agda/HyperKitty/DerivationIterations.agda

-- HyperKitty 20x Semantic Derivation Engine
-- Canonical transformation obligations for invariant preservation proofs
--
-- Architecture: 20 DISTINCT derivation obligations, NOT sequential copies.
-- Each applies a semantically unique transformation to the source invariant.
-- All obligations are independent; none depend on others.
-- This enables 20 parallel verification paths via HOL/Lean/Agda cross-prover correspondence.
--
-- Status: GENERATED_UNVERIFIED (obligations only, no proofs)
-- Source: constraint-inversion-engine.xsl fc:iteration-transform() array (lines 179-182)

module HyperKitty.DerivationIterations where

open import Data.Fin using (Fin; zero; suc)
open import Data.Vec using (Vec; []; _∷_)
open import Data.Nat using (ℕ; zero; suc)
open import Data.Bool using (Bool; true; false)
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; subst; cong; cong₂)
open import Relation.Unary using (Pred)
open import Level using (Level)

open import HyperKitty.Core

-- ============ CANONICAL REGISTRY AND SEMANTICS ============

-- InvariantRegistry: a mapping of canonical invariants from FormalizationMachine XSLT
record InvariantRegistry : Set where
  field
    id : String
    kind : String                -- PROHIBITION, TECHNOLOGY, BOOLEAN_ALGEBRA, etc.
    polarity : String            -- POSITIVE, NEGATIVE, NEUTRAL
    normalized_expr : String     -- Canonical form from fc:normalized-expression
    inverted_expr : String       -- Constraint-inverted form from fc:inverted-expression
    source_class : String        -- SPECIFIED, FORMAL, or UNRESOLVED

-- Semantics: interpretation function over normalized invariants
-- Semantics(inv) : InvariantRegistry → Set
-- (Type: an invariant is semantically valid if its semantics function is provable)
postulate
  Semantics : InvariantRegistry → Set

-- ============ SEMANTIC TRANSFORMATION FUNCTIONS ============
-- Each transform applies a distinct logical/algebraic operation to the source invariant.
-- Transforms preserve or alter invariant semantics in measurable, independent ways.

-- 1. Identity preservation: application of identity operation
postulate
  apply_identity : InvariantRegistry → InvariantRegistry
  identity_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_identity inv)

-- 2. Double negation stability: apply ¬¬ℓ = ℓ
postulate
  apply_double_negation : InvariantRegistry → InvariantRegistry
  double_negation_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_double_negation inv)

-- 3. Conjunction left projection: extract left component of (A ∧ B)
postulate
  apply_conjunction_left_projection : InvariantRegistry → InvariantRegistry
  conjunction_left_semantics : ∀ (inv : InvariantRegistry) → Semantics (apply_conjunction_left_projection inv) → Semantics inv

-- 4. Conjunction right projection: extract right component of (A ∧ B)
postulate
  apply_conjunction_right_projection : InvariantRegistry → InvariantRegistry
  conjunction_right_semantics : ∀ (inv : InvariantRegistry) → Semantics (apply_conjunction_right_projection inv) → Semantics inv

-- 5. Implication closure: ((A → B) ∧ A) → B
postulate
  apply_implication_closure : InvariantRegistry → InvariantRegistry
  implication_closure_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_implication_closure inv)

-- 6. Contrapositive check: (A → B) ≡ (¬B → ¬A)
postulate
  apply_contrapositive : InvariantRegistry → InvariantRegistry
  contrapositive_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_contrapositive inv)

-- 7. Reflexive equality: a ≡ a
postulate
  apply_reflexive_equality : InvariantRegistry → InvariantRegistry
  reflexive_equality_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_reflexive_equality inv)

-- 8. Symmetric equality: (a ≡ b) → (b ≡ a)
postulate
  apply_symmetric_equality : InvariantRegistry → InvariantRegistry
  symmetric_equality_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_symmetric_equality inv)

-- 9. Transitive equality: ((a ≡ b) ∧ (b ≡ c)) → (a ≡ c)
postulate
  apply_transitive_equality : InvariantRegistry → InvariantRegistry
  transitive_equality_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_transitive_equality inv)

-- 10. Substitution preservation: equals can be substituted in propositions
postulate
  apply_substitution_preservation : InvariantRegistry → InvariantRegistry
  substitution_preservation_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_substitution_preservation inv)

-- 11. Domain restriction: enforce domain membership preconditions
postulate
  apply_domain_restriction : InvariantRegistry → InvariantRegistry
  domain_restriction_semantics : ∀ (inv : InvariantRegistry) → Semantics inv → Semantics (apply_domain_restriction inv)

-- 12. Codomain preservation: image closure under function applications
postulate
  apply_codomain_preservation : InvariantRegistry → InvariantRegistry
  codomain_preservation_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_codomain_preservation inv)

-- 13. State transition preservation: invariants hold across state machine transitions
postulate
  apply_state_transition_preservation : InvariantRegistry → InvariantRegistry
  state_transition_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_state_transition_preservation inv)

-- 14. Graph edge preservation: invariants hold over all reachable graph nodes
postulate
  apply_graph_edge_preservation : InvariantRegistry → InvariantRegistry
  graph_edge_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_graph_edge_preservation inv)

-- 15. Topological order preservation: DAG invariants under topological sort
postulate
  apply_topological_order_preservation : InvariantRegistry → InvariantRegistry
  topological_order_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_topological_order_preservation inv)

-- 16. Refinement strengthening: P → Q where P is a refinement of Q
postulate
  apply_refinement_strengthening : InvariantRegistry → InvariantRegistry
  refinement_strengthening_semantics : ∀ (inv : InvariantRegistry) → Semantics (apply_refinement_strengthening inv) → Semantics inv

-- 17. Refinement weakening check: verify Q when proving P, where P refines Q
postulate
  apply_refinement_weakening_check : InvariantRegistry → InvariantRegistry
  refinement_weakening_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_refinement_weakening_check inv)

-- 18. Rejection monotonicity: if φ rejects a state, no refining process accepts it
postulate
  apply_rejection_monotonicity : InvariantRegistry → InvariantRegistry
  rejection_monotonicity_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_rejection_monotonicity inv)

-- 19. Acceptance soundness: acceptance criteria preserve source invariants
postulate
  apply_acceptance_soundness : InvariantRegistry → InvariantRegistry
  acceptance_soundness_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_acceptance_soundness inv)

-- 20. Cross-prover correspondence: HOL ≡ Lean ≡ Agda semantics
postulate
  apply_cross_prover_correspondence : InvariantRegistry → InvariantRegistry
  cross_prover_correspondence_semantics : ∀ (inv : InvariantRegistry) → Semantics inv = Semantics (apply_cross_prover_correspondence inv)

-- ============ DERIVATION OBLIGATION TYPE HIERARCHY ============

-- DerivationObligation: a tagged invariant transformation requirement
data DerivationObligation (inv : InvariantRegistry) : Set where
  -- Obligation 1: Identity preserves semantics exactly
  identity_preservation : DerivationObligation inv

  -- Obligation 2: Double negation does not alter semantics
  double_negation_stability : DerivationObligation inv

  -- Obligation 3: Left conjunct can be extracted from compound invariants
  conjunction_left_projection : DerivationObligation inv

  -- Obligation 4: Right conjunct can be extracted from compound invariants
  conjunction_right_projection : DerivationObligation inv

  -- Obligation 5: Implication is closed under modus ponens
  implication_closure : DerivationObligation inv

  -- Obligation 6: Contrapositive equivalence holds
  contrapositive_check : DerivationObligation inv

  -- Obligation 7: Reflexivity of equality
  reflexive_equality : DerivationObligation inv

  -- Obligation 8: Symmetry of equality
  symmetric_equality : DerivationObligation inv

  -- Obligation 9: Transitivity of equality
  transitive_equality : DerivationObligation inv

  -- Obligation 10: Substitution of equals in predicates
  substitution_preservation : DerivationObligation inv

  -- Obligation 11: Domain membership is enforced
  domain_restriction : DerivationObligation inv

  -- Obligation 12: Codomain closure across all images
  codomain_preservation : DerivationObligation inv

  -- Obligation 13: Invariants survive state transitions
  state_transition_preservation : DerivationObligation inv

  -- Obligation 14: Invariants hold at all graph-reachable states
  graph_edge_preservation : DerivationObligation inv

  -- Obligation 15: DAG topological ordering preserves invariants
  topological_order_preservation : DerivationObligation inv

  -- Obligation 16: Type refinements strengthen source invariants
  refinement_strengthening : DerivationObligation inv

  -- Obligation 17: Type weakening does not violate source invariants
  refinement_weakening_check : DerivationObligation inv

  -- Obligation 18: Rejection is monotonic under program refinement
  rejection_monotonicity : DerivationObligation inv

  -- Obligation 19: Acceptance criteria are sound w.r.t. source invariants
  acceptance_soundness : DerivationObligation inv

  -- Obligation 20: Semantics are unified across HOL, Lean, and Agda
  cross_prover_correspondence : DerivationObligation inv

-- ============ THEOREM STUBS: SEMANTIC PRESERVATION PROOFS ============
-- Each theorem proves that a distinct transformation preserves or entails the source semantics.
-- NO PROOFS: all theorems are admitted as obligations for parallel external verification.

-- Theorem 1: Identity transformation
theorem theorem_identity_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_identity inv) :=
  identity_semantics inv

-- Theorem 2: Double negation stability
theorem theorem_double_negation_stability (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_double_negation inv) :=
  double_negation_semantics inv

-- Theorem 3: Conjunction left projection
theorem theorem_conjunction_left_projection (inv : InvariantRegistry) :
  Semantics (apply_conjunction_left_projection inv) → Semantics inv :=
  conjunction_left_semantics inv

-- Theorem 4: Conjunction right projection
theorem theorem_conjunction_right_projection (inv : InvariantRegistry) :
  Semantics (apply_conjunction_right_projection inv) → Semantics inv :=
  conjunction_right_semantics inv

-- Theorem 5: Implication closure
theorem theorem_implication_closure (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_implication_closure inv) :=
  implication_closure_semantics inv

-- Theorem 6: Contrapositive check
theorem theorem_contrapositive_check (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_contrapositive inv) :=
  contrapositive_semantics inv

-- Theorem 7: Reflexive equality
theorem theorem_reflexive_equality (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_reflexive_equality inv) :=
  reflexive_equality_semantics inv

-- Theorem 8: Symmetric equality
theorem theorem_symmetric_equality (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_symmetric_equality inv) :=
  symmetric_equality_semantics inv

-- Theorem 9: Transitive equality
theorem theorem_transitive_equality (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_transitive_equality inv) :=
  transitive_equality_semantics inv

-- Theorem 10: Substitution preservation
theorem theorem_substitution_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_substitution_preservation inv) :=
  substitution_preservation_semantics inv

-- Theorem 11: Domain restriction
theorem theorem_domain_restriction (inv : InvariantRegistry) :
  Semantics inv → Semantics (apply_domain_restriction inv) :=
  domain_restriction_semantics inv

-- Theorem 12: Codomain preservation
theorem theorem_codomain_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_codomain_preservation inv) :=
  codomain_preservation_semantics inv

-- Theorem 13: State transition preservation
theorem theorem_state_transition_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_state_transition_preservation inv) :=
  state_transition_semantics inv

-- Theorem 14: Graph edge preservation
theorem theorem_graph_edge_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_graph_edge_preservation inv) :=
  graph_edge_semantics inv

-- Theorem 15: Topological order preservation
theorem theorem_topological_order_preservation (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_topological_order_preservation inv) :=
  topological_order_semantics inv

-- Theorem 16: Refinement strengthening
theorem theorem_refinement_strengthening (inv : InvariantRegistry) :
  Semantics (apply_refinement_strengthening inv) → Semantics inv :=
  refinement_strengthening_semantics inv

-- Theorem 17: Refinement weakening check
theorem theorem_refinement_weakening_check (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_refinement_weakening_check inv) :=
  refinement_weakening_semantics inv

-- Theorem 18: Rejection monotonicity
theorem theorem_rejection_monotonicity (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_rejection_monotonicity inv) :=
  rejection_monotonicity_semantics inv

-- Theorem 19: Acceptance soundness
theorem theorem_acceptance_soundness (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_acceptance_soundness inv) :=
  acceptance_soundness_semantics inv

-- Theorem 20: Cross-prover correspondence
theorem theorem_cross_prover_correspondence (inv : InvariantRegistry) :
  Semantics inv = Semantics (apply_cross_prover_correspondence inv) :=
  cross_prover_correspondence_semantics inv

-- ============ DERIVATION REGISTRY ============
-- Index-based lookup of derivation obligations for parallel verification

get_derivation : ∀ (idx : Fin 20) (inv : InvariantRegistry) → DerivationObligation inv
get_derivation Fin.zero inv = identity_preservation inv
get_derivation (Fin.suc Fin.zero) inv = double_negation_stability inv
get_derivation (Fin.suc (Fin.suc Fin.zero)) inv = conjunction_left_projection inv
get_derivation (Fin.suc (Fin.suc (Fin.suc Fin.zero))) inv = conjunction_right_projection inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))) inv = implication_closure inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))) inv = contrapositive_check inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))) inv = reflexive_equality inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))) inv = symmetric_equality inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))) inv = transitive_equality inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))) inv = substitution_preservation inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))))) inv = domain_restriction inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))))) inv = codomain_preservation inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))))))) inv = state_transition_preservation inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))))))) inv = graph_edge_preservation inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))))))))) inv = topological_order_preservation inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))))))))) inv = refinement_strengthening inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))))))))))) inv = refinement_weakening_check inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))))))))))) inv = rejection_monotonicity inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero)))))))))))))))))) inv = acceptance_soundness inv
get_derivation (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc (Fin.suc Fin.zero))))))))))))))))))) inv = cross_prover_correspondence inv

-- ============ INDEPENDENCE PROPERTIES ============
-- Each obligation is semantically independent: no derivation depends on solving another.

-- Observation 1: Each transformation function is orthogonal
-- (i.e., apply_identity inv ≠ apply_double_negation inv in general)

-- Observation 2: Each theorem has a distinct type signature
-- (identity_preservation vs double_negation_stability, etc.)

-- Observation 3: Enabling parallel verification
-- All 20 obligations can be discharged independently via HOL, Lean, Agda verification
-- without awaiting results from siblings.

-- ============ DOCUMENTATION ============

-- NOTE: Status = GENERATED_UNVERIFIED
-- - All 20 transformation functions are postulated (not implemented)
-- - All 20 semantic theorems are postulated (not proved)
-- - All 20 obligations are declared but unresolved
-- - This structure enables:
--   * Type checking to validate obligation structure
--   * Cross-prover correspondence checking (HOL ↔ Lean ↔ Agda)
--   * Parallel external verification (no sequential dependencies)
--   * Evidence collection for formal proof receipts

-- NOTE: Agda-iteration-multiplicity = 20
-- Derived from constraint-inversion-engine.xsl parameter $agda-iteration-multiplicity
-- Each iteration is a distinct DerivationObligation, not a loop iteration.

-- NOTE: Authority boundary
-- Per constraint-inversion-engine.xsl AuthorityBoundary:
-- - Rule AUTH-007: "Twenty generated iterations are obligations, not twenty verified proofs."
-- This module embodies that boundary: obligations are declared, not discharged.