Step-by-step derivation
S001 · implies
Before
source operation=incexc-general-alternating-sum;original snapshot and mathematics retained
After
explicit ordered source output fields=[family_size, terms, union_size, alternating_sum, equal, 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 indexed finite set family in SUPPLIED outer index order=[[8, 9, 10]]
After
within-set numeric normalization=[[8, 9, 10]];duplicates INSIDE a set rejected;EQUAL sets at DIFFERENT indices RETAINED;integer labels may be negative
Conditions: none
Normalize elements within each set only.Outer family positions are inclusion-exclusion indices and must not be semantically reordered or deduplicated.Empty sets are legitimate indexed members,and element labels are not cardinalities.
Step identifiers and dependencies
Rule: incexc_indexed_family_not_deduplicated_or_semantic_reordered · Uses: G001, S001
S003 · calculation
Before
inclusion mask1 signed intersection cardinality := mul(1, 3)
After
inclusion mask1 signed intersection cardinality = 3
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
inclusion mask1 alternating prefix sum := sum(0, 3)
After
inclusion mask1 alternating prefix sum = 3
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
next NONEMPTY NUMERIC bitmask=1;ORIGINAL family indices=[0];actual selected original sets=[[8, 9, 10]]
After
FULL intersection members=[8, 9, 10];size=3;ODD order sign=1;alternating prefix=3;zero-size terms retained
Conditions: none
Every nonempty index subset appears once in numeric-mask order,not cardinality or lexicographic order.Odd intersections add and even ones subtract.Retain zero-size intersections;an intermediate alternating sum is not yet asserted to be a cardinality.
Step identifiers and dependencies
Rule: incexc_every_index_submask_intersection_parity_source_order · Uses: G001, S004
S006 · calculation
Before
inclusion COMPLETE actual union cardinality := sum(1, 1, 1)
After
inclusion COMPLETE actual union cardinality = 3
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 · implies
Before
ACTUAL union element=8;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1
After
only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1
Conditions: none
Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated.
Step identifiers and dependencies
Rule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S006
S008 · implies
Before
ACTUAL union element=9;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1
After
only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1
Conditions: none
Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated.
Step identifiers and dependencies
Rule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S007
S009 · implies
Before
ACTUAL union element=10;ALL containing ORIGINAL family indices=[0];indexed multiplicity r=1
After
only nonempty subsets of these containing indices contribute to this element;pair subsets not containing the first such index with subsets adding it:opposite signs cancel except its positive singleton;total coefficient1
Conditions: none
Fix one containing index.Pair each nonempty subset omitting it with the subset adding it;their signs cancel.The missing empty subset leaves its positive singleton unpaired,giving coefficient1.Equal sets at different indices increase r and must not be deduplicated.
Step identifiers and dependencies
Rule: incexc_each_element_full_coefficient_not_one_sum_coincidence · Uses: G001, S008
S010 · implies
Before
COMPLETE indexed family size=1;ALL1 nonempty index subsets accounted for;ACTUAL union=[8, 9, 10]
After
each covered element has coefficient1 and any uncovered element coefficient0;FULL alternating sum=3=actual union size=3;empty family yields empty union/no terms/0
Conditions: none
Sum the pointwise coefficient certificates to get the entire finite union size,rather than inferring a theorem from one numerical coincidence.Uncovered points have no containing indices and contribute0;the empty family gives0 directly.
Step identifiers and dependencies
Rule: incexc_full_pointwise_identity_all_finite_indexed_sets · Uses: G001, S009