sp-acj4-344-complex contingent

Summary

Given

\(x \in \mathbb{C},\quad x \neq 0,\quad \frac{1}{x} \neq 0 \lor \operatorname{Re}\left(x\right) > 0,\quad x \neq 0 \lor \operatorname{Re}\left(-x\right) > 0\)

Claim

\(\displaystyle \frac{1}{x}^{x} \overset{?}{=} x^{-x}\)

Expected

contingent true at \(x = 1\); false at \(x = -\frac{1}{2}\)

equal at some admissible points, different at others

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

witness (false): \(x = -\frac{1}{2}\)

SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0 -> pow _0 neg _0

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 (nonzero x) ∧ (or (nonzero (/ x)) (> (re x) 0)) ∧ (or (nonzero x) (> (re (- x)) 0)) ⊢ (= (pow (/ x) x) (pow x (- x))) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → (1/x)**x - 1/x**x · ask → None 0.326 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
# 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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → (1/x)**x - 1/x**x · ask → None 0.046 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
# 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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → (1/x)**x - 1/x**x · refine → (1/x)**x - 1/x**x 0.159 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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', complex=True)
Assum = And(Q.complex(x), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
refine	`refine(Residual, Assum)`	→	`(1/x)x - 1/xx`
answer			`null`

sympy-refine no verdict null difference → (1/x)**x - 1/x**x · refine → (1/x)**x - 1/x**x 0.095 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
refine	`refine(Residual, Assum)`	→	`(1/x)x - 1/xx`
answer			`null`

sympy-simplify declined correctly null difference → (1/x)**x - 1/x**x · simplify → (1/x)**x - 1/x**x 0.027 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(-x) > 0)
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
simplify	`simplify(Residual)`	→	`(1/x)x - 1/xx`
answer			`null`

sympy-simplify no verdict null difference → (1/x)**x - 1/x**x · simplify → (1/x)**x - 1/x**x 0.028 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(-x) > 0)
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
simplify	`simplify(Residual)`	→	`(1/x)x - 1/xx`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.069 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine declined correctly null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.028 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify declined correctly null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.036 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.102 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.029 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-simplify no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.037 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

sympy-simplify no verdict null difference → (1/x)**x - 1/x**x · simplify → (1/x)**x - 1/x**x 0.028 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(-x) > 0)
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
simplify	`simplify(Residual)`	→	`(1/x)x - 1/xx`
answer			`null`

sympy-refine no verdict null difference → (1/x)**x - 1/x**x · refine → (1/x)**x - 1/x**x 0.16 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
refine	`refine(Residual, Assum)`	→	`(1/x)x - 1/xx`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (nonzero x) ∧ (or (nonzero (/ x)) (> (re x) 0)) ∧ (or (nonzero x) (> (re (- x)) 0)) ⊢ (= (pow (/ x) x) (pow x (- x))) null

sympy-ask no verdict null difference → (1/x)**x - 1/x**x · ask → None 0.324 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ∧ Or((~Q.zero(x)), (re((-x)) > 0)) ⊢ Eq(Pow((1/x), x), Pow(x, (-x))) 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), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0), Or(~Q.zero(x), re(-x) > 0))
Residual = Pow(1 / x, x) - Pow(x, -x)
# 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	`Pow(1 / x, x) - Pow(x, -x)`	→	`(1/x)x - 1/xx`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.069 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.028 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (nonzero x) ∧ (or (nonzero (/ x)) (> (re x) 0)) ∧ (or (nonzero x) (> (re (- x)) 0)) ⊢ (= (pow (/ x) x) (pow x (- x))) null

wolfram-simplify no verdict null difference → (x^(-1))^x - x^(-x) · result → (x^(-1))^x - x^(-x) 0.037 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ∧ ((x != 0) || (Re[(-x)] > 0)) ⊢ (Power[(1/x), x] == Power[x, (-x)]) 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]) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0))) && (((x != 0) || (Re[(-x)] > 0)));
  Residual = ((1/x)^x) - (x^(-x));
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

Source (complex-branch.sexp)

(case sp-acj4-344-complex (category simplipy-complex-branch)
  (note "SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0  ->  pow _0 neg _0")
  (var x complex)
  (assume (nonzero x))
  (assume (or (nonzero (/ x)) (> (re x) 0)))
  (assume (or (nonzero x) (> (re (- x)) 0)))
  (ask (= (pow (/ x) x) (pow x (- x))))
  (expect contingent)
  (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]"))
  (witness false (x -1/2) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))