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===Analysis of contingent propositions=== | ===Analysis of contingent propositions=== | ||
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{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
| [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (26) | | [[Image:Logical Graph (P (Q)) (P (R)).jpg|500px]] || (26) | ||
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{| align="center" cellpadding="8" style="text-align:center" | {| align="center" cellpadding="8" style="text-align:center" | ||
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math> | | <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}</math> | ||
|} | |} | ||
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{| align="center" cellpadding="8" style="text-align:center" | {| align="center" cellpadding="8" style="text-align:center" | ||
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| <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math> | | <math>\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}</math> | ||
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{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
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|} | |} | ||
| − | + | ====Equation 1 : Proof 1==== | |
{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
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| − | + | ====Equation 1 : Proof 2==== | |
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| + | =====Single Image Version===== | ||
{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
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|} | |} | ||
| − | + | =====Serial Image Version===== | |
{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
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| (31) | | (31) | ||
|} | |} | ||
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{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
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| − | ====Proof 3==== | + | ====Equation 1 : Proof 3==== |
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=====Variant 2===== | =====Variant 2===== | ||
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| + | {| align="center" cellpadding="8" | ||
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| + | {| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center" | ||
| + | |- | ||
| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-00.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-01.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Cast P.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-02.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Domination.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-03.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-04.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Emptiness.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-05.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-06.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Cast Q.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-07.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-08.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Domination.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-09.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-10.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Spike.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-11.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-12.jpg|500px]] | ||
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| + | | [[Image:Equational Inference Bar -- Cast R.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-13.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-14.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Emptiness.jpg|500px]] | ||
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| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-15.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Spike.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-16.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- Cancellation.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Proof (P (Q)) (P (R)) = (P (Q R)) 3-17.jpg|500px]] | ||
| + | |- | ||
| + | | [[Image:Equational Inference Bar -- QED.jpg|500px]] | ||
| + | |} | ||
| + | | (40) | ||
| + | |} | ||
| + | |||
| + | ===Praeclarum Theorema : Proof by CAST=== | ||
{| align="center" cellpadding="8" | {| align="center" cellpadding="8" | ||
Revision as of 14:00, 27 August 2009
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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
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| \(\text{Figure 22-a. Conjunction}~ pq : X \to \mathbb{B}\) |
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| \(\text{Figure 22-b. Enlargement}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
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| \(\text{Figure 22-c. Difference}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\) |
Series 2
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| \(\text{Figure 24-1. Proposition}~ pq : X \to \mathbb{B}\) |
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| \(\text{Figure 24-2. Tacit Extension}~ \varepsilon (pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \varepsilon (pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
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| \(\text{Figure 25-1. Enlargement Map}~ \operatorname{E}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{E}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \end{array}\) |
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| \(\text{Figure 25-2. Difference Map}~ \operatorname{D}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{D}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \texttt{(} \operatorname{d}p \texttt{)} \texttt{(} \operatorname{d}q \texttt{)} \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{~} \texttt{(} \operatorname{d}p \texttt{)} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{(} \operatorname{d}q \texttt{)} \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(}q \texttt{)} & \cdot & \texttt{~} \texttt{~} \operatorname{d}p \texttt{~} \texttt{~} \operatorname{d}q \texttt{~} \texttt{~} \end{array}\) |
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| \(\text{Figure 26-1. Tangent Map}~ \operatorname{d}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{d}(pq) & = & p & \cdot & q & \cdot & \texttt{(} \operatorname{d}p \texttt{,} \operatorname{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & 0 \end{array}\) |
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| \(\text{Figure 26-2. Remainder Map}~ \operatorname{r}(pq) : \operatorname{E}X \to \mathbb{B}\) |
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\(\begin{array}{rcccccc} \operatorname{r}(pq) & = & p & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \operatorname{d}p ~ \operatorname{d}q \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \operatorname{d}p ~ \operatorname{d}q \end{array}\) |
Propositional Equation Reasoning Systems
Analysis of contingent propositions
)_(P_(R)).jpg) |
(26) |
)_(P_(R)).jpg) |
(27) |
| \(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q \texttt{))~(} p \texttt{~(} r \texttt{))}\) |
).jpg) |
(28) |
| \(\text{Venn Diagram for}~ \texttt{(} p \texttt{~(} q ~ r \texttt{))}\) |
)_(P_(R))_%3D_(P_(Q_R)).jpg) |
(29) |
Equation 1 : Proof 1
)_(P_(R))_%3D_(P_(Q_R))_Proof_1.jpg)
|
(30) |
Equation 1 : Proof 2
Single Image Version
)_(P_(R))_%3D_(P_(Q_R))_Proof_2a_Alt.jpg)
|
(31) |
Serial Image Version
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(31) |
)_(P_(R))_%3D_(P_(Q_R))_2-1-0.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-1.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-2.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-3.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-4.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-5.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-6.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-7.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-8.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-1-9.jpg)
|
(33) |
)_(P_(R))_%3D_(P_(Q_R))_2-2-0.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-1.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-2.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-3.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-4.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-5.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-6.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-7.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-8.jpg)
)_(P_(R))_%3D_(P_(Q_R))_2-2-9.jpg)
Equation 1 : Proof 3
|
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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 |
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Variant 2
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Praeclarum Theorema : Proof by CAST
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