Original question and givens
The ordered rational values=[15/16, -16/17, 31/18] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).
G001 — The ordered rational values=[15/16, -16/17, 31/18] denote a complete nonempty finite real set. Duplicate listed values do not change its bounds; retain supplied order in the output. Empty, unbounded and extended-real sets are outside this slice. Give (values, supremum, infimum, supremum_attained, infimum_attained, certificate).
Step-by-step derivation
S001 · implies
Before
source operation=supinf-finite-extrema;original snapshot and mathematics retained
After
explicit ordered source output fields=[values, supremum, infimum, supremum_attained, infimum_attained, certificate]
Conditions: none
Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · implies
Before
ORIGINAL complete ordered nonempty finite real-value list=[15/16, -16/17, 31/18]
After
initialize upper/lower extremum candidates at ORIGINAL first value;duplicates retained and do not change set bounds
Conditions: none
This is the supplied complete finite set list,not a sample of an unspecified set.Duplicates do not alter bounds but remain in original output order;do not sort the data first.
Step identifiers and dependencies
Rule: supinf_complete_nonempty_finite_set_not_samples_or_sorting · Uses: G001, S001
S003 · calculation
Before
supinf original value1 minus previous maximum := sub(-16/17, 15/16)
After
supinf original value1 minus previous maximum = -511/272
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · calculation
Before
supinf original value1 minus previous minimum := sub(-16/17, 15/16)
After
supinf original value1 minus previous minimum = -511/272
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
ORIGINAL next index=1;value=-16/17;previous extremum values=[15/16, 15/16];differences=[-511/272, -511/272]
After
prefix maximum=15/16 attained at original index0;prefix minimum=-16/17 attained at original index1
Conditions: none
Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted.
Step identifiers and dependencies
Rule: supinf_every_value_extremum_prefix_invariant · Uses: G001, S004
S006 · calculation
Before
supinf original value2 minus previous maximum := sub(31/18, 15/16)
After
supinf original value2 minus previous maximum = 113/144
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · calculation
Before
supinf original value2 minus previous minimum := sub(31/18, -16/17)
After
supinf original value2 minus previous minimum = 815/306
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S006
S008 · implies
Before
ORIGINAL next index=2;value=31/18;previous extremum values=[15/16, -16/17];differences=[113/144, 815/306]
After
prefix maximum=31/18 attained at original index2;prefix minimum=-16/17 attained at original index1
Conditions: none
Compare every original value,preserving prefix maximum/minimum invariants and updating witnesses only for strict improvement.No original value is omitted.
Step identifiers and dependencies
Rule: supinf_every_value_extremum_prefix_invariant · Uses: G001, S007
S009 · implies
Before
ALL original values processed;actual maximum witness=[2, 31/18];actual minimum witness=[1, -16/17]
After
EVERY listed value between-16/17 and31/18;ANY upper bound must>=actual maximum in the set;ANY lower bound must<=actual minimum;sup=31/18,inf=-16/17,BOTH attained
Conditions: none
Because the maximum is itself in the set,no upper bound can be smaller;the minimum similarly bounds every lower bound above.This proves both bounding and leastness/greatestness.Attainment uses finiteness and nonemptiness and is not inferred for arbitrary bounded sets.
Step identifiers and dependencies
Rule: supinf_bounds_and_least_greatest_by_actual_attaining_points · Uses: G001, S008
Result
([15/16, -16/17, 31/18], 31/18, -16/17, Yes, Yes, finite_nonempty_set_extrema)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.