Cactus Rules
MyWikiBiz, Author Your Legacy — Sunday December 29, 2024
Jump to navigationJump to search
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 1
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
With an eye toward the aims of the NKS Forum, I've begun to work out
a translation of the "elementary cellular automaton rules" (ECAR's),
in effect, just the boolean functions of abstract type q : B^3 -> B,
into cactus language, and I'll post a selection of my working notes
here. By way of the briefest possible reminder, this cactus syntax,
in its existential interpretation and its traverse-string redaction,
uses just two series of k-adic connectives, first, the concatenation
of k expressions is read as their k-adic logical conjunction, second,
a bracket of the form (e_1, ..., e_k) is read to say that exactly one
of the k expressions e_1, ..., e_k is false. I may sometimes refer to
this bracket as a k-adic "boundary operator" or a k-place "cactus lobe".
Reference Material:
http://atlas.wolfram.com/
http://atlas.wolfram.com/01/01/
http://atlas.wolfram.com/01/01/views/3/TableView.html
http://atlas.wolfram.com/01/01/views/87/TableView.html
http://atlas.wolfram.com/01/01/views/172/TableView.html
Incidental Musement:
http://www.pinball.com/games/cactus/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
One of the first things I note is that several whole families
of otherwise enigmatic and obscurely expressed rules take on
remarkably simple and transparently related expressions in
the cactus syntax.
For example, Table 1 exhibits the cactus syntax for
an especially interesting family of ECAR's, that is,
boolean maps of the concrete shape [p, q, r] -> [q],
or the abstract type q_j : B^3 -> B.
Table 1. A Family of Propositional Forms On Three Variables
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_22 | q_00010110 | 0 0 0 1 0 1 1 0 | ((p), (q), (r)) |
| | | | |
| q_41 | q_00101001 | 0 0 1 0 1 0 0 1 | ((p), (q), r ) |
| | | | |
| q_73 | q_01001001 | 0 1 0 0 1 0 0 1 | ((p), q , (r)) |
| | | | |
| q_134 | q_10000110 | 1 0 0 0 0 1 1 0 | ((p), q , r ) |
| | | | |
| q_97 | q_01100001 | 0 1 1 0 0 0 0 1 | ( p , (q), (r)) |
| | | | |
| q_146 | q_10010010 | 1 0 0 1 0 0 1 0 | ( p , (q), r ) |
| | | | |
| q_148 | q_10010100 | 1 0 0 1 0 1 0 0 | ( p , q , (r)) |
| | | | |
| q_104 | q_01101000 | 0 1 1 0 1 0 0 0 | ( p , q , r ) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_233 | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| | | | |
| q_214 | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), r )) |
| | | | |
| q_182 | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), q , (r))) |
| | | | |
| q_121 | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), q , r )) |
| | | | |
| q_158 | q_10011110 | 1 0 0 1 1 1 1 0 | (( p , (q), (r))) |
| | | | |
| q_109 | q_01101101 | 0 1 1 0 1 1 0 1 | (( p , (q), r )) |
| | | | |
| q_107 | q_01101011 | 0 1 1 0 1 0 1 1 | (( p , q , (r))) |
| | | | |
| q_151 | q_10010111 | 1 0 0 1 0 1 1 1 | (( p , q , r )) |
| | | | |
o---------o------------o-----------------o-------------------o
I invite the Reader to compare these expressions with their
corresponding numbers, the same boolean functions expressed
in terms of operators from the set {And, Or, Xor, Not}, for
example, as shown in the "Wolfram Atlas of Simple Programs":
http://atlas.wolfram.com/01/01/views/172/TableView.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Here are the parse-graph portraits of the family of cacti
that we examined last time, listed in complementary pairs.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p , q , r ) ` | ` ` ` ` | `(( p , q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_104 ` ` ` | ` ` ` ` | ` ` ` q_151 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` | q r ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q , r ) ` | ` ` ` ` | `(((p), q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_134 ` ` ` | ` ` ` ` | ` ` ` q_121 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` p | r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q), r ) ` | ` ` ` ` | `(( p ,(q), r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_146 ` ` ` | ` ` ` ` | ` ` ` q_109 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p , q ,(r)) ` | ` ` ` ` | `(( p , q ,(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_148 ` ` ` | ` ` ` ` | ` ` ` q_107 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o ` ` ` ` |
| ` ` ` p q ` ` ` ` | ` ` ` ` | ` ` ` | | r ` ` ` |
| ` ` ` o o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q), r ) ` | ` ` ` ` | `(((p),(q), r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_41` ` ` ` | ` ` ` ` | ` ` ` q_214 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` | q | ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q ,(r)) ` | ` ` ` ` | `(((p), q ,(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_73` ` ` ` | ` ` ` ` | ` ` ` q_182 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o o ` ` ` |
| ` ` ` ` q r ` ` ` | ` ` ` ` | ` ` ` p | | ` ` ` |
| ` ` ` ` o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q),(r)) ` | ` ` ` ` | `(( p ,(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_97` ` ` ` | ` ` ` ` | ` ` ` q_158 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` | | | ` ` ` |
| ` ` ` o o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q),(r)) ` | ` ` ` ` | `(((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_22` ` ` ` | ` ` ` ` | ` ` ` q_233 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
As I work through the 256 ECAR's or functions q_j : B^3 -> B,
I will keep an updated copy of my worksheet as an attachment
to the first posting on this thread at the NKS Forum website:
Re: http://forum.wolframscience.com/showthread.php?s=&postid=810#post810
In: http://forum.wolframscience.com/showthread.php?s=&threadid=256
The interested reader is invited to help check this work,
as errors are almost inevitable in this type of exercise.
Plus, I can't always get expressions that are as elegant
as I might like, and it may be that other eyes would see
forms more economical than the ones that strike me first.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 4
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Given the novelty of the cactus calculus, it is probably
wise to run through a representative sample of the forms
just set down, to note some principles of interpretation,
and to pick up a few clues as to their ordinary language
renderings. Throughout the rest of this reading it will
be good to recall that "truth", or a boolean valaue of 1,
is represented by a blank string or a blank-labeled node,
while "falsity", or a boolean value of 0, is rendered as
the string "()" or an unlabeled terminal edge, a "spike".
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p , q , r ) ` | ` ` ` ` | `(( p , q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_104 ` ` ` | ` ` ` ` | ` ` ` q_151 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
The function q_104 : B^3 -> B is a basic 3-lobe,
interpreted as the "just one false" operator on
three boolean variables, and the function q_151
is its boolean complement or its exact negation.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` | q r ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q , r ) ` | ` ` ` ` | `(((p), q , r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_134 ` ` ` | ` ` ` ` | ` ` ` q_121 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
The operation of q_134 can be understood by asking
what happens if p is true, in effect, if the label
"p" disappears, leaving only its supporting spike.
That spike, the unique false argument on the lobe,
punctures the lobe beneath, if you will, and what
abides is the statement "q r", that is, "q and r".
On the other hand, if p is (), then the branch (p)
appears to be (()), which reduces to true, and so
it disappears instead, leaving just (q, r), which
is tantamount to stating that q is not equal to r.
In sum the cases are: p q r, (p) q (r), (p)(q) r.
Once again, q_121 is just the complement of q_134.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` | | | ` ` ` |
| ` ` ` o o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q),(r)) ` | ` ` ` ` | `(((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_22` ` ` ` | ` ` ` ` | ` ` ` q_233 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
The rest of this gang can be dispatched by the same method.
But I want to single out for special mention the form q_22,
the "just one true" operator that is especially handy when
the time comes to specify a partition of the universe into
a number of mutually exclusive and exhaustive territories,
here envisioned to salute the flags p, q, r, respectively.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 5
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
So long as we're seeing the sights at Cactus Junction,
we might as well take a gander at a computational way
to assay the import of any ole cactus expression that
comes down the pike. Way out here, and elsewhere, too,
the computational clarification of a formal expression
is claimed to yield its canonical or its "normal" form.
Finer distinctions can be weighed, of course, and there
is always the problem of just how, exactly, and, indeed,
even whether such forms will be forthcoming from a given
cut of syntax for a given objective domain, or any other
wide open space. But the notion of a "normal form" is
cast in the right direction, and so it'll do for now.
By way of example, let's examine the subtype of cactoid expression
that is typified by q_97 and its complement q_158, and that hardly
got its just deserts in the way of attention the last time around.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o o ` ` ` |
| ` ` ` ` q r ` ` ` | ` ` ` ` | ` ` ` p | | ` ` ` |
| ` ` ` ` o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ( p ,(q),(r)) ` | ` ` ` ` | `(( p ,(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_97` ` ` ` | ` ` ` ` | ` ` ` q_158 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
Cactus forms of the generic shape (g, (s_1), ..., (s_k))
are those that arise when we have a "genus and species"
or a "pie chart" arrangement of logical features, where
g is the genus and the k species are s_1 through s_k,
or g is the whole pie and the slices are the s_j.
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` s_1 ` s_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `g` ` `|` ` `|` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o-----o-...-o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
We can reason out the meaning of all such expressions
by using the case analysis tactic that we used before.
If g is true, then it's just like "g" wasn't there at
all, and the expression comes down to the case below:
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` s_1 ` ` s_k ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o` ` ` `o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `|` ` ` `|` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `o--...--o` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
But this expresses the "just one true" condition that partitions
the remaining space, that is to say, the space where g is true,
into k sectors where each of the s_j in its own turn is true.
On the other hand, in the case that g is false, we are left
with a (k+1)-lobe that is known to bear this one bare spike:
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` s_1 ` s_k ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o` ` `o` ` `o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `|` ` `|` ` `|` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `o-----o-...-o` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` \ ` ` ` ` / ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` ` ` `/` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ ` ` / ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `@` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
If that expression as a whole is going to turn out to be true,
then there can be only one expression that evaluates to false
on its argument list, and since we already have it in custody,
we know that the remaining arguments, (s_1), ..., (s_k), will
all have to be true. In effect, the spike collapses the lobe
to a node, leaving a conjunction of the negations of the s_j.
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` `s_1` ` `s_k` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` o `...` o ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` `\` | `/` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` \ | / ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` `\|/` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
In summation, we have the following interpretation:
If g is true, then exactly one of the s_j is true;
if g is false, then all of the s_j are false, too.
That is not yet a method that would be amenable to
computational routine, but it does get us part way.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 6
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Within each space of boolean functions {f : B^k -> B},
altogether ranking a cardinality of 2^(2^k) functions,
there are several standard subsets of cardinality 2^k
that rate special mention and study. One such subset
is the space of linear functions, known algebraically
as the set of "homomorphisms" {hom : B^k -> B} or the
"dual space" X*, because it is dual to the coordinate
space X of "points" or "vectors" in B^k.
In the present setting, where k = 3, we may expect to find
2^3 = 8 linear functions of the abstract type h : B^3 -> B.
Table 2 shows the q_j that are linear functions, together
with their boolean complements or their logical negations.
Table 2. Linear Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_0 | q_00000000 | 0 0 0 0 0 0 0 0 | ( ) |
| | | | |
| q_240 | q_11110000 | 1 1 1 1 0 0 0 0 | p |
| | | | |
| q_204 | q_11001100 | 1 1 0 0 1 1 0 0 | q |
| | | | |
| q_170 | q_10101010 | 1 0 1 0 1 0 1 0 | r |
| | | | |
| q_60 | q_00111100 | 0 0 1 1 1 1 0 0 | (p , q) |
| | | | |
| q_90 | q_01011010 | 0 1 0 1 1 0 1 0 | (p , r) |
| | | | |
| q_102 | q_01100110 | 0 1 1 0 0 1 1 0 | (q , r) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | (p , (q , r)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_255 | q_11111111 | 1 1 1 1 1 1 1 1 | (( )) |
| | | | |
| q_15 | q_00001111 | 0 0 0 0 1 1 1 1 | (p) |
| | | | |
| q_51 | q_00110011 | 0 0 1 1 0 0 1 1 | (q) |
| | | | |
| q_85 | q_01010101 | 0 1 0 1 0 1 0 1 | (r) |
| | | | |
| q_195 | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , q)) |
| | | | |
| q_165 | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , r)) |
| | | | |
| q_153 | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , r)) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , (q , r))) |
| | | | |
o---------o------------o-----------------o-------------------o
The Figures that follow give a representative selection
of the corresponding cacti in all their greenest glory.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `( )` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_0` ` ` ` | ` ` ` ` | ` ` ` q_255 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` `(p)` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_240 ` ` ` | ` ` ` ` | ` ` ` q_15` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , q)` ` ` | ` ` ` ` | ` ` ((p , q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_60` ` ` ` | ` ` ` ` | ` ` ` q_195 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o---o ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` p `\ /` ` ` |
| ` ` ` ` o---o ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p `\ /` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p , (q , r)) ` | ` ` ` ` | `((p , (q , r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_150 ` ` ` | ` ` ` ` | ` ` ` q_105 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
Beannachtaí na Féile Pádraig oraibh go leir!
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 7
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Had I been thinking ahead, I might have mentioned this first,
but now that aspects of algebra and geometry have intruded on
our logical paradise, in the guise of the dual space X*, let's
give belated notice to one family of propositions that have been
basic to our enterprise all along, whether we noticed them or not.
In a k-dimensional universe of discourse X% = [x_1, ..., x_k] the
position space X = <|x_1, ..., x_k|> is isomorphic to B^k and the
proposition space X^ = (X -> B) = {f : X -> B} bears the abstract
type B^k -> B. In algebra and geometry, as a rule, one tends to
take position spaces and function spaces together in pairs, and
so we assign the universe X% a "stereotype" of <B^k, B^k -> B>,
or B^k +-> B, for short. I like to think of these spaces as
the "paint layer" X and "draw layer" X^ of the universe X%.
What I need to make a point of at this point is that the k-set
of logical features !X! = {x_1, ..., x_k} that we invoke as the
basis of the universe of discourse also constitutes an important
family of propositions x_j : B^k -> B, for j = 1 to k. These are
called by any one of several different names: "basic propositions",
"coordinate projections", or "simple propositions".
Table 0 accords this family of simple propositions their
formal recognition, for the present case of 3 dimensions.
Table 0. Simple Propositions
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Of course, we've already seen this 3-set of basic propositions
numbered among the (2^3)-set of linear propositions in Table 2.
Additional discussion of these underpinnings can be found here:
| Jon Awbrey, "Differential Logic and Dynamic Systems"
| http://stderr.org/pipermail/inquiry/2003-May/thread.html#478
| http://stderr.org/pipermail/inquiry/2003-June/thread.html#553
Especially:
DLOG D2. http://stderr.org/pipermail/inquiry/2003-May/000480.html
DLOG D5. http://stderr.org/pipermail/inquiry/2003-May/000483.html
With that out of the way, I'll try to
get back to the main event next time.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 8
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In any k-dimensional universe of discourse X% = [x_1, ..., x_k]
there are two other (2^k)-clans of propositions that ordinarily
merit special attention. These are the "positive" propositions
and the "singular" propositions, tabulated for the present case
k = 3 in Tables 3 and 4, respectively, as usual throwing in the
logical complements just for good measure.
Table 3. Positive Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_255 | q_11111111 | 1 1 1 1 1 1 1 1 | (( )) |
| | | | |
| q_240 | q_11110000 | 1 1 1 1 0 0 0 0 | p |
| | | | |
| q_204 | q_11001100 | 1 1 0 0 1 1 0 0 | q |
| | | | |
| q_170 | q_10101010 | 1 0 1 0 1 0 1 0 | r |
| | | | |
| q_192 | q_11000000 | 1 1 0 0 0 0 0 0 | p q |
| | | | |
| q_160 | q_10100000 | 1 0 1 0 0 0 0 0 | p r |
| | | | |
| q_136 | q_10001000 | 1 0 0 0 1 0 0 0 | q r |
| | | | |
| q_128 | q_10000000 | 1 0 0 0 0 0 0 0 | p q r |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_0 | q_00000000 | 0 0 0 0 0 0 0 0 | ( ) |
| | | | |
| q_15 | q_00001111 | 0 0 0 0 1 1 1 1 | (p) |
| | | | |
| q_51 | q_00110011 | 0 0 1 1 0 0 1 1 | (q) |
| | | | |
| q_85 | q_01010101 | 0 1 0 1 0 1 0 1 | (r) |
| | | | |
| q_63 | q_00111111 | 0 0 1 1 1 1 1 1 | (p q) |
| | | | |
| q_95 | q_01011111 | 0 1 0 1 1 1 1 1 | (p r) |
| | | | |
| q_119 | q_01110111 | 0 1 1 1 0 1 1 1 | (q r) |
| | | | |
| q_127 | q_01111111 | 0 1 1 1 1 1 1 1 | (p q r) |
| | | | |
o---------o------------o-----------------o-------------------o
Table 4. Singular Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_1 | q_00000001 | 0 0 0 0 0 0 0 1 | (p) (q) (r) |
| | | | |
| q_2 | q_00000010 | 0 0 0 0 0 0 1 0 | (p) (q) r |
| | | | |
| q_4 | q_00000100 | 0 0 0 0 0 1 0 0 | (p) q (r) |
| | | | |
| q_8 | q_00001000 | 0 0 0 0 1 0 0 0 | (p) q r |
| | | | |
| q_16 | q_00010000 | 0 0 0 1 0 0 0 0 | p (q) (r) |
| | | | |
| q_32 | q_00100000 | 0 0 1 0 0 0 0 0 | p (q) r |
| | | | |
| q_64 | q_01000000 | 0 1 0 0 0 0 0 0 | p q (r) |
| | | | |
| q_128 | q_10000000 | 1 0 0 0 0 0 0 0 | p q r |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_254 | q_11111110 | 1 1 1 1 1 1 1 0 | ((p) (q) r)) |
| | | | |
| q_253 | q_11111101 | 1 1 1 1 1 1 0 1 | ((p) (q) r ) |
| | | | |
| q_251 | q_11111011 | 1 1 1 1 1 0 1 1 | ((p) q (r)) |
| | | | |
| q_247 | q_11110111 | 1 1 1 1 0 1 1 1 | ((p) q r ) |
| | | | |
| q_239 | q_11101111 | 1 1 1 0 1 1 1 1 | ( p (q) (r)) |
| | | | |
| q_223 | q_11011111 | 1 1 0 1 1 1 1 1 | ( p (q) r ) |
| | | | |
| q_191 | q_10111111 | 1 0 1 1 1 1 1 1 | ( p q (r)) |
| | | | |
| q_127 | q_01111111 | 0 1 1 1 1 1 1 1 | ( p q r ) |
| | | | |
o---------o------------o-----------------o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 9
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In the language of cacti, as in Peirce's existential graphs,
the implication p => q takes the form (p (q)), which can be
parsed in a revealing manner as "not p without q". Thus it
forms the counterpoint to its counter-exemplary form, p (q),
which may be parsed as "p without q", or just "p and not q".
The parse-graph of (p (q)) is a particular type of tree,
that my school of thought in graph theory nomenclates as
a "painted and rooted tree" (PART). The symbols from the
alphabet !X! of logical marks, in our case, "p", "q", "r",
are called "paints" as a way of signifying that one can put
as many of them as one likes on a node, or none at all, and
that there is no requirement to use all of the paints of the
given palette !X! on any particular graph. In my etchings,
the root node is singled out with the amphora sign "@".
The graph of a simple implication can be drawn in any way that
a free rooted tree can be, but it is frequently convenient to
portray it as we see below, partly because of how often we
find ourselves linking implications in stepwise series.
o-------------------------------------------------o
| |
| p q |
| o-----------o |
| \ |
| \ |
| \ |
| \ |
| \ |
| @ |
| |
o-------------------------------------------------o
| ( p ( q )) |
o-------------------------------------------------o
Table 5 shows a number of ECAR's that have the form
of simple implications or their logical complements.
Table 5. Variations on a Theme of Implication
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_207 | q_11001111 | 1 1 0 0 1 1 1 1 | (p (q)) |
| | | | |
| q_175 | q_10101111 | 1 0 1 0 1 1 1 1 | (p (r)) |
| | | | |
| q_187 | q_10111011 | 1 0 1 1 1 0 1 1 | (q (r)) |
| | | | |
| q_243 | q_11110011 | 1 1 1 1 0 0 1 1 | ((p) q) |
| | | | |
| q_245 | q_11110101 | 1 1 1 1 0 1 0 1 | ((p) r) |
| | | | |
| q_221 | q_11011101 | 1 1 0 1 1 1 0 1 | ((q) r) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_48 | q_00110000 | 0 0 1 1 0 0 0 0 | p (q) |
| | | | |
| q_80 | q_01010000 | 0 1 0 1 0 0 0 0 | p (r) |
| | | | |
| q_68 | q_01000100 | 0 1 0 0 0 1 0 0 | q (r) |
| | | | |
| q_12 | q_00001100 | 0 0 0 0 1 1 0 0 | (p) q |
| | | | |
| q_10 | q_00001010 | 0 0 0 0 1 0 1 0 | (p) r |
| | | | |
| q_34 | q_00100010 | 0 0 1 0 0 0 1 0 | (q) r |
| | | | |
o---------o------------o-----------------o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 10
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 6. More Variations on a Theme of Implication
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_176 | q_10110000 | 1 0 1 1 0 0 0 0 | p (q (r)) |
| | | | |
| q_208 | q_11010000 | 1 1 0 1 0 0 0 0 | p (r (q)) |
| | | | |
| q_11 | q_00001011 | 0 0 0 0 1 0 1 1 | (p) (q (r)) |
| | | | |
| q_13 | q_00001101 | 0 0 0 0 1 1 0 1 | (p) (r (q)) |
| | | | |
| q_140 | q_10001100 | 1 0 0 0 1 1 0 0 | q (p (r)) |
| | | | |
| q_196 | q_11000100 | 1 1 0 0 0 1 0 0 | q (r (p)) |
| | | | |
| q_35 | q_00100011 | 0 0 1 0 0 0 1 1 | (q) (p (r)) |
| | | | |
| q_49 | q_00110001 | 0 0 1 1 0 0 0 1 | (q) (r (p)) |
| | | | |
| q_138 | q_10001010 | 1 0 0 0 1 0 1 0 | r (p (q)) |
| | | | |
| q_162 | q_10100010 | 1 0 1 0 0 0 1 0 | r (q (p)) |
| | | | |
| q_69 | q_01000101 | 0 1 0 0 0 1 0 1 | (r) (p (q)) |
| | | | |
| q_81 | q_01010001 | 0 1 0 1 0 0 0 1 | (r) (q (p)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_79 | q_01001111 | 0 1 0 0 1 1 1 1 | ( p (q (r))) |
| | | | |
| q_47 | q_00101111 | 0 0 1 0 1 1 1 1 | ( p (r (q))) |
| | | | |
| q_244 | q_11110100 | 1 1 1 1 0 1 0 0 | ((p) (q (r))) |
| | | | |
| q_242 | q_11110010 | 1 1 1 1 0 0 1 0 | ((p) (r (q))) |
| | | | |
| q_115 | q_01110011 | 0 1 1 1 0 0 1 1 | ( q (p (r))) |
| | | | |
| q_59 | q_00111011 | 0 0 1 1 1 0 1 1 | ( q (r (p))) |
| | | | |
| q_220 | q_11011100 | 1 1 0 1 1 1 0 0 | ((q) (p (r))) |
| | | | |
| q_206 | q_11001110 | 1 1 0 0 1 1 1 0 | ((q) (r (p))) |
| | | | |
| q_117 | q_01110101 | 0 1 1 1 0 1 0 1 | ( r (p (q))) |
| | | | |
| q_93 | q_01011101 | 0 1 0 1 1 1 0 1 | ( r (q (p))) |
| | | | |
| q_186 | q_10111010 | 1 0 1 1 1 0 1 0 | ((r) (p (q))) |
| | | | |
| q_174 | q_10101110 | 1 0 1 0 1 1 1 0 | ((r) (q (p))) |
| | | | |
o---------o------------o-----------------o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 11
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 7. Conjunctive Implications and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_139 | q_10001011 | 1 0 0 0 1 0 1 1 | (p (q))(q (r)) |
| | | | |
| q_141 | q_10001101 | 1 0 0 0 1 1 0 1 | (p (r))(r (q)) |
| | | | |
| q_177 | q_10110001 | 1 0 1 1 0 0 0 1 | (q (r))(r (p)) |
| | | | |
| q_163 | q_10100011 | 1 0 1 0 0 0 1 1 | (q (p))(p (r)) |
| | | | |
| q_197 | q_11000101 | 1 1 0 0 0 1 0 1 | (r (p))(p (q)) |
| | | | |
| q_209 | q_11010001 | 1 1 0 1 0 0 0 1 | (r (q))(q (p)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_116 | q_01110100 | 0 1 1 1 0 1 0 0 | ((p (q))(q (r))) |
| | | | |
| q_114 | q_01110010 | 0 1 1 1 0 0 1 0 | ((p (r))(r (q))) |
| | | | |
| q_78 | q_01001110 | 0 1 0 0 1 1 1 0 | ((q (r))(r (p))) |
| | | | |
| q_92 | q_01011100 | 0 1 0 1 1 1 0 0 | ((q (p))(p (r))) |
| | | | |
| q_58 | q_00111010 | 0 0 1 1 1 0 1 0 | ((r (p))(p (q))) |
| | | | |
| q_46 | q_00101110 | 0 0 1 0 1 1 1 0 | ((r (q))(q (p))) |
| | | | |
o---------o------------o-----------------o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 12
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
In the language of cacti, unlike Peirce's alpha graphs,
it is possible to represent the logical functions that
correspond to the difference in truth value and the
equality in truth value of two logical variables
in forms that mention each variable only once.
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , q)` ` ` | ` ` ` ` | ` ` ((p , q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_60` ` ` ` | ` ` ` ` | ` ` ` q_195 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
We have already noted the initial variations on the themes
of difference and equality among the forms in Table 2 that
gave the linear propositions and their logical complements.
Table 8 enumerates a few more variations along these lines.
Table 8. More Variations on Difference and Equality
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_96 | q_01100000 | 0 1 1 0 0 0 0 0 | p (q , r) |
| | | | |
| q_72 | q_01001000 | 0 1 0 0 1 0 0 0 | q (p , r) |
| | | | |
| q_40 | q_00101000 | 0 0 1 0 1 0 0 0 | r (p , q) |
| | | | |
| q_144 | q_10010000 | 1 0 0 1 0 0 0 0 | p ((q , r)) |
| | | | |
| q_132 | q_10000100 | 1 0 0 0 0 1 0 0 | q ((p , r)) |
| | | | |
| q_130 | q_10000010 | 1 0 0 0 0 0 1 0 | r ((p , q)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_6 | q_00000110 | 0 0 0 0 0 1 1 0 | (p) (q , r) |
| | | | |
| q_18 | q_00010010 | 0 0 0 1 0 0 1 0 | (q) (p , r) |
| | | | |
| q_20 | q_00010100 | 0 0 0 1 0 1 0 0 | (r) (p , q) |
| | | | |
| q_9 | q_00001001 | 0 0 0 0 1 0 0 1 | (p) ((q , r)) |
| | | | |
| q_33 | q_00100001 | 0 0 1 0 0 0 0 1 | (q) ((p , r)) |
| | | | |
| q_65 | q_01000001 | 0 1 0 0 0 0 0 1 | (r) ((p , q)) |
| | | | |
o=========o============o=================o===================o
| | | | |
| q_159 | q_10011111 | 1 0 0 1 1 1 1 1 | (p (q , r)) |
| | | | |
| q_183 | q_10110111 | 1 0 1 1 0 1 1 1 | (q (p , r)) |
| | | | |
| q_215 | q_11010111 | 1 1 0 1 0 1 1 1 | (r (p , q)) |
| | | | |
| q_111 | q_01101111 | 0 1 1 0 1 1 1 1 | (p ((q , r))) |
| | | | |
| q_123 | q_01111011 | 0 1 1 1 1 0 1 1 | (q ((p , r))) |
| | | | |
| q_125 | q_01111101 | 0 1 1 1 1 1 0 1 | (r ((p , q))) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_249 | q_11111001 | 1 1 1 1 1 0 0 1 | ((p) (q , r)) |
| | | | |
| q_237 | q_11101101 | 1 1 1 0 1 1 0 1 | ((q) (p , r)) |
| | | | |
| q_235 | q_11101011 | 1 1 1 0 1 0 1 1 | ((r) (p , q)) |
| | | | |
| q_246 | q_11110110 | 1 1 1 1 0 1 1 0 | ((p) ((q , r))) |
| | | | |
| q_222 | q_11011110 | 1 1 0 1 1 1 1 0 | ((q) ((p , r))) |
| | | | |
| q_190 | q_10111110 | 1 0 1 1 1 1 1 0 | ((r) ((p , q))) |
| | | | |
o---------o------------o-----------------o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 13
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 9. Conjunctive Differences and Equalities
o---------o------------o-----------------o--------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o--------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o--------------------o
| | | | |
| q_24 | q_00011000 | 0 0 0 1 1 0 0 0 | (p, q) (p, r) |
| | | | |
| q_36 | q_00100100 | 0 0 1 0 0 1 0 0 | (p, q) (q, r) |
| | | | |
| q_66 | q_01000010 | 0 1 0 0 0 0 1 0 | (p, r) (q, r) |
| | | | |
| q_129 | q_10000001 | 1 0 0 0 0 0 0 1 | ((p, q))((q, r)) |
| | | | |
o---------o------------o-----------------o--------------------o
| | | | |
| q_231 | q_11100111 | 1 1 1 0 0 1 1 1 | ( (p, q) (p, r) ) |
| | | | |
| q_219 | q_11011011 | 1 1 0 1 1 0 1 1 | ( (p, q) (q, r) ) |
| | | | |
| q_189 | q_10111101 | 1 0 1 1 1 1 0 1 | ( (p, r) (q, r) ) |
| | | | |
| q_126 | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q))((q, r))) |
| | | | |
o---------o------------o-----------------o--------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 14
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I will explain my concept of "thematization"
or "thematic extension" after I copy out the
series of Tables that is formed on its basis.
In the meantime, here is a general exposition:
| Jon Awbrey, "Differential Logic and Dynamic Systems"
| DLOG D28. http://suo.ieee.org/ontology/msg04826.html
| DLOG D29. http://suo.ieee.org/ontology/msg04827.html
| DLOG D30. http://suo.ieee.org/ontology/msg04828.html
| DLOG D31. http://suo.ieee.org/ontology/msg04829.html
| DLOG D32. http://suo.ieee.org/ontology/msg04830.html
| DLOG D33. http://suo.ieee.org/ontology/msg04832.html
In order to make the pattern of their construction
more evident, I have left the expressions of the
thematic extensions in their unreduced forms.
Table 10. Thematic Extensions: [q, r] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_15 | q_00001111 | 0 0 0 0 1 1 1 1 | ((p , ( ) )) |
| | | | |
| q_30 | q_00011110 | 0 0 0 1 1 1 1 0 | ((p , (q) (r) )) |
| | | | |
| q_45 | q_00101101 | 0 0 1 0 1 1 0 1 | ((p , (q) r )) |
| | | | |
| q_60 | q_00111100 | 0 0 1 1 1 1 0 0 | ((p , (q) )) |
| | | | |
| q_75 | q_01001011 | 0 1 0 0 1 0 1 1 | ((p , q (r) )) |
| | | | |
| q_90 | q_01011010 | 0 1 0 1 1 0 1 0 | ((p , (r) )) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , (q , r) )) |
| | | | |
| q_120 | q_01111000 | 0 1 1 1 1 0 0 0 | ((p , (q r) )) |
| | | | |
| q_135 | q_10000111 | 1 0 0 0 0 1 1 1 | ((p , q r )) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | ((p , ((q , r)) )) |
| | | | |
| q_165 | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , r )) |
| | | | |
| q_180 | q_10110100 | 1 0 1 1 0 1 0 0 | ((p , (q (r)) )) |
| | | | |
| q_195 | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , q )) |
| | | | |
| q_210 | q_11010010 | 1 1 0 1 0 0 1 0 | ((p , ((q) r) )) |
| | | | |
| q_225 | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q) (r)) )) |
| | | | |
| q_240 | q_11110000 | 1 1 1 1 0 0 0 0 | ((p , )) |
| | | | |
o---------o------------o-----------------o---------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 15
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 11. Thematic Extensions: [p, r] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_51 | q_00110011 | 0 0 1 1 0 0 1 1 | ((q , ( ) )) |
| | | | |
| q_54 | q_00110110 | 0 0 1 1 0 1 1 0 | ((q , (p) (r) )) |
| | | | |
| q_57 | q_00111001 | 0 0 1 1 1 0 0 1 | ((q , (p) r )) |
| | | | |
| q_60 | q_00111100 | 0 0 1 1 1 1 0 0 | ((q , (p) )) |
| | | | |
| q_99 | q_01100011 | 0 1 1 0 0 0 1 1 | ((q , p (r) )) |
| | | | |
| q_102 | q_01100110 | 0 1 1 0 0 1 1 0 | ((q , (r) )) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((q , (p , r) )) |
| | | | |
| q_108 | q_01101100 | 0 1 1 0 1 1 0 0 | ((q , (p r) )) |
| | | | |
| q_147 | q_10010011 | 1 0 0 1 0 0 1 1 | ((q , p r )) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | ((q , ((p , r)) )) |
| | | | |
| q_153 | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , r )) |
| | | | |
| q_156 | q_10011100 | 1 0 0 1 1 1 0 0 | ((q , (p (r)) )) |
| | | | |
| q_195 | q_11000011 | 1 1 0 0 0 0 1 1 | ((q , p )) |
| | | | |
| q_198 | q_11000110 | 1 1 0 0 0 1 1 0 | ((q , ((p) r) )) |
| | | | |
| q_201 | q_11001001 | 1 1 0 0 1 0 0 1 | ((q , ((p) (r)) )) |
| | | | |
| q_204 | q_11001100 | 1 1 0 0 1 1 0 0 | ((q , )) |
| | | | |
o---------o------------o-----------------o---------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 16
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 12. Thematic Extensions: [p, q] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_85 | q_01010101 | 0 1 0 1 0 1 0 1 | ((r , ( ) )) |
| | | | |
| q_86 | q_01010110 | 0 1 0 1 0 1 1 0 | ((r , (p) (q) )) |
| | | | |
| q_89 | q_01011001 | 0 1 0 1 1 0 0 1 | ((r , (p) q )) |
| | | | |
| q_90 | q_01011010 | 0 1 0 1 1 0 1 0 | ((r , (p) )) |
| | | | |
| q_101 | q_01100101 | 0 1 1 0 0 1 0 1 | ((r , p (q) )) |
| | | | |
| q_102 | q_01100110 | 0 1 1 0 0 1 1 0 | ((r , (q) )) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((r , (p , q) )) |
| | | | |
| q_106 | q_01101010 | 0 1 1 0 1 0 1 0 | ((r , (p q) )) |
| | | | |
| q_149 | q_10010101 | 1 0 0 1 0 1 0 1 | ((r , p q )) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | ((r , ((p , q)) )) |
| | | | |
| q_153 | q_10011001 | 1 0 0 1 1 0 0 1 | ((r , q )) |
| | | | |
| q_154 | q_10011010 | 1 0 0 1 1 0 1 0 | ((r , (p (q)) )) |
| | | | |
| q_165 | q_10100101 | 1 0 1 0 0 1 0 1 | ((r , p )) |
| | | | |
| q_166 | q_10100110 | 1 0 1 0 0 1 1 0 | ((r , ((p) q) )) |
| | | | |
| q_169 | q_10101001 | 1 0 1 0 1 0 0 1 | ((r , ((p) (q)) )) |
| | | | |
| q_170 | q_10101010 | 1 0 1 0 1 0 1 0 | ((r , )) |
| | | | |
o---------o------------o-----------------o---------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 17
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 13. Differences & Equalities Conjoined with Implications
o---------o------------o-----------------o---------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_44 | q_00101100 | 0 0 1 0 1 1 0 0 | (p, q) (p (r)) |
| | | | |
| q_52 | q_00110100 | 0 0 1 1 0 1 0 0 | (p, q) ((p) r) |
| | | | |
| q_56 | q_00111000 | 0 0 1 1 1 0 0 0 | (p, q) (q (r)) |
| | | | |
| q_28 | q_00011100 | 0 0 0 1 1 1 0 0 | (p, q) ((q) r) |
| | | | |
| q_131 | q_10000011 | 1 0 0 0 0 0 1 1 | ((p, q)) (p (r)) |
| | | | |
| q_193 | q_11000001 | 1 1 0 0 0 0 0 1 | ((p, q)) ((p) r) |
| | | | |
| | | | |
| q_74 | q_01001010 | 0 1 0 0 1 0 1 0 | (p, r) (p (q)) |
| | | | |
| q_82 | q_01010010 | 0 1 0 1 0 0 1 0 | (p, r) ((p) q) |
| | | | |
| q_26 | q_00011010 | 0 0 0 1 1 0 1 0 | (p, r) (q (r)) |
| | | | |
| q_88 | q_01011000 | 0 1 0 1 1 0 0 0 | (p, r) ((q) r) |
| | | | |
| q_133 | q_10000101 | 1 0 0 0 0 1 0 1 | ((p, r)) (p (q)) |
| | | | |
| q_161 | q_10100001 | 1 0 1 0 0 0 0 1 | ((p, r)) ((p) q) |
| | | | |
| | | | |
| q_70 | q_01000110 | 0 1 0 0 0 1 1 0 | (q, r) (p (q)) |
| | | | |
| q_98 | q_01100010 | 0 1 1 0 0 0 1 0 | (q, r) ((p) q) |
| | | | |
| q_38 | q_00100110 | 0 0 1 0 0 1 1 0 | (q, r) (p (r)) |
| | | | |
| q_100 | q_01100100 | 0 1 1 0 0 1 0 0 | (q, r) ((p) r) |
| | | | |
| q_137 | q_10001001 | 1 0 0 0 1 0 0 1 | ((q, r)) (p (q)) |
| | | | |
| q_145 | q_10010001 | 1 0 0 1 0 0 0 1 | ((q, r)) ((p) q) |
| | | | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_211 | q_11010011 | 1 1 0 1 0 0 1 1 | ((p, q) (p (r))) |
| | | | |
| q_203 | q_11001011 | 1 1 0 0 1 0 1 1 | ((p, q) ((p) r)) |
| | | | |
| q_199 | q_11000111 | 1 1 0 0 0 1 1 1 | ((p, q) (q (r))) |
| | | | |
| q_227 | q_11100011 | 1 1 1 0 0 0 1 1 | ((p, q) ((q) r)) |
| | | | |
| q_124 | q_01111100 | 0 1 1 1 1 1 0 0 | (((p, q)) (p (r))) |
| | | | |
| q_62 | q_00111110 | 0 0 1 1 1 1 1 0 | (((p, q)) ((p) r)) |
| | | | |
| | | | |
| q_181 | q_10110101 | 1 0 1 1 0 1 0 1 | ((p, r) (p (q))) |
| | | | |
| q_173 | q_10101101 | 1 0 1 0 1 1 0 1 | ((p, r) ((p) q)) |
| | | | |
| q_229 | q_11100101 | 1 1 1 0 0 1 0 1 | ((p, r) (q (r))) |
| | | | |
| q_167 | q_10100111 | 1 0 1 0 0 1 1 1 | ((p, r) ((q) r)) |
| | | | |
| q_122 | q_01111010 | 0 1 1 1 1 0 1 0 | (((p, r)) (p (q))) |
| | | | |
| q_94 | q_01011110 | 0 1 0 1 1 1 1 0 | (((p, r)) ((p) q)) |
| | | | |
| | | | |
| q_185 | q_10111001 | 1 0 1 1 1 0 0 1 | ((q, r) (p (q))) |
| | | | |
| q_157 | q_10011101 | 1 0 0 1 1 1 0 1 | ((q, r) ((p) q)) |
| | | | |
| q_217 | q_11011001 | 1 1 0 1 1 0 0 1 | ((q, r) (p (r))) |
| | | | |
| q_155 | q_10011011 | 1 0 0 1 1 0 1 1 | ((q, r) ((p) r)) |
| | | | |
| q_118 | q_01110110 | 0 1 1 1 0 1 1 0 | (((q, r)) (p (q))) |
| | | | |
| q_110 | q_01101110 | 0 1 1 0 1 1 1 0 | (((q, r)) ((p) q)) |
| | | | |
o---------o------------o-----------------o---------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 18
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 14 shows the propositions q_i : B^3 -> B whose "fibers of truth",
that is, whose pre-images of 1, have the form of a single point in B^3
together with the three points that make up its immediate neighborhood.
Here I use the alternative syntax "x + y" for the exclusive-or (x , y).
Table 14. Proximal Propositions
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_23 | q_00010111 | 0 0 0 1 0 1 1 1 | (p)(q)(r) + ((p),(q),(r)) |
| | | | |
| q_43 | q_00101011 | 0 0 1 0 1 0 1 1 | (p)(q) r + ((p),(q), r ) |
| | | | |
| q_77 | q_01001101 | 0 1 0 0 1 1 0 1 | (p) q (r) + ((p), q ,(r)) |
| | | | |
| q_142 | q_10001110 | 1 0 0 0 1 1 1 0 | (p) q r + ((p), q , r ) |
| | | | |
| q_113 | q_01110001 | 0 1 1 1 0 0 0 1 | p (q)(r) + ( p ,(q),(r)) |
| | | | |
| q_178 | q_10110010 | 1 0 1 1 0 0 1 0 | p (q) r + ( p ,(q), r ) |
| | | | |
| q_212 | q_11010100 | 1 1 0 1 0 1 0 0 | p q (r) + ( p , q ,(r)) |
| | | | |
| q_232 | q_11101000 | 1 1 1 0 1 0 0 0 | p q r + ( p , q , r ) |
| | | | |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 19
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 15. Differences and Equalities between Simples and Boundaries
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_152 | q_10011000 | 1 0 0 1 1 0 0 0 | p + ( p , q , r ) |
| | | | |
| q_164 | q_10100100 | 1 0 1 0 0 1 0 0 | q + ( p , q , r ) |
| | | | |
| q_194 | q_11000010 | 1 1 0 0 0 0 1 0 | r + ( p , q , r ) |
| | | | |
| q_230 | q_11100110 | 1 1 1 0 0 1 1 0 | p + ((p), (q), (r)) |
| | | | |
| q_218 | q_11011010 | 1 1 0 1 1 0 1 0 | q + ((p), (q), (r)) |
| | | | |
| q_188 | q_10111100 | 1 0 1 1 1 1 0 0 | r + ((p), (q), (r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_103 | q_01100111 | 0 1 1 0 0 1 1 1 | p = ( p , q , r ) |
| | | | |
| q_91 | q_01011011 | 0 1 0 1 1 0 1 1 | q = ( p , q , r ) |
| | | | |
| q_61 | q_00111101 | 0 0 1 1 1 1 0 1 | r = ( p , q , r ) |
| | | | |
| q_25 | q_00011001 | 0 0 0 1 1 0 0 1 | p = ((p), (q), (r)) |
| | | | |
| q_37 | q_00100101 | 0 0 1 0 0 1 0 1 | q = ((p), (q), (r)) |
| | | | |
| q_67 | q_01000011 | 0 1 0 0 0 0 1 1 | r = ((p), (q), (r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 20
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 16. Paisley Propositions
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_216 | q_11011000 | 1 1 0 1 1 0 0 0 | (p, q)(p, r) + p q |
| | | | |
| q_184 | q_10111000 | 1 0 1 1 1 0 0 0 | (p, q)(p, r) + p r |
| | | | |
| q_228 | q_11100100 | 1 1 1 0 0 1 0 0 | (p, q)(q, r) + p q |
| | | | |
| q_172 | q_10101100 | 1 0 1 0 1 1 0 0 | (p, q)(q, r) + q r |
| | | | |
| q_226 | q_11100010 | 1 1 1 0 0 0 1 0 | (p, r)(q, r) + p r |
| | | | |
| q_202 | q_11001010 | 1 1 0 0 1 0 1 0 | (p, r)(q, r) + q r |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_39 | q_00100111 | 0 0 1 0 0 1 1 1 | (p, q)(p, r) = p q |
| | | | |
| q_71 | q_01000111 | 0 1 0 0 0 1 1 1 | (p, q)(p, r) = p r |
| | | | |
| q_27 | q_00011011 | 0 0 0 1 1 0 1 1 | (p, q)(q, r) = p q |
| | | | |
| q_83 | q_01010011 | 0 1 0 1 0 0 1 1 | (p, q)(q, r) = q r |
| | | | |
| q_29 | q_00011101 | 0 0 0 1 1 1 0 1 | (p, r)(q, r) = p r |
| | | | |
| q_53 | q_00110101 | 0 0 1 1 0 1 0 1 | (p, r)(q, r) = q r |
| | | | |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 21
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 17 gives another way of writing the "paisley propositions"
that makes their symmetry class more manifest. The venn diagram
that follows the Table may provide an idea of why I chose to dub
them that, at least, until I can think of a Greek or Latin label.
Table 17. Paisley Propositions
o---------o------------o-----------------o------------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o------------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o------------------------------o
| | | | |
| q_216 | q_11011000 | 1 1 0 1 1 0 0 0 | p + pq + pqr + (p, q, r) |
| | | | |
| q_184 | q_10111000 | 1 0 1 1 1 0 0 0 | p + pr + pqr + (p, q, r) |
| | | | |
| q_228 | q_11100100 | 1 1 1 0 0 1 0 0 | q + pq + pqr + (p, q, r) |
| | | | |
| q_172 | q_10101100 | 1 0 1 0 1 1 0 0 | q + qr + pqr + (p, q, r) |
| | | | |
| q_226 | q_11100010 | 1 1 1 0 0 0 1 0 | r + pr + pqr + (p, q, r) |
| | | | |
| q_202 | q_11001010 | 1 1 0 0 1 0 1 0 | r + qr + pqr + (p, q, r) |
| | | | |
o---------o------------o-----------------o------------------------------o
| | | | |
| q_39 | q_00100111 | 0 0 1 0 0 1 1 1 | 1 + p + pq + pqr + (p, q, r) |
| | | | |
| q_71 | q_01000111 | 0 1 0 0 0 1 1 1 | 1 + p + pr + pqr + (p, q, r) |
| | | | |
| q_27 | q_00011011 | 0 0 0 1 1 0 1 1 | 1 + q + pq + pqr + (p, q, r) |
| | | | |
| q_83 | q_01010011 | 0 1 0 1 0 0 1 1 | 1 + q + qr + pqr + (p, q, r) |
| | | | |
| q_29 | q_00011101 | 0 0 0 1 1 1 0 1 | 1 + r + pr + pqr + (p, q, r) |
| | | | |
| q_53 | q_00110101 | 0 0 1 1 0 1 0 1 | 1 + r + qr + pqr + (p, q, r) |
| | | | |
o---------o------------o-----------------o------------------------------o
o-------------------------------------------------o
| |
| |
| o-------------o |
| /%%%%%%%%%%%%%%%\ |
| /%%%%%%%%%%%%%%%%%\ |
| /%%%%%%%%%%%%%%%%%%%\ |
| /%%%%%%%%%%%%%%%%%%%%%\ |
| o%%%%%%%%%%%%%%%%%%%%%%%o |
| |%%%%%%%%%% P %%%%%%%%%%| |
| |%%%%%%%%%%%%%%%%%%%%%%%| |
| |%%%%%%%%%%%%%%%%%%%%%%%| |
| o---o---------o%%%o---------o---o |
| / \%%%%%%%%%\%/ / \ |
| / \%%%%%%%%%o / \ |
| / \%%%%%%%/%\ / \ |
| / \%%%%%/%%%\ / \ |
| o o---o-----o---o o |
| | |%%%%%| | |
| | |%%%%%| | |
| | Q |%%%%%| R | |
| o o%%%%%o o |
| \ \%%%/ / |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_216. p + p q + p q r + (p, q, r)
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 22
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I'm puzzled by the blind-spot that prevented me
from seeing this very simple and natural family
of propositions, especially since I had already
counted a third of their number. At any rate,
here they be, and modulo the usual number of
corrections I think that these complete the
set of 256 propositions on three variables.
Table 18. Desultory Junctions and Their Complements
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_224 | q_11100000 | 1 1 1 0 0 0 0 0 | p ((q)(r)) |
| | | | |
| q_200 | q_11001000 | 1 1 0 0 1 0 0 0 | q ((p)(r)) |
| | | | |
| q_168 | q_10101000 | 1 0 1 0 1 0 0 0 | r ((p)(q)) |
| | | | |
| q_14 | q_00001110 | 0 0 0 0 1 1 1 0 | (p) ((q)(r)) |
| | | | |
| q_50 | q_00110010 | 0 0 1 1 0 0 1 0 | (q) ((p)(r)) |
| | | | |
| q_84 | q_01010100 | 0 1 0 1 0 1 0 0 | (r) ((p)(q)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_31 | q_00011111 | 0 0 0 1 1 1 1 1 | (p ((q)(r))) |
| | | | |
| q_55 | q_00110111 | 0 0 1 1 0 1 1 1 | (q ((p)(r))) |
| | | | |
| q_87 | q_01010111 | 0 1 0 1 0 1 1 1 | (r ((p)(q))) |
| | | | |
| q_241 | q_11110001 | 1 1 1 1 0 0 0 1 | ((p) ((q)(r))) |
| | | | |
| q_205 | q_11001101 | 1 1 0 0 1 1 0 1 | ((q) ((p)(r))) |
| | | | |
| q_171 | q_10101011 | 1 0 1 0 1 0 1 1 | ((r) ((p)(q))) |
| | | | |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 23
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
For ease of viewing, I am placing
copies of the Cactus Rules Table
at a couple of other sites:
Table 256. http://stderr.org/pipermail/inquiry/2004-April/001314.html
Table 256. http://suo.ieee.org/ontology/msg05512.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 24a
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Here is a set of representative cactus graphs
for the 256 propositions on three variables.
To make some cactus graphs easier to draw in Ascii,
I will occasionally be forced to "stretch a point",
drawing the root node "@" as @=@, @=@=@, and so on,
and the regular nodes "o" as o=o, o=o=o, and so on.
(I will keep adding to this after Easter,
but right now I've got spikes in my eyes.)
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `( )` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_0` ` ` ` | ` ` ` ` | ` ` ` q_255 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o o o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p)(q)(r) ` ` | ` ` ` ` | ` `((p)(q)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_1` ` ` ` | ` ` ` ` | ` ` ` q_254 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p)(q) r` ` ` | ` ` ` ` | ` `((p)(q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_2` ` ` ` | ` ` ` ` | ` ` ` q_253 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p) (q)` ` ` | ` ` ` ` | ` ` ((p) (q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_3` ` ` ` | ` ` ` ` | ` ` ` q_252 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o q ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ q ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p) q (r) ` ` | ` ` ` ` | ` `((p) q (r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_4` ` ` ` | ` ` ` ` | ` ` ` q_251 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p) (r)` ` ` | ` ` ` ` | ` ` ((p) (r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_5` ` ` ` | ` ` ` ` | ` ` ` q_250 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o-o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p)(q, r) ` ` | ` ` ` ` | ` `((p)(q, r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_6` ` ` ` | ` ` ` ` | ` ` ` q_249 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p `q r` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p `q r` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p) (q r) ` ` | ` ` ` ` | ` `((p) (q r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_7` ` ` ` | ` ` ` ` | ` ` ` q_248 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o q r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ q r ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p) q r` ` ` | ` ` ` ` | ` ` ((p) q r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_8` ` ` ` | ` ` ` ` | ` ` ` q_247 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o---o ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o---o ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p `\`/` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p)((q, r))` ` | ` ` ` ` | ` ((p)((q, r))) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_9` ` ` ` | ` ` ` ` | ` ` ` q_246 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (p) r ` ` ` | ` ` ` ` | ` ` `((p) r)` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_10 ` ` ` | ` ` ` ` | ` ` ` q_245 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p ` | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o ` o q ` ` |
| ` ` ` p ` | ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o q ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p) (q (r))` ` | ` ` ` ` | ` ((p) (q (r))) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_11 ` ` ` | ` ` ` ` | ` ` ` q_244 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o q ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ q ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (p) q ` ` ` | ` ` ` ` | ` ` `((p) q)` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_12 ` ` ` | ` ` ` ` | ` ` ` q_243 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` p ` | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o ` o r ` ` |
| ` ` ` p ` | ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o r ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p) ((q) r)` ` | ` ` ` ` | ` ((p) ((q) r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_13 ` ` ` | ` ` ` ` | ` ` ` q_242 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` o ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o ` o ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p `\`/` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p)((q)(r))` ` | ` ` ` ` | ` ((p)((q)(r))) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_14 ` ` ` | ` ` ` ` | ` ` ` q_241 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `(p)` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_15 ` ` ` | ` ` ` ` | ` ` ` q_240 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `p (q)(r) ` ` | ` ` ` ` | ` ` (p (q)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_16 ` ` ` | ` ` ` ` | ` ` ` q_239 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(q) (r)` ` ` | ` ` ` ` | ` ` ((q) (r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_17 ` ` ` | ` ` ` ` | ` ` ` q_238 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p r q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o o ` ` ` |
| ` ` ` p r q ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p, r)(q) ` ` | ` ` ` ` | ` `((p, r)(q))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_18 ` ` ` | ` ` ` ` | ` ` ` q_237 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `p r` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `p r` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p r) (q) ` ` | ` ` ` ` | ` `((p r) (q))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_19 ` ` ` | ` ` ` ` | ` ` ` q_236 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p, q)(r) ` ` | ` ` ` ` | ` `((p, q)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_20 ` ` ` | ` ` ` ` | ` ` ` q_235 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `p q` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `p q` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p q) (r) ` ` | ` ` ` ` | ` `((p q) (r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_21 ` ` ` | ` ` ` ` | ` ` ` q_234 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` | | | ` ` ` |
| ` ` ` o o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q),(r)) ` | ` ` ` ` | `(((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_22 ` ` ` | ` ` ` ` | ` ` ` q_233 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `p`q r` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `o o o` ` |
| ` ` ` ` `p`q r` ` | ` ` ` ` | ` `p`q`r`| | |` ` |
| ` ` ` ` `o o o` ` | ` ` ` ` | ` `o`o`o o-o-o` ` |
| ` `p`q`r`| | |` ` | ` ` ` ` | ` ` \|/ ` \`/ ` ` |
| ` `o`o`o o-o-o` ` | ` ` ` ` | ` ` `o-----o` ` ` |
| ` ` \|/ ` \`/ ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` `o-----o` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `( (p) (q) (r)` ` | ` ` ` ` | (( (p) (q) (r)` ` |
| `,((p),(q),(r)))` | ` ` ` ` | `,((p),(q),(r)))) |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_23 ` ` ` | ` ` ` ` | ` ` ` q_232 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `p`q`p`r` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `o-o`o-o` ` ` |
| ` ` `p`q`p`r` ` ` | ` ` ` ` | ` ` ` \|`|/ ` ` ` |
| ` ` `o-o`o-o` ` ` | ` ` ` ` | ` ` ` `o=o` ` ` ` |
| ` ` ` \|`|/ ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` `@=@` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q) (p, r) ` | ` ` ` ` | `((p, q) (p, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_24 ` ` ` | ` ` ` ` | ` ` ` q_231 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o o o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | | | ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` o o o ` ` |
| ` ` ` p `\`/` ` ` | ` ` ` ` | ` ` ` ` | | | ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((` ` ` p ` ` ` ` | ` ` ` ` | `(` ` ` p ` ` ` ` |
| `,((p),(q),(r)))) | ` ` ` ` | `,((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_25 ` ` ` | ` ` ` ` | ` ` ` q_230 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o q ` ` |
| ` ` ` p r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o q ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, r)(q (r)) ` | ` ` ` ` | `((p, r)(q (r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_26 ` ` ` | ` ` ` ` | ` ` ` q_229 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p q q r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `p q` \| |/ ` ` | ` ` ` ` | ` ` ` `p q q r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `p q` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`p` ` ` q ` ` ` | ` ` ` ` | `(`p` ` ` q ` ` ` |
| `,(p, q) (q, r))) | ` ` ` ` | `,(p, q) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_27 ` ` ` | ` ` ` ` | ` ` ` q_228 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o r ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o r ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q)((q) r) ` | ` ` ` ` | `((p, q)((q) r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_28 ` ` ` | ` ` ` ` | ` ` ` q_227 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p r q r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `p r` \| |/ ` ` | ` ` ` ` | ` ` ` `p r q r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `p r` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`p` ` ` r ` ` ` | ` ` ` ` | `(`p` ` ` r ` ` ` |
| `,(p, r) (q, r))) | ` ` ` ` | `,(p, r) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_29 ` ` ` | ` ` ` ` | ` ` ` q_226 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p `\`/` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p, (q) (r)))` | ` ` ` ` | ` `(p, (q) (r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_30 ` ` ` | ` ` ` ` | ` ` ` q_225 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` p @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p ((q)(r))) ` | ` ` ` ` | ` ` p ((q)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_31 ` ` ` | ` ` ` ` | ` ` ` q_224 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 24b
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` p o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `p (q) r` ` ` | ` ` ` ` | ` ` (p (q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_32 ` ` ` | ` ` ` ` | ` ` ` q_223 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` p ` r ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` o---o ` ` ` ` |
| ` ` p ` r ` ` ` ` | ` ` ` ` | ` ` `\`/` q ` ` ` |
| ` ` o---o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `\`/` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p, r))(q)` ` | ` ` ` ` | ` (((p, r))(q)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_33 ` ` ` | ` ` ` ` | ` ` ` q_222 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (q) r ` ` ` | ` ` ` ` | ` ` `((q) r)` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_34 ` ` ` | ` ` ` ` | ` ` ` q_221 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` r ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` r ` ` ` ` ` | ` ` ` ` | ` ` ` | ` q ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` p o ` o ` ` ` |
| ` ` ` | ` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` p o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p (r))(q)` ` | ` ` ` ` | ` `((p (r))(q)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_35 ` ` ` | ` ` ` ` | ` ` ` q_220 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `p`q`q`r` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `o-o`o-o` ` ` |
| ` ` `p`q`q`r` ` ` | ` ` ` ` | ` ` ` \|`|/ ` ` ` |
| ` ` `o-o`o-o` ` ` | ` ` ` ` | ` ` ` `o=o` ` ` ` |
| ` ` ` \|`|/ ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` `@=@` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q) (q, r) ` | ` ` ` ` | `((p, q) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_36 ` ` ` | ` ` ` ` | ` ` ` q_219 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o o o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | | | ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` o o o ` ` |
| ` ` ` q `\`/` ` ` | ` ` ` ` | ` ` ` ` | | | ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((` ` ` q ` ` ` ` | ` ` ` ` | `(` ` ` q ` ` ` ` |
| `,((p),(q),(r)))) | ` ` ` ` | `,((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_37 ` ` ` | ` ` ` ` | ` ` ` q_218 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` q r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o p ` ` |
| ` ` ` q r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o p ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (q, r)(p (r)) ` | ` ` ` ` | `((q, r)(p (r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_38 ` ` ` | ` ` ` ` | ` ` ` q_217 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p q p r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `p q` \| |/ ` ` | ` ` ` ` | ` ` ` `p q p r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `p q` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`p` ` ` q ` ` ` | ` ` ` ` | `(`p` ` ` q ` ` ` |
| `,(p, q) (p, r))) | ` ` ` ` | `,(p, q) (p, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_39 ` ` ` | ` ` ` ` | ` ` ` q_216 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p, q) r` ` ` | ` ` ` ` | ` `((p, q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_40 ` ` ` | ` ` ` ` | ` ` ` q_215 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o o ` ` ` ` |
| ` ` ` p q ` ` ` ` | ` ` ` ` | ` ` ` | | r ` ` ` |
| ` ` ` o o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | | r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p),(q), r ) ` | ` ` ` ` | `(((p),(q), r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_41` ` ` ` | ` ` ` ` | ` ` ` q_214 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` `p q` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `p q` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ r ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p q) r` ` ` | ` ` ` ` | ` ` ((p q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_42 ` ` ` | ` ` ` ` | ` ` ` q_213 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `p`q` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `o o` ` ` |
| ` ` ` ` `p`q` ` ` | ` ` ` ` | ` `p`q` `| |`r` ` |
| ` ` ` ` `o o` ` ` | ` ` ` ` | ` `o`o` `o-o-o` ` |
| ` `p`q` `| | r` ` | ` ` ` ` | ` ` \|` ` \`/ ` ` |
| ` `o`o` `o-o-o` ` | ` ` ` ` | ` `r`o-----o` ` ` |
| ` ` \|` ` \`/ ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` `r`o-----o` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `( (p) (q)` r ` ` | ` ` ` ` | (( (p) (q)` r ` ` |
| `,((p),(q), r ))` | ` ` ` ` | `,((p),(q), r ))) |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_43 ` ` ` | ` ` ` ` | ` ` ` q_212 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o p ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o p ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q)(p (r)) ` | ` ` ` ` | `((p, q)(p (r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_44 ` ` ` | ` ` ` ` | ` ` ` q_211 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` | ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` o---o r ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p, (q) r))` ` | ` ` ` ` | ` `(p, (q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_45 ` ` ` | ` ` ` ` | ` ` ` q_210 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` q o ` o r ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` q o ` o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p) q) ((q) r)) | ` ` ` ` | `((p) q) ((q) r)` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_46 ` ` ` | ` ` ` ` | ` ` ` q_209 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` o r ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` p @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p ((q) r))` ` | ` ` ` ` | ` ` p ((q) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_47 ` ` ` | ` ` ` ` | ` ` ` q_208 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` p (q) ` ` ` | ` ` ` ` | ` ` `(p (q))` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_48 ` ` ` | ` ` ` ` | ` ` ` q_207 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` | ` q ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` r o ` o ` ` ` |
| ` ` ` | ` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` r o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p) r) (q)` ` | ` ` ` ` | ` (((p) r) (q)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_49 ` ` ` | ` ` ` ` | ` ` ` q_206 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` p ` r ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` o ` o ` ` ` ` |
| ` ` p ` r ` ` ` ` | ` ` ` ` | ` ` `\`/` q ` ` ` |
| ` ` o ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `\`/` q ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p) (r)) (q) ` | ` ` ` ` | `(((p) (r)) (q))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_50 ` ` ` | ` ` ` ` | ` ` ` q_205 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `(q)` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_51 ` ` ` | ` ` ` ` | ` ` ` q_204 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o r ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o r ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q)((p) r) ` | ` ` ` ` | `((p, q)((p) r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_52 ` ` ` | ` ` ` ` | ` ` ` q_203 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p r q r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `q r` \| |/ ` ` | ` ` ` ` | ` ` ` `p r q r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `q r` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`q` ` ` r ` ` ` | ` ` ` ` | `(`q` ` ` r ` ` ` |
| `,(p, r) (q, r))) | ` ` ` ` | `,(p, r) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_53 ` ` ` | ` ` ` ` | ` ` ` q_202 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q `\`/` ` ` | ` ` ` ` | ` ` ` ` p ` r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((q, (p)(r))) ` | ` ` ` ` | ` `(q, (p)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_54 ` ` ` | ` ` ` ` | ` ` ` q_201 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` o q ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (((p)(r)) q)` ` | ` ` ` ` | ` `((p)(r)) q ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_55 ` ` ` | ` ` ` ` | ` ` ` q_200 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o q ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o q ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, q)(q (r)) ` | ` ` ` ` | `((p, q)(q (r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_56 ` ` ` | ` ` ` ` | ` ` ` q_199 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` | ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` o---o r ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((q, (p) r))` ` | ` ` ` ` | ` `(q, (p) r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_57 ` ` ` | ` ` ` ` | ` ` ` q_198 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` p ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` q ` p ` ` ` |
| ` ` p o ` o r ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (q)) ((p) r)) | ` ` ` ` | `(p (q)) ((p) r)` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_58 ` ` ` | ` ` ` ` | ` ` ` q_197 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` o r ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o q ` ` ` | ` ` ` ` | ` ` ` ` o r ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(((p) r) q)` ` | ` ` ` ` | ` ` ((p) r) q ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_59 ` ` ` | ` ` ` ` | ` ` ` q_196 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , q)` ` ` | ` ` ` ` | ` ` ((p , q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_60` ` ` ` | ` ` ` ` | ` ` ` q_195 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r `\`/` ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((r, (p, q, r ))) | ` ` ` ` | `(r, (p, q, r ))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_61 ` ` ` | ` ` ` ` | ` ` ` q_194 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` p ` q ` p ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o---o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\`/` `/` ` ` | ` ` ` ` | ` ` p ` q ` p ` ` |
| ` ` ` o ` o r ` ` | ` ` ` ` | ` ` o---o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\`/` `/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p, q))((p) r)) | ` ` ` ` | `((p, q))((p) r)` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_62 ` ` ` | ` ` ` ` | ` ` ` q_193 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p q` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `p q` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (p q) ` ` ` | ` ` ` ` | ` ` ` `p q` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_63 ` ` ` | ` ` ` ` | ` ` ` q_192 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 24c
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p q o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` p q @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` p q (r) ` ` ` | ` ` ` ` | ` `(p q (r))` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_64 ` ` ` | ` ` ` ` | ` ` ` q_191 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` p ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` o---o ` ` ` ` |
| ` ` p ` q ` ` ` ` | ` ` ` ` | ` ` `\`/` r ` ` ` |
| ` ` o---o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `\`/` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p, q))(r)` ` | ` ` ` ` | ` (((p, q))(r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_65 ` ` ` | ` ` ` ` | ` ` ` q_190 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `p`r`q`r` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` `o-o`o-o` ` ` |
| ` ` `p`r`q`r` ` ` | ` ` ` ` | ` ` ` \|`|/ ` ` ` |
| ` ` `o-o`o-o` ` ` | ` ` ` ` | ` ` ` `o=o` ` ` ` |
| ` ` ` \|`|/ ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` `@=@` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, r) (q, r) ` | ` ` ` ` | `((p, r) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_66 ` ` ` | ` ` ` ` | ` ` ` q_189 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o o o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | | | ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` o o o ` ` |
| ` ` ` r `\`/` ` ` | ` ` ` ` | ` ` ` ` | | | ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((` ` ` r ` ` ` ` | ` ` ` ` | `(` ` ` r ` ` ` ` |
| `,((p),(q),(r)))) | ` ` ` ` | `,((p),(q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_67 ` ` ` | ` ` ` ` | ` ` ` q_188 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` q o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` q @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q (r) ` ` ` | ` ` ` ` | ` ` `(q (r))` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_68 ` ` ` | ` ` ` ` | ` ` ` q_187 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` q ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` q ` ` ` ` ` | ` ` ` ` | ` ` ` | ` r ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` p o ` o ` ` ` |
| ` ` ` | ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` p o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p (q))(r)` ` | ` ` ` ` | ` `((p (q))(r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_69 ` ` ` | ` ` ` ` | ` ` ` q_186 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` q r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o p ` ` |
| ` ` ` q r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o p ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (q, r)(p (q)) ` | ` ` ` ` | `((q, r)(p (q)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_70 ` ` ` | ` ` ` ` | ` ` ` q_185 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p q p r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `p r` \| |/ ` ` | ` ` ` ` | ` ` ` `p q p r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `p r` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`p` ` ` r ` ` ` | ` ` ` ` | `(`p` ` ` r ` ` ` |
| `,(p, q) (p, r))) | ` ` ` ` | `,(p, q) (p, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_71 ` ` ` | ` ` ` ` | ` ` ` q_184 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o q ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ q ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p, r) q` ` ` | ` ` ` ` | ` `((p, r) q) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_72 ` ` ` | ` ` ` ` | ` ` ` q_183 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` | q | ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` | q | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p), q ,(r)) ` | ` ` ` ` | `(((p), q ,(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_73` ` ` ` | ` ` ` ` | ` ` ` q_182 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` p r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o p ` ` |
| ` ` ` p r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o p ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, r)(p (q)) ` | ` ` ` ` | `((p, r)(p (q)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_74 ` ` ` | ` ` ` ` | ` ` ` q_181 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` | ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` o---o q ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p, q (r))) ` | ` ` ` ` | ` ` (p, q (r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_75 ` ` ` | ` ` ` ` | ` ` ` q_180 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` `p r` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `p r` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` o q ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ q ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p r) q` ` ` | ` ` ` ` | ` ` ((p r) q) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_76 ` ` ` | ` ` ` ` | ` ` ` q_179 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `p` `r` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` `o` `o` ` |
| ` ` ` ` `p` `r` ` | ` ` ` ` | ` `p`r` `| q |` ` |
| ` ` ` ` `o` `o` ` | ` ` ` ` | ` `o`o` `o-o-o` ` |
| ` `p`r` `| q |` ` | ` ` ` ` | ` ` \|` ` \ / ` ` |
| ` `o`o` `o-o-o` ` | ` ` ` ` | ` `q`o-----o` ` ` |
| ` ` \|` ` \ / ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` `q`o-----o` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `( (p)` q `(r)` ` | ` ` ` ` | (( (p)` q `(r)` ` |
| `,((p), q ,(r)))` | ` ` ` ` | `,((p), q ,(r)))) |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_77 ` ` ` | ` ` ` ` | ` ` ` q_178 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` r o ` o q ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` r o ` o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p) r) (q (r))) | ` ` ` ` | `((p) r) (q (r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_78 ` ` ` | ` ` ` ` | ` ` ` q_177 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` q o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` q o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` p @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p (q (r)))` ` | ` ` ` ` | ` ` p (q (r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_79 ` ` ` | ` ` ` ` | ` ` ` q_176 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` p (r) ` ` ` | ` ` ` ` | ` ` `(p (r))` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_80 ` ` ` | ` ` ` ` | ` ` ` q_175 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` | ` r ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` q o ` o ` ` ` |
| ` ` ` | ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` q o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p) q)(r) ` ` | ` ` ` ` | ` (((p) q)(r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_81 ` ` ` | ` ` ` ` | ` ` ` q_174 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` p r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o q ` ` |
| ` ` ` p r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o q ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, r)((p) q) ` | ` ` ` ` | `((p, r)((p) q))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_82 ` ` ` | ` ` ` ` | ` ` ` q_173 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p q q r` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `o-o o-o` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` `q r` \| |/ ` ` | ` ` ` ` | ` ` ` `p q q r` ` |
| ` ` `o---o=o` ` ` | ` ` ` ` | ` ` ` `o-o o-o` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` `q r` \| |/ ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `o---o=o` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((`q` ` ` r ` ` ` | ` ` ` ` | `(`q` ` ` r ` ` ` |
| `,(p, q) (q, r))) | ` ` ` ` | `,(p, q) (q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_83 ` ` ` | ` ` ` ` | ` ` ` q_172 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` p ` q ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` o ` o ` ` ` ` |
| ` ` p ` q ` ` ` ` | ` ` ` ` | ` ` `\`/` r ` ` ` |
| ` ` o ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` `\`/` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p)(q))(r)` ` | ` ` ` ` | ` (((p)(q))(r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_84 ` ` ` | ` ` ` ` | ` ` ` q_171 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `(r)` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_85 ` ` ` | ` ` ` ` | ` ` ` q_170 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p ` q ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r `\`/` ` ` | ` ` ` ` | ` ` ` ` p ` q ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((r, (p)(q))) ` | ` ` ` ` | ` `(r, (p)(q))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_86 ` ` ` | ` ` ` ` | ` ` ` q_169 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` o r ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (((p)(q)) r)` ` | ` ` ` ` | ` `((p)(q)) r ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_87 ` ` ` | ` ` ` ` | ` ` ` q_168 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` p r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o r ` ` |
| ` ` ` p r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o r ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (p, r)((q) r) ` | ` ` ` ` | `((p, r)((q) r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_88 ` ` ` | ` ` ` ` | ` ` ` q_167 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r ` | ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` o---o q ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((r, (p) q))` ` | ` ` ` ` | ` `(r, (p) q) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_89 ` ` ` | ` ` ` ` | ` ` ` q_166 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p , r)` ` ` | ` ` ` ` | ` ` ((p , r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_90 ` ` ` | ` ` ` ` | ` ` ` q_165 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q `\`/` ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((q, (p, q, r)))` | ` ` ` ` | `(q, (p, q, r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_91 ` ` ` | ` ` ` ` | ` ` ` q_164 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r ` p ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` r ` p ` ` ` |
| ` ` p o ` o q ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (r)) ((p) q)) | ` ` ` ` | `(p (r)) ((p) q)` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_92 ` ` ` | ` ` ` ` | ` ` ` q_163 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` p ` ` ` ` |
| ` ` ` ` o q ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o r ` ` ` | ` ` ` ` | ` ` ` ` o q ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(((p) q) r)` ` | ` ` ` ` | ` ` ((p) q) r ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_93 ` ` ` | ` ` ` ` | ` ` ` q_162 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` p ` r ` p ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o---o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\`/` `/` ` ` | ` ` ` ` | ` ` p ` r ` p ` ` |
| ` ` ` o ` o q ` ` | ` ` ` ` | ` ` o---o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\`/` `/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p, r))((p) q)) | ` ` ` ` | `((p, r))((p) q)` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_94 ` ` ` | ` ` ` ` | ` ` ` q_161 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `p r` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `p r` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (p r) ` ` ` | ` ` ` ` | ` ` ` `p r` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_95 ` ` ` | ` ` ` ` | ` ` ` q_160 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 24d
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `p (q, r) ` ` | ` ` ` ` | ` ` (p (q, r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_96 ` ` ` | ` ` ` ` | ` ` ` q_159 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o o ` ` ` |
| ` ` ` ` q r ` ` ` | ` ` ` ` | ` ` ` p | | ` ` ` |
| ` ` ` ` o o ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p | | ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p, (q),(r)) ` | ` ` ` ` | ` ((p, (q),(r)))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_97 ` ` ` | ` ` ` ` | ` ` ` q_158 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` q r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o q ` ` |
| ` ` ` q r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o q ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (q, r)((p) q) ` | ` ` ` ` | `((q, r)((p) q))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_98 ` ` ` | ` ` ` ` | ` ` ` q_157 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` | ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` o---o p ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o p ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((q, p (r))) ` | ` ` ` ` | ` ` (q, p (r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` `q_99 ` ` ` | ` ` ` ` | ` ` ` q_156 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` p ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` ` ` p ` ` ` | ` ` ` ` | ` ` ` q r | ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` o-o o r ` ` |
| ` ` ` q r | ` ` ` | ` ` ` ` | ` ` ` `\|/` ` ` ` |
| ` ` ` o-o o r ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\|/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (q, r)((p) r) ` | ` ` ` ` | `((q, r)((p) r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_100 ` ` ` | ` ` ` ` | ` ` ` q_155 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r ` | ` ` ` | ` ` ` ` | ` ` ` ` ` q ` ` ` |
| ` ` ` o---o p ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o p ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((r, p (q))) ` | ` ` ` ` | ` ` (r, p (q))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_101 ` ` ` | ` ` ` ` | ` ` ` q_154 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(q , r)` ` ` | ` ` ` ` | ` ` ((q , r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_102 ` ` ` | ` ` ` ` | ` ` ` q_153 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` p q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o-o-o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p `\`/` ` ` | ` ` ` ` | ` ` ` ` p q r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o-o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p, (p, q, r)))` | ` ` ` ` | `(p, (p, q, r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_103 ` ` ` | ` ` ` ` | ` ` ` q_152 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` (p, q, r) ` ` | ` ` ` ` | ` `((p, q, r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_104 ` ` ` | ` ` ` ` | ` ` ` q_151 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o---o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p `\ /` ` ` | ` ` ` ` | ` ` ` ` q ` r ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` o---o ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` p `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p, (q, r))) ` | ` ` ` ` | ` `(p, (q, r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_105 ` ` ` | ` ` ` ` | ` ` ` q_150 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `p q` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r ` | ` ` ` | ` ` ` ` | ` ` ` ` `p q` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` r ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((r, (p q)))` ` | ` ` ` ` | ` `(r, (p q)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_106 ` ` ` | ` ` ` ` | ` ` ` q_149 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p q | ` ` ` | ` ` ` ` | ` ` ` ` ` r ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` p q | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p, q, (r)))` | ` ` ` ` | ` ` (p, q, (r)) ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_107 ` ` ` | ` ` ` ` | ` ` ` q_148 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `p r` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` | ` ` ` | ` ` ` ` | ` ` ` ` `p r` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((q, (p r)))` ` | ` ` ` ` | ` `(q, (p r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_108 ` ` ` | ` ` ` ` | ` ` ` q_147 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p | r ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` p | r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\ /` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p, (q), r)) ` | ` ` ` ` | ` `(p, (q), r)` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_109 ` ` ` | ` ` ` ` | ` ` ` q_146 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` p ` q ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o ` o---o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\` `\`/` ` ` | ` ` ` ` | ` ` p ` q ` r ` ` |
| ` ` q o ` o ` ` ` | ` ` ` ` | ` ` o ` o---o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\` `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` q o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p) q)((q, r))) | ` ` ` ` | `((p) q)((q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_110 ` ` ` | ` ` ` ` | ` ` ` q_145 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` p @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `(p ((q, r))) ` | ` ` ` ` | ` ` p ((q, r))` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_111 ` ` ` | ` ` ` ` | ` ` ` q_144 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` `q r` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `q r` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` p @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `p (q r)` ` ` | ` ` ` ` | ` ` (p (q r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_112 ` ` ` | ` ` ` ` | ` ` ` q_143 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` `q`r` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` `o`o` ` |
| ` ` ` ` ` `q`r` ` | ` ` ` ` | ` `q`r` `p | |` ` |
| ` ` ` ` ` `o`o` ` | ` ` ` ` | ` `o`o` `o-o-o` ` |
| ` `q`r` `p | |` ` | ` ` ` ` | ` ` \|` ` \ / ` ` |
| ` `o`o` `o-o-o` ` | ` ` ` ` | ` `p`o-----o` ` ` |
| ` ` \|` ` \ / ` ` | ` ` ` ` | ` ` ` \ ` / ` ` ` |
| ` `p`o-----o` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` \ ` / ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `(` p `(q)`(r)` ` | ` ` ` ` | ((` p `(q)`(r)` ` |
| `,( p ,(q),(r)))` | ` ` ` ` | `,( p ,(q),(r)))) |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_113 ` ` ` | ` ` ` ` | ` ` ` q_142 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` r ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` r ` q ` ` ` |
| ` ` p o ` o r ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o r ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (r)) (r (q))) | ` ` ` ` | `(p (r)) (r (q))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_114 ` ` ` | ` ` ` ` | ` ` ` q_141 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` r ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` r ` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` q o ` ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` q @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p (r)) q)` ` | ` ` ` ` | ` ` (p (r)) q ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_115 ` ` ` | ` ` ` ` | ` ` ` q_140 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` q ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | ` | ` ` ` | ` ` ` ` | ` ` ` q ` r ` ` ` |
| ` ` p o ` o q ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` | ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o q ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (q)) (q (r))) | ` ` ` ` | `(p (q)) (q (r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_116 ` ` ` | ` ` ` ` | ` ` ` q_139 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` q ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` q ` ` ` ` |
| ` ` ` p o ` ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` r o ` ` ` ` | ` ` ` ` | ` ` ` p o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` r @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` `((p (q)) r)` ` | ` ` ` ` | ` ` (p (q)) r ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_117 ` ` ` | ` ` ` ` | ` ` ` q_138 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` q ` q ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o ` o---o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\` `\`/` ` ` | ` ` ` ` | ` ` q ` q ` r ` ` |
| ` ` p o ` o ` ` ` | ` ` ` ` | ` ` o ` o---o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\` `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (q))((q, r))) | ` ` ` ` | `(p (q))((q, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_118 ` ` ` | ` ` ` ` | ` ` ` q_137 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `q r` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `q r` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` (q r) ` ` ` | ` ` ` ` | ` ` ` `q r` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_119 ` ` ` | ` ` ` ` | ` ` ` q_136 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` `q r` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` | ` ` ` | ` ` ` ` | ` ` ` ` `q r` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` o ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` | ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ((p, (q r)))` ` | ` ` ` ` | ` `(p, (q r)) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_120 ` ` ` | ` ` ` ` | ` ` ` q_135 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` | q r ` ` ` | ` ` ` ` | ` ` ` p ` ` ` ` ` |
| ` ` ` o-o-o ` ` ` | ` ` ` ` | ` ` ` o ` ` ` ` ` |
| ` ` ` `\ /` ` ` ` | ` ` ` ` | ` ` ` | q r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o-o-o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| `(((p), q, r))` ` | ` ` ` ` | ` ((p), q, r) ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_121 ` ` ` | ` ` ` ` | ` ` ` q_134 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` q ` p ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o ` o---o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\` `\`/` ` ` | ` ` ` ` | ` ` q ` p ` r ` ` |
| ` ` p o ` o ` ` ` | ` ` ` ` | ` ` o ` o---o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\` `\`/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` p o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ((p (q))((p, r))) | ` ` ` ` | `(p (q))((p, r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_122 ` ` ` | ` ` ` ` | ` ` ` q_133 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` r ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` o q ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ q ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (((p, r)) q)` ` | ` ` ` ` | ` `((p, r)) q ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_123 ` ` ` | ` ` ` ` | ` ` ` q_132 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` p ` q ` r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o---o ` o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\`/` `/` ` ` | ` ` ` ` | ` ` p ` q ` r ` ` |
| ` ` ` o ` o p ` ` | ` ` ` ` | ` ` o---o ` o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\`/` `/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o p ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p, q))(p (r))) | ` ` ` ` | `((p, q))(p (r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_124 ` ` ` | ` ` ` ` | ` ` ` q_131 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` q ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` o---o ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` ` p ` q ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o---o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` o r ` ` ` | ` ` ` ` | ` ` ` ` o ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ r ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` (((p, q)) r)` ` | ` ` ` ` | ` `((p, q)) r ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_125 ` ` ` | ` ` ` ` | ` ` ` q_130 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` p q ` q r ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` o-o ` o-o ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` `\| ` |/` ` ` | ` ` ` ` | ` ` p q ` q r ` ` |
| ` ` ` o ` o ` ` ` | ` ` ` ` | ` ` o-o ` o-o ` ` |
| ` ` ` `\`/` ` ` ` | ` ` ` ` | ` ` `\| ` |/` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` o ` o ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` `\`/` ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| (((p,q)) ((q,r))) | ` ` ` ` | `((p,q)) ((q,r))` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_126 ` ` ` | ` ` ` ` | ` ` ` q_129 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o-------------------o ` ` ` ` o-------------------o
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` p q r ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` o ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
| ` ` ` ` @ ` ` ` ` | ` ` ` ` | ` ` ` ` @ ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` `(p q r)` ` ` | ` ` ` ` | ` ` ` p q r ` ` ` |
o-------------------o ` ` ` ` o-------------------o
| ` ` ` q_127 ` ` ` | ` ` ` ` | ` ` ` q_128 ` ` ` |
o-------------------o ` ` ` ` o-------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 24e
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
I'm attaching here a text file copy of the current set
of cactus graphs for propositions on three variables,
and I have placed additional copies at the following
two sites:
CR 24. http://stderr.org/pipermail/inquiry/2004-April/001322.html
CR 24. http://suo.ieee.org/ontology/msg05518.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Note 25
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 256. Propositional Forms on Three Variables
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_0 | q_00000000 | 0 0 0 0 0 0 0 0 | ( ) |
| | | | |
| q_1 | q_00000001 | 0 0 0 0 0 0 0 1 | (p) (q) (r) |
| | | | |
| q_2 | q_00000010 | 0 0 0 0 0 0 1 0 | (p) (q) r |
| | | | |
| q_3 | q_00000011 | 0 0 0 0 0 0 1 1 | (p) (q) |
| | | | |
| q_4 | q_00000100 | 0 0 0 0 0 1 0 0 | (p) q (r) |
| | | | |
| q_5 | q_00000101 | 0 0 0 0 0 1 0 1 | (p) (r) |
| | | | |
| q_6 | q_00000110 | 0 0 0 0 0 1 1 0 | (p) (q , r) |
| | | | |
| q_7 | q_00000111 | 0 0 0 0 0 1 1 1 | (p) (q r) |
| | | | |
| q_8 | q_00001000 | 0 0 0 0 1 0 0 0 | (p) q r |
| | | | |
| q_9 | q_00001001 | 0 0 0 0 1 0 0 1 | (p) ((q , r)) |
| | | | |
| q_10 | q_00001010 | 0 0 0 0 1 0 1 0 | (p) r |
| | | | |
| q_11 | q_00001011 | 0 0 0 0 1 0 1 1 | (p) (q (r)) |
| | | | |
| q_12 | q_00001100 | 0 0 0 0 1 1 0 0 | (p) q |
| | | | |
| q_13 | q_00001101 | 0 0 0 0 1 1 0 1 | (p) ((q) r) |
| | | | |
| q_14 | q_00001110 | 0 0 0 0 1 1 1 0 | (p) ((q) (r)) |
| | | | |
| q_15 | q_00001111 | 0 0 0 0 1 1 1 1 | (p) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_16 | q_00010000 | 0 0 0 1 0 0 0 0 | p (q) (r) |
| | | | |
| q_17 | q_00010001 | 0 0 0 1 0 0 0 1 | (q) (r) |
| | | | |
| q_18 | q_00010010 | 0 0 0 1 0 0 1 0 | (p , r) (q) |
| | | | |
| q_19 | q_00010011 | 0 0 0 1 0 0 1 1 | (p r) (q) |
| | | | |
| q_20 | q_00010100 | 0 0 0 1 0 1 0 0 | (p , q) (r) |
| | | | |
| q_21 | q_00010101 | 0 0 0 1 0 1 0 1 | (p q) (r) |
| | | | |
| q_22 | q_00010110 | 0 0 0 1 0 1 1 0 | ((p), (q), (r)) |
| | | | |
| q_23 | q_00010111 | 0 0 0 1 0 1 1 1 | (p)(q)(r) + ((p),(q),(r)) |
| | | | |
| q_24 | q_00011000 | 0 0 0 1 1 0 0 0 | (p, q) (p, r) |
| | | | |
| q_25 | q_00011001 | 0 0 0 1 1 0 0 1 | p = ((p), (q), (r)) |
| | | | |
| q_26 | q_00011010 | 0 0 0 1 1 0 1 0 | (p, r) (q (r)) |
| | | | |
| q_27 | q_00011011 | 0 0 0 1 1 0 1 1 | (p, q)(q, r) = p q |
| | | | |
| q_28 | q_00011100 | 0 0 0 1 1 1 0 0 | (p, q)((q) r) |
| | | | |
| q_29 | q_00011101 | 0 0 0 1 1 1 0 1 | (p, r)(q, r) = p r |
| | | | |
| q_30 | q_00011110 | 0 0 0 1 1 1 1 0 | ((p , (q) (r))) |
| | | | |
| q_31 | q_00011111 | 0 0 0 1 1 1 1 1 | (p ((q) (r))) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_32 | q_00100000 | 0 0 1 0 0 0 0 0 | p (q) r |
| | | | |
| q_33 | q_00100001 | 0 0 1 0 0 0 0 1 | ((p , r)) (q) |
| | | | |
| q_34 | q_00100010 | 0 0 1 0 0 0 1 0 | (q) r |
| | | | |
| q_35 | q_00100011 | 0 0 1 0 0 0 1 1 | (p (r)) (q) |
| | | | |
| q_36 | q_00100100 | 0 0 1 0 0 1 0 0 | (p, q) (q, r) |
| | | | |
| q_37 | q_00100101 | 0 0 1 0 0 1 0 1 | q = ((p), (q), (r)) |
| | | | |
| q_38 | q_00100110 | 0 0 1 0 0 1 1 0 | (q, r) (p (r)) |
| | | | |
| q_39 | q_00100111 | 0 0 1 0 0 1 1 1 | (p, q)(p, r) = p q |
| | | | |
| q_40 | q_00101000 | 0 0 1 0 1 0 0 0 | (p , q) r |
| | | | |
| q_41 | q_00101001 | 0 0 1 0 1 0 0 1 | ((p), (q), r) |
| | | | |
| q_42 | q_00101010 | 0 0 1 0 1 0 1 0 | (p q) r |
| | | | |
| q_43 | q_00101011 | 0 0 1 0 1 0 1 1 | (p)(q) r + ((p),(q), r ) |
| | | | |
| q_44 | q_00101100 | 0 0 1 0 1 1 0 0 | (p, q) (p (r)) |
| | | | |
| q_45 | q_00101101 | 0 0 1 0 1 1 0 1 | ((p , (q) r)) |
| | | | |
| q_46 | q_00101110 | 0 0 1 0 1 1 1 0 | ((r (q))(q (p))) |
| | | | |
| q_47 | q_00101111 | 0 0 1 0 1 1 1 1 | (p ((q) r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_48 | q_00110000 | 0 0 1 1 0 0 0 0 | p (q) |
| | | | |
| q_49 | q_00110001 | 0 0 1 1 0 0 0 1 | ((p) r) (q) |
| | | | |
| q_50 | q_00110010 | 0 0 1 1 0 0 1 0 | ((p) (r)) (q) |
| | | | |
| q_51 | q_00110011 | 0 0 1 1 0 0 1 1 | (q) |
| | | | |
| q_52 | q_00110100 | 0 0 1 1 0 1 0 0 | (p, q)((p) r) |
| | | | |
| q_53 | q_00110101 | 0 0 1 1 0 1 0 1 | (p, r)(q, r) = q r |
| | | | |
| q_54 | q_00110110 | 0 0 1 1 0 1 1 0 | ((q , (p) (r))) |
| | | | |
| q_55 | q_00110111 | 0 0 1 1 0 1 1 1 | (((p) (r)) q) |
| | | | |
| q_56 | q_00111000 | 0 0 1 1 1 0 0 0 | (p, q) (q (r)) |
| | | | |
| q_57 | q_00111001 | 0 0 1 1 1 0 0 1 | ((q , (p) r)) |
| | | | |
| q_58 | q_00111010 | 0 0 1 1 1 0 1 0 | ((r (p))(p (q))) |
| | | | |
| q_59 | q_00111011 | 0 0 1 1 1 0 1 1 | (((p) r) q) |
| | | | |
| q_60 | q_00111100 | 0 0 1 1 1 1 0 0 | (p , q) |
| | | | |
| q_61 | q_00111101 | 0 0 1 1 1 1 0 1 | r = ( p , q , r ) |
| | | | |
| q_62 | q_00111110 | 0 0 1 1 1 1 1 0 | (((p, q)) ((p) r)) |
| | | | |
| q_63 | q_00111111 | 0 0 1 1 1 1 1 1 | (p q) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_64 | q_01000000 | 0 1 0 0 0 0 0 0 | p q (r) |
| | | | |
| q_65 | q_01000001 | 0 1 0 0 0 0 0 1 | ((p , q)) (r) |
| | | | |
| q_66 | q_01000010 | 0 1 0 0 0 0 1 0 | (p, r) (q, r) |
| | | | |
| q_67 | q_01000011 | 0 1 0 0 0 0 1 1 | r = ((p), (q), (r)) |
| | | | |
| q_68 | q_01000100 | 0 1 0 0 0 1 0 0 | q (r) |
| | | | |
| q_69 | q_01000101 | 0 1 0 0 0 1 0 1 | (p (q)) (r) |
| | | | |
| q_70 | q_01000110 | 0 1 0 0 0 1 1 0 | (q, r) (p (q)) |
| | | | |
| q_71 | q_01000111 | 0 1 0 0 0 1 1 1 | (p, q)(p, r) = p r |
| | | | |
| q_72 | q_01001000 | 0 1 0 0 1 0 0 0 | (p , r) q |
| | | | |
| q_73 | q_01001001 | 0 1 0 0 1 0 0 1 | ((p), q , (r)) |
| | | | |
| q_74 | q_01001010 | 0 1 0 0 1 0 1 0 | (p, r) (p (q)) |
| | | | |
| q_75 | q_01001011 | 0 1 0 0 1 0 1 1 | ((p , q (r))) |
| | | | |
| q_76 | q_01001100 | 0 1 0 0 1 1 0 0 | (p r) q |
| | | | |
| q_77 | q_01001101 | 0 1 0 0 1 1 0 1 | (p) q (r) + ((p), q ,(r)) |
| | | | |
| q_78 | q_01001110 | 0 1 0 0 1 1 1 0 | ((q (r))(r (p))) |
| | | | |
| q_79 | q_01001111 | 0 1 0 0 1 1 1 1 | (p (q (r))) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_80 | q_01010000 | 0 1 0 1 0 0 0 0 | p (r) |
| | | | |
| q_81 | q_01010001 | 0 1 0 1 0 0 0 1 | ((p) q) (r) |
| | | | |
| q_82 | q_01010010 | 0 1 0 1 0 0 1 0 | (p, r)((p) q) |
| | | | |
| q_83 | q_01010011 | 0 1 0 1 0 0 1 1 | (p, q)(q, r) = q r |
| | | | |
| q_84 | q_01010100 | 0 1 0 1 0 1 0 0 | ((p) (q)) (r) |
| | | | |
| q_85 | q_01010101 | 0 1 0 1 0 1 0 1 | (r) |
| | | | |
| q_86 | q_01010110 | 0 1 0 1 0 1 1 0 | ((r , (p) (q))) |
| | | | |
| q_87 | q_01010111 | 0 1 0 1 0 1 1 1 | (((p) (q)) r) |
| | | | |
| q_88 | q_01011000 | 0 1 0 1 1 0 0 0 | (p, r)((q) r) |
| | | | |
| q_89 | q_01011001 | 0 1 0 1 1 0 0 1 | ((r , (p) q)) |
| | | | |
| q_90 | q_01011010 | 0 1 0 1 1 0 1 0 | (p , r) |
| | | | |
| q_91 | q_01011011 | 0 1 0 1 1 0 1 1 | q = ( p , q , r ) |
| | | | |
| q_92 | q_01011100 | 0 1 0 1 1 1 0 0 | ((q (p))(p (r))) |
| | | | |
| q_93 | q_01011101 | 0 1 0 1 1 1 0 1 | (((p) q) r) |
| | | | |
| q_94 | q_01011110 | 0 1 0 1 1 1 1 0 | (((p, r)) ((p) q)) |
| | | | |
| q_95 | q_01011111 | 0 1 0 1 1 1 1 1 | (p r) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_96 | q_01100000 | 0 1 1 0 0 0 0 0 | p (q , r) |
| | | | |
| q_97 | q_01100001 | 0 1 1 0 0 0 0 1 | (p , (q), (r)) |
| | | | |
| q_98 | q_01100010 | 0 1 1 0 0 0 1 0 | (q, r)((p) q) |
| | | | |
| q_99 | q_01100011 | 0 1 1 0 0 0 1 1 | ((q , p (r))) |
| | | | |
| q_100 | q_01100100 | 0 1 1 0 0 1 0 0 | (q, r)((p) r) |
| | | | |
| q_101 | q_01100101 | 0 1 1 0 0 1 0 1 | ((r , p (q))) |
| | | | |
| q_102 | q_01100110 | 0 1 1 0 0 1 1 0 | (q , r) |
| | | | |
| q_103 | q_01100111 | 0 1 1 0 0 1 1 1 | p = ( p , q , r ) |
| | | | |
| q_104 | q_01101000 | 0 1 1 0 1 0 0 0 | (p , q , r) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , (q , r))) |
| | | | |
| q_106 | q_01101010 | 0 1 1 0 1 0 1 0 | ((r , (p q))) |
| | | | |
| q_107 | q_01101011 | 0 1 1 0 1 0 1 1 | ((p , q , (r))) |
| | | | |
| q_108 | q_01101100 | 0 1 1 0 1 1 0 0 | ((q , (p r))) |
| | | | |
| q_109 | q_01101101 | 0 1 1 0 1 1 0 1 | ((p , (q), r)) |
| | | | |
| q_110 | q_01101110 | 0 1 1 0 1 1 1 0 | (((p) q)((q, r))) |
| | | | |
| q_111 | q_01101111 | 0 1 1 0 1 1 1 1 | (p ((q , r))) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_112 | q_01110000 | 0 1 1 1 0 0 0 0 | p (q r) |
| | | | |
| q_113 | q_01110001 | 0 1 1 1 0 0 0 1 | p (q)(r) + ( p ,(q),(r)) |
| | | | |
| q_114 | q_01110010 | 0 1 1 1 0 0 1 0 | ((p (r))(r (q))) |
| | | | |
| q_115 | q_01110011 | 0 1 1 1 0 0 1 1 | ((p (r)) q) |
| | | | |
| q_116 | q_01110100 | 0 1 1 1 0 1 0 0 | ((p (q))(q (r))) |
| | | | |
| q_117 | q_01110101 | 0 1 1 1 0 1 0 1 | ((p (q)) r) |
| | | | |
| q_118 | q_01110110 | 0 1 1 1 0 1 1 0 | (((q, r))(p (q))) |
| | | | |
| q_119 | q_01110111 | 0 1 1 1 0 1 1 1 | (q r) |
| | | | |
| q_120 | q_01111000 | 0 1 1 1 1 0 0 0 | ((p , (q r))) |
| | | | |
| q_121 | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), q , r)) |
| | | | |
| q_122 | q_01111010 | 0 1 1 1 1 0 1 0 | (((p, r))(p (q))) |
| | | | |
| q_123 | q_01111011 | 0 1 1 1 1 0 1 1 | (((p , r)) q) |
| | | | |
| q_124 | q_01111100 | 0 1 1 1 1 1 0 0 | (((p, q))(p (r))) |
| | | | |
| q_125 | q_01111101 | 0 1 1 1 1 1 0 1 | (((p , q)) r) |
| | | | |
| q_126 | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q)) ((q, r))) |
| | | | |
| q_127 | q_01111111 | 0 1 1 1 1 1 1 1 | (p q r) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_128 | q_10000000 | 1 0 0 0 0 0 0 0 | p q r |
| | | | |
| q_129 | q_10000001 | 1 0 0 0 0 0 0 1 | ((p, q)) ((q, r)) |
| | | | |
| q_130 | q_10000010 | 1 0 0 0 0 0 1 0 | ((p , q)) r |
| | | | |
| q_131 | q_10000011 | 1 0 0 0 0 0 1 1 | ((p, q)) (p (r)) |
| | | | |
| q_132 | q_10000100 | 1 0 0 0 0 1 0 0 | ((p , r)) q |
| | | | |
| q_133 | q_10000101 | 1 0 0 0 0 1 0 1 | ((p, r)) (p (q)) |
| | | | |
| q_134 | q_10000110 | 1 0 0 0 0 1 1 0 | ((p), q , r) |
| | | | |
| q_135 | q_10000111 | 1 0 0 0 0 1 1 1 | ((p , q r)) |
| | | | |
| q_136 | q_10001000 | 1 0 0 0 1 0 0 0 | q r |
| | | | |
| q_137 | q_10001001 | 1 0 0 0 1 0 0 1 | ((q, r)) (p (q)) |
| | | | |
| q_138 | q_10001010 | 1 0 0 0 1 0 1 0 | (p (q)) r |
| | | | |
| q_139 | q_10001011 | 1 0 0 0 1 0 1 1 | (p (q))(q (r)) |
| | | | |
| q_140 | q_10001100 | 1 0 0 0 1 1 0 0 | (p (r)) q |
| | | | |
| q_141 | q_10001101 | 1 0 0 0 1 1 0 1 | (p (r))(r (q)) |
| | | | |
| q_142 | q_10001110 | 1 0 0 0 1 1 1 0 | (p) q r + ((p), q , r ) |
| | | | |
| q_143 | q_10001111 | 1 0 0 0 1 1 1 1 | (p (q r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_144 | q_10010000 | 1 0 0 1 0 0 0 0 | p ((q , r)) |
| | | | |
| q_145 | q_10010001 | 1 0 0 1 0 0 0 1 | ((p) q)((q, r)) |
| | | | |
| q_146 | q_10010010 | 1 0 0 1 0 0 1 0 | (p , (q), r) |
| | | | |
| q_147 | q_10010011 | 1 0 0 1 0 0 1 1 | ((q , p r)) |
| | | | |
| q_148 | q_10010100 | 1 0 0 1 0 1 0 0 | (p , q , (r)) |
| | | | |
| q_149 | q_10010101 | 1 0 0 1 0 1 0 1 | ((r , p q)) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | (p , (q , r)) |
| | | | |
| q_151 | q_10010111 | 1 0 0 1 0 1 1 1 | ((p , q , r)) |
| | | | |
| q_152 | q_10011000 | 1 0 0 1 1 0 0 0 | p + ( p , q , r ) |
| | | | |
| q_153 | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , r)) |
| | | | |
| q_154 | q_10011010 | 1 0 0 1 1 0 1 0 | ((r , (p (q)))) |
| | | | |
| q_155 | q_10011011 | 1 0 0 1 1 0 1 1 | ((q, r)((p) r)) |
| | | | |
| q_156 | q_10011100 | 1 0 0 1 1 1 0 0 | ((q , (p (r)))) |
| | | | |
| q_157 | q_10011101 | 1 0 0 1 1 1 0 1 | ((q, r)((p) q)) |
| | | | |
| q_158 | q_10011110 | 1 0 0 1 1 1 1 0 | ((p , (q), (r))) |
| | | | |
| q_159 | q_10011111 | 1 0 0 1 1 1 1 1 | (p (q , r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_160 | q_10100000 | 1 0 1 0 0 0 0 0 | p r |
| | | | |
| q_161 | q_10100001 | 1 0 1 0 0 0 0 1 | ((p, r)) ((p) q) |
| | | | |
| q_162 | q_10100010 | 1 0 1 0 0 0 1 0 | ((p) q) r |
| | | | |
| q_163 | q_10100011 | 1 0 1 0 0 0 1 1 | (q (p))(p (r)) |
| | | | |
| q_164 | q_10100100 | 1 0 1 0 0 1 0 0 | q + ( p , q , r ) |
| | | | |
| q_165 | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , r)) |
| | | | |
| q_166 | q_10100110 | 1 0 1 0 0 1 1 0 | ((r ,((p) q))) |
| | | | |
| q_167 | q_10100111 | 1 0 1 0 0 1 1 1 | ((p, r)((q) r)) |
| | | | |
| q_168 | q_10101000 | 1 0 1 0 1 0 0 0 | ((p) (q)) r |
| | | | |
| q_169 | q_10101001 | 1 0 1 0 1 0 0 1 | ((r ,((p) (q)))) |
| | | | |
| q_170 | q_10101010 | 1 0 1 0 1 0 1 0 | r |
| | | | |
| q_171 | q_10101011 | 1 0 1 0 1 0 1 1 | (((p) (q)) (r)) |
| | | | |
| q_172 | q_10101100 | 1 0 1 0 1 1 0 0 | (p, q)(q, r) + q r |
| | | | |
| q_173 | q_10101101 | 1 0 1 0 1 1 0 1 | ((p, r)((p) q)) |
| | | | |
| q_174 | q_10101110 | 1 0 1 0 1 1 1 0 | (((p) q) (r)) |
| | | | |
| q_175 | q_10101111 | 1 0 1 0 1 1 1 1 | (p (r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_176 | q_10110000 | 1 0 1 1 0 0 0 0 | p (q (r)) |
| | | | |
| q_177 | q_10110001 | 1 0 1 1 0 0 0 1 | (q (r))(r (p)) |
| | | | |
| q_178 | q_10110010 | 1 0 1 1 0 0 1 0 | p (q) r + ( p ,(q), r ) |
| | | | |
| q_179 | q_10110011 | 1 0 1 1 0 0 1 1 | ((p r) q) |
| | | | |
| q_180 | q_10110100 | 1 0 1 1 0 1 0 0 | ((p , (q (r)))) |
| | | | |
| q_181 | q_10110101 | 1 0 1 1 0 1 0 1 | ((p, r) (p (q))) |
| | | | |
| q_182 | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), q , (r))) |
| | | | |
| q_183 | q_10110111 | 1 0 1 1 0 1 1 1 | ((p , r) q |
| | | | |
| q_184 | q_10111000 | 1 0 1 1 1 0 0 0 | (p, q)(p, r) + p r |
| | | | |
| q_185 | q_10111001 | 1 0 1 1 1 0 0 1 | ((q, r) (p (q))) |
| | | | |
| q_186 | q_10111010 | 1 0 1 1 1 0 1 0 | ((p (q)) (r)) |
| | | | |
| q_187 | q_10111011 | 1 0 1 1 1 0 1 1 | (q (r)) |
| | | | |
| q_188 | q_10111100 | 1 0 1 1 1 1 0 0 | r + ((p), (q), (r)) |
| | | | |
| q_189 | q_10111101 | 1 0 1 1 1 1 0 1 | ((p, r) (q, r)) |
| | | | |
| q_190 | q_10111110 | 1 0 1 1 1 1 1 0 | (((p , q)) (r)) |
| | | | |
| q_191 | q_10111111 | 1 0 1 1 1 1 1 1 | (p q (r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_192 | q_11000000 | 1 1 0 0 0 0 0 0 | p q |
| | | | |
| q_193 | q_11000001 | 1 1 0 0 0 0 0 1 | ((p, q)) ((p) r) |
| | | | |
| q_194 | q_11000010 | 1 1 0 0 0 0 1 0 | r + ( p , q , r ) |
| | | | |
| q_195 | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , q)) |
| | | | |
| q_196 | q_11000100 | 1 1 0 0 0 1 0 0 | ((p) r) q |
| | | | |
| q_197 | q_11000101 | 1 1 0 0 0 1 0 1 | (r (p))(p (q)) |
| | | | |
| q_198 | q_11000110 | 1 1 0 0 0 1 1 0 | ((q ,((p) r))) |
| | | | |
| q_199 | q_11000111 | 1 1 0 0 0 1 1 1 | ((p, q) (q (r))) |
| | | | |
| q_200 | q_11001000 | 1 1 0 0 1 0 0 0 | ((p) (r)) q |
| | | | |
| q_201 | q_11001001 | 1 1 0 0 1 0 0 1 | ((q ,((p) (r)))) |
| | | | |
| q_202 | q_11001010 | 1 1 0 0 1 0 1 0 | (p, r)(q, r) + q r |
| | | | |
| q_203 | q_11001011 | 1 1 0 0 1 0 1 1 | ((p, q) ((p) r)) |
| | | | |
| q_204 | q_11001100 | 1 1 0 0 1 1 0 0 | q |
| | | | |
| q_205 | q_11001101 | 1 1 0 0 1 1 0 1 | (((p) (r)) (q)) |
| | | | |
| q_206 | q_11001110 | 1 1 0 0 1 1 1 0 | (((p) r) (q)) |
| | | | |
| q_207 | q_11001111 | 1 1 0 0 1 1 1 1 | (p (q)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_208 | q_11010000 | 1 1 0 1 0 0 0 0 | p ((q) r) |
| | | | |
| q_209 | q_11010001 | 1 1 0 1 0 0 0 1 | (r (q))(q (p)) |
| | | | |
| q_210 | q_11010010 | 1 1 0 1 0 0 1 0 | ((p ,((q) r))) |
| | | | |
| q_211 | q_11010011 | 1 1 0 1 0 0 1 1 | ((p, q) (p (r))) |
| | | | |
| q_212 | q_11010100 | 1 1 0 1 0 1 0 0 | p q (r) + ( p , q ,(r)) |
| | | | |
| q_213 | q_11010101 | 1 1 0 1 0 1 0 1 | ((p q) r) |
| | | | |
| q_214 | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), r)) |
| | | | |
| q_215 | q_11010111 | 1 1 0 1 0 1 1 1 | ((p , q) r) |
| | | | |
| q_216 | q_11011000 | 1 1 0 1 1 0 0 0 | (p, q)(p, r) + p q |
| | | | |
| q_217 | q_11011001 | 1 1 0 1 1 0 0 1 | ((q, r) (p (r))) |
| | | | |
| q_218 | q_11011010 | 1 1 0 1 1 0 1 0 | q + ((p), (q), (r)) |
| | | | |
| q_219 | q_11011011 | 1 1 0 1 1 0 1 1 | ((p, q) (q, r)) |
| | | | |
| q_220 | q_11011100 | 1 1 0 1 1 1 0 0 | ((p (r)) (q)) |
| | | | |
| q_221 | q_11011101 | 1 1 0 1 1 1 0 1 | ((q) r) |
| | | | |
| q_222 | q_11011110 | 1 1 0 1 1 1 1 0 | (((p , r)) (q)) |
| | | | |
| q_223 | q_11011111 | 1 1 0 1 1 1 1 1 | (p (q) r) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_224 | q_11100000 | 1 1 1 0 0 0 0 0 | p ((q) (r)) |
| | | | |
| q_225 | q_11100001 | 1 1 1 0 0 0 0 1 | (p, (q) (r)) |
| | | | |
| q_226 | q_11100010 | 1 1 1 0 0 0 1 0 | (p, r)(q, r) + p r |
| | | | |
| q_227 | q_11100011 | 1 1 1 0 0 0 1 1 | ((p, q)((q) r)) |
| | | | |
| q_228 | q_11100100 | 1 1 1 0 0 1 0 0 | (p, q)(q, r) + p q |
| | | | |
| q_229 | q_11100101 | 1 1 1 0 0 1 0 1 | ((p, r) (q (r))) |
| | | | |
| q_230 | q_11100110 | 1 1 1 0 0 1 1 0 | p + ((p), (q), (r)) |
| | | | |
| q_231 | q_11100111 | 1 1 1 0 0 1 1 1 | ((p, q) (p, r)) |
| | | | |
| q_232 | q_11101000 | 1 1 1 0 1 0 0 0 | p q r + ( p , q , r ) |
| | | | |
| q_233 | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| | | | |
| q_234 | q_11101010 | 1 1 1 0 1 0 1 0 | ((p q) (r)) |
| | | | |
| q_235 | q_11101011 | 1 1 1 0 1 0 1 1 | ((p, q) (r)) |
| | | | |
| q_236 | q_11101100 | 1 1 1 0 1 1 0 0 | ((p r) (q)) |
| | | | |
| q_237 | q_11101101 | 1 1 1 0 1 1 0 1 | ((p, r) (q)) |
| | | | |
| q_238 | q_11101110 | 1 1 1 0 1 1 1 0 | ((q) (r)) |
| | | | |
| q_239 | q_11101111 | 1 1 1 0 1 1 1 1 | (p (q) (r)) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_240 | q_11110000 | 1 1 1 1 0 0 0 0 | p |
| | | | |
| q_241 | q_11110001 | 1 1 1 1 0 0 0 1 | ((p) ((q) (r))) |
| | | | |
| q_242 | q_11110010 | 1 1 1 1 0 0 1 0 | ((p) ((q) r)) |
| | | | |
| q_243 | q_11110011 | 1 1 1 1 0 0 1 1 | ((p) q) |
| | | | |
| q_244 | q_11110100 | 1 1 1 1 0 1 0 0 | ((p) (q (r))) |
| | | | |
| q_245 | q_11110101 | 1 1 1 1 0 1 0 1 | ((p) r) |
| | | | |
| q_246 | q_11110110 | 1 1 1 1 0 1 1 0 | ((p) ((q, r))) |
| | | | |
| q_247 | q_11110111 | 1 1 1 1 0 1 1 1 | ((p) q r) |
| | | | |
| q_248 | q_11111000 | 1 1 1 1 1 0 0 0 | ((p) (q r)) |
| | | | |
| q_249 | q_11111001 | 1 1 1 1 1 0 0 1 | ((p) (q, r)) |
| | | | |
| q_250 | q_11111010 | 1 1 1 1 1 0 1 0 | ((p) (r)) |
| | | | |
| q_251 | q_11111011 | 1 1 1 1 1 0 1 1 | ((p) q (r)) |
| | | | |
| q_252 | q_11111100 | 1 1 1 1 1 1 0 0 | ((p) (q)) |
| | | | |
| q_253 | q_11111101 | 1 1 1 1 1 1 0 1 | ((p) (q) r) |
| | | | |
| q_254 | q_11111110 | 1 1 1 1 1 1 1 0 | ((p) (q) (r)) |
| | | | |
| q_255 | q_11111111 | 1 1 1 1 1 1 1 1 | (( )) |
| | | | |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Work Area 1
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o---o-----o---o o |
| | | | | |
| | | | | |
| | Q | | R | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
Figure 0. Null Universe
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\`````````\`/`````````/`````\```````|
|``````/```````\`````````o`````````/```````\``````|
|`````/`````````\```````/`\```````/`````````\`````|
|````/```````````\`````/```\`````/```````````\````|
|```o`````````````o---o-----o---o`````````````o```|
|```|`````````````````|`````|`````````````````|```|
|```|`````````````````|`````|`````````````````|```|
|```|``````` Q ```````|`````|``````` R ```````|```|
|```o`````````````````o`````o`````````````````o```|
|````\`````````````````\```/`````````````````/````|
|`````\`````````````````\`/`````````````````/`````|
|``````\`````````````````o`````````````````/``````|
|```````\```````````````/`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Figure 1. Full Universe
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Work Area 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 1. Boundaries and Their Complements
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_22 | q_00010110 | 0 0 0 1 0 1 1 0 | ((p), (q), (r)) |
| | | | |
| q_41 | q_00101001 | 0 0 1 0 1 0 0 1 | ((p), (q), r ) |
| | | | |
| q_73 | q_01001001 | 0 1 0 0 1 0 0 1 | ((p), q , (r)) |
| | | | |
| q_134 | q_10000110 | 1 0 0 0 0 1 1 0 | ((p), q , r ) |
| | | | |
| q_97 | q_01100001 | 0 1 1 0 0 0 0 1 | ( p , (q), (r)) |
| | | | |
| q_146 | q_10010010 | 1 0 0 1 0 0 1 0 | ( p , (q), r ) |
| | | | |
| q_148 | q_10010100 | 1 0 0 1 0 1 0 0 | ( p , q , (r)) |
| | | | |
| q_104 | q_01101000 | 0 1 1 0 1 0 0 0 | ( p , q , r ) |
| | | | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_233 | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| | | | |
| q_214 | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), r )) |
| | | | |
| q_182 | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), q , (r))) |
| | | | |
| q_121 | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), q , r )) |
| | | | |
| q_158 | q_10011110 | 1 0 0 1 1 1 1 0 | (( p , (q), (r))) |
| | | | |
| q_109 | q_01101101 | 0 1 1 0 1 1 0 1 | (( p , (q), r )) |
| | | | |
| q_107 | q_01101011 | 0 1 1 0 1 0 1 1 | (( p , q , (r))) |
| | | | |
| q_151 | q_10010111 | 1 0 0 1 0 1 1 1 | (( p , q , r )) |
| | | | |
o---------o------------o-----------------o---------------------------o
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |```````````P```````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| /`````\ \`/ /`````\ |
| /```````\ o /```````\ |
| /`````````\ / \ /`````````\ |
| /```````````\ / \ /```````````\ |
| o```````````` o---o-----o---o`````````````o |
| |`````````````````| |`````````````````| |
| |`````````````````| |`````````````````| |
| |``````` Q ```````| |``````` R ```````| |
| o`````````````````o o`````````````````o |
| \`````````````````\ /`````````````````/ |
| \`````````````````\ /`````````````````/ |
| \`````````````````o`````````````````/ |
| \```````````````/ \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_22. ((p),(q),(r))
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| /`````\`````````\ /`````````/`````\ |
| /```````\`````````o`````````/```````\ |
| /`````````\```````/`\```````/`````````\ |
| /```````````\`````/```\`````/```````````\ |
| o```````````` o---o-----o---o`````````````o |
| |`````````````````| |`````````````````| |
| |`````````````````| |`````````````````| |
| |``````` Q ```````| |``````` R ```````| |
| o`````````````````o o`````````````````o |
| \`````````````````\ /`````````````````/ |
| \`````````````````\ /`````````````````/ |
| \`````````````````o`````````````````/ |
| \```````````````/ \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_25. p + ((p),(q),(r))
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \ \ /`````````/`````\ |
| / \ o`````````/```````\ |
| / \ / \```````/`````````\ |
| / \ / \`````/```````````\ |
| o o---o-----o---o`````````````o |
| | |`````|`````````````````| |
| | |`````|`````````````````| |
| | Q |`````|``````` R ```````| |
| o o`````o`````````````````o |
| \ \```/`````````````````/ |
| \ \`/`````````````````/ |
| \ o`````````````````/ |
| \ / \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_42. p + q + ((p),(q),(r))
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ /`````````/ \ |
| / \`````````o`````````/ \ |
| / \```````/ \```````/ \ |
| / \`````/ \`````/ \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_104. (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \ \`/ / \ |
| / \ o / \ |
| / \ /`\ / \ |
| / \ /```\ / \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_152. p + (p, q, r)
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| P |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/ \ \ /`````````/ \```````|
|``````/ \ o`````````/ \``````|
|`````/ \ / \```````/ \`````|
|````/ \ / \`````/ \````|
|```o o---o-----o---o o```|
|```| |`````| |```|
|```| |`````| |```|
|```| Q |`````| R |```|
|```o o`````o o```|
|````\ \```/ /````|
|`````\ \`/ /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_41. ((p),(q), r)
o---------o------------o-----------------o---------------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_216 | | 1 1 0 1 1 0 0 0 | |
| | | | |
| q_217 | | 1 1 0 1 1 0 0 1 | p + ((p),(q), r) |
| | | | |
| q_131 | | 1 0 0 0 0 0 1 1 | r + ((p),(q), r) |
| | | | |
o---------o------------o-----------------o---------------------------o
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |```````````P```````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| /`````\`````````\`/ /`````\ |
| /```````\`````````o /```````\ |
| /`````````\```````/`\ /`````````\ |
| /```````````\`````/```\ /```````````\ |
| o```````````` o---o-----o---o`````````````o |
| |`````````````````| |`````````````````| |
| |`````````````````| |`````````````````| |
| |``````` Q ```````| |``````` R ```````| |
| o`````````````````o o`````````````````o |
| \`````````````````\ /`````````````````/ |
| \`````````````````\ /`````````````````/ |
| \`````````````````o`````````````````/ |
| \```````````````/ \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_214. pq + ((p),(q),(r))
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/ \`````````\`/ / \```````|
|``````/ \`````````o / \``````|
|`````/ \```````/`\ / \`````|
|````/ \`````/```\ / \````|
|```o o---o-----o---o o```|
|```| |`````| |```|
|```| |`````| |```|
|```| Q |`````| R |```|
|```o o`````o o```|
|````\ \```/ /````|
|`````\ \`/ /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_217. p + ((p),(q), r)
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| P |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/ \ \ / /`````\```````|
|``````/ \ o /```````\``````|
|`````/ \ /`\ /`````````\`````|
|````/ \ /```\ /```````````\````|
|```o o---o-----o---o`````````````o```|
|```| | |`````````````````|```|
|```| | |`````````````````|```|
|```| Q | |``````` R ```````|```|
|```o o o`````````````````o```|
|````\ \ /`````````````````/````|
|`````\ \ /`````````````````/`````|
|``````\ o`````````````````/``````|
|```````\ /`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_131. r + ((p),(q), r)
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Work Area 3
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \ \`/ / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_24. (p, q) (p, r)
q_24. p + p q r + (p, q, r)
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|```````````P```````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/ \ \`/ / \```````|
|``````/ \ o / \``````|
|`````/ \ / \ / \`````|
|````/ \ / \ / \````|
|```o o---o-----o---o o```|
|```| |`````| |```|
|```| |`````| |```|
|```| Q |`````| R |```|
|```o o`````o o```|
|````\ \```/ /````|
|`````\ \`/ /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_25.
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/ \ \`/ /`````\```````|
|``````/ \ o /```````\``````|
|`````/ \ / \ /`````````\`````|
|````/ \ / \ /```````````\````|
|```o o---o-----o---o`````````````o```|
|```| |`````|`````````````````|```|
|```| |`````|`````````````````|```|
|```| Q |`````|``````` R ```````|```|
|```o o`````o`````````````````o```|
|````\ \```/`````````````````/````|
|`````\ \`/`````````````````/`````|
|``````\ o`````````````````/``````|
|```````\ /`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_27.
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|`````````` P ``````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\ \`/ / \```````|
|``````/```````\ o / \``````|
|`````/`````````\ / \ / \`````|
|````/```````````\ / \ / \````|
|```o`````````````o---o-----o---o o```|
|```|`````````````````|`````| |```|
|```|`````````````````|`````| |```|
|```|``````` Q ```````|`````| R |```|
|```o`````````````````o`````o o```|
|````\`````````````````\```/ /````|
|`````\`````````````````\`/ /`````|
|``````\`````````````````o /``````|
|```````\```````````````/`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_29.
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| Q |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/`````\`````````\ / / \```````|
|``````/```````\`````````o / \``````|
|`````/`````````\```````/ \ / \`````|
|````/```````````\`````/ \ / \````|
|```o`````````````o---o-----o---o o```|
|```|`````````````````|`````| |```|
|```|`````````````````|`````| |```|
|```|````````P````````|`````| R |```|
|```o`````````````````o`````o o```|
|````\`````````````````\```/ /````|
|`````\`````````````````\`/ /`````|
|``````\`````````````````o /``````|
|```````\```````````````/`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_113.
o---------o------------o-----------------o---------------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------------o
| | | | |
| q_97 | q_01100001 | 0 1 1 0 0 0 0 1 | ( p , (q), (r)) |
| | | | |
| q_225 | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q) (r)) )) |
| | | | |
o---------o------------o-----------------o---------------------------o
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| P |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/ \`````````\ /`````````/ \```````|
|``````/ \`````````o`````````/ \``````|
|`````/ \```````/ \```````/ \`````|
|````/ \`````/ \`````/ \````|
|```o o---o-----o---o o```|
|```| | | |```|
|```| | | |```|
|```| Q | | R |```|
|```o o o o```|
|````\ \ / /````|
|`````\ \ / /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Genus and Species q_97. (p, (q),(r))
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| P |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/ \`````````\ /`````````/ \```````|
|``````/ \`````````o`````````/ \``````|
|`````/ \```````/`\```````/ \`````|
|````/ \`````/```\`````/ \````|
|```o o---o-----o---o o```|
|```| | | |```|
|```| | | |```|
|```| Q | | R |```|
|```o o o o```|
|````\ \ / /````|
|`````\ \ / /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
Thematic Extension q_225. ((p, ((q)(r)) ))
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Work Area 4
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o---------o------------o-----------------o---------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o---------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_112 | q_01110000 | 0 1 1 1 0 0 0 0 | p (q r) |
| | | | |
| q_76 | q_01001100 | 0 1 0 0 1 1 0 0 | q (p r) |
| | | | |
| q_42 | q_00101010 | 0 0 1 0 1 0 1 0 | r (p q) |
| | | | |
| q_7 | q_00000111 | 0 0 0 0 0 1 1 1 | (p) (q r) |
| | | | |
| q_19 | q_00010011 | 0 0 0 1 0 0 1 1 | (p r) (q) |
| | | | |
| q_21 | q_00010101 | 0 0 0 1 0 1 0 1 | (p q) (r) |
| | | | |
o---------o------------o-----------------o---------------------o
| | | | |
| q_143 | q_10001111 | 1 0 0 0 1 1 1 1 | (p (q r)) |
| | | | |
| q_179 | q_10110011 | 1 0 1 1 0 0 1 1 | (q (p r)) |
| | | | |
| q_213 | q_11010101 | 1 1 0 1 0 1 0 1 | (r (p q)) |
| | | | |
| q_248 | q_11111000 | 1 1 1 1 1 0 0 0 | ((p) (q r)) |
| | | | |
| q_236 | q_11101100 | 1 1 1 0 1 1 0 0 | ((q) (p r)) |
| | | | |
| q_234 | q_11101010 | 1 1 1 0 1 0 1 0 | ((r) (p q)) |
| | | | |
o---------o------------o-----------------o---------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Tables Formatted for NKS
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Table 0. Simple Propositions
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 1. A Family of Propositional Forms On Three Variables
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_22 | q_00010110 | 0 0 0 1 0 1 1 0 | ((p), (q), (r)) |
| | | | |
| q_41 | q_00101001 | 0 0 1 0 1 0 0 1 | ((p), (q), r ) |
| | | | |
| q_73 | q_01001001 | 0 1 0 0 1 0 0 1 | ((p), q , (r)) |
| | | | |
| q_134 | q_10000110 | 1 0 0 0 0 1 1 0 | ((p), q , r ) |
| | | | |
| q_97 | q_01100001 | 0 1 1 0 0 0 0 1 | ( p , (q), (r)) |
| | | | |
| q_146 | q_10010010 | 1 0 0 1 0 0 1 0 | ( p , (q), r ) |
| | | | |
| q_148 | q_10010100 | 1 0 0 1 0 1 0 0 | ( p , q , (r)) |
| | | | |
| q_104 | q_01101000 | 0 1 1 0 1 0 0 0 | ( p , q , r ) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_233 | q_11101001 | 1 1 1 0 1 0 0 1 | (((p), (q), (r))) |
| | | | |
| q_214 | q_11010110 | 1 1 0 1 0 1 1 0 | (((p), (q), r )) |
| | | | |
| q_182 | q_10110110 | 1 0 1 1 0 1 1 0 | (((p), q , (r))) |
| | | | |
| q_121 | q_01111001 | 0 1 1 1 1 0 0 1 | (((p), q , r )) |
| | | | |
| q_158 | q_10011110 | 1 0 0 1 1 1 1 0 | (( p , (q), (r))) |
| | | | |
| q_109 | q_01101101 | 0 1 1 0 1 1 0 1 | (( p , (q), r )) |
| | | | |
| q_107 | q_01101011 | 0 1 1 0 1 0 1 1 | (( p , q , (r))) |
| | | | |
| q_151 | q_10010111 | 1 0 0 1 0 1 1 1 | (( p , q , r )) |
| | | | |
o---------o------------o-----------------o-------------------o
Table 2. Linear Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 | L_2 | L_3 | L_4 |
| | | | |
| Decimal | Binary | Vector | Cactus |
o---------o------------o-----------------o-------------------o
| | p : 1 1 1 1 0 0 0 0 | |
| | q : 1 1 0 0 1 1 0 0 | |
| | r : 1 0 1 0 1 0 1 0 | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_0 | q_00000000 | 0 0 0 0 0 0 0 0 | ( ) |
| | | | |
| q_240 | q_11110000 | 1 1 1 1 0 0 0 0 | p |
| | | | |
| q_204 | q_11001100 | 1 1 0 0 1 1 0 0 | q |
| | | | |
| q_170 | q_10101010 | 1 0 1 0 1 0 1 0 | r |
| | | | |
| q_60 | q_00111100 | 0 0 1 1 1 1 0 0 | (p , q) |
| | | | |
| q_90 | q_01011010 | 0 1 0 1 1 0 1 0 | (p , r) |
| | | | |
| q_102 | q_01100110 | 0 1 1 0 0 1 1 0 | (q , r) |
| | | | |
| q_150 | q_10010110 | 1 0 0 1 0 1 1 0 | (p , (q , r)) |
| | | | |
o---------o------------o-----------------o-------------------o
| | | | |
| q_255 | q_11111111 | 1 1 1 1 1 1 1 1 | (( )) |
| | | | |
| q_15 | q_00001111 | 0 0 0 0 1 1 1 1 | (p) |
| | | | |
| q_51 | q_00110011 | 0 0 1 1 0 0 1 1 | (q) |
| | | | |
| q_85 | q_01010101 | 0 1 0 1 0 1 0 1 | (r) |
| | | | |
| q_195 | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , q)) |
| | | | |
| q_165 | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , r)) |
| | | | |
| q_153 | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , r)) |
| | | | |
| q_105 | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , (q , r))) |
| | | | |
o---------o------------o-----------------o-------------------o
Table 3. Positive Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_255 ` | q_11111111 | 1 1 1 1 1 1 1 1 | ` ` ` (( )) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ` `p` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_11001100 | 1 1 0 0 1 1 0 0 | ` ` ` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_192 ` | q_11000000 | 1 1 0 0 0 0 0 0 | ` `p` ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_160 ` | q_10100000 | 1 0 1 0 0 0 0 0 | ` `p` ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_136 ` | q_10001000 | 1 0 0 0 1 0 0 0 | ` ` ` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_128 ` | q_10000000 | 1 0 0 0 0 0 0 0 | ` `p` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_0 ` ` | q_00000000 | 0 0 0 0 0 0 0 0 | ` ` ` `( )` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_15` ` | q_00001111 | 0 0 0 0 1 1 1 1 | ` (p) ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_51` ` | q_00110011 | 0 0 1 1 0 0 1 1 | ` ` ` `(q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_85` ` | q_01010101 | 0 1 0 1 0 1 0 1 | ` ` ` ` ` ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_63` ` | q_00111111 | 0 0 1 1 1 1 1 1 | ` (p` ` q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_95` ` | q_01011111 | 0 1 0 1 1 1 1 1 | ` (p` ` ` ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_119 ` | q_01110111 | 0 1 1 1 0 1 1 1 | ` ` ` `(q ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_127 ` | q_01111111 | 0 1 1 1 1 1 1 1 | ` (p` ` q ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 4. Singular Propositions and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_1 ` ` | q_00000001 | 0 0 0 0 0 0 0 1 | ` (p) `(q)` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_2 ` ` | q_00000010 | 0 0 0 0 0 0 1 0 | ` (p) `(q)` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_4 ` ` | q_00000100 | 0 0 0 0 0 1 0 0 | ` (p) ` q ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_8 ` ` | q_00001000 | 0 0 0 0 1 0 0 0 | ` (p) ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_16` ` | q_00010000 | 0 0 0 1 0 0 0 0 | ` `p` `(q)` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_32` ` | q_00100000 | 0 0 1 0 0 0 0 0 | ` `p` `(q)` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_64` ` | q_01000000 | 0 1 0 0 0 0 0 0 | ` `p` ` q ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_128 ` | q_10000000 | 1 0 0 0 0 0 0 0 | ` `p` ` q ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_254 ` | q_11111110 | 1 1 1 1 1 1 1 0 | `((p) `(q)` `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_253 ` | q_11111101 | 1 1 1 1 1 1 0 1 | `((p) `(q)` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_251 ` | q_11111011 | 1 1 1 1 1 0 1 1 | `((p) ` q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_247 ` | q_11110111 | 1 1 1 1 0 1 1 1 | `((p) ` q ` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_239 ` | q_11101111 | 1 1 1 0 1 1 1 1 | `( p` `(q)` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_223 ` | q_11011111 | 1 1 0 1 1 1 1 1 | `( p` `(q)` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_191 ` | q_10111111 | 1 0 1 1 1 1 1 1 | `( p` ` q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_127 ` | q_01111111 | 0 1 1 1 1 1 1 1 | `( p` ` q ` `r )` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 5. Variations on a Theme of Implication
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_207 ` | q_11001111 | 1 1 0 0 1 1 1 1 | ` (p ` (q)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_175 ` | q_10101111 | 1 0 1 0 1 1 1 1 | ` (p` ` ` ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_187 ` | q_10111011 | 1 0 1 1 1 0 1 1 | ` ` ` `(q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_243 ` | q_11110011 | 1 1 1 1 0 0 1 1 | `((p) ` q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_245 ` | q_11110101 | 1 1 1 1 0 1 0 1 | `((p) ` ` ` `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_221 ` | q_11011101 | 1 1 0 1 1 1 0 1 | ` ` ` ((q) ` r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_48` ` | q_00110000 | 0 0 1 1 0 0 0 0 | ` `p` `(q)` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_80` ` | q_01010000 | 0 1 0 1 0 0 0 0 | ` `p` ` ` ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_68` ` | q_01000100 | 0 1 0 0 0 1 0 0 | ` ` ` ` q ` (r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_12` ` | q_00001100 | 0 0 0 0 1 1 0 0 | ` (p) ` q ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_10` ` | q_00001010 | 0 0 0 0 1 0 1 0 | ` (p) ` ` ` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_34` ` | q_00100010 | 0 0 1 0 0 0 1 0 | ` ` ` `(q)` `r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 6. More Variations on a Theme of Implication
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_176 ` | q_10110000 | 1 0 1 1 0 0 0 0 | ` `p` `(q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_208 ` | q_11010000 | 1 1 0 1 0 0 0 0 | ` `p` `(r ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_11` ` | q_00001011 | 0 0 0 0 1 0 1 1 | ` (p) `(q ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_13` ` | q_00001101 | 0 0 0 0 1 1 0 1 | ` (p) `(r ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_140 ` | q_10001100 | 1 0 0 0 1 1 0 0 | ` `q` `(p ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_196 ` | q_11000100 | 1 1 0 0 0 1 0 0 | ` `q` `(r ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_35` ` | q_00100011 | 0 0 1 0 0 0 1 1 | ` (q) `(p ` (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_49` ` | q_00110001 | 0 0 1 1 0 0 0 1 | ` (q) `(r ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_138 ` | q_10001010 | 1 0 0 0 1 0 1 0 | ` `r` `(p ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_162 ` | q_10100010 | 1 0 1 0 0 0 1 0 | ` `r` `(q ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_69` ` | q_01000101 | 0 1 0 0 0 1 0 1 | ` (r) `(p ` (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_81` ` | q_01010001 | 0 1 0 1 0 0 0 1 | ` (r) `(q ` (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_79` ` | q_01001111 | 0 1 0 0 1 1 1 1 | `( p` `(q ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_47` ` | q_00101111 | 0 0 1 0 1 1 1 1 | `( p` `(r ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_244 ` | q_11110100 | 1 1 1 1 0 1 0 0 | `((p) `(q ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_242 ` | q_11110010 | 1 1 1 1 0 0 1 0 | `((p) `(r ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_115 ` | q_01110011 | 0 1 1 1 0 0 1 1 | `( q` `(p ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_59` ` | q_00111011 | 0 0 1 1 1 0 1 1 | `( q` `(r ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_220 ` | q_11011100 | 1 1 0 1 1 1 0 0 | `((q) `(p ` (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_206 ` | q_11001110 | 1 1 0 0 1 1 1 0 | `((q) `(r ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_117 ` | q_01110101 | 0 1 1 1 0 1 0 1 | `( r` `(p ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_93` ` | q_01011101 | 0 1 0 1 1 1 0 1 | `( r` `(q ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_186 ` | q_10111010 | 1 0 1 1 1 0 1 0 | `((r) `(p ` (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_174 ` | q_10101110 | 1 0 1 0 1 1 1 0 | `((r) `(q ` (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 7. Conjunctive Implications and Their Complements
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_139 ` | q_10001011 | 1 0 0 0 1 0 1 1 | ` (p (q))(q (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_141 ` | q_10001101 | 1 0 0 0 1 1 0 1 | ` (p (r))(r (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_177 ` | q_10110001 | 1 0 1 1 0 0 0 1 | ` (q (r))(r (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_163 ` | q_10100011 | 1 0 1 0 0 0 1 1 | ` (q (p))(p (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_197 ` | q_11000101 | 1 1 0 0 0 1 0 1 | ` (r (p))(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_209 ` | q_11010001 | 1 1 0 1 0 0 0 1 | ` (r (q))(q (p))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_116 ` | q_01110100 | 0 1 1 1 0 1 0 0 | `((p (q))(q (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_114 ` | q_01110010 | 0 1 1 1 0 0 1 0 | `((p (r))(r (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_78` ` | q_01001110 | 0 1 0 0 1 1 1 0 | `((q (r))(r (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_92` ` | q_01011100 | 0 1 0 1 1 1 0 0 | `((q (p))(p (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_58` ` | q_00111010 | 0 0 1 1 1 0 1 0 | `((r (p))(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_46` ` | q_00101110 | 0 0 1 0 1 1 1 0 | `((r (q))(q (p))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 8. More Variations on Difference and Equality
o---------o------------o-----------------o-------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_96` ` | q_01100000 | 0 1 1 0 0 0 0 0 | ` `p ` (q , `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_72` ` | q_01001000 | 0 1 0 0 1 0 0 0 | ` `q` `(p , `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_40` ` | q_00101000 | 0 0 1 0 1 0 0 0 | ` `r` `(p , `q) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_144 ` | q_10010000 | 1 0 0 1 0 0 0 0 | ` `p` ((q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_132 ` | q_10000100 | 1 0 0 0 0 1 0 0 | ` `q` ((p , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_130 ` | q_10000010 | 1 0 0 0 0 0 1 0 | ` `r` ((p , `q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_6 ` ` | q_00000110 | 0 0 0 0 0 1 1 0 | ` (p) `(q , `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_18` ` | q_00010010 | 0 0 0 1 0 0 1 0 | ` (q) `(p , `r) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_20` ` | q_00010100 | 0 0 0 1 0 1 0 0 | ` (r) `(p , `q) ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_9 ` ` | q_00001001 | 0 0 0 0 1 0 0 1 | ` (p) ((q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_33` ` | q_00100001 | 0 0 1 0 0 0 0 1 | ` (q) ((p , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_65` ` | q_01000001 | 0 1 0 0 0 0 0 1 | ` (r) ((p , `q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o=========o============o=================o===================o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_159 ` | q_10011111 | 1 0 0 1 1 1 1 1 | ` (p` `(q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_183 ` | q_10110111 | 1 0 1 1 0 1 1 1 | ` (q` `(p , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_215 ` | q_11010111 | 1 1 0 1 0 1 1 1 | ` (r` `(p , `q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_111 ` | q_01101111 | 0 1 1 0 1 1 1 1 | ` (p` ((q , `r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_123 ` | q_01111011 | 0 1 1 1 1 0 1 1 | ` (q` ((p , `r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_125 ` | q_01111101 | 0 1 1 1 1 1 0 1 | ` (r` ((p , `q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_249 ` | q_11111001 | 1 1 1 1 1 0 0 1 | `((p) `(q , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_237 ` | q_11101101 | 1 1 1 0 1 1 0 1 | `((q) `(p , `r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_235 ` | q_11101011 | 1 1 1 0 1 0 1 1 | `((r) `(p , `q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_246 ` | q_11110110 | 1 1 1 1 0 1 1 0 | `((p) ((q , `r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_222 ` | q_11011110 | 1 1 0 1 1 1 1 0 | `((q) ((p , `r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| q_190 ` | q_10111110 | 1 0 1 1 1 1 1 0 | `((r) ((p , `q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o-------------------o
Table 9. Conjunctive Differences and Equalities
o---------o------------o-----------------o--------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_24` ` | q_00011000 | 0 0 0 1 1 0 0 0 | ` (p, q)` (p, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_36` ` | q_00100100 | 0 0 1 0 0 1 0 0 | ` (p, q)` (q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_66` ` | q_01000010 | 0 1 0 0 0 0 1 0 | ` (p, r)` (q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_129 ` | q_10000001 | 1 0 0 0 0 0 0 1 | `((p, q))((q, r)) `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_231 ` | q_11100111 | 1 1 1 0 0 1 1 1 | ( (p, q)` (p, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_219 ` | q_11011011 | 1 1 0 1 1 0 1 1 | ( (p, q)` (q, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_189 ` | q_10111101 | 1 0 1 1 1 1 0 1 | ( (p, r)` (q, r) ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
| q_126 ` | q_01111110 | 0 1 1 1 1 1 1 0 | (((p, q))((q, r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o--------------------o
Table 10. Thematic Extensions: [q, r] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_15` ` | q_00001111 | 0 0 0 0 1 1 1 1 | ((p , ` `( )` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_30` ` | q_00011110 | 0 0 0 1 1 1 1 0 | ((p , `(q) (r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_45` ` | q_00101101 | 0 0 1 0 1 1 0 1 | ((p , `(q)` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_60` ` | q_00111100 | 0 0 1 1 1 1 0 0 | ((p , `(q)` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_75` ` | q_01001011 | 0 1 0 0 1 0 1 1 | ((p , ` q `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_90` ` | q_01011010 | 0 1 0 1 1 0 1 0 | ((p , ` ` `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | ((p , `(q , r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_120 ` | q_01111000 | 0 1 1 1 1 0 0 0 | ((p , `(q ` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_135 ` | q_10000111 | 1 0 0 0 0 1 1 1 | ((p , ` q ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ((p , ((q , r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_165 ` | q_10100101 | 1 0 1 0 0 1 0 1 | ((p , ` ` ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_180 ` | q_10110100 | 1 0 1 1 0 1 0 0 | ((p , `(q `(r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_195 ` | q_11000011 | 1 1 0 0 0 0 1 1 | ((p , ` q ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_210 ` | q_11010010 | 1 1 0 1 0 0 1 0 | ((p , ((q)` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_225 ` | q_11100001 | 1 1 1 0 0 0 0 1 | ((p , ((q) (r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_240 ` | q_11110000 | 1 1 1 1 0 0 0 0 | ((p , ` ` ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
Table 11. Thematic Extensions: [p, r] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_51` ` | q_00110011 | 0 0 1 1 0 0 1 1 | ((q , ` `( )` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_54` ` | q_00110110 | 0 0 1 1 0 1 1 0 | ((q , `(p) (r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_57` ` | q_00111001 | 0 0 1 1 1 0 0 1 | ((q , `(p)` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_60` ` | q_00111100 | 0 0 1 1 1 1 0 0 | ((q , `(p)` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_99` ` | q_01100011 | 0 1 1 0 0 0 1 1 | ((q , ` p `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_102 ` | q_01100110 | 0 1 1 0 0 1 1 0 | ((q , ` ` `(r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | ((q , `(p , r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_108 ` | q_01101100 | 0 1 1 0 1 1 0 0 | ((q , `(p ` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_147 ` | q_10010011 | 1 0 0 1 0 0 1 1 | ((q , ` p ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ((q , ((p , r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_153 ` | q_10011001 | 1 0 0 1 1 0 0 1 | ((q , ` ` ` r ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_156 ` | q_10011100 | 1 0 0 1 1 1 0 0 | ((q , `(p `(r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_195 ` | q_11000011 | 1 1 0 0 0 0 1 1 | ((q , ` p ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_198 ` | q_11000110 | 1 1 0 0 0 1 1 0 | ((q , ((p)` r)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_201 ` | q_00000000 | 1 1 0 0 1 0 0 1 | ((q , ((p) (r)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_204 ` | q_00000000 | 1 1 0 0 1 1 0 0 | ((q , ` ` ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
Table 12. Thematic Extensions: [p, q] -> [p, q, r]
o---------o------------o-----------------o---------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_85` ` | q_01010101 | 0 1 0 1 0 1 0 1 | ((r , ` `( )` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_86` ` | q_01010110 | 0 1 0 1 0 1 1 0 | ((r , `(p) (q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_89` ` | q_01011001 | 0 1 0 1 1 0 0 1 | ((r , `(p)` q ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_90` ` | q_01011010 | 0 1 0 1 1 0 1 0 | ((r , `(p)` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_101 ` | q_01100101 | 0 1 1 0 0 1 0 1 | ((r , ` p `(q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_102 ` | q_01100110 | 0 1 1 0 0 1 1 0 | ((r , ` ` `(q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_105 ` | q_01101001 | 0 1 1 0 1 0 0 1 | ((r , `(p , q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_106 ` | q_01101010 | 0 1 1 0 1 0 1 0 | ((r , `(p ` q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_149 ` | q_10010101 | 1 0 0 1 0 1 0 1 | ((r , ` p ` q ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_150 ` | q_10010110 | 1 0 0 1 0 1 1 0 | ((r , ((p , q)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_153 ` | q_10011001 | 1 0 0 1 1 0 0 1 | ((r , ` ` ` q ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_154 ` | q_10011010 | 1 0 0 1 1 0 1 0 | ((r , `(p `(q)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_165 ` | q_10100101 | 1 0 1 0 0 1 0 1 | ((r , ` p ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_166 ` | q_10100110 | 1 0 1 0 0 1 1 0 | ((r , ((p)` q)` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_169 ` | q_10101001 | 1 0 1 0 1 0 0 1 | ((r , ((p) (q)) ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_170 ` | q_10101010 | 1 0 1 0 1 0 1 0 | ((r , ` ` ` ` ` ))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
Table 13. Differences & Equalities Conjoined with Implications
o---------o------------o-----------------o---------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_44` ` | q_00101100 | 0 0 1 0 1 1 0 0 | ` (p, q)` `(p (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_52` ` | q_00110100 | 0 0 1 1 0 1 0 0 | ` (p, q)` `((p) r)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_56` ` | q_00111000 | 0 0 1 1 1 0 0 0 | ` (p, q)` `(q (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_28` ` | q_00011100 | 0 0 0 1 1 1 0 0 | ` (p, q)` `((q) r)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_131 ` | q_10000011 | 1 0 0 0 0 0 1 1 | `((p, q)) `(p (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_193 ` | q_11000001 | 1 1 0 0 0 0 0 1 | `((p, q)) `((p) r)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_74` ` | q_01001010 | 0 1 0 0 1 0 1 0 | ` (p, r)` `(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_82` ` | q_01010010 | 0 1 0 1 0 0 1 0 | ` (p, r)` `((p) q)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_26` ` | q_00011010 | 0 0 0 1 1 0 1 0 | ` (p, r)` `(q (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_88` ` | q_01011000 | 0 1 0 1 1 0 0 0 | ` (p, r)` `((q) r)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_133 ` | q_10000101 | 1 0 0 0 0 1 0 1 | `((p, r)) `(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_161 ` | q_10100001 | 1 0 1 0 0 0 0 1 | `((p, r)) `((p) q)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_70` ` | q_01000110 | 0 1 0 0 0 1 1 0 | ` (q, r)` `(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_98` ` | q_01100010 | 0 1 1 0 0 0 1 0 | ` (q, r)` `((p) q)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_38` ` | q_00100110 | 0 0 1 0 0 1 1 0 | ` (q, r)` `(p (r))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_100 ` | q_01100100 | 0 1 1 0 0 1 0 0 | ` (q, r)` `((p) r)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_137 ` | q_10001001 | 1 0 0 0 1 0 0 1 | `((q, r)) `(p (q))` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_145 ` | q_10010001 | 1 0 0 1 0 0 0 1 | `((q, r)) `((p) q)` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_211 ` | q_11010011 | 1 1 0 1 0 0 1 1 | `((p, q)` `(p (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_203 ` | q_11001011 | 1 1 0 0 1 0 1 1 | `((p, q)` `((p) r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_199 ` | q_11000111 | 1 1 0 0 0 1 1 1 | `((p, q)` `(q (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_227 ` | q_11100011 | 1 1 1 0 0 0 1 1 | `((p, q)` `((q) r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_124 ` | q_01111100 | 0 1 1 1 1 1 0 0 | (((p, q)) `(p (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_62` ` | q_00111110 | 0 0 1 1 1 1 1 0 | (((p, q)) `((p) r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_181 ` | q_10110101 | 1 0 1 1 0 1 0 1 | `((p, r)` `(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_173 ` | q_10101101 | 1 0 1 0 1 1 0 1 | `((p, r)` `((p) q)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_229 ` | q_11100101 | 1 1 1 0 0 1 0 1 | `((p, r)` `(q (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_167 ` | q_10100111 | 1 0 1 0 0 1 1 1 | `((p, r)` `((q) r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_122 ` | q_01111010 | 0 1 1 1 1 0 1 0 | (((p, r)) `(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_94` ` | q_01011110 | 0 1 0 1 1 1 1 0 | (((p, r)) `((p) q)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_185 ` | q_10111001 | 1 0 1 1 1 0 0 1 | `((q, r)` `(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_157 ` | q_10011101 | 1 0 0 1 1 1 0 1 | `((q, r)` `((p) q)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_217 ` | q_11011001 | 1 1 0 1 1 0 0 1 | `((q, r)` `(p (r))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_155 ` | q_10011011 | 1 0 0 1 1 0 1 1 | `((q, r)` `((p) r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_118 ` | q_01110110 | 0 1 1 1 0 1 1 0 | (((q, r)) `(p (q))) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
| q_110 ` | q_01101110 | 0 1 1 0 1 1 1 0 | (((q, r)) `((p) q)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------o
Table 14. Proximal Propositions
o---------o------------o-----------------o---------------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_23` ` | q_00010111 | 0 0 0 1 0 1 1 1 | (p)(q)(r) + ((p),(q),(r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_43` ` | q_00101011 | 0 0 1 0 1 0 1 1 | (p)(q) r `+ ((p),(q), r ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_77` ` | q_01001101 | 0 1 0 0 1 1 0 1 | (p) q (r) + ((p), q ,(r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_142 ` | q_10001110 | 1 0 0 0 1 1 1 0 | (p) q `r `+ ((p), q , r ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_113 ` | q_01110001 | 0 1 1 1 0 0 0 1 | `p (q)(r) + ( p ,(q),(r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_178 ` | q_10110010 | 1 0 1 1 0 0 1 0 | `p (q) r `+ ( p ,(q), r ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_212 ` | q_11010100 | 1 1 0 1 0 1 0 0 | `p `q (r) + ( p , q ,(r)) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_232 ` | q_11101000 | 1 1 1 0 1 0 0 0 | `p `q `r `+ ( p , q , r ) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/```````````````\````````````````|
|```````````````/`````````````````\```````````````|
|``````````````/```````````````````\``````````````|
|`````````````/`````````````````````\`````````````|
|````````````o```````````````````````o````````````|
|````````````|```````````P```````````|````````````|
|````````````|```````````````````````|````````````|
|````````````|```````````````````````|````````````|
|````````o---o---------o```o---------o---o````````|
|```````/`````\ \`/ /`````\```````|
|``````/```````\ o /```````\``````|
|`````/`````````\ / \ /`````````\`````|
|````/```````````\ / \ /```````````\````|
|```o```````````` o---o-----o---o`````````````o```|
|```|`````````````````| |`````````````````|```|
|```|`````````````````| |`````````````````|```|
|```|``````` Q ```````| |``````` R ```````|```|
|```o`````````````````o o`````````````````o```|
|````\`````````````````\ /`````````````````/````|
|`````\`````````````````\ /`````````````````/`````|
|``````\`````````````````o`````````````````/``````|
|```````\```````````````/`\```````````````/```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_23. (p)(q)(r) + ((p),(q),(r))
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ /`````````/ \ |
| / \`````````o`````````/ \ |
| / \```````/`\```````/ \ |
| / \`````/```\`````/ \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_232. p q r + (p, q, r)
Table 15. Differences and Equalities between Simples and Boundaries
o---------o------------o-----------------o---------------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_152 ` | q_10011000 | 1 0 0 1 1 0 0 0 | `p + ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_164 ` | q_10100100 | 1 0 1 0 0 1 0 0 | `q + ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_194 ` | q_11000010 | 1 1 0 0 0 0 1 0 | `r + ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_230 ` | q_11100110 | 1 1 1 0 0 1 1 0 | `p + ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_218 ` | q_11011010 | 1 1 0 1 1 0 1 0 | `q + ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_188 ` | q_10111100 | 1 0 1 1 1 1 0 0 | `r + ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_103 ` | q_01100111 | 0 1 1 0 0 1 1 1 | `p = ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_91` ` | q_01011011 | 0 1 0 1 1 0 1 1 | `q = ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_61` ` | q_00111101 | 0 0 1 1 1 1 0 1 | `r = ( p ,` q , `r )` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_25` ` | q_00011001 | 0 0 0 1 1 0 0 1 | `p = ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_37` ` | q_00100101 | 0 0 1 0 0 1 0 1 | `q = ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_67` ` | q_01000011 | 0 1 0 0 0 0 1 1 | `r = ((p),`(q), (r))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \ \`/ / \ |
| / \ o / \ |
| / \ /`\ / \ |
| / \ /```\ / \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_152. p + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| /`````\ \ /`````````/ \ |
| /```````\ o`````````/ \ |
| /`````````\ /`\```````/ \ |
| /```````````\ /```\`````/ \ |
| o`````````````o---o-----o---o o |
| |`````````````````| | | |
| |`````````````````| | | |
| |``````` Q ```````| | R | |
| o`````````````````o o o |
| \`````````````````\ / / |
| \`````````````````\ / / |
| \`````````````````o / |
| \```````````````/ \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_164. q + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ / /`````\ |
| / \`````````o /```````\ |
| / \```````/`\ /`````````\ |
| / \`````/```\ /```````````\ |
| o o---o-----o---o`````````````o |
| | | |`````````````````| |
| | | |`````````````````| |
| | Q | |``````` R ```````| |
| o o o`````````````````o |
| \ \ /`````````````````/ |
| \ \ /`````````````````/ |
| \ o`````````````````/ |
| \ / \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_194. r + (p, q, r)
o-------------------------------------------------o
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
|`````````````````o-------------o`````````````````|
|````````````````/ \````````````````|
|```````````````/ \```````````````|
|``````````````/ \``````````````|
|`````````````/ \`````````````|
|````````````o o````````````|
|````````````| P |````````````|
|````````````| |````````````|
|````````````| |````````````|
|````````o---o---------o o---------o---o````````|
|```````/ \ \ / / \```````|
|``````/ \ o / \``````|
|`````/ \ /`\ / \`````|
|````/ \ /```\ / \````|
|```o o---o-----o---o o```|
|```| | | |```|
|```| | | |```|
|```| Q | | R |```|
|```o o o o```|
|````\ \ / /````|
|`````\ \ / /`````|
|``````\ o /``````|
|```````\ /`\ /```````|
|````````o-------------o```o-------------o````````|
|`````````````````````````````````````````````````|
|`````````````````````````````````````````````````|
o-------------------------------------------------o
q_129. ((p, q))((q, r))
Table 16. Paisley Propositions
o---------o------------o-----------------o---------------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_216 ` | q_11011000 | 1 1 0 1 1 0 0 0 | ` (p, q)(p, r)` + `p q` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_184 ` | q_10111000 | 1 0 1 1 1 0 0 0 | ` (p, q)(p, r)` + `p r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_228 ` | q_11100100 | 1 1 1 0 0 1 0 0 | ` (p, q)(q, r)` + `p q` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_172 ` | q_10101100 | 1 0 1 0 1 1 0 0 | ` (p, q)(q, r)` + `q r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_226 ` | q_11100010 | 1 1 1 0 0 0 1 0 | ` (p, r)(q, r)` + `p r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_202 ` | q_11001010 | 1 1 0 0 1 0 1 0 | ` (p, r)(q, r)` + `q r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_39` ` | q_00100111 | 0 0 1 0 0 1 1 1 | ` (p, q)(p, r)` = `p q` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_71` ` | q_01000111 | 0 1 0 0 0 1 1 1 | ` (p, q)(p, r)` = `p r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_27` ` | q_00011011 | 0 0 0 1 1 0 1 1 | ` (p, q)(q, r)` = `p q` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_83` ` | q_01010011 | 0 1 0 1 0 0 1 1 | ` (p, q)(q, r)` = `q r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_29` ` | q_00011101 | 0 0 0 1 1 1 0 1 | ` (p, r)(q, r)` = `p r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_53` ` | q_00110101 | 0 0 1 1 0 1 0 1 | ` (p, r)(q, r)` = `q r` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
Table 17. Paisley Propositions
o---------o------------o-----------------o------------------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o------------------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o------------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_216 ` | q_11011000 | 1 1 0 1 1 0 0 0 | ` p + pq + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_184 ` | q_10111000 | 1 0 1 1 1 0 0 0 | ` p + pr + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_228 ` | q_11100100 | 1 1 1 0 0 1 0 0 | ` q + pq + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_172 ` | q_10101100 | 1 0 1 0 1 1 0 0 | ` q + qr + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_226 ` | q_11100010 | 1 1 1 0 0 0 1 0 | ` r + pr + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_202 ` | q_11001010 | 1 1 0 0 1 0 1 0 | ` r + qr + pqr + (p, q, r)` `|
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o------------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_39` ` | q_00100111 | 0 0 1 0 0 1 1 1 | 1 + p + pq + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_71` ` | q_01000111 | 0 1 0 0 0 1 1 1 | 1 + p + pr + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_27` ` | q_00011011 | 0 0 0 1 1 0 1 1 | 1 + q + pq + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_83` ` | q_01010011 | 0 1 0 1 0 0 1 1 | 1 + q + qr + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_29` ` | q_00011101 | 0 0 0 1 1 1 0 1 | 1 + r + pr + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| q_53` ` | q_00110101 | 0 0 1 1 0 1 0 1 | 1 + r + qr + pqr + (p, q, r) |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
o---------o------------o-----------------o------------------------------o
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-------------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/%%%%%%%%%%%%%%%\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/%%%%%%%%%%%%%%%%%%%\` ` ` ` ` ` ` |
| ` ` ` ` ` ` /%%%%%%%%%%%%%%%%%%%%%\ ` ` ` ` ` ` |
| ` ` ` ` ` `o%%%%%%%%%%%%%%%%%%%%%%%o` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%% P %%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` ` ` `|%%%%%%%%%%%%%%%%%%%%%%%|` ` ` ` ` ` |
| ` ` ` `o---o---------o%%%o---------o---o` ` ` ` |
| ` ` ` / ` ` \%%%%%%%%%\%/ ` ` ` ` / ` ` \ ` ` ` |
| ` ` `/` ` ` `\%%%%%%%%%o` ` ` ` `/` ` ` `\` ` ` |
| ` ` / ` ` ` ` \%%%%%%%/%\ ` ` ` / ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` `\%%%%%/%%%\` ` `/` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` o---o-----o---o ` ` ` ` ` ` o ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` ` ` ` ` ` |%%%%%| ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` `Q` ` ` ` |%%%%%| ` ` ` `R` ` ` ` | ` |
| ` o ` ` ` ` ` ` ` ` o%%%%%o ` ` ` ` ` ` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\%%%/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \%/ ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` /`\ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
q_216. p + p q + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \ \`/ / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_24. (p, q)(p, r)
q_24. p + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \`````````\`/ / \ |
| / \`````````o / \ |
| / \```````/`\ / \ |
| / \`````/```\ / \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_216. (p, q)(p, r) + p q
q_216. p + p q + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |`````````` P ``````````| |
| |```````````````````````| |
| |```````````````````````| |
| o---o---------o```o---------o---o |
| / \ \`/`````````/ \ |
| / \ o`````````/ \ |
| / \ /`\```````/ \ |
| / \ /```\`````/ \ |
| o o---o-----o---o o |
| | |`````| | |
| | |`````| | |
| | Q |`````| R | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_184. (p, q)(p, r) + p r
q_184. p + p r + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| /`````\ \ /`````````/ \ |
| /```````\ o`````````/ \ |
| /`````````\ / \```````/ \ |
| /```````````\ / \`````/ \ |
| o`````````````o---o-----o---o o |
| |`````````````````| | | |
| |`````````````````| | | |
| |``````` Q ```````| | R | |
| o`````````````````o o o |
| \`````````````````\ / / |
| \`````````````````\ / / |
| \`````````````````o / |
| \```````````````/ \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_36. (p, q)(q, r)
q_36. q + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| /`````\`````````\ /`````````/ \ |
| /```````\`````````o`````````/ \ |
| /`````````\```````/`\```````/ \ |
| /```````````\`````/```\`````/ \ |
| o`````````````o---o-----o---o o |
| |`````````````````| | | |
| |`````````````````| | | |
| |``````` Q ```````| | R | |
| o`````````````````o o o |
| \`````````````````\ / / |
| \`````````````````\ / / |
| \`````````````````o / |
| \```````````````/ \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_228. (p, q)(q, r) + p q
q_228. q + p q + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| /`````\ \ /`````````/ \ |
| /```````\ o`````````/ \ |
| /`````````\ /`\```````/ \ |
| /```````````\ /```\`````/ \ |
| o`````````````o---o-----o---o o |
| |`````````````````|`````| | |
| |`````````````````|`````| | |
| |``````` Q ```````|`````| R | |
| o`````````````````o`````o o |
| \`````````````````\```/ / |
| \`````````````````\`/ / |
| \`````````````````o / |
| \```````````````/ \ / |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_172. (p, q)(q, r) + q r
q_172. q + q r + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ / /`````\ |
| / \`````````o /```````\ |
| / \```````/ \ /`````````\ |
| / \`````/ \ /```````````\ |
| o o---o-----o---o`````````````o |
| | | |`````````````````| |
| | | |`````````````````| |
| | Q | |``````` R ```````| |
| o o o`````````````````o |
| \ \ /`````````````````/ |
| \ \ /`````````````````/ |
| \ o`````````````````/ |
| \ / \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_66. (p, r)(q, r)
q_66. r + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ /`````````/`````\ |
| / \`````````o`````````/```````\ |
| / \```````/`\```````/`````````\ |
| / \`````/```\`````/```````````\ |
| o o---o-----o---o`````````````o |
| | | |`````````````````| |
| | | |`````````````````| |
| | Q | |``````` R ```````| |
| o o o`````````````````o |
| \ \ /`````````````````/ |
| \ \ /`````````````````/ |
| \ o`````````````````/ |
| \ / \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_226. (p, r)(q, r) + p r
q_266. r + p r + p q r + (p, q, r)
o-------------------------------------------------o
| |
| |
| o-------------o |
| / \ |
| / \ |
| / \ |
| / \ |
| o o |
| | P | |
| | | |
| | | |
| o---o---------o o---------o---o |
| / \`````````\ / /`````\ |
| / \`````````o /```````\ |
| / \```````/`\ /`````````\ |
| / \`````/```\ /```````````\ |
| o o---o-----o---o`````````````o |
| | |`````|`````````````````| |
| | |`````|`````````````````| |
| | Q |`````|``````` R ```````| |
| o o`````o`````````````````o |
| \ \```/`````````````````/ |
| \ \`/`````````````````/ |
| \ o`````````````````/ |
| \ / \```````````````/ |
| o-------------o o-------------o |
| |
| |
o-------------------------------------------------o
q_202. (p, r)(q, r) + q r
q_202. r + q r + p q r + (p, q, r)
Table 18. Desultory Junctions and Their Complements
o---------o------------o-----------------o---------------------------o
| L_1 ` ` | L_2 ` ` ` `| L_3 ` ` ` ` ` ` | L_4 ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| Decimal | Binary` ` `| Vector` ` ` ` ` | Cactus` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` `p : 1 1 1 1 0 0 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `q : 1 1 0 0 1 1 0 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` `r : 1 0 1 0 1 0 1 0 | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_224 ` | q_11100000 | 1 1 1 0 0 0 0 0 | ` ` ` `p` `((q)(r)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_200 ` | q_11001000 | 1 1 0 0 1 0 0 0 | ` ` ` `q` `((p)(r)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_168 ` | q_10101000 | 1 0 1 0 1 0 0 0 | ` ` ` `r` `((p)(q)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_14` ` | q_00001110 | 0 0 0 0 1 1 1 0 | ` ` ` (p) `((q)(r)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_50` ` | q_00110010 | 0 0 1 1 0 0 1 0 | ` ` ` (q) `((p)(r)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_84` ` | q_01010100 | 0 1 0 1 0 1 0 0 | ` ` ` (r) `((p)(q)) ` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_31` ` | q_00011111 | 0 0 0 1 1 1 1 1 | ` ` ` (p` `((q)(r)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_55` ` | q_00110111 | 0 0 1 1 0 1 1 1 | ` ` ` (q` `((p)(r)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_87` ` | q_01010111 | 0 1 0 1 0 1 1 1 | ` ` ` (r` `((p)(q)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_241 ` | q_11110001 | 1 1 1 1 0 0 0 1 | ` ` `((p) `((q)(r)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_205 ` | q_11001101 | 1 1 0 0 1 1 0 1 | ` ` `((q) `((p)(r)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
| q_171 ` | q_10101011 | 1 0 1 0 1 0 1 1 | ` ` `((r) `((p)(q)))` ` ` |
| ` ` ` ` | ` ` ` ` ` `| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------o------------o-----------------o---------------------------o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules -- Discussion
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Discussion Note 1
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Just by way of incidental kibitzing,
I notice that Rule 73 has the form of
a "genus and species" or "pie-chart"
proposition, where q is the genus
and p and r are the species.
The cactus expression and
cactus graph are as follows:
o-------------------o
| ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` |
| ` ` ` p ` r ` ` ` |
| ` ` ` o ` o ` ` ` |
| ` ` ` | q | ` ` ` |
| ` ` ` o-o-o ` ` ` |
| ` ` ` `\ /` ` ` ` |
| ` ` ` ` @ ` ` ` ` |
o-------------------o
| ` ((p), q ,(r)) ` |
o-------------------o
| ` ` ` q_73` ` ` ` |
o-------------------o
See the discussion in and
around Cactus Rules Note 5.
http://forum.wolframscience.com/showthread.php?s=&postid=830#post830
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Discussion Note 2
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
CR. Cactus Rules
Ontology List
01. http://suo.ieee.org/ontology/msg05486.html
02. http://suo.ieee.org/ontology/msg05487.html
03. http://suo.ieee.org/ontology/msg05488.html
04. http://suo.ieee.org/ontology/msg05489.html
05. http://suo.ieee.org/ontology/msg05490.html
06. http://suo.ieee.org/ontology/msg05491.html
07. http://suo.ieee.org/ontology/msg05492.html
08. http://suo.ieee.org/ontology/msg05493.html
09. http://suo.ieee.org/ontology/msg05494.html
10. http://suo.ieee.org/ontology/msg05495.html
11. http://suo.ieee.org/ontology/msg05496.html
12. http://suo.ieee.org/ontology/msg05498.html
13. http://suo.ieee.org/ontology/msg05499.html
14. http://suo.ieee.org/ontology/msg05500.html
15. http://suo.ieee.org/ontology/msg05501.html
16. http://suo.ieee.org/ontology/msg05502.html
17. http://suo.ieee.org/ontology/msg05503.html
18. http://suo.ieee.org/ontology/msg05507.html
19. http://suo.ieee.org/ontology/msg05508.html
20. http://suo.ieee.org/ontology/msg05509.html
21. http://suo.ieee.org/ontology/msg05510.html
22. http://suo.ieee.org/ontology/msg05511.html
23. http://suo.ieee.org/ontology/msg05512.html
24. http://suo.ieee.org/ontology/msg05518.html
25.
Inquiry List
01. http://stderr.org/pipermail/inquiry/2004-March/001265.html
02. http://stderr.org/pipermail/inquiry/2004-March/001266.html
03. http://stderr.org/pipermail/inquiry/2004-March/001267.html
04. http://stderr.org/pipermail/inquiry/2004-March/001268.html
05. http://stderr.org/pipermail/inquiry/2004-March/001269.html
06. http://stderr.org/pipermail/inquiry/2004-March/001270.html
07. http://stderr.org/pipermail/inquiry/2004-March/001271.html
08. http://stderr.org/pipermail/inquiry/2004-March/001272.html
09. http://stderr.org/pipermail/inquiry/2004-March/001273.html
10. http://stderr.org/pipermail/inquiry/2004-March/001274.html
11. http://stderr.org/pipermail/inquiry/2004-March/001275.html
12. http://stderr.org/pipermail/inquiry/2004-March/001277.html
13. http://stderr.org/pipermail/inquiry/2004-March/001278.html
14. http://stderr.org/pipermail/inquiry/2004-March/001279.html
15. http://stderr.org/pipermail/inquiry/2004-March/001280.html
16. http://stderr.org/pipermail/inquiry/2004-March/001281.html
17. http://stderr.org/pipermail/inquiry/2004-March/001290.html
18. http://stderr.org/pipermail/inquiry/2004-April/001305.html
19. http://stderr.org/pipermail/inquiry/2004-April/001306.html
20. http://stderr.org/pipermail/inquiry/2004-April/001307.html
21. http://stderr.org/pipermail/inquiry/2004-April/001308.html
22. http://stderr.org/pipermail/inquiry/2004-April/001312.html
23. http://stderr.org/pipermail/inquiry/2004-April/001314.html
24. http://stderr.org/pipermail/inquiry/2004-April/001322.html
25.
NKS Forum
00. http://forum.wolframscience.com/showthread.php?s=&threadid=256
01. http://forum.wolframscience.com/showthread.php?s=&postid=810#post810
02. http://forum.wolframscience.com/showthread.php?s=&postid=818#post818
03. http://forum.wolframscience.com/showthread.php?s=&postid=826#post826
04. http://forum.wolframscience.com/showthread.php?s=&postid=829#post829
05. http://forum.wolframscience.com/showthread.php?s=&postid=830#post830
06. http://forum.wolframscience.com/showthread.php?s=&postid=831#post831
07. http://forum.wolframscience.com/showthread.php?s=&postid=832#post832
08. http://forum.wolframscience.com/showthread.php?s=&postid=834#post834
09. http://forum.wolframscience.com/showthread.php?s=&postid=835#post835
10. http://forum.wolframscience.com/showthread.php?s=&postid=838#post838
11. http://forum.wolframscience.com/showthread.php?s=&postid=840#post840
12. http://forum.wolframscience.com/showthread.php?s=&postid=841#post841
13. http://forum.wolframscience.com/showthread.php?s=&postid=842#post842
14. http://forum.wolframscience.com/showthread.php?s=&postid=843#post843
15. http://forum.wolframscience.com/showthread.php?s=&postid=844#post844
16. http://forum.wolframscience.com/showthread.php?s=&postid=845#post845
17. http://forum.wolframscience.com/showthread.php?s=&postid=854#post854
18. http://forum.wolframscience.com/showthread.php?s=&postid=891#post891
19. http://forum.wolframscience.com/showthread.php?s=&postid=894#post894
20. http://forum.wolframscience.com/showthread.php?s=&postid=897#post897
21. http://forum.wolframscience.com/showthread.php?s=&postid=898#post898
22. http://forum.wolframscience.com/showthread.php?s=&postid=902#post902
23. http://forum.wolframscience.com/showthread.php?s=&postid=909#post909
24a. http://forum.wolframscience.com/showthread.php?s=&postid=927#post927
24b. http://forum.wolframscience.com/showthread.php?s=&postid=928#post928
24c. http://forum.wolframscience.com/showthread.php?s=&postid=929#post929
24d. http://forum.wolframscience.com/showthread.php?s=&postid=933#post933
24e. http://forum.wolframscience.com/showthread.php?s=&postid=934#post934
25.
CR. Cactus Rules -- Discussion
01. http://forum.wolframscience.com/showthread.php?s=&postid=901#post901
02.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
