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R0014-07771 · q-20261012-advance-20261016-r07-surjective-0021
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.
G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 2, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.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
n=5;m=3;images=[2, 2, 1, 0, 1]
After
all original inputs and outputs form a typed total table;labels start at0
Conditions: none
First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity.
Rule: finite_surjection_typed_scope · Uses: G001
Before
phase=original;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[2, 2, 1, 0, 1]
After
fiber=[3];missing=No
Conditions: none
For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.
Rule: surjection_complete_target_fiber · Uses: G001, S001
Before
phase=original;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[2, 2, 1, 0, 1]
After
fiber=[2, 4];missing=No
Conditions: none
For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.
Rule: surjection_complete_target_fiber · Uses: G001, S002
Before
phase=original;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[2, 2, 1, 0, 1]
After
fiber=[0, 1];missing=No
Conditions: none
For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.
Rule: surjection_complete_target_fiber · Uses: G001, S003
Before
phase=original;targets=[0, 1, 2];fibers=[[3], [2, 4], [0, 1]]
After
missing=[];surjective=Yes
Conditions: none
Surjectivity is relative to the declared codomain:every target needs a preimage.An empty codomain has no targets to test and satisfies the condition;an empty domain with a nonempty codomain misses every target.
Rule: surjection_all_declared_targets · Uses: G001, S004
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, 2, 1, 0, 1], [[3], [2, 4], [0, 1]], [], Yes)
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: 390fa0b0e59121706f46ac609fadfdc0535b564cde41013b299e5c608d920b7ecandidate record SHA-256: 637fbdf30af66c4976ae88e170e673d59d2f6dbf1ab65e969cfdf99cefdaf349semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED