sp-acj4-234-cond2 true

Summary

Given

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

Claim

\(\displaystyle \frac{1}{x^{x}} = \frac{1}{x}^{x}\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0 -> pow inv _0 _0 [weakest assumption found: integer]

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 no verdict null 0.0 2026-10-13 17:21:18 ba9dc4def822 (integer x) ∧ (or (nonzero x) (> (re x) 0)) ∧ (nonzero (pow x x)) ∧ (nonzero x) ∧ (or (nonzero (/ x)) (> (re x) 0)) ⊢ (= (/ (pow x x)) (pow (/ x) x)) null

sympy

sympy-ask solved true difference → 0 · ask → True 0.011 2026-10-13 17:21:19 ba9dc4def822 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

sympy-ask solved true difference → 0 · ask → True 0.013 2026-10-12 21:54:02 ee702ae2fe06 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.035 2026-10-13 17:22:30 ba9dc4def822 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.038 2026-10-12 21:54:16 ee702ae2fe06 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.007 2026-10-13 17:24:01 ba9dc4def822 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(x) > 0)
# dropped (not a unary fact about one symbol): ~Q.zero(Pow(x, x))
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.008 2026-10-12 21:54:43 ee702ae2fe06 Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(x) > 0)
# dropped (not a unary fact about one symbol): ~Q.zero(Pow(x, x))
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

wolfram

wolfram-fullsimplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.029 2026-10-13 17:24:42 ba9dc4def822 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.027 2026-10-13 17:24:31 ba9dc4def822 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.029 2026-10-13 17:24:17 ba9dc4def822 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

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

wolfram-fullsimplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.029 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.028 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.03 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.008 2026-10-13 16:42:18 3ba0dfac4bca Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): Or(~Q.zero(x), re(x) > 0)
# dropped (not a unary fact about one symbol): ~Q.zero(Pow(x, x))
# dropped (not a unary fact about one symbol): Or(~Q.zero(1 / x), re(x) > 0)
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-refine solved true difference → 0 · refine → 0 0.037 2026-10-13 16:40:40 3ba0dfac4bca Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

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

sympy-ask solved true difference → 0 · ask → True 0.012 2026-10-13 16:39:28 3ba0dfac4bca Q.integer(x) ∧ Or((~Q.zero(x)), (re(x) > 0)) ∧ (~Q.zero(Pow(x, x))) ∧ (~Q.zero(x)) ∧ Or((~Q.zero((1/x))), (re(x) > 0)) ⊢ Eq((1/Pow(x, x)), Pow((1/x), x)) true

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', integer=True)
Assum = And(Q.integer(x), Or(~Q.zero(x), re(x) > 0), ~Q.zero(Pow(x, x)), ~Q.zero(x), Or(~Q.zero(1 / x), re(x) > 0))
Residual = 1 / Pow(x, x) - Pow(1 / 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	`1 / Pow(x, x) - Pow(1 / x, x)`	→	`0`
ask	`Q.zero(Residual)`	→	`True`
answer			`true`

wolfram-fullsimplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.029 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.027 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

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

wolfram-simplify solved true difference → -(x^(-1))^x + x^(-x) · result → 0 0.029 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Integers] ∧ ((x != 0) || (Re[x] > 0)) ∧ (Power[x, x] != 0) ∧ (x != 0) ∧ (((1/x) != 0) || (Re[x] > 0)) ⊢ ((1/Power[x, x]) == Power[(1/x), x]) 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, Integers]) && (((x != 0) || (Re[x] > 0))) && ((Power[x, x] != 0)) && ((x != 0)) && ((((1/x) != 0) || (Re[x] > 0)));
  Residual = (1/x^x) - ((1/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) - ((1/x)^x)`	→	`-(x^(-1))^x + x^(-x)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Source (conditional.sexp)

(case sp-acj4-234-cond2 (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[234]: inv pow _0 _0  ->  pow inv _0 _0  [weakest assumption found: integer]")
  (var x integer)
  (assume (or (nonzero x) (> (re x) 0)))
  (assume (nonzero (pow x x)))
  (assume (nonzero x))
  (assume (or (nonzero (/ x)) (> (re x) 0)))
  (ask (= (/ (pow x x)) (pow (/ x) x)))
  (expect true)
  (lean-proof "obtain ⟨n, rfl⟩ := hd0; have hn : (n:ℂ) ≠ 0 := h2.2; refine ⟨⟨⟨h1.1.1, h1.2⟩, hn, (AE.pow.D_iff _ _).mpr (Or.inl (by simpa using hn))⟩, ?_⟩; rw [AE.inv_def, AE.inv_def, AE.cond_pow_intCast' hn, AE.cond_pow_intCast' (inv_ne_zero hn), inv_zpow]"))