sp-acj4-372-real true

category simplipy-real

Summary

Given

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

Claim

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

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5) -> pow cosh _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) ∧ (nonzero (cosh x)) ⊢ (= (surd (cosh x) -5) (pow (cosh x) -1/5)) null

sympy

sympy-ask solved true difference → 0 · ask → True 0.011 2026-10-13 17:21:19 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

sympy-ask solved true difference → 0 · ask → True 0.012 2026-10-12 21:54:02 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.036 2026-10-13 17:22:30 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.037 2026-10-12 21:54:16 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.009 2026-10-13 17:24:01 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.01 2026-10-12 21:54:43 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

wolfram

wolfram-fullsimplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.025 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.022 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.024 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

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

wolfram-fullsimplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.025 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.022 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.025 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.01 2026-10-13 16:42:18 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.036 2026-10-13 16:40:40 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

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

sympy-ask solved true difference → 0 · ask → True 0.011 2026-10-13 16:39:28 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(real_root(cosh(x), (-5)), Pow(cosh(x), Rational(-1, 5))) true

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.zero(cosh(x)))
Residual = real_root(cosh(x), -5) - Pow(cosh(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(cosh(x), -5) - Pow(cosh(x), Rational(-1, 5))`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

wolfram-fullsimplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.025 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.022 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

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

wolfram-simplify solved true difference → -Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1) · result → 0 0.025 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ (Cosh[x] != 0) ⊢ (Surd[Cosh[x], (-5)] == Power[Cosh[x], (-1/5)]) true

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]) && ((Cosh[x] != 0));
  Residual = (Surd[Cosh[x], -5]) - (Cosh[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[Cosh[x], -5]) - (Cosh[x]^(-5^(-1)))`	→	`-Cosh[x]^(-1/5) + Surd[Cosh[x], 5]^(-1)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Source (real.sexp)

(case sp-acj4-372-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[372] + rules.json[240]: rootn cosh _0 (-5)  ->  pow cosh _0 -0.2")
  (var x real)
  (assume (nonzero (cosh x)))
  (ask (= (surd (cosh x) -5) (pow (cosh x) -1/5)))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hc := Real.cosh_pos r; rw [AE.cosh_ofReal, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num]; refine ⟨⟨⟨trivial, by rw [AE.surd.D_iff]; exact ⟨by simp, AE.IsInt.ofInt _, by norm_num, Or.inl (by simp; positivity), Or.inl (by exact_mod_cast hc.ne')⟩⟩, trivial, by rw [AE.pow.D_iff]; exact Or.inl (by exact_mod_cast hc.ne')⟩, ?_⟩; rw [AE.surd_ofReal_nonneg hc.le (by norm_num), show ((-1:ℂ)/5) = (((((-5:ℤ):ℝ))⁻¹ : ℝ) : ℂ) by push_cast; ring, AE.pow_ofReal hc.le (Or.inl hc.ne')]"))