Step-by-step derivation
S001 · implies
Before
source operation=subseq-composed-index;original snapshot and mathematics retained
After
explicit ordered source output fields=[j, inner_index, source_index, composed_step, strictly_increasing, certificate]
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 · implies
Before
outer index k:ORIGINAL nonnegative start=1;positive integer step=2;F(q)=start+step*q for EVERY natural q>=0
After
F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(2)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix
Conditions: none
A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values.
Step identifiers and dependencies
Rule: subseq_affine_all_natural_index_order_not_value_monotonicity · Uses: G001, S001
S003 · implies
Before
inner index l:ORIGINAL nonnegative start=0;positive integer step=5;F(q)=start+step*q for EVERY natural q>=0
After
F(q)>=start>=0 and for ANY q<r,F(r)-F(q)=(5)*(r-q)>0;strictly increasing on ALL naturals,not just the evaluated prefix
Conditions: none
A positive integer step preserves the order of every pair of natural indices,not merely injectivity or an arbitrary reordering.This universal mathematical law is not truncated by an evaluated-output safety range and says nothing about monotonicity of source values.
Step identifiers and dependencies
Rule: subseq_affine_all_natural_index_order_not_value_monotonicity · Uses: G001, S002
S004 · calculation
Before
subsequence inner step times ORIGINAL j := mul(5, 17)
After
subsequence inner step times ORIGINAL j = 85
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 · calculation
Before
subsequence ORIGINAL inner index l(j) := sum(0, 85)
After
subsequence ORIGINAL inner index l(j) = 85
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, S004
S006 · calculation
Before
subsequence outer step times INNER result := mul(2, 85)
After
subsequence outer step times INNER result = 170
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 · calculation
Before
subsequence ORIGINAL composed index k(l(j)) := sum(1, 170)
After
subsequence ORIGINAL composed index k(l(j)) = 171
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, S006
S008 · calculation
Before
subsequence composed positive step := mul(2, 5)
After
subsequence composed positive step = 10
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 outer k(i)=1+2i;inner l(j)=0+5j;actual j=17;inner value=85
After
outer(inner(j))=171;for EVERY j,composed consecutive difference=10>0;exact evaluated fields within SAFE integer range
Conditions: none
Apply inner to j first,then feed that index into outer,not the reverse.The composed difference is the product of positive steps for every natural j.The SAFE check concerns explicitly evaluated numeric fields,not the mathematical domain of the infinite index law.
Step identifiers and dependencies
Rule: subseq_composition_inner_then_outer_all_j_positive_step · Uses: G001, S008