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R0013-07228 · q-20261011-advance-20261015-r03-domain-codomain-range-0053
Fin(2)={0,…,1} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, after checking the whole original table. Give (old output, new table, new range, disappeared values, added values, m).
G001 — Fin(2)={0,…,1} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0] are indexed from 0. All entries must be typed, even unused ones. Replace at=0 by new value=2, 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(2)→Fin(3);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(2)
After
image[0]=0∈Fin(3)
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(2)
After
image[1]=0∈Fin(3)
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
old table=[0, 0];at=0;new value=2
After
old output=0;new table=[2, 0]
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, S003
Before
old range=[0];new range=[0, 2]
After
lost=[];added=[2];declared m=3
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, S004
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.(0, [2, 0], [0, 2], [], [2], 3)
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: 36ba2bc8e58b83cce28ff255264a4068941390fe510eef09162b6307906ca4a3candidate record SHA-256: 51ab861ec59fcfaa7eae7d7b2e6782e8bd8f143ab1c5e4bcc3c9ff0d56fb8c80semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED