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Currently Argument Clinic doesn't support the __init__ methods, mainly because the C implementation of this method should return int, not PyObject *. In some cases it is possible to wrap a function generated by Argument Clinic (as in Modules/_pickle.c). But not always, because the __init__ method is called with the self, args, and kwargs arguments, while Argument Clinic can generate function without kwargs.

Oh! I thought it worked fine, because I've seen people convert both an __init__ and a __new__. But I guess they wrapped them.
It's an easy change, I can try to do that today.

> * As is probably expected, __init__ and __call__ procs
> can't be converted.
I should have a fix in for __init__ and __call__ later today. (Issue #20294.)
For the record, you could convert them, you just had to wrap the parsing function
to deal with the return type difference, and *args / **kwargs might be tricky too.
> * sqlite3.Connection.{execute, executemany, executescript} don't use
> PyArg_Parse*.
In the next few days I'm going to add support for "*args" and "**kwargs", and
then you'll be able to convert these functions. (Issue #20291.) Again, even
if Argument Clinic doesn't do any parsing, it's still helpful to have it
generate signatures.
> * The clinic version of 'sqlite3.adapt' has an argument string of "O|OO".
> The first optional parameter defaults to 'pysqlite_PrepareProtocolType'
> in C and 'sqlite3.ProtocolType' in Python. However, I don't know how to
> represent the Python value because Python default values are 'eval'd by
> clinic *and* the 'sqlite3' module is not imported for the eval.
You'll be able to do this soon:
parameter: object(c_default='pysqlite_PrepareProtocolType') = ProtocolType
This feature isn't checked in yet, I'm waiting for a review. It's part of
#20226, which I guess you already noticed.
> * The clinic version of 'sqlite3.Cursor.fetchmany' has an arguments string
> of "|i". The first parameter defaults to
> '((pysqlite_Cursor *)self)->arraysize' in C and 'self.arraysize' in
> Python. I don't know how to express the Python initialization.
You can't. How about you use a default of -1 and then:
 if (maxrows == -1)
 maxrows = self->arraysize
> * I didn't convert 'sqlite3.Cursor.setinputsizes' and
> 'sqlite3.Cursor.setoutputsize' because they share a docstring and
> simple no-op method def.
I'd prefer it if you converted them anyway. Converting them means they'll
have signature information which is an unconditional good.
> * I didn't convert 'sqlite.connect' because it would have required
> packaing the arguments back up to pass to 'PyObject_Call'.
Once I add the ability to pass in *args and **kwargs, you'll be able to
convert sqlite.connect. 
I think I responded to all your other comments when I reviewed the patch.
Thanks!

Oops, sorry, that last comment was intended for a different issue. Please ignore, and sorry for the spam!

Here's a patch. Wasn't as easy as I thought, it kind of took all day.
Changes include:
* __init__ and __new__ always take kwargs.
 * if the function signature doesn't accept keyword arguments,
 it calls _PyArg_NoKeywords().
* __init__ returns int, not PyObject *.
* __init__ and __new__ should support all argument parsing
 scenarios (unpack tuple, positional only, etc).
Pre-exiting behavior:
* The C function basename chosen for __new__ is just the name of
 the class, it doesn't end in __new__.
* The methoddef #define is suppressed.

Rietveld doesn't like the patch and I'm having a hard time finding a changeset on which I can get it to apply cleanly. May I suggest avoiding `--git` on `hg diff` unless copying/moving/renaming a file?

Whoops, that doesn't apply cleanly. Here's an updated patch.

Output looks good and I didn't see anything that scared me in the patch, though I can't pretend to fully understand 100% of it. Builds on Windows, pickle tests pass, and nothing else is affected (that's checked in so far). Looks good to me, as far as I can see.

I have reviewed this as best I am able. I'll be honest that a lot of clinic.py makes my eyes cross; I'm used to webdev templating, which is inverted from AC (flow control inside the template). Not complaining, it's a complicated subject. I do like the centralization of the templates.
I didn't see anything that jumped out at me. I tried out the new conversions on socketmodule and like the output, and none of the existing conversions mutated. So LGTM!
It took me a while to figure out how to make __new__ a class method though, decorator support should probably be documented (or else a hint like '__new__ must be a @classmethod!')

New changeset 2e32462e4832 by Larry Hastings in branch 'default':
Issue #20294: Argument Clinic now supports argument parsing for __new__ and
http://hg.python.org/cpython/rev/2e32462e4832 

Checked in. Thanks for bringing the problem to my attention, Serhiy, and thanks for the reviews you two!
rmsr: If I had a stronger background in templating, maybe Argument Clinic would be cleaner inside. As it is I'm beating the problem to death with sheer force of will.

Thank you Larry, now I can use Argument Clinic in the __init__ methods. But there is one problem.
Docstring generated for the __init__ method contains the "__init__" name in the signature. Therefore it can't be used as class docstring. On other hand, the compiler complains of not used __init__ docstring.

That problem will be irrelevant, once builtin classes support signatures (#20189).