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R0010-05552 · q-20261008-advance-20261012-r07-function-mapping-0002
The complete total map f:Fin(1)→Fin(1) has images=[0] indexed from 0; Fin(1)={0,…,0}. Every table entry must lie in the target type. List every [i,f(i)] in source-index order, using [] for an empty source.
G001 — The complete total map f:Fin(1)→Fin(1) has images=[0] indexed from 0; Fin(1)={0,…,0}. Every table entry must lie in the target type. List every [i,f(i)] in source-index order, using [] for an empty source.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
f:Fin(1)→Fin(1);source indices start at0
After
complete total map,not a partial program function
Conditions: none
Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal input.
Rule: total_finite_mapping_scope · Uses: G001
Before
f input=0∈Fin(1)
After
f(0)=0∈Fin(1)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_mapping_entry · Uses: G001, S001
Before
[0]
After
[[0, 0]]
Conditions: none
List[i,f(i)] for every source index;do not invent graph points for unused codomain elements.An empty source has an empty graph.
Rule: complete_finite_function_graph · 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.[[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: 0c62fc1ab38a254b3f9b9e276372d08b27f370206cd47dbc32170b08ebea75efcandidate record SHA-256: 79c8a7c2520dd50136ea37497b0e768296a9e5ed8201c1c2ceb31e1ca8409e3bsemantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED