Temperament addition: Difference between revisions
Cmloegcmluin (talk | contribs) →Algebraic explanation: get rough version out |
Cmloegcmluin (talk | contribs) →Algebraic explanation: refine |
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=== Algebraic explanation=== | === Algebraic explanation=== | ||
This explanation relies on comparing the results of the multivector and matrix approaches to temperament arithmetic, and showing algebraically how the matrix approach can only achieve the same answer as the multivector approach on the condition that it keeps all but one vector between the added matrices the same, that is, not only are the temperaments addable, but their | This explanation relies on comparing the results of the multivector and matrix approaches to temperament arithmetic, and showing algebraically how the matrix approach can only achieve the same answer as the multivector approach on the condition that it keeps all but one vector between the added matrices the same, that is, not only are the temperaments addable, but their <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> appears explicitly in the added matrices. | ||
To compare results, we eventually get both approaches into a multivector form. With the multivector approach, we wedge the vector set first and then add the resultant multivectors to get a new multivector. With the matrix approach, we treat the vector set as a matrix and add first, then treat the resultant matrix as a vector set and wedge those vectors to get a new multivector. | To compare results, we eventually get both approaches into a multivector form. With the multivector approach, we wedge the vector set first and then add the resultant multivectors to get a new multivector. With the matrix approach, we treat the vector set as a matrix and add first, then treat the resultant matrix as a vector set and wedge those vectors to get a new multivector. | ||
The diagrams below are organized into a 2×2 layout. The left part shows the multivector approach, and the right part shows the matrix approach. The top part shows how the results of two approaches match when the | The diagrams below are organized into a 2×2 layout. The left part shows the multivector approach, and the right part shows the matrix approach. The top part shows how the results of two approaches match when the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> is successfully explicit (and in these cases, the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> vectors are highlighted in green and the <span style="color: #B6321C;"><math>L_{\text{ind}}</math></span> vectors are highlighted in red), and the bottom part shows how the results fail to match when it is not. Successful matches are highlighted in yellow and failures to match are highlighted in blue. | ||
This first diagram demonstrates this situation for a d=3, g=2 case. | This first diagram demonstrates this situation for a <math>d=3, g=2</math> case. | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|+ | |+ | ||
! | ! | ||
| Line 784: | Line 784: | ||
! | ! | ||
! | ! | ||
| colspan="11" rowspan="1" |multivector approach | | colspan="11" rowspan="1" |'''multivector approach''' | ||
! | ! | ||
| colspan="11" rowspan="1" |matrix approach | | colspan="11" rowspan="1" |'''matrix approach''' | ||
! | ! | ||
|- | |- | ||
| Line 798: | Line 798: | ||
|- | |- | ||
! rowspan="5" | | ! rowspan="5" | | ||
| colspan="1" rowspan="5" |explicit | | colspan="1" rowspan="5" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
{{bra|{{vector|<math>a</math> <math>b</math> <math>c</math>}}}} | |||
! rowspan="5" | | ! rowspan="5" | | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
! rowspan="5" | | ! rowspan="5" | | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
| colspan="1" rowspan="2" |+ | | colspan="1" rowspan="2" |<math>+</math> | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
| colspan="1" rowspan="2" |= | | colspan="1" rowspan="2" |<math>=</math> | ||
|2a | |<math>2a</math> | ||
|2b | |<math>2b</math> | ||
|2c | |<math>2c</math> | ||
! rowspan="5" | | ! rowspan="5" | | ||
|- | |- | ||
|d | | style="background-color: #E7BBB3;"|<math>d</math> | ||
|e | | style="background-color: #E7BBB3;"|<math>e</math> | ||
|f | | style="background-color: #E7BBB3;"|<math>f</math> | ||
|g | | style="background-color: #E7BBB3;"|<math>g</math> | ||
|h | | style="background-color: #E7BBB3;"|<math>h</math> | ||
|i | | style="background-color: #E7BBB3;"|<math>i</math> | ||
|d | | style="background-color: #E7BBB3;"|<math>d</math> | ||
|e | | style="background-color: #E7BBB3;"|<math>e</math> | ||
|f | | style="background-color: #E7BBB3;"|<math>f</math> | ||
|g | | style="background-color: #E7BBB3;"|<math>g</math> | ||
|h | | style="background-color: #E7BBB3;"|<math>h</math> | ||
|i | | style="background-color: #E7BBB3;"|<math>i</math> | ||
|d+g | |<math>d+g</math> | ||
|e+h | |<math>e+h</math> | ||
|f+i | |<math>f+i</math> | ||
|- | |- | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="3" | | | colspan="3" | | ||
| Line 850: | Line 850: | ||
| colspan="3" | | | colspan="3" | | ||
| | | | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
|- | |- | ||
|bf-ce | |<math>bf-ce</math> | ||
|af-cd | |<math>af-cd</math> | ||
|ae-bd | |<math>ae-bd</math> | ||
| + | |<math> +</math> | ||
|bi-ch | |<math>bi-ch</math> | ||
|ai-cg | |<math>ai-cg</math> | ||
|ah-bg | |<math>ah-bg</math> | ||
|= | |<math>=</math> | ||
|bf-ce + bi-ch | |<math>bf-ce+bi-ch</math> | ||
|af-cd + ai-cg | |<math>af-cd+ai-cg</math> | ||
|ae-bd + ah-bg | |<math>ae-bd+ah-bg</math> | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
| rowspan="2" | | | rowspan="2" | | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
| rowspan="2" | | | rowspan="2" | | ||
|2b(f+i) - 2c(e+h) | |<math>2b(f+i)-2c(e+h)</math> | ||
|2a(f+i) - 2c(d+g) | |<math>2a(f+i)-2c(d+g)</math> | ||
|2a(e+h) - 2b(d+g) | |<math>2a(e+h)-2b(d+g)</math> | ||
|- | |- | ||
| colspan="3" | | | colspan="3" | | ||
| Line 875: | Line 875: | ||
| colspan="3" | | | colspan="3" | | ||
| | | | ||
|b(f+i) - c(e+h) | | style="background-color: LightYellow;"|<math>b(f+i)-c(e+h)</math> | ||
|a(f+i) - c(d+g) | | style="background-color: LightYellow;"|<math>a(f+i)-c(d+g)</math> | ||
|a(e+h) - b(d+g) | | style="background-color: LightYellow;"|<math>a(e+h)-b(d+g)</math> | ||
|b(f+i) - c(e+h) | | style="background-color: LightYellow;"|<math>b(f+i)-c(e+h)</math> | ||
|a(f+i) - c(d+g) | | style="background-color: LightYellow;"|<math>a(f+i)-c(d+g)</math> | ||
|a(e+h) - b(d+g) | | style="background-color: LightYellow;"|<math>a(e+h)-b(d+g)</math> | ||
|- | |- | ||
! | ! | ||
| Line 891: | Line 891: | ||
|- | |- | ||
! rowspan="5" | | ! rowspan="5" | | ||
| rowspan="5" |hidden | | rowspan="5" |hidden <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
! rowspan="5" | | ! rowspan="5" | | ||
|a | |<math>a</math> | ||
|b | |<math>b</math> | ||
|c | |<math>c</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
|j | |<math>j</math> | ||
|k | |<math>k</math> | ||
|l | |<math>l</math> | ||
| | | | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
! rowspan="5" | | ! rowspan="5" | | ||
|a | |<math>a</math> | ||
|b | |<math>b</math> | ||
|c | |<math>c</math> | ||
| colspan="1" rowspan="2" |+ | | colspan="1" rowspan="2" |<math>+</math> | ||
|j | |<math>j</math> | ||
|k | |<math>k</math> | ||
|l | |<math>l</math> | ||
| colspan="1" rowspan="2" |= | | colspan="1" rowspan="2" |<math>=</math> | ||
|a+j | |<math>a+j</math> | ||
|b+k | |<math>b+k</math> | ||
|c+l | |<math>c+l</math> | ||
! rowspan="5" | | ! rowspan="5" | | ||
|- | |- | ||
|d | |<math>d</math> | ||
|e | |<math>e</math> | ||
|f | |<math>f</math> | ||
|g | |<math>g</math> | ||
|h | |<math>h</math> | ||
|i | |<math>i</math> | ||
| | |<math></math> | ||
|d | |<math>d</math> | ||
|e | |<math>e</math> | ||
|f | |<math>f</math> | ||
|g | |<math>g</math> | ||
|h | |<math>h</math> | ||
|i | |<math>i</math> | ||
|d+g | |<math>d+g</math> | ||
|e+h | |<math>e+h</math> | ||
|f+i | |<math>f+i</math> | ||
|- | |- | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="3" | | | colspan="3" | | ||
| Line 942: | Line 942: | ||
| colspan="3" | | | colspan="3" | | ||
| | | | ||
| colspan="3" rowspan="1" |∧ | | colspan="3" rowspan="1" |<math>∧</math> | ||
|- | |- | ||
|bf-ce | |<math>bf-ce</math> | ||
|af-cd | |<math>af-cd</math> | ||
|ae-bd | |<math>ae-bd</math> | ||
| + | |<math>+</math> | ||
|ki-lh | |<math>ki-lh</math> | ||
|ji-lg | |<math>ji-lg</math> | ||
|jh-kg | |<math>jh-kg</math> | ||
|= | |<math>=</math> | ||
|bf - ce + ki - lh | |<math>bf-ce+ki-lh</math> | ||
|af - cd + ji - lg | |<math>af-cd+ji-lg</math> | ||
|ae - bd + jh - kg | |<math>ae-bd+jh-kg</math> | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
| rowspan="2" | | | rowspan="2" | | ||
| colspan="3" rowspan="2" | | | colspan="3" rowspan="2" | | ||
| rowspan="2" | | | rowspan="2" | | ||
|(b+k)(f+i) - (c+l)(e+h) | |<math>(b+k)(f+i)-(c+l)(e+h)</math> | ||
|(a+j)(f+i) - (c+l)(d+g) | |<math>(a+j)(f+i)-(c+l)(d+g)</math> | ||
|(a+j)(e+h) - (b+k)(d+g) | |<math>(a+j)(e+h)-(b+k)(d+g)</math> | ||
|- | |- | ||
| colspan="3" | | | colspan="3" | | ||
| Line 967: | Line 967: | ||
| colspan="3" | | | colspan="3" | | ||
| | | | ||
|bf - ce + ki - lh | | style="background-color: LightBlue;"|<math>bf-ce+ki-lh</math> | ||
|af - cd + ji - lg | | style="background-color: LightBlue;"|<math>af-cd+ji-lg</math> | ||
|ae - bd + jh - kg | | style="background-color: LightBlue;"|<math>ae-bd+jh-kg</math> | ||
|bf + bi + kf + ki - ce - ch - le - lh | | style="background-color: LightBlue;"|<math>bf+bi+kf+ki-ce-ch-le-lh</math> | ||
|af + ai + jf + ji - cd - cg - ld - lg | | style="background-color: LightBlue;"|<math>af+ai+jf+ji-cd-cg-ld-lg</math> | ||
|ae + ah + je + jh - bd - bg - kd - kg | | style="background-color: LightBlue;"|<math>ae+ah+je+jh-bd-bg-kd-kg</math> | ||
|- | |- | ||
! | ! | ||
| Line 982: | Line 982: | ||
! | ! | ||
|} | |} | ||
This second diagram demonstrates this situation for a d=5, g=3 case. | This second diagram demonstrates this situation for a <math>d=5, g=3</math> case. One pair of the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> vectors are explicitly matching, but not the other, which isn't enough. | ||
{| class="wikitable" | {| class="wikitable center-all" | ||
|+ | |+ | ||
! | ! | ||
| Line 996: | Line 996: | ||
! colspan="2" | | ! colspan="2" | | ||
! | ! | ||
| colspan="32" rowspan="1" |multivector approach | | colspan="32" rowspan="1" |'''multivector approach''' | ||
! | ! | ||
| colspan="22" rowspan="1" |matrix approach | | colspan="22" rowspan="1" |'''matrix approach''' | ||
! | ! | ||
|- | |- | ||
| Line 1,010: | Line 1,010: | ||
|- | |- | ||
! rowspan="7" | | ! rowspan="7" | | ||
| colspan="1" rowspan="7" |explicit | | colspan="1" rowspan="7" |explicit <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
⟨{{vector|<math>a</math> <math>b</math> <math>c</math> <math>d</math> <math>e</math>}} | |||
| | {{vector|<math>f</math> <math>g</math> <math>h</math> <math>i</math> <math>j</math>}}] | ||
| style="background-color: #BED5BA;"|<math>r_1</math> | |||
! rowspan="7" | | ! rowspan="7" | | ||
| colspan="2" rowspan="1" |a | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>a</math> | ||
| colspan="2" rowspan="1" |b | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>b</math> | ||
| colspan="2" rowspan="1" |c | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>c</math> | ||
| colspan="2" rowspan="1" |d | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>d</math> | ||
| colspan="2" rowspan="1" |e | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>e</math> | ||
| rowspan="3" | | | rowspan="3" | | ||
| colspan="2" rowspan="1" |a | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>a</math> | ||
| colspan="2" rowspan="1" |b | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>b</math> | ||
| colspan="2" rowspan="1" |c | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>c</math> | ||
| colspan="2" rowspan="1" |d | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>d</math> | ||
| colspan="2" rowspan="1" |e | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>e</math> | ||
| rowspan="3" | | | rowspan="3" | | ||
| colspan="10" rowspan="3" | | | colspan="10" rowspan="3" | | ||
! rowspan="7" | | ! rowspan="7" | | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
|d | | style="background-color: #BED5BA;"|<math>d</math> | ||
|e | | style="background-color: #BED5BA;"|<math>e</math> | ||
| colspan="1" rowspan="3" |+ | | colspan="1" rowspan="3" |+ | ||
|a | | style="background-color: #BED5BA;"|<math>a</math> | ||
|b | | style="background-color: #BED5BA;"|<math>b</math> | ||
|c | | style="background-color: #BED5BA;"|<math>c</math> | ||
|d | | style="background-color: #BED5BA;"|<math>d</math> | ||
|e | | style="background-color: #BED5BA;"|<math>e</math> | ||
| colspan="1" rowspan="3" |= | | colspan="1" rowspan="3" |<math>=</math> | ||
| colspan="2" rowspan="1" |2a | | colspan="2" rowspan="1" |<math>2a</math> | ||
| colspan="2" rowspan="1" |2b | | colspan="2" rowspan="1" |<math>2b</math> | ||
| colspan="2" rowspan="1" |2c | | colspan="2" rowspan="1" |<math>2c</math> | ||
| colspan="2" rowspan="1" |2d | | colspan="2" rowspan="1" |<math>2d</math> | ||
| colspan="2" rowspan="1" |2e | | colspan="2" rowspan="1" |<math>2e</math> | ||
! rowspan="7" | | ! rowspan="7" | | ||
|- | |- | ||
| | | style="background-color: #BED5BA;"|<math>r_2</math> | ||
| colspan="2" rowspan="1" |f | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>f</math> | ||
| colspan="2" rowspan="1" |g | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>g</math> | ||
| colspan="2" rowspan="1" |h | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>h</math> | ||
| colspan="2" rowspan="1" |i | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>i</math> | ||
| colspan="2" rowspan="1" |j | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>j</math> | ||
| colspan="2" rowspan="1" |f | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>f</math> | ||
| colspan="2" rowspan="1" |g | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>g</math> | ||
| colspan="2" rowspan="1" |h | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>h</math> | ||
| colspan="2" rowspan="1" |i | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>i</math> | ||
| colspan="2" rowspan="1" |j | | style="background-color: #BED5BA;" colspan="2" rowspan="1" |<math>j</math> | ||
|f | | style="background-color: #BED5BA;" |<math>f</math> | ||
|g | | style="background-color: #BED5BA;" |<math>g</math> | ||
|h | | style="background-color: #BED5BA;" |<math>h</math> | ||
|i | | style="background-color: #BED5BA;" |<math>i</math> | ||
|j | | style="background-color: #BED5BA;" |<math>j</math> | ||
|f | | style="background-color: #BED5BA;" |<math>f</math> | ||
|g | | style="background-color: #BED5BA;" |<math>g</math> | ||
|h | | style="background-color: #BED5BA;" |<math>h</math> | ||
|i | | style="background-color: #BED5BA;" |<math>i</math> | ||
|j | | style="background-color: #BED5BA;" |<math>j</math> | ||
| colspan="2" rowspan="1" |2f | | colspan="2" rowspan="1" |<math>2f</math> | ||
| colspan="2" rowspan="1" |2g | | colspan="2" rowspan="1" |<math>2g</math> | ||
| colspan="2" rowspan="1" |2h | | colspan="2" rowspan="1" |<math>2h</math> | ||
| colspan="2" rowspan="1" |2i | | colspan="2" rowspan="1" |<math>2i</math> | ||
| colspan="2" rowspan="1" |2j | | colspan="2" rowspan="1" |<math>2j</math> | ||
|- | |- | ||
| | | style="background-color: #E7BBB3;"|<math>r_3</math> | ||
| colspan="2" rowspan="1" |k | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>k</math> | ||
| colspan="2" rowspan="1" |l | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>l</math> | ||
| colspan="2" rowspan="1" |m | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>m</math> | ||
| colspan="2" rowspan="1" |n | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>n</math> | ||
| colspan="2" rowspan="1" |o | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>o</math> | ||
| colspan="2" rowspan="1" |p | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>p</math> | ||
| colspan="2" rowspan="1" |q | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>q</math> | ||
| colspan="2" rowspan="1" |r | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>r</math> | ||
| colspan="2" rowspan="1" |s | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>s</math> | ||
| colspan="2" rowspan="1" |t | | style="background-color: #E7BBB3;" colspan="2" rowspan="1" |<math>t</math> | ||
|k | | style="background-color: #E7BBB3;"|<math>k</math> | ||
|l | | style="background-color: #E7BBB3;"|<math>l</math> | ||
|m | | style="background-color: #E7BBB3;"|<math>m</math> | ||
|n | | style="background-color: #E7BBB3;"|<math>n</math> | ||
|o | | style="background-color: #E7BBB3;"|<math>o</math> | ||
|p | | style="background-color: #E7BBB3;"|<math>p</math> | ||
|q | | style="background-color: #E7BBB3;"|<math>q</math> | ||
|r | | style="background-color: #E7BBB3;"|<math>r</math> | ||
|s | | style="background-color: #E7BBB3;"|<math>s</math> | ||
|t | | style="background-color: #E7BBB3;"|<math>t</math> | ||
| colspan="2" rowspan="1" |k+p | | colspan="2" rowspan="1" |<math>k+p</math> | ||
| colspan="2" rowspan="1" |l+q | | colspan="2" rowspan="1" |<math>l+q</math> | ||
| colspan="2" rowspan="1" |m+r | | colspan="2" rowspan="1" |<math>m+r</math> | ||
| colspan="2" rowspan="1" |n+s | | colspan="2" rowspan="1" |<math>n+s</math> | ||
| colspan="2" rowspan="1" |o+t | | colspan="2" rowspan="1" |<math>o+t</math> | ||
|- | |- | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="10" | | | colspan="10" | | ||
| Line 1,114: | Line 1,113: | ||
| colspan="5" | | | colspan="5" | | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
|- | |- | ||
| | |<math>r_1∧r_2</math> | ||
| rowspan="2" |ag-bf | | rowspan="2" |<math>ag-bf</math> | ||
| rowspan="2" |ah-cf | | rowspan="2" |<math>ah-cf</math> | ||
| rowspan="2" |ai-df | | rowspan="2" |<math>ai-df</math> | ||
| rowspan="2" |aj-ef | | rowspan="2" |<math>aj-ef</math> | ||
| rowspan="2" |bh-cg | | rowspan="2" |<math>bh-cg</math> | ||
| rowspan="2" |bi-dg | | rowspan="2" |<math>bi-dg</math> | ||
| rowspan="2" |bj-eg | | rowspan="2" |<math>bj-eg</math> | ||
| rowspan="2" |ci-dh | | rowspan="2" |<math>ci-dh</math> | ||
| rowspan="2" |cj-eh | | rowspan="2" |<math>cj-eh</math> | ||
| rowspan="2" |dj-ei | | rowspan="2" |<math>dj-ei</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
| rowspan="2" |ag-bf | | rowspan="2" |<math>ag-bf</math> | ||
| rowspan="2" |ah-cf | | rowspan="2" |<math>ah-cf</math> | ||
| rowspan="2" |ai-df | | rowspan="2" |<math>ai-df</math> | ||
| rowspan="2" |aj-ef | | rowspan="2" |<math>aj-ef</math> | ||
| rowspan="2" |bh-cg | | rowspan="2" |<math>bh-cg</math> | ||
| rowspan="2" |bi-dg | | rowspan="2" |<math>bi-dg</math> | ||
| rowspan="2" |bj-eg | | rowspan="2" |<math>bj-eg</math> | ||
| rowspan="2" |ci-dh | | rowspan="2" |<math>ci-dh</math> | ||
| rowspan="2" |cj-eh | | rowspan="2" |<math>cj-eh</math> | ||
| rowspan="2" |dj-ei | | rowspan="2" |<math>dj-ei</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
| colspan="10" rowspan="2" | | | colspan="10" rowspan="2" | | ||
| Line 1,144: | Line 1,143: | ||
| colspan="5" rowspan="3" | | | colspan="5" rowspan="3" | | ||
| rowspan="3" | | | rowspan="3" | | ||
|4ag-4bf | |<math>4ag-4bf</math> | ||
|4ah-4cf | |<math>4ah-4cf</math> | ||
|4ai-4df | |<math>4ai-4df</math> | ||
|4aj-4ef | |<math>4aj-4ef</math> | ||
|4bh-4cg | |<math>4bh-4cg</math> | ||
|4bi-4dg | |<math>4bi-4dg</math> | ||
|4bj-4eg | |<math>4bj-4eg</math> | ||
|4ci-4dh | |<math>4ci-4dh</math> | ||
|4cj-4eh | |<math>4cj-4eh</math> | ||
|4dj-4ei | |<math>4dj-4ei</math> | ||
|- | |- | ||
|simplify | |simplify <math>r_1∧r_2</math> if necessary | ||
|ag-bf | |<math>ag-bf</math> | ||
|ah-cf | |<math>ah-cf</math> | ||
|ai-df | |<math>ai-df</math> | ||
|aj-ef | |<math>aj-ef</math> | ||
|bh-cg | |<math>bh-cg</math> | ||
|bi-dg | |<math>bi-dg</math> | ||
|bj-eg | |<math>bj-eg</math> | ||
|ci-dh | |<math>ci-dh</math> | ||
|cj-eh | |<math>cj-eh</math> | ||
|dj-ei | |<math>dj-ei</math> | ||
|- | |- | ||
|( | |<math>(r_1∧r_2)∧r_3</math> | ||
|k(bh-cg) | |<math>k(bh-cg)\\-l(ah-cf)\\+m(ag-bf)</math> | ||
- | |<math>k(bi-dg)\\-l(ai-df)\\+n(ag-bf)</math> | ||
|<math>k(bj-eg)\\-l(aj-ef)\\+o(ag-bf)</math> | |||
l(ah-cf) | |<math>k(ci-dh)\\-m(ai-df)\\+n(ah-cf)</math> | ||
|<math>k(cj-eh)\\-m(aj-ef)\\+o(ah-cf)</math> | |||
+ | |<math>k(dj-ei)\\-n(aj-ef)\\+o(ai-df)</math> | ||
|<math>l(ci-dh)\\-m(bi-dg)\\+n(bh-cg)</math> | |||
m(ag-bf) | |<math>l(cj-eh)\\-m(bj-eg)\\+o(bh-cg)</math> | ||
|k(bi-dg) | |<math>l(dj-ei)\\-n(bj-eg)\\+o(bi-dg)</math> | ||
- | |<math>m(dj-ei)\\-n(cj-eh)\\+o(ci-dh)</math> | ||
|<math>+</math> | |||
l(ai-df) | |<math>p(bh-cg)\\-q(ah-cf)\\+r(ag-bf)</math> | ||
|<math>p(bi-dg)\\-q(ai-df)\\+s(ag-bf)</math> | |||
+ | |<math>p(bj-eg)\\-q(aj-ef)\\+t(ag-bf)</math> | ||
|<math>p(ci-dh)\\-r(ai-df)\\+s(ah-cf)</math> | |||
n(ag-bf) | |<math>p(cj-eh)\\-r(aj-ef)\\+t(ah-cf)</math> | ||
|k(bj-eg) | |<math>p(dj-ei)\\-s(aj-ef)\\+t(ai-df)</math> | ||
- | |<math>q(ci-dh)\\-r(bi-dg)\\+s(bh-cg)</math> | ||
|<math>q(cj-eh)\\-r(bj-eg)\\+t(bh-cg)</math> | |||
l(aj-ef) | |<math>q(dj-ei)\\-s(bj-eg)\\+t(bi-dg)</math> | ||
|<math>r(dj-ei)\\-s(cj-eh)\\+t(ci-dh)</math> | |||
+ | |<math>=</math> | ||
| style="background-color: LightYellow;"|<math>(k+p)(bh-cg)\\-(l+q)(ah-cf)\\+(m+r)(ag-bf)</math> | |||
o(ag-bf) | | style="background-color: LightYellow;"|<math>(k+p)(bi-dg)\\-(l+q)(ai-df)\\+(n+s)(ag-bf)</math> | ||
|k(ci-dh) | | style="background-color: LightYellow;"|<math>(k+p)(bj-eg)\\-(l+q)(aj-ef)\\+(o+t)(ag-bf)</math> | ||
- | | style="background-color: LightYellow;"|<math>(k+p)(ci-dh)\\-(m+r)(ai-df)\\+(n+s)(ah-cf)</math> | ||
| style="background-color: LightYellow;"|<math>(k+p)(cj-eh)\\-(m+r)(aj-ef)\\+(o+t)(ah-cf)</math> | |||
m(ai-df) | | style="background-color: LightYellow;"|<math>(k+p)(dj-ei)\\-(n+s)(aj-ef)\\+(o+t)(ai-df)</math> | ||
| style="background-color: LightYellow;"|<math>(l+q)(ci-dh)\\-(m+r)(bi-dg)\\+(n+s)(bh-cg)</math> | |||
+ | | style="background-color: LightYellow;"|<math>(l+q)(cj-eh)\\-(m+r)(bj-eg)\\+(o+t)(bh-cg)</math> | ||
| style="background-color: LightYellow;"|<math>(l+q)(dj-ei)\\-(n+s)(bj-eg)\\+(o+t)(bi-dg)</math> | |||
n(ah-cf) | | style="background-color: LightYellow;"|<math>(m+r)(dj-ei)\\-(n+s)(cj-eh)\\+(o+t)(ci-dh)</math> | ||
|k(cj-eh) | | style="background-color: LightYellow;"|<math>(k+p)(bh-cg)\\-(l+q)(ah-cf)\\+(m+r)(ag-bf)</math> | ||
- | | style="background-color: LightYellow;"|<math>(k+p)(bi-dg)\\-(l+q)(ai-df)\\+(n+s)(ag-bf)</math> | ||
| style="background-color: LightYellow;"|<math>(k+p)(bj-eg)\\-(l+q)(aj-ef)\\+(o+t)(ag-bf)</math> | |||
m(aj-ef) | | style="background-color: LightYellow;"|<math>(k+p)(ci-dh)\\-(m+r)(ai-df)\\+(n+s)(ah-cf)</math> | ||
| style="background-color: LightYellow;"|<math>(k+p)(cj-eh)\\-(m+r)(aj-ef)\\+(o+t)(ah-cf)</math> | |||
+ | | style="background-color: LightYellow;"|<math>(k+p)(dj-ei)\\-(n+s)(aj-ef)\\+(o+t)(ai-df)</math> | ||
| style="background-color: LightYellow;"|<math>(l+q)(ci-dh)\\-(m+r)(bi-dg)\\+(n+s)(bh-cg)</math> | |||
o(ah-cf) | | style="background-color: LightYellow;"|<math>(l+q)(cj-eh)\\-(m+r)(bj-eg)\\+(o+t)(bh-cg)</math> | ||
|k(dj-ei) | | style="background-color: LightYellow;"|<math>(l+q)(dj-ei)\\-(n+s)(bj-eg)\\+(o+t)(bi-dg)</math> | ||
- | | style="background-color: LightYellow;"|<math>(m+r)(dj-ei)\\-(n+s)(cj-eh)\\+(o+t)(ci-dh)</math> | ||
n(aj-ef) | |||
+ | |||
o(ai-df) | |||
|l(ci-dh) | |||
- | |||
m(bi-dg) | |||
+ | |||
n(bh-cg) | |||
|l(cj-eh) | |||
- | |||
m(bj-eg) | |||
+ | |||
o(bh-cg) | |||
|l(dj-ei) | |||
- | |||
n(bj-eg) | |||
+ | |||
o(bi-dg) | |||
|m(dj-ei) | |||
- | |||
n(cj-eh) | |||
+ | |||
o(ci-dh) | |||
| + | |||
|p(bh-cg)-q(ah-cf)+r(ag-bf) | |||
|p(bi-dg)-q(ai-df)+s(ag-bf) | |||
|p(bj-eg)-q(aj-ef)+t(ag-bf) | |||
|p(ci-dh)-r(ai-df)+s(ah-cf) | |||
|p(cj-eh)-r(aj-ef)+t(ah-cf) | |||
|p(dj-ei)-s(aj-ef)+t(ai-df) | |||
|q(ci-dh)-r(bi-dg)+s(bh-cg) | |||
|q(cj-eh)-r(bj-eg)+t(bh-cg) | |||
|q(dj-ei)-s(bj-eg)+t(bi-dg) | |||
|r(dj-ei)-s(cj-eh)+t(ci-dh) | |||
|= | |||
|(k+p)(bh-cg)-(l+q)(ah-cf)+(m+r)(ag-bf) | |||
|(k+p)(bi-dg)-(l+q)(ai-df)+(n+s)(ag-bf) | |||
|(k+p)(bj-eg)-(l+q)(aj-ef)+(o+t)(ag-bf) | |||
|(k+p)(ci-dh)-(m+r)(ai-df)+(n+s)(ah-cf) | |||
|(k+p)(cj-eh)-(m+r)(aj-ef)+(o+t)(ah-cf) | |||
|(k+p)(dj-ei)-(n+s)(aj-ef)+(o+t)(ai-df) | |||
|(l+q)(ci-dh)-(m+r)(bi-dg)+(n+s)(bh-cg) | |||
|(l+q)(cj-eh)-(m+r)(bj-eg)+(o+t)(bh-cg) | |||
|(l+q)(dj-ei)-(n+s)(bj-eg)+(o+t)(bi-dg) | |||
|(m+r)(dj-ei)-(n+s)(cj-eh)+(o+t)(ci-dh) | |||
|(k+p)(bh-cg)-(l+q)(ah-cf)+(m+r)(ag-bf) | |||
|(k+p)(bi-dg)-(l+q)(ai-df)+(n+s)(ag-bf) | |||
|(k+p)(bj-eg)-(l+q)(aj-ef)+(o+t)(ag-bf) | |||
|(k+p)(ci-dh)-(m+r)(ai-df)+(n+s)(ah-cf) | |||
|(k+p)(cj-eh)-(m+r)(aj-ef)+(o+t)(ah-cf) | |||
|(k+p)(dj-ei)-(n+s)(aj-ef)+(o+t)(ai-df) | |||
|(l+q)(ci-dh)-(m+r)(bi-dg)+(n+s)(bh-cg) | |||
|(l+q)(cj-eh)-(m+r)(bj-eg)+(o+t)(bh-cg) | |||
|(l+q)(dj-ei)-(n+s)(bj-eg)+(o+t)(bi-dg) | |||
|(m+r)(dj-ei)-(n+s)(cj-eh)+(o+t)(ci-dh) | |||
|- | |- | ||
! | ! | ||
| Line 1,291: | Line 1,220: | ||
|- | |- | ||
! rowspan="7" | | ! rowspan="7" | | ||
| rowspan="7" |hidden | | rowspan="7" |hidden <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> | ||
| | |<math>r_1</math> | ||
! rowspan="7" | | ! rowspan="7" | | ||
| colspan="2" rowspan="1" |a | | colspan="2" rowspan="1" |<math>a</math> | ||
| colspan="2" rowspan="1" |b | | colspan="2" rowspan="1" |<math>b</math> | ||
| colspan="2" rowspan="1" |c | | colspan="2" rowspan="1" |<math>c</math> | ||
| colspan="2" rowspan="1" |d | | colspan="2" rowspan="1" |<math>d</math> | ||
| colspan="2" rowspan="1" |e | | colspan="2" rowspan="1" |<math>e</math> | ||
| rowspan="3" | | | rowspan="3" | | ||
| colspan="2" rowspan="1" |a | | colspan="2" rowspan="1" |<math>a</math> | ||
| colspan="2" rowspan="1" |b | | colspan="2" rowspan="1" |<math>b</math> | ||
| colspan="2" rowspan="1" |c | | colspan="2" rowspan="1" |<math>c</math> | ||
| colspan="2" rowspan="1" |d | | colspan="2" rowspan="1" |<math>d</math> | ||
| colspan="2" rowspan="1" |e | | colspan="2" rowspan="1" |<math>e</math> | ||
| rowspan="3" | | | rowspan="3" | | ||
| colspan="10" rowspan="3" | | | colspan="10" rowspan="3" | | ||
! rowspan="7" | | ! rowspan="7" | | ||
|a | |<math>a</math> | ||
|b | |<math>b</math> | ||
|c | |<math>c</math> | ||
|d | |<math>d</math> | ||
|e | |<math>e</math> | ||
| colspan="1" rowspan="3" |+ | | colspan="1" rowspan="3" |<math>+</math> | ||
|a | |<math>a</math> | ||
|b | |<math>b</math> | ||
|c | |<math>c</math> | ||
|d | |<math>d</math> | ||
|e | |<math>e</math> | ||
| colspan="1" rowspan="3" |= | | colspan="1" rowspan="3" |<math>=</math> | ||
| colspan="2" rowspan="1" |2a | | colspan="2" rowspan="1" |<math>2a</math> | ||
| colspan="2" rowspan="1" |2b | | colspan="2" rowspan="1" |<math>2b</math> | ||
| colspan="2" rowspan="1" |2c | | colspan="2" rowspan="1" |<math>2c</math> | ||
| colspan="2" rowspan="1" |2d | | colspan="2" rowspan="1" |<math>2d</math> | ||
| colspan="2" rowspan="1" |2e | | colspan="2" rowspan="1" |<math>2e</math> | ||
! rowspan="7" | | ! rowspan="7" | | ||
|- | |- | ||
| | |<math>r_2</math> | ||
| colspan="2" rowspan="1" |f | | colspan="2" rowspan="1" |<math>f</math> | ||
| colspan="2" rowspan="1" |g | | colspan="2" rowspan="1" |<math>g</math> | ||
| colspan="2" rowspan="1" |h | | colspan="2" rowspan="1" |<math>h</math> | ||
| colspan="2" rowspan="1" |i | | colspan="2" rowspan="1" |<math>i</math> | ||
| colspan="2" rowspan="1" |j | | colspan="2" rowspan="1" |<math>j</math> | ||
| colspan="2" rowspan="1" |u | | colspan="2" rowspan="1" |<math>u</math> | ||
| colspan="2" rowspan="1" |v | | colspan="2" rowspan="1" |<math>v</math> | ||
| colspan="2" rowspan="1" |w | | colspan="2" rowspan="1" |<math>w</math> | ||
| colspan="2" rowspan="1" |x | | colspan="2" rowspan="1" |<math>x</math> | ||
| colspan="2" rowspan="1" |y | | colspan="2" rowspan="1" |<math>y</math> | ||
|f | |<math>f</math> | ||
|g | |<math>g</math> | ||
|h | |<math>h</math> | ||
|i | |<math>i</math> | ||
|j | |<math>j</math> | ||
|u | |<math>u</math> | ||
|v | |<math>v</math> | ||
|w | |<math>w</math> | ||
|x | |<math>x</math> | ||
|y | |<math>y</math> | ||
| colspan="2" rowspan="1" |f+u | | colspan="2" rowspan="1" |<math>f+u</math> | ||
| colspan="2" rowspan="1" |g+v | | colspan="2" rowspan="1" |<math>g+v</math> | ||
| colspan="2" rowspan="1" |w+h | | colspan="2" rowspan="1" |<math>w+h</math> | ||
| colspan="2" rowspan="1" |i+x | | colspan="2" rowspan="1" |<math>i+x</math> | ||
| colspan="2" rowspan="1" |j+y | | colspan="2" rowspan="1" |<math>j+y</math> | ||
|- | |- | ||
| | |<math>r_3</math> | ||
| colspan="2" rowspan="1" |k | | colspan="2" rowspan="1" |<math>k</math> | ||
| colspan="2" rowspan="1" |l | | colspan="2" rowspan="1" |<math>l</math> | ||
| colspan="2" rowspan="1" |m | | colspan="2" rowspan="1" |<math>m</math> | ||
| colspan="2" rowspan="1" |n | | colspan="2" rowspan="1" |<math>n</math> | ||
| colspan="2" rowspan="1" |o | | colspan="2" rowspan="1" |<math>o</math> | ||
| colspan="2" rowspan="1" |p | | colspan="2" rowspan="1" |<math>p</math> | ||
| colspan="2" rowspan="1" |q | | colspan="2" rowspan="1" |<math>q</math> | ||
| colspan="2" rowspan="1" |r | | colspan="2" rowspan="1" |<math>r</math> | ||
| colspan="2" rowspan="1" |s | | colspan="2" rowspan="1" |<math>s</math> | ||
| colspan="2" rowspan="1" |t | | colspan="2" rowspan="1" |<math>t</math> | ||
|k | |<math>k</math> | ||
|l | |<math>l</math> | ||
|m | |<math>m</math> | ||
|n | |<math>n</math> | ||
|o | |<math>o</math> | ||
|p | |<math>p</math> | ||
|q | |<math>q</math> | ||
|r | |<math>r</math> | ||
|s | |<math>s</math> | ||
|t | |<math>t</math> | ||
| colspan="2" rowspan="1" |k+p | | colspan="2" rowspan="1" |<math>k+p</math> | ||
| colspan="2" rowspan="1" |l+q | | colspan="2" rowspan="1" |<math>l+q</math> | ||
| colspan="2" rowspan="1" |m+r | | colspan="2" rowspan="1" |<math>m+r</math> | ||
| colspan="2" rowspan="1" |n+s | | colspan="2" rowspan="1" |<math>n+s</math> | ||
| colspan="2" rowspan="1" |o+t | | colspan="2" rowspan="1" |<math>o+t</math> | ||
|- | |- | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
| | | | ||
| colspan="10" | | | colspan="10" | | ||
| Line 1,391: | Line 1,320: | ||
| colspan="5" | | | colspan="5" | | ||
| | | | ||
| colspan="10" rowspan="1" |∧ | | colspan="10" rowspan="1" |<math>∧</math> | ||
|- | |- | ||
| | |<math>r_1∧r_2</math> | ||
| rowspan="2" |ag-bf | | rowspan="2" |<math>ag-bf</math> | ||
| rowspan="2" |ah-cf | | rowspan="2" |<math>ah-cf</math> | ||
| rowspan="2" |ai-df | | rowspan="2" |<math>ai-df</math> | ||
| rowspan="2" |aj-ef | | rowspan="2" |<math>aj-ef</math> | ||
| rowspan="2" |bh-cg | | rowspan="2" |<math>bh-cg</math> | ||
| rowspan="2" |bi-dg | | rowspan="2" |<math>bi-dg</math> | ||
| rowspan="2" |bj-eg | | rowspan="2" |<math>bj-eg</math> | ||
| rowspan="2" |ci-dh | | rowspan="2" |<math>ci-dh</math> | ||
| rowspan="2" |cj-eh | | rowspan="2" |<math>cj-eh</math> | ||
| rowspan="2" |dj-ei | | rowspan="2" |<math>dj-ei</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
| rowspan="2" |av-bu | | rowspan="2" |<math>av-bu</math> | ||
| rowspan="2" |aw-cu | | rowspan="2" |<math>aw-cu</math> | ||
| rowspan="2" |ax-du | | rowspan="2" |<math>ax-du</math> | ||
| rowspan="2" |ay-eu | | rowspan="2" |<math>ay-eu</math> | ||
| rowspan="2" |bw-cv | | rowspan="2" |<math>bw-cv</math> | ||
| rowspan="2" |bx-dv | | rowspan="2" |<math>bx-dv</math> | ||
| rowspan="2" |by-ev | | rowspan="2" |<math>by-ev</math> | ||
| rowspan="2" |cx-dw | | rowspan="2" |<math>cx-dw</math> | ||
| rowspan="2" |cy-ew | | rowspan="2" |<math>cy-ew</math> | ||
| rowspan="2" |dy-ex | | rowspan="2" |<math>dy-ex</math> | ||
| rowspan="2" | | | rowspan="2" | | ||
| colspan="10" rowspan="2" | | | colspan="10" rowspan="2" | | ||
| Line 1,421: | Line 1,350: | ||
| colspan="5" rowspan="3" | | | colspan="5" rowspan="3" | | ||
| rowspan="3" | | | rowspan="3" | | ||
|2a(g+v) - 2b(f+u) | |<math>2a(g+v)\\-2b(f+u)</math> | ||
|2a(w+h) | |<math>2a(w+h)\\-2c(f+u)</math> | ||
|<math>2a(i+x)\\-2d(f+u)</math> | |||
- | |<math>2a(j+y)\\-2e(f+u)</math> | ||
|<math>2b(w+h)\\-2c(g+v)</math> | |||
2c(f+u) | |<math>2b(i+x)\\-2d(g+v)</math> | ||
|2a(i+x) | |<math>2b(j+y)\\-2e(g+v)</math> | ||
|<math>2c(i+x)\\-2d(w+h)</math> | |||
- | |<math>2c(j+y)\\-2e(w+h)</math> | ||
|<math>2d(j+y)\\-2e(i+x)</math> | |||
2d(f+u) | |||
|2a(j+y) | |||
- | |||
2e(f+u) | |||
|2b(w+h) | |||
- | |||
2c(g+v) | |||
|2b(i+x) | |||
- | |||
2d(g+v) | |||
|2b(j+y) | |||
- | |||
2e(g+v) | |||
|2c(i+x) | |||
- | |||
2d(w+h) | |||
|2c(j+y) | |||
- | |||
2e(w+h) | |||
|2d(j+y) | |||
- | |||
2e(i+x) | |||
|- | |- | ||
|simplify( | |simplify <math>(r_1∧r_2)</math> if necessary | ||
|a(g+v)-b(f+u) | |<math>a(g+v)\\-b(f+u)</math> | ||
|a(w+h)- c(f+u) | |<math>a(w+h)\\-c(f+u)</math> | ||
|a(i+x)-d(f+u) | |<math>a(i+x)\\-d(f+u)</math> | ||
|a(j+y)-e(f+u) | |<math>a(j+y)\\-e(f+u)</math> | ||
|b(w+h)-c(g+v) | |<math>b(w+h)\\-c(g+v)</math> | ||
|b(i+x)-d(g+v) | |<math>b(i+x)\\-d(g+v)</math> | ||
|b(j+y)-e(g+v) | |<math>b(j+y)\\-e(g+v)</math> | ||
|c(i+x)-d(w+h) | |<math>c(i+x)\\-d(w+h)</math> | ||
|c(j+y)-e(w+h) | |<math>c(j+y)\\-e(w+h)</math> | ||
|d(j+y)-e(i+x) | |<math>d(j+y)\\-e(i+x)</math> | ||
|- | |- | ||
|( | |<math>(r_1∧r_2)∧r_3</math> | ||
|k(bh-cg) | |<math>k(bh-cg)\\-l(ah-cf)\\+m(ag-bf)</math> | ||
- | |<math>k(bi-dg)\\-l(ai-df)\\+n(ag-bf)</math> | ||
|<math>k(bj-eg)\\-l(aj-ef)\\+o(ag-bf)</math> | |||
l(ah-cf) | |<math>k(ci-dh)\\-m(ai-df)\\+n(ah-cf)</math> | ||
|<math>k(cj-eh)\\-m(aj-ef)\\+o(ah-cf)</math> | |||
+ | |<math>k(dj-ei)\\-n(aj-ef)\\+o(ai-df)</math> | ||
|<math>l(ci-dh)\\-m(bi-dg)\\+n(bh-cg)</math> | |||
m(ag-bf) | |<math>l(cj-eh)\\-m(bj-eg)\\+o(bh-cg)</math> | ||
|k(bi-dg) | |<math>l(dj-ei)\\-n(bj-eg)\\+o(bi-dg)</math> | ||
- | |<math>m(dj-ei)\\-n(cj-eh)\\+o(ci-dh)</math> | ||
|<math>+</math> | |||
l(ai-df) | |<math>p(bw-cv)\\-q(aw-cu)\\+r(av-bu)</math> | ||
|<math>p(bx-dv)\\-q(ax-du)\\+s(av-bu)</math> | |||
+ | |<math>p(by-ev)\\-q(ay-eu)\\+t(av-bu)</math> | ||
|<math>p(cx-dw)\\-r(ax-du)\\+s(aw-cu)</math> | |||
n(ag-bf) | |<math>p(cy-ew)\\-r(ay-eu)\\+t(aw-cu)</math> | ||
|k(bj-eg) | |<math>p(dy-ex)\\-s(ay-eu)\\+t(ax-du)</math> | ||
- | |<math>q(cx-dw)\\-r(bx-dv)\\+s(bw-cv)</math> | ||
|<math>q(cy-ew)\\-r(by-ev)\\+t(bw-cv)</math> | |||
l(aj-ef) | |<math>q(dy-ex)\\-s(by-ev)\\+t(bw-cv)</math> | ||
|<math>r(dy-ex)\\-s(cy-ew)\\+t(cx-dw)</math> | |||
+ | |<math>=</math> | ||
| style="background-color: LightBlue;"|<math>k(bh-cg)\\-l(ah-cf)\\+m(ag-bf)\\+p(bw-cv)\\-q(aw-cu)\\+r(av-bu)</math> | |||
o(ag-bf) | | style="background-color: LightBlue;"|<math>k(bi-dg)\\-l(ai-df)\\+n(ag-bf)\\+p(bx-dv)\\-q(ax-du)\\+s(av-bu)</math> | ||
|k(ci-dh) | | style="background-color: LightBlue;"|<math>k(bj-eg)\\-l(aj-ef)\\+o(ag-bf)\\+p(by-ev)\\-q(ay-eu)\\+t(av-bu)</math> | ||
- | | style="background-color: LightBlue;"|<math>k(ci-dh)\\-m(ai-df)\\+n(ah-cf)\\+p(cx-dw)\\-r(ax-du)\\+s(aw-cu)</math> | ||
| style="background-color: LightBlue;"|<math>k(cj-eh)\\-m(aj-ef)\\+o(ah-cf)\\+p(cy-ew)\\-r(ay-eu)\\+t(aw-cu)</math> | |||
m(ai-df) | | style="background-color: LightBlue;"|<math>k(dj-ei)\\-n(aj-ef)\\+o(ai-df)\\+p(dy-ex)\\-s(ay-eu)\\+t(ax-du)</math> | ||
| style="background-color: LightBlue;"|<math>l(ci-dh)\\-m(bi-dg)\\+n(bh-cg)\\+q(cx-dw)\\-r(bx-dv)\\+s(bw-cv)</math> | |||
+ | | style="background-color: LightBlue;"|<math>l(cj-eh)\\-m(bj-eg)\\+o(bh-cg)\\+q(cy-ew)\\-r(by-ev)\\+t(bw-cv)</math> | ||
| style="background-color: LightBlue;"|<math>l(dj-ei)\\-n(bj-eg)\\+o(bi-dg)\\+q(dy-ex)\\-s(by-ev)\\+t(bw-cv)</math> | |||
n(ah-cf) | | style="background-color: LightBlue;"|<math>m(dj-ei)\\-n(cj-eh)\\+o(ci-dh)\\+r(dy-ex)\\-s(cy-ew)\\+t(cx-dw)</math> | ||
|k(cj-eh) | | style="background-color: LightBlue;"|<math>(k+p)\\(b(w+h)-c(g+v))\\-(l+q)\\(a(w+h)-c(f+u))\\+(m+r)\\(a(g+v)-b(f+u))</math> | ||
- | | style="background-color: LightBlue;"|<math>(k+p)\\(b(i+x)-d(g+v))\\-(l+q)\\(a(i+x)-d(f+u))\\+(n+s)\\(a(g+v)-b(f+u))</math> | ||
| style="background-color: LightBlue;"|<math>(k+p)\\(b(j+y)-e(g+v))\\-(l+q)\\(a(j+y)-e(f+u))\\+(o+t)\\(a(g+v)-b(f+u))</math> | |||
m(aj-ef) | | style="background-color: LightBlue;"|<math>(k+p)\\(c(i+x)-d(w+h))\\-(m+r)\\(a(i+x)-d(f+u))\\+(n+s)\\(a(w+h)-c(f+u))</math> | ||
| style="background-color: LightBlue;"|<math>(k+p)\\(c(j+y)-e(w+h))\\-(m+r)\\(a(j+y)-e(f+u))\\+(o+t)\\(a(w+h)-c(f+u))</math> | |||
+ | | style="background-color: LightBlue;"|<math>(k+p)\\(d(j+y)-e(i+x))\\-(n+s)\\(a(j+y)-e(f+u))\\+(o+t)\\(a(i+x)-d(f+u))</math> | ||
| style="background-color: LightBlue;"|<math>(l+q)\\(c(i+x)-d(w+h))\\-(m+r)\\(b(i+x)-d(g+v))\\+(n+s)\\(b(w+h)-c(g+v))</math> | |||
o(ah-cf) | | style="background-color: LightBlue;"|<math>(l+q)\\(c(j+y)-e(w+h))\\-(m+r)\\(b(j+y)-e(g+v))\\+(o+t)\\(b(w+h)-c(g+v))</math> | ||
|k(dj-ei) | | style="background-color: LightBlue;"|<math>(l+q)\\(d(j+y)-e(i+x))\\-(n+s)\\(b(j+y)-e(g+v))\\+(o+t)\\(b(i+x)-d(g+v))</math> | ||
- | | style="background-color: LightBlue;"|<math>(m+r)\\(d(j+y)-e(i+x))\\-(n+s)\\(c(j+y)-e(w+h))\\+(o+t)\\(c(i+x)-d(w+h))</math> | ||
n(aj-ef) | |||
+ | |||
o(ai-df) | |||
|l(ci-dh) | |||
- | |||
m(bi-dg) | |||
+ | |||
n(bh-cg) | |||
|l(cj-eh) | |||
- | |||
m(bj-eg) | |||
+ | |||
o(bh-cg) | |||
|l(dj-ei) | |||
- | |||
n(bj-eg) | |||
+ | |||
o(bi-dg) | |||
|m(dj-ei) | |||
- | |||
n(cj-eh) | |||
+ | |||
o(ci-dh) | |||
|< | |||
|p(bw-cv) | |||
- | |||
q(aw-cu) | |||
+ | |||
r(av-bu) | |||
|p(bx-dv) | |||
- | |||
q(ax-du) | |||
+ | |||
|p(by-ev) | |||
- | |||
q(ay-eu) | |||
+ | |||
t(av-bu) | |||
|p(cx-dw) | |||
- | |||
r(ax-du) | |||
+ | |||
s(aw-cu) | |||
|p(cy-ew) | |||
- | |||
r(ay-eu) | |||
+ | |||
t(aw-cu) | |||
|p(dy-ex) | |||
- | |||
s(ay-eu) | |||
+ | |||
t(ax-du) | |||
|q(cx-dw) - | |||
r(bx-dv) | |||
+ | |||
s(bw-cv) | |||
|q(cy-ew) | |||
- | |||
r(by-ev) + | |||
t(bw-cv) | |||
|q(dy-ex) - | |||
s(by-ev) | |||
+ | |||
t(bw-cv) | |||
|r(dy-ex) - | |||
s(cy-ew) | |||
+ | |||
t(cx-dw) | |||
|= | |||
|k(bh-cg) | |||
- | |||
l(ah-cf) | |||
+ | |||
+ | |||
p(bw-cv) | |||
- | |||
q(aw-cu) | |||
+ | |||
r(av-bu) | |||
|k(bi-dg) | |||
- | |||
l(ai-df) | |||
+ | |||
+ | |||
p(bx-dv) | |||
- | |||
q(ax-du) | |||
+ | |||
s(av-bu) | |||
|k(bj-eg) | |||
- | |||
l(aj-ef) | |||
+ | |||
+ | |||
p(by-ev) | |||
- | |||
q(ay-eu) | |||
+ | |||
t(av-bu) | |||
|k(ci-dh) | |||
- | |||
m(ai-df) | |||
+ | |||
+ | |||
p(cx-dw) | |||
- | |||
r(ax-du) | |||
+ | |||
s(aw-cu) | |||
|k(cj-eh) | |||
- | |||
m(aj-ef) | |||
+ | |||
+ | |||
p(cy-ew) | |||
- | |||
r(ay-eu) | |||
+ | |||
t(aw-cu) | |||
|k(dj-ei) | |||
- | |||
n(aj-ef) | |||
+ | |||
+ | |||
p(dy-ex) | |||
- | |||
s(ay-eu) | |||
+ | |||
t(ax-du) | |||
|l(ci-dh) | |||
- | |||
m(bi-dg) | |||
+ | |||
+ | |||
q(cx-dw) | |||
- | |||
r(bx-dv) | |||
+ | |||
s(bw-cv) | |||
|l(cj-eh) | |||
- | |||
m(bj-eg) | |||
+ | |||
+ | |||
q(cy-ew) | |||
- | |||
r(by-ev) | |||
+ | |||
t(bw-cv) | |||
|l(dj-ei) | |||
- | |||
n(bj-eg) | |||
+ | |||
+ | |||
q(dy-ex) | |||
- | |||
s(by-ev) | |||
+ | |||
t(bw-cv) | |||
|m(dj-ei) | |||
- | |||
n(cj-eh) | |||
+ | |||
+ | |||
r(dy-ex) | |||
- | |||
s(cy-ew) | |||
+ | |||
t(cx-dw) | |||
|(k+p)(b(w+h)-c(g+v)) - (l+q)(a(w+h)- c(f+u)) + (m+r)(a(g+v)-b(f+u)) | |||
| | |||
|(k+p)(b(j+y)-e(g+v)) - (l+q)(a(j+y)-e(f+u)) + (o+t)(a(g+v)-b(f+u)) | |||
|(k+p)(c(i+x)-d(w+h)) - (m+r)(a(i+x)-d(f+u)) + (n+s)(a(w+h)- c(f+u)) | |||
|(k+p)(c(j+y)-e(w+h)) - (m+r)(a(j+y)-e(f+u)) + (o+t)(a(w+h)- c(f+u)) | |||
|(k+p)(d(j+y)-e(i+x)) - (n+s)(a(j+y)-e(f+u)) + (o+t)(a(i+x)-d(f+u)) | |||
|(l+q)(c(i+x)-d(w+h)) - (m+r)(b(i+x)-d(g+v)) + (n+s)(b(w+h)-c(g+v)) | |||
|(l+q)(c(j+y)-e(w+h)) - (m+r)(b(j+y)-e(g+v)) + (o+t)(b(w+h)-c(g+v)) | |||
|(l+q)(d(j+y)-e(i+x)) - (n+s)(b(j+y)-e(g+v)) + (o+t)(b(i+x)-d(g+v)) | |||
|(m+r)(d(j+y)-e(i+x)) - (n+s)(c(j+y)-e(w+h)) + (o+t)(c(i+x)-d(w+h)) | |||
|- | |- | ||
! | ! | ||
| Line 1,860: | Line 1,425: | ||
! | ! | ||
|} | |} | ||
These two examples are by no means a proof, but meditation on the patterns in the variables is at least fairly convincing. | These two examples are by no means a proof, but meditation on the patterns in the variables is at least fairly convincing. | ||
===Sintel's proof of the <span style="color: #B6321C;">linear-independence</span> conjecture=== | ===Sintel's proof of the <span style="color: #B6321C;">linear-independence</span> conjecture=== | ||