# Question 32549: Integer Membership

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[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260919-integer-foundations/#q-20260919-integer-membership-0024)

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.

R0064-32549 · q-20260919-integer-membership-0024



## Original question and givens

Does -23 belong to the integer set Z?


- G001 — Does -23 belong to the integer set Z?





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





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-integer-membership, no-required-operation, no-required-domain, []]

After

derived = [[mathematical_membership_YesNo_STRING]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001





S002 · impliesBefore

source roles = [-23, Yes]

After

derived = [Yes, -23/1, Yes]

Conditions: none

FIRST verify the raw string and Boolean flag against the declared ASCII syntax,THEN classify the exact value.The frozen No cases are genuine half-integers.A parser rejecting string1.0 does NOT prove real1.0 is outside Z.Ambiguous out-of-slice cases are not false mathematical No answers.

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





S003 · impliesBefore

source roles = [-23, -23/1]

After

derived = [-23, Yes]

Conditions: none

A reduced denominator1 means the value IS the integer n.Positive integers,zero and negative integers all belong to Z.This is not an appearance guess or a rewrite of original raw data.

Step identifiers and dependenciesRule: Integer_EXACT_value_has_denominator1_signed_integer_definition · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.





## Result

Yes

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: 118e45b6cc3ec20168ed295bcd463b803525beeee5150d0469a02aa1e7071a64
- candidate record SHA-256: 5aeaca3c8863cf798e63d823b6b963409d567af3fc4564d1e963a14cdb2aafa6
- semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
- question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260919-integer-membership-0024.json)