First f, then g, then h: Fin(0)=∅→Fin(0)→Fin(0)→Fin(0); complete f=[], g=[], h=[]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.
G001 — First f, then g, then h: Fin(0)=∅→Fin(0)→Fin(0)→Fin(0); complete f=[], g=[], h=[]. Give ([n,m,k,l], complete (h∘g)∘f table, complete h∘(g∘f) table, every [x,f(x),g(f(x)),h(g(f(x)))] row, are the two tables equal?). All cells, including unused cells, must be typed.
Definition and scope
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.
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation.
Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation.
Require the WHOLE declared table length to equal its source size and EVERY cell to lie in Fin(m),including unused cells.Even an empty outer source cannot waive this complete table typing obligation.
Compute HG then F on one side,and GF then H on the other,comparing EVERY original source.Even an empty outer source requires ALL unused cells typed first.This is exact checking of the given finite tables,not a new formal certificate for every function identity.
S005 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
Result
([0, 0, 0, 0], [], [], [], Yes)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
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.