Algebraic Proof Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the mathematical ontology of algebraic proof. Algebraic proof establishes the truth of a mathematical statement through logically valid transformations, definitions, identities, equations, implications and previously established results. This record is designed for semantic retrieval, proof reasoning, ontology linking, knowledge-graph construction and AI consumption within the Kireth Research Semantic Library.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Algebraic Proof Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Algebra"
 SPECIALISATION = "Mathematical Proof / Symbolic Algebra"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/algebra-ontologies/algebraic-proof-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = AlgebraicProof
 SYMBOLIC_FORMS = ["P => Q", "LHS = RHS", "therefore", "QED"]
 ENTITY_CLASS = FormalReasoningStructure
 PRIMARY_CONTEXT = Algebra
 DEFINITION = "An algebraic proof is a logically valid sequence of algebraic statements and transformations establishing that a proposition follows from accepted assumptions, definitions, axioms or previously proved results."
 SEMANTIC_ROLE = "Connects premises to a conclusion through verifiable algebraic inference."
}

SCOPE {
 INCLUDE = [
  AlgebraicProof,
  DirectProof,
  ProofByContradiction,
  ProofByContrapositive,
  ProofByCases,
  IdentityProof,
  EquationProof,
  InequalityProof,
  EquivalenceProof,
  Derivation,
  Lemma,
  Theorem,
  Premise,
  Conclusion,
  LogicalImplication,
  AlgebraicTransformation,
  Justification,
  Validity
 ]

 EXCLUDE_AS_PRIMARY = [
  NumericalVerificationOnly,
  EmpiricalEvidence,
  HeuristicArgument,
  InformalAssertionWithoutJustification
 ]

 DISAMBIGUATION_RULE =
  "When proof occurs with algebra, equations, identities, implications, transformations, theorems, lemmas or symbolic derivations, interpret it as formal or semi-formal mathematical proof."
}

CORE_CONCEPTS {

 AlgebraicProof {
  definition = "A finite or recursively structured sequence of valid statements leading from premises to a conclusion."
  components = [
   Premises,
   IntermediateStatements,
   Justifications,
   Conclusion
  ]
 }

 Premise {
  definition = "An accepted starting statement, assumption, definition, axiom or previously proved result."
 }

 Conclusion {
  definition = "The proposition established by the proof."
 }

 DerivationStep {
  definition = "A transformation or inference justified by an accepted algebraic or logical rule."
 }

 Justification {
  definition = "The reason a derivation step is valid."
  examples = [
   DistributiveLaw,
   Substitution,
   AdditionOfEquals,
   MultiplicationOfEquals,
   Factorisation,
   Expansion,
   Simplification,
   Definition,
   PreviouslyProvedTheorem
  ]
 }

 IdentityProof {
  definition = "A proof that two expressions are identically equal over a stated domain."
 }

 EquivalenceProof {
  definition = "A proof that two propositions or algebraic conditions imply one another."
  form = "P <=> Q"
 }

 DirectProof {
  definition = "A proof deriving the conclusion directly from the premises."
 }

 ProofByContradiction {
  definition = "A proof assuming the negation of the desired conclusion and deriving a contradiction."
 }

 ProofByContrapositive {
  definition = "A proof of P=>Q by proving not-Q => not-P."
 }
}

RELATIONS {
 AlgebraicProof --starts_from--> Premise
 AlgebraicProof --contains--> DerivationStep
 DerivationStep --requires--> Justification
 AlgebraicProof --establishes--> Conclusion
 AlgebraicProof --can_use--> AlgebraicIdentity
 AlgebraicProof --can_use--> Substitution
 AlgebraicProof --can_use--> Simplification
 AlgebraicProof --can_use--> Expansion
 AlgebraicProof --can_use--> Factorisation
 AlgebraicProof --can_use--> EquationTransformation
 AlgebraicProof --can_use--> LogicalInference
 IdentityProof --is_special_case_of--> AlgebraicProof
 EquivalenceProof --is_special_case_of--> AlgebraicProof
 DirectProof --is_special_case_of--> AlgebraicProof
 ProofByContradiction --is_special_case_of--> AlgebraicProof
 ProofByContrapositive --is_special_case_of--> AlgebraicProof
}

OPERATORS {
 PROVE(PREMISES,GOAL) :=
  construct_valid_derivation(PREMISES,GOAL)

 VALID_STEP(S_i,S_j,RULE) :=
  RULE justifies transition from S_i to S_j

 VERIFY_PROOF(PROOF) :=
  every_step_is_valid
  AND conclusion_matches_goal

 PROVE_IDENTITY(LHS,RHS,D) :=
  derive(LHS ≡ RHS over domain D)

 PROVE_EQUIVALENCE(P,Q) :=
  prove(P=>Q) AND prove(Q=>P)

 CONTRADICTION_PROOF(PREMISES,GOAL) :=
  assume(NOT GOAL)
  derive(False)

 CONTRAPOSITIVE_PROOF(P,Q) :=
  prove(NOT Q => NOT P)

 NORMALISE_PROOF_STEP(E) :=
  convert_to_explicit_justified_algebraic_form(E)
}

DEFINITIONS {
 D1 = "A proof establishes a conclusion by valid inference from accepted premises."
 D2 = "Every nontrivial proof step must have a justification."
 D3 = "Algebraic manipulation is valid only when it preserves the relevant equality, implication or equivalence."
 D4 = "An identity proof must establish universal equality over the stated domain."
 D5 = "An equation-solving derivation may require tracking equivalence versus one-way implication."
 D6 = "Proof by contradiction derives an impossibility from the negation of the goal."
 D7 = "Proof by contrapositive proves P=>Q through not-Q=>not-P."
 D8 = "A finite collection of numerical examples does not constitute a general algebraic proof."
}

AXIOMS {
 A1_REFLEXIVITY:
  "A=A"

 A2_SYMMETRY_OF_EQUALITY:
  IF A=B
  THEN B=A

 A3_TRANSITIVITY_OF_EQUALITY:
  IF A=B AND B=C
  THEN A=C

 A4_SUBSTITUTION_OF_EQUALS:
  IF A=B
  THEN admissible replacement of A by B preserves truth

 A5_ADDITION_PROPERTY:
  IF A=B
  THEN A+C=B+C

 A6_MULTIPLICATION_PROPERTY:
  IF A=B
  THEN AC=BC

 A7_LOGICAL_MODUS_PONENS:
  IF P AND (P=>Q)
  THEN Q

 A8_CONTRADICTION:
  IF assumption leads to P AND NOT P
  THEN assumption is false
}

REASONING {

 R1_DIRECT_PROOF {
  INPUT = [premises, goal]
  START = premises
  SELECT = relevant definitions and identities
  APPLY = valid algebraic transformations
  CHAIN = intermediate statements
  IF goal_reached
   RETURN ValidDirectProof
  ELSE
   RETURN IncompleteProof
 }

 R2_IDENTITY_PROOF {
  INPUT = [LHS,RHS,domain]
  METHOD_OPTIONS = [
   transform_LHS_to_RHS,
   transform_RHS_to_LHS,
   transform_both_to_common_form,
   show_simplify(LHS-RHS)=0
  ]
  VERIFY = domain_conditions
  RETURN = IdentityProofStatus
 }

 R3_EQUATION_DERIVATION {
  INPUT = equation
  FOR each transformation
   CLASSIFY = equivalence_preserving_or_implication_only
   RECORD = restrictions
  RETURN = logically_annotated_derivation
 }

 R4_CONTRADICTION {
  INPUT = [premises,goal]
  ASSUME = NOT goal
  DERIVE = consequences
  IF contradiction_found
   RETURN ProofByContradiction
 }

 R5_CONTRAPOSITIVE {
  INPUT = proposition "P=>Q"
  TARGET = "NOT Q => NOT P"
  PROVE = target
  RETURN = ProofByContrapositive
 }

 R6_EQUIVALENCE {
  INPUT = [P,Q]
  PROVE_1 = P=>Q
  PROVE_2 = Q=>P
  IF both_valid
   RETURN "P <=> Q"
 }

 R7_PROOF_VERIFICATION {
  INPUT = proof_sequence
  FOR each step
   CHECK = premise_or_valid_rule_application
   CHECK = domain_and_side_conditions
  CHECK = final_statement_matches_goal
  RETURN = ProofValidity
 }
}

CONSTRAINTS {
 C1 = "Every proof step must follow from earlier statements or accepted rules."
 C2 = "Algebraic transformations must preserve the intended logical relation."
 C3 = "Division by an expression requires proof or assumption that the divisor is nonzero."
 C4 = "Squaring both sides may introduce extraneous solutions and is not automatically equivalence-preserving."
 C5 = "Taking square roots may require sign conditions."
 C6 = "Cancelling factors requires nonzero conditions."
 C7 = "Identity proofs must respect the common domain of all expressions."
 C8 = "Examples can support intuition but do not prove a universal statement."
 C9 = "Circular reasoning is invalid."
 C10 = "The conclusion must match the proposition actually established."
}

SEMANTIC_EQUIVALENCE {
 "AlgebraicProof" <=> "ValidAlgebraicDerivation"
 "IdentityProof" <=> "ProofOfUniversalAlgebraicEquality"
 "EquivalenceProof" <=> "ProofOfBothImplicationDirections"
 "ProofByContrapositive(P=>Q)" <=> "ProofOf(not-Q=>not-P)"
 "ProofByContradiction" <=> "NegatedGoalLeadsToContradiction"
}

VALIDATION {
 V1 = "Every derivation step has a valid justification."
 V2 = "Dividing by a variable requires a nonzero condition."
 V3 = "Squaring can create extraneous solutions and must be tracked."
 V4 = "Transforming both sides by reversible operations preserves equivalence."
 V5 = "An identity proof must hold throughout the stated domain."
 V6 = "A contradiction proof must derive an actual logical contradiction."
 V7 = "A proof cannot rely solely on numerical examples for a universal claim."
 V8 = "The final statement must establish the stated goal."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 PROOF_DEFINITION = 0.99
 STEP_VALIDITY_REQUIREMENT = 0.99
 DOMAIN_TRACKING = 0.99
 EQUIVALENCE_TRACKING = 0.99
 CONTRADICTION_REASONING = 0.99
 CONTRAPOSITIVE_REASONING = 0.99
 IDENTITY_PROOF_REASONING = 0.99

 POLICY =
  "Reduce confidence when assumptions, domain conditions, inference rules or transformation reversibility are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MathematicalProofReasoning,
  SymbolicAlgebraReasoning,
  ProofVerification,
  IdentityReasoning,
  EquationReasoning,
  LogicalInference,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "algebraic proof" -> AlgebraicProof
  "prove algebraically" -> AlgebraicProof
  "show that" -> ProofGoal
  "identity proof" -> IdentityProof
  "proof by contradiction" -> ProofByContradiction
  "proof by contrapositive" -> ProofByContrapositive
  "prove equivalent" -> EquivalenceProof
  "derivation" -> AlgebraicProof
 }

 INTERPRETATION_ORDER = [
  identify_goal,
  identify_premises,
  identify_domain_and_assumptions,
  select_proof_strategy,
  retrieve_relevant_definitions_and_identities,
  construct_justified_steps,
  track_equivalence_and_implication_direction,
  track_domain_restrictions,
  verify_each_step,
  verify_goal_reached,
  return_proof_structure
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("proof")
  AND algebraic_context
  THEN activate(AlgebraicProofOntology)

 E2:
  IF goal_is_identity
  THEN prefer(identity_proof_methods)

 E3:
  IF direct_route_available
  THEN prefer(DirectProof)

 E4:
  IF proposition_has_form(P=>Q)
  AND contrapositive_is_simpler
  THEN permit(ProofByContrapositive)

 E5:
  IF contradiction_strategy_selected
  THEN assume_negated_goal
  AND derive_contradiction

 E6:
  IF algebraic_step_divides_by_expression
  THEN require(nonzero_condition)

 E7:
  IF step_is_not_reversible
  THEN mark_relation_as_implication_not_equivalence

 E8:
  IF request=="verify proof"
  THEN validate_every_step
  AND validate_final_goal
}

RECURSION {
 ENTRY = AlgebraicProof

 EXPAND(AlgebraicProof) -> [
  Proposition,
  Premise,
  Conclusion,
  LogicalImplication,
  AlgebraicExpression,
  Equation,
  AlgebraicIdentity,
  Substitution,
  Simplification,
  Expansion,
  Factorisation,
  DomainRestriction
 ]

 EXPAND(IdentityProof) -> [
  AlgebraicIdentity,
  EquivalentExpression,
  Simplification,
  Expansion,
  Factorisation,
  Domain
 ]

 EXPAND(EquationProof) -> [
  Equation,
  Transformation,
  Equivalence,
  Implication,
  SolutionSet
 ]

 EXPAND(ProofByContradiction) -> [
  Negation,
  Assumption,
  Contradiction,
  LogicalInference
 ]

 STOP_CONDITION =
  "Stop when the conclusion is validly derived from the premises, every required side condition is resolved, and the proof has been verified."
}

ROUTING {
 PARENT_ONTOLOGY = "Algebra Ontology"

 IMMEDIATE_CONTEXTS = [
  "Algebraic Identity Ontology",
  "Equation Ontology",
  "Simplification Ontology",
  "Substitution Ontology"
 ]

 RELATED_ONTOLOGIES = [
  "Algebraic Expression Ontology",
  "Equation Ontology",
  "Linear Equation Ontology",
  "Algebraic Identity Ontology",
  "Factorisation Ontology",
  "Expansion Ontology",
  "Simplification Ontology",
  "Substitution Ontology",
  "Polynomial Ontology",
  "Rational Expression Ontology"
 ]

 ROUTE_RULES = [
  "Algebraic Proof -> Algebraic Identity for identity proofs",
  "Algebraic Proof -> Equation for derivational equality reasoning",
  "Algebraic Proof -> Simplification for canonical-form proof steps",
  "Algebraic Proof -> Substitution for replacement reasoning",
  "Algebraic Proof -> Factorisation or Expansion for symbolic transformation",
  "Algebraic Proof -> Rational Expression for domain-sensitive algebraic steps"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible proof strategies, operators or relations"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve logical validity as the primary criterion.",
  "Preserve explicit justification of algebraic steps.",
  "Preserve domain and nonzero side conditions.",
  "Preserve distinction between equivalence and one-way implication.",
  "Preserve distinction between proof and numerical verification.",
  "Preserve non-circular reasoning.",
  "Preserve alignment between premises, derivation and conclusion."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An algebraic proof is a logically valid sequence of justified algebraic statements establishing a conclusion from accepted premises."

 DIRECT_PROOF_SUMMARY =
  "premises -> justified algebraic steps -> conclusion"

 IDENTITY_PROOF_SUMMARY =
  "Prove LHS ≡ RHS over the stated domain."

 EQUIVALENCE_SUMMARY =
  "P <=> Q requires proof of both P=>Q and Q=>P."

 CONTRAPOSITIVE_SUMMARY =
  "To prove P=>Q, it is equivalent to prove not-Q=>not-P."

 CONTRADICTION_SUMMARY =
  "Assume the negation of the goal and derive an impossibility."

 VALIDITY_SUMMARY =
  "Every step must be justified and all side conditions preserved."
}

FLOW {
 INPUT
 -> detect_proof_context
 -> identify_proposition_or_goal
 -> identify_premises_and_assumptions
 -> identify_domain
 -> select_proof_strategy
 -> retrieve_relevant_definitions_identities_and_theorems
 -> construct_first_valid_step
 -> recursively_construct_justified_derivation
 -> track_equivalence_or_implication
 -> track_domain_and_nonzero_conditions
 -> detect_contradiction_when_strategy_requires
 -> verify_each_step
 -> verify_final_conclusion
 -> link_related_ontologies
 -> assign_confidence
 -> produce_semantic_proof_output
 -> END
}

SEMANTIC_SUMMARY {
 "Algebraic proof establishes mathematical truth by connecting premises to a conclusion through justified algebraic and logical steps. Valid proofs must preserve the appropriate equality, implication or equivalence relation, respect domain restrictions and side conditions, and avoid unsupported or circular reasoning. Algebraic identities, substitution, simplification, expansion and factorisation serve as common proof mechanisms."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-ALGEBRA-ALGEBRAIC-PROOF-ONTOLOGY-V1"
}

}

END_FLOW