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*From*: Stephan Merz <stephan.merz@xxxxxxxxx>*Date*: Mon, 21 Nov 2022 17:00:35 +0100*References*: <57a329dc-6578-4235-ad56-a579e35c2234n@googlegroups.com> <1986E5A8-0341-431E-AF85-8443FFDAD787@gmail.com> <5ae6ca5e-e749-4373-afce-13ceea4655a1n@googlegroups.com>

Hi Andrew, you are right that the SUFFICES keyword is optional in the grammar. However, its presence changes the meaning of the step. Consider the following example of a step claiming that some predicate P is provable. <1>x. P <2>1. SUFFICES Q BY ... \* proof establishing ASSUME Q PROVE P <2>. QED BY ... \* proof establishing Q Step <2>1 reduces the proof of P to that of Q, i.e. its proof requires inferring P provided Q is provable. The current goal is now no longer P, but Q, and the QED step must prove that Q holds. On the other hand, consider the following: <1>x. P <2>1. Q BY ... \* proof establishing Q <2>. QED BY ... \* proof establishing P (possibly using Q) Here, step <2>1 just asserts Q as a fact, and the BY clause must allow it to be proved. The current goal is still P, and therefore the QED step must establish P. It may use the fact Q by invoking <2>1 in the BY clause for the QED step. So, while both proofs are syntactically legal, their logical meaning is different. At the end of the day, the same arguments must be provided in both proofs, but in a different order, and one or the other may feel more natural in a given context. In proof assistant jargon, the first approach is called "backward chaining", and the second one is called "forward chaining". The latter is the default in TLAPS. Also, this discussion is actually unrelated to the original question about WITNESS: that keyword is intended for instantiating existential quantification, and it doesn't make sense to write SUFFICES WITNESS e. Regards, Stephan
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**Follow-Ups**:**Re: [tlaplus] WITNESS BY***From:*Andrew Helwer

**References**:**[tlaplus] WITNESS BY***From:*Jaco van de Pol

**Re: [tlaplus] WITNESS BY***From:*Stephan Merz

**Re: [tlaplus] WITNESS BY***From:*Andrew Helwer

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