← suite simplipy

sp-acj4-378-real true

category simplipy-f64-edge

Summary

Given
\(x \in \mathbb{R},\quad e^{x} \neq 0\)
Claim
\(\displaystyle \operatorname{surd}\left(e^{x}, -5\right) = e^{\frac{x}{-5}}\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[378]: rootn exp _0 (-5) -> exp / _0 (-5)

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(real x) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -5) (exp (/ x -5)))null
sympy
sympy-asksolvedtruedifference → 0 · ask → True0.0142026-10-13 17:21:19 ba9dc4def822Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
askQ.zero(Residual)→True
answertrue
sympy-asksolvedtruedifference → 0 · ask → True0.0162026-10-12 21:54:02 ee702ae2fe06Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
askQ.zero(Residual)→True
answertrue
sympy-refinesolvedtruedifference → 0 · refine → 00.0382026-10-13 17:22:30 ba9dc4def822Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
refinerefine(Residual, Assum)→0
answertrue
sympy-refinesolvedtruedifference → 0 · refine → 00.042026-10-12 21:54:16 ee702ae2fe06Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
refinerefine(Residual, Assum)→0
answertrue
sympy-simplifysolvedtruedifference → 0 · simplify → 00.0122026-10-13 17:24:01 ba9dc4def822Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
simplifysimplify(Residual)→0
answertrue
sympy-simplifysolvedtruedifference → 0 · simplify → 00.0132026-10-12 21:54:43 ee702ae2fe06Q.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
simplifysimplify(Residual)→0
answertrue
wolfram
wolfram-fullsimplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0262026-10-13 17:24:42 ba9dc4def822Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0232026-10-13 17:24:31 ba9dc4def822Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0252026-10-13 17:24:17 ba9dc4def822Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultSimplify[Residual, Assum]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0262026-10-13 16:43:00 3ba0dfac4bcaElement[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0232026-10-13 16:42:50 3ba0dfac4bcaElement[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0252026-10-13 16:42:35 3ba0dfac4bcaElement[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultSimplify[Residual, Assum]→0
answertrue
sympy-simplifysolvedtruedifference → 0 · simplify → 00.0122026-10-13 16:42:18 3ba0dfac4bcaQ.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
# dropped (not a unary fact about one symbol): ~Q.zero(exp(x))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
simplifysimplify(Residual)→0
answertrue
sympy-refinesolvedtruedifference → 0 · refine → 00.0392026-10-13 16:40:40 3ba0dfac4bcaQ.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
refinerefine(Residual, Assum)→0
answertrue
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(real x) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -5) (exp (/ x -5)))null
sympy-asksolvedtruedifference → 0 · ask → True0.0142026-10-13 16:39:28 3ba0dfac4bcaQ.real(x) ∧ (~Q.zero(exp(x))) ⊢ Eq(real_root(exp(x), (-5)), exp((x/(-5))))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', real=True)
Assum = And(Q.real(x), ~Q.zero(exp(x)))
Residual = real_root(exp(x), -5) - exp(x / -5)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencereal_root(exp(x), -5) - exp(x / -5)→0
askQ.zero(Residual)→True
answertrue
wolfram-fullsimplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0262026-10-12 21:55:24 ee702ae2fe06Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0232026-10-12 21:55:14 ee702ae2fe06Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultRefine[Residual, Assum]→0
answertrue
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(real x) ∧ (nonzero (exp x)) ⊢ (= (surd (exp x) -5) (exp (/ x -5)))null
wolfram-simplify solvedtruedifference → -E^(-1/5*x) + Surd[E^x, 5]^(-1) · result → 00.0252026-10-12 21:55:00 ee702ae2fe06Element[x, Reals] ∧ (Exp[x] != 0) ⊢ (Surd[Exp[x], (-5)] == Exp[(x/(-5))])true
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Exp[x] != 0));
  Residual = (Surd[Exp[x], -5]) - (Exp[x/-5]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Surd[Exp[x], -5]) - (Exp[x/-5])→-E^(-1/5*x) + Surd[E^x, 5]^(-1)
resultSimplify[Residual, Assum]→0
answertrue

Source (f64-edge.sexp)

(case sp-acj4-378-real (category simplipy-f64-edge)
  (note "SimpliPy acj-4 (MIT) rules_real.json[378]: rootn exp _0 (-5)  ->  exp / _0 (-5)")
  (var x real)
  (assume (nonzero (exp x)))
  (ask (= (surd (exp x) -5) (exp (/ x -5))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; refine ⟨(AE.surd.D_iff _ _).2 ⟨by simp [AE.exp_ofReal], (⟨-5, by norm_num⟩ : ∃ n : ℤ, (-5:ℂ) = n), by norm_num, Or.inl (by rw [AE.exp_ofReal, Complex.ofReal_re]; exact (Real.exp_pos r).le), Or.inl h0.2⟩, ?_⟩; rw [AE.exp_ofReal, show (-5:ℂ) = ((-5:ℤ):ℂ) by norm_num, AE.surd_ofReal_nonneg (Real.exp_pos r).le (by norm_num), ← Real.exp_mul, show (r:ℂ) * AE.inv ((-5:ℤ):ℂ) = ((r * ((-5:ℤ):ℝ)⁻¹ : ℝ) : ℂ) by push_cast; simp, AE.exp_ofReal]"))