Defining Cardinality with TLA

[marta zhango]

How can I define the cardinality of a set as the number of elements in a set S

using the notation |S| using TLA.

[marta zhango]

Like this for instance ?

Cardinality (S) = |S| == Count ( e \in S )

[marta zhango]

Have thought something like this too

Cardinality(set) == (+ e \in DOMAIN set: IF set[e] THEN 1 ELSE 0)

[Markus Kuppe]

https://github.com/tlaplus/tlaplus/blob/03c7bf4f35ccf1217ac2c53e5bf60d1220c8b0d8/tlatools/org.lamport.tlatools/src/tla2sany/StandardModules/FiniteSets.tla#L21-L27

[marta zhango]

Is the line

IN  CS[S]

part of the definition of cardinality ?

[Stephan Merz]

As Markus pointed out, cardinality of finite sets is defined in a standard module of TLA+. A collection of lemmas about finite sets and cardinality, useful for actually reasoning about these notions, is available at https://github.com/tlaplus/tlapm/blob/main/library/FiniteSetTheorems.tla (included in the distribution of TLAPS).

Stephan

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[Stephan Merz]

P.S.: Proofs of these theorems checked by TLAPS are given in https://github.com/tlaplus/tlapm/blob/main/library/FiniteSetTheorems_proofs.tla.

[marta zhango]

That is what I needed to see as I am new to all this, with no idea where to start.

[Lee]

Hi Marta,

If you want to know more about Cardinality please see this page by Hillel Wayne (https://learntla.com/core/operators.html?highlight=cardinality) . Since you said you're new, I can also recommend reading his entire tutorial page and blog posts in addition to his book (https://www.amazon.se/-/en/Hillel-Wayne/dp/1484238281). Other resources when you're done working these are:

- Specifying Systems by Leslie Lamport (which is only TLA+) 

- http://lamport.azurewebsites.net/tla/learning.html - great way to get into how to think about this

Regards

Lee

[marta zhango]

How can I declare or define a symbol to mean Cardinality such as |S|

Something like this maybe, to say that |S| means or in equivalent to Cardinality(S)

|S| == Cardinality (S)

Does one also use the keyword LET or something else ?

[Felipe Oliveira Carvalho]

That's not currently possible because the TLA+ doesn't support
user-defined syntax constructs this advanced.

--
Felipe

> To view this discussion on the web visit https://groups.google.com/d/msgid/tlaplus/a98df958-c39e-4597-bcab-f08e8b5040e9n%40googlegroups.com.

[Hillel Wayne]

You can define S^#, though!

S^# == Cardinality(S)

H

[marta zhango]

Ok, So I can define.  But just not with | 

Is that right ?