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R0013-07252 · q-20261011-advance-20261015-r03-domain-codomain-range-0027
Fin(0)=∅ is the source and Fin(2)={0,…,1} the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.
G001 — Fin(0)=∅ is the source and Fin(2)={0,…,1} the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, 1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.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(2);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
target candidate=0;inside Fin(2)=Yes
After
preimage=[];size/status=0
Conditions: none
A legal but unhit codomain value has empty preimage of size0.An outside-codomain candidate is not a legal empty-preimage query in this typed interface.
Rule: typed_preimage_codomain_guard · Uses: G001, S001
Before
target candidate=1;inside Fin(2)=Yes
After
preimage=[];size/status=0
Conditions: none
A legal but unhit codomain value has empty preimage of size0.An outside-codomain candidate is not a legal empty-preimage query in this typed interface.
Rule: typed_preimage_codomain_guard · 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.([Yes, Yes], [[], []], [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: d4d0e090c088db91537e89e941f44b6025988c9e017323ba926e9530497104eccandidate record SHA-256: e1e35e0cc067616c8c5fa7d087f268969ba15c0b1abd72774cfff652afe59f93semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED