# Question 5077: Complete Finite Powerset

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0009-05077 · q-20261008-advance-20261011-r04-set-0077



## Original question and givens

Finite integer set A=[0]. 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=[0]. 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.





## 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-set](https://ocme.evemisslab.com/en/mko/mko-set/)





## Step-by-step derivation

S001 · impliesBefore

raw integer list=[0]

After

extensional set=[0]

Conditions: none

A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source.

Step identifiers and dependenciesRule: set_extensionality_normalization · Uses: G001





S002 · impliesBefore

no elements processed

After

[[]]

Conditions: none

The empty set has one subset,the empty set itself;its power set is not an empty list.

Step identifiers and dependenciesRule: power_set_empty_initial · Uses: G001, S001





S003 · impliesBefore

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.

Step identifiers and dependenciesRule: power_set_include_omit_branch · Uses: G001, S002





S004 · impliesBefore

(2)^1

After

power-set cardinality = 2

Conditions: none

Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity.

Step identifiers and dependenciesRule: integer_power · Uses: G001, S003





S005 · impliesBefore

[[], [0]]

After

all 2 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.

Step identifiers and dependenciesRule: finite_power_set_completeness · Uses: G001, S004









## Reverse checks


- 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]]

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: 05543fdbc84847fd4e3e2349d76b949bd94339d7b0998732634f53248485fbf9
- candidate record SHA-256: bda510b7ab3966c97898eff35c5f8d0915b696ec66786417444407654790b358
- semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
- question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r04-set-0077.json)