sp-acj4-403-condf contingent

category simplipy-conditional

Summary

Given

\(x \in \mathbb{R},\quad x < 0,\quad x \neq 0,\quad \frac{1}{x} \neq 0\)

Claim

\(\displaystyle \sin\left(-\log x\right) \overset{?}{=} \sin\left(\log\left(\frac{1}{x}\right)\right)\)

Expected

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

equal at some admissible points, different at others

witness (true): \(x = -e^{\frac{\pi}{2}}\)

witness (false): \(x = -1\)

SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0 -> sin log inv _0 [false under: negative]; relabelled contingent: true at x = (- (exp (/ pi 2))) (a probe the random samples miss)

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) ∧ (negative x) ∧ (nonzero x) ∧ (nonzero (/ x)) ⊢ (= (sin (- (log x))) (sin (log (/ x)))) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → -sin(log(1/x)) - sin(log(x)) · ask → None 0.14 2026-10-13 17:21:19 ba9dc4def822 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -sin(log(1/x)) - sin(log(x)) · ask → None 0.036 2026-10-12 21:54:02 ee702ae2fe06 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → -sin(log(1/x)) - sin(log(x)) · refine → -sin(log(1/x)) - sin(log(x)) 0.041 2026-10-13 17:22:30 ba9dc4def822 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
refine	`refine(Residual, Assum)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

sympy-refine no verdict null difference → -sin(log(1/x)) - sin(log(x)) · refine → -sin(log(1/x)) - sin(log(x)) 0.047 2026-10-12 21:54:16 ee702ae2fe06 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
refine	`refine(Residual, Assum)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

sympy-simplify declined correctly null difference → -sin(log(1/x)) - sin(log(x)) · simplify → -sin(log(1/x)) - sin(log(x)) 0.051 2026-10-13 17:24:01 ba9dc4def822 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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, negative=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
simplify	`simplify(Residual)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

sympy-simplify no verdict null difference → -sin(log(1/x)) - sin(log(x)) · simplify → -sin(log(1/x)) - sin(log(x)) 0.056 2026-10-12 21:54:43 ee702ae2fe06 Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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, negative=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
simplify	`simplify(Residual)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.088 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine declined correctly null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → (-I)*Sinh[Pi - I*Log[-x^(-1)]] - I*Sinh[Pi - I*Log[-x]] 0.022 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[-Log[x]]) - (Sin[Log[1/x]])`	→	`-Sin[Log[x^(-1)]] - Sin[Log[x]]`
result	`Refine[Residual, Assum]`	→	`(-I)Sinh[Pi - ILog[-x^(-1)]] - ISinh[Pi - ILog[-x]]`
answer			`null`

wolfram-simplify declined correctly null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.028 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[-Log[x]]) - (Sin[Log[1/x]])`	→	`-Sin[Log[x^(-1)]] - Sin[Log[x]]`
result	`Simplify[Residual, Assum]`	→	`-Sin[Log[x^(-1)]] - Sin[Log[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 → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.089 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → (-I)*Sinh[Pi - I*Log[-x^(-1)]] - I*Sinh[Pi - I*Log[-x]] 0.023 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[-Log[x]]) - (Sin[Log[1/x]])`	→	`-Sin[Log[x^(-1)]] - Sin[Log[x]]`
result	`Refine[Residual, Assum]`	→	`(-I)Sinh[Pi - ILog[-x^(-1)]] - ISinh[Pi - ILog[-x]]`
answer			`null`

wolfram-simplify no verdict null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.028 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

sympy-simplify no verdict null difference → -sin(log(1/x)) - sin(log(x)) · simplify → -sin(log(1/x)) - sin(log(x)) 0.076 2026-10-13 16:42:18 3ba0dfac4bca Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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, negative=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
simplify	`simplify(Residual)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

sympy-refine no verdict null difference → -sin(log(1/x)) - sin(log(x)) · refine → -sin(log(1/x)) - sin(log(x)) 0.042 2026-10-13 16:40:40 3ba0dfac4bca Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
refine	`refine(Residual, Assum)`	→	`-sin(log(1/x)) - sin(log(x))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (real x) ∧ (negative x) ∧ (nonzero x) ∧ (nonzero (/ x)) ⊢ (= (sin (- (log x))) (sin (log (/ x)))) null

sympy-ask no verdict null difference → -sin(log(1/x)) - sin(log(x)) · ask → None 0.138 2026-10-13 16:39:28 3ba0dfac4bca Q.real(x) ∧ Q.negative(x) ∧ (~Q.zero(x)) ∧ (~Q.zero((1/x))) ⊢ Eq(sin((-log(x))), sin(log((1/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.negative(x), ~Q.zero(x), ~Q.zero(1 / x))
Residual = sin(-log(x)) - sin(log(1 / 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	`sin(-log(x)) - sin(log(1 / x))`	→	`-sin(log(1/x)) - sin(log(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.089 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

wolfram-refine no verdict null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → (-I)*Sinh[Pi - I*Log[-x^(-1)]] - I*Sinh[Pi - I*Log[-x]] 0.023 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Sin[-Log[x]]) - (Sin[Log[1/x]])`	→	`-Sin[Log[x^(-1)]] - Sin[Log[x]]`
result	`Refine[Residual, Assum]`	→	`(-I)Sinh[Pi - ILog[-x^(-1)]] - ISinh[Pi - ILog[-x]]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (real x) ∧ (negative x) ∧ (nonzero x) ∧ (nonzero (/ x)) ⊢ (= (sin (- (log x))) (sin (log (/ x)))) null

wolfram-simplify no verdict null difference → -Sin[Log[x^(-1)]] - Sin[Log[x]] · result → -Sin[Log[x^(-1)]] - Sin[Log[x]] 0.028 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ (Element[x, Reals] && x < 0) ∧ (x != 0) ∧ ((1/x) != 0) ⊢ (Sin[(-Log[x])] == Sin[Log[(1/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]) && ((Element[x, Reals] && x < 0)) && ((x != 0)) && (((1/x) != 0));
  Residual = (Sin[-Log[x]]) - (Sin[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

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

Source (conditional.sexp)

(case sp-acj4-403-condf (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[403]: sin neg log _0  ->  sin log inv _0  [false under: negative]; relabelled contingent: true at x = (- (exp (/ pi 2))) (a probe the random samples miss)")
  (var x real)
  (assume (negative x))
  (assume (nonzero x))
  (assume (nonzero (/ x)))
  (ask (= (sin (- (log x))) (sin (log (/ x)))))
  (expect contingent)
  (witness true (x (- (exp (/ pi 2)))) (lean-proof "exact AE.w3_c403_w1"))
  (witness false (x -1) (lean-proof "exact AE.w3_c403_w2")))