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User:Jon Awbrey/GRAPHICS

MyWikiBiz, Author Your Legacy — Sunday February 17, 2019
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Animations

Riffs 1 to 60

Animation Riff 60 x 0.16.gif

Rotes 1 to 60

Animation Rote 60 x 0.16.gif

Image Gallery

Reduction 6:1

Cacti

Image Scale
Rooted Node Big.jpg 117 px

20 px
Rooted Edge Big.jpg 117 px

20 px

Riffs

Integer Riff Scale
1    
2 Riff 2 Big.jpg 117 px

20 px
3 Riff 3 Big.jpg 240 px

40 px
4 Riff 4 Big.jpg 233 px

40 px
5 Riff 5 Big.jpg 384 px

65 px
6 Riff 6 Big.jpg 384 px

65 px
7 Riff 7 Big.jpg 377 px

65 px
8 Riff 8 Big.jpg 384 px

65 px
9 Riff 9 Big.jpg 240 px

40 px
10 Riff 10 Big.jpg 528 px

90 px
11 Riff 11 Big.jpg 528 px

90 px
12 Riff 12 Big.jpg 384 px

65 px
13 Riff 13 Big.jpg 384 px

65 px
14 Riff 14 Big.jpg 521 px

90 px
15 Riff 15 Big.jpg 528 px

90 px
16 Riff 16 Big.jpg 377 px

65 px
17 Riff 17 Big.jpg 521 px

90 px
18 Riff 18 Big.jpg 384 px

65 px
19 Riff 19 Big.jpg 528 px

90 px
20 Riff 20 Big.jpg 528 px

90 px
21 Riff 21 Big.jpg 521 px

90 px
22 Riff 22 Big.jpg 672 px

115 px
23 Riff 23 Big.jpg 384 px

65 px
24 Riff 24 Big.jpg 672 px

115 px
25 Riff 25 Big.jpg 384 px

65 px
26 Riff 26 Big.jpg 528 px

90 px
27 Riff 27 Big.jpg 384 px

65 px
28 Riff 28 Big.jpg 521 px

90 px
29 Riff 29 Big.jpg 528 px

90 px
30 Riff 30 Big.jpg 672 px

115 px
31 Riff 31 Big.jpg 672 px

115 px
32 Riff 32 Big.jpg 528 px

90 px
33 Riff 33 Big.jpg 672 px

115 px
34 Riff 34 Big.jpg 665 px

115 px
35 Riff 35 Big.jpg 521 px

90 px
36 Riff 36 Big.jpg 384 px

65 px
37 Riff 37 Big.jpg 384 px

65 px
38 Riff 38 Big.jpg 672 px

115 px
39 Riff 39 Big.jpg 672 px

115 px
40 Riff 40 Big.jpg 816 px

135 px
41 Riff 41 Big.jpg 528 px

90 px
42 Riff 42 Big.jpg 665 px

115 px
43 Riff 43 Big.jpg 521 px

90 px
44 Riff 44 Big.jpg 672 px

115 px
45 Riff 45 Big.jpg 528 px

90 px
46 Riff 46 Big.jpg 528 px

90 px
47 Riff 47 Big.jpg 528 px

90 px
48 Riff 48 Big.jpg 384 px

65 px
49 Riff 49 Big.jpg 377 px

65 px
50 Riff 50 Big.jpg 528 px

90 px
51 Riff 51 Big.jpg 665 px

115 px
52 Riff 52 Big.jpg 528 px

90 px
53 Riff 53 Big.jpg 521 px

90 px
54 Riff 54 Big.jpg 528 px

90 px
55 Riff 55 Big.jpg 672 px

115 px
56 Riff 56 Big.jpg 809 px

135 px
57 Riff 57 Big.jpg 672 px

115 px
58 Riff 58 Big.jpg 672 px

115 px
59 Riff 59 Big.jpg 665 px

115 px
60 Riff 60 Big.jpg 672 px

115 px
64 Riff 64 Big.jpg 384 px

65 px
81 Riff 81 Big.jpg 377 px

65 px
105 Riff 105 Big.jpg 665 px

115 px
128 Riff 128 Big.jpg 521 px

90 px
165 Riff 165 Big.jpg 816 px

135 px
256 Riff 256 Big.jpg 528 px

90 px
512 Riff 512 Big.jpg 384 px

65 px
2010 Riff 2010 Big.jpg 1104 px

185 px
2500 Riff 2500 Big.jpg 528 px

90 px
65536 Riff 65536 Big.jpg 521 px

90 px
802701 Riff 802701 Big.jpg 1248 px

210 px
123456789 Riff 123456789 Big.jpg 1296 px

220 px

Rotes

Integer Rote Scale
1 Rote 1 Big.jpg 117 px

20 px
2 Rote 2 Big.jpg 233 px

40 px
3 Rote 3 Big.jpg 233 px

40 px
4 Rote 4 Big.jpg 377 px

65 px
5 Rote 5 Big.jpg 233 px

40 px
6 Rote 6 Big.jpg 473 px

80 px
7 Rote 7 Big.jpg 377 px

65 px
8 Rote 8 Big.jpg 377 px

65 px
9 Rote 9 Big.jpg 473 px

80 px
10 Rote 10 Big.jpg 473 px

80 px
11 Rote 11 Big.jpg 233 px

40 px
12 Rote 12 Big.jpg 617 px

105 px
13 Rote 13 Big.jpg 473 px

80 px
14 Rote 14 Big.jpg 617 px

105 px
15 Rote 15 Big.jpg 473 px

80 px
16 Rote 16 Big.jpg 521 px

90 px
17 Rote 17 Big.jpg 377 px

65 px
18 Rote 18 Big.jpg 713 px

120 px
19 Rote 19 Big.jpg 377 px

65 px
20 Rote 20 Big.jpg 617 px

105 px
21 Rote 21 Big.jpg 617 px

105 px
22 Rote 22 Big.jpg 473 px

80 px
23 Rote 23 Big.jpg 473 px

80 px
24 Rote 24 Big.jpg 617 px

105 px
25 Rote 25 Big.jpg 473 px

80 px
26 Rote 26 Big.jpg 713 px

120 px
27 Rote 27 Big.jpg 473 px

80 px
28 Rote 28 Big.jpg 761 px

130 px
29 Rote 29 Big.jpg 473 px

80 px
30 Rote 30 Big.jpg 713 px

120 px
31 Rote 31 Big.jpg 233 px

40 px
32 Rote 32 Big.jpg 377 px

65 px
33 Rote 33 Big.jpg 473 px

80 px
34 Rote 34 Big.jpg 617 px

105 px
35 Rote 35 Big.jpg 617 px

105 px
36 Rote 36 Big.jpg 857 px

145 px
37 Rote 37 Big.jpg 617 px

105 px
38 Rote 38 Big.jpg 617 px

105 px
39 Rote 39 Big.jpg 713 px

120 px
40 Rote 40 Big.jpg 617 px

105 px
41 Rote 41 Big.jpg 473 px

80 px
42 Rote 42 Big.jpg 857 px

145 px
43 Rote 43 Big.jpg 617 px

105 px
44 Rote 44 Big.jpg 617 px

105 px
45 Rote 45 Big.jpg 713 px

120 px
46 Rote 46 Big.jpg 713 px

120 px
47 Rote 47 Big.jpg 473 px

80 px
48 Rote 48 Big.jpg 617 px

105 px
49 Rote 49 Big.jpg 473 px

80 px
50 Rote 50 Big.jpg 713 px

120 px
51 Rote 51 Big.jpg 617 px

105 px
52 Rote 52 Big.jpg 857 px

145 px
53 Rote 53 Big.jpg 521 px

90 px
54 Rote 54 Big.jpg 713 px

120 px
55 Rote 55 Big.jpg 473 px

80 px
56 Rote 56 Big.jpg 761 px

130 px
57 Rote 57 Big.jpg 617 px

105 px
58 Rote 58 Big.jpg 713 px

120 px
59 Rote 59 Big.jpg 377 px

65 px
60 Rote 60 Big.jpg 905 px

155 px
64 Rote 64 Big.jpg 617 px

105 px
81 Rote 81 Big.jpg 617 px

105 px
105 Rote 105 Big.jpg 857 px

145 px
128 Rote 128 Big.jpg 521 px

90 px
165 Rote 165 Big.jpg 713 px

120 px
256 Rote 256 Big.jpg 521 px

90 px
512 Rote 512 Big.jpg 617 px

105 px
2010 Rote 2010 Big.jpg 1145 px

190 px
2500 Rote 2500 Big.jpg 1001 px

170 px
65536 Rote 65536 Big.jpg 665 px

115 px
802701 Rote 802701 Big.jpg 1961 px

330 px
123456789 Rote 123456789 Big.jpg 2048 px

345 px

Reduction 10:1

Cacti

Image Scale
Rooted Node Big.jpg 117 px

12 px
Rooted Edge Big.jpg 117 px

12 px

Riffs

Integer Riff Scale
1    
2 Riff 2 Big.jpg 117 px

12 px
3 Riff 3 Big.jpg 240 px

24 px
4 Riff 4 Big.jpg 233 px

24 px
5 Riff 5 Big.jpg 384 px

38 px
6 Riff 6 Big.jpg 384 px

38 px
7 Riff 7 Big.jpg 377 px

38 px
8 Riff 8 Big.jpg 384 px

38 px
9 Riff 9 Big.jpg 240 px

24 px
10 Riff 10 Big.jpg 528 px

54 px
11 Riff 11 Big.jpg 528 px

54 px
12 Riff 12 Big.jpg 384 px

38 px
13 Riff 13 Big.jpg 384 px

38 px
14 Riff 14 Big.jpg 521 px

54 px
15 Riff 15 Big.jpg 528 px

54 px
16 Riff 16 Big.jpg 377 px

38 px
17 Riff 17 Big.jpg 521 px

54 px
18 Riff 18 Big.jpg 384 px

38 px
19 Riff 19 Big.jpg 528 px

54 px
20 Riff 20 Big.jpg 528 px

54 px
21 Riff 21 Big.jpg 521 px

54 px
22 Riff 22 Big.jpg 672 px

68 px
23 Riff 23 Big.jpg 384 px

38 px
24 Riff 24 Big.jpg 672 px

68 px
25 Riff 25 Big.jpg 384 px

38 px
26 Riff 26 Big.jpg 528 px

54 px
27 Riff 27 Big.jpg 384 px

38 px
28 Riff 28 Big.jpg 521 px

54 px
29 Riff 29 Big.jpg 528 px

54 px
30 Riff 30 Big.jpg 672 px

68 px
31 Riff 31 Big.jpg 672 px

68 px
32 Riff 32 Big.jpg 528 px

54 px
33 Riff 33 Big.jpg 672 px

68 px
34 Riff 34 Big.jpg 665 px

68 px
35 Riff 35 Big.jpg 521 px

54 px
36 Riff 36 Big.jpg 384 px

38 px
37 Riff 37 Big.jpg 384 px

38 px
38 Riff 38 Big.jpg 672 px

68 px
39 Riff 39 Big.jpg 672 px

68 px
40 Riff 40 Big.jpg 816 px

82 px
41 Riff 41 Big.jpg 528 px

54 px
42 Riff 42 Big.jpg 665 px

68 px
43 Riff 43 Big.jpg 521 px

54 px
44 Riff 44 Big.jpg 672 px

68 px
45 Riff 45 Big.jpg 528 px

54 px
46 Riff 46 Big.jpg 528 px

54 px
47 Riff 47 Big.jpg 528 px

54 px
48 Riff 48 Big.jpg 384 px

38 px
49 Riff 49 Big.jpg 377 px

38 px
50 Riff 50 Big.jpg 528 px

54 px
51 Riff 51 Big.jpg 665 px

68 px
52 Riff 52 Big.jpg 528 px

54 px
53 Riff 53 Big.jpg 521 px

54 px
54 Riff 54 Big.jpg 528 px

54 px
55 Riff 55 Big.jpg 672 px

68 px
56 Riff 56 Big.jpg 809 px

82 px
57 Riff 57 Big.jpg 672 px

68 px
58 Riff 58 Big.jpg 672 px

68 px
59 Riff 59 Big.jpg 665 px

68 px
60 Riff 60 Big.jpg 672 px

68 px
64 Riff 64 Big.jpg 384 px

38 px
81 Riff 81 Big.jpg 377 px

38 px
105 Riff 105 Big.jpg 665 px

68 px
128 Riff 128 Big.jpg 521 px

54 px
165 Riff 165 Big.jpg 816 px

82 px
256 Riff 256 Big.jpg 528 px

54 px
512 Riff 512 Big.jpg 384 px

38 px
2010 Riff 2010 Big.jpg 1104 px

110 px
2500 Riff 2500 Big.jpg 528 px

54 px
65536 Riff 65536 Big.jpg 521 px

54 px
802701 Riff 802701 Big.jpg 1248 px

125 px
123456789 Riff 123456789 Big.jpg 1296 px

130 px

Rotes

Integer Rote Scale
1 Rote 1 Big.jpg 117 px

12 px
2 Rote 2 Big.jpg 233 px

24 px
3 Rote 3 Big.jpg 233 px

24 px
4 Rote 4 Big.jpg 377 px

38 px
5 Rote 5 Big.jpg 233 px

24 px
6 Rote 6 Big.jpg 473 px

48 px
7 Rote 7 Big.jpg 377 px

38 px
8 Rote 8 Big.jpg 377 px

38 px
9 Rote 9 Big.jpg 473 px

48 px
10 Rote 10 Big.jpg 473 px

48 px
11 Rote 11 Big.jpg 233 px

24 px
12 Rote 12 Big.jpg 617 px

62 px
13 Rote 13 Big.jpg 473 px

48 px
14 Rote 14 Big.jpg 617 px

62 px
15 Rote 15 Big.jpg 473 px

48 px
16 Rote 16 Big.jpg 521 px

54 px
17 Rote 17 Big.jpg 377 px

38 px
18 Rote 18 Big.jpg 713 px

72 px
19 Rote 19 Big.jpg 377 px

38 px
20 Rote 20 Big.jpg 617 px

62 px
21 Rote 21 Big.jpg 617 px

62 px
22 Rote 22 Big.jpg 473 px

48 px
23 Rote 23 Big.jpg 473 px

48 px
24 Rote 24 Big.jpg 617 px

62 px
25 Rote 25 Big.jpg 473 px

48 px
26 Rote 26 Big.jpg 713 px

72 px
27 Rote 27 Big.jpg 473 px

48 px
28 Rote 28 Big.jpg 761 px

76 px
29 Rote 29 Big.jpg 473 px

48 px
30 Rote 30 Big.jpg 713 px

72 px
31 Rote 31 Big.jpg 233 px

24 px
32 Rote 32 Big.jpg 377 px

38 px
33 Rote 33 Big.jpg 473 px

48 px
34 Rote 34 Big.jpg 617 px

62 px
35 Rote 35 Big.jpg 617 px

62 px
36 Rote 36 Big.jpg 857 px

86 px
37 Rote 37 Big.jpg 617 px

62 px
38 Rote 38 Big.jpg 617 px

62 px
39 Rote 39 Big.jpg 713 px

72 px
40 Rote 40 Big.jpg 617 px

62 px
41 Rote 41 Big.jpg 473 px

48 px
42 Rote 42 Big.jpg 857 px

86 px
43 Rote 43 Big.jpg 617 px

62 px
44 Rote 44 Big.jpg 617 px

62 px
45 Rote 45 Big.jpg 713 px

72 px
46 Rote 46 Big.jpg 713 px

72 px
47 Rote 47 Big.jpg 473 px

48 px
48 Rote 48 Big.jpg 617 px

62 px
49 Rote 49 Big.jpg 473 px

48 px
50 Rote 50 Big.jpg 713 px

72 px
51 Rote 51 Big.jpg 617 px

62 px
52 Rote 52 Big.jpg 857 px

86 px
53 Rote 53 Big.jpg 521 px

54 px
54 Rote 54 Big.jpg 713 px

72 px
55 Rote 55 Big.jpg 473 px

48 px
56 Rote 56 Big.jpg 761 px

76 px
57 Rote 57 Big.jpg 617 px

62 px
58 Rote 58 Big.jpg 713 px

72 px
59 Rote 59 Big.jpg 377 px

38 px
60 Rote 60 Big.jpg 905 px

90 px
64 Rote 64 Big.jpg 617 px

62 px
81 Rote 81 Big.jpg 617 px

62 px
105 Rote 105 Big.jpg 857 px

86 px
128 Rote 128 Big.jpg 521 px

54 px
165 Rote 165 Big.jpg 713 px

72 px
256 Rote 256 Big.jpg 521 px

54 px
512 Rote 512 Big.jpg 617 px

62 px
2010 Rote 2010 Big.jpg 1145 px

115 px
2500 Rote 2500 Big.jpg 1001 px

100 px
65536 Rote 65536 Big.jpg 665 px

68 px
802701 Rote 802701 Big.jpg 1961 px

196 px
123456789 Rote 123456789 Big.jpg 2048 px

205 px

Riffs in Numerical Order

\(\text{Riffs in Numerical Order}\!\)

 


\(1\!\)


\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

Riff 2 Big.jpg


\(\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

Riff 3 Big.jpg


\(\text{p}_\text{p}\!\)


\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

Riff 4 Big.jpg


\(\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

Riff 5 Big.jpg


\(\text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

Riff 6 Big.jpg


\(\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

Riff 7 Big.jpg


\(\text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

Riff 8 Big.jpg


\(\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

Riff 9 Big.jpg


\(\text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

Riff 10 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

Riff 11 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

Riff 12 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

Riff 13 Big.jpg


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

Riff 14 Big.jpg


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

Riff 15 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

Riff 16 Big.jpg


\(\text{p}^{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

Riff 17 Big.jpg


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

Riff 18 Big.jpg


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

Riff 19 Big.jpg


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

Riff 20 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

Riff 21 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

Riff 22 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

Riff 23 Big.jpg


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

Riff 24 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

Riff 25 Big.jpg


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

Riff 26 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

Riff 27 Big.jpg


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

Riff 28 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

Riff 29 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

Riff 30 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

Riff 31 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

Riff 32 Big.jpg


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

Riff 33 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

Riff 34 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

Riff 35 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

Riff 36 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

Riff 37 Big.jpg


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

Riff 38 Big.jpg


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

Riff 39 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

Riff 40 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

Riff 41 Big.jpg


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

Riff 42 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

Riff 43 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

Riff 44 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

Riff 45 Big.jpg


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

Riff 46 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

Riff 47 Big.jpg


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

Riff 48 Big.jpg


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

Riff 49 Big.jpg


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

Riff 50 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

Riff 51 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

Riff 52 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

Riff 53 Big.jpg


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

Riff 54 Big.jpg


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

Riff 55 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

Riff 56 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

Riff 57 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

Riff 58 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

Riff 59 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

Riff 60 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Rotes in Numerical Order

Rote 1 Big.jpg


\(1\!\)


\(\begin{array}{l} \varnothing \\ 1 \end{array}\)

Rote 2 Big.jpg


\(\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 \\ 2 \end{array}\)

Rote 3 Big.jpg


\(\text{p}_\text{p}\!\)


\(\begin{array}{l} 2\!:\!1 \\ 3 \end{array}\)

Rote 4 Big.jpg


\(\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 \\ 4 \end{array}\)

Rote 5 Big.jpg


\(\text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 \\ 5 \end{array}\)

Rote 6 Big.jpg


\(\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 \\ 6 \end{array}\)

Rote 7 Big.jpg


\(\text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 4\!:\!1 \\ 7 \end{array}\)

Rote 8 Big.jpg


\(\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 \\ 8 \end{array}\)

Rote 9 Big.jpg


\(\text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 2\!:\!2 \\ 9 \end{array}\)

Rote 10 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!1 \\ 10 \end{array}\)

Rote 11 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 5\!:\!1 \\ 11 \end{array}\)

Rote 12 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 \\ 12 \end{array}\)

Rote 13 Big.jpg


\(\text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 6\!:\!1 \\ 13 \end{array}\)

Rote 14 Big.jpg


\(\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 4\!:\!1 \\ 14 \end{array}\)

Rote 15 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 3\!:\!1 \\ 15 \end{array}\)

Rote 16 Big.jpg


\(\text{p}^{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!4 \\ 16 \end{array}\)

Rote 17 Big.jpg


\(\text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 7\!:\!1 \\ 17 \end{array}\)

Rote 18 Big.jpg


\(\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!2 \\ 18 \end{array}\)

Rote 19 Big.jpg


\(\text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 8\!:\!1 \\ 19 \end{array}\)

Rote 20 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 3\!:\!1 \\ 20 \end{array}\)

Rote 21 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 4\!:\!1 \\ 21 \end{array}\)

Rote 22 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 5\!:\!1 \\ 22 \end{array}\)

Rote 23 Big.jpg


\(\text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 9\!:\!1 \\ 23 \end{array}\)

Rote 24 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 2\!:\!1 \\ 24 \end{array}\)

Rote 25 Big.jpg


\(\text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 3\!:\!2 \\ 25 \end{array}\)

Rote 26 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 6\!:\!1 \\ 26 \end{array}\)

Rote 27 Big.jpg


\(\text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!3 \\ 27 \end{array}\)

Rote 28 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 4\!:\!1 \\ 28 \end{array}\)

Rote 29 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 10\!:\!1 \\ 29 \end{array}\)

Rote 30 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 30 \end{array}\)

Rote 31 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}_\text{p}}}}\!\)


\(\begin{array}{l} 11\!:\!1 \\ 31 \end{array}\)

Rote 32 Big.jpg


\(\text{p}^{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!5 \\ 32 \end{array}\)

Rote 33 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 5\!:\!1 \\ 33 \end{array}\)

Rote 34 Big.jpg


\(\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 7\!:\!1 \\ 34 \end{array}\)

Rote 35 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 4\!:\!1 \\ 35 \end{array}\)

Rote 36 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!2 \\ 36 \end{array}\)

Rote 37 Big.jpg


\(\text{p}_{\text{p}^\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 12\!:\!1 \\ 37 \end{array}\)

Rote 38 Big.jpg


\(\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 8\!:\!1 \\ 38 \end{array}\)

Rote 39 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 6\!:\!1 \\ 39 \end{array}\)

Rote 40 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 3\!:\!1 \\ 40 \end{array}\)

Rote 41 Big.jpg


\(\text{p}_{\text{p}_{\text{p} \text{p}_\text{p}}}\!\)


\(\begin{array}{l} 13\!:\!1 \\ 41 \end{array}\)

Rote 42 Big.jpg


\(\text{p} \text{p}_\text{p} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!1 ~~ 4\!:\!1 \\ 42 \end{array}\)

Rote 43 Big.jpg


\(\text{p}_{\text{p} \text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 14\!:\!1 \\ 43 \end{array}\)

Rote 44 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 5\!:\!1 \\ 44 \end{array}\)

Rote 45 Big.jpg


\(\text{p}_\text{p}^\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 2\!:\!2 ~~ 3\!:\!1 \\ 45 \end{array}\)

Rote 46 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 9\!:\!1 \\ 46 \end{array}\)

Rote 47 Big.jpg


\(\text{p}_{\text{p}_\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 15\!:\!1 \\ 47 \end{array}\)

Rote 48 Big.jpg


\(\text{p}^{\text{p}^\text{p}} \text{p}_\text{p}\!\)


\(\begin{array}{l} 1\!:\!4 ~~ 2\!:\!1 \\ 48 \end{array}\)

Rote 49 Big.jpg


\(\text{p}_{\text{p}^\text{p}}^\text{p}\!\)


\(\begin{array}{l} 4\!:\!2 \\ 49 \end{array}\)

Rote 50 Big.jpg


\(\text{p} \text{p}_{\text{p}_\text{p}}^\text{p}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 3\!:\!2 \\ 50 \end{array}\)

Rote 51 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}_{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 7\!:\!1 \\ 51 \end{array}\)

Rote 52 Big.jpg


\(\text{p}^\text{p} \text{p}_{\text{p} \text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 6\!:\!1 \\ 52 \end{array}\)

Rote 53 Big.jpg


\(\text{p}_{\text{p}^{\text{p}^\text{p}}}\!\)


\(\begin{array}{l} 16\!:\!1 \\ 53 \end{array}\)

Rote 54 Big.jpg


\(\text{p} \text{p}_\text{p}^{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 2\!:\!3 \\ 54 \end{array}\)

Rote 55 Big.jpg


\(\text{p}_{\text{p}_\text{p}} \text{p}_{\text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 3\!:\!1 ~~ 5\!:\!1 \\ 55 \end{array}\)

Rote 56 Big.jpg


\(\text{p}^{\text{p}_\text{p}} \text{p}_{\text{p}^\text{p}}\!\)


\(\begin{array}{l} 1\!:\!3 ~~ 4\!:\!1 \\ 56 \end{array}\)

Rote 57 Big.jpg


\(\text{p}_\text{p} \text{p}_{\text{p}^{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 2\!:\!1 ~~ 8\!:\!1 \\ 57 \end{array}\)

Rote 58 Big.jpg


\(\text{p} \text{p}_{\text{p} \text{p}_{\text{p}_\text{p}}}\!\)


\(\begin{array}{l} 1\!:\!1 ~~ 10\!:\!1 \\ 58 \end{array}\)

Rote 59 Big.jpg


\(\text{p}_{\text{p}_{\text{p}_{\text{p}^\text{p}}}}\!\)


\(\begin{array}{l} 17\!:\!1 \\ 59 \end{array}\)

Rote 60 Big.jpg


\(\text{p}^\text{p} \text{p}_\text{p} \text{p}_{\text{p}_\text{p}}\!\)


\(\begin{array}{l} 1\!:\!2 ~~ 2\!:\!1 ~~ 3\!:\!1 \\ 60 \end{array}\)

Miscellaneous Examples (Reduction 8:1)

\(\text{Integers, Riffs, Rotes}\!\)
\(\text{Integer}\!\) \(\text{Riff}\!\) \(\text{Rote}\!\)
\(1\!\)   Rote 1 Big.jpg
\(2\!\) Riff 2 Big.jpg Rote 2 Big.jpg
\(3\!\) Riff 3 Big.jpg Rote 3 Big.jpg
\(4\!\) Riff 4 Big.jpg Rote 4 Big.jpg
\(2010\!\) Riff 2010 Big.jpg Rote 2010 Big.jpg
\(2011\!\) Riff 2011 Big.jpg Rote 2011 Big.jpg
\(2500\!\) Riff 2500 Big.jpg Rote 2500 Big.jpg
\(802701\!\) Riff 802701 Big.jpg Rote 802701 Big.jpg
\(123456789\!\) Riff 123456789 Big.jpg Rote 123456789 Big.jpg

Cactus Graphs

Hi Res


Rooted Node Big.jpg 117 px → 20 px
Rooted Edge Big.jpg 117 px → 20 px
Cactus A Big.jpg 117 px → 20 px
Cactus (A) Big.jpg 117 px → 20 px
Cactus ABC Big.jpg 290 px → 50 px
Cactus ((A)(B)(C)) Big.jpg 386 px → 65 px
Cactus (A)B Big.jpg 204 px → 35 px
Cactus (A(B)) Big.jpg 348 px → 60 px
Cactus (A,B) Big.jpg 386 px → 65 px
Cactus ((A,B)) Big.jpg 386 px → 65 px
Cactus (A,B,C) Big.jpg 386 px → 65 px
Cactus ((A),(B),(C)) Big.jpg 386 px → 65 px
Cactus ((A,B,C)) Big.jpg 386 px → 65 px
Cactus (((A),(B),(C))) Big.jpg 386 px → 65 px
Cactus (A,(B),(C)) Big.jpg 386 px → 65 px
Cactus (((A),B,C)) Big.jpg 386 px → 65 px
Cactus (A,(B,C)) Big.jpg 530 px → 90 px
Cactus (X,(A),(B),(C)) Big.jpg 530 px → 90 px


Lo Res


Cactus Graph Node Connective.jpg
Cactus Graph Lobe Connective.jpg
Cactus Graph Lobe Rule.jpg
Cactus Graph Spike Rule.jpg


Differential Logic

ASCII Graphics

Series 1

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /                 o                 \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |                 |%%%%%|                 |   |
|   |        P        |%%%%%|        Q        |   |
|   |                 |%%%%%|                 |   |
|   o                 o%%%%%o                 o   |
|    \                 \%%%/                 /    |
|     \                 \%/                 /     |
|      \                 o                 /      |
|       \               / \               /       |
|        o-------------o   o-------------o        |
|                                                 |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
Figure 22-a.  Conjunction pq : X -> B

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o.->-.o                 o   |
|   |    p(q)(dp)dq   |%\%/%|  (p)q dp(dq)    |   |
|   | o---------------|->o<-|---------------o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        |                        |
|                        o                        |
|                  (p)(q) dp dq                   |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Ef =              p  q   (dp)(dq)               |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-b.  Enlargement E[pq] : EX -> B

o-------------------------------------------------o
|                                                 |
|                                                 |
|        o-------------o   o-------------o        |
|       /               \ /               \       |
|      /        P        o        Q        \      |
|     /                 /%\                 \     |
|    /                 /%%%\                 \    |
|   o                 o%%%%%o                 o   |
|   |       (dp)dq    |%%%%%|    dp(dq)       |   |
|   | o<--------------|->o<-|-------------->o |   |
|   |                 |%%^%%|                 |   |
|   o                 o%%|%%o                 o   |
|    \                 \%|%/                 /    |
|     \                 \|/                 /     |
|      \                 o                 /      |
|       \               /|\               /       |
|        o-------------o | o-------------o        |
|                        |                        |
|                        |                        |
|                        v                        |
|                        o                        |
|                      dp dq                      |
|                                                 |
o-------------------------------------------------o
|  f =                  p q                       |
o-------------------------------------------------o
|                                                 |
| Df =              p  q  ((dp)(dq))              |
|                                                 |
|           +       p (q)  (dp) dq                |
|                                                 |
|           +      (p) q    dp (dq)               |
|                                                 |
|           +      (p)(q)   dp  dq                |
|                                                 |
o-------------------------------------------------o
Figure 22-c.  Difference D[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o                                    |
|           /                     \                                   |
|          /                       \                                  |
|         /                         \                                 |
|        /                           \                                |
|       /                             \                               |
|      /                               \                              |
|     /                                 \                             |
|    o                                   o                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                 G                 |                            |
|    |                                   |                            |
|    |                                   |                            |
|    |                                   |                            |
|    o                                   o                            |
|     \                                 /                             |
|      \                               /                              |
|       \                           T /                               |
|        \             o<------------/-------------o                  |
|         \                         /                                 |
|          \                       /                                  |
|           \                     /                                   |
|            o-------------------o                                    |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 23.  Elements of a Cybernetic System

Series 2

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /                       o                       \          |
|         /                       /%\                       \         |
|        /                       /%%%\                       \        |
|       /                       /%%%%%\                       \       |
|      /                       /%%%%%%%\                       \      |
|     /                       /%%%%%%%%%\                       \     |
|    o                       o%%%%%%%%%%%o                       o    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |          P            |%%%%%%%%%%%|            Q          |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    |                       |%%%%%%%%%%%|                       |    |
|    o                       o%%%%%%%%%%%o                       o    |
|     \                       \%%%%%%%%%/                       /     |
|      \                       \%%%%%%%/                       /      |
|       \                       \%%%%%/                       /       |
|        \                       \%%%/                       /        |
|         \                       \%/                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-1.  Proposition pq : X -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o<-----------------o----------------->o          |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 24-2.  Tacit Extension !e![pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o (dp) (dq) o                       o    |
|    |                       |  o-->--o  |                       |    |
|    |                       |   \   /   |                       |    |
|    |             (dp) dq   |    \ /    |   dp (dq)             |    |
|    |          o----------------->o<-----------------o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  |                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-1.  Enlargement E[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |             (dp) dq   |           |   dp (dq)             |    |
|    |          o<---------------->o<---------------->o          |    |
|    |                       |     ^     |                       |    |
|    |                       |     |     |                       |    |
|    |                       |     |     |                       |    |
|    o                       o     |     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                       |                       /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                               dp | dq                               |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 25-2.  Difference Map D[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /   o   \                       \      |
|     /                       /   ^ ^   \                       \     |
|    o                       o   /   \   o                       o    |
|    |                       |  /     \  |                       |    |
|    |                       | /       \ |                       |    |
|    |                       |/         \|                       |    |
|    |                   (dp)/ dq     dp \(dq)                   |    |
|    |                      /|           |\                      |    |
|    |                     / |           | \                     |    |
|    |                    /  |           |  \                    |    |
|    o                   /   o           o   \                   o    |
|     \                 v     \  dp dq  /     v                 /     |
|      \               o<--------------------->o               /      |
|       \                       \     /                       /       |
|        \                       \   /                       /        |
|         \                       \ /                       /         |
|          \                       o                       /          |
|           \                     / \                     /           |
|            o-------------------o   o-------------------o            |
|                                                                     |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-1.  Differential or Tangent d[pq] : EX -> B

o---------------------------------------------------------------------o
|                                                                     |
|   X                                                                 |
|            o-------------------o   o-------------------o            |
|           /                     \ /                     \           |
|          /  P                    o                    Q  \          |
|         /                       / \                       \         |
|        /                       /   \                       \        |
|       /                       /     \                       \       |
|      /                       /       \                       \      |
|     /                       /         \                       \     |
|    o                       o           o                       o    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |   dp dq   |                       |    |
|    |            o<------------------------------->o            |    |
|    |                       |           |                       |    |
|    |                       |           |                       |    |
|    |                       |     o     |                       |    |
|    o                       o     ^     o                       o    |
|     \                       \    |    /                       /     |
|      \                       \   |   /                       /      |
|       \                       \  |  /                       /       |
|        \                       \ | /                       /        |
|         \                       \|/                       /         |
|          \                    dp | dq                    /          |
|           \                     /|\                     /           |
|            o-------------------o | o-------------------o            |
|                                  |                                  |
|                                  |                                  |
|                                  |                                  |
|                                  v                                  |
|                                  o                                  |
|                                                                     |
o---------------------------------------------------------------------o
Figure 26-2.  Remainder r[pq] : EX -> B

JPEG Graphics

Series 1

Field Picture PQ Conjunction.jpg
\(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\)
Field Picture PQ Enlargement Conjunction.jpg
\(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

Field Picture PQ Difference Conjunction.jpg
\(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

Series 2

Field Picture PQ Conjunction.jpg
\(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\)
Field Picture PQ Tacit Extension Conjunction.jpg
\(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

Field Picture PQ Enlargement Conjunction.jpg
\(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\)

Field Picture PQ Difference Conjunction.jpg
\(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\)

Field Picture PQ Differential Conjunction.jpg
\(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\)

Field Picture PQ Remainder Conjunction.jpg
\(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\)

\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\)

Propositional Equation Reasoning Systems

Analysis of contingent propositions

Logical Graph (P (Q)) (P (R)).jpg (26)
Venn Diagram (P (Q)) (P (R)).jpg (27)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}\)
Venn Diagram (P (Q R)).jpg (28)
\(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\)
Logical Graph (P (Q)) (P (R)) = (P (Q R)).jpg (29)

Equation 1 : Proof 1

Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 1.jpg (30)

Equation 1 : Proof 2

Single Image Version
Logical Graph (P (Q)) (P (R)) = (P (Q R)) Proof 2a Alt.jpg (31)
Serial Image Version
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-0.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-1.jpg
Equational Inference Bar -- Cast P.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-2.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-3.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-4.jpg
Equational Inference Bar -- Cast Q.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-5.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-6.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-7.jpg
Equational Inference Bar -- Cast R.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-8.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-1-9.jpg
Equational Inference Bar -- DNF.jpg
(31)
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-0.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-1.jpg
Equational Inference Bar -- Cast P.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-2.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-3.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-4.jpg
Equational Inference Bar -- Cast Q.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-5.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-6.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-7.jpg
Equational Inference Bar -- Cast R.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-8.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 2-2-9.jpg
Equational Inference Bar -- DNF.jpg
(33)

Equation 1 : Proof 3

  • Variant 1
  1. start
  2. cast p
  3. dom
  4. can
  5. empty
  6. can
  7. cast q
  8. dom
  9. can
  10. dom
  11. spike
  12. can
  13. cast r
  14. can
  15. empty
  16. spike
  17. can
  • Variant 2
  1. start
  2. cast p
  3. dom
  4. can
  5. empty
  6. can
  7. cast q
  8. can
  9. dom
  10. can
  11. spike
  12. can
  13. cast r
  14. can
  15. empty
  16. spike
  17. can
Variant 1
o-----------------------------------------------------------o
| Equation E_1.  Proof 3.                                   |
o-----------------------------------------------------------o
|                                                        1  |
|               q o   o r   q o r                           |
|                 |   |       |                             |
|               p o   o p   p o                             |
|                  \ /        |                             |
|                   o---------o                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        o                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        |                                  |
|                        @                                  |
|                                                           |
o==================================< CAST "p" >=============o
|                                                        2  |
|          q   r    q r    q   r    qr                      |
|          o   o     o     o o o o   o o                    |
|          |   |     |     |/  |/    |/                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        3  |
|          q   r    q r                                     |
|          o   o     o       o   o     o                    |
|          |   |     |      /   /     /                     |
|          o   o     o     o   o     o                      |
|           \ /      |      \ /      |                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        4  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o       o-------o                      |
|             \     /         \     /                       |
|              \   /           \   /                        |
|               \ /             \ /                         |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness >============o
|                                                        5  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o           o                          |
|             \     /            |                          |
|              \   /             |                          |
|               \ /              |                          |
|                o               o                          |
|                |               |                          |
|                |               |                          |
|                |               |                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        6  |
|          q   r    q r                                     |
|          o   o     o                                      |
|          |   |     |                                      |
|          o   o     o                                      |
|           \ /      |                                      |
|            o-------o                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|                o                                          |
|                |                                          |
|                |                                          |
|                |                                          |
|              p o---------------o---o p                    |
|                 \             /                           |
|                  \           /                            |
|                   \         /                             |
|                    \       /                              |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "q" >=============o
|                                                        7  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o r                        |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                        8  |
|                      o         o                          |
|          r     r     |   r     |                          |
|      o   o     o     o   o     o                          |
|      |   |     |     |   |     |                          |
|      o   o     o     o   o     o                          |
|       \ /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                        9  |
|          r     r         r                                |
|          o     o         o                                |
|          |     |         |                                |
|          o     o     o   o     o                          |
|         /      |      \ /      |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Domination >===========o
|                                                       10  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o     o         o                          |
|         /      |      \        |                          |
|        o-------o       o-------o                          |
|         \     /         \     /                           |
|          \   /           \   /                            |
|           \ /             \ /                             |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Spike >================o
|                                                       11  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o           o                              |
|         \     /            |                              |
|          \   /             |                              |
|           \ /              |                              |
|            o               o                              |
|            |               |                              |
|            |               |                              |
|            |               |                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       12  |
|          r     r                                          |
|          o     o                                          |
|          |     |                                          |
|          o     o                                          |
|         /      |                                          |
|        o-------o                                          |
|         \     /                                           |
|          \   /                                            |
|           \ /                                             |
|            o                                              |
|            |                                              |
|            |                                              |
|            |                                              |
|          q o---------------o---o q                        |
|             \             /                               |
|              \           /                                |
|               \         /                                 |
|                \       /                                  |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< CAST "r" >=============o
|                                                       13  |
|                      o     o                              |
|                      |     |                              |
|      o     o         o     o                              |
|      |     |         |     |                              |
|      o     o         o     o                              |
|     /      |        /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       14  |
|                      o     o                              |
|                     /      |                              |
|    o-------o       o-------o                              |
|     \     /         \     /                               |
|      \   /           \   /                                |
|       \ /             \ /                                 |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Emptiness & Spike >====o
|                                                       15  |
|        o               o                              16  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|        o               o                                  |
|        |               |                                  |
|        |               |                                  |
|        |               |                                  |
|      r o---------------o---o r                            |
|         \             /                                   |
|          \           /                                    |
|           \         /                                     |
|            \       /                                      |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< Cancellation >=========o
|                                                       17  |
|          r o-------o---o r                                |
|             \     /                                       |
|              \   /                                        |
|               \ /                                         |
|              q o-------o---o q                            |
|                 \     /                                   |
|                  \   /                                    |
|                   \ /                                     |
|                  p o-------o---o p                        |
|                     \     /                               |
|                      \   /                                |
|                       \ /                                 |
|                        @                                  |
|                                                           |
o==================================< QED >==================o

(40)
Variant 2
Proof (P (Q)) (P (R)) = (P (Q R)) 3-00.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-01.jpg
Equational Inference Bar -- Cast P.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-04.jpg
Equational Inference Bar -- Emptiness.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-05.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-06.jpg
Equational Inference Bar -- Cast Q.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-07.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg
Equational Inference Bar -- Domination.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-10.jpg
Equational Inference Bar -- Spike.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-11.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-12.jpg
Equational Inference Bar -- Cast R.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-13.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-14.jpg
Equational Inference Bar -- Emptiness.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-15.jpg
Equational Inference Bar -- Spike.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-16.jpg
Equational Inference Bar -- Cancellation.jpg
Proof (P (Q)) (P (R)) = (P (Q R)) 3-17.jpg
Equational Inference Bar -- QED.jpg
(40)

Praeclarum Theorema : Proof by CAST

Proof Praeclarum Theorema CAST 00.jpg
Proof Praeclarum Theorema CAST 01.jpg
Equational Inference Bar -- Cast A.jpg
Proof Praeclarum Theorema CAST 02.jpg
Equational Inference Bar -- Domination.jpg
Proof Praeclarum Theorema CAST 03.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 04.jpg
Equational Inference Bar -- Domination.jpg
Proof Praeclarum Theorema CAST 05.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 06.jpg
Equational Inference Bar -- Cast D.jpg
Proof Praeclarum Theorema CAST 07.jpg
Equational Inference Bar -- Domination.jpg
Proof Praeclarum Theorema CAST 08.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 09.jpg
Equational Inference Bar -- Domination.jpg
Proof Praeclarum Theorema CAST 10.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 11.jpg
Equational Inference Bar -- Cast B.jpg
Proof Praeclarum Theorema CAST 12.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 13.jpg
Equational Inference Bar -- Domination.jpg
Proof Praeclarum Theorema CAST 14.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 15.jpg
Equational Inference Bar -- Cast C.jpg
Proof Praeclarum Theorema CAST 16.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 17.jpg
Equational Inference Bar -- Cancellation.jpg
Proof Praeclarum Theorema CAST 18.jpg
Equational Inference Bar -- QED.jpg
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