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R0013-07277 · q-20261011-advance-20261015-r03-domain-codomain-range-0077
Source=Fin(1)={0,…,0}, codomain=Fin(1)={0,…,0}; each input must have exactly one data row. Candidate raw rows=[[0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.
G001 — Source=Fin(1)={0,…,0}, codomain=Fin(1)={0,…,0}; each input must have exactly one data row. Candidate raw rows=[[0, 0]]. Give ([n,m], row-endpoint typing mask, raw occurrence counts for each legal input, missing inputs, repeated inputs, total-table eligibility, ordered images or not-total-table). Count by input before rejecting an outside output.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
declared types Fin(1)→Fin(1);0-based source labels
After
domain,codomain and actual range remain distinct
Conditions: none
Fin(0) is empty;a complete total table is not a partial program function,and its actual range need not equal its declared codomain.This audits finite data,not general infinite-set theorems.
Rule: typed_domain_codomain_data_scope · Uses: G001
Before
raw row position=0;input=0;output=0
After
both ends typed=Yes
Conditions: none
Check both endpoint types row by row,but count every raw occurrence of a legal input even if its output is outside the codomain.
Rule: raw_candidate_table_row_type · Uses: G001, S001
Before
source input=0;raw row count=1
After
exactly one row=Yes
Conditions: none
The one-row protocol requires exactly one raw row per source.Even identical repeated values violate that format,but do not imply the deduplicated mathematical relation is multivalued.
Rule: one_raw_row_per_input_audit · Uses: G001, S002
Before
row typing=[Yes];input counts=[1]
After
complete one-row table=Yes;ordered images=[0]
Conditions: none
Only both conditions permit orderedImages by source index.An empty source with no rows is a complete empty table;do not construct a table for an illegal candidate.
Rule: candidate_table_protocol_not_relation_multivaluedness · Uses: G001, S003
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.([1, 1], [Yes], [1], [], [], Yes, [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: 269e87d05fdfe949060efc80e5fb005c8ef3bab2a448d25a583bb0ef68f2b6dacandidate record SHA-256: 97bb19a41f98682aac78f9e18988d87ae85b52702f250fb0bec7f1e1c86a5a6csemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED