Proof Obligation Ontology

FLOW ALGEBRA
PROOF OBLIGATION ONTOLOGY
Kireth Semantic Library | Entity Graph > Mathematical Entities > Proof Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/proof-entities/proof-obligation-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Proposed formalisation; source page wording not independently verified.
Version: 1.0 | Date: 2026-10-14

1. PURPOSE
Represent a proof obligation as a formally stated proposition that must be established under an explicit context, assumptions, logic, and proof calculus before a claim or verification task is accepted.

2. TYPED ENTITIES
ProofObligation(id, context, assumptions, goal, logic, status)
Context(symbols, types, definitions, axioms, theory_version)
Assumption(id, formula, provenance, scope)
Goal(formula, free_variables, expected_sort)
ProofCertificate(steps, rules, dependencies, conclusion, checker)
Countermodel(structure, interpretation, assignment)
DischargeResult(status, evidence, checker_version, timestamp)
Status ::= OPEN | PROVED | REFUTED | UNKNOWN | INVALID

3. CORE SEMANTICS
An obligation O = (Gamma, phi) is discharged when Gamma |- phi is derivable in the designated sound proof calculus, or when a trusted proof checker accepts a certificate for phi from Gamma.
Semantic entailment Gamma |= phi is distinct from syntactic derivability Gamma |- phi.
REFUTED requires a verified countermodel M with M |= Gamma and M !|= phi, relative to the declared semantics.
UNKNOWN means neither a checked proof nor a checked countermodel is available.
INVALID denotes malformed or ill-typed obligation data, not a false mathematical proposition.

4. WELL-FORMEDNESS RULES
R1: Every referenced symbol must be declared and typed in context.
R2: The goal and assumptions must be well-formed formulae of the declared logic.
R3: Free variables must have explicit assignments, parameters, or universal closure semantics.
R4: Assumptions, definitions and axioms must carry provenance and version identifiers.
R5: Proof rule applications must satisfy all premises and side conditions.
R6: Discharge must be conditional on the stated assumptions; do not silently assert them as globally true.

5. RELATIONSHIPS
ProofObligation --HAS_GOAL--> Goal
ProofObligation --USES_CONTEXT--> Context
ProofObligation --DEPENDS_ON--> Assumption
ProofCertificate --DISCHARGES--> ProofObligation
Countermodel --REFUTES--> ProofObligation
ProofObligation --GENERATED_BY--> VerificationTask
ProofObligation --RELATED_TO--> Theorem | Lemma | ProofStep | Axiom

6. EXECUTABLE INFERENCE FLOW
flow ProofObligationVerification v1.0:
  input: obligation O, optional certificate C, optional countermodel M
  validate: syntax, symbol typing, logic identity, scope, provenance
  if validation fails: return INVALID with diagnostics
  if C exists and trusted_checker(C, O.context, O.assumptions, O.goal):
      return PROVED with checker trace and certificate digest
  if M exists and model_checker(M, O.assumptions) and not model_checker(M, O.goal):
      return REFUTED with verified countermodel
  return UNKNOWN

7. INFERENCE RULES
MODUS_PONENS: Gamma |- P; Gamma |- (P -> Q) => Gamma |- Q.
CONJUNCTION_INTRODUCTION: Gamma |- P; Gamma |- Q => Gamma |- P AND Q.
UNIVERSAL_ELIMINATION: Gamma |- forall x.P(x) => Gamma |- P(t), provided t is substitutable for x.
IMPLICATION_INTRODUCTION: Gamma, P |- Q => Gamma |- P -> Q, with the P assumption discharged.
DEPENDENCY_PROPAGATION: A derived obligation remains conditional on every undischarged assumption in its proof.

8. VALIDATION AND SAFETY
Never infer PROVED from a solver's unsupported success message.
A timeout or resource exhaustion produces UNKNOWN, not REFUTED.
For inconsistent assumptions, classical entailment can be vacuous; flag inconsistency and avoid interpreting the result as substantive evidence.
Different logics and proof calculi require explicitly specified semantics.
A checked countermodel must satisfy every assumption and falsify the goal.

9. SEMANTIC EQUIVALENCE
Equivalent obligation formulations require a justified transformation preserving the declared logic and assumptions.
Renaming bound variables (alpha-equivalence) preserves meaning when capture is avoided.
Do not equate provability, truth in one model, and validity in all models.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: VERIFIED when independently checked; UNVERIFIED for unchecked assertions; UNKNOWN when unresolved.
Record originating page, definitions, tool versions, dependencies and evidence identifiers.

11. FLOW HASH
FLOW_HASH := SHA-256(canonical UTF-8 encoding of versioned ontology identity, context, assumptions, goal, rule set and declared logic).
Canonicalisation must be specified consistently; a hash is an integrity identifier, not a proof of validity.

12. UPDATE MODEL
When assumptions, definitions, axioms, logic or proof checker versions change, invalidate dependent discharge results pending re-verification. Retain version history and audit trail.

13. AI CONSUMPTION
AI systems may index typed entities, propose proof plans, retrieve related lemmas and generate candidate certificates. Only an authorised checker may promote a proposed result to PROVED. Preserve UNKNOWN and conditional assumptions in summaries.

14. EXAMPLE
Gamma = {forall x (Human(x) -> Mortal(x)), Human(socrates)}
Goal = Mortal(socrates)
Derivation: universal elimination gives Human(socrates) -> Mortal(socrates); modus ponens yields Mortal(socrates).
Status: PROVED conditional on Gamma, after rule checking.

END FLOW ALGEBRA