Monotonic Sequence Ontology

FLOW ALGEBRA
MONOTONIC SEQUENCE ONTOLOGY
Kireth Semantic Library — Mathematical Entities / Sequence and Series Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/sequence-and-series-entities/monotonic-sequence-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB
Source status: Formal mathematical synthesis; source page not independently verified.

01. PURPOSE
flow "MonotonicSequenceOntology" v1.0
  objective: represent, classify, validate and reason about monotonic sequences.
  scope: finite and infinite sequences over ordered codomains, especially real numbers.
  exclusions: monotonic functions are related but not identical to monotonic sequences.

02. ENTITY SCHEMA
entity MonotonicSequence extends Sequence:
  id: URI
  index_set: ordered discrete domain (e.g. {0,...,N} or natural numbers)
  codomain: ordered set (D, <=)
  term_rule: n -> a_n
  direction: NONDECREASING | NONINCREASING | STRICTLY_INCREASING | STRICTLY_DECREASING | CONSTANT
  bounded_above: bool?
  bounded_below: bool?
  limit: value?
  evidence: proof | certified symbolic rule | finite enumeration | observation
  confidence: [0,1]
  provenance: source reference

03. PRECONDITIONS
require index_set has a successor relation where comparisons are made.
require codomain admits a specified order relation.
require term_rule defined at each relevant index.
for partial orders, comparability must be established for consecutive terms.

04. DEFINITIONS
nondecreasing(a) := forall n with n+1 in I: a_n <= a_(n+1).
nonincreasing(a) := forall n with n+1 in I: a_n >= a_(n+1).
strictly_increasing(a) := forall n with n+1 in I: a_n < a_(n+1).
strictly_decreasing(a) := forall n with n+1 in I: a_n > a_(n+1).
constant(a) := forall n,m in I: a_n = a_m.
monotone(a) := nondecreasing(a) OR nonincreasing(a).
strictly_monotone(a) := strictly_increasing(a) OR strictly_decreasing(a).
For singleton or empty comparison ranges, predicates may hold vacuously.

05. ADJACENT-TERM CRITERIA
on linearly ordered codomain and consecutive index domain:
  nondecreasing(a) iff forall n: a_(n+1)-a_n >= 0 (numeric codomain).
  nonincreasing(a) iff forall n: a_(n+1)-a_n <= 0.
  strictly_increasing(a) iff forall n: a_(n+1)-a_n > 0.
  strictly_decreasing(a) iff forall n: a_(n+1)-a_n < 0.
For all m<n, adjacent inequalities imply a_m <= a_n (or reverse).

06. CLASSIFICATION
constant => nondecreasing AND nonincreasing.
strictly_increasing => nondecreasing.
strictly_decreasing => nonincreasing.
strictly increasing and strictly decreasing cannot both hold with two distinct indices.
A sequence can be nonmonotone yet eventually monotone.

07. EVENTUAL MONOTONICITY
eventually_nondecreasing(a) := exists N: forall n>=N, a_n <= a_(n+1).
eventually_nonincreasing(a) := exists N: forall n>=N, a_n >= a_(n+1).
eventual properties do not imply global monotonicity.

08. BOUNDEDNESS
bounded_above(a) := exists M: forall n, a_n <= M.
bounded_below(a) := exists m: forall n, m <= a_n.
bounded(a) := bounded_above(a) AND bounded_below(a).
Finite real sequences are bounded.
An infinite nondecreasing real sequence is bounded below by its first term.
An infinite nonincreasing real sequence is bounded above by its first term.

09. MONOTONE CONVERGENCE THEOREM
For a real sequence indexed by natural numbers:
  nondecreasing AND bounded_above => converges to sup{a_n:n in I}.
  nonincreasing AND bounded_below => converges to inf{a_n:n in I}.
  nondecreasing AND unbounded_above => diverges to +infinity (extended reals).
  nonincreasing AND unbounded_below => diverges to -infinity (extended reals).
Do not infer a finite limit from monotonicity alone.

10. EXAMPLES
sequence a_n=n (n>=0): strictly increasing, unbounded above, no finite limit.
sequence a_n=1/(n+1) (n>=0): strictly decreasing, bounded, limit 0.
sequence a_n=5: constant, nondecreasing, nonincreasing, limit 5.
sequence a_n=(-1)^n: nonmonotone, bounded, divergent.
sequence a_n=1-2^(-n): strictly increasing, bounded above by 1, limit 1.

11. OPERATIONS AND TRANSFORMATIONS
If a,b nondecreasing, then a+b nondecreasing where sums defined.
If a nondecreasing and c>0, then c*a nondecreasing.
If a nondecreasing and c<0, then c*a nonincreasing.
If a nondecreasing and f order-preserving on its range, then f(a_n) nondecreasing.
If f strictly order-preserving and a strictly increasing, then f(a_n) strictly increasing.
Product of nondecreasing sequences need not be nondecreasing without additional sign conditions.
Subsequence of a nondecreasing sequence, with increasing selected indices, is nondecreasing.

12. RELATIONSHIP GRAPH
MonotonicSequence --IS_A--> Sequence
MonotonicSequence --HAS_PROPERTY--> Monotonicity
MonotonicSequence --HAS_INDEX_SET--> OrderedIndexSet
MonotonicSequence --HAS_CODOMAIN--> OrderedSet
MonotonicSequence --MAY_HAVE--> UpperBound
MonotonicSequence --MAY_HAVE--> LowerBound
MonotonicSequence --MAY_CONVERGE_TO--> Limit
MonotonicSequence --RELATED_TO--> BoundedSequence
MonotonicSequence --RELATED_TO--> ConvergentSequence
MonotonicSequence --RELATED_TO--> RecursiveSequence
MonotonicSequence --RELATED_TO--> InfiniteSequence

13. SEMANTIC EQUIVALENCE
"monotonically increasing" commonly means nondecreasing; resolve convention explicitly.
"strictly increasing" means a_n < a_(n+1), not merely <=.
"monotonically decreasing" commonly means nonincreasing; resolve convention explicitly.
"non-decreasing sequence" == "nondecreasing sequence".
Never equate strict and non-strict monotonicity.

14. INFERENCE RULES
rule R1: if forall adjacent indices a_n<=a_(n+1), infer NONDECREASING.
rule R2: if forall adjacent indices a_n<a_(n+1), infer STRICTLY_INCREASING.
rule R3: if nondecreasing and bounded_above and real-valued and infinite, infer FINITE_CONVERGENCE.
rule R4: if nonincreasing and bounded_below and real-valued and infinite, infer FINITE_CONVERGENCE.
rule R5: if constant, infer NONDECREASING and NONINCREASING.
rule R6: if exists i,j with a_i<a_(i+1) and a_j>a_(j+1), infer NOT_MONOTONE.
rule R7: if strictly increasing, infer NONDECREASING.
rule R8: if order-preserving f and nondecreasing a, infer NONDECREASING(f o a).

15. VALIDATION
check V1: sequence indices and successor comparisons are well-typed.
check V2: all required terms exist.
check V3: ordering relation is declared and compatible with codomain.
check V4: strictness is not inferred from weak inequalities.
check V5: convergence theorem uses complete ordered reals (or justified generalisation).
check V6: finite sampling is not accepted as proof for an infinite sequence.
check V7: examples are consistent with declared index origin.
check V8: constant sequences can have both weak monotonic directions.

16. TEST CASES
test A: [1,2,2,4] => nondecreasing true; strictly increasing false.
test B: [4,3,2,1] => strictly decreasing true.
test C: [2,2,2] => constant true; both weak directions true.
test D: [1,3,2] => monotone false.
test E: a_n=n => strictly increasing true; bounded above false.
test F: a_n=1/(n+1) => strictly decreasing true; limit 0.

17. CONFIDENCE
mathematical_theorem: 1.0 when hypotheses formally established.
symbolic_proof: 1.0 after verified proof checking.
finite_enumeration: 1.0 only for fully enumerated finite domain.
numerical_sampling: provisional; never sufficient for universal infinite claim.
unknown: report UNKNOWN, not false.

18. PROVENANCE
record source_url, source_status, authors, version, creation_date.
record every inferred claim with rule_id, premises and proof/evidence.
record whether statements apply to finite or infinite sequences.

19. FLOW_HASH
canonicalise: UTF-8 NFC; stable field order; normalised identifiers.
FLOW_HASH := SHA-256(canonicalised ontology payload excluding hash field).
Hash must be computed by implementation; do not invent digest values.

20. UPDATE_MODEL
on source revision: compare definition, assumptions, classifications and relationships.
on theorem revision: rerun inference validation and test suite.
on graph update: refresh affected nodes and dependent edges.
maintain version history and immutable provenance records.

21. AI_CONSUMPTION
input: symbolic sequence rule, index set, codomain, assumptions, evidence.
output: classification, direction, strictness, bounds, convergence, explanations.
require: uncertainty and unproven claims remain explicit.
require: distinguish observed finite prefix from global proof.
expose: structured entities, typed edges, inference trace, confidence, provenance.

22. TERMINATION
assert required_sections_present == true.
assert mathematical_consistency_checks == passed.
assert unverified_source_claims_are_not_presented_as_verified == true.
END FLOW ALGEBRA