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

My version proves this "BY OBVIOUS", but it may be because I'm using a pre-release version. "BY Z3" may work for the official release of TLAPS, assuming that you have Z3 installed. The proof (and perhaps the spec) would become easier if C and D were just sets of natural numbers instead of one-field records.

Stephan

> On 18 Mar 2019, at 12:29, haroldas.giedra@xxxxxxxxx wrote:
>
>
>
> The example I gave came from the problem I want to solve. I want to prove such lemma:
>
> VARIABLE C, D
>
> LEMMA lemma5 ==
>      C \subseteq [s : Nat]
>      /\
>      D = {[s|->x.s] : x \in C}
>      =>
>      C = {[s|->x.s] : x \in D}
>
> Is it possible to prove such lemma in TLA Proof System ?
>
> Harold
>
> On Monday, 18 March 2019 10:07:54 UTC+2, Stephan Merz  wrote:
>> 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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