Mathematical Induction Ontology

KIRETH RESEARCH — SEMANTIC LIBRARY
AI CORE → ONTOLOGIES → MATHEMATICAL ONTOLOGIES → MATHEMATICAL LOGIC ONTOLOGIES
PROOF THEORY → MATHEMATICAL INDUCTION ONTOLOGY

Canonical URL:
https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/proof-theory/mathematical-induction-ontology.html

FLOW_ALGEBRA MathematicalInductionOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN MathematicalInductionOntology := {
    MathematicalInduction,
    Induction,
    Proof,
    ProofTheory,
    NaturalNumbers,
    WellOrderedSet,
    Predicate,
    Proposition,
    Property,
    BaseCase,
    InductiveHypothesis,
    InductiveStep,
    Successor,
    InductionPrinciple,
    WeakInduction,
    StrongInduction,
    CompleteInduction,
    StructuralInduction,
    WellFoundedInduction,
    Recursion,
    RecursiveDefinition,
    MinimalCounterexample,
    Theorem,
    Derivation,
    ProofStep,
    UniversalStatement,
    Quantifier,
    PeanoArithmetic,
    InductionAxiomSchema
}

DEFINE ENTITY_TYPES := {
    INDUCTION_METHOD,
    PROOF_METHOD,
    DOMAIN,
    PREDICATE,
    BASE_CASE,
    INDUCTIVE_HYPOTHESIS,
    INDUCTIVE_STEP,
    SUCCESSOR,
    PROOF,
    PROOF_STEP,
    THEOREM,
    AXIOM_SCHEMA,
    RECURSIVE_STRUCTURE,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    PROVES,
    HAS_BASE_CASE,
    HAS_INDUCTIVE_HYPOTHESIS,
    HAS_INDUCTIVE_STEP,
    APPLIES_TO,
    HOLDS_FOR,
    IMPLIES_NEXT,
    PRECEDES,
    DEPENDS_ON,
    DERIVES,
    ASSUMES,
    DISCHARGES,
    GENERALISES_TO,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF,
    EQUIVALENT_TO
}

SET MathematicalInduction.type = INDUCTION_METHOD
SET MathematicalInduction.parent = Proof
SET MathematicalInduction.context = ProofTheory
SET MathematicalInduction.role =
    "establish a universally quantified property over an inductively generated or well-founded domain by proving initial cases and a propagation condition"

RULE standard_induction:
    Predicate(P)
    ∧ BaseCase(P(n0))
    ∧ FOR_ALL k >= n0 [
        P(k) -> P(k+1)
    ]
    -> FOR_ALL n >= n0 [
        P(n)
    ]

RULE base_case:
    Goal(∀n>=n0 P(n))
    -> REQUIRE Proof(P(n0))

RULE inductive_hypothesis:
    Goal(∀n>=n0 P(n))
    -> Assume(P(k))
       FOR arbitrary k >= n0

RULE inductive_step:
    Assume(P(k))
    ∧ DERIVES(P(k+1))
    -> IMPLIES_NEXT(P(k),P(k+1))

RULE induction_conclusion:
    BaseCase(P(n0))
    ∧ InductiveStep(P(k)->P(k+1))
    ∧ k arbitrary_in_domain
    -> GENERALISES_TO(∀n>=n0 P(n))

RULE strong_induction:
    BaseCasesEstablishedUpTo(k0)
    ∧ FOR_ALL n > k0 [
        (FOR_ALL j [k0 <= j < n -> P(j)])
        -> P(n)
    ]
    -> FOR_ALL n >= k0 [
        P(n)
    ]

RULE weak_strong_equivalence_naturals:
    Domain=NaturalNumbers
    -> WeakInduction
       EQUIVALENT_IN_PROVING_POWER_TO
       StrongInduction
       UNDER standard_foundational_assumptions

RULE minimal_counterexample:
    WellOrderedDomain(D)
    ∧ Assume(EXISTS n IN D [¬P(n)])
    -> EXISTS least m IN D [¬P(m)]

RULE induction_via_no_minimal_counterexample:
    BaseCase(P(n0))
    ∧ FOR_ALL n>n0 [
        (FOR_ALL j<n [P(j)]) -> P(n)
    ]
    -> NOT EXISTS minimal_counterexample
    -> FOR_ALL n>=n0 P(n)

RULE structural_induction:
    InductivelyDefinedStructure(S)
    ∧ Property(P)
    ∧ P holds_for_all_base_constructors
    ∧ FOR_ALL constructor C [
        P holds_for_immediate_components
        -> P holds_for C(components)
    ]
    -> FOR_ALL x IN S [P(x)]

RULE well_founded_induction:
    WellFoundedRelation(<,D)
    ∧ FOR_ALL x IN D [
        (FOR_ALL y<x [P(y)])
        -> P(x)
    ]
    -> FOR_ALL x IN D [P(x)]

RULE induction_schema_arithmetic:
    Formula(P(n))
    -> [
        P(0)
        ∧ ∀n(P(n)->P(n+1))
    ]
    -> ∀n P(n)

DEFINE STANDARD_INDUCTION_MODEL := {
    target: "∀n ≥ n₀, P(n)",
    components: {
        BaseCase:
            "prove P(n₀)",
        InductiveHypothesis:
            "assume P(k) for an arbitrary admissible k",
        InductiveStep:
            "using the hypothesis, prove P(k+1)",
        Conclusion:
            "infer P(n) for every n ≥ n₀"
    }
}

DEFINE STRONG_INDUCTION_MODEL := {
    target: "∀n ≥ n₀, P(n)",
    inductive_hypothesis:
        "assume P(j) for every j with n₀ ≤ j ≤ k",
    inductive_step:
        "prove P(k+1)",
    interpretation:
        "the next case may use all earlier established cases"
}

DEFINE STRUCTURAL_INDUCTION_MODEL := {
    domain:
        "recursively or inductively generated structures",
    base_cases:
        "primitive constructors",
    inductive_cases:
        "show constructors preserve the target property when it holds for their immediate substructures"
}

DEFINE WELL_FOUNDED_INDUCTION_MODEL := {
    domain: D,
    relation: "<",
    requirement: WellFoundedRelation,
    step:
        "(∀y<x P(y)) -> P(x)",
    conclusion:
        "∀x∈D P(x)"
}

DEFINE CRITICAL_DISTINCTIONS := {
    induction_vs_deductive_pattern_guessing: {
        mathematical_induction:
            "a deductively valid proof principle",
        empirical_induction:
            "generalisation from observed examples",
        equivalence: FALSE
    },
    base_case_vs_inductive_step: {
        base_case:
            "anchors the property at the initial point",
        inductive_step:
            "propagates truth from established cases to later cases"
    },
    induction_vs_recursion: {
        induction:
            "proof principle",
        recursion:
            "definition or construction principle",
        relation:
            "closely connected but conceptually distinct"
    }
}

DEFINE EXAMPLE_1 := {
    theorem:
        "1 + 2 + ... + n = n(n+1)/2 for n ≥ 1",
    base_case:
        "n=1: 1 = 1(2)/2",
    inductive_hypothesis:
        "assume 1+...+k = k(k+1)/2",
    inductive_step:
        "1+...+k+(k+1) = k(k+1)/2 + (k+1) = (k+1)(k+2)/2",
    conclusion:
        "formula holds for all n ≥ 1"
}

DEFINE EXAMPLE_2 := {
    theorem:
        "2^n ≥ n+1 for n ≥ 0",
    base_case:
        "2^0 = 1 ≥ 1",
    hypothesis:
        "2^k ≥ k+1",
    step:
        "2^(k+1)=2·2^k ≥ 2(k+1) ≥ k+2",
    conclusion:
        "true for all n ≥ 0"
}

DEFINE EXAMPLE_3 := {
    method: StrongInduction,
    theorem_type:
        "every integer n≥2 can be factored into primes",
    step:
        "if n is not prime, write n=ab with 2≤a,b<n and use the strong inductive hypothesis on a and b"
}

DEFINE PROOF_PLANNING_PIPELINE :=
    UniversalGoal
    -> IdentifyInductiveDomain
    -> IdentifyStartingIndexOrBaseConstructors
    -> ChooseInductionVariant
    -> FormulatePropertyP
    -> GenerateBaseCaseGoals
    -> GenerateInductiveHypothesis
    -> GenerateInductiveStepGoal
    -> ProveAllComponents
    -> ApplyInductionPrinciple
    -> ReturnUniversalConclusion

DEFINE STANDARD_PROOF_PIPELINE :=
    Goal(∀n>=n0 P(n))
    -> Prove(P(n0))
    -> IntroduceArbitrary(k>=n0)
    -> Assume(P(k))
    -> Derive(P(k+1))
    -> DischargeInductiveHypothesisAsRequired
    -> ApplyInductionPrinciple
    -> Conclude(∀n>=n0 P(n))

DEFINE STRONG_PROOF_PIPELINE :=
    Goal(∀n>=n0 P(n))
    -> EstablishRequiredInitialCases
    -> IntroduceArbitrary(k)
    -> Assume(FOR_ALL j [n0<=j<=k -> P(j)])
    -> Derive(P(k+1))
    -> ApplyStrongInduction
    -> ConcludeUniversalGoal

DEFINE VERIFICATION_PIPELINE :=
    CandidateInductionProof
    -> VerifyDomainSupportsChosenInductionPrinciple
    -> VerifyBaseCases
    -> VerifyInductiveVariableIsArbitrary
    -> VerifyInductiveHypothesisScope
    -> VerifyStepActuallyDerivesRequiredNextCase
    -> DetectCircularUseOfTarget
    -> VerifyStartingIndexCoverage
    -> ApplyInductionRule
    -> Return ValidProof OR InvalidProof

DEFINE INDUCTION_GRAPH_MODEL := {
    base_node: P(n0),
    generic_node: P(k),
    successor_node: P(k+1),
    propagation_edge:
        "P(k) --IMPLIES_NEXT--> P(k+1)",
    domain_chain:
        "n0 -> n0+1 -> n0+2 -> ...",
    coverage:
        "base anchor plus valid propagation reaches every admissible finite successor stage"
}

DEFINE ONTOLOGY_RELATIONS := {
    MathematicalInduction IS_SUBTYPE_OF Proof,
    MathematicalInduction IS_COMPONENT_OF ProofTheory,
    MathematicalInduction HAS_BASE_CASE BaseCase,
    MathematicalInduction HAS_INDUCTIVE_HYPOTHESIS InductiveHypothesis,
    MathematicalInduction HAS_INDUCTIVE_STEP InductiveStep,
    WeakInduction IS_SUBTYPE_OF MathematicalInduction,
    StrongInduction IS_SUBTYPE_OF MathematicalInduction,
    StructuralInduction IS_SUBTYPE_OF MathematicalInduction,
    WellFoundedInduction IS_GENERALISATION_OF MathematicalInduction,
    InductionPrinciple DEPENDS_ON DomainStructure,
    PeanoArithmetic CONTAINS InductionAxiomSchema,
    Recursion IS_RELATED_TO Induction
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "mathematical induction" <=> MathematicalInduction,
    "proof by induction" <=> MathematicalInduction,
    "ordinary induction" <=> WeakInduction,
    "weak induction" <=> WeakInduction,
    "strong induction" <=> StrongInduction,
    "complete induction" <=> StrongInduction,
    "structural induction" <=> StructuralInduction,
    "well-founded induction" <=> WellFoundedInduction,
    "induction hypothesis" <=> InductiveHypothesis,
    "induction step" <=> InductiveStep
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation, generation and verification of mathematical induction proofs",
    preserve_target_property: TRUE,
    preserve_domain: TRUE,
    preserve_starting_index: TRUE,
    preserve_base_cases: TRUE,
    preserve_inductive_variable: TRUE,
    preserve_inductive_hypothesis_scope: TRUE,
    preserve_inductive_step: TRUE,
    distinguish_weak_strong_structural_well_founded_induction: TRUE,
    detect_missing_base_cases: TRUE,
    detect_invalid_inductive_step: TRUE,
    detect_circular_reasoning: TRUE,
    verify_domain_coverage: TRUE,
    support_recursive_induction_planning: TRUE,
    reasoning_mode: "symbolic + recursive + structural + proof-theoretic"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    standard_induction_mapping: 0.99,
    strong_induction_mapping: 0.99,
    structural_induction_mapping: 0.99,
    well_founded_induction_mapping: 0.99,
    base_case_verification: 0.99,
    inductive_step_verification: domain_dependent,
    automated_induction_proof_search: search_space_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        base_case,
        inductive_hypothesis,
        inductive_step,
        arbitrary_inductive_variable,
        universal_conclusion,
        domain_coverage
    },
    extensible_layers: {
        strong_induction,
        structural_induction,
        well_founded_induction,
        transfinite_induction,
        course_of_values_induction,
        induction_in_type_theory,
        automated_induction_tactics,
        proof_assistant_induction_schemas
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + STANDARD_INDUCTION_MODEL
    + STRONG_INDUCTION_MODEL
    + STRUCTURAL_INDUCTION_MODEL
    + WELL_FOUNDED_INDUCTION_MODEL
    + CRITICAL_DISTINCTIONS
    + PROOF_PLANNING_PIPELINE
    + STANDARD_PROOF_PIPELINE
    + STRONG_PROOF_PIPELINE
    + VERIFICATION_PIPELINE
    + INDUCTION_GRAPH_MODEL
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: MathematicalInductionOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/proof-theory/mathematical-induction-ontology.html",
    parent_domain: ProofTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: MathematicalInduction,
    standard_induction_enabled: TRUE,
    strong_induction_enabled: TRUE,
    structural_induction_enabled: TRUE,
    well_founded_induction_enabled: TRUE,
    proof_verification_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    flow_complete: TRUE
}

END_FLOW