sp-acj4-368-real true

category simplipy-real

Summary

Given

\(x \in \mathbb{R},\quad \arccos x \in \mathbb{R},\quad \arccos x \neq 0\)

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[368] + rules.json[236]: 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 x) ∧ (real (acos x)) ∧ (nonzero (acos x)) ⊢ (= (surd (acos x) -5) (pow (acos x) -1/5)) null

sympy

sympy-ask no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · ask → None 0.327 2026-10-13 17:21:19 ba9dc4def822 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · ask → None 0.041 2026-10-12 21:54:02 ee702ae2fe06 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · refine → -1/acos(x)**(1/5) + sign(acos(x))/Abs(acos(x))**(1/5) 0.072 2026-10-13 17:22:30 ba9dc4def822 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
refine	`refine(Residual, Assum)`	→	`-1/acos(x)(1/5) + sign(acos(x))/Abs(acos(x))(1/5)`
answer			`null`

sympy-refine no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · refine → -1/acos(x)**(1/5) + sign(acos(x))/Abs(acos(x))**(1/5) 0.061 2026-10-12 21:54:16 ee702ae2fe06 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
refine	`refine(Residual, Assum)`	→	`-1/acos(x)(1/5) + sign(acos(x))/Abs(acos(x))(1/5)`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · simplify → Piecewise((-1/acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)**(1/5) + … 0.107 2026-10-13 17:24:01 ba9dc4def822 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
# dropped (not a unary fact about one symbol): Q.real(acos(x))
# dropped (not a unary fact about one symbol): ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-1/acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)(1/5) + acos(x)/Abs(acos(x)**(6/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · simplify → Piecewise((-1/acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)**(1/5) + … 0.106 2026-10-12 21:54:43 ee702ae2fe06 Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
# dropped (not a unary fact about one symbol): Q.real(acos(x))
# dropped (not a unary fact about one symbol): ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-1/acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)(1/5) + acos(x)/Abs(acos(x)**(6/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

wolfram

wolfram-fullsimplify no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.622 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.028 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.06 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-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 → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.621 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.028 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.06 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · simplify → Piecewise((-1/acos(x)**(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)**(1/5) + … 0.108 2026-10-13 16:42:18 3ba0dfac4bca Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
# dropped (not a unary fact about one symbol): Q.real(acos(x))
# dropped (not a unary fact about one symbol): ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-1/acos(x)(1/5), Eq(acos(x), 0) & Eq(im(acos(x)), 0)), (-1/acos(x)(1/5) + acos(x)/Abs(acos(x)**(6/5)), Eq(im(acos(x)), 0)), (0, True))`
answer			`null`

sympy-refine no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · refine → -1/acos(x)**(1/5) + sign(acos(x))/Abs(acos(x))**(1/5) 0.074 2026-10-13 16:40:40 3ba0dfac4bca Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/acos(x)**(1/5)`
refine	`refine(Residual, Assum)`	→	`-1/acos(x)(1/5) + sign(acos(x))/Abs(acos(x))(1/5)`
answer			`null`

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

sympy-ask no verdict null difference → Piecewise((sign(acos(x))/Abs(acos(x))**(1/5), Eq(im(acos(x)), 0)), (acos(x)**(-1/5), True… · ask → None 0.325 2026-10-13 16:39:28 3ba0dfac4bca Q.real(x) ∧ Q.real(acos(x)) ∧ (~Q.zero(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', real=True)
Assum = And(Q.real(x), Q.real(acos(x)), ~Q.zero(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((sign(acos(x))/Abs(acos(x))(1/5), Eq(im(acos(x)), 0)), (acos(x)(-1/5), True)) - 1/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]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.625 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

wolfram-refine no verdict null difference → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.027 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (real x) ∧ (real (acos x)) ∧ (nonzero (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]^(-1) · result → -ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1) 0.06 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ Element[ArcCos[x], Reals] ∧ (ArcCos[x] != 0) ⊢ (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, Reals]) && (Element[ArcCos[x], Reals]) && ((ArcCos[x] != 0));
  Residual = (Surd[ArcCos[x], -5]) - (ArcCos[x]^(-5^(-1)));
  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]^(-5^(-1)))`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`-ArcCos[x]^(-1/5) + Surd[ArcCos[x], 5]^(-1)`
answer			`null`

Source (real.sexp)

(case sp-acj4-368-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[368] + rules.json[236]: rootn acos _0 (-5)  ->  pow acos _0 -0.2")
  (var x real)
  (assume (real (acos x)))
  (assume (nonzero (acos x)))
  (ask (= (surd (acos x) -5) (pow (acos x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.mem_Icc_of_acos_real (by simpa using h0.2); rw [AE.acos_ofReal hI.1 hI.2]; have ha : 0 ≤ Real.arccos r := Real.arccos_nonneg r; have hne : Real.arccos r ≠ 0 := (by have := h1.2; rw [AE.acos_ofReal hI.1 hI.2] at this; intro h; exact this (by rw [h]; simp)); rw [show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨by simp, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simpa using ha), Or.inl (by exact_mod_cast hne)⟩⟩, by simp, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hne)⟩, ?_⟩; rw [AE.surd_ofReal_nonneg ha (by norm_num), AE.pow_ofReal ha (Or.inl hne)]"))