Mathematical Analysis Ontology

flow "MathematicalAnalysisOntology" v1.0

# ============================================================
# MATHEMATICAL ANALYSIS ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/foundations-of-calculus/mathematical-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Foundations of Calculus"
ENTITY := "Mathematical Analysis"

# ---------- CORE ENTITY ----------
MATHEMATICAL_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    purpose: "Rigorous study of limits, continuity, convergence, differentiation, integration, infinite processes and related structures."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    REAL_NUMBERS,
    COMPLEX_NUMBERS,
    FUNCTION,
    SEQUENCE,
    SERIES,
    LIMIT,
    CONTINUITY,
    CONVERGENCE,
    DERIVATIVE,
    INTEGRAL,
    METRIC,
    NORM,
    COMPLETENESS,
    COMPACTNESS,
    CONNECTEDNESS
}

# ---------- DISCIPLINE STRUCTURE ----------
MATHEMATICAL_ANALYSIS
    -> develops_from(CALCULUS)
    -> formalises(LIMIT_PROCESSES)
    -> studies(FUNCTIONS)
    -> studies(SEQUENCES)
    -> studies(SERIES)
    -> studies(CONTINUITY)
    -> studies(DIFFERENTIATION)
    -> studies(INTEGRATION)

MATHEMATICAL_ANALYSIS
    -> contains(REAL_ANALYSIS)
    -> contains(COMPLEX_ANALYSIS)
    -> contains(FUNCTIONAL_ANALYSIS)
    -> relates_to(MEASURE_THEORY)
    -> relates_to(TOPOLOGY)
    -> relates_to(DIFFERENTIAL_EQUATIONS)

# ---------- NUMBER SYSTEM FOUNDATIONS ----------
REAL_NUMBERS
    -> possess(ORDER_STRUCTURE)
    -> possess(FIELD_STRUCTURE)
    -> possess(COMPLETENESS)

COMPLETENESS
    -> supports(CONVERGENCE_THEOREMS)
    -> supports(SUPREMUM_PROPERTY)
    -> supports(CAUCHY_SEQUENCE_CONVERGENCE)

COMPLEX_NUMBERS
    -> extend(REAL_NUMBERS)
    -> support(COMPLEX_ANALYSIS)

# ---------- LIMITS ----------
limit(f(x), x -> a) = L

iff

forall epsilon > 0
    exists delta > 0
        such_that
        0 < abs(x-a) < delta
        -> abs(f(x)-L) < epsilon

LIMIT
    -> formalises(APPROACH_BEHAVIOUR)
    -> underpins(CONTINUITY)
    -> underpins(DIFFERENTIATION)
    -> underpins(INTEGRATION)
    -> underpins(CONVERGENCE)

# ---------- SEQUENCES ----------
SEQUENCE := ordered_map(NATURAL_NUMBERS -> VALUE_SPACE)

converges(a_n, L)
    iff
    forall epsilon > 0
        exists N
            such_that
            n >= N
            -> distance(a_n, L) < epsilon

CAUCHY_SEQUENCE
    -> terms_become(ARBITRARILY_CLOSE)

COMPLETE_SPACE
    -> ensures(EVERY_CAUCHY_SEQUENCE_CONVERGES_WITHIN_SPACE)

# ---------- SERIES ----------
SERIES := sum(a_n, n=1..infinity)

PARTIAL_SUM(n)
    := sum(a_k, k=1..n)

SERIES_CONVERGES
    iff
    sequence(PARTIAL_SUM(n))
        -> converges_to(FINITE_LIMIT)

ABSOLUTE_CONVERGENCE
    -> implies(CONVERGENCE)

CONVERGENCE
    -/> necessarily_implies(ABSOLUTE_CONVERGENCE)

# ---------- CONTINUITY ----------
continuous(f, a)
    iff
    limit(f(x), x -> a) = f(a)

CONTINUITY
    -> preserves(LOCAL_LIMIT_BEHAVIOUR)

UNIFORM_CONTINUITY
    -> strengthens(CONTINUITY)
    -> uses_single_delta_for_all_points_given_epsilon

DIFFERENTIABLE_AT(f, a)
    -> implies(CONTINUOUS_AT(f, a))

CONTINUOUS_AT(f, a)
    -/> necessarily_implies(DIFFERENTIABLE_AT(f, a))

# ---------- DIFFERENTIATION ----------
DERIVATIVE(f, x)
    := limit(
        (f(x+h)-f(x))/h,
        h -> 0
    )

DERIVATIVE
    -> measures(LOCAL_RATE_OF_CHANGE)
    -> supports(LOCAL_APPROXIMATION)
    -> supports(OPTIMISATION)

DIFFERENTIATION
    -> studied_for(EXISTENCE)
    -> studied_for(REGULARITY)
    -> studied_for(HIGHER_ORDER_BEHAVIOUR)

# ---------- INTEGRATION ----------
RIEMANN_INTEGRAL
    := limit_of(RIEMANN_SUMS)

LEBESGUE_INTEGRAL
    -> extends(INTEGRATION_FRAMEWORK)
    -> integrates_via(MEASURE)

INTEGRATION
    -> measures(ACCUMULATION)
    -> studies(EXISTENCE)
    -> studies(CONVERGENCE)
    -> studies(INTERCHANGE_OF_LIMIT_OPERATIONS)

# ---------- METRIC STRUCTURE ----------
METRIC(d)
    -> maps(PAIR_OF_POINTS, NONNEGATIVE_REAL)
    -> satisfies(NONNEGATIVITY)
    -> satisfies(IDENTITY_OF_INDISCERNIBLES)
    -> satisfies(SYMMETRY)
    -> satisfies(TRIANGLE_INEQUALITY)

METRIC_SPACE
    -> provides(DISTANCE_STRUCTURE)
    -> defines(OPEN_SETS)
    -> defines(CONVERGENCE)
    -> defines(CONTINUITY)

# ---------- NORM STRUCTURE ----------
NORM
    -> measures(VECTOR_MAGNITUDE)
    -> induces(METRIC)

NORMED_SPACE
    -> combines(VECTOR_SPACE, NORM)

BANACH_SPACE
    := COMPLETE_NORMED_VECTOR_SPACE

# ---------- TOPOLOGICAL PROPERTIES ----------
COMPACTNESS
    -> supports(FINITE_SUBCOVER_PROPERTY)
    -> strengthens(GLOBAL_CONTROL_OF_LOCAL_PROPERTIES)

CONNECTEDNESS
    -> prevents(SEPARATION_INTO_DISJOINT_NONEMPTY_OPEN_COMPONENTS)

CONTINUOUS_IMAGE_OF_COMPACT_SET
    -> is(COMPACT)

CONTINUOUS_IMAGE_OF_CONNECTED_SET
    -> is(CONNECTED)

# ---------- FUNDAMENTAL ANALYTIC RELATIONS ----------
LIMIT
    -> defines(CONTINUITY)

LIMIT
    -> defines(DERIVATIVE)

LIMIT
    -> determines(SEQUENCE_CONVERGENCE)

COMPLETENESS
    -> governs(CAUCHY_CONVERGENCE)

METRIC
    -> defines(DISTANCE)

DISTANCE
    -> supports(LIMIT_AND_CONTINUITY)

MEASURE
    -> supports(LEBESGUE_INTEGRATION)

MATHEMATICAL_ANALYSIS
    -> seeks(EXISTENCE)
    -> seeks(UNIQUENESS)
    -> seeks(CONVERGENCE)
    -> seeks(STABILITY)
    -> seeks(REGULARITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "analysis" ~= "mathematical analysis when mathematical context is explicit",
    "approach behaviour" ~= "limit behaviour",
    "sequence tends to L" ~= "sequence converges to L",
    "complete normed space" ~= "Banach space",
    "local rate of change" ~= "derivative",
    "infinite sum convergence" ~= "convergence of its sequence of partial sums"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    convergence_requires_defined_topological_or_metric_context: true,
    differentiability_implies_continuity: true,
    continuity_does_not_imply_differentiability: true,
    absolute_convergence_implies_convergence: true,
    convergence_does_not_always_imply_absolute_convergence: true,
    completeness_is_space_dependent: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    discipline_structure: 0.99,
    limit_relations: 0.99,
    convergence_relations: 0.99,
    continuity_relations: 0.99,
    metric_and_norm_relations: 0.98,
    semantic_equivalence: 0.96
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(query.entity)
    -> determine(ambient_space)
    -> determine(structure: metric | norm | topology | measure)
    -> resolve(definitions)
    -> traverse(dependencies)
    -> test(existence_conditions)
    -> test(convergence_conditions)
    -> apply(relevant_theorems)
    -> validate(logical_implications)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed mathematical semantic graph",
    primary_entity: MATHEMATICAL_ANALYSIS,
    preserve: [
        quantifier_structure,
        convergence_conditions,
        implication_direction,
        space_dependencies,
        definitions,
        theorem_relations
    ],
    infer_through: [
        limits,
        sequences,
        series,
        continuity,
        differentiation,
        integration,
        metric_structure,
        completeness
    ],
    avoid: [
        assuming_every_cauchy_sequence_converges_in_incomplete_spaces,
        reversing_one_way_implications,
        assuming_pointwise_and_uniform_convergence_are_equivalent,
        assuming_continuity_implies_differentiability,
        assuming_convergence_implies_absolute_convergence
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    validate(new_information)
    -> identify(affected_structure)
    -> compare(existing_relations)
    -> preserve(mathematical_invariants)
    -> update(entities_and_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LIMIT
    + CONVERGENCE
    + CONTINUITY
    + DERIVATIVE
    + INTEGRATION
    + METRIC
    + COMPLETENESS
    + SEMANTIC_EQUIVALENCE
)

# ---------- EXECUTION ----------
INPUT(query)
    -> MATHEMATICAL_ANALYSIS
    -> STRUCTURE_IDENTIFICATION
    -> LIMIT_ANALYSIS
    -> CONVERGENCE_ANALYSIS
    -> CONTINUITY_ANALYSIS
    -> DIFFERENTIAL_ANALYSIS
    -> INTEGRAL_ANALYSIS
    -> SPACE_PROPERTY_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW