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R0009-05229 · q-20261008-advance-20261011-r07-tends-to-relation-0079
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=[0, 1]. 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=[0, 1]. 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.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
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.
Rule: complete_principal_map_row · Uses: G001
Before
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.
Rule: complete_principal_map_row · Uses: G001, S001
Before
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.
Rule: complete_principal_map_row · Uses: G001, S002
Before
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.
Rule: complete_principal_map_row · Uses: G001, S003
Before
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.
Rule: complete_principal_map_row · Uses: G001, S004
Before
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.
Rule: complete_principal_map_row · Uses: G001, S005
Before
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.
Rule: complete_principal_map_row · Uses: G001, S006
Before
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.
Rule: complete_principal_map_row · Uses: G001, S007
Before
Fin(8)→Fin(8);S=[0, 1];full map=[7, 6, 5, 4, 3, 2, 1, 0]
After
principal(S)={A:S⊆A};empty generators allowed,no NeBotConditions: 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.
Rule: finite_principal_generator_convention · Uses: G001, S008
Before
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.
Rule: complete_after_map_row · Uses: G001, S009
Before
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.
Rule: complete_after_map_row · Uses: G001, S010
Before
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.
Rule: complete_after_map_row · Uses: G001, S011
Before
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.
Rule: complete_after_map_row · Uses: G001, S012
Before
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.
Rule: complete_after_map_row · Uses: G001, S013
Before
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.
Rule: complete_after_map_row · Uses: G001, S014
Before
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.
Rule: complete_after_map_row · Uses: G001, S015
Before
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.
Rule: complete_after_map_row · Uses: G001, S016
Before
x=0;f(x)=7
After
g(f(x))=0
Conditions: none
The first route composes at each source element,retaining the direction of both maps.
Rule: principal_direct_composition_row · Uses: G001, S017
Before
x=1;f(x)=6
After
g(f(x))=1
Conditions: none
The first route composes at each source element,retaining the direction of both maps.
Rule: principal_direct_composition_row · Uses: G001, S018
Before
distinct intermediate image=6
After
g(y)=1
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.
Rule: principal_image_then_map_row · Uses: G001, S019
Before
distinct intermediate image=7
After
g(y)=0
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.
Rule: principal_image_then_map_row · Uses: G001, S020
Before
direct composite image=[0, 1];image then map=[0, 1]
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.
Rule: principal_composition_image_identity · Uses: G001, S021
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.([0, 1], [0, 1])
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: 77f405e97a63383cabcb4c6c894558c466fac79872258ebe54a7c0118fda9f60candidate record SHA-256: da3ef19e9f3472caf03d4b9b828acc10a8bd12dcca2044cf9473c14d9d66025asemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED