Uniform Convergence Ontology

flow "UniformConvergenceOntology" v1.0

# ============================================================
# UNIFORM CONVERGENCE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
UNIFORM_CONVERGENCE := {
    type: FunctionSequenceConvergenceMode,
    domain: RealAnalysis,
    purpose: "Represent convergence of a sequence of functions to a limiting function with a single index threshold that works uniformly for every point in the domain."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FUNCTION_SEQUENCE,
    LIMIT_FUNCTION,
    UNIFORM_CONVERGENCE,
    POINTWISE_CONVERGENCE,
    SUPREMUM_NORM,
    UNIFORM_CAUCHY_CRITERION,
    CONTINUITY,
    INTEGRATION,
    DIFFERENTIATION,
    SERIES_OF_FUNCTIONS,
    WEIERSTRASS_M_TEST,
    UNIFORM_LIMIT_THEOREM,
    INTERCHANGE_OF_LIMITS
}

# ---------- BASIC DEFINITION ----------
let
    f_n: D -> R

and
    f: D -> R

f_n
    -> uniformly_converges_to(f)

iff

for_every epsilon > 0
exists N
such_that

for_all n >= N
and
for_all x in D:

abs(
    f_n(x)-f(x)
)
    < epsilon

UNIFORM_CONVERGENCE
    -> one_N_for_all_x

# ---------- QUANTIFIER STRUCTURE ----------
UNIFORM:
    for_every epsilon
    exists N
    for_all x
    for_all n>=N

POINTWISE:
    for_every x
    for_every epsilon
    exists N(x,epsilon)
    for_all n>=N

QUANTIFIER_ORDER
    -> distinguishes(UNIFORM_FROM_POINTWISE)

# ---------- SUPREMUM CHARACTERISATION ----------
if
    sup_x_in_D(
        abs(f_n(x)-f(x))
    )
    -> 0
then

f_n
    -> uniformly_converges_to(f)

and conversely
under(extended_supremum_interpretation)

UNIFORM_ERROR
    := sup_x_in_D(
        abs(f_n(x)-f(x))
    )

UNIFORM_CONVERGENCE
    <-> UNIFORM_ERROR -> 0

# ---------- SUPREMUM NORM ----------
if
    functions_are_bounded_on(D)

||g||_infinity
    :=
sup_x_in_D(
    abs(g(x))
)

then

f_n -> f uniformly

iff

||f_n-f||_infinity
    -> 0

UNIFORM_CONVERGENCE
    -> NORM_CONVERGENCE_IN(SUP_NORM_SPACE)

# ---------- POINTWISE RELATION ----------
UNIFORM_CONVERGENCE
    -> implies(POINTWISE_CONVERGENCE)

POINTWISE_CONVERGENCE
    -/> generally_implies(UNIFORM_CONVERGENCE)

UNIFORM
    -> stronger_than(POINTWISE)

# ---------- CLASSIC NONUNIFORM EXAMPLE ----------
on
    D=[0,1]

f_n(x)
    := x^n

pointwise_limit:
    f(x)
        = 0 for 0 <= x < 1
        = 1 for x = 1

f_n
    -> pointwise_converges_to(f)

but

f_n
    -/> uniformly_converges_to(f)

DISCONTINUOUS_LIMIT
    -> incompatible_with(UNIFORM_LIMIT_OF_CONTINUOUS_FUNCTIONS)

# ---------- UNIFORM CAUCHY CRITERION ----------
f_n
    -> uniformly_Cauchy

iff

for_every epsilon > 0
exists N
such_that

m,n >= N
and
x in D

implies

abs(
    f_n(x)-f_m(x)
)
    < epsilon

if
    codomain_complete
then

UNIFORM_CAUCHY
    -> UNIFORM_CONVERGENCE

# ---------- UNIFORM LIMIT THEOREM ----------
if
    each f_n
        -> continuous_on(D)

and

f_n
    -> uniformly_converges_to(f)

then

f
    -> continuous_on(D)

UNIFORM_CONVERGENCE
    -> preserves(CONTINUITY)

# ---------- INTERCHANGE OF FUNCTION LIMIT AND POINT LIMIT ----------
if
    f_n -> f uniformly
    near relevant domain
    AND
    x -> a
    AND
    suitable_limits_exist
then

lim_(x->a) f(x)
    =
lim_(n->infinity)(
    lim_(x->a) f_n(x)
)

under(STANDARD_INTERCHANGE_HYPOTHESES)

UNIFORMITY
    -> controls(LIMIT_INTERCHANGE)

# ---------- INTEGRATION ----------
if
    f_n
        -> Riemann_integrable_on([a,b])

and

f_n
    -> uniformly_converges_to(f)

then

f
    -> Riemann_integrable

and

integral_from_a_to_b(
    f_n(x) dx
)
    ->
integral_from_a_to_b(
    f(x) dx
)

UNIFORM_CONVERGENCE
    -> permits(INTEGRAL_LIMIT_INTERCHANGE)
       under(STANDARD_RIEMANN_CONTEXT)

# ---------- DIFFERENTIATION CAUTION ----------
f_n
    -> uniformly_converges_to(f)

alone

-/> guarantees(
    f_n_prime -> f_prime
)

DIFFERENTIATION
    -> requires(STRONGER_CONDITIONS)

# ---------- DERIVATIVE LIMIT THEOREM ----------
one standard form:

if
    f_n differentiable_on([a,b])

and
    f_n_prime
        -> uniformly_converges_to(g)

and
    exists x_0 in [a,b]
    such_that
    f_n(x_0)
        -> finite_limit

then

f_n
    -> uniformly_converges_to(f)

and

f
    -> differentiable

with

f_prime
    = g

DERIVATIVE_INTERCHANGE
    -> requires(CONTROL_OF_DERIVATIVES)

# ---------- SERIES OF FUNCTIONS ----------
FUNCTION_SERIES
    := sum_from_n_1_to_infinity(
        f_n(x)
    )

PARTIAL_SUM_FUNCTION
    := S_N(x)
       = sum_from_n_1_to_N(f_n(x))

FUNCTION_SERIES
    -> uniformly_convergent

iff

S_N
    -> uniformly_converges_to(S)

# ---------- WEIERSTRASS M-TEST ----------
if
    abs(f_n(x))
    <= M_n

for_all x in D

and

sum(M_n)
    -> converges

then

sum(f_n(x))
    -> converges_uniformly_and_absolutely_on(D)

WEIERSTRASS_M_TEST
    -> sufficient_condition_for(UNIFORM_CONVERGENCE)

# ---------- UNIFORM ABSOLUTE CONVERGENCE ----------
if
    sum(
        sup_x abs(f_n(x))
    )
    < infinity

then

FUNCTION_SERIES
    -> UNIFORMLY_CONVERGENT

and

-> ABSOLUTELY_CONVERGENT_POINTWISE

SUP_NORM_SUMMABILITY
    -> strong_uniform_control

# ---------- POWER SERIES ----------
POWER_SERIES
    := sum(
        c_n*(x-a)^n
    )

inside any closed interval
    |x-a| <= r < R

POWER_SERIES
    -> uniformly_convergent

R
    -> RADIUS_OF_CONVERGENCE

UNIFORM_CONVERGENCE_ON_COMPACT_SUBINTERVALS
    -> permits(TERM_BY_TERM_DIFFERENTIATION_AND_INTEGRATION)

# ---------- UNIFORM CONVERGENCE ON COMPACT SETS ----------
LOCAL_UNIFORM_CONVERGENCE
    := uniform_convergence_on_every_compact_subset

LOCAL_UNIFORM
    -> weaker_than(GLOBAL_UNIFORM)
       on_noncompact_domains

LOCAL_UNIFORM_CONVERGENCE
    -> important_in(ANALYSIS)

# ---------- PRESERVATION OF BOUNDEDNESS ----------
if
    f_n -> f uniformly

and
    some f_N bounded
    with uniform_error_finite
then

f
    -> bounded

if
    all f_n uniformly_bounded_by_common_M
then

f
    -> bounded_by(M)

UNIFORM_BOUNDS
    -> pass_to(LIMIT)

# ---------- ERROR CONTROL ----------
if
    ||f_n-f||_infinity
    <= epsilon_n

and

epsilon_n -> 0

then

f_n
    -> uniformly_converges_to(f)

ERROR_BOUND
    -> explicit_uniform_convergence_certificate

# ---------- NEGATION ----------
f_n
    -/> uniformly_converges_to(f)

iff

exists epsilon_0 > 0
such_that

for_every N
exists n >= N
exists x_n in D

with

abs(
    f_n(x_n)-f(x_n)
)
    >= epsilon_0

NONUNIFORMITY
    -> witnessed_by(POINTS_DEPENDING_ON_n)

# ---------- APPLICATIONS ----------
UNIFORM_CONVERGENCE
    -> applied_to(FUNCTION_SEQUENCES)
    -> applied_to(FUNCTION_SERIES)
    -> applied_to(POWER_SERIES)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(APPROXIMATION_THEORY)
    -> applied_to(NUMERICAL_ANALYSIS)
    -> applied_to(FUNCTIONAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
FUNCTION_SEQUENCE
    -> may_converge(UNIFORMLY)

UNIFORM_CONVERGENCE
    -> implies(POINTWISE_CONVERGENCE)

SUPREMUM_NORM
    -> metrises(UNIFORM_CONVERGENCE_FOR_BOUNDED_FUNCTIONS)

UNIFORM_CONVERGENCE
    + CONTINUITY_OF_TERMS
    -> CONTINUITY_OF_LIMIT

UNIFORM_CONVERGENCE
    -> supports(INTEGRAL_LIMIT_INTERCHANGE)

WEIERSTRASS_M_TEST
    -> proves(UNIFORM_CONVERGENCE_OF_FUNCTION_SERIES)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "uniform convergence" ~= "function convergence with a common index threshold valid at every domain point",
    "uniform limit" ~= "limit function obtained through uniform convergence",
    "supremum-norm convergence" ~= "uniform convergence for bounded real or complex functions under the sup norm",
    "uniform Cauchy" ~= "function sequence whose terms become uniformly close over the whole domain",
    "Weierstrass M-test" ~= "uniform convergence criterion based on a convergent scalar majorant series",
    "locally uniform convergence" ~= "uniform convergence on every compact subset"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    quantifier_order_is_essential: true,
    uniform_convergence_implies_pointwise_convergence: true,
    pointwise_convergence_does_not_generally_imply_uniform_convergence: true,
    uniform_limit_of_continuous_functions_is_continuous: true,
    uniform_convergence_alone_does_not_justify_derivative_interchange: true,
    integral_interchange_requires_relevant_integrability_context: true,
    supremum_norm_characterisation_requires_well_defined_supremum_or_extended_interpretation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    quantifier_structure: 0.99,
    supremum_relation: 0.99,
    continuity_preservation: 0.99,
    integration_relation: 0.99,
    m_test_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function_sequence_and_domain)
    -> identify(candidate_limit_function)
    -> compare(pointwise_and_uniform_quantifiers)
    -> compute_or_bound(supremum_error)
    -> test(uniform_cauchy_property_if_useful)
    -> determine(uniform_convergence)
    -> apply(continuity_preservation_if_hypotheses_hold)
    -> analyse(integral_or_derivative_interchange_if_requested)
    -> apply(weierstrass_m_test_for_function_series_if_valid)
    -> distinguish(global_and_local_uniform_convergence)
    -> validate(all_domain_and_regularities)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed uniform-convergence semantic graph",
    primary_entity: UNIFORM_CONVERGENCE,
    preserve: [
        function_sequence,
        domain,
        limit_function,
        epsilon,
        common_index_threshold,
        supremum_error,
        continuity_status,
        convergence_mode
    ],
    infer_through: [
        pointwise_convergence,
        sup_norm_convergence,
        uniform_cauchy_criteria,
        continuity_preservation,
        integration,
        function_series,
        weierstrass_m_test,
        local_uniform_convergence
    ],
    avoid: [
        swapping_uniform_and_pointwise quantifiers,
        assuming_pointwise_convergence_is uniform,
        differentiating_term_by_term_from uniform convergence alone,
        applying_m_test_without a common scalar majorant,
        ignoring_domain_in supremum estimates,
        confusing_local_uniform_with global uniform convergence
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + UNIFORM_CONVERGENCE
    + POINTWISE_CONVERGENCE
    + SUPREMUM_NORM
    + UNIFORM_CAUCHY_CRITERION
    + UNIFORM_LIMIT_THEOREM
    + WEIERSTRASS_M_TEST
    + FUNCTION_SERIES
    + LOCAL_UNIFORM_CONVERGENCE
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> UNIFORM_CONVERGENCE
    -> FUNCTION_SEQUENCE_DOMAIN_IDENTIFICATION
    -> LIMIT_FUNCTION_IDENTIFICATION
    -> QUANTIFIER_ANALYSIS
    -> SUPREMUM_ERROR_ANALYSIS
    -> UNIFORM_CAUCHY_ANALYSIS
    -> CONVERGENCE_CLASSIFICATION
    -> CONTINUITY_PRESERVATION_ANALYSIS
    -> LIMIT_INTERCHANGE_ANALYSIS
    -> FUNCTION_SERIES_TEST_ANALYSIS
    -> LOCAL_GLOBAL_UNIFORM_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW