Nodes#
A deduction presents a logical argument, which consists of an inferential structure, plus mathematical content. The nodes of a Proofscape deduction are where the content is represented.
In a Proofscape diagram, the inferential structure of the argument is displayed by the arrows, while the content is displayed on the nodes.
Recalling that every deduction has the form:
preamble {
node_definitions
meson_script
}
it is time now to discuss the node_definitions
part.
Labels and languages#
The basic steps when defining a node are:
Defining a node:
choose its type,
give it a name, and
write its label(s).
For a first example, consider a representation of Theorem 1 from Hilbert’s Zahlbericht, where we define just one node:
1deduc Thm {
2
3 asrt C {
4
5 de = "
6 In jedem Körper $k$ gibt es eine Zahl $\theta$ derart, daß alle
7 anderen Zahlen des Körpers ganze rationale Funktionen von $\theta$
8 mit rationalen Koeffizienten sind.
9 "
10
11 en = "
12 In every number field $k$ there is a number $\theta$ such that all
13 other numbers of the field are polynomials in $\theta$ with rational
14 coefficients.
15 "
16
17 }
18
19 meson = "C"
20}
The general form of a node definition is:
NODETYPE NAME MODIFIERS {
ASSIGNMENTS
NODE_DEFINITIONS
}
In the one node defined in Listing 6, the NODETYPE
is asrt
(which stands for “assertion”), the NAME
is C
, and there are no
MODIFIERS
. There are two
ASSIGNMENTS
, which assign strings to the fields de
and en
.
This is where we define the text that will be rendered on the node as its
label. In general a node definition
may contain nested NODE_DEFINITIONS
(see for example Existential nodes),
but in this case we do not have any.
Since Hilbert’s Zahlbericht is in German, we have written the original
theorem statement in German under the de
field. Since English is the
lingua franca for presentday mathematical communication, we have
provided a translation under the en
field.
Fields and assignments#
In our discussion of Listing 6, we have referred to de
and
en
as “fields.” Indeed, anything defined with an =
sign within a node
or deduction definition is called a field.
The whole expression having a field on the left, an =
sign in the middle,
and a value on the right, is called an assignment.
In general, the righthand side or value of an assignment may be any valid Proofscapeflavored JSON. For a given field, however, the allowed forms for the value may be more restricted.
Current node fields#
Proofscape is a relatively new system, and is expected to evolve over time. At present, there is a small set of accepted node fields, which have arisen out of those applications that have so far been contemplated. As new applications emerge, new fields may be added, and existing ones may be taken away. For now, users should consider the set of node fields to be unstable.
The current fields for nodes are as follows.
Field 
Value 
Description 


string 
German language lable. See TeX labels. 

string 
English language lable. See TeX labels. 

string 
French language lable. See TeX labels. 

string 
Latin language lable. See TeX labels. 

string 
Russian language lable. See TeX labels. 

string 
Purely symbolic lable. See TeX labels. 

string 
See PDF labels. 

string 
Representation in the syntax of SymPy. See Other syntaxes. 

string 
Representation in the syntax of Lean. See Other syntaxes. 

list of strings or libpaths 
See Cf references. 
TeX labels#
TeX labels, as the name suggests, consist of words mixed with TeX math modes, as would be typed in any TeX (or LaTeX) document.
TeX labels are recorded in fields whose names
(such as de
, en
, fr
, la
, ru
)
reflect the natural language
being used; however, when a label happens to be purely symbolic, it should
instead be recorded under the sy
field, to indicate that it is suitable for
use with all natural languages.
Where to record TeX labels:
Use
sy
when the label is purely symbolic.Use an appropriate language code (
de
,en
, etc.) otherwise.
Manual formatting#
When a multiline string is used to define a TeX label, line breaks and indentation are honored, to allow manual formatting according to the following rules:
Manual formatting for TeX labels:
Leading and trailing whitespace is ignored.
The first nonempty line sets the base indentation level.
Subsequent lines are indented relative to the first nonempty line.
Linebreaks within TeX math mode are ignored.
Linebreaks outside of TeX math mode are preserved.
For example, a node defined as follows,
1asrt C {
2 en = "
3 Catalan's equation
4 $x^p  y^q = 1$
5 has no nontrivial solutions
6 with primes $p, q \geq 3$.
7 "
8}
will appear as in Fig. 1
Manual formatting should be employed to give each node a good aspect ratio: neither too flat and wide, nor too tall and thin.
Internal links#
In a TeX label for a node A
, you can create an internal link to any other
node B
with the syntax
[TEXT](DOTTED_PATH)
where TEXT
is what you want to appear as the link text, and the
DOTTED_PATH
is any valid reference to the node B
from the context
where node A
is found.
For example, the (incomplete) proof in Listing 9 uses an internal link to refer to an equation displayed in the theorem statement.
1deduc Thm {
2
3 supp S {
4 en = "Suppose [certain assumptions...]"
5 }
6
7 asrt C {
8 en = "
9 The equation
10 (17) $\quad x^a  y^b = p$
11 does not hold.
12 "
13 }
14
15 meson = "Suppose S. Then C."
16}
17
18deduc Pf of Thm.C {
19
20 supp S10 {
21 en = "
22 Suppose $x, y$ satisfied
23 equation [(17)](Thm.C).
24 "
25 }
26
27 # more node definitions...
28
29 meson = "
30 Suppose S10
31 !
32 Thm.C
33 "
34}
Graphically, an internal link will appear in a node label just like a link in
a web page. For the example of Listing 9, the conclusion
node Thm.C
from the theorem would appear as in Fig. 2, while the
supposition node Pf.S10
from the proof that references this equation would appear as
in Fig. 3.
Hovering the mouse over an internal link causes the referenced node to be highlighted if it is currently visible on screen; clicking the link puts the referenced node on screen if it is not already there.
PDF labels#
UNDER CONSTRUCTION…
Other syntaxes#
UNDER CONSTRUCTION…
Cf references#
Not every assignment in a node defines a label, and the purpose of the cf
field is instead to note “comparisons” or “cross references.”
The idea is simply to note similarities. It could be that a theorem in one work corresponds to a theorem in another development of the same subject. It could be that a step in one proof is similar to a step in another proof.
The format of the value of the cf field is expected to evolve in later versions of proofscape, but for now is very simple: It is a list of strings or libpaths, in which each entry gives the libpath of another node to which the given one should be compared. Strings are also acceptable in place of actual libpaths, in order to resolve potential circular import issues.
Example:
1from ..Thm5 import Pf as Pf5
2
3deduc Thm6 {
4
5 #...
6
7}
8
9deduc Pf of Thm6.C {
10
11 intr I10 {
12 en = "
13 Let $F(x, y)$ be the polynomial obtained by
14 substituting $x + y$ for $t$ in $f(t)$.
15 "
16
17 cf = [
18 Pf5.I30,
19 ]
20 }
21
22 # ...
23
24 meson = "Let I10. ..."
25}
In Listing 10 we can imagine that, in order to prove “Theorem 6”
of some development,
we introduce a polynomial \(F(x, y)\) based on a given one \(f(t)\), and that a
similar trick was used on node I30
in the proof of this work’s Theorem 5.
In order to note the similarity, we record the cf
field on lines 1719.
In future versions of Proofscape, the data format for the cf
field may evolve to allow more
information to be recorded for each comparison, such as:
keywords, indicating categories to which the comparison belongs, such as “corresponding theorem” or “similar technique”.
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