← suite simplipy

sp-acj4-124-condf contingent

category simplipy-conditional

Summary

Given
\(x \in \mathbb{C},\quad x \in i\mathbb{R},\quad \cosh x \neq 0\)
Claim
\(\displaystyle \left|\arcsin\left(\tanh x\right)\right| \overset{?}{=} \arccos\left(\frac{1}{\cosh x}\right)\)
Expected
contingent true at \(x = 2 \, \pi \, i\); false at \(x = i\)

equal at some admissible points, different at others

witness (true): \(x = 2 \, \pi \, i\)

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

SimpliPy acj-4 (MIT) rules_real.json[124]: abs asin tanh !0 -> acos inv cosh !0 [false under: imaginary]; relabelled contingent: true at x = (* 2 pi i) (a probe the random samples miss)

Lean proof: included

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

systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
none
always-nonedeclined correctlynull0.02026-10-13 17:21:18 ba9dc4def822(imaginary x) ∧ (nonzero (cosh x)) ⊢ (= (abs (asin (tanh x))) (acos (/ (cosh x))))null (correct: contingent)
sympy
sympy-askdeclined correctlynulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · ask → None0.1782026-10-13 17:21:19 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), Q.imaginary(x), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · ask → None0.0462026-10-12 21:54:02 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
askQ.zero(Residual)→None
answernull
sympy-refinedeclined correctlynulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · refine → Abs(asin(tanh(x))) - acos(1/cosh(x))0.0612026-10-13 17:22:30 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), Q.imaginary(x), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
refinerefine(Residual, Assum)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
sympy-refineno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · refine → Abs(asin(tanh(x))) - acos(1/cosh(x))0.0622026-10-12 21:54:16 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
refinerefine(Residual, Assum)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
sympy-simplifydeclined correctlynulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · simplify → Abs(asin(tanh(x))) - acos(1/cosh(x))0.1032026-10-13 17:24:01 ba9dc4def822Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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, imaginary=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
simplifysimplify(Residual)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
sympy-simplifyno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · simplify → Abs(asin(tanh(x))) - acos(1/cosh(x))0.1012026-10-12 21:54:43 ee702ae2fe06Q.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
simplifysimplify(Residual)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
wolfram
wolfram-fullsimplify declined correctlynulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]2.9242026-10-13 17:24:42 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultFullSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
wolfram-refine declined correctlynulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0222026-10-13 17:24:31 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultRefine[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
wolfram-simplify declined correctlynulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0592026-10-13 17:24:17 ba9dc4def822Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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]) && ((Re[x] == 0 && x != 0)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]2.9282026-10-13 16:43:00 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultFullSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
wolfram-refine no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0222026-10-13 16:42:50 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultRefine[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
wolfram-simplify no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0592026-10-13 16:42:35 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
sympy-simplifyno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · simplify → Abs(asin(tanh(x))) - acos(1/cosh(x))0.1032026-10-13 16:42:18 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
simplifysimplify(Residual)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
sympy-refineno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · refine → Abs(asin(tanh(x))) - acos(1/cosh(x))0.0612026-10-13 16:40:40 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
refinerefine(Residual, Assum)→Abs(asin(tanh(x))) - acos(1/cosh(x))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(imaginary x) ∧ (nonzero (cosh x)) ⊢ (= (abs (asin (tanh x))) (acos (/ (cosh x))))null
sympy-askno verdictnulldifference → Abs(asin(tanh(x))) - acos(1/cosh(x)) · ask → None0.1762026-10-13 16:39:28 3ba0dfac4bcaQ.complex(x) ∧ Q.imaginary(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(Abs(asin(tanh(x))), acos((1/cosh(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), ~Q.zero(cosh(x)))
Residual = Abs(asin(tanh(x))) - acos(1 / cosh(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')
differenceAbs(asin(tanh(x))) - acos(1 / cosh(x))→Abs(asin(tanh(x))) - acos(1/cosh(x))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]2.9342026-10-12 21:55:24 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultFullSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
wolfram-refine no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0222026-10-12 21:55:14 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultRefine[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(imaginary x) ∧ (nonzero (cosh x)) ⊢ (= (abs (asin (tanh x))) (acos (/ (cosh x))))null
wolfram-simplify no verdictnulldifference → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]] · result → Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]0.0592026-10-12 21:55:00 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] == 0 && x != 0) ∧ (Cosh[x] != 0) ⊢ (Abs[ArcSin[Tanh[x]]] == ArcCos[(1/Cosh[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)) && ((Cosh[x] != 0));
  Residual = (Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Abs[ArcSin[Tanh[x]]]) - (ArcCos[1/Cosh[x]])→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
resultSimplify[Residual, Assum]→Abs[ArcSin[Tanh[x]]] - ArcCos[Sech[x]]
answernull

Source (conditional.sexp)

(case sp-acj4-124-condf (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[124]: abs asin tanh !0  ->  acos inv cosh !0  [false under: imaginary]; relabelled contingent: true at x = (* 2 pi i) (a probe the random samples miss)")
  (var x complex)
  (assume (imaginary x))
  (assume (nonzero (cosh x)))
  (ask (= (abs (asin (tanh x))) (acos (/ (cosh x)))))
  (expect contingent)
  (witness true (x (* 2 pi i)) (lean-proof "exact AE.w3_c124_w1"))
  (witness false (x i) (lean-proof "exact AE.w3_c124_w2")))