Explore related knowledge and questions →
← Question derivation index · Original question batch
R0014-07727 · q-20261012-advance-20261016-r07-surjective-0052
Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).
G001 — Finite types start at 0: Fin(3)={0,…,2}→Fin(2), complete images=[0, 1, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Delete original input at=1 and keep the original codomain. Give (original images, original missing targets, retained original labels, remaining images, later missing targets, later surjectivity).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=3;m=2;images=[0, 1, 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];images on labels=[0, 1, 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, S001
Before
phase=original;target=1;source labels=[0, 1, 2];images on labels=[0, 1, 1]
After
fiber=[1, 2];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;targets=[0, 1];fibers=[[0], [1, 2]]
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, S003
Before
deleted original input=1;old labels=[0, 1, 2]
After
kept labels=[0, 2];remaining images=[0, 1];codomain Fin(2)
Conditions: none
Delete the specified original input without changing the codomain or renumbering surviving labels.The remaining images list is ordered by retained labels,not a declaration of relabeled inputs.
Rule: delete_input_keeps_original_labels_and_codomain · Uses: G001, S004
Before
phase=after deletion;target=0;source labels=[0, 2];images on labels=[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, S005
Before
phase=after deletion;target=1;source labels=[0, 2];images on labels=[0, 1]
After
fiber=[2];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, S006
Before
phase=after deletion;targets=[0, 1];fibers=[[0], [2]]
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, S007
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.([0, 1, 1], [], [0, 2], [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: 216c0597c1a1ab9fb8211dc2c5cdf0f9f0802bf167b0cbc903494c78961c2155candidate record SHA-256: 66d6fb05afb015fe70faa070cc584e476114b0b2483e20b8868783fbc85fb78fsemantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705Lean: NOT_RUN · human review: NOT_PERFORMED