# Question 4757: Simultaneous Variable Swap [Return to catalogue page 24](https://ocme.evemisslab.com/en/questions/page/24/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261011-r03-variable-expression-foundations/#q-20261008-advance-20261011-r03-variable-expression-0032) 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. R0008-04757 · q-20261008-advance-20261011-r03-variable-expression-0032 ## Original question and givens All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(7)x+(-6)y, x=-1, y=9. Swap x and y simultaneously; give (original value, swapped value) as integers. - G001 — All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(7)x+(-6)y, x=-1, y=9. Swap x and y simultaneously; give (original value, swapped value) as integers. ## 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-variable-expression](https://ocme.evemisslab.com/en/mko/mko-variable-expression/) ## Step-by-step derivation S001 · impliesBefore typed rational expression and explicitly given environment After evaluation,not solving an equation Conditions: none Evaluating an expression is not solving an equation;substitute only the supplied environment without inventing roots or changing bindings. Step identifiers and dependenciesRule: expression_environment_scope · Uses: G001 S002 · calculationBefore original first term := mul(7, -1) After original first term = -7 Conditions: none Multiply the first coefficient by the current environment's first component. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore original second term := mul(-6, 9) After original second term = -54 Conditions: none Multiply the second coefficient by the second component of that same environment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original full value := sum(-7, -54) After original full value = -61 Conditions: none Add the two terms to evaluate this environment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore simultaneously swapped first term := mul(7, 9) After simultaneously swapped first term = 63 Conditions: none Multiply the first coefficient by the current environment's first component. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore simultaneously swapped second term := mul(-6, -1) After simultaneously swapped second term = 6 Conditions: none Multiply the second coefficient by the second component of that same environment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore simultaneously swapped full value := sum(63, 6) After simultaneously swapped full value = 69 Conditions: none Add the two terms to evaluate this environment. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore old environment=(x=-1,y=9) After new environment=(x=9,y=-1);old unchanged Conditions: none Both new components read the old environment;assigning x←y before letting y read the modified x would not be a genuine swap. Step identifiers and dependenciesRule: simultaneous_substitution_snapshot · Uses: G001, S007 ## Reverse checks - 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 — Recompute simultaneously swapped full value with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute simultaneously swapped second term with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute simultaneously swapped first term with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original full value with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original second term with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original first term with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (-61, 69) 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: 61c03ba3744cde6c98fc0275e51f4bc68e3a43305ae3df81806038c6e87de92d - candidate record SHA-256: 19dac94e09437a9059b55644d0b311c28d035f867552b93ddadb0548623934dc - semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289 - question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261011-r03-variable-expression-0032.json)