# Question 5238: Two Finite Principal-Filter Composition Images [Return to catalogue page 27](https://ocme.evemisslab.com/en/questions/page/27/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261011-r07-tends-to-relation-foundations/#q-20261008-advance-20261011-r07-tends-to-relation-0088) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0009-05238 · q-20261008-advance-20261011-r07-tends-to-relation-0088 ## Original question and givens The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets. - G001 — The complete total map f:Fin(8)→Fin(8) has images=[7, 6, 5, 4, 3, 2, 1, 0] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[2, 3]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(8)→Fin(8) has after=[7, 6, 5, 4, 3, 2, 1, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets. ## 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. - [mko-tends-to-relation](https://ocme.evemisslab.com/en/mko/mko-tends-to-relation/) ## Step-by-step derivation S001 · impliesBefore source index=0∈Fin(8) After image=7∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001 S002 · impliesBefore source index=1∈Fin(8) After image=6∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S001 S003 · impliesBefore source index=2∈Fin(8) After image=5∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S002 S004 · impliesBefore source index=3∈Fin(8) After image=4∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S003 S005 · impliesBefore source index=4∈Fin(8) After image=3∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S004 S006 · impliesBefore source index=5∈Fin(8) After image=2∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S005 S007 · impliesBefore source index=6∈Fin(8) After image=1∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S006 S008 · impliesBefore source index=7∈Fin(8) After image=0∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_principal_map_row · Uses: G001, S007 S009 · impliesBefore Fin(8)→Fin(8);S=[2, 3];full map=[7, 6, 5, 4, 3, 2, 1, 0] After principal(S)={A:S⊆A};empty generators allowed,no NeBot Conditions: none The declared convention allows empty S;its family of supersets then contains the empty set,and no proper-filter or NeBot condition is added.This is a finite principal-generator model,not a general sequence or point limit. Step identifiers and dependenciesRule: finite_principal_generator_convention · Uses: G001, S008 S010 · impliesBefore source index=0∈Fin(8) After image=7∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S009 S011 · impliesBefore source index=1∈Fin(8) After image=6∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S010 S012 · impliesBefore source index=2∈Fin(8) After image=5∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S011 S013 · impliesBefore source index=3∈Fin(8) After image=4∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S012 S014 · impliesBefore source index=4∈Fin(8) After image=3∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S013 S015 · impliesBefore source index=5∈Fin(8) After image=2∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S014 S016 · impliesBefore source index=6∈Fin(8) After image=1∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S015 S017 · impliesBefore source index=7∈Fin(8) After image=0∈Fin(8) Conditions: none Every entry of the complete table must be legal;an entry being unused later does not waive total-map typing. Step identifiers and dependenciesRule: complete_after_map_row · Uses: G001, S016 S018 · impliesBefore x=2;f(x)=5 After g(f(x))=2 Conditions: none The first route composes at each source element,retaining the direction of both maps. Step identifiers and dependenciesRule: principal_direct_composition_row · Uses: G001, S017 S019 · impliesBefore x=3;f(x)=4 After g(f(x))=3 Conditions: none The first route composes at each source element,retaining the direction of both maps. Step identifiers and dependenciesRule: principal_direct_composition_row · Uses: G001, S018 S020 · impliesBefore distinct intermediate image=4 After g(y)=3 Conditions: none The second route deduplicates f(S) before applying g;a total function assigns the same successor to the same intermediate value,so deduplication loses no final image. Step identifiers and dependenciesRule: principal_image_then_map_row · Uses: G001, S019 S021 · impliesBefore distinct intermediate image=5 After g(y)=2 Conditions: none The second route deduplicates f(S) before applying g;a total function assigns the same successor to the same intermediate value,so deduplication loses no final image. Step identifiers and dependenciesRule: principal_image_then_map_row · Uses: G001, S020 S022 · impliesBefore direct composite image=[2, 3];image then map=[2, 3] After (g∘f)(S)=g(f(S)) in this complete finite model Conditions: none Both routes give the same final image;empty S gives empty images on both routes.No general analytic limit or additional nonemptiness premise is introduced. Step identifiers and dependenciesRule: principal_composition_image_identity · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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. ## Result ([2, 3], [2, 3]) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: f566b2e889ac450fa4f19493421c357394f51eec7476ebc0571e548fa0640b88 - candidate record SHA-256: ccfe92d2f23e6cf7595fa8bbe7134c89eabcd1f388de009e2aadd4e80c5f1449 - semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56 - question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r07-tends-to-relation-0088.json)