demo-log-02 contingent

category branch-cuts

Summary

Given

\(z, w \in \mathbb{C},\quad z \neq 0,\quad w \neq 0\)

Claim

\(\displaystyle \log\left(z \, w\right) \overset{?}{=} \log z + \log w\)

Expected

contingent true at \(z = 1, w = 1\); false at \(z = -1, w = -1\)

equal at some admissible points, different at others

witness (true): \(z = 1, w = 1\)

witness (false): \(z = -1, w = -1\)

Lean proof: included

Answers (latest run per system; open code for the reproducer that ran)

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

none

always-none declined correctly null 0.0 2026-10-13 17:20:05 df65d778fc5c (nonzero z) ∧ (nonzero w) ⊢ (= (log (* z w)) (+ (log z) (log w))) null (correct: contingent)

sympy

sympy-ask declined correctly null difference → -log(w) - log(z) + log(w*z) · ask → None 0.136 2026-10-13 17:20:46 df65d778fc5c Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
# 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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → -log(w) - log(z) + log(w*z) · ask → None 0.037 2026-10-12 20:21:28 d7af4def9aaa Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
# 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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine declined correctly null difference → -log(w) - log(z) + log(w*z) · refine → -log(w) - log(z) + log(w*z) 0.037 2026-10-13 17:20:49 df65d778fc5c Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
refine	`refine(Residual, Assum)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

sympy-refine no verdict null difference → -log(w) - log(z) + log(w*z) · refine → -log(w) - log(z) + log(w*z) 0.039 2026-10-12 20:21:30 d7af4def9aaa Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
refine	`refine(Residual, Assum)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

sympy-simplify declined correctly null difference → -log(w) - log(z) + log(w*z) · simplify → -log(w) - log(z) + log(w*z) 0.026 2026-10-13 17:20:51 df65d778fc5c Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True, zero=False)
w = Symbol('w', complex=True, zero=False)
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
simplify	`simplify(Residual)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

sympy-simplify no verdict null difference → -log(w) - log(z) + log(w*z) · simplify → -log(w) - log(z) + log(w*z) 0.027 2026-10-12 20:21:32 d7af4def9aaa Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True, zero=False)
w = Symbol('w', complex=True, zero=False)
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
simplify	`simplify(Residual)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

wolfram

wolfram-fullsimplify declined correctly null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.053 2026-10-13 17:20:59 df65d778fc5c Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`FullSimplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-refine declined correctly null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.021 2026-10-13 17:20:56 df65d778fc5c Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Refine[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-simplify declined correctly null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.023 2026-10-13 17:20:52 df65d778fc5c Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Simplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

Older runs (11)

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

wolfram-fullsimplify no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.054 2026-10-13 16:39:09 03f5149718d0 Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`FullSimplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-refine no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.022 2026-10-13 16:39:06 03f5149718d0 Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Refine[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-simplify no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.023 2026-10-13 16:39:02 03f5149718d0 Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Simplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

sympy-simplify no verdict null difference → -log(w) - log(z) + log(w*z) · simplify → -log(w) - log(z) + log(w*z) 0.027 2026-10-13 16:39:01 03f5149718d0 Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True, zero=False)
w = Symbol('w', complex=True, zero=False)
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
simplify	`simplify(Residual)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

sympy-refine no verdict null difference → -log(w) - log(z) + log(w*z) · refine → -log(w) - log(z) + log(w*z) 0.046 2026-10-13 16:38:59 03f5149718d0 Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
refine	`refine(Residual, Assum)`	→	`-log(w) - log(z) + log(w*z)`
answer			`null`

always-none no verdict null 0.0 2026-10-13 16:38:56 03f5149718d0 (nonzero z) ∧ (nonzero w) ⊢ (= (log (* z w)) (+ (log z) (log w))) null

sympy-ask no verdict null difference → -log(w) - log(z) + log(w*z) · ask → None 0.135 2026-10-13 16:38:56 03f5149718d0 Q.complex(z) ∧ Q.complex(w) ∧ (~Q.zero(z)) ∧ (~Q.zero(w)) ⊢ Eq(log((z*w)), (log(z) + log(w))) 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()

z = Symbol('z', complex=True)
w = Symbol('w', complex=True)
Assum = And(Q.complex(z), Q.complex(w), ~Q.zero(z), ~Q.zero(w))
Residual = log(z * w) - (log(z) + log(w))
# 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(z * w) - (log(z) + log(w))`	→	`-log(w) - log(z) + log(w*z)`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.053 2026-10-12 20:21:41 d7af4def9aaa Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`FullSimplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-refine no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.022 2026-10-12 20:21:37 d7af4def9aaa Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Refine[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

wolfram-simplify no verdict null difference → -Log[w] - Log[z] + Log[w*z] · result → -Log[w] - Log[z] + Log[w*z] 0.023 2026-10-12 20:21:34 d7af4def9aaa Element[z, Complexes] ∧ Element[w, Complexes] ∧ (z != 0) ∧ (w != 0) ⊢ (Log[(z*w)] == (Log[z] + Log[w])) 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[z, Complexes]) && (Element[w, Complexes]) && ((z != 0)) && ((w != 0));
  Residual = (Log[z*w]) - (Log[z] + Log[w]);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`(Log[z*w]) - (Log[z] + Log[w])`	→	`-Log[w] - Log[z] + Log[w*z]`
result	`Simplify[Residual, Assum]`	→	`-Log[w] - Log[z] + Log[w*z]`
answer			`null`

always-none no verdict null 0.0 2026-10-12 20:21:33 d7af4def9aaa (nonzero z) ∧ (nonzero w) ⊢ (= (log (* z w)) (+ (log z) (log w))) null

Source (branch.sexp)

(case demo-log-02 (category branch-cuts)
  (var z complex) (var w complex)
  (assume (nonzero z)) (assume (nonzero w))
  (ask (= (log (* z w)) (+ (log z) (log w))))
  (expect contingent)
  (witness true (z 1) (w 1)
    (lean-proof "simp [AE.nonzero, AE.nonzero.D, AE.log]"))
  (witness false (z -1) (w -1)
    (lean-proof "refine ⟨by simp [AE.nonzero, AE.nonzero.D], by simp [AE.nonzero, AE.nonzero.D], by norm_num, fun h => ?_⟩; have h1 := congrArg Complex.im h; simp [AE.log, Complex.log_neg_one] at h1; linarith [Real.pi_pos]")))