sp-acj4-180-real contingent
category simplipy-f64-edge
Summary
Given
\(x \in \mathbb{R},\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq i,\quad \sinh\left(\operatorname{arcosh}\left(x\right)\right) \neq -i,\quad \cosh\left(\operatorname{arcosh}\left(x\right)\right) \neq 0\)
Claim
\(\displaystyle \arctan\left(\sinh\left(\operatorname{arcosh}\left(x\right)\right)\right) \overset{?}{=} \arcsin\left(\tanh\left(\operatorname{arcosh}\left(x\right)\right)\right)\)
Expected
contingent true at \(x = 1\); false at \(x = -2\)
equal at some admissible points, different at others
witness (true): \(x = 1\)
witness (false): \(x = -2\)
SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0
Lean proof: included
Answers (latest run per system; open code for the reproducer that ran)
| system | outcome | answer | steps (from the trace) | s | run | query (as sent) | detail | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| none | |||||||||||||||||||
| always-none | declined correctly | null | 0.0 | 2026-10-13 17:21:18 ba9dc4def822 | (real x) ∧ (!= (sinh (acosh x)) i) ∧ (!= (sinh (acosh x)) (- i)) ∧ (nonzero (cosh (acosh x))) ⊢ (= (atan (sinh (acosh x))) (asin (tanh (acosh x)))) | null (correct: contingent) | |||||||||||||
| sympy | |||||||||||||||||||
| sympy-ask | declined correctly | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · ask → None | 1.638 | 2026-10-13 17:21:19 ba9dc4def822 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null (correct: contingent) | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-ask | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · ask → None | 0.057 | 2026-10-12 21:54:02 ee702ae2fe06 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-refine | declined correctly | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · refine → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.052 | 2026-10-13 17:22:30 ba9dc4def822 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null (correct: contingent) | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-refine | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · refine → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.054 | 2026-10-12 21:54:16 ee702ae2fe06 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-simplify | declined correctly | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · simplify → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.053 | 2026-10-13 17:24:01 ba9dc4def822 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null (correct: contingent) | ||||||||||||
codesetup — 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): Ne(sinh(acosh(x)), I, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sinh(acosh(x)), -I, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(acosh(x)))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-simplify | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · simplify → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.054 | 2026-10-12 21:54:43 ee702ae2fe06 | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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): Ne(sinh(acosh(x)), I, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sinh(acosh(x)), -I, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(acosh(x)))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| wolfram | |||||||||||||||||||
| wolfram-fullsimplify | declined correctly | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcCsc[x/(Sqrt[-1 + x]*Sqrt[1 + x])] + ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]] | 0.228 | 2026-10-13 17:24:42 ba9dc4def822 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null (correct: contingent) | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | declined correctly | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.028 | 2026-10-13 17:24:31 ba9dc4def822 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null (correct: contingent) | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-simplify | declined correctly | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.039 | 2026-10-13 17:24:17 ba9dc4def822 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null (correct: contingent) | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
Older runs (11)
| system | outcome | answer | steps (from the trace) | s | run | query (as sent) | detail | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| wolfram-fullsimplify | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcCsc[x/(Sqrt[-1 + x]*Sqrt[1 + x])] + ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]] | 0.211 | 2026-10-13 16:43:00 3ba0dfac4bca | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.028 | 2026-10-13 16:42:50 3ba0dfac4bca | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-simplify | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.039 | 2026-10-13 16:42:35 3ba0dfac4bca | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| sympy-simplify | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · simplify → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.052 | 2026-10-13 16:42:18 3ba0dfac4bca | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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): Ne(sinh(acosh(x)), I, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sinh(acosh(x)), -I, evaluate=False)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(acosh(x)))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| sympy-refine | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · refine → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) | 0.053 | 2026-10-13 16:40:40 3ba0dfac4bca | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| always-none | no verdict | null | 0.0 | 2026-10-13 16:39:28 3ba0dfac4bca | (real x) ∧ (!= (sinh (acosh x)) i) ∧ (!= (sinh (acosh x)) (- i)) ∧ (nonzero (cosh (acosh x))) ⊢ (= (atan (sinh (acosh x))) (asin (tanh (acosh x)))) | null | |||||||||||||
| sympy-ask | no verdict | null | difference → -asin(sqrt(x - 1)*sqrt(x + 1)/x) + atan(sqrt(x - 1)*sqrt(x + 1)) · ask → None | 1.613 | 2026-10-13 16:39:28 3ba0dfac4bca | Q.real(x) ∧ Ne(sinh(acosh(x)), I) ∧ Ne(sinh(acosh(x)), (-I)) ∧ (~Q.zero(cosh(acosh(x)))) ⊢ Eq(atan(sinh(acosh(x))), asin(tanh(acosh(x)))) | null | ||||||||||||
codesetup — 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), Ne(sinh(acosh(x)), I, evaluate=False), Ne(sinh(acosh(x)), -I, evaluate=False), ~Q.zero(cosh(acosh(x))))
Residual = atan(sinh(acosh(x))) - asin(tanh(acosh(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')
| |||||||||||||||||||
| wolfram-fullsimplify | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcCsc[x/(Sqrt[-1 + x]*Sqrt[1 + x])] + ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]] | 0.211 | 2026-10-12 21:55:24 ee702ae2fe06 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = FullSimplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| wolfram-refine | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.028 | 2026-10-12 21:55:14 ee702ae2fe06 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Refine[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
| always-none | no verdict | null | 0.0 | 2026-10-12 21:55:00 ee702ae2fe06 | (real x) ∧ (!= (sinh (acosh x)) i) ∧ (!= (sinh (acosh x)) (- i)) ∧ (nonzero (cosh (acosh x))) ⊢ (= (atan (sinh (acosh x))) (asin (tanh (acosh x)))) | null | |||||||||||||
| wolfram-simplify | no verdict | null | difference → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] · result → -ArcSin[(Sqrt[(-1 + x)/(1 + x)]*(1 + x))/x] + ArcTan[Sqrt[(-1 + x)/(1 + x)]*(1 + x)] | 0.039 | 2026-10-12 21:55:00 ee702ae2fe06 | Element[x, Reals] ∧ (Sinh[ArcCosh[x]] != I) ∧ (Sinh[ArcCosh[x]] != (-I)) ∧ (Cosh[ArcCosh[x]] != 0) ⊢ (ArcTan[Sinh[ArcCosh[x]]] == ArcSin[Tanh[ArcCosh[x]]]) | null | ||||||||||||
codesetup — 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]) && ((Sinh[ArcCosh[x]] != I)) && ((Sinh[ArcCosh[x]] != (-I))) && ((Cosh[ArcCosh[x]] != 0));
Residual = (ArcTan[Sinh[ArcCosh[x]]]) - (ArcSin[Tanh[ArcCosh[x]]]);
Result = Simplify[Residual, Assum];
Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
{Residual, Result, Answer}
)
| |||||||||||||||||||
Source (f64-edge.sexp)
(case sp-acj4-180-real (category simplipy-f64-edge) (note "SimpliPy acj-4 (MIT) rules_real.json[180]: atan sinh acosh _0 -> asin tanh acosh _0") (var x real) (assume (!= (sinh (acosh x)) i)) (assume (!= (sinh (acosh x)) (- i))) (assume (nonzero (cosh (acosh x)))) (ask (= (atan (sinh (acosh x))) (asin (tanh (acosh x))))) (expect contingent) (witness true (x 1) (lean-proof "norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")) (witness false (x -2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_f180w2 | exact AE.w2_f180_hI | exact AE.w2_f180_hnI | exact AE.w2_f180_atanD | exact AE.w2_f180_cosh | exact AE.w2_f180_tanhD | norm_num [AE.abs_eq, AE.pow, AE.w2_asin_zero, AE.w2_asin_one, AE.w2_acos_zero, AE.w2_acos_one, AE.w2_atan_zero, AE.w2_atanh_zero, AE.w2_asinh_zero, AE.w2_acosh_one, AE.surd, AE.w2_zero_ne_I, AE.w2_isInt_three, AE.w2_isInt_five, AE.w2_isInt_nine, AE.w2_isInt_neg_five, AE.w2_odd_three, AE.w2_odd_five, AE.w2_odd_nine, AE.w2_odd_neg_five, Real.pi_pos, Complex.exp_re, Real.exp_pos]")))