Create Field and Scalar interfaces

src/Data/NumIdr/Multiply.idr

@@ -1,70 +0,0 @@
-module Data.NumIdr.Multiply
-
-%default total
-
-
-infixr 9 *.
-infixr 10 ^
-
-||| A generalized multiplication/application operator. This interface is
-||| necessary since the standard multiplication operator is homogenous.
-|||
-||| All instances of this interface must collectively satisfy these axioms:
-||| * If `(x *. y) *. z` is defined, then `x *. (y *. z)` is defined and equal.
-||| * If `x *. (y *. z)` is defined, then `(x *. y) *. z` is defined and equal.
-public export
-interface Mult a b c | a,b where
-  (*.) : a -> b -> c
-
-public export
-Mult' : Type -> Type
-Mult' a = Mult a a a
-
-
-||| An interface for monoids using the `*.` operator.
-|||
-||| An instance of this interface must satisfy:
-||| * `x *. identity == x`
-||| * `identity *. x == x`
-public export
-interface Mult' a => MultMonoid a where
-  identity : a
-
-||| An interface for groups using the `*.` operator.
-|||
-||| An instance of this interface must satisfy:
-||| * `x *. inverse x == identity`
-||| * `inverse x *. x == identity`
-public export
-interface MultMonoid a => MultGroup a where
-  inverse : a -> a
-
-
-namespace Semigroup
-  ||| Multiplication forms a semigroup
-  public export
-  [Mult] Mult' a => Semigroup a where
-    (<+>) = (*.)
-
-namespace Monoid
-  ||| Multiplication with an identity element forms a monoid
-  public export
-  [Mult] MultMonoid a => Monoid a using Semigroup.Mult where
-    neutral = identity
-
-
-||| Raise a multiplicative value (e.g. a matrix or a transformation) to a natural
-||| number power.
-public export
-power : MultMonoid a => Nat -> a -> a
-power 0 _ = identity
-power 1 x = x
-power (S n@(S _)) x = x *. power n x
-
-||| Raise a multiplicative value (e.g. a matrix or a transformation) to a natural
-||| number power.
-|||
-||| This is the operator form of `power`.
-public export %inline
-(^) : MultMonoid a => a -> Nat -> a
-a ^ n = power n a