Foundation Primitive Recognition Calculus Completion Conservativity Completion C

Certificates and artifacts

A completion is a bridge from native data to display data, with certificates that vouch for display predicates. In plain terms, native data is the underlying record, display data is what gets shown, and a certificate is a witness that a displayed item really satisfies the property it claims to satisfy. The framework's machine-checked library of formal theorems proves a headline result: a completion is conservative for a predicate exactly when it has no uncertified display artifacts. An artifact is a display datum that satisfies the predicate but carries no certificate, a gap between what is shown and what is vouched for.

The equivalence is proved as completion_conservativity_headline in the framework's library. It states, for any completion interface, that conservativity and artifact-freeness are the same condition. The proof runs through the earlier conservative_iff_no_artifact theorem, which unfolds the definitions and shows the two directions directly. The same headline also proves the base case: the identity completion, where display and certificate are both just the datum itself, is conservative for every predicate, because the datum itself serves as its own certificate.

The result composes. The library proves that product completions preserve conservativity: if two component displays are each conservative for their predicates, then the paired display is conservative for the paired predicate, with paired certificates. It also proves function completions preserve conservativity pointwise: if each coordinate display is conservative, then a whole function display is conservative for the pointwise predicate, with coordinatewise certificates. These are separate theorems, product_completion_headline and function_completion_headline, proved in the same file.

What the headline does not claim is equally precise. It does not assert that every completion is conservative; it characterizes when one is. It does not claim that artifacts never occur, only that conservativity is equivalent to their absence. It does not prove that any particular physical or mathematical system is conservative; it establishes the logical equivalence and the identity base case, leaving the question of which actual completions are conservative to be settled by the specific certificates at hand.

/-- **Completion conservativity headline.** A completion is conservative exactly
when it has no uncertified display artifacts; the identity completion is
conservative for every predicate. -/
theorem completion_conservativity_headline (N D Cert : Type*) (C : Completion N D Cert) :
    (∀ P : D → Prop, ConservativeFor C P ↔ ¬ ArtifactFor C P)
      ∧ (∀ P : N → Prop, ConservativeFor (identityCompletion N) P) :=
  ⟨fun P => conservative_iff_no_artifact C P, identity_conservative N⟩

theorem identity_conservative (N : Type*) (P : N → Prop) :
    ConservativeFor (identityCompletion N) P := by
  intro d _
  exact ⟨d, rfl⟩

/-- **Product completion headline.** Certificate-preserving completion is stable
under products, so multi-field display objects can be certified componentwise. -/
theorem product_completion_headline
    {N₁ D₁ Cert₁ N₂ D₂ Cert₂ : Type*}
    (C₁ : Completion N₁ D₁ Cert₁) (C₂ : Completion N₂ D₂ Cert₂)
    (P₁ : D₁ → Prop) (P₂ : D₂ → Prop) :
    ConservativeFor C₁ P₁ → ConservativeFor C₂ P₂ →
      ConservativeFor (productCompletion C₁ C₂) (ProductPredicate P₁ P₂) :=
  product_conservative C₁ C₂ P₁ P₂

/-- **Function-space completion headline.** Certificate-preserving completion is
stable under pointwise finite/function displays, so finite vectors and finite
fields can be certified coordinatewise. -/
theorem function_completion_headline
    {I N D Cert : Type*} (C : Completion N D Cert) (P : D → Prop) :
    ConservativeFor C P →
      ConservativeFor (functionCompletion I N D Cert C) (AllPredicate P) :=
  function_conservative C P

What this page does not claim

Every completion is conservative for every predicate. Artifacts never occur in any completion. Any particular physical system is conservative; the theorem only characterizes when one is.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/CompletionConservativity.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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