u       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~     NoneHaskell semiringsMITmail@doisinkidney.com experimentalNone /23456IThe " .https://ncatlab.org/nlab/show/max-plus+algebraArctic0" or max-plus semiring. It is a semiring where:  =   = -"  =   = Note that we can't use  from   % because annihilation needs to hold: -"  x = x  -" = -"Taking -" to be " would break the above law. Using  " to represent it follows the law.The " /https://ncatlab.org/nlab/show/tropical+semiringTropical0" or min-plus semiring. It is a semiring where:  = !  = "  =   = Note that we can't use   from   % because annihilation needs to hold: "  x = x  " = "Taking " to be "" would break the above law. Using  " to represent it follows the law. Monoid under . Analogous to   , but uses the  constraint, rather than #.  Monoid under . Analogous to  , but uses the  constraint, rather than #. A class for semirings with a concept of "negative infinity". It's important that this isn't regarded as the same as "bounded": x    should probably equal  . A negative infinite value%Test if a value is negative infinity.xA class for semirings with a concept of "infinity". It's important that this isn't regarded as the same as "bounded": x   should probably equal .A positive infinite value%Test if a value is positive infinity.iUseful for operations where zeroes may need to be discarded: for instance in sparse matrix calculations.$ if x is .A  5https://en.wikipedia.org/wiki/Semiring#Star_semirings Star semiring adds one operation,  to a  , such that it follows the law:  x =   x   x =    x  xNFor the semiring of types, this is equivalent to a list. When looking at the % and ; classes as (near-) semirings, this is equivalent to the  operation.Another operation,  , can be defined in relation to :  x = x   xBThis should be recognizable as a non-empty list on types, or the  operation in .A  &https://en.wikipedia.org/wiki/SemiringSemiring% is like the the combination of two  s. The first is called ; it has the identity element /, and it is commutative. The second is called ; it has identity element , and it must distribute over .LawsNormal & laws (a  b)  c = a  (b  c)   a = a   = a (a  b)  c = a  (b  c)   a = a   = aCommutativity of  a  b = b  aDistribution of  over  a  (b  c) = (a  b)  (a  c) (a  b)  c = (a  c)  (b  c) Annihilation   a = a   = %An ordered semiring follows the laws: x ' y => x  z ' y  z x ' y => x  z ' y  z  ' z ( x ' y => x  z ' y  z ( z  x ' z  yThe identity of .The identity of .8An associative binary operation, which distributes over .-An associative, commutative binary operation.#Takes the sum of the elements of a ). Analogous to * on numbers, or + on ,s. add [1..5]15add [False, False]Falseadd [False, True]Trueadd [True, undefined]True'Takes the product of the elements of a ). Analogous to - on numbers, or . on ,s. mul [1..5]120mul [True, True]Truemul [True, False]Falsemul [False, undefined]FalseThis is not a true semiring. In particular, it requires the underlying monoid to be commutative, and even then, it is only a near semiring. It is, however, extremely useful. For instance, this type:  forall a. / (/ a)hIs a valid encoding of church numerals, with addition and multiplication being their semiring variants.The (->)& instance is analogous to the one for &.(getMin . foldMap Min) [1..10]1.0(getMax . foldMap Max) [1..10]10.0A polynomial in x: can be defined as a list of its coefficients, where the i!th element is the coefficient of x^i7. This is the semiring for such a list. Adapted from  Nhttps://pdfs.semanticscholar.org/702d/348c32133997e992db362a19697d5607ab32.pdfhere. 0  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ '    0   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~76None234IAThe free semiring. Adapted from PureScript's version, available  Zhttps://pursuit.purescript.org/packages/purescript-semirings/3.0.0/docs/Data.Semiring.Freehere9. Only a valid semiring if treated as a multiset, as in: Free [[1],[0]] == Free [[0],[1]]True"Run a .#Run a -, interpreting it in the underlying semiring.%%Extremely slow. For testing purposes.  !"#12$%&' !"# !"#  !"#12$%&'19 None2345T[.hAdds positive and negative infinity to a type. Useful for expressing detectable infinity in types like 3, etc.2[Adds positive infinity to a type. Useful for expressing detectable infinity in types like 3, etc.5[Adds negative infinity to a type. Useful for expressing detectable infinity in types like 3, etc.PNot distributive.Q,Only lawful when used with positive numbers.RDoesn't follow  annihilateL or  mulDistribR.,4./01234567567889:;<=>?@ABCDEFGHIJKLMNOPQRST ./01234567 567234./01%4./01234567567889:;<=>?@ABCDEFGHIJKLMNOPQRST"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone2345Ip(Adds a star operation to integral types. () = () () = ()  =   =   0 = 1  _ = s*Adds a star operation to fractional types. () = () () = ()  =   =  $ x = if x < 1 then 1 / (1 - x) else vDUseful for optimizing multiplication, or working with large numbers. () = (9) x  y = -(: (; (-x) + ; (-y)))  =   = 0y ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis3 paper. Apparently used for probabilistic parsing. () =  () = ()  =   = | ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis paper. () =  x  y =  0 (x 9 y < 1)  =   = Positive numbers only. () = = () = >  =   = $Useful for some constraint problems. () =  () = !  =   = "Only expects positive numbers&pqrstuvwxyz{|}~?pqrstuvwxyz{|}~|}~yz{vwxstupqrpqrstuvwxyz{|}~?(Some functions for generating tests for s.MITmail@doisinkidney.com experimentalNoneT0Typealias for ternary laws. Can be used like so: -smallCheck 6 (ternaryLaws :: TernaryLaws Int)/Typealias for binary laws. Can be used like so: +smallCheck 8 (binaryLaws :: BinaryLaws Int).Typealias for unary laws. Can be used like so: *smallCheck 10 (unaryLaws :: UnaryLaws Int)Plus is associative. (x  y)  z = x  (y  z)Multiplication is associative. (x  y)  z = x  (y  z)Plus is commutative. x  y = y  x Multiplication distributes left. x  (y  z) = x  y  x  z!Multiplication distributes right. (x  y)  z = x  z  y  zAdditive identity. x   =   x = xMultiplicative identity. x   =   x = xRight annihilation of  by .   x = Left annihilation of  by . x   =  $A test for all three unary laws for s (, , , and ). #A test for the unary laws for near-s (, , and ). ,A test for all of the ternary laws for near-s (, , ). &A test for all of the binary laws for s (just ). 'A test for all of the ternary laws for s (, , , ).The star law for s.  x =   x   x =    x  xThe plus law for s.  x = x   x The laws for s (, ).Addition law for ordered s. x ' y => x  z ' y  z ( z  x ' z  yMultiplication law for ordered s. x ' y => x  z ' y  z ( z  x ' z  yLaws for ordered s (, ).Law for result of  operation. x @  =  @ x =  xZero is zero law.   = $ The laws for  s (, ).                    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