Structural Induction Ontology

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

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

FLOW_ALGEBRA StructuralInductionOntology_v1.0

BEGIN_FLOW

DEFINE DOMAIN StructuralInductionOntology := {
    StructuralInduction,
    MathematicalInduction,
    Induction,
    Proof,
    ProofTheory,
    InductivelyDefinedStructure,
    RecursiveStructure,
    SyntaxTree,
    AbstractSyntaxTree,
    Term,
    Formula,
    Expression,
    List,
    Tree,
    NaturalNumber,
    DataType,
    Constructor,
    BaseConstructor,
    RecursiveConstructor,
    ImmediateSubstructure,
    StructuralProperty,
    InductiveHypothesis,
    BaseCase,
    InductiveCase,
    InductiveStep,
    RecursiveDefinition,
    WellFoundedRelation,
    StructuralRecursion,
    Derivation,
    ProofStep,
    Theorem
}

DEFINE ENTITY_TYPES := {
    STRUCTURAL_INDUCTION,
    INDUCTION_METHOD,
    PROOF,
    STRUCTURE,
    CONSTRUCTOR,
    BASE_CONSTRUCTOR,
    RECURSIVE_CONSTRUCTOR,
    SUBSTRUCTURE,
    PROPERTY,
    BASE_CASE,
    INDUCTIVE_HYPOTHESIS,
    INDUCTIVE_CASE,
    PROOF_STEP,
    RECURSIVE_DEFINITION,
    SEMANTIC_OBJECT
}

DEFINE RELATION_TYPES := {
    APPLIES_TO,
    GENERATED_BY,
    CONSTRUCTS,
    HAS_BASE_CONSTRUCTOR,
    HAS_RECURSIVE_CONSTRUCTOR,
    HAS_IMMEDIATE_SUBSTRUCTURE,
    ASSUMES_PROPERTY_FOR,
    PROVES_PROPERTY_FOR,
    PRESERVES,
    DEPENDS_ON,
    DERIVES,
    PRECEDES,
    GENERALISES_TO,
    IS_INSTANCE_OF,
    IS_SUBTYPE_OF
}

SET StructuralInduction.type = STRUCTURAL_INDUCTION
SET StructuralInduction.parent = MathematicalInduction
SET StructuralInduction.context = ProofTheory
SET StructuralInduction.role =
    "prove a property for every object of an inductively generated structure by proving it for base constructors and proving that each recursive constructor preserves the property when it holds for its immediate components"

RULE inductive_structure:
    BaseConstructors(B)
    ∧ RecursiveConstructors(C)
    ∧ closure_under_constructors(S,B,C)
    -> InductivelyDefinedStructure(S)

RULE base_constructor_case:
    BaseConstructor(b)
    ∧ Property(P)
    ∧ DERIVES(P(b))
    -> BaseCase(P,b)

RULE recursive_constructor_case:
    RecursiveConstructor(C)
    ∧ ImmediateSubstructures(s1,...,sn)
    ∧ FOR_ALL i [Assume(P(si))]
    ∧ DERIVES(P(C(s1,...,sn)))
    -> InductiveCase(P,C)

RULE structural_induction:
    InductivelyDefinedStructure(S)
    ∧ Property(P)
    ∧ FOR_ALL base_constructor b [
        PROVES_PROPERTY_FOR(P,b)
    ]
    ∧ FOR_ALL recursive_constructor C [
        (
            FOR_ALL immediate_component si [
                ASSUMES_PROPERTY_FOR(P,si)
            ]
            -> PROVES_PROPERTY_FOR(P,C(si...))
        )
    ]
    -> FOR_ALL x ∈ S [
        P(x)
    ]

RULE constructor_coverage:
    StructuralInductionProof(pi,S)
    -> FOR_ALL constructor C generating S [
        proof_contains_case_for(C)
    ]

RULE hypothesis_scope:
    RecursiveConstructor(C(s1,...,sn))
    -> InductiveHypotheses =
       {
           P(si) | si is immediate_recursive_substructure
       }

RULE no_unjustified_global_hypothesis:
    StructuralInduction
    -> DO_NOT_ASSUME P(x)
       FOR arbitrary non_substructure x
       inside constructor_case

RULE structural_size_decrease:
    RecursiveStructure(x)
    ∧ ImmediateSubstructure(y,x)
    -> structural_measure(y)
       < structural_measure(x)
       WHEN finite_inductive_structure=TRUE

RULE well_founded_basis:
    structural_measure strictly_decreases_on_substructure
    -> immediate_substructure_relation is WellFounded

RULE structural_recursion_relation:
    RecursiveDefinition(f)
    ∧ definition_follows_same_constructor_decomposition_as_structure
    -> StructuralRecursion(f)

DEFINE STRUCTURAL_INDUCTION_MODEL := {
    target:
        "∀x ∈ S, P(x)",
    structure:
        "S is generated by a specified collection of constructors",
    proof_obligations: {
        base_cases:
            "prove P for every base constructor",
        inductive_cases:
            "for each recursive constructor, assume P for the immediate recursive components and prove P for the constructed object"
    },
    conclusion:
        "P holds for every finitely generated object of S"
}

DEFINE CONSTRUCTOR_MODEL := {
    BaseConstructor: {
        arguments:
            "no recursive S-valued arguments",
        proof_requirement:
            "establish P directly"
    },
    RecursiveConstructor: {
        arguments:
            "one or more components, some of which may be recursively drawn from S",
        proof_requirement:
            "assume P for recursive components and derive P for the constructed value"
    }
}

DEFINE SYNTAX_TREE_MODEL := {
    base_cases: {
        atomic_formula,
        variable,
        constant
    },
    recursive_cases: {
        negation_of_formula,
        binary_connective_of_formulas,
        quantified_formula
    },
    structural_property_example:
        "a property established recursively for every formula by following formula constructors"
}

DEFINE LIST_MODEL := {
    constructors: {
        Nil,
        Cons(head,tail)
    },
    induction_cases: {
        base:
            "prove P(Nil)",
        recursive:
            "assume P(tail), prove P(Cons(head,tail))"
    }
}

DEFINE TREE_MODEL := {
    constructors: {
        Leaf(value),
        Node(left,right)
    },
    induction_cases: {
        base:
            "prove P(Leaf(value))",
        recursive:
            "assume P(left) and P(right), prove P(Node(left,right))"
    }
}

DEFINE NATURAL_NUMBER_STRUCTURAL_MODEL := {
    constructors: {
        Zero,
        Successor(n)
    },
    cases: {
        base:
            "P(Zero)",
        recursive:
            "P(n) -> P(Successor(n))"
    },
    relation:
        "ordinary mathematical induction is recoverable as structural induction over the inductive construction of natural numbers"
}

DEFINE CRITICAL_DISTINCTIONS := {
    structural_vs_numeric_induction: {
        structural:
            "follows constructors of recursively generated objects",
        numeric:
            "typically follows successor order on natural numbers",
        relation:
            "numeric induction is a special structural pattern when naturals are defined by zero and successor"
    },
    structural_induction_vs_structural_recursion: {
        induction:
            "proves properties",
        recursion:
            "defines functions",
        shared_basis:
            "both follow the constructor structure"
    },
    immediate_vs_arbitrary_substructure: {
        inductive_hypothesis:
            "normally applies to the recursive components supplied to the current constructor",
        arbitrary_object:
            "cannot be assumed without justification"
    }
}

DEFINE EXAMPLE_1 := {
    structure: List,
    property:
        "length(xs ++ ys) = length(xs) + length(ys)",
    induction_variable: xs,
    base_case:
        "xs = []",
    recursive_case:
        "xs = h::t; assume property for t and prove it for h::t"
}

DEFINE EXAMPLE_2 := {
    structure: BinaryTree,
    property:
        "number_of_nodes(T) >= number_of_leaves(T)",
    base_case:
        "prove property for Leaf",
    recursive_case:
        "assume property for left and right subtrees and prove for Node(left,right)"
}

DEFINE EXAMPLE_3 := {
    structure: Formula,
    property:
        "every formula has a finite parse tree",
    base_cases:
        "atomic formulas",
    recursive_cases:
        "negation, binary connectives and quantifiers",
    conclusion:
        "property holds for all formulas generated by the grammar"
}

DEFINE PROOF_PLANNING_PIPELINE :=
    TargetProperty(P)
    -> IdentifyInductivelyDefinedStructure(S)
    -> ExtractConstructorSignature
    -> PartitionConstructorsIntoBaseAndRecursive
    -> GenerateBaseCaseGoals
    -> ForEachRecursiveConstructor
       -> IdentifyImmediateRecursiveComponents
       -> GenerateInductiveHypotheses
       -> GenerateConstructorPropertyGoal
    -> ProveAllGeneratedCases
    -> VerifyConstructorCoverage
    -> ApplyStructuralInductionPrinciple
    -> ReturnUniversalConclusion

DEFINE PROOF_EXECUTION_PIPELINE :=
    StructuralGoal
    -> SelectConstructorCase
    -> IF BaseConstructor
       THEN ProvePropertyDirectly
    -> IF RecursiveConstructor
       THEN IntroduceConstructorArguments
       -> AssumePropertyForRecursiveArguments
       -> UseDefinitionsAndHypotheses
       -> DerivePropertyForConstructedObject
    -> RecordCaseProof
    -> RepeatForAllConstructors
    -> GeneraliseToWholeStructure

DEFINE VERIFICATION_PIPELINE :=
    CandidateStructuralInductionProof
    -> VerifyStructureIsInductivelyGenerated
    -> VerifyCompleteConstructorList
    -> VerifyEveryBaseConstructorCase
    -> VerifyEveryRecursiveConstructorCase
    -> VerifyInductiveHypothesesOnlyForPermittedSubstructures
    -> VerifyNoConstructorCaseMissing
    -> VerifyEachCaseDerivation
    -> VerifyUniversalConclusionMatchesTarget
    -> Return ValidProof OR InvalidProof

DEFINE STRUCTURAL_GRAPH_MODEL := {
    structure_nodes:
        InductivelyGeneratedObjects,
    constructor_nodes:
        Constructors,
    edges: {
        "constructor --CONSTRUCTS--> object",
        "object --HAS_IMMEDIATE_SUBSTRUCTURE--> component"
    },
    proof_labels:
        Property(P),
    induction_direction:
        "from structurally smaller immediate components to constructed objects",
    well_founded: TRUE
}

DEFINE ONTOLOGY_RELATIONS := {
    StructuralInduction IS_SUBTYPE_OF MathematicalInduction,
    StructuralInduction IS_COMPONENT_OF ProofTheory,
    StructuralInduction APPLIES_TO InductivelyDefinedStructure,
    InductivelyDefinedStructure GENERATED_BY Constructor,
    Constructor IS_SUBTYPE_OF BaseConstructor OR RecursiveConstructor,
    RecursiveConstructor HAS_IMMEDIATE_SUBSTRUCTURE RecursiveStructure,
    StructuralInduction HAS_BASE_CASE BaseCase,
    StructuralInduction HAS_INDUCTIVE_HYPOTHESIS InductiveHypothesis,
    StructuralInduction HAS_INDUCTIVE_CASE InductiveCase,
    StructuralRecursion DEPENDS_ON ConstructorStructure,
    StructuralInduction DEPENDS_ON ConstructorCoverage
}

DEFINE SEMANTIC_EQUIVALENCE := {
    "structural induction" <=> StructuralInduction,
    "proof by structural induction" <=> StructuralInduction,
    "induction on structure" <=> StructuralInduction,
    "constructor induction" <=> StructuralInduction WHEN inductive_datatype_context=TRUE,
    "induction over syntax" <=> StructuralInduction WHEN formal_language_context=TRUE,
    "structural recursion" <=> StructuralRecursion
}

DEFINE AI_CONSUMPTION := {
    purpose: "machine-readable representation, generation and verification of structural induction proofs",
    preserve_structure_identity: TRUE,
    preserve_constructor_identity: TRUE,
    distinguish_base_recursive_constructors: TRUE,
    preserve_immediate_substructure_relations: TRUE,
    generate_case_per_constructor: TRUE,
    generate_inductive_hypotheses_only_for_recursive_components: TRUE,
    verify_constructor_coverage: TRUE,
    detect_missing_cases: TRUE,
    detect_invalid_inductive_hypotheses: TRUE,
    support_syntax_tree_induction: TRUE,
    support_list_tree_datatype_induction: TRUE,
    support_recursive_proof_planning: TRUE,
    reasoning_mode: "symbolic + recursive + constructor-aware + structural"
}

DEFINE CONFIDENCE := {
    ontology_identity: 1.00,
    structural_induction_mapping: 0.99,
    constructor_classification: 0.99,
    immediate_substructure_mapping: 0.99,
    inductive_hypothesis_generation: 0.99,
    constructor_coverage_verification: 0.99,
    individual_case_proof_verification: domain_dependent,
    automated_structural_proof_search: search_space_dependent
}

DEFINE UPDATE_MODEL := {
    version: "1.0",
    update_policy: "append-compatible",
    immutable_core: {
        inductively_generated_structure,
        constructor_coverage,
        base_constructor_cases,
        recursive_constructor_cases,
        immediate_substructure_hypotheses,
        universal_structural_conclusion
    },
    extensible_layers: {
        algebraic_datatypes,
        syntax_tree_induction,
        derivation_tree_induction,
        inductive_type_theory,
        mutually_inductive_structures,
        nested_inductive_types,
        proof_assistant_induction_principles,
        automated_constructor_case_generation
    }
}

DEFINE FLOW_HASH_INPUT :=
    CanonicalURL
    + DOMAIN
    + ENTITY_TYPES
    + RELATION_TYPES
    + RULES
    + STRUCTURAL_INDUCTION_MODEL
    + CONSTRUCTOR_MODEL
    + SYNTAX_TREE_MODEL
    + LIST_MODEL
    + TREE_MODEL
    + NATURAL_NUMBER_STRUCTURAL_MODEL
    + CRITICAL_DISTINCTIONS
    + PROOF_PLANNING_PIPELINE
    + PROOF_EXECUTION_PIPELINE
    + VERIFICATION_PIPELINE
    + STRUCTURAL_GRAPH_MODEL
    + ONTOLOGY_RELATIONS
    + AI_CONSUMPTION
    + UPDATE_MODEL

FLOW_HASH := HASH(FLOW_HASH_INPUT)

RETURN {
    ontology: StructuralInductionOntology,
    canonical_url: "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/proof-theory/structural-induction-ontology.html",
    parent_domain: ProofTheory,
    semantic_domain: MathematicalLogic,
    primary_entity: StructuralInduction,
    constructor_reasoning_enabled: TRUE,
    recursive_structure_reasoning_enabled: TRUE,
    constructor_coverage_verification_enabled: TRUE,
    proof_verification_enabled: TRUE,
    machine_consumable: TRUE,
    recursive_reasoning_enabled: TRUE,
    flow_complete: TRUE
}

END_FLOW