GHC.TypeLits.Normalise.SOP

SOP: Sum-of-Products, sorta

The arithmetic operation for Nat are, addition (+), subtraction (-), multiplication (*), and exponentiation (^). This means we cannot write expressions in a canonical SOP normal form. We can get rid of subtraction by working with integers, and translating a - b to a + (-1)*b. Exponentation cannot be getten rid of that way. So we define the following grammar for our canonical SOP-like normal form of arithmetic expressions:

SOP      ::= Product '+' SOP | Product
Product  ::= Symbol '*' Product | Symbol
Symbol   ::= Integer
          |  Var
          |  Var '^' Product
          |  SOP '^' ProductE

ProductE ::= SymbolE '*' ProductE | SymbolE
SymbolE  ::= Var
          |  Var '^' Product
          |  SOP '^' ProductE

So a valid SOP terms are:

x*y + y^2
(x+y)^(k*z)

, but,

(x*y)^2

is not, and should be:

x^2 * y^2

Exponents are thus not allowed to have products, so for example, the expression:

(x + 2)^(y + 2)

in valid SOP form is:

4*x*(2 + x)^y + 4*(2 + x)^y + (2 + x)^y*x^2

Also, exponents can only be integer values when the base is a variable. Although not enforced by the grammar, the exponentials are flatted as far as possible in SOP form. So:

(x^y)^z

is flattened to:

x^(y*z)

SOP types

data Symbol v c

Constructors

I Integer

Integer constant

C c

Non-integer constant

E (SOP v c) (Product v c)

Exponentiation

V v

Variable

Instances

(Eq v, Eq c) => Eq (Symbol v c)

Methods

(==) :: Symbol v c -> Symbol v c -> Bool

(/=) :: Symbol v c -> Symbol v c -> Bool

(Ord v, Ord c) => Ord (Symbol v c)

Methods

compare :: Symbol v c -> Symbol v c -> Ordering

(<) :: Symbol v c -> Symbol v c -> Bool

(<=) :: Symbol v c -> Symbol v c -> Bool

(>) :: Symbol v c -> Symbol v c -> Bool

(>=) :: Symbol v c -> Symbol v c -> Bool

max :: Symbol v c -> Symbol v c -> Symbol v c

min :: Symbol v c -> Symbol v c -> Symbol v c

(Outputable v, Outputable c) => Outputable (Symbol v c)

Methods

ppr :: Symbol v c -> SDoc

pprPrec :: Rational -> Symbol v c -> SDoc

newtype Product v c

Constructors

P

Fields

• unP :: [Symbol v c]

Instances

(Eq c, Eq v) => Eq (Product v c)

Methods

(==) :: Product v c -> Product v c -> Bool

(/=) :: Product v c -> Product v c -> Bool

(Ord v, Ord c) => Ord (Product v c)

Methods

compare :: Product v c -> Product v c -> Ordering

(<) :: Product v c -> Product v c -> Bool

(<=) :: Product v c -> Product v c -> Bool

(>) :: Product v c -> Product v c -> Bool

(>=) :: Product v c -> Product v c -> Bool

max :: Product v c -> Product v c -> Product v c

min :: Product v c -> Product v c -> Product v c

(Outputable v, Outputable c) => Outputable (Product v c)

Methods

ppr :: Product v c -> SDoc

pprPrec :: Rational -> Product v c -> SDoc

newtype SOP v c

Constructors

S

Fields

• unS :: [Product v c]

Instances

(Eq v, Eq c) => Eq (SOP v c)

Methods

(==) :: SOP v c -> SOP v c -> Bool

(/=) :: SOP v c -> SOP v c -> Bool

(Ord c, Ord v) => Ord (SOP v c)

Methods

compare :: SOP v c -> SOP v c -> Ordering

(<) :: SOP v c -> SOP v c -> Bool

(<=) :: SOP v c -> SOP v c -> Bool

(>) :: SOP v c -> SOP v c -> Bool

(>=) :: SOP v c -> SOP v c -> Bool

max :: SOP v c -> SOP v c -> SOP v c

min :: SOP v c -> SOP v c -> SOP v c

(Outputable v, Outputable c) => Outputable (SOP v c)

Methods

ppr :: SOP v c -> SDoc

pprPrec :: Rational -> SOP v c -> SDoc

Simplification

reduceExp :: (Ord v, Ord c) => Symbol v c -> Symbol v c

reduce exponentials

Performs the following rewrites:

x^0          ==>  1
0^x          ==>  0
2^3          ==>  8
(k ^ i) ^ j  ==>  k ^ (i * j)

mergeS :: (Ord v, Ord c) => Symbol v c -> Symbol v c -> Either (Symbol v c) (Symbol v c)

Merge two symbols of a Product term

Performs the following rewrites:

8 * 7    ==>  56
1 * x    ==>  x
x * 1    ==>  x
0 * x    ==>  0
x * 0    ==>  0
x * x^4  ==>  x^5
x^4 * x  ==>  x^5
y*y      ==>  y^2

mergeP :: (Eq v, Eq c) => Product v c -> Product v c -> Either (Product v c) (Product v c)

Merge two products of a SOP term

Performs the following rewrites:

2xy + 3xy  ==>  5xy
2xy + xy   ==>  3xy
xy + 2xy   ==>  3xy
xy + xy    ==>  2xy

mergeSOPAdd :: (Ord v, Ord c) => SOP v c -> SOP v c -> SOP v c

Merge two SOP terms by additions

mergeSOPMul :: (Ord v, Ord c) => SOP v c -> SOP v c -> SOP v c

Merge two SOP terms by multiplication

normaliseExp :: (Ord v, Ord c) => SOP v c -> SOP v c -> SOP v c

Expand or Simplify complex exponentials

Performs the following rewrites:

b^1              ==>  b
2^(y^2)          ==>  4^y
(x + 2)^2        ==>  x^2 + 4xy + 4
(x + 2)^(2x)     ==>  (x^2 + 4xy + 4)^x
(x + 2)^(y + 2)  ==>  4x(2 + x)^y + 4(2 + x)^y + (2 + x)^yx^2