import qualified Data.Set as Set
import qualified Data.List as List

n = 12
additionInZn a b = (a+b) `mod` n
multiplicationInZn a b = (a*b) `mod` n
theta n = n `mod` 4

q = quotientRing additionInZn [0..n-1] (kernel theta [0..n-1])

multiplicationTable = setwiseOperationTable q multiplicationInZn
additionTable = setwiseOperationTable q additionInZn

main = putStrLn $
	(latexTable additionTable q q latexRepresentationOfList "+")
	++ "\n\n" ++
	(latexTable multiplicationTable q q latexRepresentationOfList "*")


join j s = concat $ List.intersperse j s

concatTimes n s = concatMap (\c -> s) [1..n]

latexRepresentationOfList s = "\\{" ++ (join ", " $ map show s) ++ "\\}"

latexTable rows header left representation operation = 
	"\\begin{tabular}{r|" ++  (concatTimes (length header) "r") ++ "}\n" ++
	operation ++ "\n& " ++
	(join "\n& " $ map representation header) ++
	"\n\\\\ \\hline\n" ++
	body ++
	"\\end{tabular}"
	where body =
		flip concatMap (zip left rows) (\(leftColumn, columns) ->	
			representation leftColumn ++ "\n" ++
			flip concatMap columns (\c ->
				"& " ++ representation c ++ "\n")
			++ "\\\\\n")


uniqueSorted l = Set.toAscList $ Set.fromList l

cosetCombination a b operation = uniqueSorted [operation i j | i<-a, j<-b]

setwiseOperationTable ring operation =
	[[cosetCombination x y operation | x <- ring ] | y <- ring]

quotientRing addition ring ideal =
	uniqueSorted [uniqueSorted [addition i r | i<-ideal] | r<-ring]

kernel f domain = [d | d<- domain, f(d) == 0]