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R0007-04371 · q-20261008-advance-20261010-r07-quantifier-0096
U=[0, 1], V=[0, 1, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
G001 — U=[0, 1], V=[0, 1, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
complete finite U=[0, 1]
After
universal on empty domain=True;existential on empty domain=False
Conditions: none
Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain.
Rule: finite_quantifier_scope · Uses: G001
Before
x=0;y=0;relation=eq
After
Yes
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S001
Before
x=0;y=1;relation=eq
After
No
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S002
Before
x=0;y=100;relation=eq
After
No
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S003
Before
x=0;row truths=[Yes, No, No]
After
some y satisfies=Yes
Conditions: none
The inner existential needs at least one true entry;when V is empty this condition is false for every x.
Rule: nested_exists_row · Uses: G001, S004
Before
x=1;y=0;relation=eq
After
No
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S005
Before
x=1;y=1;relation=eq
After
Yes
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S006
Before
x=1;y=100;relation=eq
After
No
Conditions: none
Evaluate this ordered pair without exchanging x and y.
Rule: finite_relation_cell · Uses: G001, S007
Before
x=1;row truths=[No, Yes, No]
After
some y satisfies=Yes
Conditions: none
The inner existential needs at least one true entry;when V is empty this condition is false for every x.
Rule: nested_exists_row · Uses: G001, S008
Before
y=0;column truths=[Yes, No]
After
all x satisfy=No
Conditions: none
The inner universal requires every entry true;when U is empty it holds for each y.
Rule: nested_forall_column · Uses: G001, S009
Before
y=1;column truths=[No, Yes]
After
all x satisfy=No
Conditions: none
The inner universal requires every entry true;when U is empty it holds for each y.
Rule: nested_forall_column · Uses: G001, S010
Before
y=100;column truths=[No, No]
After
all x satisfy=No
Conditions: none
The inner universal requires every entry true;when U is empty it holds for each y.
Rule: nested_forall_column · Uses: G001, S011
Before
exists-by-x=[Yes, Yes];forall-by-y=[No, No, No]
After
(2, 0)
Conditions: none
The first count concerns x with some suitable y;the second concerns y satisfying every x.These are distinct ordered quantified conditions.
Rule: nested_quantifier_count_order · Uses: G001, S012
S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(2, 0)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 246d92a50ced5944cb5aeb2985f469061d3261f92a48ab22da47e6cf5ded5adbcandidate record SHA-256: bd5f1ff4345210e97f9fd4bed329a99413fceed8fa19b7a236cd63ea3895b0e8semantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED