Gh;      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Haskell semiringsMITmail@doisinkidney.com experimentalNone/56I  Monoid under  . Analogous to , but uses the  constraint, rather than . Monoid under  . Analogous to , but uses the  constraint, rather than .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)HAnother useful law, annihilation, may be deduced from the axioms above:    a = a    = The identity of  .The identity of  . 8An associative binary operation, which distributes over  . -An associative, commutative binary operation.KThe ( semiring uses function composition for  , and pointwise  for  . The underlying  needs to be commutative.LThe (->)& instance is analogous to the one for .MThe  for  uses the  operation for  , and normal  for  .NThe  for  uses the  operation for  , and normal  for  .[The   is  for  , and a Cartesian product for  .`  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]     X   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\] 7 6"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone5It ;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.w ;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.z  is ,   is . Positive numbers only.}  is ,   is Only expects positive numbers$tuvwxyz{|}~ tuvwxyz{|}~ }~z{|wxytuvtuvwxyz{|}~None234IAThe 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]]TrueRun a .Run a -, interpreting it in the underlying semiring.%Extremely slow. For testing purposes.  9 (Some QuickCheck properties for SemiringsMITmail@doisinkidney.com experimentalNoneTPlus is associative.Multiplication is associative.Plus is commutative. 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