sp-acj4-376-cond2 true

Summary

Given

\(x \in \mathbb{C},\quad \left|\operatorname{Im}\left(x\right)\right| < \pi,\quad e^{x} \in \mathbb{R},\quad e^{x} \neq 0\)

Claim

\(\displaystyle \operatorname{surd}\left(e^{x}, -3\right) = e^{\frac{x}{-3}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[376]: rootn exp _0 (-3) -> exp / _0 (-3) [weakest assumption found: im-lt-pi]

Lean proof: included

Answers (latest run per system; open code for the reproducer that ran)

system outcome answer steps (from the trace) s run query (as sent) detail

none

always-none no verdict null 0.0 2026-10-13 17:21:18 ba9dc4def822 (< (abs (im x)) pi) ∧ (real (exp x)) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -3) (exp (/ x -3))) null

sympy

sympy-ask no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · ask → None 0.718 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, Q.real(exp(x)), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -3) - exp(x / -3)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · ask → None 0.047 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, Q.real(exp(x)), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -3) - exp(x / -3)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine timeout null 10.0 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) no answer in time

sympy-refine no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · refine → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… 1.349 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, Q.real(exp(x)), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -3) - exp(x / -3)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
refine	`refine(Residual, Assum)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · simplify → Piecewise((exp(x - 4*re(x)/3) - exp(-x/3), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3)… 0.189 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): Q.real(exp(x))
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -3) - exp(x / -3)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
simplify	`simplify(Residual)`	→	`Piecewise((exp(x - 4re(x)/3) - exp(-x/3), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3) - exp(-x/3), True))`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · simplify → Piecewise((exp(x - 4*re(x)/3) - exp(-x/3), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3)… 0.184 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): Q.real(exp(x))
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -3) - exp(x / -3)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
simplify	`simplify(Residual)`	→	`Piecewise((exp(x - 4re(x)/3) - exp(-x/3), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3) - exp(-x/3), True))`
answer			`null`

wolfram

wolfram-fullsimplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.174 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.029 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Refine[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

wolfram-simplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.054 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

Older runs (11)

system outcome answer steps (from the trace) s run query (as sent) detail

wolfram-fullsimplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.175 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.031 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Refine[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

wolfram-simplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.054 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · simplify → Piecewise((exp(x - 4*re(x)/3) - exp(-x/3), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3)… 0.191 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): Q.real(exp(x))
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -3) - exp(x / -3)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
simplify	`simplify(Residual)`	→	`Piecewise((exp(x - 4re(x)/3) - exp(-x/3), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3) - exp(-x/3), True))`
answer			`null`

sympy-refine timeout null 10.0 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) no answer in time

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (< (abs (im x)) pi) ∧ (real (exp x)) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -3) (exp (/ x -3))) null

sympy-ask no verdict null difference → Piecewise((exp(-re(x)/3)*sign(exp(x)), Eq(exp(re(x))*sin(im(x)), 0)), (exp(x)**(-1/3), Tr… · ask → None 0.72 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ Q.real(exp(x)) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-3)), exp((x/(-3)))) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, Q.real(exp(x)), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -3) - exp(x / -3)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`real_root(exp(x), -3) - exp(x / -3)`	→	`Piecewise((exp(-re(x)/3)sign(exp(x)), Eq(exp(re(x))sin(im(x)), 0)), (exp(x)**(-1/3), True)) - exp(-x/3)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.175 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.029 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Refine[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (< (abs (im x)) pi) ∧ (real (exp x)) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -3) (exp (/ x -3))) null

wolfram-simplify no verdict null difference → -E^(-1/3*x) + Surd[E^x, 3]^(-1) · result → -E^(-1/3*x) + Surd[E^x, 3]^(-1) 0.054 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ Element[Exp[x], Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-3)] == Exp[(x/(-3))]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Abs[Im[x]] < Pi)) && (Element[Exp[x], Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -3]) - (Exp[x/-3]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[Exp[x], -3]) - (Exp[x/-3])`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-E^(-1/3*x) + Surd[E^x, 3]^(-1)`
answer			`null`

Source (conditional.sexp)

(case sp-acj4-376-cond2 (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[376]: rootn exp _0 (-3)  ->  exp / _0 (-3)  [weakest assumption found: im-lt-pi]")
  (var x complex)
  (assume (< (abs (im x)) pi))
  (assume (real (exp x)))
  (assume (nonzero (exp x)))
  (ask (= (surd (exp x) -3) (exp (/ x -3))))
  (expect true)
  (lean-proof "have hb : |x.im| < Real.pi := (by simpa [AE.abs_ofReal] using h0.2); have hb' := abs_lt.mp hb; have hs : Real.sin x.im = 0 := (by have := h1.2; simp [Complex.exp_im] at this; exact this); have hi : x.im = 0 := (Real.sin_eq_zero_iff_of_lt_of_lt hb'.1 hb'.2).mp hs; obtain ⟨a, rfl⟩ := AE.IsReal.exists hi; rw [AE.exp_ofReal]; refine ⟨(by rw [AE.surd.D_iff]; exact ⟨by simp, ⟨-3, by norm_num⟩, by norm_num, Or.inl (by rw [Complex.ofReal_re]; exact (Real.exp_pos a).le), Or.inl (by exact_mod_cast (Real.exp_pos a).ne')⟩), ?_⟩; rw [show (-3:ℂ) = ((-3:ℤ):ℂ) by norm_num, AE.surd_ofReal_nonneg (Real.exp_pos a).le (by norm_num), ← Real.exp_mul]; simp only [AE.exp_def, AE.inv_def]; push_cast; ring_nf"))