← suite simplipy

sp-acj4-162-real contingent

category simplipy-real

Summary

Given
\(x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{arcosh}\left(\operatorname{surd}\left(x, 5\right)\right) \overset{?}{=} \operatorname{arcosh}\left(x^{\frac{1}{5}}\right)\)
Expected
contingent true at \(x = 0\); false at \(x = -1\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5 -> acosh pow _0 0.2

Lean proof: included

Answers (latest run per system; open code for the reproducer that ran)

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-nonedeclined correctlynull0.02026-10-13 17:21:18 ba9dc4def822(real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null (correct: contingent)
sympy
sympy-askdeclined correctlynulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · ask → None0.0972026-10-13 17:21:19 ba9dc4def822Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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', real=True)
Assum = And(Q.real(x))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · ask → None0.0312026-10-12 21:54:02 ee702ae2fe06Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
askQ.zero(Residual)→None
answernull
sympy-refinedeclined correctlynulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0662026-10-13 17:22:30 ba9dc4def822Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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 = 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))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
refinerefine(Residual, Assum)→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
answernull
sympy-refineno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0642026-10-12 21:54:16 ee702ae2fe06Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
refinerefine(Residual, Assum)→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
answernull
sympy-simplifydeclined correctlynulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · simplify → Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))0.0752026-10-13 17:24:01 ba9dc4def822Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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', real=True)
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
simplifysimplify(Residual)→Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))
answernull
sympy-simplifyno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · simplify → Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))0.0722026-10-12 21:54:43 ee702ae2fe06Q.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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)
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
simplifysimplify(Residual)→Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))
answernull
wolfram
wolfram-fullsimplify declined correctlynulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.212026-10-13 17:24:42 ba9dc4def822Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Reals]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultFullSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
wolfram-refine declined correctlynulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0212026-10-13 17:24:31 ba9dc4def822Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Reals]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultRefine[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
wolfram-simplify declined correctlynulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0252026-10-13 17:24:17 ba9dc4def822Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Reals]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.2132026-10-13 16:43:00 3ba0dfac4bcaElement[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultFullSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
wolfram-refine no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0222026-10-13 16:42:50 3ba0dfac4bcaElement[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultRefine[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
wolfram-simplify no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0262026-10-13 16:42:35 3ba0dfac4bcaElement[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
sympy-simplifyno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · simplify → Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))0.0762026-10-13 16:42:18 3ba0dfac4bcaQ.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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)
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
simplifysimplify(Residual)→Piecewise((0, Eq(x, 0)), (-acosh(x**(1/5)) + acosh(x/Abs(x**(4/5))), True))
answernull
sympy-refineno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0672026-10-13 16:40:40 3ba0dfac4bcaQ.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
refinerefine(Residual, Assum)→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null
sympy-askno verdictnulldifference → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x)) · ask → None0.0982026-10-13 16:39:28 3ba0dfac4bcaQ.real(x) ⊢ Eq(acosh(real_root(x, 5)), acosh(Pow(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))
Residual = acosh(real_root(x, 5)) - acosh(Pow(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')
differenceacosh(real_root(x, 5)) - acosh(Pow(x, Rational(1, 5)))→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.212026-10-12 21:55:24 ee702ae2fe06Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultFullSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
wolfram-refine no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0222026-10-12 21:55:14 ee702ae2fe06Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultRefine[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null
wolfram-simplify no verdictnulldifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]0.0262026-10-12 21:55:00 ee702ae2fe06Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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]);
  Residual = (ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcCosh[Surd[x, 5]]) - (ArcCosh[x^(1/5)])→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
resultSimplify[Residual, Assum]→-ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]]
answernull

Source (real.sexp)

(case sp-acj4-162-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5  ->  acosh pow _0 0.2")
  (var x real)
  (ask (= (acosh (surd x 5)) (acosh (pow x 1/5))))
  (expect contingent)
  (witness true (x 0) (lean-proof "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]"))
  (witness false (x -1) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r162w2 | 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]")))