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[tlaplus] Re: Defining a function in terms of variables in TLA



I currently want to define the Fourier Transform on a stochastic generated seismogram h(t).
That is, define the fourier mapping upon h(t).

Thus F(h(t)) = Integral h(t) exp^{-iwt} dt

And then show the proof of the linearity of the Fourier Transform

Thus

h(t) = a f(t) + b g(t) 
F(h(t)) = a F(f(t)) + b F(g(t))





On Friday, November 22, 2024 at 6:50:59 AM UTC+12 marta zhango wrote:
I would use the <1> <2> nomenclature, and will do a mathematical proof as in a book,
not as a program check of the proof.  In a way that would make it easier to check the proof
and construct the TLAPS code in future.   I am currently have to deal with constructs that might
not be defined at this time.

Would like to see how much progress I can do in providing mathematical proof sketches.
This is my main intention.

On Friday, November 22, 2024 at 6:33:19 AM UTC+12 marta zhango wrote:
Right.  TLA+ and TLAPM is not used here because the requirements of handling industrial
applications require it to be a little more complicated.

I want to use formulas and decomposing them into smaller pieces. This is done using simple
definitions, rather than the more complex constructs of coding languages.  To formalize mathematics
and make it easier to write a proof.

This was discussed in "How to Write a Proof" by  Leslie Lamport in 1993.

Are there also other possibilities, using a variable for instance?  How would one define the time
variable?

The seismogram f(t) is a stachastic generated function of time, not necessarily correlated
with an earthquake, but a more general recording of incessant earth motion.


On Thursday, November 21, 2024 at 3:31:02 PM UTC+12 Andrew Helwer wrote:
If you're using this for writing proofs over it, you'll probably make the function a constant then define some set of assumptions over it. Like:

CONSTANT f

ASSUME f \in [Real -> Real]

ASSUME IsContinuous(f) /\ IsDifferentiable(f)

Where you define IsContinuous and IsDifferentiable to mean some definition you can use in proofs. However, if you want to formally verify the proofs then TLAPM does not really support reasoning over Reals at this time. Some researchers developed an extension for it which you can read about here, but this work has not yet been integrated into the main development branch.

Andrew Helwer

On Wednesday, November 20, 2024 at 5:28:20 AM UTC-5 marta zhango wrote:
How can I define a function of time or some other variable, say of a seismogram
in TLA ?  But without specifying the mapping.

Using a VARIABLE, LET, or something else ?

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