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R0014-07708 · q-20261012-advance-20261016-r07-surjective-0033
Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).
G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 0, 0, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Candidate original-order codomain=[2, 1, 0] may use distinct integer labels outside the original Fin(m). Give (original images, sorted new codomain, are all original outputs in it?, missing new targets or outside-codomain, surjectivity or not-evaluated).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, 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
original images=[2, 0, 0, 0, 1];candidate codomain=[0, 1, 2]
After
all original outputs eligible=Yes
Conditions: none
The new codomain may use arbitrary distinct integer labels,but must contain every original output.Do not discard outside outputs to manufacture function eligibility.
Rule: changed_codomain_full_range_guard · Uses: G001, S001
Before
phase=changed codomain;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[2, 0, 0, 0, 1]
After
fiber=[1, 2, 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, S002
Before
phase=changed codomain;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[2, 0, 0, 0, 1]
After
fiber=[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, S003
Before
phase=changed codomain;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[2, 0, 0, 0, 1]
After
fiber=[0];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, S004
Before
phase=changed codomain;targets=[0, 1, 2];fibers=[[1, 2, 3], [4], [0]]
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, S005
Before
qualified=Yes;missing/status=[]
After
surjectivity/status=Yes
Conditions: none
If an output is outside the new codomain,the specified function type is not valid,so surjectivity is unevaluated rather than False.When eligible,use all targets of the new codomain.
Rule: changed_codomain_guarded_result · Uses: G001, S006
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, 0, 0, 1], [0, 1, 2], Yes, [], 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: dd0bbdb0c20ed4ae773b424090a2e42521c2bd89ee030903261a5117a203dd7ccandidate record SHA-256: 21b23d8a23708676850076fa5d0ba4b2f8e1e49222b601e4c002fffa9c7acd81semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED