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R0010-05661 · q-20261008-advance-20261012-r04-set-membership-0086
Finite integer set U=[-10, -1, 0, 1, 10]; P(x) means x=10. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).
G001 — Finite integer set U=[-10, -1, 0, 1, 10]; P(x) means x=10. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).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
declared finite type or bounded integer universe
After
membership truth is distinct from eligibility to ask the question
Conditions: none
Being outside the type is not the same as false membership.Check only the declared finite type or integer set,not all set theory or unrestricted comprehension.
Rule: finite_membership_scope · Uses: G001
Before
predicate difference at-10 := sub(-10, 10)
After
predicate difference at-10 = -20
Conditions: none
Compute x minus the parameter as an exact integer comparison witness.
Rule: exact_rational_arithmetic · Uses: G001, S001
Before
x=-10;predicate=eq 10
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S002
Before
predicate difference at-1 := sub(-1, 10)
After
predicate difference at-1 = -11
Conditions: none
Compute x minus the parameter as an exact integer comparison witness.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
x=-1;predicate=eq 10
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S004
Before
predicate difference at0 := sub(0, 10)
After
predicate difference at0 = -10
Conditions: none
Compute x minus the parameter as an exact integer comparison witness.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
x=0;predicate=eq 10
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S006
Before
predicate difference at1 := sub(1, 10)
After
predicate difference at1 = -9
Conditions: none
Compute x minus the parameter as an exact integer comparison witness.
Rule: exact_rational_arithmetic · Uses: G001, S007
Before
x=1;predicate=eq 10
After
False
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S008
Before
predicate difference at10 := sub(10, 10)
After
predicate difference at10 = 0
Conditions: none
Compute x minus the parameter as an exact integer comparison witness.
Rule: exact_rational_arithmetic · Uses: G001, S009
Before
x=10;predicate=eq 10
After
True
Conditions: none
Evaluate every element in normalized U order;predicate success does not expand U.
Rule: finite_membership_predicate_row · Uses: G001, S010
Before
normalized U=[-10, -1, 0, 1, 10];truths=[False, False, False, False, True]
After
A={x∈U:P(x)}=[10]Conditions: none
The full extension contains exactly the successful elements of the supplied U;the truth list remains aligned with all sorted U-elements.Empty U gives two empty lists.
Rule: bounded_predicate_extension · Uses: G001, S011
S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S010 · premise_and_result_audit — Recompute predicate difference at10 with exact rational arithmetic, not rounded decimals.S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S008 · premise_and_result_audit — Recompute predicate difference at1 with exact rational arithmetic, not rounded decimals.S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S006 · premise_and_result_audit — Recompute predicate difference at0 with exact rational arithmetic, not rounded decimals.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 — Recompute predicate difference at-1 with exact rational arithmetic, not rounded decimals.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 — Recompute predicate difference at-10 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.([10], [False, False, False, False, True])
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: 54ca1a92dbc4173e19556611d93fa450e195964650f313b64b85cedb09cff24acandidate record SHA-256: 50ac757daff6118420e00ab9cc01c417eb95e5fd14b6d0cdbf370c2ffa396e0csemantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06fquestion-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fcLean: NOT_RUN · human review: NOT_PERFORMED