Step-by-step derivation
S001 · implies
Before
source operation=subseq-finite-prefix-boundary;original snapshot and mathematics retained
After
explicit ordered source output fields=[indices, finite_prefix_increasing, infinite_certificate_supplied, boundary]
Conditions: none
Retain the historical interface marker in the original snapshot while listing ordered source fields here,without JavaScript or external lookup.Field labels are not additional mathematical evidence,and No retains the source claim scope.
Step identifiers and dependencies
Rule: source_ordered_interface_explicit_without_javascript · Uses: G001
S002 · calculation
Before
subsequence finite prefix adjacent index1 difference := sub(21, 20)
After
subsequence finite prefix adjacent index1 difference = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
ORIGINAL finite adjacent positions=[0, 1];actual indices=[20, 21]
After
exact forward difference=1;strictly increasing here=Yes
Conditions: none
Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase.
Step identifiers and dependencies
Rule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S002
S004 · calculation
Before
subsequence finite prefix adjacent index2 difference := sub(22, 21)
After
subsequence finite prefix adjacent index2 difference = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
ORIGINAL finite adjacent positions=[1, 2];actual indices=[21, 22]
After
exact forward difference=1;strictly increasing here=Yes
Conditions: none
Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase.
Step identifiers and dependencies
Rule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S004
S006 · calculation
Before
subsequence finite prefix adjacent index3 difference := sub(23, 22)
After
subsequence finite prefix adjacent index3 difference = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S005
S007 · implies
Before
ORIGINAL finite adjacent positions=[2, 3];actual indices=[22, 23]
After
exact forward difference=1;strictly increasing here=Yes
Conditions: none
Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase.
Step identifiers and dependencies
Rule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S006
S008 · calculation
Before
subsequence finite prefix adjacent index4 difference := sub(24, 23)
After
subsequence finite prefix adjacent index4 difference = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S007
S009 · implies
Before
ORIGINAL finite adjacent positions=[3, 4];actual indices=[23, 24]
After
exact forward difference=1;strictly increasing here=Yes
Conditions: none
Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase.
Step identifiers and dependencies
Rule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S008
S010 · calculation
Before
subsequence finite prefix adjacent index5 difference := sub(25, 24)
After
subsequence finite prefix adjacent index5 difference = 1
Conditions: none
Substitute original roles and established intermediates exactly,retaining indices,signs,denominators and the proof's applicability conditions.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S009
S011 · implies
Before
ORIGINAL finite adjacent positions=[4, 5];actual indices=[24, 25]
After
exact forward difference=1;strictly increasing here=Yes
Conditions: none
Check every adjacent pair in original order without sorting or deduplicating to repair the prefix.Positive differences at all finite adjacent positions certify this prefix's strict increase.
Step identifiers and dependencies
Rule: subseq_finite_order_each_actual_adjacent_pair · Uses: G001, S010
S012 · implies
Before
ORIGINAL finite index list=[20, 21, 22, 23, 24, 25];ALL adjacent strict flags=[Yes, Yes, Yes, Yes, Yes]
After
this finite prefix increasing=Yes;INFINITE certificate supplied by original data=No;finite_prefix_not_infinite_subsequence_certificate
Conditions: none
The No says the original data do not specify a universal law for all later indices.A valid finite prefix can be separately extended,for example by adding1 after its last value,but that does not determine an unspecified original continuation.If the prefix already fails,its actual adjacent violation is the obstruction.
Step identifiers and dependencies
Rule: subseq_prefix_No_not_nonextendability_or_sorting · Uses: G001, S011