← suite simplipy

sp-acj4-335-cond true

category simplipy-conditional

Summary

Given
\(x \in \mathbb{C},\quad \operatorname{Im}\left(x\right) > -\pi \land \operatorname{Im}\left(x\right) \le \pi,\quad e^{x} \neq 0 \lor \operatorname{Re}\left(\pi\right) > 0\)
Claim
\(\displaystyle \left(e^{x}\right)^{\pi} = e^{\pi \, x}\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi -> exp * np.pi _0 [weakest assumption found: im-principal]

Lean proof: included

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

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-noneno verdictnull0.02026-10-13 17:21:18 ba9dc4def822(and (> (im x) (- pi)) (<= (im x) pi)) ∧ (or (nonzero (exp x)) (> (re pi) 0)) ⊢ (= (pow (exp x) pi) (exp (* pi x)))null
sympy
sympy-askno verdictnulldifference → exp(x)**pi - exp(pi*x) · ask → None0.372026-10-13 17:21:19 ba9dc4def822Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → exp(x)**pi - exp(pi*x) · ask → None0.1422026-10-12 21:54:02 ee702ae2fe06Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → exp(x)**pi - exp(pi*x) · refine → exp(x)**pi - exp(pi*x)0.1782026-10-13 17:22:30 ba9dc4def822Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
refinerefine(Residual, Assum)→exp(x)**pi - exp(pi*x)
answernull
sympy-refineno verdictnulldifference → exp(x)**pi - exp(pi*x) · refine → exp(x)**pi - exp(pi*x)0.0552026-10-12 21:54:16 ee702ae2fe06Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
refinerefine(Residual, Assum)→exp(x)**pi - exp(pi*x)
answernull
sympy-simplifyno verdictnulldifference → exp(x)**pi - exp(pi*x) · simplify → exp(x)**pi - exp(pi*x)0.0342026-10-13 17:24:01 ba9dc4def822Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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)
# dropped (not a unary fact about one symbol): And(im(x) > -pi, im(x) <= pi)
# dropped (not a unary fact about one symbol): Or(~Q.zero(exp(x)), re(pi) > 0)
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
simplifysimplify(Residual)→exp(x)**pi - exp(pi*x)
answernull
sympy-simplifyno verdictnulldifference → exp(x)**pi - exp(pi*x) · simplify → exp(x)**pi - exp(pi*x)0.0352026-10-12 21:54:43 ee702ae2fe06Q.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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)
# dropped (not a unary fact about one symbol): And(im(x) > -pi, im(x) <= pi)
# dropped (not a unary fact about one symbol): Or(~Q.zero(exp(x)), re(pi) > 0)
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
simplifysimplify(Residual)→exp(x)**pi - exp(pi*x)
answernull
wolfram
wolfram-fullsimplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0252026-10-13 17:24:42 ba9dc4def822Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0232026-10-13 17:24:31 ba9dc4def822Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0242026-10-13 17:24:17 ba9dc4def822Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultSimplify[Residual, Assum]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0252026-10-13 16:43:00 3ba0dfac4bcaElement[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0232026-10-13 16:42:50 3ba0dfac4bcaElement[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0252026-10-13 16:42:35 3ba0dfac4bcaElement[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultSimplify[Residual, Assum]→0
answertrue
sympy-simplifyno verdictnulldifference → exp(x)**pi - exp(pi*x) · simplify → exp(x)**pi - exp(pi*x)0.0322026-10-13 16:42:18 3ba0dfac4bcaQ.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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)
# dropped (not a unary fact about one symbol): And(im(x) > -pi, im(x) <= pi)
# dropped (not a unary fact about one symbol): Or(~Q.zero(exp(x)), re(pi) > 0)
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
simplifysimplify(Residual)→exp(x)**pi - exp(pi*x)
answernull
sympy-refineno verdictnulldifference → exp(x)**pi - exp(pi*x) · refine → exp(x)**pi - exp(pi*x)0.1792026-10-13 16:40:40 3ba0dfac4bcaQ.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
refinerefine(Residual, Assum)→exp(x)**pi - exp(pi*x)
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(and (> (im x) (- pi)) (<= (im x) pi)) ∧ (or (nonzero (exp x)) (> (re pi) 0)) ⊢ (= (pow (exp x) pi) (exp (* pi x)))null
sympy-askno verdictnulldifference → exp(x)**pi - exp(pi*x) · ask → None0.3692026-10-13 16:39:28 3ba0dfac4bcaQ.complex(x) ∧ And((im(x) > (-pi)), (im(x) <= pi)) ∧ Or((~Q.zero(exp(x))), (re(pi) > 0)) ⊢ Eq(Pow(exp(x), pi), exp((pi*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), And(im(x) > -pi, im(x) <= pi), Or(~Q.zero(exp(x)), re(pi) > 0))
Residual = Pow(exp(x), pi) - exp(pi * 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')
differencePow(exp(x), pi) - exp(pi * x)→exp(x)**pi - exp(pi*x)
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0252026-10-12 21:55:24 ee702ae2fe06Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0232026-10-12 21:55:14 ee702ae2fe06Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultRefine[Residual, Assum]→0
answertrue
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(and (> (im x) (- pi)) (<= (im x) pi)) ∧ (or (nonzero (exp x)) (> (re pi) 0)) ⊢ (= (pow (exp x) pi) (exp (* pi x)))null
wolfram-simplify solvedtruedifference → -E^(Pi*x) + (E^x)^Pi · result → 00.0252026-10-12 21:55:00 ee702ae2fe06Element[x, Complexes] ∧ ((Im[x] > (-Pi)) && (Im[x] <= Pi)) ∧ ((Exp[x] != 0) || (Re[Pi] > 0)) ⊢ (Power[Exp[x], Pi] == Exp[(Pi*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, Complexes]) && (((Im[x] > (-Pi)) && (Im[x] <= Pi))) && (((Exp[x] != 0) || (Re[Pi] > 0)));
  Residual = (Exp[x]^Pi) - (Exp[Pi*x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Exp[x]^Pi) - (Exp[Pi*x])→-E^(Pi*x) + (E^x)^Pi
resultSimplify[Residual, Assum]→0
answertrue

Source (conditional.sexp)

(case sp-acj4-335-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[335]: pow exp _0 np.pi  ->  exp * np.pi _0  [weakest assumption found: im-principal]")
  (var x complex)
  (assume (and (> (im x) (- pi)) (<= (im x) pi)))
  (assume (or (nonzero (exp x)) (> (re pi) 0)))
  (ask (= (pow (exp x) pi) (exp (* pi x))))
  (expect true)
  (lean-proof "simp at h0; refine ⟨(AE.pow.D_iff _ _).mpr (Or.inl (Complex.exp_ne_zero x)), ?_⟩; rw [AE.cond_pow_exp_of_im h0.1 h0.2]"))