Question 5227: Two Finite Principal-Filter Composition Images

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0009-05227 · q-20261008-advance-20261011-r07-tends-to-relation-0077

Original question and givens

The complete total map f:Fin(4)→Fin(3) has images=[0, 0, 1, 2] indexed from 0; Fin(4)={0,…,3}. Every table entry must lie in the target type. S=[0, 1, 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(3)→Fin(8) has after=[0, 0, 0]. Give ((g∘f)(S), g(f(S))) as complete increasing sets.

G001 — The complete total map f:Fin(4)→Fin(3) has images=[0, 0, 1, 2] indexed from 0; Fin(4)={0,…,3}. Every table entry must lie in the target type. S=[0, 1, 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(3)→Fin(8) has after=[0, 0, 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

Step-by-step derivation

S001 · implies

Before

source index=0∈Fin(4)

After

image=0∈Fin(3)

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 dependencies

Rule: complete_principal_map_row · Uses: G001

S002 · implies

Before

source index=1∈Fin(4)

After

image=0∈Fin(3)

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 dependencies

Rule: complete_principal_map_row · Uses: G001, S001

S003 · implies

Before

source index=2∈Fin(4)

After

image=1∈Fin(3)

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 dependencies

Rule: complete_principal_map_row · Uses: G001, S002

S004 · implies

Before

source index=3∈Fin(4)

After

image=2∈Fin(3)

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 dependencies

Rule: complete_principal_map_row · Uses: G001, S003

S005 · implies

Before

Fin(4)→Fin(3);S=[0, 1, 2, 3];full map=[0, 0, 1, 2]

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, S004

S006 · implies

Before

source index=0∈Fin(3)

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 dependencies

Rule: complete_after_map_row · Uses: G001, S005

S007 · implies

Before

source index=1∈Fin(3)

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 dependencies

Rule: complete_after_map_row · Uses: G001, S006

S008 · implies

Before

source index=2∈Fin(3)

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 dependencies

Rule: complete_after_map_row · Uses: G001, S007

S009 · implies

Before

x=0;f(x)=0

After

g(f(x))=0

Conditions: none

The first route composes at each source element,retaining the direction of both maps.

Step identifiers and dependencies

Rule: principal_direct_composition_row · Uses: G001, S008

S010 · implies

Before

x=1;f(x)=0

After

g(f(x))=0

Conditions: none

The first route composes at each source element,retaining the direction of both maps.

Step identifiers and dependencies

Rule: principal_direct_composition_row · Uses: G001, S009

S011 · implies

Before

x=2;f(x)=1

After

g(f(x))=0

Conditions: none

The first route composes at each source element,retaining the direction of both maps.

Step identifiers and dependencies

Rule: principal_direct_composition_row · Uses: G001, S010

S012 · implies

Before

x=3;f(x)=2

After

g(f(x))=0

Conditions: none

The first route composes at each source element,retaining the direction of both maps.

Step identifiers and dependencies

Rule: principal_direct_composition_row · Uses: G001, S011

S013 · implies

Before

distinct intermediate image=0

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.

Step identifiers and dependencies

Rule: principal_image_then_map_row · Uses: G001, S012

S014 · implies

Before

distinct intermediate image=1

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.

Step identifiers and dependencies

Rule: principal_image_then_map_row · Uses: G001, S013

S015 · implies

Before

distinct intermediate image=2

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.

Step identifiers and dependencies

Rule: principal_image_then_map_row · Uses: G001, S014

S016 · implies

Before

direct composite image=[0];image then map=[0]

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, S015

Reverse checks

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

([0], [0])

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: 8fdd13039d24d2325e525fbdc8e4d9626edf9de8bb5622da37c330cbcee279b4
candidate record SHA-256: 3abecb203284b01765432e03f478aea684e6f51ca0997719111cf46da5c439e8
semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
Lean: NOT_RUN · human review: NOT_PERFORMED

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