-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/


-- | Tensor and Group typeclasses
--   
--   Typeclasses for Groups (Monoids with an <a>invert</a> operation) and
--   Tensors.
@package data-tensor
@version 0.1.0.0


-- | Typeclasses for Group and Tensor, extending <a>Monoid</a>.
module Data.Tensor

-- | A group is a monoid with an invert operation. Intuition:
--   <a>&gt;&lt;</a> is to <a>&lt;&gt;</a> what subtraction is to addition;
--   <a>invert</a> turns a value into its complement (see Laws below), and
--   corresponds with unary minus in addition.
--   
--   Laws:
--   
--   <pre>
--   a &gt;&lt; b == a &lt;&gt; (invert b)
--   a &gt;&lt; mempty == a
--   a &gt;&lt; a == mempty
--   a &lt;&gt; (invert a) == mempty
--   invert mempty == mempty
--   </pre>
class Monoid a => Group a where a >< b = a <> invert b invert x = mempty >< x

-- | Dual to <a>&lt;&gt;</a>.
(><) :: Group a => a -> a -> a

-- | "Negation": convert an operand into its dual.
invert :: Group a => a -> a

-- | Tensor allows us to define a relationship between two types, the
--   second one forming a Group. The intuition is that the first type
--   models something like a "location", and the second (the group) models
--   the relative distance between two locations. Examples of Tensors
--   include date/time values (point in time) and timespans; positions in a
--   vector space and displacement vectors; pitches and intervals in music.
--   
--   Tensor provides three operations: <a>?&lt;&gt;</a> ("tensor
--   addition"), adding a "distance" to a "location"; <a>?&gt;&lt;</a>
--   ("tensor subtraction"), undoing the effect of adding a "distance" to a
--   "location", and <a>&gt;?&lt;</a>, getting the "distance" between two
--   "locations".
--   
--   Laws:
--   
--   <pre>
--   a ?&lt;&gt; (b &gt;?&lt; a) == b
--   a ?&lt;&gt; (x &lt;&gt; y) == a ?&lt;&gt; x ?&lt;&gt; y
--   a ?&gt;&lt; b == a ?&lt;&gt; (invert b)
--   a ?&lt;&gt; (x &gt;&lt; y) == a ?&lt;&gt; x ?&gt;&lt; y
--   </pre>
class Group b => Tensor a b where a ?>< b = a ?<> invert b
(?<>) :: Tensor a b => a -> b -> a
(?><) :: Tensor a b => a -> b -> a
(>?<) :: Tensor a b => a -> a -> b
instance GHC.Num.Num a => Data.Tensor.Group (Data.Monoid.Sum a)
instance Data.Tensor.Group a => Data.Tensor.Tensor a a