Completeness Ontology

flow "CompletenessOntology" v1.0

# ============================================================
# COMPLETENESS ONTOLOGY — REAL ANALYSIS — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/real-analysis/completeness-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Real Analysis"
ENTITY := "Completeness"

# ---------- CORE ENTITY ----------
COMPLETENESS := {
    type: MathematicalSpaceProperty,
    domain: RealAnalysis,
    purpose: "Represent the property that limiting objects implied by internal Cauchy or order structure remain within the mathematical space."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    COMPLETENESS,
    COMPLETE_SPACE,
    REAL_NUMBERS,
    CAUCHY_SEQUENCE,
    CONVERGENCE,
    LIMIT,
    METRIC_SPACE,
    NORMED_SPACE,
    BANACH_SPACE,
    SUPREMUM,
    INFIMUM,
    LEAST_UPPER_BOUND_PROPERTY,
    MONOTONE_CONVERGENCE,
    COMPLETION,
    DEDekIND_COMPLETENESS
}

# ---------- METRIC COMPLETENESS ----------
let
    (X,d)
        := METRIC_SPACE

X
    -> COMPLETE

iff

every_Cauchy_sequence_in(X)
    -> converges_to(point_in_X)

METRIC_COMPLETENESS
    -> no_missing_Cauchy_limits

# ---------- CAUCHY SEQUENCE ----------
sequence
    := (x_n)

x_n
    -> CAUCHY

iff

for_every epsilon > 0
exists N
such_that

m,n >= N
    -> d(x_m,x_n) < epsilon

CAUCHY_PROPERTY
    -> defined_without(KNOWN_LIMIT)

# ---------- CONVERGENCE IMPLIES CAUCHY ----------
in every metric space:

CONVERGENT_SEQUENCE
    -> CAUCHY_SEQUENCE

CAUCHY_SEQUENCE
    -> CONVERGENT_SEQUENCE
       iff(SPACE_COMPLETE)

COMPLETENESS
    -> supplies(MISSING_REVERSE_IMPLICATION)

# ---------- REAL NUMBER COMPLETENESS ----------
REAL_NUMBERS
    -> COMPLETE

R
    -> complete_under(STANDARD_METRIC)

d(x,y)
    := abs(x-y)

every_real_Cauchy_sequence
    -> converges_to(REAL_NUMBER)

# ---------- RATIONAL INCOMPLETENESS ----------
RATIONAL_NUMBERS
    -> NOT_COMPLETE

exists
    rational_Cauchy_sequence
    whose_limit_in_R
        -> IRRATIONAL

example:
    rational_approximations_to(sqrt(2))

Q
    -> dense_subspace_of(R)
    -/> closed_in(R)

# ---------- ORDER COMPLETENESS ----------
LEAST_UPPER_BOUND_PROPERTY:

if
    A subset_of R
    AND A nonempty
    AND A bounded_above
then

sup(A)
    -> exists_in(R)

ORDER_COMPLETENESS
    -> fundamental_property_of(REAL_NUMBERS)

# ---------- GREATEST LOWER BOUND ----------
if
    A nonempty
    AND bounded_below
then

inf(A)
    -> exists_in(R)

INFIMUM_PROPERTY
    -> dual_to(SUPREMUM_PROPERTY)

# ---------- DEDEKIND COMPLETENESS ----------
DEDEKIND_CUT
    -> partitions(RATIONAL_NUMBERS)
       according_to(ORDER_CONDITIONS)

REAL_NUMBERS
    -> may_be_constructed_from(DEDEKIND_CUTS)

DEDEKIND_COMPLETENESS
    ~= ORDER_COMPLETENESS
       for(ARCHIMEDEAN_ORDERED_FIELDS_IN_STANDARD_CONTEXT)

# ---------- CAUCHY COMPLETENESS ----------
CAUCHY_COMPLETENESS
    := every_Cauchy_sequence_converges

for R:

CAUCHY_COMPLETENESS
    <-> LEAST_UPPER_BOUND_PROPERTY
       given(ORDERED_FIELD_STRUCTURE)

COMPLETENESS_FORMS
    -> equivalent_under(APPROPRIATE_REAL_FIELD_ASSUMPTIONS)

# ---------- MONOTONE CONVERGENCE PROPERTY ----------
if
    sequence increasing
    AND bounded_above
then

sequence
    -> converges

if
    sequence decreasing
    AND bounded_below
then

sequence
    -> converges

MONOTONE_CONVERGENCE_PROPERTY
    -> equivalent_to(COMPLETENESS)
       in(REAL_NUMBER_CONTEXT)

# ---------- NESTED INTERVAL PROPERTY ----------
let

I_n
    := [a_n,b_n]

with

I_1 superset I_2 superset ...

and
    each I_n nonempty_closed_bounded

then

intersection_from_n_1_to_infinity(I_n)
    -> NONEMPTY

NESTED_INTERVAL_PROPERTY
    -> equivalent_completeness_form
       under(STANDARD_REAL_CONTEXT)

if
    length(I_n) -> 0
then

intersection
    -> SINGLE_POINT

# ---------- BOLZANO-WEIERSTRASS RELATION ----------
every_bounded_sequence_in_R
    -> has(CONVERGENT_SUBSEQUENCE)

BOLZANO_WEIERSTRASS_PROPERTY
    -> related_to(COMPLETENESS_AND_COMPACTNESS)

in R^n:
    bounded_sequence
        -> convergent_subsequence

# ---------- COMPLETION ----------
for incomplete metric space X:

COMPLETION
    := complete_metric_space(X_hat)

with
    isometric_embedding(
        X -> X_hat
    )

such_that

image(X)
    -> dense_in(X_hat)

COMPLETION
    -> fills(MISSING_LIMIT_POINTS)

# ---------- UNIQUENESS OF COMPLETION ----------
COMPLETION
    -> unique_up_to(ISOMETRY_FIXING_DENSE_COPY)
       under(STANDARD_COMPLETION_THEOREM)

different_constructions
    -> represent(SAME_COMPLETE_STRUCTURE_UP_TO_ISOMETRY)

# ---------- Q COMPLETION ----------
RATIONAL_NUMBERS
    -> completion_under(ABSOLUTE_VALUE_METRIC)
    -> REAL_NUMBERS

R
    -> COMPLETION_OF(Q)

REAL_NUMBERS
    -> may_be_constructed_as(EQUIVALENCE_CLASSES_OF_CAUCHY_SEQUENCES_OF_RATIONALS)

# ---------- NORMED SPACE COMPLETENESS ----------
NORMED_SPACE
    := (X,||.||)

metric:
    d(x,y)
        := ||x-y||

X
    -> COMPLETE

iff

every_norm_Cauchy_sequence
    -> norm_converges_in(X)

COMPLETE_NORMED_SPACE
    := BANACH_SPACE

# ---------- BANACH SPACE ----------
BANACH_SPACE
    -> COMPLETE_NORMED_VECTOR_SPACE

COMPLETENESS
    -> supports(INFINITE_SUMS)
    -> supports(OPERATOR_LIMITS)
    -> supports(FIXED_POINT_METHODS)

# ---------- HILBERT SPACE ----------
HILBERT_SPACE
    -> COMPLETE_INNER_PRODUCT_SPACE

INNER_PRODUCT
    -> induces(NORM)

HILBERT_COMPLETENESS
    -> completeness_under(INNER_PRODUCT_NORM)

# ---------- CLOSED SUBSPACE ----------
if
    X complete_metric_space
    AND
    A closed_subset_of(X)
then

A
    -> COMPLETE
       under(RESTRICTED_METRIC)

CLOSED_SUBSET_OF_COMPLETE_SPACE
    -> COMPLETE

# ---------- COMPLETE SUBSPACE CLOSEDNESS ----------
if
    X metric_space
    AND
    A subset_of X
    is complete_under_restricted_metric
then

A
    -> CLOSED_IN(X)

COMPLETE_SUBSPACE
    -> closed_in(AMBIENT_METRIC_SPACE)

# ---------- COMPACTNESS RELATION ----------
COMPACT_METRIC_SPACE
    -> COMPLETE

COMPACTNESS
    -> implies(COMPLETENESS_IN_METRIC_CONTEXT)

COMPLETE
    -/> implies(COMPACT)

COMPACT_METRIC_SPACE
    <-> COMPLETE_AND_TOTALLY_BOUNDED

# ---------- ABSOLUTELY CONVERGENT SERIES ----------
in Banach space X:

if
    sum(
        ||x_n||
    )
    < infinity
then

partial_sums
    -> CAUCHY

COMPLETENESS
    -> ensures(
        sum(x_n)
        converges_in_X
    )

BANACH_COMPLETENESS
    -> converts(ABSOLUTE_NORM_SUMMABILITY_TO_CONVERGENCE)

# ---------- FIXED POINT ----------
BANACH_FIXED_POINT_THEOREM
    -> requires(COMPLETE_METRIC_SPACE)

CONTRACTION
    -> generates(CAUCHY_ITERATION)

COMPLETENESS
    -> ensures(ITERATIVE_LIMIT_EXISTS_IN_SPACE)

# ---------- DIFFERENTIAL EQUATIONS ----------
COMPLETE_FUNCTION_SPACE
    -> allows(CAUCHY_LIMIT_OF_APPROXIMATIONS_TO_REMAIN_IN_SPACE)

EXISTENCE_PROOFS
    -> often_use(COMPLETENESS)

PICARD_ITERATION
    -> uses(COMPLETE_FUNCTION_SPACE)
       in(STANDARD_FIXED_POINT_ARGUMENTS)

# ---------- APPLICATIONS ----------
COMPLETENESS
    -> applied_to(REAL_ANALYSIS)
    -> applied_to(FUNCTIONAL_ANALYSIS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(FIXED_POINT_THEORY)
    -> applied_to(NUMERICAL_ANALYSIS)
    -> applied_to(METRIC_GEOMETRY)

# ---------- RELATION GRAPH ----------
CAUCHY_SEQUENCE
    + COMPLETE_SPACE
    -> CONVERGENT_SEQUENCE

REAL_NUMBERS
    -> have(ORDER_COMPLETENESS)

LEAST_UPPER_BOUND_PROPERTY
    -> supports(MONOTONE_CONVERGENCE)

INCOMPLETE_SPACE
    -> has(COMPLETION)

NORMED_SPACE
    + COMPLETENESS
    -> BANACH_SPACE

INNER_PRODUCT_SPACE
    + COMPLETENESS
    -> HILBERT_SPACE

COMPACT_METRIC_SPACE
    -> COMPLETE

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "completeness" ~= "property that every Cauchy sequence converges within the space",
    "complete metric space" ~= "metric space containing limits of all its Cauchy sequences",
    "least upper bound property" ~= "order-completeness property of the real numbers",
    "Banach space" ~= "complete normed vector space",
    "Hilbert space" ~= "complete inner-product space",
    "completion" ~= "complete space containing an isometric dense copy of the original space"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    completeness_depends_on_metric_or_uniform_structure: true,
    cauchy_implies_convergence_only_in_complete_space: true,
    rational_numbers_are_incomplete_under_standard_metric: true,
    real_numbers_are_complete_under_standard_metric: true,
    complete_does_not_imply_compact_in_general: true,
    compact_metric_space_is_complete: true,
    equivalent_completeness_forms_require_context_specific_assumptions: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    cauchy_relation: 0.99,
    real_order_completeness: 0.99,
    completion_relation: 0.99,
    banach_hilbert_relation: 0.99,
    compactness_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(space_and_metric_or_norm)
    -> identify(cauchy_sequences_or_order_structure)
    -> select(metric_or_order_completeness_definition)
    -> test(whether_required_limits_remain_in_space)
    -> classify(complete_or_incomplete)
    -> construct_or_identify(completion_if_requested)
    -> analyse(closed_subspace_or_compactness_relations)
    -> connect_to(banach_hilbert_or_fixed_point_context_if_relevant)
    -> verify(equivalence_theorem_assumptions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed completeness semantic graph",
    primary_entity: COMPLETENESS,
    preserve: [
        space,
        metric,
        norm,
        order_structure,
        cauchy_property,
        limit_location,
        closedness,
        completion
    ],
    infer_through: [
        real_number_completeness,
        least_upper_bounds,
        cauchy_sequences,
        monotone_convergence,
        completions,
        banach_spaces,
        hilbert_spaces,
        compact_metric_spaces
    ],
    avoid: [
        assuming_cauchy_implies convergence in incomplete space,
        confusing_completeness_with compactness,
        claiming_rationals_complete,
        using_order_completeness_outside ordered-field context without justification,
        ignoring_metric_or_norm dependence,
        assuming_all_completeness formulations equivalent in arbitrary spaces
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(completeness_context)
    -> compare(existing_relations)
    -> preserve(space_metric_order_and_limit_invariants)
    -> update(affected_entities)
    -> update(affected_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + COMPLETENESS
    + CAUCHY_SEQUENCE
    + REAL_NUMBERS
    + LEAST_UPPER_BOUND_PROPERTY
    + MONOTONE_CONVERGENCE
    + COMPLETION
    + BANACH_SPACE
    + HILBERT_SPACE
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> COMPLETENESS
    -> SPACE_STRUCTURE_IDENTIFICATION
    -> CAUCHY_ORDER_ANALYSIS
    -> COMPLETENESS_DEFINITION_SELECTION
    -> LIMIT_LOCATION_ANALYSIS
    -> COMPLETE_INCOMPLETE_CLASSIFICATION
    -> COMPLETION_ANALYSIS
    -> CLOSEDNESS_COMPACTNESS_ANALYSIS
    -> BANACH_HILBERT_FIXED_POINT_ANALYSIS
    -> EQUIVALENCE_HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW