- class C a where
- subtract :: C a => a -> a -> a
- sum :: C a => [a] -> a
- sum1 :: C a => [a] -> a
- elementAdd :: C x => (v -> x) -> T (v, v) x
- elementSub :: C x => (v -> x) -> T (v, v) x
- elementNeg :: C x => (v -> x) -> T v x
- (<*>.+) :: C x => T (v, v) (x -> a) -> (v -> x) -> T (v, v) a
- (<*>.-) :: C x => T (v, v) (x -> a) -> (v -> x) -> T (v, v) a
- (<*>.-$) :: C x => T v (x -> a) -> (v -> x) -> T v a
- propAssociative :: (Eq a, C a) => a -> a -> a -> Bool
- propCommutative :: (Eq a, C a) => a -> a -> Bool
- propIdentity :: (Eq a, C a) => a -> Bool
- propInverse :: (Eq a, C a) => a -> Bool

# Class

Additive a encapsulates the notion of a commutative group, specified by the following laws:

a + b === b + a (a + b) + c === a + (b + c) zero + a === a a + negate a === 0

Typical examples include integers, dollars, and vectors.

C Double | |

C Float | |

C Int | |

C Int8 | |

C Int16 | |

C Int32 | |

C Int64 | |

C Integer | |

C Word | |

C Word8 | |

C Word16 | |

C Word32 | |

C Word64 | |

C T | |

C T | |

C T | |

C T | |

C v => C [v] | |

Integral a => C (Ratio a) | |

(Ord a, C a) => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

(C a, C a, C a) => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

(C a, C a) => C (T a) | |

C a => C (T a) | |

(Eq a, C a) => C (T a) | |

(Eq a, C a) => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C a => C (T a) | |

C v => C (b -> v) | |

(C v0, C v1) => C (v0, v1) | |

(Ord i, Eq v, C v) => C (Map i v) | |

(Ord a, C b) => C (T a b) | |

(C u, C a) => C (T u a) | |

C v => C (T a v) | |

(Ord i, C a) => C (T i a) | |

C v => C (T a v) | |

(C v0, C v1, C v2) => C (v0, v1, v2) |

subtract :: C a => a -> a -> aSource

`subtract`

is `(-)`

with swapped operand order.
This is the operand order which will be needed in most cases
of partial application.

# Complex functions

Sum up all elements of a list. An empty list yields zero.

This function is inappropriate for number types like Peano.
Maybe we should make `sum`

a method of Additive.
This would also make `lengthLeft`

and `lengthRight`

superfluous.

Sum up all elements of a non-empty list. This avoids including a zero which is useful for types where no universal zero is available.

# Instance definition helpers

elementAdd :: C x => (v -> x) -> T (v, v) xSource

Instead of baking the add operation into the element function,
we could use higher rank types
and pass a generic `uncurry (+)`

to the run function.
We do not do so in order to stay Haskell 98
at least for parts of NumericPrelude.

elementSub :: C x => (v -> x) -> T (v, v) xSource

elementNeg :: C x => (v -> x) -> T v xSource

(<*>.+) :: C x => T (v, v) (x -> a) -> (v -> x) -> T (v, v) aSource

addPair :: (Additive.C a, Additive.C b) => (a,b) -> (a,b) -> (a,b) addPair = Elem.run2 $ Elem.with (,) <*>.+ fst <*>.+ snd

# Instances for atomic types

propAssociative :: (Eq a, C a) => a -> a -> a -> BoolSource

propCommutative :: (Eq a, C a) => a -> a -> BoolSource

propIdentity :: (Eq a, C a) => a -> BoolSource

propInverse :: (Eq a, C a) => a -> BoolSource