Dear Thomas,
Thank you so much for your elaborated reply.
I will go through Section 5.5.1.1 that you referred to
Thanks again!
Yassmeen
On Fri, Oct 25, 2019 at 1:22 PM Thomas Tuerk <tho...@tuerk-brechen.de>
wrote:
> Dear Yassmeen,
>
> set comprehensions in HOL are a bit cryptic. They are explained in Section
> 5.5.1.1 of the Description manual. In short, something like {t | p} is
> syntactic sugar that is turned by the parser into GSPEC(\(x1 ,...,xn).
> (t,p)). The trick is figuring out, which variables x1, ... xn to use. There
> are rules for that, which for non-trivial cases gives sometimes unexpected
> results. See excerpt from Sec. 5.5.1.1 below. For making it unambiguous,
> there is also the form {t | (x1, ..., xn) | p} where the variables are
> given explicitly. For complicated sets, I would always use the explicit
> form and I would not be surprised if L is free in one of the forms below
> and free in the other. In my opinion the syntax {t | p} should only be used
> in trivial cases, because it can easily lead to human readers
> misunderstanding the meaning otherwise. I for one are confused by
> complicated examples like the comprehensions you use. I need essentially to
> run the parser and look at the result before knowing what the meaning is.
> The following trace might be useful to show more information when
> pretty-printing the parsed comprehensions:
>
> set_trace "pp_unambiguous_comprehensions" 1
>
>
> Excerpt from Section 5.5.1.1
>
> The normal interpretation of { t | p } is the set of all ts such that
> p. In HOL , such syntax parses to: GSPEC(\(x 1 ,...,x n ).(t,p)) where x 1
> , ... , x n are those
> free variables that occur in both t and p if both have at least one free
> variable. If t or p
> has no free variables, then x 1 , ... , x n are taken to be the free
> variables of the other term.
> If both terms have free variables, but there is no overlap, then an error
> results. The
> order in which the variables are listed in the variable structure of the
> paired abstraction
> is an unspecified function of the structure of t (it is approximately left
> to right).
>
> I hope this helps.
>
> Thomas
>
>
> On 25.10.19 18:05, Yassmeen Derhalli wrote:
>
> Hi,
> I have this proof goal, where I need to prove that two sets are equal.
>
> {{f i | i IN if a IN {b | b IN L} then s1 a else s2 a} |
> a |
> a IN if j IN L then s3 j else s4 j} =
> {{f i | i IN if a IN {b | b IN L} then s1 a else s2 a} |
> a IN if j IN L then s3 j else s4 j}
>
> I have two questions regarding this proof, first, what is the difference
> between the first format when we use "|a | a IN...." (the left side of the
> goal) and the second format "...|a IN..." (the right side of the goal)?
> Secondly, are these sets equal? because it seems that HOL treats set L in
> the right side of the goal as a dummy variable of the set.
> I need help in clarifying the difference between both formats and some
> clues about how to proceed with the proof if the sets are equal.
>
> Best Regards,
> Yassmeen Elderhalli
>
>
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