# Question 32541: Integer Membership

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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.

R0064-32541 · q-20260919-integer-membership-0016



## Original question and givens

Does 15.5 belong to the integer set Z?


- G001 — Does 15.5 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 = [15.5, No]

After

derived = [No, 31/2, No]

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 · calculationBefore

Integer nonmember ORIGINAL floor times reduced denominator := mul(15, 2)

After

Integer nonmember ORIGINAL floor times reduced denominator = 30

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

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





S004 · calculationBefore

Integer nonmember ORIGINAL numerator minus floor product := sub(31, 30)

After

Integer nonmember ORIGINAL numerator minus floor product = 1

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

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





S005 · calculationBefore

Integer nonmember ORIGINAL positive proper remainder fraction := div(1, 2)

After

Integer nonmember ORIGINAL positive proper remainder fraction = 1/2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

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





S006 · calculationBefore

Integer nonmember ORIGINAL value minus lower integer := sub(31/2, 15)

After

Integer nonmember ORIGINAL value minus lower integer = 1/2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005





S007 · calculationBefore

Integer nonmember ORIGINAL upper integer minus value := sub(16, 31/2)

After

Integer nonmember ORIGINAL upper integer minus value = 1/2

Conditions: none

Use ORIGINAL operands/order for exact integer/rational arithmetic.RAW bounds are bank-specific;derived values are not SAFE-recapped or approximated.Do not import legacy caps into BASE integers.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006





S008 · impliesBefore

source roles = [15.5, 31/2, 15, 2]

After

derived = [30, 1, 1/2, 1/2, 1/2, k < value < k+1]

Conditions: none

Reduce to n/d and write n=kd+r with0<r<d.The value lies STRICTLY between consecutive integers k,k+1,so cannot be an integer.These five have denominator2 and odd numerators;this does NOT classify every decimal representation as noninteger.

Step identifiers and dependenciesRule: Integer_NONmember_EXACT_between_consecutive_integers_NOT_all_decimals · Uses: G001, S007









## Reverse checks


- S008 · 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.
- S007 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL upper integer minus value with exact rational arithmetic, not rounded decimals.
- S006 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL value minus lower integer with exact rational arithmetic, not rounded decimals.
- S005 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL positive proper remainder fraction with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL numerator minus floor product with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute Integer nonmember ORIGINAL floor times reduced denominator with exact rational arithmetic, not rounded decimals.
- 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

No

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: 297d76c1933e1982f7f13a6a9306482b41ee727b1e385f481385b27c23b72700
- candidate record SHA-256: 6c44f7ace0de7091bfc1702d9c35e1dff7503f6e4f387c5dea28307a2591ddcd
- 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-0016.json)