sp-acj4-370-real true
category simplipy-real
Summary
Given
\(x \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \in \mathbb{R},\quad \operatorname{arcosh}\left(x\right) \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(\operatorname{arcosh}\left(x\right), -5\right) = \operatorname{arcosh}\left(x\right)^{-\frac{1}{5}}\)
Expected
true equal at every admissible point
the two sides are equal at every admissible point
SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _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 (acosh x)) ∧ (nonzero (acosh x)) ⊢ (= (surd (acosh x) -5) (pow (acosh x) -1/5)) | null | |||||||||||||
| sympy | |||||||||||||||||||
| sympy-ask | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · ask → None | 0.183 | 2026-10-13 17:21:19 ba9dc4def822 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-ask | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · ask → None | 0.039 | 2026-10-12 21:54:02 ee702ae2fe06 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-refine | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · refine → -1/acosh(x)**(1/5) + sign(acosh(x))/Abs(acosh(x))**(1/5) | 0.071 | 2026-10-13 17:22:30 ba9dc4def822 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-refine | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · refine → -1/acosh(x)**(1/5) + sign(acosh(x))/Abs(acosh(x))**(1/5) | 0.059 | 2026-10-12 21:54:16 ee702ae2fe06 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-simplify | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · simplify → Piecewise((-1/acosh(x)**(1/5), Eq(acosh(x), 0) & Eq(im(acosh(x)), 0)), (-1/acosh(x)**(1/5… | 0.104 | 2026-10-13 17:24:01 ba9dc4def822 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x))
# dropped (not a unary fact about one symbol): ~Q.zero(acosh(x))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-simplify | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · simplify → Piecewise((-1/acosh(x)**(1/5), Eq(acosh(x), 0) & Eq(im(acosh(x)), 0)), (-1/acosh(x)**(1/5… | 0.103 | 2026-10-12 21:54:43 ee702ae2fe06 | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x))
# dropped (not a unary fact about one symbol): ~Q.zero(acosh(x))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| wolfram | |||||||||||||||||||
| wolfram-fullsimplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.421 | 2026-10-13 17:24:42 ba9dc4def822 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.023 | 2026-10-13 17:24:31 ba9dc4def822 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-simplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.065 | 2026-10-13 17:24:17 ba9dc4def822 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
Older runs (11)
| system | outcome | answer | steps (from the trace) | s | run | query (as sent) | detail | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| wolfram-fullsimplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.418 | 2026-10-13 16:43:00 3ba0dfac4bca | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.023 | 2026-10-13 16:42:50 3ba0dfac4bca | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-simplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.064 | 2026-10-13 16:42:35 3ba0dfac4bca | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| sympy-simplify | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · simplify → Piecewise((-1/acosh(x)**(1/5), Eq(acosh(x), 0) & Eq(im(acosh(x)), 0)), (-1/acosh(x)**(1/5… | 0.105 | 2026-10-13 16:42:18 3ba0dfac4bca | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x))
# dropped (not a unary fact about one symbol): ~Q.zero(acosh(x))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| sympy-refine | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · refine → -1/acosh(x)**(1/5) + sign(acosh(x))/Abs(acosh(x))**(1/5) | 0.072 | 2026-10-13 16:40:40 3ba0dfac4bca | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| always-none | no verdict | null | 0.0 | 2026-10-13 16:39:28 3ba0dfac4bca | (real x) ∧ (real (acosh x)) ∧ (nonzero (acosh x)) ⊢ (= (surd (acosh x) -5) (pow (acosh x) -1/5)) | null | |||||||||||||
| sympy-ask | no verdict | null | difference → Piecewise((sign(acosh(x))/Abs(acosh(x))**(1/5), Eq(im(acosh(x)), 0)), (acosh(x)**(-1/5), … · ask → None | 0.181 | 2026-10-13 16:39:28 3ba0dfac4bca | Q.real(x) ∧ Q.real(acosh(x)) ∧ (~Q.zero(acosh(x))) ⊢ Eq(real_root(acosh(x), (-5)), Pow(acosh(x), Rational(-1, 5))) | null | ||||||||||||
codesetup — 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(acosh(x)), ~Q.zero(acosh(x)))
Residual = real_root(acosh(x), -5) - Pow(acosh(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')
| |||||||||||||||||||
| wolfram-fullsimplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.422 | 2026-10-12 21:55:24 ee702ae2fe06 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.023 | 2026-10-12 21:55:14 ee702ae2fe06 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| always-none | no verdict | null | 0.0 | 2026-10-12 21:55:00 ee702ae2fe06 | (real x) ∧ (real (acosh x)) ∧ (nonzero (acosh x)) ⊢ (= (surd (acosh x) -5) (pow (acosh x) -1/5)) | null | |||||||||||||
| wolfram-simplify | no verdict | null | difference → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) · result → -ArcCosh[x]^(-1/5) + Surd[ArcCosh[x], 5]^(-1) | 0.064 | 2026-10-12 21:55:00 ee702ae2fe06 | Element[x, Reals] ∧ Element[ArcCosh[x], Reals] ∧ (ArcCosh[x] != 0) ⊢ (Surd[ArcCosh[x], (-5)] == Power[ArcCosh[x], (-1/5)]) | null | ||||||||||||
codesetup — 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[ArcCosh[x], Reals]) && ((ArcCosh[x] != 0));
Residual = (Surd[ArcCosh[x], -5]) - (ArcCosh[x]^(-5^(-1)));
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
Source (real.sexp)
(case sp-acj4-370-real (category simplipy-real) (note "SimpliPy acj-4 (MIT) rules_real.json[370] + rules.json[238]: rootn acosh _0 (-5) -> pow acosh _0 -0.2") (var x real) (assume (real (acosh x))) (assume (nonzero (acosh x))) (ask (= (surd (acosh x) -5) (pow (acosh x) -1/5))) (expect true) (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hI := AE.one_le_of_acosh_real (by simpa using h0.2); rw [AE.acosh_ofReal hI]; have ha : 0 ≤ Real.arcosh r := AE.arcosh_nonneg hI; have hne : Real.arcosh r ≠ 0 := (by have := h1.2; rw [AE.acosh_ofReal hI] 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)]"))