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R0013-07246 · q-20261011-advance-20261015-r03-domain-codomain-range-0071
Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[4, 2, 5] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=7, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).
G001 — Fin(3)={0,…,2} is the source and Fin(8)={0,…,7} the codomain; complete typed images=[4, 2, 5] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=7, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).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(3)→Fin(8);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
source index=0∈Fin(3)
After
image[0]=4∈Fin(8)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: typed_complete_original_table_row · Uses: G001, S001
Before
source index=1∈Fin(3)
After
image[1]=2∈Fin(8)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: typed_complete_original_table_row · Uses: G001, S002
Before
source index=2∈Fin(3)
After
image[2]=5∈Fin(8)
Conditions: none
First type-check every cell of the complete original table,not only cells queried later.
Rule: typed_complete_original_table_row · Uses: G001, S003
Before
old table=[4, 2, 5];at=0;new value=7
After
old output=4;new table=[7, 2, 5]
Conditions: none
Replace only the designated original cell and retain all others.An identical replacement changes nothing;the original source data are not mutated in place.
Rule: typed_single_output_replacement · Uses: G001, S004
Before
old range=[2, 4, 5];new range=[2, 5, 7]
After
lost=[4];added=[7];declared m=8
Conditions: none
An old value remains if another input still hits it;a replacement value already hit is not new.The declared codomain size m does not change.
Rule: typed_replacement_range_difference · 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.(4, [7, 2, 5], [2, 5, 7], [4], [7], 8)
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: f43d7bfa94a769262bacca2a7066b0e254a47edad9caf9a5a84da63e7bc905cbcandidate record SHA-256: 508e23b3865745b4e8bf943317f7a0e8a32e89fd6b07bf571382d04799c1e743semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED