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R0013-07201 · q-20261011-advance-20261015-r03-domain-codomain-range-0001
Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).
G001 — Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).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(0)→Fin(0);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
complete graph=[]
After
range=[];unused codomain=[];range size=0
Conditions: none
The range is the deduplicated output set,while declared n and m still fix domain and codomain sizes.A smaller range does not shrink the codomain.
Rule: typed_graph_range_inventory · Uses: G001, S001
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, 0], [], [], [], 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: ee55dfe058504dca1327b24b06c9b32593d89cd77896d510694257edc464b939candidate record SHA-256: ba15a34db26953c2c1894de0b25c393ebfcb93c06aa96fc37cfc6517a2c9fd5fsemantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED