module Tests.Cube
where

import Prelude hiding (LT)
import Test.QuickCheck

import Cardinal
import Comparisons
import Cube
import FunctionValues
import Misc (all_equal)
import Tests.FunctionValues ()
import Tetrahedron (b0, b1, b2, b3, c, fv,
                    v0, v1, v2, v3, volume)

instance Arbitrary Cube where
    arbitrary = do
      (Positive h') <- arbitrary :: Gen (Positive Double)
      i' <- choose (coordmin, coordmax)
      j' <- choose (coordmin, coordmax)
      k' <- choose (coordmin, coordmax)
      fv' <- arbitrary :: Gen FunctionValues
      return (Cube h' i' j' k' fv')
        where
          coordmin = -268435456 -- -(2^29 / 2)
          coordmax = 268435456  -- +(2^29 / 2)


-- Quickcheck tests.

-- | Since the grid size is necessarily positive, all tetrahedrons
--   (which comprise cubes of positive volume) must have positive volume
--   as well.
prop_all_volumes_positive :: Cube -> Bool
prop_all_volumes_positive cube =
    null nonpositive_volumes
    where
      ts = tetrahedrons cube
      volumes = map volume ts
      nonpositive_volumes = filter (<= 0) volumes

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron0_volumes_exact :: Cube -> Bool
prop_tetrahedron0_volumes_exact cube =
    volume (tetrahedron0 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube


-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron1_volumes_exact :: Cube -> Bool
prop_tetrahedron1_volumes_exact cube =
    volume (tetrahedron1 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron2_volumes_exact :: Cube -> Bool
prop_tetrahedron2_volumes_exact cube =
    volume (tetrahedron2 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron3_volumes_exact :: Cube -> Bool
prop_tetrahedron3_volumes_exact cube =
    volume (tetrahedron3 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron4_volumes_exact :: Cube -> Bool
prop_tetrahedron4_volumes_exact cube =
    volume (tetrahedron4 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron5_volumes_exact :: Cube -> Bool
prop_tetrahedron5_volumes_exact cube =
    volume (tetrahedron5 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron6_volumes_exact :: Cube -> Bool
prop_tetrahedron6_volumes_exact cube =
    volume (tetrahedron6 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron7_volumes_exact :: Cube -> Bool
prop_tetrahedron7_volumes_exact cube =
    volume (tetrahedron7 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron8_volumes_exact :: Cube -> Bool
prop_tetrahedron8_volumes_exact cube =
    volume (tetrahedron8 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron9_volumes_exact :: Cube -> Bool
prop_tetrahedron9_volumes_exact cube =
    volume (tetrahedron9 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron10_volumes_exact :: Cube -> Bool
prop_tetrahedron10_volumes_exact cube =
    volume (tetrahedron10 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron11_volumes_exact :: Cube -> Bool
prop_tetrahedron11_volumes_exact cube =
    volume (tetrahedron11 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron12_volumes_exact :: Cube -> Bool
prop_tetrahedron12_volumes_exact cube =
    volume (tetrahedron12 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron13_volumes_exact :: Cube -> Bool
prop_tetrahedron13_volumes_exact cube =
    volume (tetrahedron13 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron14_volumes_exact :: Cube -> Bool
prop_tetrahedron14_volumes_exact cube =
    volume (tetrahedron14 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron15_volumes_exact :: Cube -> Bool
prop_tetrahedron15_volumes_exact cube =
    volume (tetrahedron15 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron16_volumes_exact :: Cube -> Bool
prop_tetrahedron16_volumes_exact cube =
    volume (tetrahedron16 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron17_volumes_exact :: Cube -> Bool
prop_tetrahedron17_volumes_exact cube =
    volume (tetrahedron17 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron18_volumes_exact :: Cube -> Bool
prop_tetrahedron18_volumes_exact cube =
    volume (tetrahedron18 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron19_volumes_exact :: Cube -> Bool
prop_tetrahedron19_volumes_exact cube =
    volume (tetrahedron19 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron20_volumes_exact :: Cube -> Bool
prop_tetrahedron20_volumes_exact cube =
    volume (tetrahedron20 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron21_volumes_exact :: Cube -> Bool
prop_tetrahedron21_volumes_exact cube =
    volume (tetrahedron21 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron22_volumes_exact :: Cube -> Bool
prop_tetrahedron22_volumes_exact cube =
    volume (tetrahedron22 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | In fact, since all of the tetrahedra are identical, we should
--   already know their volumes. There's 24 tetrahedra to a cube, so
--   we'd expect the volume of each one to be (1/24)*h^3.
prop_tetrahedron23_volumes_exact :: Cube -> Bool
prop_tetrahedron23_volumes_exact cube =
    volume (tetrahedron23 cube) ~~= (1/24)*(delta^(3::Int))
    where
      delta = h cube

-- | All tetrahedron should have their v0 located at the center of the cube.
prop_v0_all_equal :: Cube -> Bool
prop_v0_all_equal cube = (v0 t0) == (v0 t1)
    where
      t0 = head (tetrahedrons cube) -- Doesn't matter which two we choose.
      t1 = head $ tail (tetrahedrons cube)


-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron0_volumes_positive :: Cube -> Bool
prop_tetrahedron0_volumes_positive cube =
    volume (tetrahedron0 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron1_volumes_positive :: Cube -> Bool
prop_tetrahedron1_volumes_positive cube =
    volume (tetrahedron1 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron2_volumes_positive :: Cube -> Bool
prop_tetrahedron2_volumes_positive cube =
    volume (tetrahedron2 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron3_volumes_positive :: Cube -> Bool
prop_tetrahedron3_volumes_positive cube =
    volume (tetrahedron3 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron4_volumes_positive :: Cube -> Bool
prop_tetrahedron4_volumes_positive cube =
    volume (tetrahedron4 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron5_volumes_positive :: Cube -> Bool
prop_tetrahedron5_volumes_positive cube =
    volume (tetrahedron5 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron6_volumes_positive :: Cube -> Bool
prop_tetrahedron6_volumes_positive cube =
    volume (tetrahedron6 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron7_volumes_positive :: Cube -> Bool
prop_tetrahedron7_volumes_positive cube =
    volume (tetrahedron7 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron8_volumes_positive :: Cube -> Bool
prop_tetrahedron8_volumes_positive cube =
    volume (tetrahedron8 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron9_volumes_positive :: Cube -> Bool
prop_tetrahedron9_volumes_positive cube =
    volume (tetrahedron9 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron10_volumes_positive :: Cube -> Bool
prop_tetrahedron10_volumes_positive cube =
    volume (tetrahedron10 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron11_volumes_positive :: Cube -> Bool
prop_tetrahedron11_volumes_positive cube =
    volume (tetrahedron11 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron12_volumes_positive :: Cube -> Bool
prop_tetrahedron12_volumes_positive cube =
    volume (tetrahedron12 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron13_volumes_positive :: Cube -> Bool
prop_tetrahedron13_volumes_positive cube =
    volume (tetrahedron13 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron14_volumes_positive :: Cube -> Bool
prop_tetrahedron14_volumes_positive cube =
    volume (tetrahedron14 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron15_volumes_positive :: Cube -> Bool
prop_tetrahedron15_volumes_positive cube =
    volume (tetrahedron15 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron16_volumes_positive :: Cube -> Bool
prop_tetrahedron16_volumes_positive cube =
    volume (tetrahedron16 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron17_volumes_positive :: Cube -> Bool
prop_tetrahedron17_volumes_positive cube =
    volume (tetrahedron17 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron18_volumes_positive :: Cube -> Bool
prop_tetrahedron18_volumes_positive cube =
    volume (tetrahedron18 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron19_volumes_positive :: Cube -> Bool
prop_tetrahedron19_volumes_positive cube =
    volume (tetrahedron19 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron20_volumes_positive :: Cube -> Bool
prop_tetrahedron20_volumes_positive cube =
    volume (tetrahedron20 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron21_volumes_positive :: Cube -> Bool
prop_tetrahedron21_volumes_positive cube =
    volume (tetrahedron21 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron22_volumes_positive :: Cube -> Bool
prop_tetrahedron22_volumes_positive cube =
    volume (tetrahedron22 cube) > 0

-- | This pretty much repeats the prop_all_volumes_positive property,
--   but will let me know which tetrahedrons's vertices are disoriented.
prop_tetrahedron23_volumes_positive :: Cube -> Bool
prop_tetrahedron23_volumes_positive cube =
    volume (tetrahedron23 cube) > 0


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). It appears that
--   the assumptions in sections (2.6) and (2.7) have been
--   switched. From the description, one would expect 'tetrahedron0'
--   and 'tetrahedron3' to share face \<v0,v1,v2\>; however, we have
--   to use 'tetrahedron0' and 'tetahedron1' for all of the tests in
--   section (2.6). Also note that the third and fourth indices of
--   c-t1 have been switched. This is because we store the triangles
--   oriented such that their volume is positive. If T and T-tilde
--   share \<v0,v1,v2\> and v3,v3-tilde point in opposite directions,
--   one of them has to have negative volume!
prop_c0120_identity1 :: Cube -> Bool
prop_c0120_identity1 cube =
   c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t1 0 0 1 2) / 2
     where
       t0 = tetrahedron0 cube
       t1 = tetrahedron1 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
--   'prop_c0120_identity1' with tetrahedrons 1 and 2.
prop_c0120_identity2 :: Cube -> Bool
prop_c0120_identity2 cube =
   c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t2 0 0 1 2) / 2
     where
       t1 = tetrahedron1 cube
       t2 = tetrahedron2 cube

-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
--   'prop_c0120_identity1' with tetrahedrons 2 and 3.
prop_c0120_identity3 :: Cube -> Bool
prop_c0120_identity3 cube =
   c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t3 0 0 1 2) / 2
     where
       t2 = tetrahedron2 cube
       t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
--   'prop_c0120_identity1' with tetrahedrons 4 and 5.
-- prop_c0120_identity4 :: Cube -> Bool
-- prop_c0120_identity4 cube =
--     sum [trace ("c_t4_0120: " ++ (show tmp1)) tmp1,
--          trace ("c_t5_0012: " ++ (show tmp2)) tmp2,
--          trace ("c_t5_0102: " ++ (show tmp3)) tmp3,
--          trace ("c_t5_1002: " ++ (show tmp4)) tmp4,
--          trace ("c_t5_0120: " ++ (show tmp5)) tmp5,
--          trace ("c_t5_1020: " ++ (show tmp6)) tmp6,
--          trace ("c_t5_1200: " ++ (show tmp7)) tmp7,
--          trace ("c_t5_0021: " ++ (show tmp8)) tmp8,
--          trace ("c_t5_0201: " ++ (show tmp9)) tmp9,
--          trace ("c_t5_2001: " ++ (show tmp10)) tmp10,
--          trace ("c_t5_0210: " ++ (show tmp11)) tmp11,
--          trace ("c_t5_2010: " ++ (show tmp12)) tmp12,
--          trace ("c_t5_2100: " ++ (show tmp13)) tmp13] == 10
-- --   c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
--      where
--        t4 = tetrahedron4 cube
--        t5 = tetrahedron5 cube
--        tmp1 = c t4 0 1 2 0
--        tmp2 = (c t4 0 0 2 1 + c t5 0 0 1 2) / 2
--        tmp3 = (c t4 0 0 2 1 + c t5 0 1 0 2) / 2
--        tmp4 = (c t4 0 0 2 1 + c t5 1 0 0 2) / 2
--        tmp5 = (c t4 0 0 2 1 + c t5 0 1 2 0) / 2
--        tmp6 = (c t4 0 0 2 1 + c t5 1 0 2 0) / 2
--        tmp7 = (c t4 0 0 2 1 + c t5 1 2 0 0) / 2
--        tmp8 = (c t4 0 0 2 1 + c t5 0 0 2 1) / 2
--        tmp9 = (c t4 0 0 2 1 + c t5 0 2 0 1) / 2
--        tmp10 = (c t4 0 0 2 1 + c t5 2 0 0 1) / 2
--        tmp11 = (c t4 0 0 2 1 + c t5 0 2 1 0) / 2
--        tmp12 = (c t4 0 0 2 1 + c t5 2 0 1 0) / 2
--        tmp13 = (c t4 0 0 2 1 + c t5 2 1 0 0) / 2

-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- --   'prop_c0120_identity1' with tetrahedrons 5 and 6.
-- prop_c0120_identity5 :: Cube -> Bool
-- prop_c0120_identity5 cube =
--    c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t6 0 0 1 2) / 2
--      where
--        t5 = tetrahedron5 cube
--        t6 = tetrahedron6 cube


-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
-- --   'prop_c0120_identity1' with tetrahedrons 6 and 7.
-- prop_c0120_identity6 :: Cube -> Bool
-- prop_c0120_identity6 cube =
--    c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t7 0 0 1 2) / 2
--      where
--        t6 = tetrahedron6 cube
--        t7 = tetrahedron7 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
--   'prop_c0120_identity1'.
prop_c0210_identity1 :: Cube -> Bool
prop_c0210_identity1 cube =
    c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
--   'prop_c0120_identity1'.
prop_c0300_identity1 :: Cube -> Bool
prop_c0300_identity1 cube =
    c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t1 0 2 1 0) / 2
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
--   'prop_c0120_identity1'.
prop_c1110_identity :: Cube -> Bool
prop_c1110_identity cube =
    c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
--   'prop_c0120_identity1'.
prop_c1200_identity1 :: Cube -> Bool
prop_c1200_identity1 cube =
    c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t1 1 1 1 0) / 2
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
--   'prop_c0120_identity1'.
prop_c2100_identity1 :: Cube -> Bool
prop_c2100_identity1 cube =
    c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t1 2 0 1 0) / 2
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube



-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). It appears that
--   the assumptions in sections (2.6) and (2.7) have been
--   switched. From the description, one would expect 'tetrahedron0'
--   and 'tetrahedron1' to share face \<v0,v1,v3\>; however, we have
--   to use 'tetrahedron0' and 'tetahedron3' for all of the tests in
--   section (2.7). Also note that the third and fourth indices of
--   c-t3 have been switched. This is because we store the triangles
--   oriented such that their volume is positive. If T and T-tilde
--   share \<v0,v1,v2\> and v3,v3-tilde point in opposite directions,
--   one of them has to have negative volume!
prop_c0102_identity1 :: Cube -> Bool
prop_c0102_identity1 cube =
    c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t3 0 0 2 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
--   'prop_c0102_identity1'.
prop_c0201_identity1 :: Cube -> Bool
prop_c0201_identity1 cube =
    c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
--   'prop_c0102_identity1'.
prop_c0300_identity2 :: Cube -> Bool
prop_c0300_identity2 cube =
    c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t3 0 2 0 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
--   'prop_c0102_identity1'.
prop_c1101_identity :: Cube -> Bool
prop_c1101_identity cube =
    c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
--   'prop_c0102_identity1'.
prop_c1200_identity2 :: Cube -> Bool
prop_c1200_identity2 cube =
    c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t3 1 1 0 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
--   'prop_c0102_identity1'.
prop_c2100_identity2 :: Cube -> Bool
prop_c2100_identity2 cube =
    c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t3 2 0 0 1) / 2
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
--   fourth indices of c-t6 have been switched. This is because we
--   store the triangles oriented such that their volume is
--   positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
--   point in opposite directions, one of them has to have negative
--   volume! We also switch the third and fourth vertices of t6, but
--   as of now, why this works is a mystery.
prop_c3000_identity :: Cube -> Bool
prop_c3000_identity cube =
    c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
                    - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
--   'prop_c3000_identity'.
prop_c2010_identity :: Cube -> Bool
prop_c2010_identity cube =
    c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
                    - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
--   'prop_c3000_identity'.
prop_c2001_identity :: Cube -> Bool
prop_c2001_identity cube =
    c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
                    - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
--   'prop_c3000_identity'.
prop_c1020_identity :: Cube -> Bool
prop_c1020_identity cube =
    c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
                    - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
--   'prop_c3000_identity'.
prop_c1002_identity :: Cube -> Bool
prop_c1002_identity cube =
    c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
                    - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }


-- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
--   'prop_c3000_identity'.
prop_c1011_identity :: Cube -> Bool
prop_c1011_identity cube =
    c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
                    ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
      where
        t0 = tetrahedron0 cube
        t6 = (tetrahedron6 cube) { v2 = (v3 t6), v3 = (v2 t6) }



-- | Given in Sorokina and Zeilfelder, p. 78.
-- prop_cijk1_identity :: Cube -> Bool
-- prop_cijk1_identity cube =
--      and [ c t0 i j k 1 ~=
--                  (c t1 (i+1) j k 0) * ((b0 t0) (v3 t1)) +
--                  (c t1 i (j+1) k 0) * ((b1 t0) (v3 t1)) +
--                  (c t1 i j (k+1) 0) * ((b2 t0) (v3 t1)) +
--                  (c t1 i j k 1) * ((b3 t0) (v3 t1)) | i <- [0..2],
--                                                       j <- [0..2],
--                                                       k <- [0..2],
--                                                       i + j + k == 2]
--       where
--         t0 = tetrahedron0 cube
--         t1 = tetrahedron1 cube



-- | The function values at the interior should be the same for all tetrahedra.
prop_interior_values_all_identical :: Cube -> Bool
prop_interior_values_all_identical cube =
    all_equal [i0, i1, i2, i3, i4, i5, i6, i7, i8,
               i9, i10, i11, i12, i13, i14, i15, i16,
               i17, i18, i19, i20, i21, i22, i23]
    where
      i0 = eval (Tetrahedron.fv (tetrahedron0 cube)) $ I
      i1 = eval (Tetrahedron.fv (tetrahedron1 cube)) $ I
      i2 = eval (Tetrahedron.fv (tetrahedron2 cube)) $ I
      i3 = eval (Tetrahedron.fv (tetrahedron3 cube)) $ I
      i4 = eval (Tetrahedron.fv (tetrahedron4 cube)) $ I
      i5 = eval (Tetrahedron.fv (tetrahedron5 cube)) $ I
      i6 = eval (Tetrahedron.fv (tetrahedron6 cube)) $ I
      i7 = eval (Tetrahedron.fv (tetrahedron7 cube)) $ I
      i8 = eval (Tetrahedron.fv (tetrahedron8 cube)) $ I
      i9 = eval (Tetrahedron.fv (tetrahedron9 cube)) $ I
      i10 = eval (Tetrahedron.fv (tetrahedron10 cube)) $ I
      i11 = eval (Tetrahedron.fv (tetrahedron11 cube)) $ I
      i12 = eval (Tetrahedron.fv (tetrahedron12 cube)) $ I
      i13 = eval (Tetrahedron.fv (tetrahedron13 cube)) $ I
      i14 = eval (Tetrahedron.fv (tetrahedron14 cube)) $ I
      i15 = eval (Tetrahedron.fv (tetrahedron15 cube)) $ I
      i16 = eval (Tetrahedron.fv (tetrahedron16 cube)) $ I
      i17 = eval (Tetrahedron.fv (tetrahedron17 cube)) $ I
      i18 = eval (Tetrahedron.fv (tetrahedron18 cube)) $ I
      i19 = eval (Tetrahedron.fv (tetrahedron19 cube)) $ I
      i20 = eval (Tetrahedron.fv (tetrahedron20 cube)) $ I
      i21 = eval (Tetrahedron.fv (tetrahedron21 cube)) $ I
      i22 = eval (Tetrahedron.fv (tetrahedron22 cube)) $ I
      i23 = eval (Tetrahedron.fv (tetrahedron23 cube)) $ I


-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
--   This test checks the rotation works as expected.
prop_c_tilde_2100_rotation_correct :: Cube -> Bool
prop_c_tilde_2100_rotation_correct cube =
    expr1 == expr2
    where
      t0 = tetrahedron0 cube
      t6 = tetrahedron6 cube

      -- What gets computed for c2100 of t6.
      expr1 = eval (Tetrahedron.fv t6) $
          (3/8)*I +
          (1/12)*(T + R + L + D) +
          (1/64)*(FT + FR + FL + FD) +
          (7/48)*F +
          (1/48)*B +
          (1/96)*(RT + LD + LT + RD) +
          (1/192)*(BT + BR + BL + BD)

      -- What should be computed for c2100 of t6.
      expr2 = eval (Tetrahedron.fv t0) $
              (3/8)*I +
              (1/12)*(F + R + L + B) +
              (1/64)*(FT + RT + LT + BT) +
              (7/48)*T +
              (1/48)*D +
              (1/96)*(FR + FL + BR + BL) +
              (1/192)*(FD + RD + LD + BD)


-- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
--   This test checks the actual value based on the FunctionValues of the cube.
prop_c_tilde_2100_correct :: Cube -> Bool
prop_c_tilde_2100_correct cube =
    c t6 2 1 0 0 == (3/8)*int
                    + (1/12)*(f + r + l + b)
                    + (1/64)*(ft + rt + lt + bt)
                    + (7/48)*t + (1/48)*d + (1/96)*(fr + fl + br + bl)
                    + (1/192)*(fd + rd + ld + bd)
    where
      t0 = tetrahedron0 cube
      t6 = tetrahedron6 cube
      fvs = Tetrahedron.fv t0
      int = interior fvs
      f = front fvs
      r = right fvs
      l = left fvs
      b = back fvs
      ft = front_top fvs
      rt = right_top fvs
      lt = left_top fvs
      bt = back_top fvs
      t = top fvs
      d = down fvs
      fr = front_right fvs
      fl = front_left fvs
      br = back_right fvs
      bl = back_left fvs
      fd = front_down fvs
      rd = right_down fvs
      ld = left_down fvs
      bd = back_down fvs

-- Tests to check that the correct edges are incidental.
prop_t0_shares_edge_with_t1 :: Cube -> Bool
prop_t0_shares_edge_with_t1 cube =
    (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
      where
        t0 = tetrahedron0 cube
        t1 = tetrahedron1 cube

prop_t0_shares_edge_with_t3 :: Cube -> Bool
prop_t0_shares_edge_with_t3 cube =
    (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
      where
        t0 = tetrahedron0 cube
        t3 = tetrahedron3 cube

prop_t0_shares_edge_with_t6 :: Cube -> Bool
prop_t0_shares_edge_with_t6 cube =
    (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
      where
        t0 = tetrahedron0 cube
        t6 = tetrahedron6 cube

prop_t1_shares_edge_with_t2 :: Cube -> Bool
prop_t1_shares_edge_with_t2 cube =
    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
      where
        t1 = tetrahedron1 cube
        t2 = tetrahedron2 cube

prop_t1_shares_edge_with_t19 :: Cube -> Bool
prop_t1_shares_edge_with_t19 cube =
    (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
      where
        t1 = tetrahedron1 cube
        t19 = tetrahedron19 cube

prop_t2_shares_edge_with_t3 :: Cube -> Bool
prop_t2_shares_edge_with_t3 cube =
    (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
      where
        t1 = tetrahedron1 cube
        t2 = tetrahedron2 cube

prop_t2_shares_edge_with_t12 :: Cube -> Bool
prop_t2_shares_edge_with_t12 cube =
    (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
      where
        t2 = tetrahedron2 cube
        t12 = tetrahedron12 cube

prop_t3_shares_edge_with_t21 :: Cube -> Bool
prop_t3_shares_edge_with_t21 cube =
    (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
      where
        t3 = tetrahedron3 cube
        t21 = tetrahedron21 cube

prop_t4_shares_edge_with_t5 :: Cube -> Bool
prop_t4_shares_edge_with_t5 cube =
    (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
      where
        t4 = tetrahedron4 cube
        t5 = tetrahedron5 cube

prop_t4_shares_edge_with_t7 :: Cube -> Bool
prop_t4_shares_edge_with_t7 cube =
    (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
      where
        t4 = tetrahedron4 cube
        t7 = tetrahedron7 cube

prop_t4_shares_edge_with_t10 :: Cube -> Bool
prop_t4_shares_edge_with_t10 cube =
    (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
      where
        t4 = tetrahedron4 cube
        t10 = tetrahedron10 cube

prop_t5_shares_edge_with_t6 :: Cube -> Bool
prop_t5_shares_edge_with_t6 cube =
    (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
      where
        t5 = tetrahedron5 cube
        t6 = tetrahedron6 cube

prop_t5_shares_edge_with_t16 :: Cube -> Bool
prop_t5_shares_edge_with_t16 cube =
    (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
      where
        t5 = tetrahedron5 cube
        t16 = tetrahedron16 cube

prop_t6_shares_edge_with_t7 :: Cube -> Bool
prop_t6_shares_edge_with_t7 cube =
    (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
      where
        t6 = tetrahedron6 cube
        t7 = tetrahedron7 cube

prop_t7_shares_edge_with_t20 :: Cube -> Bool
prop_t7_shares_edge_with_t20 cube =
    (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
      where
        t7 = tetrahedron7 cube
        t20 = tetrahedron20 cube