Relations.hs

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----------------------------------------------------------------------------------}

module Relations where

import Data.List(transpose)
import RelatedFunctions

{--
--- Creates all possible reflexive and symetric square matrices of a given size
---
--- that means putting together the top, ones and the bottom,
--- which is reformated top, because of symetry
 _______
|1|_ top|
| |1|_  |
|   |1|_|
|botto|1| m

 _______
|1|a|b|c|
|a|1|d|e|
|b|d|1|f|
|c|e|f|1| 

--}

all_reflexive_symetric_matrices :: Integer -> [[[Bool]]]
all_reflexive_symetric_matrices n = map (new_reflexive_symetric_matrix_from_vector n) vectors
  where
    vectors = sequence $ ple vector_length
    -- the result of the division is supposed to be integer
    vector_length = ((n)*(n) - (n)) `div` 2

   
{--
--- The heart of the stuff.
--- new_reflexive_symetric_matrix_from_vector 3 [True,False,True]
> [[True,True,False],
>  [True,True,True],
>  [False,True,True]]
--}
new_reflexive_symetric_matrix_from_vector :: Integer -> [Bool] -> [[Bool]]
new_reflexive_symetric_matrix_from_vector n vector = combine bottom_with_padding top_with_padding
  where
    bottom = groupTakingLess (n-1) vector
    -- discovered by throughtful experimentation :)
    --Example:
    -- reverse $ transpose $ map reverse ["abc","de","f"]
    -- > ["a","bd","cef"]
    bottom_converted_into_rows = reverse $ transpose $ map reverse bottom
    bottom_with_last_row_ended_with_one = (init bottom_converted_into_rows) ++ (appendToAll True [last bottom_converted_into_rows])
    bottom_with_padding = [[]] ++ bottom_with_last_row_ended_with_one
    
    top = groupTakingLess (n-1) vector
    top_with_leading_ones = prependToAll True top
    top_with_padding = top_with_leading_ones ++ [[]]
    
    combine bottom top = zipWith (++) bottom top
    

-- vytvori n-prvkový vektor a udělá z něj sqrt(n)-prvkovou čtvercovou matici
matice :: Integer -> [[[Bool]]]
matice n = zmaticuj n $ sequence $ ple (n*n)

--
zmaticuj :: Integer -> [[Bool]] -> [[[Bool]]]
zmaticuj n m = map (chunk n) m

-- vygeneruje vsechny moznosti jak udělat n-prvkový vektor 
vektor :: Integer -> [[Bool]]
vektor 1 = [[True], [False]]
vektor n = map fce (vektor (n-1)) ++ map (++[False]) (vektor (n-1))
  where
    fce = (++[True])

-- to stejné, jen kratšeji
ple :: Integer -> [[Bool]]
ple 1 = [[True,False]]
ple n = [True,False] : (ple (n-1)) 


--    
reflexivni [[True]] = True
reflexivni [[False]] = False
reflexivni (b:xb) = head b && reflexivni (map tail xb)

tranzitivni :: Integer -> [[Bool]] -> (Integer, Integer) -> Bool
tranzitivni n matice (i, j)
  | i == j = True
  | a == True = all (==True) b
  | otherwise = True
    where
      a = ((sloupec j) . (radek i)) matice  
      g = radek j matice
      b = map fce [1..n]
        where fce r
                | sloupec r g == True = ((sloupec r) . (radek i)) matice
                | otherwise = True

antisymetricka :: [[Bool]] -> (Integer, Integer) -> Bool
antisymetricka matice (i, j)
  | i == j = True
  | a == True && b == True = False
  | otherwise = True
    where
        a = ((sloupec j) . (radek i)) matice  
        b = ((sloupec i) . (radek j)) matice