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R0008-04759 · q-20261008-advance-20261011-r03-variable-expression-0034
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. E(x,y)=(9)x+(-8)y, x=1, y=11. 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)=(9)x+(-8)y, x=1, y=11. Swap x and y simultaneously; give (original value, swapped value) as integers.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
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.
Rule: expression_environment_scope · Uses: G001
Before
original first term := mul(9, 1)
After
original first term = 9
Conditions: none
Multiply the first coefficient by the current environment's first component.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
original second term := mul(-8, 11)
After
original second term = -88
Conditions: none
Multiply the second coefficient by the second component of that same environment.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
original full value := sum(9, -88)
After
original full value = -79
Conditions: none
Add the two terms to evaluate this environment.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
simultaneously swapped first term := mul(9, 11)
After
simultaneously swapped first term = 99
Conditions: none
Multiply the first coefficient by the current environment's first component.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
simultaneously swapped second term := mul(-8, 1)
After
simultaneously swapped second term = -8
Conditions: none
Multiply the second coefficient by the second component of that same environment.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
simultaneously swapped full value := sum(99, -8)
After
simultaneously swapped full value = 91
Conditions: none
Add the two terms to evaluate this environment.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
old environment=(x=1,y=11)
After
new environment=(x=11,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.
Rule: simultaneous_substitution_snapshot · Uses: G001, S007
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.(-79, 91)
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: 5b4539c770f65f958278026ccd85bbf463cad35b03f6aabc587b4cc517efb246candidate record SHA-256: d8f83c8a805a4238d6c53ceb328eb41752f3570504a5ef050670ab0f817791c7semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67Lean: NOT_RUN · human review: NOT_PERFORMED