# Re: [SOCIAL NETWORK] Re: [tlaplus] Resources for learning TLA+ proofs

Dear Andrew,

I’ve been struggling since a while to get into
TLA+ proofs. There is some stuff out there
about temporal proofs, and these PTL rules.
Then there is the theorem library in the
TLA+ folders which is quite helpful.

Concerning the syntax: this is relatively
simple. Like with TLA+ itself. But because it is
so immensely expressive, the semantics is difficult.

I found these principles as a good
orientation:
1. you often have to show that an A is \in B.
The rest is easy for the prover backends.
I found this most important when doing
arithmetic proofs: you have to show for
each component of your formula that
it is in Nat or Int or Real.
2. you can split your proof into pieces and
leave the proof as „omitted“ to see what
components you need.
3. if you use BY-proofs, then you need to
„collect“ assumptions and preconditions.
You find these in the theorem library.
4. Proofs with recursive functions are extremely
hard.
5. you can look into the documentation of
the Isabelle hol prover. There are some
constructs you can copy and paste.
6. The HAVE and TAKE proofs are mainly
reformulations for the SUFFICES construct.
7. In any case, you have to be aware what facts
are visible at a certain proof step. The TLA+ toolbox
shows the expanded proof when proving, so
you can check what facts are taken.
8. Do not overload the prover with complex
constructs… better simplify and prove layer
by layer. The HIDE statement does a good job here.

Kind regards

Andreas

Am 29.03.2023 um 08:22 schrieb Andrew Helwer <andrew.helwer@xxxxxxxxx>:

﻿ Achtung, externe Mail!
Hah! Thanks Calvin. Actually I've been looking at my own tree-sitter grammar I wrote to figure out the proof syntax: https://github.com/tlaplus-community/tree-sitter-tlaplus/blob/main/grammar.js

Ironic, I can parse the proof language but I cannot write one myself (yet!)

Like most proof languages, the TLA+ proof language is sort of a mirror version of the TLA+ language itself. Each TLA+ language construct has its corresponding proof language element that is used to "destructure" that construct in proofs. For example here's a (probably not yet correct) mental model I'm building about how to handle every TLA+ language construct when it's in either the P or Q spot of a P => Q we are trying to prove:

 TLA+ Language Construct Antecedent or Consequent (P => Q)? Destructuring Element e1 /\ e2 Antecedent (P) Prove e1 => Q \/ e2 => Q e1 /\ e2 Consequent (Q) Prove P => e1 /\ P => e2 e1 \/ e2 Antecedent (P) Prove e1 => Q /\ e2 => Q e1 \/ e2 Consequent (Q) Prove P => e1 \/ P => e2 IF e1 THEN e2 ELSE e3 Antecedent (P) ??? IF e1 THEN e2 ELSE e3 Consequent (Q) Prove CASE e1 => e2 /\ CASE ~e1 => e3 \E e \in S : f(e) Antecedent (P) Prove (PICK e \in S : f(e)) => Q \E e \in S : f(e) Consequent (Q) Prove (WITNESS e \in S) => f(e) \A e \in S : f(e) Antecedent (P) Prove (PICK e \in S : f(e)) => Q \A e \in S : f(e) Consequent (Q) Prove (TAKE e \in S) => f(e)

Andrew

On Tue, Mar 28, 2023 at 10:49 AM Calvin Loncaric <c.a.loncaric@xxxxxxxxx> wrote:
When I was first learning TLAPS back in 2019 I had a lot of trouble understanding the syntax. There are lots of awe-inspiring examples, but no clear reference for the actual structure of a proof. In particular, I couldn't seem to figure out how to begin a new proof for something I cared about without copying the structure of a similar proof from somewhere else. And when I did manage to find a similar proof, I was never sure what kinds of changes to the proof steps were syntactically valid.

I wrote myself the kind of reference I wish I'd had:

Looking at it now I think it's a bit sparse, and I wouldn't recommend it as the first thing to look at when learning. Still, I hope it helps fill in gaps for you and others.

--
Calvin

On Fri, Mar 24, 2023 at 10:26 PM JosEdu Solsona <josedusolsona@xxxxxxxxx> wrote:
Some other resources i found useful are:
Specifying Systems is not really about proofs. (In fact, in the introduction we have: "However, this book is about specification; it says almost nothing about proofs." )
On Saturday, March 25, 2023 at 1:38:14 AM UTC-3 Daniel Craig wrote:
There is also Specifying Systems by Dr Lamport

On Mar 24, 2023, at 5:56 PM, Andrew Helwer <andrew...@xxxxxxxxx> wrote:

﻿
What resources exist? I know of:
• The TLA+ hyperbook (unfinished?)
• Documentation on the TLAPS INRIA website
• Specs with proofs in the tlaplus/examples repo
Any others? Those of you who have written TLA+ proofs, how did you learn?

Andrew

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