← suite simplipy

sp-acj4-162-cond true

category simplipy-conditional

Summary

Given
\(x \in \mathbb{C},\quad \operatorname{Re}\left(x\right) \ge 0,\quad x \in \mathbb{R}\)
Claim
\(\displaystyle \operatorname{arcosh}\left(\operatorname{surd}\left(x, 5\right)\right) = \operatorname{arcosh}\left(x^{\frac{1}{5}}\right)\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

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

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-noneno verdictnull0.02026-10-13 17:21:18 ba9dc4def822(>= (re x) 0) ∧ (real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null
sympy
sympy-askno verdictnulldifference → -acosh(x**(1/5)) + acosh(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · ask → None0.1172026-10-13 17:21:19 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -acosh(x**(1/5)) + acosh(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · ask → None0.0362026-10-12 21:54:02 ee702ae2fe06Q.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -acosh(x**(1/5)) + acosh(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0832026-10-13 17:22:30 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
refinerefine(Residual, Assum)→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
answernull
sympy-refineno verdictnulldifference → -acosh(x**(1/5)) + acosh(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0682026-10-12 21:54:16 ee702ae2fe06Q.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
refinerefine(Residual, Assum)→-acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))
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))0.0752026-10-13 17:24:01 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True, real=True)
# dropped (not a unary fact about one symbol): re(x) >= 0
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.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True, real=True)
# dropped (not a unary fact about one symbol): re(x) >= 0
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 solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-13 17:24:42 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
wolfram-refine solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0222026-10-13 17:24:31 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
wolfram-simplify solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-13 17:24:17 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-13 16:43:00 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
wolfram-refine solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-13 16:42:50 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
wolfram-simplify solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-13 16:42:35 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
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.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True, real=True)
# dropped (not a unary fact about one symbol): re(x) >= 0
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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · refine → -acosh(x**(1/5)) + acosh(Abs(x)**(1/5)*sign(x))0.0842026-10-13 16:40:40 3ba0dfac4bcaQ.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
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(>= (re x) 0) ∧ (real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null
sympy-askno verdictnulldifference → -acosh(x**(1/5)) + acosh(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True… · ask → None0.1172026-10-13 16:39:28 3ba0dfac4bcaQ.complex(x) ∧ (re(x) >= 0) ∧ Q.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', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, 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(Piecewise((Abs(x)**(1/5)*sign(x), Eq(im(x), 0)), (x**(1/5), True)))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-12 21:55:24 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
wolfram-refine solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0222026-10-12 21:55:14 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(>= (re x) 0) ∧ (real x) ⊢ (= (acosh (surd x 5)) (acosh (pow x 1/5)))null
wolfram-simplify solvedtruedifference → -ArcCosh[x^(1/5)] + ArcCosh[Surd[x, 5]] · result → 00.0232026-10-12 21:55:00 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ Element[x, Reals] ⊢ (ArcCosh[Surd[x, 5]] == ArcCosh[Power[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, Complexes]) && ((Re[x] >= 0)) && (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]→0
answertrue

Source (conditional.sexp)

(case sp-acj4-162-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[162] + rules.json[119]: acosh rootn _0 5  ->  acosh pow _0 0.2  [weakest assumption found: re-nonneg]")
  (var x complex)
  (assume (>= (re x) 0))
  (assume (real x))
  (ask (= (acosh (surd x 5)) (acosh (pow x 1/5))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists (by simpa using h1.2); have hr : 0 ≤ r := (by simpa using h0.2); have hs : AE.surd (↑r) 5 = AE.pow (↑r) (1/5) := (by rw [AE.surd, if_pos (by simpa using hr)]; norm_num [AE.inv_def]); rw [hs]; refine ⟨⟨⟨?_, by simp⟩, (AE.pow.D_iff _ _).mpr (Or.inr (by norm_num)), by simp⟩, rfl⟩; rw [AE.surd.D_iff]; exact ⟨by simp, ⟨5, by norm_num⟩, by norm_num, Or.inl (by simpa using hr), Or.inr (by norm_num)⟩"))