← suite simplipy

sp-acj4-417-cond true

category simplipy-conditional

Summary

Given
\(x \in \mathbb{R},\quad \left|x\right| < 1,\quad \cos\left(\arcsin\left(\left|x\right|\right)\right) \neq 0,\quad \cos\left(\arcsin x\right) \neq 0\)
Claim
\(\displaystyle \tan\left(\arcsin\left(\left|x\right|\right)\right) = \left|\tan\left(\arcsin x\right)\right|\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0 -> abs tan asin _0 [weakest assumption found: abs-lt-1]

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) ∧ (< (abs x) 1) ∧ (nonzero (cos (asin (abs x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (asin (abs x))) (abs (tan (asin x))))null
sympy
sympy-askno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · ask → None2.8542026-10-13 17:21:19 ba9dc4def822Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · ask → None0.0862026-10-12 21:54:02 ee702ae2fe06Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · refine → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)1.7622026-10-13 17:22:30 ba9dc4def822Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
refinerefine(Residual, Assum)→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
sympy-refineno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · refine → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)0.1542026-10-12 21:54:16 ee702ae2fe06Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
refinerefine(Residual, Assum)→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
sympy-simplifyno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · simplify → -Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)0.0582026-10-13 17:24:01 ba9dc4def822Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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): Abs(x) < 1
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(Abs(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
simplifysimplify(Residual)→-Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
sympy-simplifyno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · simplify → -Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)0.0592026-10-12 21:54:43 ee702ae2fe06Q.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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): Abs(x) < 1
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(Abs(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
simplifysimplify(Residual)→-Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
wolfram
wolfram-fullsimplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0362026-10-13 17:24:42 ba9dc4def822Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0322026-10-13 17:24:31 ba9dc4def822Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0352026-10-13 17:24:17 ba9dc4def822Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultSimplify[Residual, Assum]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0352026-10-13 16:43:00 3ba0dfac4bcaElement[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0332026-10-13 16:42:50 3ba0dfac4bcaElement[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0362026-10-13 16:42:35 3ba0dfac4bcaElement[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultSimplify[Residual, Assum]→0
answertrue
sympy-simplifyno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · simplify → -Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)0.0592026-10-13 16:42:18 3ba0dfac4bcaQ.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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): Abs(x) < 1
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(Abs(x))))
# dropped (not a unary fact about one symbol): ~Q.zero(cos(asin(x)))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
simplifysimplify(Residual)→-Abs(x/sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
sympy-refineno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · refine → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)1.7412026-10-13 16:40:40 3ba0dfac4bcaQ.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
refinerefine(Residual, Assum)→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(real x) ∧ (< (abs x) 1) ∧ (nonzero (cos (asin (abs x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (asin (abs x))) (abs (tan (asin x))))null
sympy-askno verdictnulldifference → -Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2) · ask → None2.8832026-10-13 16:39:28 3ba0dfac4bcaQ.real(x) ∧ (Abs(x) < 1) ∧ (~Q.zero(cos(asin(Abs(x))))) ∧ (~Q.zero(cos(asin(x)))) ⊢ Eq(tan(asin(Abs(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), Abs(x) < 1, ~Q.zero(cos(asin(Abs(x)))), ~Q.zero(cos(asin(x))))
Residual = tan(asin(Abs(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')
differencetan(asin(Abs(x))) - Abs(tan(asin(x)))→-Abs(x)/Abs(sqrt(1 - x**2)) + Abs(x)/sqrt(1 - x**2)
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0352026-10-12 21:55:24 ee702ae2fe06Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0322026-10-12 21:55:14 ee702ae2fe06Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultRefine[Residual, Assum]→0
answertrue
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(real x) ∧ (< (abs x) 1) ∧ (nonzero (cos (asin (abs x)))) ∧ (nonzero (cos (asin x))) ⊢ (= (tan (asin (abs x))) (abs (tan (asin x))))null
wolfram-simplify solvedtruedifference → Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]] · result → 00.0352026-10-12 21:55:00 ee702ae2fe06Element[x, Reals] ∧ (Abs[x] < 1) ∧ (Cos[ArcSin[Abs[x]]] != 0) ∧ (Cos[ArcSin[x]] != 0) ⊢ (Tan[ArcSin[Abs[x]]] == Abs[Tan[ArcSin[x]]])true
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]) && ((Abs[x] < 1)) && ((Cos[ArcSin[Abs[x]]] != 0)) && ((Cos[ArcSin[x]] != 0));
  Residual = (Tan[ArcSin[Abs[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[ArcSin[Abs[x]]]) - (Abs[Tan[ArcSin[x]]])→Abs[x]/Sqrt[1 - Abs[x]^2] - Abs[x/Sqrt[1 - x^2]]
resultSimplify[Residual, Assum]→0
answertrue

Source (conditional.sexp)

(case sp-acj4-417-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[417] + rules.json[257]: tan asin abs _0  ->  abs tan asin _0  [weakest assumption found: abs-lt-1]")
  (var x real)
  (assume (< (abs x) 1))
  (assume (nonzero (cos (asin (abs x)))))
  (assume (nonzero (cos (asin x))))
  (ask (= (tan (asin (abs x))) (abs (tan (asin x)))))
  (expect true)
  (lean-proof "obtain ⟨r, rfl⟩ := AE.IsReal.exists hd0; have hr : |r| < 1 := (by simpa [AE.abs_ofReal] using h0.2); have hr' := abs_lt.mp hr; simp at h1 h2; refine ⟨by simp [h1, h2], ?_⟩; rw [AE.abs_ofReal, AE.asin_ofReal (by linarith [abs_nonneg r]) hr.le, AE.asin_ofReal hr'.1.le hr'.2.le, AE.tan_ofReal, AE.tan_ofReal, AE.abs_ofReal, AE.cond_arcsin_abs, AE.cond_tan_abs_of_mem (Real.neg_pi_div_two_lt_arcsin.mpr hr'.1) (Real.arcsin_lt_pi_div_two.mpr hr'.2)]"))