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R0013-07205 · q-20261011-advance-20261015-r03-domain-codomain-range-0005
Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).
G001 — Fin(3)={0,…,2} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 1, 0] are indexed from 0. All entries must be typed, even unused ones. Give ([n,m], all ordered [x,images[x]] rows, increasing range, unused codomain, range size).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(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(3)
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(3)
After
image[1]=1∈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
source index=2∈Fin(3)
After
image[2]=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, S003
Before
complete graph=[[0, 0], [1, 1], [2, 0]]
After
range=[0, 1];unused codomain=[2];range size=2
Conditions: none
The range is the deduplicated output set,while declared n and m still fix domain and codomain sizes.A smaller range does not shrink the codomain.
Rule: typed_graph_range_inventory · 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.([3, 3], [[0, 0], [1, 1], [2, 0]], [0, 1], [2], 2)
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: 68dfd7ee540fdd37199403e40718f361624870e604be5a1885c0520a114a264dcandidate record SHA-256: 23b4b65104731bc5a73523411bf310bbd03888db103f5ac6d1c2d8daae7b3e36semantic review receipt SHA-256: e89fc6d4a50a4a7fc942c6c8939a92f835e30105a9a5850e37dd198f31c1ed59question-first witness SHA-256: 0e1039f4cf022bf35af4e5531651c9a2b7ca7a5fdacac34fb1e4bcd9a81f6299Lean: NOT_RUN · human review: NOT_PERFORMED