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R0010-05501 · q-20261008-advance-20261012-r07-function-mapping-0051
The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. Ordered queries=[-1, 0, 1]. Give (legal-source-type mask, outputs for typed queries or ill-typed); do not evaluate f outside its source type.
G001 — The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. Ordered queries=[-1, 0, 1]. Give (legal-source-type mask, outputs for typed queries or ill-typed); do not evaluate f outside its source type.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(0)→Fin(0);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
query=-1;0≤query<0=No
After
value=type mismatch
Conditions: none
Check source typing before table access;a negative index may not read the last entry,and an outside index may not acquire an invented f-value.
Rule: mapping_query_type_guard · Uses: G001, S001
Before
query=0;0≤query<0=No
After
value=type mismatch
Conditions: none
Check source typing before table access;a negative index may not read the last entry,and an outside index may not acquire an invented f-value.
Rule: mapping_query_type_guard · Uses: G001, S002
Before
query=1;0≤query<0=No
After
value=type mismatch
Conditions: none
Check source typing before table access;a negative index may not read the last entry,and an outside index may not acquire an invented f-value.
Rule: mapping_query_type_guard · 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.([No, No, No], [ill-typed, ill-typed, ill-typed])
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: bf5bdc328439f5bc42693cac9e7dec5e38c8cae73fa8ce4b5961414fb86aab3bcandidate record SHA-256: 8815e051d12281ef9810aa492c8feb34de29b97ce4b2c78d83d125220db63f18semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED