sagemathgh-36449: Autogenerate flint headers
    
We
- write down a script to auto-generate flint header files (currently at
https://github.com/videlec/flint-header-sage-autogen)
- replace most of the files in `src/sage/libs/flint` by auto-generated
files. Doing so, the custom sage code in `src/sage/libs/flint` is always
contained in files suffixed by `_sage.pxd`/`_sage.pyx`.
- deprecate the files in `src/sage/libs/arb/`

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

sagemath#35848

#### Upstream issues and patches

- flintlib/flint#1523
- flintlib/flint#1529
- flintlib/flint#1532
- flintlib/flint#1535
- flintlib/flint#1536
- cython/cython#5779
- flintlib/flint#1653
- flintlib/flint#1655
    
URL: sagemath#36449
Reported by: Vincent Delecroix
Reviewer(s): Marc Mezzarobba, Matthias Köppe, Vincent Delecroix

Author: Release Manager

src/doc/en/reference/libs/index.rst

@@ -42,10 +42,11 @@ FLINT
 .. toctree::
    :maxdepth: 1
 
-   sage/libs/flint/flint
-   sage/libs/flint/fmpz_poly
-   sage/libs/flint/arith
-   sage/libs/flint/qsieve
+   sage/libs/flint/fmpz_poly_sage
+   sage/libs/flint/fmpq_poly_sage
+   sage/libs/flint/arith_sage
+   sage/libs/flint/qsieve_sage
+   sage/libs/flint/ulong_extras_sage
 
 Giac
 ----

src/sage/algebras/quatalg/quaternion_algebra_element.pyx

@@ -46,6 +46,7 @@ from sage.libs.gmp.mpq cimport *
 from sage.libs.ntl.convert cimport mpz_to_ZZ, ZZ_to_mpz
 from sage.libs.flint.fmpz cimport *
 from sage.libs.flint.fmpz_poly cimport *
+from sage.libs.flint.fmpz_poly_sage cimport *
 from sage.libs.flint.ntl_interface cimport *
 
 # variables for holding temporary values computed in

src/sage/arith/misc.py

@@ -375,7 +375,7 @@ def bernoulli(n, algorithm='default', num_threads=1):
         if n >= 100000:
             from warnings import warn
             warn("flint is known to not be accurate for large Bernoulli numbers")
-        from sage.libs.flint.arith import bernoulli_number as flint_bernoulli
+        from sage.libs.flint.arith_sage import bernoulli_number as flint_bernoulli
         return flint_bernoulli(n)
     elif algorithm == 'pari' or algorithm == 'gp':
         from sage.libs.pari.all import pari
@@ -6230,7 +6230,7 @@ def dedekind_sum(p, q, algorithm='default'):
     - :wikipedia:`Dedekind\_sum`
     """
     if algorithm == 'default' or algorithm == 'flint':
-        from sage.libs.flint.arith import dedekind_sum as flint_dedekind_sum
+        from sage.libs.flint.arith_sage import dedekind_sum as flint_dedekind_sum
         return flint_dedekind_sum(p, q)
 
     if algorithm == 'pari':

src/sage/combinat/combinat.py

@@ -407,8 +407,8 @@ def bell_number(n, algorithm='flint', **options) -> Integer:
         return ZZ(int(ret_mp))
 
     elif algorithm == 'flint':
-        import sage.libs.flint.arith
-        return sage.libs.flint.arith.bell_number(n)
+        import sage.libs.flint.arith_sage
+        return sage.libs.flint.arith_sage.bell_number(n)
 
     elif algorithm == 'gap':
         from sage.libs.gap.libgap import libgap
@@ -576,8 +576,8 @@ def euler_number(n, algorithm='flint') -> Integer:
     if algorithm == 'maxima':
         return ZZ(maxima.euler(n))  # type:ignore
     elif algorithm == 'flint':
-        import sage.libs.flint.arith
-        return sage.libs.flint.arith.euler_number(n)
+        import sage.libs.flint.arith_sage
+        return sage.libs.flint.arith_sage.euler_number(n)
     else:
         raise ValueError("algorithm must be 'flint' or 'maxima'")
 
@@ -906,8 +906,8 @@ def stirling_number1(n, k, algorithm="gap") -> Integer:
         from sage.libs.gap.libgap import libgap
         return libgap.Stirling1(n, k).sage()
     if algorithm == 'flint':
-        import sage.libs.flint.arith
-        return sage.libs.flint.arith.stirling_number_1(n, k)
+        import sage.libs.flint.arith_sage
+        return sage.libs.flint.arith_sage.stirling_number_1(n, k)
     raise ValueError("unknown algorithm: %s" % algorithm)
 
 
@@ -1034,8 +1034,8 @@ def stirling_number2(n, k, algorithm=None) -> Integer:
         from sage.libs.gap.libgap import libgap
         return libgap.Stirling2(n, k).sage()
     if algorithm == 'flint':
-        import sage.libs.flint.arith
-        return sage.libs.flint.arith.stirling_number_2(n, k)
+        import sage.libs.flint.arith_sage
+        return sage.libs.flint.arith_sage.stirling_number_2(n, k)
     if algorithm == 'maxima':
         return ZZ(maxima.stirling2(n, k))  # type:ignore
     raise ValueError("unknown algorithm: %s" % algorithm)

src/sage/combinat/partition.py

@@ -284,7 +284,7 @@
 
 from sage.arith.misc import binomial, factorial, gcd, multinomial
 from sage.libs.pari.all import pari
-from sage.libs.flint.arith import number_of_partitions as flint_number_of_partitions
+from sage.libs.flint.arith_sage import number_of_partitions as flint_number_of_partitions
 from sage.structure.global_options import GlobalOptions
 from sage.structure.parent import Parent
 from sage.structure.unique_representation import UniqueRepresentation

src/sage/features/sagemath.py

@@ -380,8 +380,8 @@ def __init__(self):
             True
         """
         JoinFeature.__init__(self, 'sage.libs.flint',
-                             [PythonModule('sage.libs.flint.flint'),
-                              PythonModule('sage.libs.arb.arith')],
+                             [PythonModule('sage.libs.flint.arith_sage'),
+                              PythonModule('sage.libs.flint.flint_sage')],
                              spkg='sagemath_flint', type='standard')
 
 

src/sage/functions/log.py

@@ -27,7 +27,7 @@
 lazy_import('sage.symbolic.constants', 'pi', as_='const_pi')
 lazy_import('sage.rings.complex_double', 'CDF')
 
-lazy_import('sage.libs.flint.arith', 'harmonic_number', as_='_flint_harmonic_number')
+lazy_import('sage.libs.flint.arith_sage', 'harmonic_number', as_='_flint_harmonic_number')
 
 lazy_import('sage.libs.mpmath.utils', 'call', as_='_mpmath_utils_call')
 lazy_import('mpmath', 'harmonic', as_='_mpmath_harmonic')

src/sage/graphs/matchpoly.pyx

@@ -45,6 +45,8 @@ from sage.rings.integer cimport Integer
 
 from sage.libs.flint.fmpz cimport *
 from sage.libs.flint.fmpz_poly cimport *
+from sage.libs.flint.fmpz_poly_sage cimport *
+
 
 x = polygen(ZZ, 'x')
 

src/sage/interfaces/qsieve.py

@@ -1,4 +1,4 @@
 from sage.misc.superseded import deprecated_function_alias
-import sage.libs.flint.qsieve
+import sage.libs.flint.qsieve_sage
 
-qsieve = deprecated_function_alias(1, sage.libs.flint.qsieve.qsieve)
+qsieve = deprecated_function_alias(1, sage.libs.flint.qsieve_sage.qsieve)

src/sage/libs/all.py

@@ -14,6 +14,6 @@
 lazy_import('sage.libs.eclib.mwrank', 'set_precision', 'mwrank_set_precision')
 lazy_import('sage.libs.eclib.mwrank', 'initprimes', 'mwrank_initprimes')
 
-lazy_import('sage.libs.flint.qsieve', 'qsieve')
+lazy_import('sage.libs.flint.qsieve_sage', 'qsieve')
 
 lazy_import('sage.libs.giac.giac', 'libgiac')