sp-acj4-145-cond2 true

category simplipy-conditional

Summary

Given

\(x \in \mathbb{R},\quad \left|x\right| \le 1\)

Claim

\(\displaystyle \left|\sin\left(\arccos x\right)\right| = \cos\left(\arcsin x\right)\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0 -> cos asin _0 [weakest assumption found: abs-le-1]

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 no verdict null 0.0 2026-10-13 17:21:18 ba9dc4def822 (real x) ∧ (<= (abs x) 1) ⊢ (= (abs (sin (acos x))) (cos (asin x))) null

sympy

sympy-ask no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · ask → None 0.853 2026-10-13 17:21:19 ba9dc4def822 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · ask → None 0.06 2026-10-12 21:54:02 ee702ae2fe06 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · refine → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.458 2026-10-13 17:22:30 ba9dc4def822 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
refine	`refine(Residual, Assum)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

sympy-refine no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · refine → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.086 2026-10-12 21:54:16 ee702ae2fe06 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
refine	`refine(Residual, Assum)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

sympy-simplify no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · simplify → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.037 2026-10-13 17:24:01 ba9dc4def822 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
simplify	`simplify(Residual)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

sympy-simplify no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · simplify → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.038 2026-10-12 21:54:43 ee702ae2fe06 Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
simplify	`simplify(Residual)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

wolfram

wolfram-fullsimplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.03 2026-10-13 17:24:42 ba9dc4def822 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.029 2026-10-13 17:24:31 ba9dc4def822 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.03 2026-10-13 17:24:17 ba9dc4def822 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

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

wolfram-fullsimplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.033 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.032 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.032 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · simplify → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.04 2026-10-13 16:42:18 3ba0dfac4bca Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
simplify	`simplify(Residual)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

sympy-refine no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · refine → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) 0.457 2026-10-13 16:40:40 3ba0dfac4bca Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
refine	`refine(Residual, Assum)`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (real x) ∧ (<= (abs x) 1) ⊢ (= (abs (sin (acos x))) (cos (asin x))) null

sympy-ask no verdict null difference → -sqrt(1 - x**2) + Abs(sqrt(1 - x**2)) · ask → None 0.845 2026-10-13 16:39:28 3ba0dfac4bca Q.real(x) ∧ (Abs(x) <= 1) ⊢ Eq(Abs(sin(acos(x))), cos(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)
Residual = Abs(sin(acos(x))) - cos(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	`Abs(sin(acos(x))) - cos(asin(x))`	→	`-sqrt(1 - x2) + Abs(sqrt(1 - x2))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.03 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.03 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (real x) ∧ (<= (abs x) 1) ⊢ (= (abs (sin (acos x))) (cos (asin x))) null

wolfram-simplify solved true difference → -Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]] · result → 0 0.032 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Reals] ∧ (Abs[x] <= 1) ⊢ (Abs[Sin[ArcCos[x]]] == Cos[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));
  Residual = (Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Abs[Sin[ArcCos[x]]]) - (Cos[ArcSin[x]])`	→	`-Sqrt[1 - x^2] + Sqrt[Abs[1 - x^2]]`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Source (conditional.sexp)

(case sp-acj4-145-cond2 (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[145] + rules.json[113]: abs sin acos _0  ->  cos asin _0  [weakest assumption found: abs-le-1]")
  (var x real)
  (assume (<= (abs x) 1))
  (ask (= (abs (sin (acos x))) (cos (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_le.mp hr; refine ⟨by simp, ?_⟩; rw [AE.acos_ofReal hr'.1 hr'.2, AE.asin_ofReal hr'.1 hr'.2, AE.sin_ofReal, AE.cos_ofReal, AE.abs_ofReal, AE.cond_sin_arccos_eq_cos_arcsin, abs_of_nonneg (Real.cos_arcsin_nonneg r)]"))