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R0014-07797 · q-20261012-advance-20261016-r07-surjective-0097
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).
G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[1, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Ordered target candidates=[0, 1]. Give (original images, two codomain-eligibility values, fibers or outside-codomain, each least preimage or does-not-exist/outside-codomain).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=[1, 0, 0, 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
query0=0;codomain Fin(3);original images=[1, 0, 0, 0, 1]
After
eligible=Yes;fiber/status=[1, 2, 3];least/status=1
Conditions: none
First check0≤target<m.A legal target with no preimage has an empty fiber and no least preimage.An outside target is not a legal empty fiber,and negative indexing is not used.
Rule: preimage_query_type_and_empty_distinction · Uses: G001, S001
Before
query1=1;codomain Fin(3);original images=[1, 0, 0, 0, 1]
After
eligible=Yes;fiber/status=[0, 4];least/status=0
Conditions: none
First check0≤target<m.A legal target with no preimage has an empty fiber and no least preimage.An outside target is not a legal empty fiber,and negative indexing is not used.
Rule: preimage_query_type_and_empty_distinction · Uses: G001, S002
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.([1, 0, 0, 0, 1], [Yes, Yes], [[1, 2, 3], [0, 4]], [1, 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: 1a728b394e7242a763636de3124243ab876379f8cccea8992f6e35d90dd435b2candidate record SHA-256: a37a043bb5fae239c4e2f09356d54da6cdff31fd31b752089b5b2d6948ea0d02semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED