← suite simplipy

sp-acj4-190-real true

category simplipy-f64-edge

Summary

Given
\(x \in \mathbb{R},\quad \cos\left(\arctan x\right) \neq 1,\quad \cos\left(\arctan x\right) \neq -1,\quad x \neq 0\)
Claim
\(\displaystyle \operatorname{artanh}\left(\cos\left(\arctan x\right)\right) = \left|\operatorname{arsinh}\left(\frac{1}{x}\right)\right|\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0 -> abs asinh inv _0

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-noneno verdictnull0.02026-10-13 17:21:18 ba9dc4def822(real x) ∧ (!= (cos (atan x)) 1) ∧ (!= (cos (atan x)) -1) ∧ (nonzero x) ⊢ (= (atanh (cos (atan x))) (abs (asinh (/ x))))null
sympy
sympy-askno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · ask → None0.2462026-10-13 17:21:19 ba9dc4def822Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · ask → None0.0542026-10-12 21:54:02 ee702ae2fe06Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · refine → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0952026-10-13 17:22:30 ba9dc4def822Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
refinerefine(Residual, Assum)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
sympy-refineno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · refine → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0712026-10-12 21:54:16 ee702ae2fe06Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
refinerefine(Residual, Assum)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
sympy-simplifyno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · simplify → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0442026-10-13 17:24:01 ba9dc4def822Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), -1, evaluate=False)
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
simplifysimplify(Residual)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
sympy-simplifyno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · simplify → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0442026-10-12 21:54:43 ee702ae2fe06Q.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), -1, evaluate=False)
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
simplifysimplify(Residual)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
wolfram
wolfram-fullsimplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]0.6372026-10-13 17:24:42 ba9dc4def822Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultFullSimplify[Residual, Assum]→-Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]
answernull
wolfram-refine no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0852026-10-13 17:24:31 ba9dc4def822Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultRefine[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull
wolfram-simplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0992026-10-13 17:24:17 ba9dc4def822Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultSimplify[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]0.6342026-10-13 16:43:00 3ba0dfac4bcaElement[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultFullSimplify[Residual, Assum]→-Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]
answernull
wolfram-refine no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0852026-10-13 16:42:50 3ba0dfac4bcaElement[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultRefine[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull
wolfram-simplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0982026-10-13 16:42:35 3ba0dfac4bcaElement[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultSimplify[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull
sympy-simplifyno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · simplify → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0432026-10-13 16:42:18 3ba0dfac4bcaQ.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True, zero=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), 1, evaluate=False)
# dropped (not a unary fact about one symbol): Ne(cos(atan(x)), -1, evaluate=False)
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
simplifysimplify(Residual)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
sympy-refineno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · refine → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))0.0952026-10-13 16:40:40 3ba0dfac4bcaQ.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
refinerefine(Residual, Assum)→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(real x) ∧ (!= (cos (atan x)) 1) ∧ (!= (cos (atan x)) -1) ∧ (nonzero x) ⊢ (= (atanh (cos (atan x))) (abs (asinh (/ x))))null
sympy-askno verdictnulldifference → -Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1)) · ask → None0.2472026-10-13 16:39:28 3ba0dfac4bcaQ.real(x) ∧ Ne(cos(atan(x)), 1) ∧ Ne(cos(atan(x)), (-1)) ∧ (~Q.zero(x)) ⊢ Eq(atanh(cos(atan(x))), Abs(asinh((1/x))))null
code
setup — not timed
from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

x = Symbol('x', real=True)
Assum = And(Q.real(x), Ne(cos(atan(x)), 1, evaluate=False), Ne(cos(atan(x)), -1, evaluate=False), ~Q.zero(x))
Residual = atanh(cos(atan(x))) - Abs(asinh(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')
differenceatanh(cos(atan(x))) - Abs(asinh(1 / x))→-Abs(asinh(1/x)) + atanh(1/sqrt(x**2 + 1))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]0.6372026-10-12 21:55:24 ee702ae2fe06Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultFullSimplify[Residual, Assum]→-Abs[ArcCsch[x]] + ArcCoth[Sqrt[1 + x^2]]
answernull
wolfram-refine no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0852026-10-12 21:55:14 ee702ae2fe06Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultRefine[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(real x) ∧ (!= (cos (atan x)) 1) ∧ (!= (cos (atan x)) -1) ∧ (nonzero x) ⊢ (= (atanh (cos (atan x))) (abs (asinh (/ x))))null
wolfram-simplify no verdictnulldifference → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]] · result → -Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]0.0972026-10-12 21:55:00 ee702ae2fe06Element[x, Reals] ∧ (Cos[ArcTan[x]] != 1) ∧ (Cos[ArcTan[x]] != (-1)) ∧ (x != 0) ⊢ (ArcTanh[Cos[ArcTan[x]]] == Abs[ArcSinh[(1/x)]])null
code
setup — not timed
(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)
(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Reals]) && ((Cos[ArcTan[x]] != 1)) && ((Cos[ArcTan[x]] != (-1))) && ((x != 0));
  Residual = (ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(ArcTanh[Cos[ArcTan[x]]]) - (Abs[ArcSinh[1/x]])→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
resultSimplify[Residual, Assum]→-Abs[ArcSinh[x^(-1)]] + ArcTanh[1/Sqrt[1 + x^2]]
answernull

Source (f64-edge.sexp)

(case sp-acj4-190-real (category simplipy-f64-edge)
  (note "SimpliPy acj-4 (MIT) rules_real.json[190]: atanh cos atan _0  ->  abs asinh inv _0")
  (var x real)
  (assume (!= (cos (atan x)) 1))
  (assume (!= (cos (atan x)) -1))
  (assume (nonzero x))
  (ask (= (atanh (cos (atan x))) (abs (asinh (/ x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : r ≠ 0 := (by simpa using h2.2); have hS : 1 < Real.sqrt (1 + r ^ 2) := (Real.lt_sqrt zero_le_one).2 (by have := pow_pos (abs_pos.2 hr) 2; rw [sq_abs] at this; linarith); have hS0 : 0 < Real.sqrt (1 + r ^ 2) := (by linarith); refine ⟨⟨⟨⟨AE.atan.D_ofReal r, trivial⟩, (AE.atanh.D_iff _).2 ⟨h0.2, h1.2⟩⟩, ⟨h2.2, by simp⟩, by simp⟩, ?_⟩; rw [AE.atan_ofReal, AE.cos_ofReal, Real.cos_arctan, AE.atanh_ofReal (by have := one_div_pos.2 hS0; linarith) ((div_lt_one hS0).2 hS), AE.inv_ofReal, AE.asinh_ofReal, AE.abs_ofReal, AE.artanh_eq_abs_arsinh hr]"))