import Cardinal
import Comparisons
-import Cube
+import Cube hiding (i, j, k)
import FunctionValues
import Misc (all_equal)
import Tests.FunctionValues ()
volume (tetrahedron23 cube) > 0
--- | Given in Sorokina and Zeilfelder, p. 79. 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
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). 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
t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity2 with tetrahedrons 3 and 2.
+-- | 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 t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
+ c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
where
- t3 = tetrahedron3 cube
- t2 = tetrahedron2 cube
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 2 and 1.
+ 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_identity3 :: Cube -> Bool
prop_c0120_identity3 cube =
c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
where
- t2 = tetrahedron2 cube
t1 = tetrahedron1 cube
+ t2 = tetrahedron2 cube
-
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 4 and 7.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
prop_c0120_identity4 :: Cube -> Bool
prop_c0120_identity4 cube =
- c t4 0 1 2 0 ~= (c t4 0 0 2 1 + c t7 0 0 1 2) / 2
+ c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
where
- t4 = tetrahedron4 cube
- t7 = tetrahedron7 cube
+ t2 = tetrahedron2 cube
+ t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 7 and 6.
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- 'prop_c0120_identity1' with tetrahedrons 4 and 5.
prop_c0120_identity5 :: Cube -> Bool
prop_c0120_identity5 cube =
- c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
- where
- t7 = tetrahedron7 cube
- t6 = tetrahedron6 cube
-
+ c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
+ where
+ t4 = tetrahedron4 cube
+ t5 = tetrahedron5 cube
--- | Given in Sorokina and Zeilfelder, p. 79. Repeats
--- prop_c0120_identity1 with tetrahedrons 6 and 5.
+-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- -- 'prop_c0120_identity1' with tetrahedrons 5 and 6.
prop_c0120_identity6 :: Cube -> Bool
prop_c0120_identity6 cube =
c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
where
- t6 = tetrahedron6 cube
t5 = tetrahedron5 cube
+ t6 = tetrahedron6 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
+-- -- 'prop_c0120_identity1' with tetrahedrons 6 and 7.
+prop_c0120_identity7 :: Cube -> Bool
+prop_c0120_identity7 cube =
+ c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 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 t3 0 1 1 1) / 2
t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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 t3 0 2 1 0) / 2
t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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 t3 1 0 1 1) / 2
t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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 t3 1 1 1 0) / 2
t3 = tetrahedron3 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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!
+-- | 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 t3 2 0 1 0) / 2
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | Given in Sorokina and Zeilfelder, p. 79, (2.7). 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 t1 0 0 2 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 0 1 1 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 0 2 0 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 1 0 1 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 1 1 0 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79. 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,v3\> and v2,v2-tilde point
--- in opposite directions, one of them has to have negative volume!
+-- | 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 t1 2 0 0 1) / 2
t1 = tetrahedron1 cube
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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!
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)
+ 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) }
+ t6 = tetrahedron6 cube
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 1 0 - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
+ 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
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 0 1 - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
+ 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
--- | Given in Sorokina and Zeilfelder, p. 79.
+
+-- | 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 2 0 - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
+ 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
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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 0 2 - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
+ 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
--- | Given in Sorokina and Zeilfelder, p. 79.
+-- | 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)
+ 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
-- | 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
-
+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.
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
+ 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 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)
+ 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