Let a binder bind i, in every binder there is (#976)

[Rafael-SOWNet]

Closes #976. Replaces #979, which I am closing unmerged — it patched the MathS entry points, and after your "(2) would probably be most mathematically close to how we denote in math" that is the wrong place. This is (2).

What was wrong

i is the imaginary unit and the lexer decides it — NUMBER: ... | 'i' — so it never reaches the rule that makes variables and cannot be a bound name anywhere in the language. Every binder handed one did something other than bind:

written	was	is
sum(i, i, 1, 10)	unevaluated	55
sum(2i, i, 1, 3)	6i	12
integral(i, i)	-1/2 + C	i ^ 2 / 2 + C
limit(i, i, 0)	unevaluated	0
derivative(i ^ 2, i)	0	2 * i
{ i : i > 0 }	NaN	{ i : i > 0 }
lambda(i, i + 1)	InvalidArgumentParseException	the lambda

integral(i, i) is the one that shows it was never merely missing: -1/2 + C is the symbol read as the variable in one place and as the constant in another, in one answer.

Where it is done

One type, Core/Binding.cs, holding the decision; the nodes ask it, not the ways in. CalculusOperator covers the five calculus operators and ConditionalSet the set builder, so the parser, MathS.Sum and new Summationf(...) all get the same reading and none of them has to remember to.

A lambda's parameter is typed Variable, so it is the one binder that cannot be handed the unit at all — no node can read what it cannot hold. That one is read in the grammar, and the parser is regenerated from the pinned antlr-4.13.1-complete.jar; the generated diff is exactly the nine lines of the changed action.

2i is one token, so a written coefficient on the bound name arrives as a single number with nothing in it to rename. Under a binder that names i it is the writer's 2 beside the writer's i — 2k there is a product — and that is what turns 6i into 12.

Your four questions

sum(sqrt(-1) * i, i, 1, 10) → 55i. The sum's index, times the imaginary unit. That is what SymPy gives, and it works here because sqrt(-1) is Powf(-1, 1/2) at parse time and not a complex literal, so it is not a written i and is not renamed.

How SymPy resolves it — measured, 1.14: it names the constant I, so lowercase i is free, and it refuses to bind the constant (Sum(I, (I, 1, 10)) is a ValueError, Lambda(I, I + 1) a BadSignatureError). It buys consistency by giving up the notation — nobody writes ∑_I.

integral(i, i) → i ^ 2 / 2 + C, which is what integral(k, k) gives with the name changed. Under this reading there is nothing special left about it.

Should i become a Variable? I built it to find out: three lines and a regenerated lexer. It works, and it costs 53 tests — 42 LaTeX juxtaposition, 5 SymPy export, the rest round-trips — all about how i is written. This PR costs 0. It is a real option for a major version and its price is now measured rather than guessed; I did not want to spend it here.

A correction, and a new issue

I told you on #979 that "e and pi need nothing — they already work, everywhere". That was measured on too few paths and it is wrong.

derivative(e ^ 2, e)    0          should be 2 * e
derivative(pi ^ 2, pi)  0          should be 2 * pi
{ e : e > 0 }           { e : True }
limit(e, e, 0).Evaled   2.71828…   .Simplify() gets it right, and gives 0

Variable.ConstantList is keyed by name, so a variable called e is 2.718… wherever it is read — a bound e and the constant e are the same object, and nothing downstream can tell them apart. That is the opposite of i's problem and it is why i was fixable at all: a Variable named i is distinct from Complex.ImaginaryOne. Filed with the measurements as #984; the tests here pin the four wrong answers so nobody re-derives them.

Two unrelated export defects turned up while checking that the new forms round-trip, both reproducing on master with ordinary names: #985.

Measured

• Suite 7353 passed, 0 failed, 14 skipped — 29 new tests.
• Corpus 116/119, 0 wrong, 0 error, 0 timeout — unchanged.
• Public surface unchanged; PublicApi.txt needed no edit.
• Cost: one type test on the way into a binder node, which fails for the Variable nearly every bound name is. Node construction over three runs each — master 7.66 / 8.02 / 8.10 ns, this branch 7.84 / 7.85 / 8.25 ns. That is the same number.
• Round trips, LaTeX and SymPy export all check out, and LaTeX even distinguishes them: \sum_{i=1}^{3}2 i for the index against \mathrm{i} for the unit.
• BREAKING-CHANGES.md records all seven changed answers, including Let i be the loop variable when it is named as one (#976) #981's, which was never recorded.