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R0009-05085 · q-20261008-advance-20261011-r04-set-0085
Finite integer set A=[-9, 0, 1, 9]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.
G001 — Finite integer set A=[-9, 0, 1, 9]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.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
raw integer list=[-9, 0, 1, 9]
After
extensional set=[-9, 0, 1, 9]
Conditions: none
A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source.
Rule: set_extensionality_normalization · Uses: G001
Before
no elements processed
After
[[]]
Conditions: none
The empty set has one subset,the empty set itself;its power set is not an empty list.
Rule: power_set_empty_initial · Uses: G001, S001
Before
processed subset=[];next element=-9
After
omit=[];include=[-9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S002
Before
processed subset=[];next element=0
After
omit=[];include=[0]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S003
Before
processed subset=[-9];next element=0
After
omit=[-9];include=[-9, 0]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S004
Before
processed subset=[];next element=1
After
omit=[];include=[1]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S005
Before
processed subset=[-9];next element=1
After
omit=[-9];include=[-9, 1]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S006
Before
processed subset=[0];next element=1
After
omit=[0];include=[0, 1]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S007
Before
processed subset=[-9, 0];next element=1
After
omit=[-9, 0];include=[-9, 0, 1]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S008
Before
processed subset=[];next element=9
After
omit=[];include=[9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S009
Before
processed subset=[-9];next element=9
After
omit=[-9];include=[-9, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S010
Before
processed subset=[0];next element=9
After
omit=[0];include=[0, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S011
Before
processed subset=[-9, 0];next element=9
After
omit=[-9, 0];include=[-9, 0, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S012
Before
processed subset=[1];next element=9
After
omit=[1];include=[1, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S013
Before
processed subset=[-9, 1];next element=9
After
omit=[-9, 1];include=[-9, 1, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S014
Before
processed subset=[0, 1];next element=9
After
omit=[0, 1];include=[0, 1, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S015
Before
processed subset=[-9, 0, 1];next element=9
After
omit=[-9, 0, 1];include=[-9, 0, 1, 9]
Conditions: none
Split into disjoint omit/include branches for the new element;each resulting subset appears in exactly one branch.
Rule: power_set_include_omit_branch · Uses: G001, S016
Before
(2)^4
After
power-set cardinality = 16
Conditions: none
Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.
Rule: integer_power · Uses: G001, S017
Before
[[], [-9], [0], [1], [9], [-9, 0], [-9, 1], [-9, 9], [0, 1], [0, 9], [1, 9], [-9, 0, 1], [-9, 0, 9], [-9, 1, 9], [0, 1, 9], [-9, 0, 1, 9]]
After
all 16 subsets in size-then-numeric-lex order
Conditions: none
Order subsets first by size,then numeric lexicographic order,with increasing elements inside each subset.This constructs only subsets of the supplied finite A,not unrestricted comprehension.
Rule: finite_power_set_completeness · Uses: G001, S018
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.[[], [-9], [0], [1], [9], [-9, 0], [-9, 1], [-9, 9], [0, 1], [0, 9], [1, 9], [-9, 0, 1], [-9, 0, 9], [-9, 1, 9], [0, 1, 9], [-9, 0, 1, 9]]
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: 45b94a66eda282054c4b41fd8a39c4e5ed9f9ae3c4b825ef0f7bd226130dcd46candidate record SHA-256: 8df09b0e26f2c35b8b1bb23c7a42d14b2b51340b39ea09f1b78bdf14794f00dcsemantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462Lean: NOT_RUN · human review: NOT_PERFORMED