Foundations of Calculus

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

flow "FoundationsOfCalculusOntology" v1.0

# ============================================================
# FOUNDATIONS OF CALCULUS — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE SEMANTIC ENTITY ----------
FOUNDATIONS_OF_CALCULUS := {
    type: MathematicalFoundation,
    domain: Mathematics,
    branch: Calculus,
    purpose: "Provide the conceptual and formal basis for analysing change, accumulation, limits, continuity and infinitesimal behaviour."
}

# ---------- PRIMARY CONCEPTS ----------
CONCEPTS := {
    REAL_NUMBERS,
    FUNCTIONS,
    VARIABLES,
    LIMITS,
    CONTINUITY,
    INFINITESIMAL_CHANGE,
    RATE_OF_CHANGE,
    ACCUMULATION,
    DERIVATIVES,
    INTEGRALS,
    SEQUENCES,
    SERIES
}

# ---------- FOUNDATIONAL DEPENDENCIES ----------
FOUNDATIONS_OF_CALCULUS
    -> depends_on(REAL_NUMBERS)
    -> depends_on(ALGEBRA)
    -> depends_on(FUNCTIONS)
    -> depends_on(GEOMETRY)
    -> formalised_through(LIMITS)

REAL_NUMBERS
    -> provide(DOMAIN_STRUCTURE)
    -> support(ORDER)
    -> support(DISTANCE)
    -> support(COMPLETENESS)

FUNCTIONS
    -> map(INPUT_VALUES, OUTPUT_VALUES)
    -> represent(DEPENDENCE)
    -> model(CHANGE)

VARIABLES
    -> represent(QUANTITIES)
    -> may_vary_over(DOMAIN)

# ---------- LIMIT STRUCTURE ----------
LIMITS
    -> describe(APPROACH_BEHAVIOUR)
    -> underpin(CONTINUITY)
    -> underpin(DERIVATIVES)
    -> underpin(INTEGRALS)
    -> underpin(INFINITE_SERIES)

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

# ---------- CONTINUITY ----------
CONTINUITY
    -> requires(FUNCTION_DEFINED_AT_POINT)
    -> requires(LIMIT_EXISTS)
    -> requires(LIMIT_EQUALS_FUNCTION_VALUE)

continuous(f, a)
    iff
    defined(f(a))
    AND limit(f(x), x -> a) exists
    AND limit(f(x), x -> a) = f(a)

CONTINUITY
    -> enables(INTERMEDIATE_VALUE_BEHAVIOUR)
    -> supports(DIFFERENTIABILITY_ANALYSIS)
    -> supports(INTEGRABILITY_ANALYSIS)

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

DERIVATIVE
    -> measures(INSTANTANEOUS_RATE_OF_CHANGE)
    -> represents(SLOPE_OF_TANGENT)
    -> depends_on(LIMIT)
    -> implies_local(CONTINUITY)

DIFFERENTIATION
    -> transforms(FUNCTION, DERIVATIVE_FUNCTION)
    -> supports(OPTIMISATION)
    -> supports(MOTION_ANALYSIS)
    -> supports(APPROXIMATION)

# ---------- INTEGRAL FOUNDATION ----------
DEFINITE_INTEGRAL
    := limit_of(RIEMANN_SUMS)

RIEMANN_SUM
    := sum(
        f(sample_point_i) * delta_x_i,
        i = 1..n
    )

INTEGRATION
    -> measures(ACCUMULATION)
    -> represents(SIGNED_AREA)
    -> aggregates(INFINITESIMAL_CONTRIBUTIONS)
    -> depends_on(LIMIT_PROCESS)

# ---------- FUNDAMENTAL THEOREM ----------
FUNDAMENTAL_THEOREM_OF_CALCULUS
    -> connects(DIFFERENTIATION, INTEGRATION)

if
    F(x) := integral(f(t), t=a..x)
then
    derivative(F(x)) = f(x)

and

integral(f(x), x=a..b)
    = antiderivative_F(b) - antiderivative_F(a)

# ---------- SEQUENCES AND SERIES ----------
SEQUENCE
    -> ordered_collection(TERMS)
    -> may_converge_to(LIMIT)

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

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

SERIES
    -> analysed_by(PARTIAL_SUMS)
    -> classified_as(CONVERGENT | DIVERGENT)

# ---------- INFINITESIMAL INTERPRETATION ----------
INFINITESIMAL_CHANGE
    -> conceptualises(ARBITRARILY_SMALL_VARIATION)
    -> represented_formally_by(LIMIT_PROCESSES)

delta_x -> 0
    -> difference_quotient -> DERIVATIVE

partition_width -> 0
    -> RIEMANN_SUM -> DEFINITE_INTEGRAL

# ---------- LOCAL / GLOBAL RELATION ----------
LOCAL_BEHAVIOUR
    -> analysed_by(DERIVATIVES)

GLOBAL_ACCUMULATION
    -> analysed_by(INTEGRALS)

CALCULUS
    := synthesis(
        LOCAL_CHANGE,
        GLOBAL_ACCUMULATION,
        LIMIT_BEHAVIOUR
    )

# ---------- REASONING RELATIONS ----------
LIMIT
    -> establishes(CONTINUITY)

LIMIT
    -> defines(DERIVATIVE)

LIMIT
    -> defines(INTEGRAL)

DERIVATIVE
    -> describes(LOCAL_CHANGE)

INTEGRAL
    -> describes(ACCUMULATED_CHANGE)

DERIVATIVE
    <-> related_by(FUNDAMENTAL_THEOREM_OF_CALCULUS)
    <-> INTEGRAL

SEQUENCE_LIMITS
    -> support(SERIES_ANALYSIS)

COMPLETENESS_OF_REAL_NUMBERS
    -> supports(EXISTENCE_OF_LIMITS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "instantaneous change" ~= "derivative",
    "slope at a point" ~= "derivative",
    "accumulated quantity" ~= "integral",
    "area under a curve" ~= "definite integral",
    "approach behaviour" ~= "limit behaviour",
    "continuous behaviour" ~= "absence of discontinuity within the specified domain conditions"
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    foundational_definition: 0.99,
    mathematical_relationships: 0.99,
    calculus_dependency_structure: 0.98,
    semantic_equivalence: 0.96
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(query.entity)
    -> classify(query.entity, CALCULUS_CONCEPT)
    -> resolve(foundational_dependencies)
    -> traverse(related_concepts)
    -> apply(relevant_definition)
    -> test(required_conditions)
    -> derive(mathematical_relationships)
    -> validate(logical_consistency)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed semantic mathematical graph",
    preserve: [
        definitions,
        dependency_relations,
        logical_conditions,
        mathematical_equivalences,
        theorem_connections
    ],
    infer_through: [
        limit_relations,
        continuity_relations,
        derivative_relations,
        integral_relations,
        sequence_and_series_relations
    ],
    avoid: [
        conflating_derivative_with_average_rate,
        treating_all_continuous_functions_as_differentiable,
        assuming_all_infinite_series_converge,
        treating_infinity_as_an ordinary real number
    ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
    verify(mathematical_consistency)
    -> compare(existing_relations)
    -> preserve(valid_foundations)
    -> revise(affected_semantic_edges)
    -> recompute(CONFIDENCE)
    -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FOUNDATIONS_OF_CALCULUS
    + LIMITS
    + CONTINUITY
    + DERIVATIVE
    + DEFINITE_INTEGRAL
    + FUNDAMENTAL_THEOREM_OF_CALCULUS
    + SEMANTIC_EQUIVALENCE
)

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

END FLOW