← suite simplipy

sp-acj4-442-cond4 true

category simplipy-conditional

Summary

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

the two sides are equal at every admissible point

SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0 -> log inv log ?0 [weakest assumption found: integer]

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(integer x) ∧ (nonzero x) ∧ (nonzero (log x)) ∧ (nonzero (/ (log x))) ⊢ (= (- (log (log x))) (log (/ (log x))))null
sympy
sympy-askno verdictnulldifference → -log(1/log(x)) - log(log(x)) · ask → None0.3012026-10-13 17:21:19 ba9dc4def822Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
askQ.zero(Residual)→None
answernull
sympy-askno verdictnulldifference → -log(1/log(x)) - log(log(x)) · ask → None0.0392026-10-12 21:54:02 ee702ae2fe06Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
askQ.zero(Residual)→None
answernull
sympy-refineno verdictnulldifference → -log(1/log(x)) - log(log(x)) · refine → -log(1/log(x)) - log(log(x))0.0512026-10-13 17:22:30 ba9dc4def822Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
refinerefine(Residual, Assum)→-log(1/log(x)) - log(log(x))
answernull
sympy-refineno verdictnulldifference → -log(1/log(x)) - log(log(x)) · refine → -log(1/log(x)) - log(log(x))0.0472026-10-12 21:54:16 ee702ae2fe06Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
refinerefine(Residual, Assum)→-log(1/log(x)) - log(log(x))
answernull
sympy-simplifyno verdictnulldifference → -log(1/log(x)) - log(log(x)) · simplify → -log(1/log(x)) - log(log(x))0.0252026-10-13 17:24:01 ba9dc4def822Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(log(x))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / log(x))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
simplifysimplify(Residual)→-log(1/log(x)) - log(log(x))
answernull
sympy-simplifyno verdictnulldifference → -log(1/log(x)) - log(log(x)) · simplify → -log(1/log(x)) - log(log(x))0.0262026-10-12 21:54:43 ee702ae2fe06Q.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(log(x))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / log(x))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
simplifysimplify(Residual)→-log(1/log(x)) - log(log(x))
answernull
wolfram
wolfram-fullsimplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.1842026-10-13 17:24:42 ba9dc4def822Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultFullSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
wolfram-refine no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0282026-10-13 17:24:31 ba9dc4def822Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultRefine[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
wolfram-simplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0312026-10-13 17:24:17 ba9dc4def822Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
Older runs (11)
systemoutcomeanswersteps (from the trace)srunquery (as sent)detail
wolfram-fullsimplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.1852026-10-13 16:43:00 3ba0dfac4bcaElement[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultFullSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
wolfram-refine no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0292026-10-13 16:42:50 3ba0dfac4bcaElement[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultRefine[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
wolfram-simplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0312026-10-13 16:42:35 3ba0dfac4bcaElement[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
sympy-simplifyno verdictnulldifference → -log(1/log(x)) - log(log(x)) · simplify → -log(1/log(x)) - log(log(x))0.0252026-10-13 16:42:18 3ba0dfac4bcaQ.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True, zero=False)
# dropped (not a unary fact about one symbol): ~Q.zero(log(x))
# dropped (not a unary fact about one symbol): ~Q.zero(1 / log(x))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
simplifysimplify(Residual)→-log(1/log(x)) - log(log(x))
answernull
sympy-refineno verdictnulldifference → -log(1/log(x)) - log(log(x)) · refine → -log(1/log(x)) - log(log(x))0.0522026-10-13 16:40:40 3ba0dfac4bcaQ.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
refinerefine(Residual, Assum)→-log(1/log(x)) - log(log(x))
answernull
always-noneno verdictnull0.02026-10-13 16:39:28 3ba0dfac4bca(integer x) ∧ (nonzero x) ∧ (nonzero (log x)) ∧ (nonzero (/ (log x))) ⊢ (= (- (log (log x))) (log (/ (log x))))null
sympy-askno verdictnulldifference → -log(1/log(x)) - log(log(x)) · ask → None0.2992026-10-13 16:39:28 3ba0dfac4bcaQ.integer(x) ∧ (~Q.zero(x)) ∧ (~Q.zero(log(x))) ∧ (~Q.zero((1/log(x)))) ⊢ Eq((-log(log(x))), log((1/log(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', integer=True)
Assum = And(Q.integer(x), ~Q.zero(x), ~Q.zero(log(x)), ~Q.zero(1 / log(x)))
Residual = -log(log(x)) - log(1 / log(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-log(log(x)) - log(1 / log(x))→-log(1/log(x)) - log(log(x))
askQ.zero(Residual)→None
answernull
wolfram-fullsimplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.1842026-10-12 21:55:24 ee702ae2fe06Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultFullSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
wolfram-refine no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0282026-10-12 21:55:14 ee702ae2fe06Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultRefine[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull
always-noneno verdictnull0.02026-10-12 21:55:00 ee702ae2fe06(integer x) ∧ (nonzero x) ∧ (nonzero (log x)) ∧ (nonzero (/ (log x))) ⊢ (= (- (log (log x))) (log (/ (log x))))null
wolfram-simplify no verdictnulldifference → -Log[Log[x]^(-1)] - Log[Log[x]] · result → -Log[Log[x]^(-1)] - Log[Log[x]]0.0312026-10-12 21:55:00 ee702ae2fe06Element[x, Integers] ∧ (x != 0) ∧ (Log[x] != 0) ∧ ((1/Log[x]) != 0) ⊢ ((-Log[Log[x]]) == Log[(1/Log[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, Integers]) && ((x != 0)) && ((Log[x] != 0)) && (((1/Log[x]) != 0));
  Residual = (-Log[Log[x]]) - (Log[1/Log[x]]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)
difference(-Log[Log[x]]) - (Log[1/Log[x]])→-Log[Log[x]^(-1)] - Log[Log[x]]
resultSimplify[Residual, Assum]→-Log[Log[x]^(-1)] - Log[Log[x]]
answernull

Source (conditional.sexp)

(case sp-acj4-442-cond4 (category simplipy-conditional)
  (note "SimpliPy acj-4 (MIT) rules_real.json[442]: neg log log ?0  ->  log inv log ?0  [weakest assumption found: integer]")
  (var x integer)
  (assume (nonzero x))
  (assume (nonzero (log x)))
  (assume (nonzero (/ (log x))))
  (ask (= (- (log (log x))) (log (/ (log x)))))
  (expect true)
  (lean-proof "obtain ⟨n, rfl⟩ := hd0; have hn : (n:ℂ) ≠ 0 := h0.2; have hn' : n ≠ 0 := (by exact_mod_cast hn); have hre : 0 ≤ (AE.log (n:ℂ)).re := (by rw [AE.log_def, Complex.log_re]; apply Real.log_nonneg; rw [Complex.norm_intCast]; exact_mod_cast Int.one_le_abs hn'); refine ⟨⟨⟨h1.1.1, h1.2⟩, ⟨h1.1.1, h1.2⟩, h2.2⟩, ?_⟩; rw [AE.cond_log_inv_of (Or.inl hre)]"))