Fix text around least upper bound.

Fix text around least upper bound.

R=eernst@google.com

Committed: https://github.com/dart-lang/sdk/commit/94b23d3b125e9d246e07a2b43b61740759a0dace

[Lasse Reichstein Nielsen, 2017-04-05 09:25:40]

lrn@google.com changed reviewers:
+ eernst@google.com

[Lasse Reichstein Nielsen, 2017-04-05 09:38:31]

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec.tex
File docs/language/dartLangSpec.tex (right):

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7814: but it is not associative.
Only real change is this line and adding { and } around I and J in the union
expressions above. The rest is just reflowing and removing spurious whitespace.

[eernst, 2017-04-05 11:47:30]

LGTM

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec.tex
File docs/language/dartLangSpec.tex (right):

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7800: where $depth(T)$ is the number of steps in
the longest inheritance path
How lucky, `inheritance path` is never defined, so we can make it solve the
halting problem. ;-)

I believe it must be the length of the longest path to `Object` in the
superinterface graph (which is a subgraph of what I usually call "the clause
based supertype graph", it just omits all mixins and their superinterfaces).

To avoid the substantial extra verbiage, I think we can leave it as it is, but
it might be a good idea to add a comment saying these things, briefly, such that
we remind ourselves next time we look.

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7803: which must exists because $S_0$ is
$\{\code{Object}\}$.
Typo: `exist`.

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7814: but it is not associative.
On 2017/04/05 09:38:31, Lasse Reichstein Nielsen wrote:
> Only real change is this line and adding { and } around I and J in the union
> expressions above. The rest is just reflowing and removing spurious
whitespace.

I think `commutative` is commonly associated with binary operators, and a page
like http://math.stackexchange.com/questions/185471/commutative-functions
actually indicates that it's called 'symmetric' for functions. As far as I can
see, idempotent is being used for functions as well as operators.

But the text uses the full phrase `least upper bound` every time (only once does
it say that it's a 'relation', and that's being changed anyway), so I believe we
are free to decide that it's an operator, and we can then say 'The least upper
bound operator is commutative and idempotent'.

[Lasse Reichstein Nielsen, 2017-04-05 12:09:00]

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec.tex
File docs/language/dartLangSpec.tex (right):

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7800: where $depth(T)$ is the number of steps in
the longest inheritance path
I noticed the same thing, and decided not to touch it now. Adding a comment for
posterity.

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7803: which must exists because $S_0$ is
$\{\code{Object}\}$.
On 2017/04/05 11:47:30, eernst wrote:
> Typo: `exist`.

Done.

https://codereview.chromium.org/2802633002/diff/1/docs/language/dartLangSpec....
docs/language/dartLangSpec.tex:7814: but it is not associative.
Done, except using "operation" instead of "operator" (the latter, to me, feels
like it's defining something that hasn't been defined yet).

[Lasse Reichstein Nielsen, 2017-04-05 12:09:11]

Description was changed from

==========
Fix text around least upper bound.
==========

to

==========
Fix text around least upper bound.

R=eernst@google.com

Committed:
https://github.com/dart-lang/sdk/commit/94b23d3b125e9d246e07a2b43b61740759a0dace
==========

[Lasse Reichstein Nielsen, 2017-04-05 12:09:12]

Committed patchset #2 (id:20001) manually as
94b23d3b125e9d246e07a2b43b61740759a0dace (presubmit successful).