{"id":"counter-is-complete-multiset-algebra","text":"Counter provides a complete algebraic toolkit for multiset problems: construction (from iterables), measurement (frequency queries with zero-default), comparison (subtraction as containment test), and combination (intersection as element-wise min), making it the single abstraction sufficient to cover the entire multiset problem class without supplementary data structures.","truth_value":"IN","source":"","source_url":"","source_hash":"","justifications":[{"type":"SL","antecedents":["counter-universal-frequency-primitive","counter-subtraction-is-multiset-containment-test"],"outlist":[],"label":"frequency measurement + subtraction containment + intersection min = algebraically complete for multisets"}],"dependents":["counter-algebra-grounds-hash-pipeline-universality","palindrome-is-canonical-counter-pipeline-exemplar"],"metadata":{"source_type":"derived","last_reviewed":"2026-06-07T22:02:22","review_result":"invalid"},"created_at":"","updated_at":"","reviewed_at":"","verified_at":"","retracted_at":"","explanation":{"steps":[{"node":"counter-is-complete-multiset-algebra","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-universal-frequency-primitive","counter-subtraction-is-multiset-containment-test"],"label":"frequency measurement + subtraction containment + intersection min = algebraically complete for multisets"},{"node":"counter-universal-frequency-primitive","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-dominant-frequency-tool","counter-pattern-dominates-frequency-problems","counter-missing-key-returns-zero","counter-subtraction-drops-nonpositive"],"label":"Counter's built-in semantics eliminate boilerplate that manual dicts would require"},{"node":"counter-dominant-frequency-tool","truth_value":"IN","reason":"premise"},{"node":"counter-pattern-dominates-frequency-problems","truth_value":"IN","reason":"premise"},{"node":"counter-missing-key-returns-zero","truth_value":"IN","reason":"premise"},{"node":"counter-subtraction-drops-nonpositive","truth_value":"IN","reason":"premise"},{"node":"counter-subtraction-is-multiset-containment-test","truth_value":"IN","reason":"SL justification valid","antecedents":["counter-subtraction-as-subset-check","counter-sub-empty-means-containment","counter-subtraction-drops-nonpositive"],"label":"Depth-1 unification: three base beliefs describe mechanism (drops nonpositive), consequence (empty = containment), and application (subset check) of the same Counter algebraic property"},{"node":"counter-subtraction-as-subset-check","truth_value":"IN","reason":"premise"},{"node":"counter-sub-empty-means-containment","truth_value":"IN","reason":"premise"}]}}