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R0010-05504 · q-20261008-advance-20261012-r07-function-mapping-0054
The complete total map f:Fin(2)→Fin(2) has images=[0, 1] indexed from 0; Fin(2)={0,…,1}. Every table entry must lie in the target type. Ordered queries=[1, 0, 102]. 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(2)→Fin(2) has images=[0, 1] indexed from 0; Fin(2)={0,…,1}. Every table entry must lie in the target type. Ordered queries=[1, 0, 102]. 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(2)→Fin(2);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(2)
After
f(0)=0∈Fin(2)
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
f input=1∈Fin(2)
After
f(1)=1∈Fin(2)
Conditions: none
Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal.
Rule: complete_mapping_entry · Uses: G001, S002
Before
query=1;0≤query<2=Yes
After
value=1
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
Before
query=0;0≤query<2=Yes
After
value=0
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, S004
Before
query=102;0≤query<2=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, S005
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.([Yes, Yes, No], [1, 0, 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: e0654b0c26eca0647f0b9556f8a86d157458e6ea1a57bf5ef0f152d0f8ddf14dcandidate record SHA-256: 8494e2af480efe9c5fc2f409492c25d02e3517b73b2f1e2a4d8765e20e928348semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED