← suite simplipy

sp-acj4-420-cond true

category simplipy-conditional

Summary

Given
\(x \in \mathbb{C},\quad \operatorname{Re}\left(x\right) \ge 0,\quad x \neq 0,\quad \cos\left(-\log x\right) \neq 0,\quad \frac{1}{x} \neq 0,\quad \cos\left(\log\left(\frac{1}{x}\right)\right) \neq 0\)
Claim
\(\displaystyle \tan\left(-\log x\right) = \tan\left(\log\left(\frac{1}{x}\right)\right)\)
Expected
true equal at every admissible point

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0 -> tan log inv _0 [weakest assumption found: re-nonneg]

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(>= (re x) 0) ∧ (nonzero x) ∧ (nonzero (cos (- (log x)))) ∧ (nonzero (/ x)) ∧ (nonzero (cos (log (/ x)))) ⊢ (= (tan (- (log x))) (tan (log (/ x))))null
sympy
sympy-askno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · ask → None0.1682026-10-13 17:21:19 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · ask → None0.0412026-10-12 21:54:02 ee702ae2fe06Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · refine → -tan(log(1/x)) - tan(log(x))0.0512026-10-13 17:22:30 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
refinerefine(Residual, Assum)→-tan(log(1/x)) - tan(log(x))
answernull
sympy-refineno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · refine → -tan(log(1/x)) - tan(log(x))0.0532026-10-12 21:54:16 ee702ae2fe06Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
refinerefine(Residual, Assum)→-tan(log(1/x)) - tan(log(x))
answernull
sympy-simplifyno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · simplify → -tan(log(1/x)) - tan(log(x))0.0542026-10-13 17:24:01 ba9dc4def822Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True, zero=False)
# dropped (not a unary fact about one symbol): re(x) >= 0
# dropped (not a unary fact about one symbol): ~Q.zero(cos(-log(x)))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(log(1 / x)))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
simplifysimplify(Residual)→-tan(log(1/x)) - tan(log(x))
answernull
sympy-simplifyno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · simplify → -tan(log(1/x)) - tan(log(x))0.0592026-10-12 21:54:43 ee702ae2fe06Q.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True, zero=False)
# dropped (not a unary fact about one symbol): re(x) >= 0
# dropped (not a unary fact about one symbol): ~Q.zero(cos(-log(x)))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(log(1 / x)))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
simplifysimplify(Residual)→-tan(log(1/x)) - tan(log(x))
answernull
wolfram
wolfram-fullsimplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0372026-10-13 17:24:42 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0282026-10-13 17:24:31 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0362026-10-13 17:24:17 ba9dc4def822Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultSimplify[Residual, Assum]→0
answertrue
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0372026-10-13 16:43:00 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0292026-10-13 16:42:50 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultRefine[Residual, Assum]→0
answertrue
wolfram-simplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0372026-10-13 16:42:35 3ba0dfac4bcaElement[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultSimplify[Residual, Assum]→0
answertrue
sympy-simplifyno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · simplify → -tan(log(1/x)) - tan(log(x))0.0552026-10-13 16:42:18 3ba0dfac4bcaQ.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True, zero=False)
# dropped (not a unary fact about one symbol): re(x) >= 0
# dropped (not a unary fact about one symbol): ~Q.zero(cos(-log(x)))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / x)
# dropped (not a unary fact about one symbol): ~Q.zero(cos(log(1 / x)))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
simplifysimplify(Residual)→-tan(log(1/x)) - tan(log(x))
answernull
sympy-refineno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · refine → -tan(log(1/x)) - tan(log(x))0.0512026-10-13 16:40:40 3ba0dfac4bcaQ.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
refinerefine(Residual, Assum)→-tan(log(1/x)) - tan(log(x))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(>= (re x) 0) ∧ (nonzero x) ∧ (nonzero (cos (- (log x)))) ∧ (nonzero (/ x)) ∧ (nonzero (cos (log (/ x)))) ⊢ (= (tan (- (log x))) (tan (log (/ x))))null
sympy-askno verdictnulldifference → -tan(log(1/x)) - tan(log(x)) · ask → None0.1692026-10-13 16:39:28 3ba0dfac4bcaQ.complex(x) ∧ (re(x) >= 0) ∧ (~Q.zero(x)) ∧ (~Q.zero(cos((-log(x))))) ∧ (~Q.zero((1/x))) ∧ (~Q.zero(cos(log((1/x))))) ⊢ Eq(tan((-log(x))), tan(log((1/x))))null
code
setup — not timed
from sympy import *
import sympy

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

x = Symbol('x', complex=True)
Assum = And(Q.complex(x), re(x) >= 0, ~Q.zero(x), ~Q.zero(cos(-log(x))), ~Q.zero(1 / x), ~Q.zero(cos(log(1 / x))))
Residual = tan(-log(x)) - tan(log(1 / x))
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')
differencetan(-log(x)) - tan(log(1 / x))→-tan(log(1/x)) - tan(log(x))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0372026-10-12 21:55:24 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultFullSimplify[Residual, Assum]→0
answertrue
wolfram-refine solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0282026-10-12 21:55:14 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultRefine[Residual, Assum]→0
answertrue
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(>= (re x) 0) ∧ (nonzero x) ∧ (nonzero (cos (- (log x)))) ∧ (nonzero (/ x)) ∧ (nonzero (cos (log (/ x)))) ⊢ (= (tan (- (log x))) (tan (log (/ x))))null
wolfram-simplify solvedtruedifference → -Tan[Log[x^(-1)]] - Tan[Log[x]] · result → 00.0372026-10-12 21:55:00 ee702ae2fe06Element[x, Complexes] ∧ (Re[x] >= 0) ∧ (x != 0) ∧ (Cos[(-Log[x])] != 0) ∧ ((1/x) != 0) ∧ (Cos[Log[(1/x)]] != 0) ⊢ (Tan[(-Log[x])] == Tan[Log[(1/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, Complexes]) && ((Re[x] >= 0)) && ((x != 0)) && ((Cos[(-Log[x])] != 0)) && (((1/x) != 0)) && ((Cos[Log[(1/x)]] != 0));
  Residual = (Tan[-Log[x]]) - (Tan[Log[1/x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(Tan[-Log[x]]) - (Tan[Log[1/x]])→-Tan[Log[x^(-1)]] - Tan[Log[x]]
resultSimplify[Residual, Assum]→0
answertrue

Source (conditional.sexp)

(case sp-acj4-420-cond (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[420]: tan neg log _0  ->  tan log inv _0  [weakest assumption found: re-nonneg]")
  (var x complex)
  (assume (>= (re x) 0))
  (assume (nonzero x))
  (assume (nonzero (cos (- (log x)))))
  (assume (nonzero (/ x)))
  (assume (nonzero (cos (log (/ x)))))
  (ask (= (tan (- (log x))) (tan (log (/ x)))))
  (expect true)
  (lean-proof "have hre : 0 ≤ x.re := (by simpa using h0.2); have hL := AE.cond_log_inv_of_re_nonneg hre; rw [hL] at h4 ⊢; exact ⟨⟨⟨h1.2, (AE.tan.D_iff _).mpr (by simpa using h2.2)⟩, ⟨h1.2, h3.2⟩, (AE.tan.D_iff _).mpr (by simpa using h4.2)⟩, rfl⟩"))