sp-acj4-22-complex contingent

category simplipy-complex-branch

Summary

Given

\(x \in \mathbb{C},\quad \cosh x \neq 0\)

Claim

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

Expected

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

equal at some admissible points, different at others

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

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

SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0 -> abs tanh _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 (nonzero (cosh x)) ⊢ (= (tanh (abs x)) (abs (tanh x))) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → tanh(Abs(x)) - Abs(tanh(x)) · ask → None 0.095 2026-10-13 17:21:19 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null (correct: contingent)

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · ask → None 0.035 2026-10-12 21:54:02 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → tanh(Abs(x)) - Abs(tanh(x)) · refine → tanh(Abs(x)) - Abs(tanh(x)) 0.055 2026-10-13 17:22:30 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null (correct: contingent)

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
refine	`refine(Residual, Assum)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

sympy-refine no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · refine → tanh(Abs(x)) - Abs(tanh(x)) 0.058 2026-10-12 21:54:16 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
refine	`refine(Residual, Assum)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

sympy-simplify declined correctly null difference → tanh(Abs(x)) - Abs(tanh(x)) · simplify → tanh(Abs(x)) - Abs(tanh(x)) 0.095 2026-10-13 17:24:01 ba9dc4def822 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null (correct: contingent)

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
simplify	`simplify(Residual)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

sympy-simplify no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · simplify → tanh(Abs(x)) - Abs(tanh(x)) 0.091 2026-10-12 21:54:43 ee702ae2fe06 Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
simplify	`simplify(Residual)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.069 2026-10-13 17:24:42 ba9dc4def822 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null (correct: contingent)

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

wolfram-refine declined correctly null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.021 2026-10-13 17:24:31 ba9dc4def822 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null (correct: contingent)

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Refine[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

wolfram-simplify declined correctly null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.024 2026-10-13 17:24:17 ba9dc4def822 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null (correct: contingent)

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[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[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.069 2026-10-13 16:43:00 3ba0dfac4bca Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.021 2026-10-13 16:42:50 3ba0dfac4bca Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Refine[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

wolfram-simplify no verdict null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.024 2026-10-13 16:42:35 3ba0dfac4bca Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

sympy-simplify no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · simplify → tanh(Abs(x)) - Abs(tanh(x)) 0.096 2026-10-13 16:42:18 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
# dropped (not a unary fact about one symbol): ~Q.zero(cosh(x))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
simplify	`simplify(Residual)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

sympy-refine no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · refine → tanh(Abs(x)) - Abs(tanh(x)) 0.056 2026-10-13 16:40:40 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
refine	`refine(Residual, Assum)`	→	`tanh(Abs(x)) - Abs(tanh(x))`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:39:28 3ba0dfac4bca (nonzero (cosh x)) ⊢ (= (tanh (abs x)) (abs (tanh x))) null

sympy-ask no verdict null difference → tanh(Abs(x)) - Abs(tanh(x)) · ask → None 0.095 2026-10-13 16:39:28 3ba0dfac4bca Q.complex(x) ∧ (~Q.zero(cosh(x))) ⊢ Eq(tanh(Abs(x)), Abs(tanh(x))) null

code

setup — not timed

from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), ~Q.zero(cosh(x)))
Residual = tanh(Abs(x)) - Abs(tanh(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	`tanh(Abs(x)) - Abs(tanh(x))`	→	`tanh(Abs(x)) - Abs(tanh(x))`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.07 2026-10-12 21:55:24 ee702ae2fe06 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`FullSimplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

wolfram-refine no verdict null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.021 2026-10-12 21:55:14 ee702ae2fe06 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Refine[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 21:55:00 ee702ae2fe06 (nonzero (cosh x)) ⊢ (= (tanh (abs x)) (abs (tanh x))) null

wolfram-simplify no verdict null difference → -Abs[Tanh[x]] + Tanh[Abs[x]] · result → -Abs[Tanh[x]] + Tanh[Abs[x]] 0.024 2026-10-12 21:55:00 ee702ae2fe06 Element[x, Complexes] ∧ (Cosh[x] != 0) ⊢ (Tanh[Abs[x]] == Abs[Tanh[x]]) null

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[x, Complexes]) && ((Cosh[x] != 0));
  Residual = (Tanh[Abs[x]]) - (Abs[Tanh[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Tanh[Abs[x]]) - (Abs[Tanh[x]])`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
result	`Simplify[Residual, Assum]`	→	`-Abs[Tanh[x]] + Tanh[Abs[x]]`
answer			`null`

Source (complex-branch.sexp)

(case sp-acj4-22-complex (category simplipy-complex-branch)
  (note "SimpliPy acj-4 (MIT) rules_real.json[22] + rules.json[17]: tanh abs _0  ->  abs tanh _0")
  (var x complex)
  (assume (nonzero (cosh x)))
  (ask (= (tanh (abs x)) (abs (tanh x))))
  (expect contingent)
  (witness true (x 0) (lean-proof "first | (simp [AE.abs_eq]; done) | (norm_num [AE.abs_eq]; done) | (simp [AE.abs_eq, Complex.ext_iff]; done) | (norm_num [AE.abs_eq, Complex.ext_iff]; done)"))
  (witness false (x i) (lean-proof "norm_num [AE.abs_eq, AE.w1_asin_zero, AE.w1_asin_one, AE.w1_acos_one, AE.w1_acos_zero, AE.w1_acosh_one, AE.w1_atan_zero, AE.w1_atan_one, AE.w1_atanh_zero, AE.w1_asinh_zero, AE.w1_asinh_one, AE.w1_pow_one_left, AE.w1_surd_one, AE.w1_isInt_ofNat, AE.w1_isInt_neg_ofNat, AE.w1_pow_zero_right, Real.pi_pos, AE.w1_isInt_5, AE.w1_isInt_7, AE.w1_isInt_neg5, AE.w1_isInt_neg7, AE.w1_one_ne_I, AE.w1_one_ne_neg_I, AE.w1_acos_cos_one, AE.w1_asin_cos_pi4, AE.w1_cos_pi4_ne_one, AE.w1_cos_pi4_ne_neg_one, AE.w1_atanh_cos_pi4, AE.w1_tanh_abs_arsinh_one, AE.w1_norm_tanh_arsinh_one, AE.w1_norm_log_sin_one, AE.w1_norm_log_tanh_one, AE.w1_sin_one_ne, AE.w1_tanh_one_ne, AE.w1_cosh_one_ne, AE.cosh_ofReal_ne_zero, Complex.log_neg_one, AE.w1_log_inv_pi_I, AE.w1_tanh_arsinh_one', AE.w1_log_inv_pi_I', AE.w1_cos_two_ne, AE.w1_cosh_I_ne, AE.w1_cos_I_ne, AE.w1_cosh_2I_ne, AE.w1_cos_piI_ne, AE.w1_tanh_2I_ne_one, AE.w1_tanh_2I_ne_neg_one, AE.w1_neg_piI_ne, AE.w1_piI_ne_neg, AE.w1_c0, AE.w1_c6, AE.w1_c20, AE.w1_c252, AE.w1_c22, AE.w1_c253, AE.w1_c266, AE.w1_c267, AE.w1_c274, AE.w1_c278, AE.w1_pow_neg_one_neg_one, AE.w1_cos_asin_I, AE.w1_sin_acos_I, AE.w1_cos_asin_neg_I, AE.w1_sqrt2_ne_one, AE.w1_sqrt2_ne_neg_one, AE.w1_atanh_sqrt2_ne, AE.w1_c131, AE.w1_c130, AE.w1_c260, AE.w1_c261, AE.w1_log_half_ne, AE.w1_c442, AE.w1_odd_5, AE.w1_odd_7, AE.w1_c387, AE.w1_c389, AE.w1_c336, AE.w1_c177, AE.w1_ofReal_ne_I, AE.w1_ofReal_ne_neg_I, AE.w1_zero_ne_sqrt2, AE.w1_pow_ne_zero, AE.w1_c50, AE.w1_c51, AE.w1_c55, AE.w1_c56, AE.w1_c335, AE.w1_c334, AE.w1_c137, AE.w1_c173, AE.w1_c176, AE.w1_c154, AE.w1_c153, AE.w1_c132, AE.w1_c133, AE.w1_c151, AE.w1_halfI_ne_negI, AE.w1_negIhalf_ne_I, AE.w1_cos_atan_halfI_ne_one, AE.w1_cos_atan_halfI_ne_neg_one, AE.w1_c190, AE.w1_c422, AE.w1_c424, AE.w1_cosh_asinh_two_ne, AE.w1_c446a, AE.w1_c446, AE.w1_c396, AE.w1_c404, AE.w1_c414a, AE.w1_c414b, AE.w1_c414, AE.w1_c203, AE.w1_c372, AE.w1_c373, AE.w1_cosh_2I_im, AE.w1_odd_neg5, AE.w1_c344, AE.w1_c234, AE.w1_c421a, AE.w1_c421b, AE.w1_c421, AE.w1_c415a, AE.w1_c415, AE.w1_one_add_I_ne, AE.w1_one_add_I_ne_negI, AE.w1_inv_one_add_I_ne_I, AE.w1_inv_one_add_I_ne_negI, AE.w1_cosh_asinh_one_add_I_ne, AE.w1_c147, AE.w1_c165]")))