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Question 5226: Two Finite Principal-Filter Composition Images

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0009-05226 · q-20261008-advance-20261011-r07-tends-to-relation-0076

Original question and givens

The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(0)→Fin(0) has after=[]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

  • G001 — The complete total map f:Fin(0)→Fin(0) has images=[] indexed from 0; Fin(0)=∅. Every table entry must lie in the target type. S=[]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. The complete map g:Fin(0)→Fin(0) has after=[]. 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.

Step-by-step derivation

S001 · implies

Before

Fin(0)→Fin(0);S=[];full map=[]

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 dependencies

Rule: finite_principal_generator_convention · Uses: G001

S002 · implies

Before

direct composite image=[];image then map=[]

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 dependencies

Rule: principal_composition_image_identity · Uses: G001, S001

Reverse checks

  • 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

([], [])

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: be9fa2fbc415cc20f9c5526ef5fa361bebe973564e321ff2ee4701c7c8a60a2e
  • candidate record SHA-256: e6a5bca15768e023c128fe21296ac960ac6b3575befe89f5194f4455851c899c
  • semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
  • question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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