Difference between revisions of "User:Jon Awbrey/GRAPHICS"

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Revision as of 14:28, 27 August 2009

Cactus Graphs

Hi Res


Cactus Node Big Fat.jpg 117 px → 20 px
Cactus Spike Big Fat.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
(23)
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