sp-acj4-344-real contingent

Summary

Given

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

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 (real x) ∧ (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 0.356 2026-10-13 17:21:19 ba9dc4def822 Q.real(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', real=True)
Assum = And(Q.real(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 0.044 2026-10-12 21:54:02 ee702ae2fe06 Q.real(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', real=True)
Assum = And(Q.real(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 0.113 2026-10-13 17:22:30 ba9dc4def822 Q.real(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', real=True)
Assum = And(Q.real(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 0.066 2026-10-12 21:54:16 ee702ae2fe06 Q.real(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', real=True)
Assum = And(Q.real(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.real(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', real=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 0.027 2026-10-12 21:54:43 ee702ae2fe06 Q.real(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', real=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) 0.081 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ (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, Reals]) && ((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) 0.029 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ (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, Reals]) && ((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) 0.047 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ (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, Reals]) && ((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) 0.081 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ (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, Reals]) && ((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) 0.03 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ (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, Reals]) && ((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) 0.048 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ (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, Reals]) && ((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 0.027 2026-10-13 16:42:18 3ba0dfac4bca Q.real(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', real=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 0.115 2026-10-13 16:40:40 3ba0dfac4bca Q.real(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', real=True)
Assum = And(Q.real(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 (real x) ∧ (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 0.352 2026-10-13 16:39:28 3ba0dfac4bca Q.real(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', real=True)
Assum = And(Q.real(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) 0.082 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ (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, Reals]) && ((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 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ (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, Reals]) && ((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 (real x) ∧ (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) 0.048 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ (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, Reals]) && ((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 (f64-edge.sexp)

(case sp-acj4-344-real (category simplipy-f64-edge)
  (note "SimpliPy acj-4 (MIT) rules_real.json[344]: pow inv _0 _0  ->  pow _0 neg _0")
  (var x real)
  (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.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 -1/2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_f344w2 | 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]")))