sp-acj4-369-complex true

category simplipy-complex-branch

Summary

Given

\(x \in \mathbb{C},\quad \arccos x \in \mathbb{R}\)

Claim

\(\displaystyle \operatorname{surd}\left(\arccos x, 5\right) = \left(\arccos x\right)^{\frac{1}{5}}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5 -> pow acos _0 0.2

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 (real (acos x)) ⊢ (= (surd (acos x) 5) (pow (acos x) 1/5)) null

sympy

sympy-ask no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · ask → None 0.248 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
# 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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · ask → None 0.038 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
# 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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · refine → Abs(acos(x))**(1/5)*sign(acos(x)) - acos(x)**(1/5) 0.073 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
refine	`refine(Residual, Assum)`	→	`Abs(acos(x))*(1/5)sign(acos(x)) - acos(x)**(1/5)`
answer			`null`

sympy-refine no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · refine → Abs(acos(x))**(1/5)*sign(acos(x)) - acos(x)**(1/5) 0.059 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
refine	`refine(Residual, Assum)`	→	`Abs(acos(x))*(1/5)sign(acos(x)) - acos(x)**(1/5)`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · simplify → Piecewise((-acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)**(1/5) + acos… 0.092 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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): Q.real(acos(x))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)(1/5) + acos(x)/Abs(acos(x)**(4/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · simplify → Piecewise((-acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)**(1/5) + acos… 0.085 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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): Q.real(acos(x))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)(1/5) + acos(x)/Abs(acos(x)**(4/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

wolfram

wolfram-fullsimplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.29 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.022 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.028 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.3 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.022 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.028 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · simplify → Piecewise((-acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)**(1/5) + acos… 0.093 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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): Q.real(acos(x))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-acos(x)(1/5) + acos(x)/Abs(acos(x)**(4/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

sympy-refine no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · refine → Abs(acos(x))**(1/5)*sign(acos(x)) - acos(x)**(1/5) 0.073 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
refine	`refine(Residual, Assum)`	→	`Abs(acos(x))*(1/5)sign(acos(x)) - acos(x)**(1/5)`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (real (acos x)) ⊢ (= (surd (acos x) 5) (pow (acos x) 1/5)) null

sympy-ask no verdict null difference → Piecewise((Abs(acos(x))**(1/5)*sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)**(1/5), True)… · ask → None 0.25 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ Q.real(acos(x)) ⊢ Eq(real_root(acos(x), 5), Pow(acos(x), Rational(1, 5))) 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), Q.real(acos(x)))
Residual = real_root(acos(x), 5) - Pow(acos(x), Rational(1, 5))
# 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(acos(x), 5) - Pow(acos(x), Rational(1, 5))`	→	`Piecewise((Abs(acos(x))*(1/5)sign(acos(x)), Eq(im(acos(x)), 0)), (acos(x)(1/5), True)) - acos(x)(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.291 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.022 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (real (acos x)) ⊢ (= (surd (acos x) 5) (pow (acos x) 1/5)) null

wolfram-simplify no verdict null difference → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] · result → -ArcCos[x]^(1/5) + Surd[ArcCos[x], 5] 0.028 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ Element[ArcCos[x], Reals] ⊢ (Surd[ArcCos[x], 5] == Power[ArcCos[x], (1/5)]) 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]) && (Element[ArcCos[x], Reals]);
  Residual = (Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Surd[ArcCos[x], 5]) - (ArcCos[x]^(1/5))`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(1/5) + Surd[ArcCos[x], 5]`
answer			`null`

Source (complex-branch.sexp)

(case sp-acj4-369-complex (category simplipy-complex-branch)
  (note "SimpliPy acj-4 (MIT) rules_real.json[369] + rules.json[237]: rootn acos _0 5  ->  pow acos _0 0.2")
  (var x complex)
  (assume (real (acos x)))
  (ask (= (surd (acos x) 5) (pow (acos x) 1/5)))
  (expect true)
  (lean-proof "have hi : (AE.acos x).im = 0 := (AE.real_iff _).1 h0.2; have hr := AE.acos_re_nonneg hi; refine ⟨⟨⟨AE.acos.D_true x, (AE.surd.D_iff _ _).2 ⟨hi, (⟨5, by norm_num⟩ : ∃ n : ℤ, (5:ℂ) = n), by norm_num, Or.inl hr, Or.inr (by norm_num)⟩⟩, AE.acos.D_true x, (AE.pow.D_iff _ _).2 (Or.inr (by norm_num))⟩, ?_⟩; rw [AE.surd_eq_pow_of_nonneg hr]"))