Theorem Proving Ontology

AI SEARCH FLOW ALGEBRA — THEOREM PROVING ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/mathematical-logic-ontologies/formal-verification/theorem-proving-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Mathematical Logic"
SUBDOMAIN := "Formal Verification"
ONTOLOGY := "Theorem Proving Ontology"
PRIMARY_ENTITY := TheoremProving
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

TheoremProving :=
    FormalReasoning
    ∧ LogicalInference
    ∧ DerivationOfTheorem
    ∧ From(Axioms, Definitions, Assumptions, PreviouslyProvedResults)

Theorem :=
    Proposition
    ∧ ProvableWithin(FormalSystem)

FormalSystem :=
    Language ∧ Syntax ∧ Axioms ∧ InferenceRules ∧ Semantics

Proof :=
    FiniteSequence(Formulae)
    ∧ each_step_is(Axiom ∨ Assumption ∨ DerivedByValidInference)
    ∧ final_step = Theorem

ProofGoal := Γ ⊢ φ
SemanticValidity := Γ ⊨ φ
SyntacticProvability := Γ ⊢ φ

Soundness :=
    Γ ⊢ φ ⇒ Γ ⊨ φ

Completeness :=
    Γ ⊨ φ ⇒ Γ ⊢ φ

Consistency :=
    ¬∃φ : Γ ⊢ φ ∧ Γ ⊢ ¬φ

Axiom := FormulaAcceptedWithoutProof
Definition := MeaningAssignment ∧ introduces(Symbol ∨ Concept)
Lemma := ProvenProposition ∧ supports(FurtherProof)
Corollary := Theorem ∧ follows_from(PreviousResult)
Conjecture := Proposition ∧ not_yet_proved_or_refuted
Hypothesis := Assumption ∧ used_within(ProofContext)

InferenceRule := Premises → Conclusion

ModusPonens :=
    P ∧ (P → Q) ⇒ Q

ModusTollens :=
    ¬Q ∧ (P → Q) ⇒ ¬P

ConjunctionIntroduction :=
    P ∧ Q ⇒ P ∧ Q

ConjunctionElimination :=
    P ∧ Q ⇒ P

DisjunctionIntroduction :=
    P ⇒ P ∨ Q

UniversalInstantiation :=
    ∀x P(x) ⇒ P(a)

ExistentialGeneralisation :=
    P(a) ⇒ ∃x P(x)

UniversalGeneralisation :=
    P(x) under_valid_conditions ⇒ ∀x P(x)

NaturalDeduction :=
    IntroductionRules ∧ EliminationRules

SequentCalculus :=
    Sequents ∧ StructuralRules ∧ LogicalRules

Sequent := Γ ⊢ Δ

HilbertSystem :=
    AxiomSchemas ∧ InferenceRules

Resolution :=
    (A ∨ C) ∧ (¬A ∨ D) ⇒ (C ∨ D)

RefutationProof :=
    Assume(¬Goal) ∧ derive(Contradiction)

ProofByContradiction :=
    Γ ∧ ¬φ ⊢ ⊥ ⇒ Γ ⊢ φ

DirectProof :=
    Assumptions → ValidInferenceSequence → Goal

ProofByCases :=
    ExhaustiveCases
    ∧ prove(Goal, eachCase)
    ⇒ Goal

MathematicalInduction :=
    BaseCase ∧ InductiveStep ⇒ ∀n Property(n)

StructuralInduction :=
    BaseConstructors
    ∧ InductiveConstructors
    ⇒ Property(AllConstructedObjects)

WellFoundedInduction :=
    ∀x : (∀y ≺ x : P(y)) ⇒ P(x)
    ⇒ ∀x P(x)

ProofMethod := {
    DirectProof,
    Contradiction,
    Contraposition,
    Induction,
    StructuralInduction,
    CaseAnalysis,
    Resolution,
    Rewriting,
    Simplification,
    DecisionProcedure
}

AutomatedTheoremProving :=
    MachineSearch ∧ FormalLogic ∧ ProofProcedure

InteractiveTheoremProving :=
    HumanGuidance ∧ MachineCheckedInference ∧ ProofAssistant

ProofAssistant :=
    FormalEnvironment ∧ Kernel ∧ Tactics ∧ Libraries ∧ ProofChecker

AutomatedProver :=
    InputFormulae
    → SearchProofSpace
    → {Proof, Refutation, Unknown, Timeout}

ProofSearch :=
    InitialGoal
    → ApplyInferenceRules
    → GenerateSubgoals
    → ResolveSubgoals
    → ConstructProof

BackwardReasoning :=
    Goal → RequiredPremises → Subgoals

ForwardReasoning :=
    KnownFacts → ApplyRules → DerivedFacts → Goal

BidirectionalReasoning :=
    ForwardReasoning ∪ BackwardReasoning

ProofState :=
    Context ∧ Goals ∧ Constraints

Tactic :=
    ProofState → NewProofStates

Simplification :=
    RewriteRules ∧ Normalisation → SimplerEquivalentGoal

Rewriting :=
    Equality ∧ Substitution → EquivalentExpression

Unification :=
    find(Substitution σ)
    such_that σ(t1) = σ(t2)

MostGeneralUnifier := MGU(t1,t2)

Substitution :=
    Variables → Terms

Skolemisation :=
    ExistentialQuantifiers → SkolemFunctionsOrConstants

Clausification :=
    Formula
    → PrenexForm
    → Skolemisation
    → CNF
    → Clauses

CNF := Conjunction(Clauses)
Clause := Disjunction(Literals)
Literal := AtomicFormula ∨ NegatedAtomicFormula

ResolutionRefutation :=
    FormulaSet
    ∧ NegatedGoal
    → Clauses
    → Resolution
    → EmptyClause

EmptyClause := □ := Contradiction

SATBasedProving :=
    PropositionalEncoding → SATSolver

SMTBasedProving :=
    LogicalFormula
    ∧ BackgroundTheories
    → SMTSolver

SMTTheories := {
    Equality,
    Arithmetic,
    BitVectors,
    Arrays,
    UninterpretedFunctions,
    Datatypes,
    Sets,
    Strings
}

DecisionProcedure :=
    Algorithm ∧ decides(FormulaClass)

ProofKernel :=
    SmallTrustedCore ∧ validates(ProofSteps)

ProofCertificate :=
    MachineCheckableRepresentation(Proof)

CertifiedProof :=
    ProofCertificate ∧ accepted_by(TrustedProofChecker)

ProofObject :=
    StructuredRepresentation ∧ contains(Derivation)

CurryHowardCorrespondence :=
    Propositions ↔ Types
    ∧ Proofs ↔ Programs

ConstructiveProof :=
    Proof ∧ provides(WitnessOrConstruction)

ClassicalProof :=
    may_use(LawOfExcludedMiddle ∨ DoubleNegationElimination)

IntuitionisticProof :=
    ConstructiveInference

FirstOrderTheoremProving :=
    Terms ∧ Predicates ∧ Quantifiers ∧ Equality ∧ InferenceRules

HigherOrderTheoremProving :=
    QuantificationOver(Functions ∨ Predicates ∨ HigherOrderObjects)

EquationalReasoning :=
    Equations ∧ RewriteRules ∧ Congruence

Congruence :=
    x = y ⇒ f(x) = f(y)

InductiveTheoremProving :=
    RecursiveDefinitions ∧ InductionHypotheses ∧ InductiveProofSearch

TemporalTheoremProving :=
    TemporalLogic ∧ TransitionSemantics ∧ TemporalInference

ProgramVerification :=
    Program
    ∧ Specification
    → VerificationConditions
    → TheoremProving

VerificationCondition :=
    LogicalFormula ∧ must_be_proved_for(Correctness)

VCDischarge :=
    TheoremProver ⊢ VerificationCondition

HoareLogic :=
    {P} C {Q}

InvariantProof :=
    InitialCondition ⇒ Invariant
    ∧
    Invariant ∧ Transition ⇒ Invariant'

TerminationProof :=
    RankingFunction
    ∧ WellFoundedRelation
    ∧ StrictDecrease

SafetyProof :=
    SystemAxioms ⊢ SafetyProperty

LivenessProof :=
    SystemAxioms
    ∧ FairnessAssumptions
    ⊢ LivenessProperty

RefinementProof :=
    ConcreteSpecification
    ⊢ AbstractSpecificationObligations

EquivalenceProof :=
    SystemA ∧ SystemB
    ⊢ ObservationalEquivalence

ProofObligation :=
    FormalClaim
    ∧ required_for(
        Correctness
        ∨ Safety
        ∨ Liveness
        ∨ Refinement
        ∨ Consistency
    )

ProofDependency :=
    TheoremA dependsOn {Lemma1, Lemma2, ..., LemmaN}

ProofDependencyGraph :=
    G = (Theorems, DependencyEdges)

ProofReuse :=
    ExistingLemma
    → MatchGoal
    → Instantiate
    → Apply

LibraryTheorem :=
    ProvenReusableResult ∧ stored_with(Provenance)

ProofAutomation := {
    Simplifier,
    Rewriter,
    AutoTactic,
    DecisionProcedure,
    SMTIntegration,
    ResolutionEngine,
    InductionTactic
}

SearchStrategy := {
    DepthFirst,
    BreadthFirst,
    BestFirst,
    IterativeDeepening,
    GoalDirected,
    PortfolioSearch
}

ProofFailure := {
    GoalFalse,
    MissingLemma,
    InsufficientAxioms,
    SearchSpaceExplosion,
    Timeout,
    UnsupportedTheory,
    IncorrectFormalisation
}

Countermodel :=
    Model M
    ∧ M ⊨ Assumptions
    ∧ M ⊭ Goal

CountermodelRelation :=
    CountermodelExists
    ⇒ GoalNotSemanticallyEntailed

ProofResult := {
    Proved,
    Refuted,
    CountermodelFound,
    Unknown,
    Timeout
}

ProofValidation :=
    ProofObject
    → ProofChecker
    → {Accepted, Rejected}

ProofSoundnessChain :=
    Formalisation
    → Logic
    → InferenceRules
    → ProofObject
    → TrustedKernel

TrustedComputingBase := {
    Parser,
    TypeChecker,
    LogicKernel,
    ProofChecker,
    AxiomDefinitions
}

AssumptionTracking :=
    Proof → ExplicitAssumptionSet

ProofTraceability :=
    Requirement
    → FormalStatement
    → ProofObligation
    → Lemmas
    → Theorem
    → ProofCertificate

ProofProvenance := {
    Author,
    FormalSystem,
    Tool,
    ToolVersion,
    Axioms,
    Libraries,
    Timestamp,
    Source,
    TransformationHistory
}

IncrementalProofChecking :=
    ChangedDependency
    → IdentifyAffectedProofs
    → RecheckAffectedProofs

TheoremProvingOntology :=
    Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules

CoreClasses := {
    Theorem,
    Proposition,
    Axiom,
    Definition,
    Assumption,
    Lemma,
    Corollary,
    Conjecture,
    Proof,
    ProofGoal,
    ProofState,
    InferenceRule,
    Tactic,
    TheoremProver,
    ProofAssistant,
    ProofCertificate,
    Countermodel,
    FormalSystem,
    Logic,
    VerificationCondition
}

CoreRelations := {
    proves,
    derives,
    assumes,
    entails,
    implies,
    dependsOn,
    appliesRule,
    generates,
    discharges,
    refutes,
    validates,
    instantiates,
    rewrites,
    tracesTo
}

TypedRelations :=
    Proof --proves--> Theorem
    Theorem --dependsOn--> Lemma
    ProofStep --appliesRule--> InferenceRule
    TheoremProver --generates--> Proof
    TheoremProver --discharges--> VerificationCondition
    Countermodel --refutes--> SemanticEntailmentClaim
    ProofChecker --validates--> ProofCertificate
    ProofGoal --dependsOn--> Assumption
    Theorem --tracesTo--> FormalRequirement

InferenceRulesOntology := {
    Axiom(A)
        ⇒ Provable(A),

    Provable(P)
        ∧ Provable(P → Q)
        ⇒ Provable(Q),

    Γ ⊢ φ
        ∧ Sound(FormalSystem)
        ⇒ Γ ⊨ φ,

    Γ ⊨ φ
        ∧ Complete(FormalSystem)
        ⇒ Γ ⊢ φ,

    ValidProofCertificate(C)
        ∧ C proves φ
        ⇒ Verified(φ),

    Countermodel(M, Γ, φ)
        ⇒ ¬(Γ ⊨ φ),

    AllVerificationConditionsProved
        ∧ SoundVCGeneration
        ⇒ ProgramSatisfiesSpecification,

    ChangedLemma
        ⇒ Recheck(DependentTheorems)
}

SemanticMeaning :=
    Theorem
    → PropositionTrueInAllModelsOfAssumptions

SEMANTIC_EQUIVALENCE(TheoremA, TheoremB) :=
    Models(TheoremA) = Models(TheoremB)

LogicalEquivalence :=
    ⊨ (TheoremA ↔ TheoremB)

ProofEquivalence :=
    DifferentProofObjects
    ∧ same(FormalConclusion)
    ∧ compatible(Assumptions)

CanonicalProofRepresentation :=
    ParseProof
    → TypeCheck
    → NormaliseTerms
    → ResolveDependencies
    → CanonicalProofGraph

TheoremGraph :=
    G = (V,E)

V := {
    Theorems,
    Axioms,
    Definitions,
    Assumptions,
    Lemmas,
    ProofGoals,
    ProofSteps,
    InferenceRules,
    ProofCertificates,
    Countermodels
}

E := {
    proves,
    derives,
    assumes,
    entails,
    dependsOn,
    appliesRule,
    validates,
    refutes,
    tracesTo
}

CONFIDENCE(TheoremProof) :=
    weighted_mean(
        FormalisationQuality,
        LogicSoundness,
        ProofCheckerTrust,
        DependencyVerification,
        AssumptionTransparency,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectTheoremProvingConcept
    → ResolveLogic
    → IdentifyAssumptions
    → IdentifyGoal
    → IdentifyApplicableProofMethods
    → ResolveProofContext

AI_REASONING_FLOW :=
    Query
    → IdentifyFormalSystem
    → ParseGoal
    → ParseAssumptions
    → ResolveDefinitions
    → RetrieveRelevantLemmas
    → SelectInferenceRules
    → GenerateCandidateProofSteps
    → CreateSubgoals
    → ApplySimplificationAndRewriting
    → InvokeDecisionProceduresWhereApplicable
    → ResolveRemainingSubgoals
    → ConstructProofObject
    → ValidateProof
    → ProduceProofEvidence
    → ExplainDerivation

AI_CONSUMPTION := {
    ParseTheoremEntities,
    ParseTypedRelations,
    ResolveFormalSemantics,
    ResolveDefinitions,
    TrackAssumptions,
    RetrieveDependencies,
    ApplyInferenceRules,
    GenerateSubgoals,
    CompareProofStrategies,
    ValidateProofObjects,
    DetectCountermodels,
    FollowTraceability,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewProofEvidence
    → ValidateEvidence
    → ResolveAffectedTheorem
    → UpdateDependencies
    → DetectContradictions
    → RecheckDependentProofs
    → UpdateTheoremGraph
    → RecomputeProofStatus
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

ProofUpdate :=
    ValidProofCertificate
    ⇒ mark(Theorem, Proved)

CountermodelUpdate :=
    ValidCountermodel
    ⇒ mark(EntailmentClaim, Refuted)

DependencyUpdate :=
    ChangedAxiom
    ∨ ChangedDefinition
    ∨ ChangedLemma
    ⇒ invalidate_or_recheck(DependentTheorems)

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRulesOntology,
        Soundness,
        Completeness,
        ProofSearch,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    TheoremEntityResolution,
    FormalMeaning,
    AssumptionSet,
    ProofDependencies,
    ProofStatus,
    CountermodelStatus,
    ProofCertificate,
    VerificationEvidence,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF AI SEARCH FLOW ALGEBRA