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R0009-05276 · q-20261008-advance-20261011-r07-tends-to-relation-0026
The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.
G001 — The complete total map f:Fin(3)→Fin(1) has images=[0, 0, 0] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. S=[0, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. T=[0] generates the target principal filter. List all x∈S with f(x)∉T increasingly; the required direction is f(S)⊆T.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(3)
After
image=0∈Fin(1)
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(3)
After
image=0∈Fin(1)
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(3)
After
image=0∈Fin(1)
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
Fin(3)→Fin(1);S=[0, 2];full map=[0, 0, 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, S003
Before
source generator x=0∈S
After
f(x)=0
Conditions: none
Map only source generators in S;do not confuse the whole codomain with f(S).
Rule: source_generator_image_row · Uses: G001, S004
Before
source generator x=2∈S
After
f(x)=0
Conditions: none
Map only source generators in S;do not confuse the whole codomain with f(S).
Rule: source_generator_image_row · Uses: G001, S005
Before
x=0;f(x)=0;T=[0]
After
source obstruction=No
Conditions: none
The obstruction is the source index x,not its target value;include it only when f(x) is outside T.
Rule: principal_target_obstruction_row · Uses: G001, S006
Before
x=2;f(x)=0;T=[0]
After
source obstruction=No
Conditions: none
The obstruction is the source index x,not its target value;include it only when f(x) is outside T.
Rule: principal_target_obstruction_row · Uses: G001, S007
Before
f(S)=[0];T=[0];obstructions=[]
After
f(S)⊆T=Yes
Conditions: none
The direction is f(S)⊆T,from image generator to target generator;empty S makes it vacuously true.Do not reverse it to T⊆f(S).
Rule: principal_inclusion_direction · Uses: G001, S008
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.[]
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: f44783d5538c5869524ee5f57e9d98b3612e3933d684b746810e73b0a2dad9c8candidate record SHA-256: 2458071457e909c1558627947736d8af7897be5227b6dd7ab74e0515ff81a823semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED