sp-acj4-191-condf false

Summary

Given

\(x \in \mathbb{C},\quad x \in i\mathbb{R},\quad \sin\left(\arccos x\right) \neq 1,\quad \sin\left(\arccos x\right) \neq -1,\quad x \neq 0\)

Claim

\(\displaystyle \operatorname{artanh}\left(\sin\left(\arccos x\right)\right) \neq \operatorname{arcosh}\left(\left|\frac{1}{x}\right|\right)\)

Expected

false unequal at every admissible point (e.g. at \(x = i\))

the two sides differ at every admissible point

witness (false): \(x = i\)

SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0 -> acosh abs inv _0 [false under: imaginary]

Lean proof: included

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

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

none

always-none no verdict null 0.0 2026-10-13 17:21:18 ba9dc4def822 (imaginary x) ∧ (!= (sin (acos x)) 1) ∧ (!= (sin (acos x)) -1) ∧ (nonzero x) ⊢ (= (atanh (sin (acos x))) (acosh (abs (/ x)))) null

sympy

sympy-ask no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · ask → None 0.283 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · ask → None 0.048 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · refine → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.051 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

sympy-refine no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · refine → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.054 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

sympy-simplify no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · simplify → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.053 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True, imaginary=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), -1, evaluate=False)
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

sympy-simplify no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · simplify → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.05 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True, imaginary=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), -1, evaluate=False)
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

wolfram

wolfram-fullsimplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]] 2.071 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.024 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.034 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]] 2.034 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.024 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

wolfram-simplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.034 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

sympy-simplify no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · simplify → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.054 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True, imaginary=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(sin(acos(x)), -1, evaluate=False)
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

sympy-refine no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · refine → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) 0.05 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (imaginary x) ∧ (!= (sin (acos x)) 1) ∧ (!= (sin (acos x)) -1) ∧ (nonzero x) ⊢ (= (atanh (sin (acos x))) (acosh (abs (/ x)))) null

sympy-ask no verdict null difference → -acosh(1/Abs(x)) + atanh(sqrt(1 - x**2)) · ask → None 0.285 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ Q.imaginary(x) ∧ Ne(sin(acos(x)), 1) ∧ Ne(sin(acos(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(sin(acos(x))), acosh(Abs((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', complex=True)
Assum = And(Q.complex(x), Q.imaginary(x), Ne(sin(acos(x)), 1, evaluate=False), Ne(sin(acos(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(sin(acos(x))) - acosh(Abs(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	`atanh(sin(acos(x))) - acosh(Abs(1 / x))`	→	`-acosh(1/Abs(x)) + atanh(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]] 2.05 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`-ArcSech[Abs[Im[x]]] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

wolfram-refine no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.024 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (imaginary x) ∧ (!= (sin (acos x)) 1) ∧ (!= (sin (acos x)) -1) ∧ (nonzero x) ⊢ (= (atanh (sin (acos x))) (acosh (abs (/ x)))) null

wolfram-simplify no verdict null difference → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] · result → -ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]] 0.036 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Sin[ArcCos[x]] != 1) ∧ (Sin[ArcCos[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Sin[ArcCos[x]]] == ArcCosh[Abs[(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, Complexes]) && ((Re[x] == 0 && x != 0)) && ((Sin[ArcCos[x]] != 1)) && ((Sin[ArcCos[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcTanh[Sin[ArcCos[x]]]) - (ArcCosh[Abs[1/x]])`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`-ArcCosh[Abs[x]^(-1)] + ArcTanh[Sqrt[1 - x^2]]`
answer			`null`

Source (conditional.sexp)

(case sp-acj4-191-condf (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[191]: atanh sin acos _0  ->  acosh abs inv _0  [false under: imaginary]")
  (var x complex)
  (assume (imaginary x))
  (assume (!= (sin (acos x)) 1))
  (assume (!= (sin (acos x)) -1))
  (assume (nonzero x))
  (ask (= (atanh (sin (acos x))) (acosh (abs (/ x)))))
  (expect false)
  (witness false (x i))
  (lean-proof "exact AE.w2_c191 x h0 h1 h2 h3"))