Initialisation du repository de Beta

venv/lib/python3.12/site-packages/sympy/assumptions/handlers/ntheory.py

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+"""
+Handlers for keys related to number theory: prime, even, odd, etc.
+"""
+
+from sympy.assumptions import Q, ask
+from sympy.core import Add, Basic, Expr, Float, Mul, Pow, S
+from sympy.core.numbers import (ImaginaryUnit, Infinity, Integer, NaN,
+    NegativeInfinity, NumberSymbol, Rational, int_valued)
+from sympy.functions import Abs, im, re
+from sympy.ntheory import isprime
+
+from sympy.multipledispatch import MDNotImplementedError
+
+from ..predicates.ntheory import (PrimePredicate, CompositePredicate,
+    EvenPredicate, OddPredicate)
+
+
+# PrimePredicate
+
+def _PrimePredicate_number(expr, assumptions):
+    # helper method
+    exact = not expr.atoms(Float)
+    try:
+        i = int(expr.round())
+        if (expr - i).equals(0) is False:
+            raise TypeError
+    except TypeError:
+        return False
+    if exact:
+        return isprime(i)
+    # when not exact, we won't give a True or False
+    # since the number represents an approximate value
+
+@PrimePredicate.register(Expr)
+def _(expr, assumptions):
+    ret = expr.is_prime
+    if ret is None:
+        raise MDNotImplementedError
+    return ret
+
+@PrimePredicate.register(Basic)
+def _(expr, assumptions):
+    if expr.is_number:
+        return _PrimePredicate_number(expr, assumptions)
+
+@PrimePredicate.register(Mul)
+def _(expr, assumptions):
+    if expr.is_number:
+        return _PrimePredicate_number(expr, assumptions)
+    for arg in expr.args:
+        if not ask(Q.integer(arg), assumptions):
+            return None
+    for arg in expr.args:
+        if arg.is_number and arg.is_composite:
+            return False
+
+@PrimePredicate.register(Pow)
+def _(expr, assumptions):
+    """
+    Integer**Integer     -> !Prime
+    """
+    if expr.is_number:
+        return _PrimePredicate_number(expr, assumptions)
+    if ask(Q.integer(expr.exp), assumptions) and \
+            ask(Q.integer(expr.base), assumptions):
+        prime_base = ask(Q.prime(expr.base), assumptions)
+        if prime_base is False:
+            return False
+        is_exp_one = ask(Q.eq(expr.exp, 1), assumptions)
+        if is_exp_one is False:
+            return False
+        if prime_base is True and is_exp_one is True:
+            return True
+
+@PrimePredicate.register(Integer)
+def _(expr, assumptions):
+    return isprime(expr)
+
+@PrimePredicate.register_many(Rational, Infinity, NegativeInfinity, ImaginaryUnit)
+def _(expr, assumptions):
+    return False
+
+@PrimePredicate.register(Float)
+def _(expr, assumptions):
+    return _PrimePredicate_number(expr, assumptions)
+
+@PrimePredicate.register(NumberSymbol)
+def _(expr, assumptions):
+    return _PrimePredicate_number(expr, assumptions)
+
+@PrimePredicate.register(NaN)
+def _(expr, assumptions):
+    return None
+
+
+# CompositePredicate
+
+@CompositePredicate.register(Expr)
+def _(expr, assumptions):
+    ret = expr.is_composite
+    if ret is None:
+        raise MDNotImplementedError
+    return ret
+
+@CompositePredicate.register(Basic)
+def _(expr, assumptions):
+    _positive = ask(Q.positive(expr), assumptions)
+    if _positive:
+        _integer = ask(Q.integer(expr), assumptions)
+        if _integer:
+            _prime = ask(Q.prime(expr), assumptions)
+            if _prime is None:
+                return
+            # Positive integer which is not prime is not
+            # necessarily composite
+            _is_one = ask(Q.eq(expr, 1), assumptions)
+            if _is_one:
+                return False
+            if _is_one is None:
+                return None
+            return not _prime
+        else:
+            return _integer
+    else:
+        return _positive
+
+
+# EvenPredicate
+
+def _EvenPredicate_number(expr, assumptions):
+    # helper method
+    if isinstance(expr, (float, Float)):
+        if int_valued(expr):
+            return None
+        return False
+    try:
+        i = int(expr.round())
+    except TypeError:
+        return False
+    if not (expr - i).equals(0):
+        return False
+    return i % 2 == 0
+
+@EvenPredicate.register(Expr)
+def _(expr, assumptions):
+    ret = expr.is_even
+    if ret is None:
+        raise MDNotImplementedError
+    return ret
+
+@EvenPredicate.register(Basic)
+def _(expr, assumptions):
+    if expr.is_number:
+        return _EvenPredicate_number(expr, assumptions)
+
+@EvenPredicate.register(Mul)
+def _(expr, assumptions):
+    """
+    Even * Integer    -> Even
+    Even * Odd        -> Even
+    Integer * Odd     -> ?
+    Odd * Odd         -> Odd
+    Even * Even       -> Even
+    Integer * Integer -> Even if Integer + Integer = Odd
+    otherwise         -> ?
+    """
+    if expr.is_number:
+        return _EvenPredicate_number(expr, assumptions)
+    even, odd, irrational, acc = False, 0, False, 1
+    for arg in expr.args:
+        # check for all integers and at least one even
+        if ask(Q.integer(arg), assumptions):
+            if ask(Q.even(arg), assumptions):
+                even = True
+            elif ask(Q.odd(arg), assumptions):
+                odd += 1
+            elif not even and acc != 1:
+                if ask(Q.odd(acc + arg), assumptions):
+                    even = True
+        elif ask(Q.irrational(arg), assumptions):
+            # one irrational makes the result False
+            # two makes it undefined
+            if irrational:
+                break
+            irrational = True
+        else:
+            break
+        acc = arg
+    else:
+        if irrational:
+            return False
+        if even:
+            return True
+        if odd == len(expr.args):
+            return False
+
+@EvenPredicate.register(Add)
+def _(expr, assumptions):
+    """
+    Even + Odd  -> Odd
+    Even + Even -> Even
+    Odd  + Odd  -> Even
+
+    """
+    if expr.is_number:
+        return _EvenPredicate_number(expr, assumptions)
+    _result = True
+    for arg in expr.args:
+        if ask(Q.even(arg), assumptions):
+            pass
+        elif ask(Q.odd(arg), assumptions):
+            _result = not _result
+        else:
+            break
+    else:
+        return _result
+
+@EvenPredicate.register(Pow)
+def _(expr, assumptions):
+    if expr.is_number:
+        return _EvenPredicate_number(expr, assumptions)
+    if ask(Q.integer(expr.exp), assumptions):
+        if ask(Q.positive(expr.exp), assumptions):
+            return ask(Q.even(expr.base), assumptions)
+        elif ask(~Q.negative(expr.exp) & Q.odd(expr.base), assumptions):
+            return False
+        elif expr.base is S.NegativeOne:
+            return False
+
+@EvenPredicate.register(Integer)
+def _(expr, assumptions):
+    return not bool(expr.p & 1)
+
+@EvenPredicate.register_many(Rational, Infinity, NegativeInfinity, ImaginaryUnit)
+def _(expr, assumptions):
+    return False
+
+@EvenPredicate.register(NumberSymbol)
+def _(expr, assumptions):
+    return _EvenPredicate_number(expr, assumptions)
+
+@EvenPredicate.register(Abs)
+def _(expr, assumptions):
+    if ask(Q.real(expr.args[0]), assumptions):
+        return ask(Q.even(expr.args[0]), assumptions)
+
+@EvenPredicate.register(re)
+def _(expr, assumptions):
+    if ask(Q.real(expr.args[0]), assumptions):
+        return ask(Q.even(expr.args[0]), assumptions)
+
+@EvenPredicate.register(im)
+def _(expr, assumptions):
+    if ask(Q.real(expr.args[0]), assumptions):
+        return True
+
+@EvenPredicate.register(NaN)
+def _(expr, assumptions):
+    return None
+
+
+# OddPredicate
+
+@OddPredicate.register(Expr)
+def _(expr, assumptions):
+    ret = expr.is_odd
+    if ret is None:
+        raise MDNotImplementedError
+    return ret
+
+@OddPredicate.register(Basic)
+def _(expr, assumptions):
+    _integer = ask(Q.integer(expr), assumptions)
+    if _integer:
+        _even = ask(Q.even(expr), assumptions)
+        if _even is None:
+            return None
+        return not _even
+    return _integer