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-- 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 (β„•; zero; suc)
open import Data.Bool using (Bool; true; false)
open import Relation.Binary.PropositionalEquality using (_≑_; refl; sym; trans; subst; cong; congβ‚‚)
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 β†’ Set
-- (Type: an invariant is semantically valid if its semantics function is provable)
postulate
Semantics : InvariantRegistry β†’ 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 β†’ InvariantRegistry
identity_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_identity inv)
-- 2. Double negation stability: apply ¬¬ℓ = β„“
postulate
apply_double_negation : InvariantRegistry β†’ InvariantRegistry
double_negation_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_double_negation inv)
-- 3. Conjunction left projection: extract left component of (A ∧ B)
postulate
apply_conjunction_left_projection : InvariantRegistry β†’ InvariantRegistry
conjunction_left_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics (apply_conjunction_left_projection inv) β†’ Semantics inv
-- 4. Conjunction right projection: extract right component of (A ∧ B)
postulate
apply_conjunction_right_projection : InvariantRegistry β†’ InvariantRegistry
conjunction_right_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics (apply_conjunction_right_projection inv) β†’ Semantics inv
-- 5. Implication closure: ((A β†’ B) ∧ A) β†’ B
postulate
apply_implication_closure : InvariantRegistry β†’ InvariantRegistry
implication_closure_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_implication_closure inv)
-- 6. Contrapositive check: (A β†’ B) ≑ (Β¬B β†’ Β¬A)
postulate
apply_contrapositive : InvariantRegistry β†’ InvariantRegistry
contrapositive_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_contrapositive inv)
-- 7. Reflexive equality: a ≑ a
postulate
apply_reflexive_equality : InvariantRegistry β†’ InvariantRegistry
reflexive_equality_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_reflexive_equality inv)
-- 8. Symmetric equality: (a ≑ b) β†’ (b ≑ a)
postulate
apply_symmetric_equality : InvariantRegistry β†’ InvariantRegistry
symmetric_equality_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_symmetric_equality inv)
-- 9. Transitive equality: ((a ≑ b) ∧ (b ≑ c)) β†’ (a ≑ c)
postulate
apply_transitive_equality : InvariantRegistry β†’ InvariantRegistry
transitive_equality_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_transitive_equality inv)
-- 10. Substitution preservation: equals can be substituted in propositions
postulate
apply_substitution_preservation : InvariantRegistry β†’ InvariantRegistry
substitution_preservation_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_substitution_preservation inv)
-- 11. Domain restriction: enforce domain membership preconditions
postulate
apply_domain_restriction : InvariantRegistry β†’ InvariantRegistry
domain_restriction_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv β†’ Semantics (apply_domain_restriction inv)
-- 12. Codomain preservation: image closure under function applications
postulate
apply_codomain_preservation : InvariantRegistry β†’ InvariantRegistry
codomain_preservation_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_codomain_preservation inv)
-- 13. State transition preservation: invariants hold across state machine transitions
postulate
apply_state_transition_preservation : InvariantRegistry β†’ InvariantRegistry
state_transition_semantics : βˆ€ (inv : InvariantRegistry) β†’ 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 β†’ InvariantRegistry
graph_edge_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_graph_edge_preservation inv)
-- 15. Topological order preservation: DAG invariants under topological sort
postulate
apply_topological_order_preservation : InvariantRegistry β†’ InvariantRegistry
topological_order_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_topological_order_preservation inv)
-- 16. Refinement strengthening: P β†’ Q where P is a refinement of Q
postulate
apply_refinement_strengthening : InvariantRegistry β†’ InvariantRegistry
refinement_strengthening_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics (apply_refinement_strengthening inv) β†’ Semantics inv
-- 17. Refinement weakening check: verify Q when proving P, where P refines Q
postulate
apply_refinement_weakening_check : InvariantRegistry β†’ InvariantRegistry
refinement_weakening_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_refinement_weakening_check inv)
-- 18. Rejection monotonicity: if Ο† rejects a state, no refining process accepts it
postulate
apply_rejection_monotonicity : InvariantRegistry β†’ InvariantRegistry
rejection_monotonicity_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_rejection_monotonicity inv)
-- 19. Acceptance soundness: acceptance criteria preserve source invariants
postulate
apply_acceptance_soundness : InvariantRegistry β†’ InvariantRegistry
acceptance_soundness_semantics : βˆ€ (inv : InvariantRegistry) β†’ Semantics inv = Semantics (apply_acceptance_soundness inv)
-- 20. Cross-prover correspondence: HOL ≑ Lean ≑ Agda semantics
postulate
apply_cross_prover_correspondence : InvariantRegistry β†’ InvariantRegistry
cross_prover_correspondence_semantics : βˆ€ (inv : InvariantRegistry) β†’ 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) β†’ Semantics inv :=
conjunction_left_semantics inv
-- Theorem 4: Conjunction right projection
theorem theorem_conjunction_right_projection (inv : InvariantRegistry) :
Semantics (apply_conjunction_right_projection inv) β†’ 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 β†’ 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) β†’ 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 : βˆ€ (idx : Fin 20) (inv : InvariantRegistry) β†’ 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 β‰  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.