y",
"right": "Additive (x + y)"
},
{
"left": "mempty :: Additive _",
"right": "Additive zero"
}
],
"instances": [
"Newtype (Additive a) _",
"Eq a =\u003e Eq (Additive a)",
"Ord a =\u003e Ord (Additive a)",
"Bounded a =\u003e Bounded (Additive a)",
"Functor Additive",
"Invariant Additive",
"Apply Additive",
"Applicative Additive",
"Bind Additive",
"Monad Additive",
"Extend Additive",
"Comonad Additive",
"Show a =\u003e Show (Additive a)",
"Semiring a =\u003e Semigroup (Additive a)",
"Semiring a =\u003e Monoid (Additive a)"
]
},
"multiplicative": {
"name": "Multiplicative",
"info": "May be multiplied on something",
"package": "Data.Monoid",
"statements": [
{
"left": "Multiplicative x \u003c\u003e Multiplicative y",
"right": "Multiplicative (x * y)"
},
{
"left": "mempty :: Multiplicative _",
"right": "Multiplicative one"
}
],
"instances": [
"Newtype (Multiplicative a) _",
"Eq a =\u003e Eq (Multiplicative a)",
"Ord a =\u003e Ord (Multiplicative a)",
"Bounded a =\u003e Bounded (Multiplicative a)",
"Functor Multiplicative",
"Invariant Multiplicative",
"Apply Multiplicative",
"Applicative Multiplicative",
"Bind Multiplicative",
"Monad Multiplicative",
"Extend Multiplicative",
"Comonad Multiplicative",
"Show a =\u003e Show (Multiplicative a)",
"Semiring a =\u003e Semigroup (Multiplicative a)",
"Semiring a =\u003e Monoid (Multiplicative a)"
]
},
"conj": {
"name": "Multiplicative",
"info": "Monoid under conjunction",
"package": "Data.Monoid",
"statements": [
{
"left": "Conj x \u003c\u003e Conj y",
"right": "Conj (x \u0026\u0026 y)"
},
{
"left": "mempty :: Conj _",
"right": "Conj top"
}
],
"instances": [
"Newtype (Conj a) _",
"Eq a =\u003e Eq (Conj a)",
"Ord a =\u003e Ord (Conj a)",
"Bounded a =\u003e Bounded (Conj a)",
"Functor Conj",
"Invariant Conj",
"Apply Conj",
"Applicative Conj",
"Bind Conj",
"Monad Conj",
"Extend Conj",
"Comonad Conj",
"Show a =\u003e Show (Conj a)",
"HeytingAlgebra a =\u003e Semigroup (Conj a)",
"HeytingAlgebra a =\u003e Monoid (Conj a)",
"HeytingAlgebra a =\u003e Semiring (Conj a)"
]
},
"disj": {
"name": "Disj",
"info": "Monoid under disjunction",
"package": "Data.Monoid",
"statements": [
{
"left": "Conj x \u003c\u003e Conj y",
"right": "Disj (x || y)"
},
{
"left": "mempty :: Disj _",
"right": "Disj bottom"
}
],
"instances": [
"Newtype (Disj a) _",
"Eq a =\u003e Eq (Disj a)",
"Ord a =\u003e Ord (Disj a)",
"Bounded a =\u003e Bounded (Disj a)",
"Functor Disj",
"Invariant Disj",
"Apply Disj",
"Applicative Disj",
"Bind Disj",
"Monad Disj",
"Extend Disj",
"Comonad Disj",
"Show a =\u003e Show (Disj a)",
"HeytingAlgebra a =\u003e Semigroup (Disj a)",
"HeytingAlgebra a =\u003e Monoid (Disj a)",
"HeytingAlgebra a =\u003e Semiring (Disj a)"
]
},
"alternate": {
"name": "Alternate",
"info": "Monoid and Semigroup instances corresponding to Plus and Alt for f",
"package": "Data.Monoid",
"statements": [
{
"left": "Alternate fx \u003c\u003e Alternate fy",
"right": "Alternate (fx \u003c|\u003e fy)"
},
{
"left": "mempty :: Alternate _",
"right": "Alternate empty"
}
],
"instances": [
"Newtype (Alternate f a) _",
"Eq a =\u003e Eq (Alternate f)",
"Ord a =\u003e Ord (Alternate a)",
"Bounded a =\u003e Bounded (Alternate a)",
"Functor f =\u003e Functor Alternate",
"Invariant f =\u003e Invariant (Alternate f)",
"Apply f =\u003e Apply Alternate",
"Applicative f =\u003e Applicative Alternate",
"Bind f =\u003e Bind Alternate",
"Monad Additive",
"Extend Additive",
"Comonad Additive",
"Show (Alternate f a)",
"Semigroup (Alternate f a)",
"Monoid (Alternate f a)"
]
}
}
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