# Re: [tlaplus] TLA Proof System - the syntax has impact on provability ?

Hello,

theorem proving is indeed driven by syntax and unfortunately, TLAPS's automatic backends may fail to prove lemmas that are true, even if their truth is obvious to a human. Recognizing the elements of the set expression [s:{0}] requires reasoning about the extension of the set, whereas the set { [s |-> 0] } is just a singleton whose sole element is explicitly given.

I am happy to report that your lemma1 will (also) be proved by SMT in the upcoming release, as well as

LEMMA lemma3 ==
ASSUME NEW C, NEW D,
C = {[s|->0]},
D = {[s|->x.s] : x \in C}
PROVE  C = D
OBVIOUS

The proof of lemma2 currently seems to require a somewhat roundabout proof:

LEMMA lemma2 ==
ASSUME NEW C, NEW D,
C = [s : {0}],
D = {[s|->x.s] : x \in C}
PROVE  C = {[s|->x.s] : x \in D}
<1>1. C = {[s |-> 0]}  OBVIOUS
<1>2. QED  BY ONLY <1>1, D = {[s|->x.s] : x \in C}

where step <1>1 is proved by Isabelle and step <1>2 by either SMT or Isabelle (but both backends fail if the assumption about C is not hidden). Thanks for reporting this, we'll try to add a suitable rewrite rule for automating the handling of similar expressions.

Stephan

> On 17 Mar 2019, at 20:50, haroldas.giedra@xxxxxxxxx wrote:
>
>
> Hello,
>
> I have example of two lemmas which have the same meaning but slightly differs in the syntax.
>
> LEMMA lemma1 == C = {[s|->0]}
>      /\
>      D = {[s|->x.s] : x \in C}
>      =>
>      C = {[s|->x.s] : x \in D}
> OBVIOUS
>
> LEMMA lemma2 == C = [s : {0}]
>      /\
>      D = {[s|->x.s] : x \in C}
>      =>
>      C = {[s|->x.s] : x \in D}
> OBVIOUS
>
> lemma1 has been proved by "TLA - Isabelle", but "TLA - SMT", "TLA - Zenon", "TLA - Isabelle" fail to prove lemma2.
>
> I don't understand why TLA proof system fails to prove lemma2, which is different from lemma1 only in the syntax of TLA records. Meaning:
>
> C = {[s|->0]}
> and
> C = [s : {0}]
>
> which are the same for me.
>
> Harold
>
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