Foundation Primitive Recognition Calculus Completion Conservativity Function Com

Certified completions

A completion interface is a bridge between two kinds of data: native data N that a system works with internally, and display data D that gets shown to a user or another system. The bridge has two parts. A display function maps each native datum to a display datum. A certifies relation pairs certificates with display data, marking which displays are backed by evidence. The framework defines this as a structure with exactly those two fields, so any completion in the library is a concrete object with a display map and a certification relation.

Conservativity is the property that matters. A completion is conservative for a predicate P when every display datum satisfying P carries a certificate. In plainer terms: if a displayed fact looks true, the completion guarantees there is a certificate proving it. The framework's headline theorem states this cleanly: a completion is conservative exactly when it has no uncertified display artifacts, and the identity completion is conservative for every predicate. An artifact is a display datum that satisfies the predicate but carries no certificate. The equivalence is a theorem in the machine-checked library of formal theorems, proved by unfolding the definitions and applying the certificate-covered condition directly.

The function-space headline extends this to collections. When a completion is conservative for a predicate, the same completion applied pointwise to a function from an index set I is conservative for the predicate that holds when every coordinate satisfies the original predicate. This means finite vectors and finite fields can be certified coordinatewise: if each component display is backed by a certificate, the whole vector is backed by a tuple of certificates. The theorem function_completion_headline states this as a direct implication from component conservativity to function conservativity, with the proof choosing a certificate for each coordinate and packaging them into a single certificate function.

The theorem is a structural guarantee, not a claim about any particular data format. It does not say which predicates are conservative, only that conservativity composes across pointwise displays. It does not assert that any specific completion exists for a given N and D; the user must supply the completion and prove its component conservativity. The identity completion is the base case, and products compose conservativity for paired displays, but the framework does not claim that every completion is conservative or that certificates are unique. The headline is a tool: when you have a conservative completion for a predicate, you get one for free over finite families of that predicate.

/-- **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⟩

/-- **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

No claim that any specific completion is conservative without a proof of its component conservativity. No claim about the uniqueness of certificates for a given display datum. No claim that conservativity holds for infinite index sets without additional conditions.

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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