sp-acj4-412-real contingent

category simplipy-real

Summary

Given

\(x \in \mathbb{R},\quad \cos\left(\left|\arcsin x\right|\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)

Claim

\(\displaystyle \tan\left(\left|\arcsin x\right|\right) \overset{?}{=} \left|\tan\left(\arcsin x\right)\right|\)

Expected

contingent true at \(x = 0\); false at \(x = 2\)

equal at some admissible points, different at others

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

witness (false): \(x = 2\)

SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0 -> abs tan asin _0

Lean proof: included

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

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

none

always-none declined correctly null 0.0 2026-10-13 17:21:18 ba9dc4def822 (real x) ∧ (nonzero (cos (abs (asin x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (abs (asin x))) (abs (tan (asin x)))) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · ask → None 0.89 2026-10-13 17:21:19 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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', real=True)
Assum = And(Q.real(x), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · ask → None 0.079 2026-10-12 21:54:02 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · refine → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) 1.704 2026-10-13 17:22:30 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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', real=True)
Assum = And(Q.real(x), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
answer			`null`

sympy-refine no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · refine → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) 0.148 2026-10-12 21:54:16 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
answer			`null`

sympy-simplify declined correctly null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · simplify → tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2)) 0.081 2026-10-13 17:24:01 ba9dc4def822 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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', real=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(Abs(asin(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2))`
answer			`null`

sympy-simplify no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · simplify → tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2)) 0.087 2026-10-12 21:54:43 ee702ae2fe06 Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(Abs(asin(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2))`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]] 1.06 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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, Reals]) && ((Cos[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`FullSimplify[Residual, Assum]`	→	`-(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]]`
answer			`null`

wolfram-refine declined correctly null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.026 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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, Reals]) && ((Cos[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Refine[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
answer			`null`

wolfram-simplify declined correctly null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.032 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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, Reals]) && ((Cos[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[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 → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]] 1.059 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`FullSimplify[Residual, Assum]`	→	`-(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.026 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Refine[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
answer			`null`

wolfram-simplify no verdict null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.032 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
answer			`null`

sympy-simplify no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · simplify → tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2)) 0.082 2026-10-13 16:42:18 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(Abs(asin(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
simplify	`simplify(Residual)`	→	`tan(Abs(asin(x))) - Abs(x/sqrt(1 - x**2))`
answer			`null`

sympy-refine no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · refine → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) 1.665 2026-10-13 16:40:40 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
refine	`refine(Residual, Assum)`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (real x) ∧ (nonzero (cos (abs (asin x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (abs (asin x))) (abs (tan (asin x)))) null

sympy-ask no verdict null difference → tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2)) · ask → None 0.906 2026-10-13 16:39:28 3ba0dfac4bca Q.real(x) ∧ (~Q.zero(cos(Abs(asin(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(Abs(asin(x))), Abs(tan(asin(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), ~Q.zero(cos(Abs(asin(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(Abs(asin(x))) - Abs(tan(asin(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	`tan(Abs(asin(x))) - Abs(tan(asin(x)))`	→	`tan(Abs(asin(x))) - Abs(x)/Abs(sqrt(1 - x**2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]] 1.072 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`FullSimplify[Residual, Assum]`	→	`-(Abs[x]/Sqrt[Abs[-1 + x^2]]) + Tan[Abs[ArcSin[x]]]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.026 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Refine[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (real x) ∧ (nonzero (cos (abs (asin x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (abs (asin x))) (abs (tan (asin x)))) null

wolfram-simplify no verdict null difference → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] · result → -Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]] 0.032 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ (Cos[Abs[ArcSin[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[Abs[ArcSin[x]]] == Abs[Tan[ArcSin[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[Abs[ArcSin[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tan[Abs[ArcSin[x]]]) - (Abs[Tan[ArcSin[x]]])`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[x/Sqrt[1 - x^2]] + Tan[Abs[ArcSin[x]]]`
answer			`null`

Source (real.sexp)

(case sp-acj4-412-real (category simplipy-real)
  (note "SimpliPy acj-4 (MIT) rules_real.json[412] + rules.json[253]: tan abs asin _0  ->  abs tan asin _0")
  (var x real)
  (assume (nonzero (cos (abs (asin x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (abs (asin x))) (abs (tan (asin 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 2) (lean-proof "constructorm* _ ∧ _ <;> first | exact AE.w2_r412w2 | exact AE.w2_r412_cos | exact AE.w2_r412_tanD | 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, AE.w2_cos_asin_two]")))