In fact, even for arbitrary functions with both positive and negative values,
if the partial sum ever reached +/-Inf, further adding finite values on the
other side cannot “pull” the result back to “normal”. For instance, suppose at
some moment the partial sum is +inf, adding any normal reals will not change
it, it’s then not allowed to add -Inf on it, because under the textbook
definition of `+`, PosInf + NegInf is unspecified. Thus, in some sense the
suminf of possible limiting values has a very different behavior with the real
version.
Chun
Inviato da iPad
> Il giorno 20 set 2019, alle ore 01:33, Chun Tian (binghe)
> <binghe.l...@gmail.com> ha scritto:
>
> Hi,
>
> The extreal version of `suminf` is only “correct” and equivalent with the
> real version when the involved function is positive - the partial sum is
> mono-increasing. Its relatively simple definition serves the current need (in
> measure and probability theories). Fully mimicking the real version requires
> a (full) porting of seqTheory (and limTheory, netsTheory, eventually
> metricTheory) to extreals, which is an almost impossible task in my opinion.
>
> Besides, I think it’s also not needed to do so, because if the sum limit
> never reaches +Inf, one can just reduce the work back to the ‘suminf’ of
> reals. On the hand, If the partial sum ever reached +Inf, assuming it’s
> monotonic, then the limit value must be also +Inf, it’s reduced to a finite
> sum.
>
> Chun
>
>> Il giorno 20 set 2019, alle ore 01:00, Norrish, Michael (Data61, Acton)
>> <michael.norr...@data61.csiro.au> ha scritto:
>>
>> The definition of suminf over the reals (in seqTheory) is completely
>> different; is it clear that the extreal version can't mimic the original?
>>
>> Michael
>>
>> On 18/9/19, 06:18, "Chun Tian (binghe)" <binghe.l...@gmail.com> wrote:
>>
>> Hi,
>>
>> I occasionally found that HOL's current definition of ext_suminf
>> (extrealTheory) has a bug:
>>
>> [ext_suminf_def] Definition
>> ⊢ ∀f. suminf f = sup (IMAGE (λn. ∑ f (count n)) 𝕌(:num))
>>
>> The purpose of `suminf f` is to calculate an infinite sum: f(0) + f(1) +
>> .... To make the resulting summation "meaningful", all lemmas about
>> ext_suminf assume that (!n. 0 <= f n) (f is positive). This also indeed how
>> it's used in all applications. For instance, one lemma says, if f is
>> positive, so is `suminf f`:
>>
>> [ext_suminf_pos] Theorem
>> ⊢ ∀f. (∀n. 0 ≤ f n) ⇒ 0 ≤ suminf f
>>
>> However, I found that the above lemma can be proved even without knowing
>> anything of `f`, see the following proofs:
>>
>> Theorem ext_suminf_pos_bug :
>> !f. 0 <= ext_suminf f
>> Proof
>> RW_TAC std_ss [ext_suminf_def]
>>>> MATCH_MP_TAC le_sup_imp'
>>>> REWRITE_TAC [IN_IMAGE, IN_UNIV]
>>>> Q.EXISTS_TAC `0` >> BETA_TAC
>>>> REWRITE_TAC [COUNT_ZERO, EXTREAL_SUM_IMAGE_EMPTY]
>> QED
>>
>> where [le_sup_imp'] is a version of [le_sup_imp] before applying IN_APP:
>>
>> [le_sup_imp'] Theorem
>>
>> ⊢ ∀p x. x ∈ p ⇒ x ≤ sup p
>>
>> For the reason, ext_suminf is actually calculating the sup of the
>> following values:
>>
>> 0
>> f(0)
>> f(0) + f(1)
>> f(0) + f(1) + f(2)
>> ...
>>
>> The first line (0) comes from `count 0 = {}` where `0 IN univ(:num)`, then
>> SUM_IMAGE f {} = 0.
>>
>> Now consider f = (\n. -1), the above list of values are: 0, -1, -2, -3
>> .... But since the sup of a set containing 0 is at least 0, the `suminf f`
>> in this case will be 0 (instead of the correct value -1). This is why `0 <=
>> suminf f` holds for whatever f.
>>
>> I believe Isabelle's suminf (in Extended_Real.thy) has the same problem
>> (but I can't find its definition, correct me if I'm wrong here), i.e. the
>> following theorem holds even without the "assumes":
>>
>> lemma suminf_0_le:
>> fixes f :: "nat ⇒ ereal"
>> assumes "⋀n. 0 ≤ f n"
>> shows "0 ≤ (∑n. f n)"
>> using suminf_upper[of f 0, OF assms]
>> by simp
>>
>> The solution is quite obvious. I'm going to fix HOL's definition of
>> ext_suminf with the following one:
>>
>> val ext_suminf_def = Define
>> `ext_suminf f = sup (IMAGE (\n. SIGMA f (count (SUC n))) UNIV)`;
>>
>> That is, using `SUC n` intead of `n` to eliminate the fake base case (0).
>> I believe this change only causes minor incompatibilities.
>>
>> Any comment?
>>
>> Regards,
>>
>> Chun Tian
>>
>>
>>
>>
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