Real Analysis

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "RealAnalysisOntology" v1.0

# ============================================================
# REAL ANALYSIS — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
REAL_ANALYSIS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: MathematicalAnalysis,
    purpose: "Study the real numbers, sequences, limits, continuity, differentiation, integration, convergence and the rigorous foundations of calculus."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    REAL_NUMBERS,
    ORDER,
    COMPLETENESS,
    SEQUENCE,
    LIMIT,
    CONVERGENCE,
    CAUCHY_SEQUENCE,
    CONTINUITY,
    UNIFORM_CONTINUITY,
    DIFFERENTIABILITY,
    RIEMANN_INTEGRAL,
    SERIES,
    FUNCTION_SEQUENCE,
    UNIFORM_CONVERGENCE,
    COMPACTNESS,
    CONNECTEDNESS
}

# ---------- DISCIPLINE STRUCTURE ----------
REAL_ANALYSIS
    -> branch_of(MATHEMATICAL_ANALYSIS)
    -> provides(RIGOROUS_FOUNDATIONS_OF_CALCULUS)
    -> studies(REAL_VALUED_FUNCTIONS)
    -> studies(SEQUENCES_AND_SERIES)
    -> studies(LIMIT_PROCESSES)
    -> studies(TOPOLOGICAL_PROPERTIES_OF_REAL_SPACES)

# ---------- REAL NUMBERS ----------
REAL_NUMBERS
    := R

R
    -> ORDERED_FIELD
    -> COMPLETE_ORDERED_FIELD

REAL_NUMBER_SYSTEM
    -> supports(ARITHMETIC)
    -> supports(ORDER)
    -> supports(LIMITS)

# ---------- ORDER ----------
for
    x,y in R

x < y
    -> ORDER_RELATION

ORDER
    -> compatible_with(ADDITION)
    -> compatible_with(MULTIPLICATION_BY_POSITIVE_NUMBERS)

ORDER_STRUCTURE
    -> supports(INTERVALS_AND_BOUNDS)

# ---------- BOUNDED SET ----------
A subset_of R

A
    -> bounded_above
       if
       exists M
       such_that
       x <= M
       for_all x in A

A
    -> bounded_below
       if
       exists m
       such_that
       m <= x
       for_all x in A

# ---------- SUPREMUM / INFIMUM ----------
SUPREMUM(A)
    := least_upper_bound

INFIMUM(A)
    := greatest_lower_bound

COMPLETENESS_AXIOM:
    every_nonempty_subset_of_R_bounded_above
        -> has(SUPREMUM_IN_R)

COMPLETENESS
    -> distinguishes(RATIONALS_FROM_REALS)

# ---------- ARCHIMEDEAN PROPERTY ----------
for every real x
exists natural n
such_that

n > x

ARCHIMEDEAN_PROPERTY
    -> prevents(INFINITELY_LARGE_REAL_NUMBERS)

# ---------- DENSITY ----------
RATIONAL_NUMBERS
    -> dense_in(REAL_NUMBERS)

IRRATIONAL_NUMBERS
    -> dense_in(REAL_NUMBERS)

between any two distinct reals:
    -> exists(RATIONAL)
    -> exists(IRRATIONAL)

# ---------- SEQUENCE ----------
SEQUENCE
    := function(
        NATURAL_NUMBERS -> REAL_NUMBERS
    )

x_n
    -> nth_term

SEQUENCE
    -> fundamental_object_for(CONVERGENCE)

# ---------- SEQUENCE LIMIT ----------
x_n
    -> converges_to(L)

iff

for_every epsilon > 0
exists N
such_that

n >= N
    -> abs(x_n-L) < epsilon

LIMIT
    -> unique_when_exists

# ---------- DIVERGENCE ----------
if
    no finite L satisfies(limit_definition)
then

SEQUENCE
    -> DIVERGENT

DIVERGENCE
    -> may_include(OSCILLATION)
    -> may_include(UNBOUNDED_GROWTH)

# ---------- CAUCHY SEQUENCE ----------
x_n
    -> CAUCHY

iff

for_every epsilon > 0
exists N
such_that

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

in R:

CAUCHY_SEQUENCE
    <-> CONVERGENT_SEQUENCE

CAUCHY_COMPLETENESS
    -> equivalent_form_of(REAL_COMPLETENESS)

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

sequence
    -> converges_to(SUPREMUM_OF_TERMS)

if
    sequence decreasing
    AND bounded_below
then

sequence
    -> converges_to(INFIMUM_OF_TERMS)

MONOTONE_CONVERGENCE_THEOREM
    -> uses(COMPLETENESS)

# ---------- BOLZANO-WEIERSTRASS ----------
every_bounded_sequence_in_R^n
    -> has(CONVERGENT_SUBSEQUENCE)

BOLZANO_WEIERSTRASS
    -> sequential_compactness_property_of(BOUNDED_SEQUENCES)

# ---------- FUNCTION LIMIT ----------
for
    f: D -> R

limit_x_to_a f(x)
    = L

iff

for_every epsilon > 0
exists delta > 0
such_that

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

FUNCTION_LIMIT
    -> independent_of(FUNCTION_VALUE_AT_a)

# ---------- CONTINUITY ----------
f
    -> continuous_at(a)

iff

limit_x_to_a f(x)
    = f(a)

epsilon_delta_form:

for_every epsilon > 0
exists delta > 0
such_that

abs(x-a) < delta
    -> abs(f(x)-f(a)) < epsilon

# ---------- SEQUENTIAL CONTINUITY ----------
f continuous_at(a)

iff

for every sequence x_n -> a
with x_n in domain

f(x_n)
    -> f(a)

SEQUENTIAL_CHARACTERISATION
    -> equivalent_in(METRIC_SPACES)

# ---------- UNIFORM CONTINUITY ----------
f
    -> uniformly_continuous_on(D)

iff

for_every epsilon > 0
exists delta > 0
such_that

for_all x,y in D:

abs(x-y) < delta
    -> abs(f(x)-f(y)) < epsilon

delta
    -> independent_of(BASE_POINT)

UNIFORM_CONTINUITY
    -> stronger_than(POINTWISE_CONTINUITY)

# ---------- HEINE-CANTOR ----------
if
    K compact
    AND f continuous_on(K)
then

f
    -> uniformly_continuous_on(K)

HEINE_CANTOR_THEOREM
    -> compactness_strengthens(CONTINUITY)

# ---------- EXTREME VALUE THEOREM ----------
if
    K nonempty_compact
    AND f continuous_on(K)
then

f
    -> attains(MAXIMUM)
    -> attains(MINIMUM)

CONTINUITY
    + COMPACTNESS
    -> EXTREME_VALUE_EXISTENCE

# ---------- INTERMEDIATE VALUE THEOREM ----------
if
    f continuous_on([a,b])
then

for any value y
between f(a) and f(b)

exists c in [a,b]
such_that

f(c)=y

CONTINUOUS_IMAGE_OF_INTERVAL
    -> INTERVAL

# ---------- DIFFERENTIABILITY ----------
f
    -> differentiable_at(a)

if

f_prime(a)
    :=
limit_h_to_0(
    [f(a+h)-f(a)]/h
)

exists_finitely

DIFFERENTIABILITY
    -> implies(CONTINUITY)

CONTINUITY
    -/> implies(DIFFERENTIABILITY)

# ---------- MEAN VALUE THEOREM ----------
if
    f continuous_on([a,b])
    AND differentiable_on((a,b))
then

exists c in (a,b)

such_that

f_prime(c)
    =
[f(b)-f(a)]/(b-a)

MEAN_VALUE_THEOREM
    -> links(LOCAL_DERIVATIVE, GLOBAL_CHANGE)

# ---------- ROLLE THEOREM ----------
if
    f continuous_on([a,b])
    AND differentiable_on((a,b))
    AND f(a)=f(b)
then

exists c in (a,b)
such_that

f_prime(c)=0

ROLLE_THEOREM
    -> special_case_of(MEAN_VALUE_THEOREM)

# ---------- RIEMANN INTEGRAL ----------
for bounded function f on [a,b]:

PARTITION
    := a=x_0 < x_1 < ... < x_n=b

RIEMANN_SUM
    :=
sum(
    f(x_i_star)
    * (x_i-x_(i-1))
)

f
    -> RIEMANN_INTEGRABLE

if
    RIEMANN_SUMS
        -> common_limit
       as(MESH_SIZE -> 0)

# ---------- FUNDAMENTAL THEOREM OF CALCULUS ----------
if
    f continuous
then

F(x)
    :=
integral_from_a_to_x(
    f(t) dt
)

implies

F_prime(x)
    = f(x)

and if
    G_prime=f
then

integral_from_a_to_b(
    f(x) dx
)
    =
G(b)-G(a)

FTC
    -> connects(DIFFERENTIATION, INTEGRATION)

# ---------- INFINITE SERIES ----------
SERIES
    := sum_from_n_1_to_infinity(a_n)

PARTIAL_SUM
    := S_N
       = sum_from_n_1_to_N(a_n)

SERIES_CONVERGES
    iff

S_N
    -> finite_limit

SERIES
    -> analysed_through(PARTIAL_SUM_SEQUENCE)

# ---------- ABSOLUTE CONVERGENCE ----------
if
    sum(abs(a_n))
        converges
then

sum(a_n)
    -> converges

ABSOLUTE_CONVERGENCE
    -> implies(CONVERGENCE)

CONVERSE
    -/> always_true

# ---------- FUNCTION SEQUENCE ----------
f_n
    -> sequence_of_functions

POINTWISE_CONVERGENCE:
    for each x:
        f_n(x) -> f(x)

UNIFORM_CONVERGENCE:
    sup_x(
        abs(f_n(x)-f(x))
    )
    -> 0

UNIFORM_CONVERGENCE
    -> stronger_than(POINTWISE_CONVERGENCE)

# ---------- PRESERVATION UNDER UNIFORM CONVERGENCE ----------
if
    each f_n continuous
    AND f_n -> f uniformly
then

f
    -> continuous

UNIFORM_LIMIT
    -> preserves(CONTINUITY)

# ---------- INTERCHANGE OF LIMIT AND INTEGRAL ----------
if
    f_n -> f uniformly
    on [a,b]
    AND each f_n Riemann_integrable
then

integral(f_n)
    -> integral(f)

UNIFORM_CONVERGENCE
    -> supports(INTEGRAL_LIMIT_INTERCHANGE)

# ---------- DIFFERENTIATION OF FUNCTION SEQUENCES ----------
UNIFORM_CONVERGENCE_OF_FUNCTIONS
    -/> enough_for(DERIVATIVE_LIMIT_INTERCHANGE)

DERIVATIVE_INTERCHANGE
    -> requires(STRONGER_CONDITIONS)

REAL_ANALYSIS
    -> tracks(EXACT_HYPOTHESES_FOR_LIMIT_OPERATIONS)

# ---------- COMPACTNESS ----------
in R^n:

K compact
    iff
    K closed_and_bounded

HEINE_BOREL_THEOREM
    -> characterises(COMPACT_SUBSETS_OF_R^n)

COMPACTNESS
    -> supports(FINITE_SUBCOVER_PROPERTY)
    -> supports(SEQUENTIAL_COMPACTNESS)

# ---------- CONNECTEDNESS ----------
INTERVAL
    -> connected_subset_of(R)

CONNECTED_SUBSETS_OF_R
    -> intervals

CONTINUOUS_IMAGE_OF_CONNECTED_SET
    -> connected

CONNECTEDNESS
    -> underlies(INTERMEDIATE_VALUE_THEOREM)

# ---------- APPLICATIONS ----------
REAL_ANALYSIS
    -> foundation_of(CALCULUS)
    -> applied_to(DIFFERENTIAL_EQUATIONS)
    -> applied_to(PROBABILITY)
    -> applied_to(OPTIMISATION)
    -> applied_to(NUMERICAL_ANALYSIS)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(FUNCTIONAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
REAL_NUMBERS
    -> have(COMPLETENESS)

COMPLETENESS
    -> supports(SEQUENCE_CONVERGENCE_THEOREMS)

LIMIT
    -> defines(CONTINUITY)

LIMIT
    -> defines(DERIVATIVE)

RIEMANN_SUM_LIMIT
    -> defines(RIEMANN_INTEGRAL)

COMPACTNESS
    + CONTINUITY
    -> yields(EXTREME_VALUE_AND_UNIFORM_CONTINUITY_RESULTS)

UNIFORM_CONVERGENCE
    -> controls(LIMIT_OPERATION_INTERCHANGE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "real analysis" ~= "rigorous analysis of real numbers, limits, continuity, differentiation and integration",
    "least upper bound property" ~= "order-completeness property of the real numbers",
    "Cauchy completeness" ~= "every Cauchy sequence converges in the space",
    "uniform convergence" ~= "function convergence uniform over the whole domain",
    "Heine-Borel theorem" ~= "closed-and-bounded characterisation of compact subsets of finite-dimensional Euclidean space",
    "fundamental theorem of calculus" ~= "theorem connecting differentiation and definite integration"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    completeness_is_fundamental_to_real_number_limit_theory: true,
    differentiability_implies_continuity_but_not_conversely: true,
    uniform_continuity_is_stronger_than ordinary continuity: true,
    uniform_convergence_is_stronger_than pointwise convergence: true,
    heine_borel_closed_bounded_characterisation_is_specific_to_finite_dimensional_euclidean_space: true,
    limit_interchanges_require_exact_hypotheses: true,
    theorem_statements_require_domain_and_regularities_to_be_preserved: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    discipline_structure: 0.99,
    completeness_relation: 0.99,
    sequence_limit_relation: 0.99,
    continuity_differentiability_relation: 0.99,
    integration_relation: 0.99,
    compactness_convergence_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(real_analysis_object)
    -> classify(number_set_sequence_series_function_or_set)
    -> identify(domain_and_boundedness)
    -> determine(relevant_limit_or_convergence_structure)
    -> select(epsilon_delta_cauchy_or_topological_definition)
    -> test(theorem_hypotheses)
    -> apply(completeness_compactness_or_continuity_results_if_valid)
    -> analyse(differentiation_or_integration_if_requested)
    -> distinguish(pointwise_uniform_and_norm_like_convergence)
    -> validate(limit_interchanges)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed real-analysis semantic graph",
    primary_entity: REAL_ANALYSIS,
    preserve: [
        real_number_structure,
        domain,
        order,
        boundedness,
        convergence_mode,
        continuity_conditions,
        differentiability_conditions,
        integration_conditions
    ],
    infer_through: [
        completeness,
        sequences,
        cauchy_sequences,
        limits,
        continuity,
        derivatives,
        riemann_integrals,
        compactness,
        uniform_convergence
    ],
    avoid: [
        assuming_bounded_sequence_converges_without subsequence,
        confusing_pointwise_and_uniform convergence,
        claiming_continuity_implies differentiability,
        applying_heine_borel_outside finite-dimensional Euclidean context,
        interchanging_limits_derivatives_or_integrals_without hypotheses,
        ignoring_domain_conditions
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + REAL_ANALYSIS
    + REAL_NUMBERS
    + COMPLETENESS
    + SEQUENCE
    + CONTINUITY
    + DIFFERENTIABILITY
    + RIEMANN_INTEGRAL
    + UNIFORM_CONVERGENCE
    + COMPACTNESS
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> REAL_ANALYSIS
    -> OBJECT_IDENTIFICATION
    -> DOMAIN_BOUNDEDNESS_ANALYSIS
    -> LIMIT_CONVERGENCE_ANALYSIS
    -> DEFINITION_SELECTION
    -> THEOREM_HYPOTHESIS_VALIDATION
    -> COMPLETENESS_COMPACTNESS_ANALYSIS
    -> CONTINUITY_DIFFERENTIABILITY_ANALYSIS
    -> INTEGRATION_ANALYSIS
    -> CONVERGENCE_MODE_ANALYSIS
    -> LIMIT_INTERCHANGE_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW