sp-acj4-373-cond contingent

category simplipy-conditional

Summary

Given

\(x \in \mathbb{C},\quad \operatorname{Re}\left(x\right) > 0,\quad \cosh x \in \mathbb{R}\)

Claim

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

Expected

contingent true at \(x = 1\); false at \(x = \frac{1}{2} + \pi \, i\)

equal at some admissible points, different at others

witness (true): \(x = 1\)

witness (false): \(x = \frac{1}{2} + \pi \, i\)

SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5 -> pow cosh _0 0.2 [weakest assumption found: re-pos]; relabelled contingent: false at x = (+ 1/2 (* pi i)) (a probe the random samples miss)

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 declined correctly null 0.0 2026-10-13 17:21:18 ba9dc4def822 (> (re x) 0) ∧ (real (cosh x)) ⊢ (= (surd (cosh x) 5) (pow (cosh x) 1/5)) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · ask → None 0.197 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null (correct: contingent)

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) > 0, Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · ask → None 0.054 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) > 0, Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine timeout null 10.0 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) no answer in time

sympy-refine no verdict null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · refine → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… 1.219 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) > 0, Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
refine	`refine(Residual, Assum)`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
answer			`null`

sympy-simplify declined correctly null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · simplify → Piecewise((-cosh(x)**(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))*sinh(re(x)), 0)), (-cosh(x)**(… 0.34 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null (correct: contingent)

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): re(x) > 0
# dropped (not a unary fact about one symbol): Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-cosh(x)*(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))sinh(re(x)), 0)), (-cosh(x)(1/5) + cosh(x)/Abs(cosh(x)(4/5)), Eq(sin(im(x))*sinh(re(x)), 0)), (0, True))`
answer			`null`

sympy-simplify no verdict null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · simplify → Piecewise((-cosh(x)**(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))*sinh(re(x)), 0)), (-cosh(x)**(… 0.349 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): re(x) > 0
# dropped (not a unary fact about one symbol): Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-cosh(x)*(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))sinh(re(x)), 0)), (-cosh(x)(1/5) + cosh(x)/Abs(cosh(x)(4/5)), Eq(sin(im(x))*sinh(re(x)), 0)), (0, True))`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.108 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null (correct: contingent)

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine declined correctly null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.022 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null (correct: contingent)

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify declined correctly null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.031 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null (correct: contingent)

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.108 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.023 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.031 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

sympy-simplify no verdict null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · simplify → Piecewise((-cosh(x)**(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))*sinh(re(x)), 0)), (-cosh(x)**(… 0.347 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): re(x) > 0
# dropped (not a unary fact about one symbol): Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
simplify	`simplify(Residual)`	→	`Piecewise((-cosh(x)*(1/5), Eq(cosh(x), 0) & Eq(sin(im(x))sinh(re(x)), 0)), (-cosh(x)(1/5) + cosh(x)/Abs(cosh(x)(4/5)), Eq(sin(im(x))*sinh(re(x)), 0)), (0, True))`
answer			`null`

sympy-refine timeout null 10.0 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) no answer in time

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (> (re x) 0) ∧ (real (cosh x)) ⊢ (= (surd (cosh x) 5) (pow (cosh x) 1/5)) null

sympy-ask no verdict null difference → Piecewise((Abs(cosh(x))**(1/5)*sign(cosh(x)), Eq(sin(im(x))*sinh(re(x)), 0)), (cosh(x)**(… · ask → None 0.199 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ (re(x) > 0) ∧ Q.real(cosh(x)) ⊢ Eq(real_root(cosh(x), 5), Pow(cosh(x), Rational(1, 5))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) > 0, Q.real(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))`	→	`Piecewise((Abs(cosh(x))*(1/5)sign(cosh(x)), Eq(sin(im(x))sinh(re(x)), 0)), (cosh(x)(1/5), True)) - cosh(x)*(1/5)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.108 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.022 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (> (re x) 0) ∧ (real (cosh x)) ⊢ (= (surd (cosh x) 5) (pow (cosh x) 1/5)) null

wolfram-simplify no verdict null difference → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] · result → -Cosh[x]^(1/5) + Surd[Cosh[x], 5] 0.031 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] > 0) ∧ Element[Cosh[x], Reals] ⊢ (Surd[Cosh[x], 5] == Power[Cosh[x], (1/5)]) null

code

setup — not timed

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

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Re[x] > 0)) && (Element[Cosh[x], Reals]);
  Residual = (Surd[Cosh[x], 5]) - (Cosh[x]^(1/5));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

Source (conditional.sexp)

(case sp-acj4-373-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[373] + rules.json[241]: rootn cosh _0 5  ->  pow cosh _0 0.2  [weakest assumption found: re-pos]; relabelled contingent: false at x = (+ 1/2 (* pi i)) (a probe the random samples miss)")
  (var x complex)
  (assume (> (re x) 0))
  (assume (real (cosh x)))
  (ask (= (surd (cosh x) 5) (pow (cosh x) 1/5)))
  (expect contingent)
  (witness true (x 1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_c373w1 | exact AE.w2_cosh_one_ne | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos, AE.w2_ccosh_one_ne, AE.w2_ccosh_one_im, AE.w2_ccosh_one_re, Real.cosh_pos]"))
  (witness false (x (+ 1/2 (* pi i))) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_c373w2 | exact AE.w2_real_cosh_xpi | exact AE.w2_c373_surdD | exact AE.w2_c373_powD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))