Notation: mathematics, programming languages
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meaning | mathematics | Haskell | Coq |
---|---|---|---|
C | flip | ||
K | const | ||
identity | I | id ($) |
|
fixed point combinators | Y | fix | fix |
composition of functions | ∘ B (combinator) |
. | |
λ-abstraction | λxy (λ-calculus) λx.y (λ-calculus) ↦ |
\ x -> y | fun x => y |
not equal | ≠ | /= | <> (Prop) |
equal | = | == | = (Prop) |
lower or equal | ≤ | <= | <= |
greater or equal | ≥ | >= | >= |
is of type | name : type | name :: type | name : type |
logical negation | ¬ | not | ~ (Prop) |
material implication | → ⊃ |
-> | |
function type constructor | → | -> | -> |
categorical product | a × b | (a, b) | a * b |
conjunction | ∧ | && | /\ (Prop) |
categorical sum | + | Either | + (Type) |
disjunction | ∨ | || | \/ (Prop) |
verum | ⊤ | True | True (Prop) |
falsum, absurd, contradiction | ⊥ | False | False (Prop) |
empty set | ∅ | Empty_set | |
singleton set | {*} {∙} |
() | unit |
inhabitant of the singleton set | * ∙ |
() | tt |
universal quantification | ∀ | forall | forall |
existential quantification | ∃ | forall (before data constructor) | exists ex sig sigT |
constructing a list from elements | ⟨x, y, z⟩ | [x, y, z] | |
empty list | [] | nil | |
constructing a list from a head and a tail | x : xs | x :: xs | |
concatenation of lists | ⧺ | ++ | app |
tuple constructor | ⟨x, y, z⟩ | (x, y, z) | (x, y, z) |
natural numbers | ℕ | nat | |
covariant Hom functor | Hom(a,−) | Reader a | |
contravariant Hom functor | Hom(−,a) | Writer a | |
functor | Functor | ||
lax monoidal functor | Applicative | ||
monad | Monad | ||
neutral element of a monad | η | return | |
binary operation of a monad | μ | join | |
double arrow | ⇒ | => | => |