sp-acj4-171-cond contingent

category simplipy-conditional

Summary

Given

\(x \in \mathbb{C},\quad \left|\operatorname{Im}\left(x\right)\right| < \pi,\quad \cosh x \neq 0\)

Claim

\(\displaystyle \arcsin\left(\sin\left(\tanh x\right)\right) \overset{?}{=} \tanh x\)

Expected

contingent true at \(x = 0\); false at \(x = \frac{1}{2} + \frac{\pi}{2} \, i\)

equal at some admissible points, different at others

witness (true): \(x = 0\)

witness (false): \(x = \frac{1}{2} + \frac{\pi}{2} \, i\)

SimpliPy acj-4 (MIT) rules_real.json[171] + rules.json[125]: asin sin tanh _0 -> tanh _0 [weakest assumption found: im-lt-pi]; relabelled contingent: false at x = (+ 1/2 (* (/ pi 2) i)) (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 (< (abs (im x)) pi) ∧ (nonzero (cosh x)) ⊢ (= (asin (sin (tanh x))) (tanh x)) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → -tanh(x) + asin(sin(tanh(x))) · ask → None 0.137 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · ask → None 0.038 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → -tanh(x) + asin(sin(tanh(x))) · refine → -tanh(x) + asin(sin(tanh(x))) 0.043 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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', complex=True)
Assum = And(Q.complex(x), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
refine	`refine(Residual, Assum)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

sympy-refine no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · refine → -tanh(x) + asin(sin(tanh(x))) 0.057 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
refine	`refine(Residual, Assum)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

sympy-simplify declined correctly null difference → -tanh(x) + asin(sin(tanh(x))) · simplify → -tanh(x) + asin(sin(tanh(x))) 0.115 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
simplify	`simplify(Residual)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

sympy-simplify no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · simplify → -tanh(x) + asin(sin(tanh(x))) 0.107 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(x)) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
simplify	`simplify(Residual)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.577 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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, Complexes]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`FullSimplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

wolfram-refine declined correctly null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.028 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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, Complexes]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Refine[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

wolfram-simplify declined correctly null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.033 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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, Complexes]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Simplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[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 → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.579 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`FullSimplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

wolfram-refine no verdict null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.029 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Refine[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

wolfram-simplify no verdict null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.034 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Simplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

sympy-simplify no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · simplify → -tanh(x) + asin(sin(tanh(x))) 0.117 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(x)) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): Abs(im(x)) < pi
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
simplify	`simplify(Residual)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

sympy-refine no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · refine → -tanh(x) + asin(sin(tanh(x))) 0.044 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
refine	`refine(Residual, Assum)`	→	`-tanh(x) + asin(sin(tanh(x)))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (< (abs (im x)) pi) ∧ (nonzero (cosh x)) ⊢ (= (asin (sin (tanh x))) (tanh x)) null

sympy-ask no verdict null difference → -tanh(x) + asin(sin(tanh(x))) · ask → None 0.131 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ (Abs(im(x)) < pi) ∧ (~Q.zero(cosh(x))) ⊢ Eq(asin(sin(tanh(x))), tanh(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), Abs(im(x)) < pi, ~Q.zero(cosh(x)))
Residual = asin(sin(tanh(x))) - tanh(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	`asin(sin(tanh(x))) - tanh(x)`	→	`-tanh(x) + asin(sin(tanh(x)))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.574 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`FullSimplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

wolfram-refine no verdict null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.029 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Refine[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (< (abs (im x)) pi) ∧ (nonzero (cosh x)) ⊢ (= (asin (sin (tanh x))) (tanh x)) null

wolfram-simplify no verdict null difference → ArcSin[Sin[Tanh[x]]] - Tanh[x] · result → ArcSin[Sin[Tanh[x]]] - Tanh[x] 0.033 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Abs[Im[x]] < Pi) ∧ (Cosh[x] != 0) ⊢ (ArcSin[Sin[Tanh[x]]] == Tanh[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]) && ((Abs[Im[x]] < Pi)) && ((Cosh[x] != 0));
  Residual = (ArcSin[Sin[Tanh[x]]]) - (Tanh[x]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(ArcSin[Sin[Tanh[x]]]) - (Tanh[x])`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
result	`Simplify[Residual, Assum]`	→	`ArcSin[Sin[Tanh[x]]] - Tanh[x]`
answer			`null`

Source (conditional.sexp)

(case sp-acj4-171-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[171] + rules.json[125]: asin sin tanh _0  ->  tanh _0  [weakest assumption found: im-lt-pi]; relabelled contingent: false at x = (+ 1/2 (* (/ pi 2) i)) (a probe the random samples miss)")
  (var x complex)
  (assume (< (abs (im x)) pi))
  (assume (nonzero (cosh x)))
  (ask (= (asin (sin (tanh x))) (tanh x)))
  (expect contingent)
  (witness true (x 0) (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 (+ 1/2 (* (/ pi 2) i))) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_c171w2 | exact AE.w2_c171_cosh | exact AE.w2_c171_tanhD | exact AE.w2_c171_lt | 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]")))