Copyright | (c) Justin Le 2017 |
---|---|

License | BSD3 |

Maintainer | justin@jle.im |

Stability | experimental |

Portability | non-portable |

Safe Haskell | None |

Language | Haskell2010 |

Offers full functionality for implicit-graph back-propagation. The
intended usage is to write a `BPOp`

, which is a normal Haskell
function from `BVar`

s to a result `BVar`

. These `BVar`

s can be
manipulated using their `Num`

/ `Fractional`

/ `Floating`

instances.

The library can then perform back-propagation on the function (using
`backprop`

or `grad`

) by using an implicitly built graph.

This should actually be powerful enough for most use cases, but falls short for a couple of situations:

- If the result of a function on
`BVar`

s is used twice (like`z`

in`let z = x * y in z + z`

), this will allocate a new redundant graph node for every usage site of`z`

. You can explicitly*force*`z`

, but only using an explicit graph description using Numeric.Backprop. - This can't handle sum types, like Numeric.Backprop can. You can
never pattern match on the constructors of a value inside a
`BVar`

. I'm not sure if this is a fundamental limitation (I suspect it might be) or if I just can't figure out how to implement it. Suggestions welcome!

As a comparison, this module offers functionality and an API very
similar to Numeric.AD.Mode.Reverse from the *ad* library, except for
the fact that it can handle *heterogeneous* values.

- type BPOp rs a = forall s. Prod (BVar s rs) rs -> BVar s rs a
- data BVar :: Type -> [Type] -> Type -> Type
- type Op as a = forall m. Monad m => OpM m as a
- type OpB s as a = OpM (ST s) as a
- data Prod k f a :: forall k. (k -> *) -> [k] -> * where
- type Tuple = Prod * I
- newtype I a :: * -> * = I {
- getI :: a

- backprop :: (Known Length rs, Every Num rs, Num a) => BPOp rs a -> Tuple rs -> (a, Tuple rs)
- grad :: (Known Length rs, Every Num rs, Num a) => BPOp rs a -> Tuple rs -> Tuple rs
- eval :: (Known Length rs, Every Num rs, Num a) => BPOp rs a -> Tuple rs -> a
- backprop' :: Prod Summer rs -> Prod Unity rs -> BPOp rs a -> Tuple rs -> (a, Tuple rs)
- grad' :: Prod Summer rs -> Prod Unity rs -> BPOp rs a -> Tuple rs -> Tuple rs
- constVar :: a -> BVar s rs a
- liftB :: OpB s as a -> Prod (BVar s rs) as -> BVar s rs a
- (.$) :: OpB s as a -> Prod (BVar s rs) as -> BVar s rs a
- liftB1 :: OpB s '[a] b -> BVar s rs a -> BVar s rs b
- liftB2 :: OpB s '[a, b] c -> BVar s rs a -> BVar s rs b -> BVar s rs c
- liftB3 :: OpB s '[a, b, c] d -> BVar s rs a -> BVar s rs b -> BVar s rs c -> BVar s rs d
- partsVar :: forall s rs bs a. (Known Length bs, Every Num bs) => Iso' a (Tuple bs) -> BVar s rs a -> Prod (BVar s rs) bs
- withParts :: forall s rs bs a r. (Known Length bs, Every Num bs) => Iso' a (Tuple bs) -> BVar s rs a -> (Prod (BVar s rs) bs -> r) -> r
- splitVars :: forall s rs as. (Known Length as, Every Num as) => BVar s rs (Tuple as) -> Prod (BVar s rs) as
- gSplit :: forall s rs as a. (Generic a, Code a ~ '[as], Known Length as, Every Num as) => BVar s rs a -> Prod (BVar s rs) as
- gTuple :: (Generic a, Code a ~ '[as]) => Iso' a (Tuple as)
- partsVar' :: forall s rs bs a. Prod Summer bs -> Prod Unity bs -> Iso' a (Tuple bs) -> BVar s rs a -> Prod (BVar s rs) bs
- withParts' :: forall s rs bs a r. Prod Summer bs -> Prod Unity bs -> Iso' a (Tuple bs) -> BVar s rs a -> (Prod (BVar s rs) bs -> r) -> r
- splitVars' :: forall s rs as. Prod Summer as -> Prod Unity as -> BVar s rs (Tuple as) -> Prod (BVar s rs) as
- gSplit' :: forall s rs as a. (Generic a, Code a ~ '[as]) => Prod Summer as -> Prod Unity as -> BVar s rs a -> Prod (BVar s rs) as
- op1 :: Num a => (forall s. AD s (Forward a) -> AD s (Forward a)) -> Op '[a] a
- op2 :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a -> Reverse s a) -> Op '[a, a] a
- op3 :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a -> Reverse s a -> Reverse s a) -> Op '[a, a, a] a
- opN :: (Num a, Known Nat n) => (forall s. Reifies s Tape => Vec n (Reverse s a) -> Reverse s a) -> Op (Replicate n a) a
- op1' :: (a -> (b, Maybe b -> a)) -> Op '[a] b
- op2' :: (a -> b -> (c, Maybe c -> (a, b))) -> Op '[a, b] c
- op3' :: (a -> b -> c -> (d, Maybe d -> (a, b, c))) -> Op '[a, b, c] d
- pattern (:>) :: forall k f a b. f a -> f b -> Prod k f ((:) k a ((:) k b ([] k)))
- only :: f a -> Prod k f ((:) k a ([] k))
- head' :: Prod k f ((:<) k a as) -> f a
- pattern (::<) :: forall a as. a -> Tuple as -> Tuple ((:<) * a as)
- only_ :: a -> Tuple ((:) * a ([] *))
- newtype Summer a = Summer {
- runSummer :: [a] -> a

- newtype Unity a = Unity {
- getUnity :: a

- summers :: (Every Num as, Known Length as) => Prod Summer as
- unities :: (Every Num as, Known Length as) => Prod Unity as
- summers' :: Every Num as => Length as -> Prod Summer as
- unities' :: Every Num as => Length as -> Prod Unity as

# Types

## Backprop types

type BPOp rs a = forall s. Prod (BVar s rs) rs -> BVar s rs a Source #

An operation on `BVar`

s that can be backpropagated. A value of type:

`BPOp`

rs a

takes a bunch of `BVar`

s containg `rs`

and uses them to (purely) produce
a `BVar`

containing an `a`

.

foo ::`BPOp`

'[ Double, Double ] Double foo (x`:<`

y`:<`

'Ø') = x + sqrt y

`BPOp`

here is related to `BPOpI`

from the normal
explicit-graph backprop module Numeric.Backprop.

data BVar :: Type -> [Type] -> Type -> Type Source #

The basic unit of manipulation inside `BP`

(or inside an
implicit-graph backprop function). Instead of directly working with
values, you work with `BVar`

s contating those values. When you work
with a `BVar`

, the *backprop* library can keep track of what values
refer to which other values, and so can perform back-propagation to
compute gradients.

A

refers to a value of type `BVar`

s rs a`a`

, with an environment
of values of the types `rs`

. The phantom parameter `s`

is used to
ensure that stray `BVar`

s don't leak outside of the backprop process.

(That is, if you're using implicit backprop, it ensures that you interact
with `BVar`

s in a polymorphic way. And, if you're using explicit
backprop, it ensures that a

never leaves the `BVar`

s rs a

that it was created in.)`BP`

s rs

`BVar`

s have `Num`

, `Fractional`

, `Floating`

, etc. instances, so they
can be manipulated using polymorphic functions and numeric functions in
Haskell. You can add them, subtract them, etc., in "implicit" backprop
style.

(However, note that if you directly manipulate `BVar`

s using those
instances or using `liftB`

, it delays evaluation, so every usage site
has to re-compute the result/create a new node. If you want to re-use
a `BVar`

you created using `+`

or `-`

or `liftB`

, use
`bindVar`

to force it first. See documentation for
`bindVar`

for more details.)

Floating a => Floating (BVar s rs a) Source # | See note for |

Fractional a => Fractional (BVar s rs a) Source # | See note for |

Num a => Num (BVar s rs a) Source # | Note that if you use the |

type Op as a = forall m. Monad m => OpM m as a Source #

An

describes a differentiable function from `Op`

as a`as`

to `a`

.

For example, a value of type

`Op`

'[Int, Bool] Double

is a function from an `Int`

and a `Bool`

, returning a `Double`

. It can
be differentiated to give a *gradient* of an `Int`

and a `Bool`

if given
a total derivative for the `Double`

. If we call `Bool`

\(2\), then,
mathematically, it is akin to a:

\[ f : \mathbb{Z} \times 2 \rightarrow \mathbb{R} \]

See `runOp`

, `gradOp`

, and `gradOpWith`

for examples on how to run it,
and `Op`

for instructions on creating it.

This type is abstracted over using the pattern synonym with constructor
`Op`

, so you can create one from scratch with it. However, it's
simplest to create it using `op2'`

, `op1'`

, `op2'`

, and `op3'`

helper
smart constructors And, if your function is a numeric function, they
can even be created automatically using `op1`

, `op2`

, `op3`

, and `opN`

with a little help from Numeric.AD from the *ad* library.

Note that this type is a *subset* or *subtype* of `OpM`

(and also of
`OpB`

). So, if a function ever expects an

(or a `OpM`

m as
a`OpB`

), you can always provide an

instead.`Op`

as a

Many functions in this library will expect an

(or
an `OpM`

m as a

), and in all of these cases, you can
provide an `OpB`

s as a

.`Op`

as a

type OpB s as a = OpM (ST s) as a Source #

A subclass of `OpM`

(and superclass of `Op`

), representing `Op`

s that
the *backprop* library uses to perform backpropation.

An

`OpB`

s rs a

represents a differentiable function that takes a tuple of `rs`

and
produces an a `a`

, which can be run on

s and also inside `BVar`

s

s. For example, an `BP`

s

takes an `OpB`

s '[ Int, Double ] Bool`Int`

and
a `Double`

and produces a `Bool`

, and does it in a differentiable way.

`OpB`

is a *superset* of `Op`

, so, if you see any function
that expects an `OpB`

(like `opVar'`

and
`~$`

, for example), you can give them an `Op`

, as well.

You can think of `OpB`

as a superclass/parent class of `Op`

in this
sense, and of `Op`

as a subclass of `OpB`

.

## Tuple types

See Numeric.Backprop for a mini-tutorial on `Prod`

and
`Tuple`

data Prod k f a :: forall k. (k -> *) -> [k] -> * where #

Witness ØC ØC (Prod k f (Ø k)) | |

IxFunctor1 k [k] (Index k) (Prod k) | |

IxFoldable1 k [k] (Index k) (Prod k) | |

IxTraversable1 k [k] (Index k) (Prod k) | |

Functor1 [k] k (Prod k) | |

Foldable1 [k] k (Prod k) | |

Traversable1 [k] k (Prod k) | |

TestEquality k f => TestEquality [k] (Prod k f) | |

BoolEquality k f => BoolEquality [k] (Prod k f) | |

Eq1 k f => Eq1 [k] (Prod k f) | |

Ord1 k f => Ord1 [k] (Prod k f) | |

Show1 k f => Show1 [k] (Prod k f) | |

Read1 k f => Read1 [k] (Prod k f) | |

(Known [k] (Length k) as, Every k (Known k f) as) => Known [k] (Prod k f) as | |

(Witness p q (f a1), Witness s t (Prod a f as)) => Witness (p, s) (q, t) (Prod a f ((:<) a a1 as)) | |

ListC ((<$>) Constraint * Eq ((<$>) * k f as)) => Eq (Prod k f as) | |

(ListC ((<$>) Constraint * Eq ((<$>) * k f as)), ListC ((<$>) Constraint * Ord ((<$>) * k f as))) => Ord (Prod k f as) | |

ListC ((<$>) Constraint * Show ((<$>) * k f as)) => Show (Prod k f as) | |

type WitnessC ØC ØC (Prod k f (Ø k)) | |

type KnownC [k] (Prod k f) as | |

type WitnessC (p, s) (q, t) (Prod a f ((:<) a a1 as)) | |

# back-propagation

backprop :: (Known Length rs, Every Num rs, Num a) => BPOp rs a -> Tuple rs -> (a, Tuple rs) Source #

# Var manipulation

liftB :: OpB s as a -> Prod (BVar s rs) as -> BVar s rs a Source #

Apply `OpB`

over a `Prod`

of `BVar`

s, as inputs. Provides
"implicit-graph" back-propagation, with deferred evaluation.

If you had an

, this function will expect a 3-Prod
of a `OpB`

s '[a, b, c] d

, a `BVar`

s rs a

, and a `BVar`

s rs b

, and the
result will be a `BVar`

s rs c

:`BVar`

s rs d

myOp ::`OpB`

s '[a, b, c] d x ::`BVar`

s rs a y ::`BVar`

s rs b z ::`BVar`

s rs c x :< y :< z :< Ø ::`Prod`

(`BVar`

s rs) '[a, b, c]`liftB`

myOp (x :< y :< z :< Ø) ::`BVar`

s rs d

Note that `OpB`

is a superclass of `Op`

, so you can provide any `Op`

here, as well (like those created by `op1`

, `op2`

, `constOp`

, `op0`

etc.)

`liftB`

has an infix alias, `.$`

, so the above example can also be
written as:

myOp`.$`

(x :< y :< z :< Ø) ::`BVar`

s rs d

to let you pretend that you're applying the `myOp`

function to three
inputs.

The result is a new *deferred* `BVar`

. This should be fine in most
cases, unless you use the result in more than one location. This will
cause evaluation to be duplicated and multiple redundant graph nodes to
be created. If you need to use it in two locations, you should use
`opVar`

instead of `liftB`

, or use `bindVar`

:

`opVar`

o xs =`bindVar`

(`liftB`

o xs)

`liftB`

can be thought of as a "deferred evaluation" version of `opVar`

.

(.$) :: OpB s as a -> Prod (BVar s rs) as -> BVar s rs a infixr 5 Source #

Infix synonym for `liftB`

, which lets you pretend that you're applying
`OpB`

s as if they were functions:

myOp ::`OpB`

s '[a, b, c] d x ::`BVar`

s rs a y ::`BVar`

s rs b z ::`BVar`

s rs c x :< y :< z :< Ø ::`Prod`

(`BVar`

s rs) '[a, b, c] myOp`.$`

(x :< y :< z :< Ø) ::`BVar`

s rs d

Note that `OpB`

is a superclass of `Op`

, so you can pass in any `Op`

here, as well (like those created by `op1`

, `op2`

, `constOp`

, `op0`

etc.)

See the documentation for `liftB`

for all the caveats of this usage.

`.$`

can also be thought of as a "deferred evaluation" version of `~$`

:

o`~$`

xs =`bindVar`

(o`.$`

xs)

liftB1 :: OpB s '[a] b -> BVar s rs a -> BVar s rs b Source #

Convenient wrapper over `liftB`

that takes an `OpB`

with one argument
and a single `BVar`

argument. Lets you not have to type out the entire
`Prod`

.

`liftB1`

o x =`liftB`

o (x`:<`

'Ø') myOp ::`Op`

'[a] b x ::`BVar`

s rs a`liftB1`

myOp x ::`BVar`

s rs b

Note that `OpB`

is a superclass of `Op`

, so you can pass in an `Op`

here
(like one made with `op1`

) as well.

See the documentation for `liftB`

for caveats and potential problematic
situations with this.

liftB2 :: OpB s '[a, b] c -> BVar s rs a -> BVar s rs b -> BVar s rs c Source #

Convenient wrapper over `liftB`

that takes an `OpB`

with two arguments
and two `BVar`

arguments. Lets you not have to type out the entire
`Prod`

.

`liftB2`

o x y =`liftB`

o (x`:<`

y`:<`

'Ø') myOp ::`Op`

'[a, b] c x ::`BVar`

s rs a y ::`BVar`

s rs b`liftB2`

myOp x y ::`BVar`

s rs c

Note that `OpB`

is a superclass of `Op`

, so you can pass in an `Op`

here
(like one made with `op2`

) as well.

See the documentation for `liftB`

for caveats and potential problematic
situations with this.

liftB3 :: OpB s '[a, b, c] d -> BVar s rs a -> BVar s rs b -> BVar s rs c -> BVar s rs d Source #

Convenient wrapper over `liftB`

that takes an `OpB`

with three arguments
and three `BVar`

arguments. Lets you not have to type out the entire
`Prod`

.

`liftB3`

o x y z =`liftB`

o (x`:<`

y`:<`

z`:<`

'Ø') myOp ::`Op`

'[a, b, c] d x ::`BVar`

s rs a y ::`BVar`

s rs b z ::`BVar`

s rs c`liftB3`

myOp x y z ::`BVar`

s rs d

Note that `OpB`

is a superclass of `Op`

, so you can pass in an `Op`

here
(like one made with `op3`

) as well.

See the documentation for `liftB`

for caveats and potential problematic
situations with this.

## As Parts

partsVar :: forall s rs bs a. (Known Length bs, Every Num bs) => Iso' a (Tuple bs) -> BVar s rs a -> Prod (BVar s rs) bs Source #

Use an `Iso`

(or compatible `Iso`

from the lens
library) to "pull out" the parts of a data type and work with each part
as a `BVar`

.

If there is an isomorphism between a `b`

and a

(that is, if
an `Tuple`

as`a`

is just a container for a bunch of `as`

), then it lets you break
out the `as`

inside and work with those.

data Foo = F Int Bool fooIso ::`Iso'`

Foo (Tuple '[Int, Bool]) fooIso =`iso`

(\(F i b) -> i ::< b ::< Ø) (\(i ::< b ::< Ø) -> F i b )`partsVar`

fooIso ::`BVar`

rs Foo ->`Prod`

(`BVar`

s rs) '[Int, Bool] stuff ::`BPOp`

s '[Foo] a stuff (foo :< Ø) = case`partsVar`

fooIso foo of i ::< Ø - -- now, i is a`BVar`

pointing to the`Int`

inside foo -- and b is a`BVar`

pointing to the`Bool`

inside foo -- you can do stuff with the i and b here

You can use this to pass in product types as the environment to a `BP`

,
and then break out the type into its constituent products.

Note that for a type like `Foo`

, `fooIso`

can be generated automatically
with `Generic`

from GHC.Generics and
`Generic`

from Generics.SOP and *generics-sop*, using the
`gTuple`

iso. See `gSplit`

for more information.

Also, if you are literally passing a tuple (like

) then you can give in the identity
isomorphism (`BP`

s '[Tuple '[Int, Bool]`id`

) or use `splitVars`

.

At the moment, this implicit `partsVar`

is less efficient than the
explicit `partsVar`

, but this might change in the
future.

withParts :: forall s rs bs a r. (Known Length bs, Every Num bs) => Iso' a (Tuple bs) -> BVar s rs a -> (Prod (BVar s rs) bs -> r) -> r Source #

A continuation-based version of `partsVar`

. Instead of binding the
parts and using it in the rest of the block, provide a continuation to
handle do stuff with the parts inside.

Building on the example from `partsVar`

:

data Foo = F Int Bool fooIso ::`Iso'`

Foo (Tuple '[Int, Bool]) fooIso =`iso`

(\(F i b) -> i ::< b ::< Ø) (\(i ::< b ::< Ø) -> F i b ) stuff ::`BPOp`

s '[Foo] a stuff (foo :< Ø) =`withParts`

fooIso foo $ \case i :< b :< Ø -> -- now, i is a`BVar`

pointing to the`Int`

inside foo -- and b is a`BVar`

pointing to the`Bool`

inside foo -- you can do stuff with the i and b here

Mostly just a stylistic alternative to `partsVar`

.

splitVars :: forall s rs as. (Known Length as, Every Num as) => BVar s rs (Tuple as) -> Prod (BVar s rs) as Source #

Split out a `BVar`

of a tuple into a tuple (`Prod`

) of `BVar`

s.

-- the environment is a single Int-Bool tuple, tup stuff ::`BPOp`

s '[ Tuple '[Int, Bool] ] a stuff (tup :< Ø) = case`splitVar`

tup of i :< b :< Ø <-`splitVars`

tup -- now, i is a`BVar`

pointing to the`Int`

inside tup -- and b is a`BVar`

pointing to the`Bool`

inside tup -- you can do stuff with the i and b here

Note that

`splitVars`

=`partsVar`

`id`

gSplit :: forall s rs as a. (Generic a, Code a ~ '[as], Known Length as, Every Num as) => BVar s rs a -> Prod (BVar s rs) as Source #

Using `Generic`

from GHC.Generics and
`Generic`

from Generics.SOP, *split* a `BVar`

containing
a product type into a tuple (`Prod`

) of `BVar`

s pointing to each value
inside.

Building on the example from `partsVar`

:

import qualified Generics.SOP as SOP data Foo = F Int Bool deriving Generic instance SOP.Generic Foo`gSplit`

::`BVar`

rs Foo ->`Prod`

(`BVar`

s rs) '[Int, Bool] stuff ::`BPOp`

s '[Foo] a stuff (foo :< Ø) = case`gSplit`

foo of i ::< Ø - -- now, i is a`BVar`

pointing to the`Int`

inside foo -- and b is a`BVar`

pointing to the`Bool`

inside foo -- you can do stuff with the i and b here

Because `Foo`

is a straight up product type, `gSplit`

can use
GHC.Generics and take out the items inside.

Note that

`gSplit`

=`splitVars`

`gTuple`

gTuple :: (Generic a, Code a ~ '[as]) => Iso' a (Tuple as) Source #

An `Iso`

between a type that is a product type, and a tuple that
contains all of its components. Uses Generics.SOP and the
`Generic`

typeclass.

`>>>`

`import qualified Generics.SOP as SOP`

`>>>`

`data Foo = A Int Bool deriving Generic`

`>>>`

`instance SOP.Generic Foo`

`>>>`

10 ::< True ::< Ø`view gTuple (A 10 True)`

`>>>`

A 15 False`review gTuple (15 ::< False ::< Ø)`

partsVar' :: forall s rs bs a. Prod Summer bs -> Prod Unity bs -> Iso' a (Tuple bs) -> BVar s rs a -> Prod (BVar s rs) bs Source #

withParts' :: forall s rs bs a r. Prod Summer bs -> Prod Unity bs -> Iso' a (Tuple bs) -> BVar s rs a -> (Prod (BVar s rs) bs -> r) -> r Source #

splitVars' :: forall s rs as. Prod Summer as -> Prod Unity as -> BVar s rs (Tuple as) -> Prod (BVar s rs) as Source #

gSplit' :: forall s rs as a. (Generic a, Code a ~ '[as]) => Prod Summer as -> Prod Unity as -> BVar s rs a -> Prod (BVar s rs) as Source #

# Op

op2 :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a -> Reverse s a) -> Op '[a, a] a Source #

op3 :: Num a => (forall s. Reifies s Tape => Reverse s a -> Reverse s a -> Reverse s a -> Reverse s a) -> Op '[a, a, a] a Source #

opN :: (Num a, Known Nat n) => (forall s. Reifies s Tape => Vec n (Reverse s a) -> Reverse s a) -> Op (Replicate n a) a Source #

op1' :: (a -> (b, Maybe b -> a)) -> Op '[a] b Source #

Create an `Op`

of a function taking one input, by giving its explicit
derivative. The function should return a tuple containing the result of
the function, and also a function taking the derivative of the result
and return the derivative of the input.

If we have

\[ \eqalign{ f &: \mathbb{R} \rightarrow \mathbb{R}\cr y &= f(x)\cr z &= g(y) } \]

Then the derivative \( \frac{dz}{dx} \), it would be:

\[ \frac{dz}{dx} = \frac{dz}{dy} \frac{dy}{dx} \]

If our `Op`

represents \(f\), then the second item in the resulting
tuple should be a function that takes \(\frac{dz}{dy}\) and returns
\(\frac{dz}{dx}\).

If the input is `Nothing`

, then \(\frac{dz}{dy}\) should be taken to be
\(1\).

As an example, here is an `Op`

that squares its input:

square :: Num a =>`Op`

'[a] a square =`op1'`

$ \x -> (x*x, \case Nothing -> 2 * x Just d -> 2 * d * x )

Remember that, generally, end users shouldn't directly construct `Op`

s;
they should be provided by libraries or generated automatically.

For numeric functions, single-input `Op`

s can be generated automatically
using `op1`

.

op2' :: (a -> b -> (c, Maybe c -> (a, b))) -> Op '[a, b] c Source #

Create an `Op`

of a function taking two inputs, by giving its explicit
gradient. The function should return a tuple containing the result of
the function, and also a function taking the derivative of the result
and return the derivative of the input.

If we have

\[ \eqalign{ f &: \mathbb{R}^2 \rightarrow \mathbb{R}\cr z &= f(x, y)\cr k &= g(z) } \]

Then the gradient \( \left< \frac{\partial k}{\partial x}, \frac{\partial k}{\partial y} \right> \) would be:

\[ \left< \frac{\partial k}{\partial x}, \frac{\partial k}{\partial y} \right> = \left< \frac{dk}{dz} \frac{\partial z}{dx}, \frac{dk}{dz} \frac{\partial z}{dy} \right> \]

If our `Op`

represents \(f\), then the second item in the resulting
tuple should be a function that takes \(\frac{dk}{dz}\) and returns
\( \left< \frac{\partial k}{dx}, \frac{\partial k}{dx} \right> \).

If the input is `Nothing`

, then \(\frac{dk}{dz}\) should be taken to be
\(1\).

As an example, here is an `Op`

that multiplies its inputs:

mul :: Num a =>`Op`

'[a, a] a mul =`op2'`

$ \x y -> (x*y, \case Nothing -> (y , x ) Just d -> (d*y, x*d) )

Remember that, generally, end users shouldn't directly construct `Op`

s;
they should be provided by libraries or generated automatically.

For numeric functions, two-input `Op`

s can be generated automatically
using `op2`

.

# Utility

pattern (:>) :: forall k f a b. f a -> f b -> Prod k f ((:) k a ((:) k b ([] k))) infix 6 #

Construct a two element Prod. Since the precedence of (:>) is higher than (:<), we can conveniently write lists like:

`>>>`

`a :< b :> c`

Which is identical to:

`>>>`

`a :< b :< c :< Ø`

Instructions on how to "sum" a list of values of a given type.
Basically used as an explicit witness for a `Num`

instance.

For most types, the only meaningful value of type

is
`Summer`

a

. However, using `Summer`

`sum`

`Summer`

lets us use `BP`

with types
that are *not* instances of `Num`

. Any type can be used, as long as you
provide a way to "sum" it!

For most of the functions in this library, you can completely ignore
this, as they will be generated automatically. You only need to work
with this directly if you want to use custom types that *aren't*
instances of `Num`

with this library.

If 'Num a' is satisfied, one can create the canonical `Summer`

using

.`known`

:: `Num`

a => `Summer`

a

A canonical "unity" (the multiplicative identity) for a given type.
Basically used as an explicit witness for a `Num`

instance.

For most types, the only meaningful value of type

is
`Unity`

a

. However, using `Unity`

1'`Unity`

lets us use `BP`

with types
that are *not* instances of `Num`

. Any type can be used, as long as you
provide a way to get a multiplicative identity in it!

For most of the functions in this library, you can completely ignore
this, as they will be generated automatically. You only need to work
with this directly if you want to use custom types that *aren't*
instances of `Num`

with this library.

If 'Num a' is satisfied, one can create the canonical `Unity`

using

.`known`

:: `Num`

a => `Unity`

a