Foundation Primitive Recognition Calculus Grow Eta Completion M0a Mk Eq Mk Of Eq

The completion map

A ledger, a discrete record of events, often needs to represent quantities that are not exact ratios, like the square root of two. The framework builds its real numbers by a standard completion: it takes all infinite sequences of ratios that settle down in a precise way, then identifies two sequences when their terms eventually become indistinguishable. The declaration mk_eq_mk_of_equiv is the bridge that makes this identification work: it states that if two such sequences are equivalent, then they name the same completed real number.

This is a structural theorem about the construction, not a claim about any particular value. It does not say which real number a given sequence converges to, nor does it assert that every real number arises this way. It only guarantees that the map from sequences to completed reals respects the equivalence relation, which is exactly what is needed for the later definition of the embedding from the framework's rationals to be well-defined. The proof is a one-line application of the quotient construction's soundness property.

In Recognition Science, this theorem is a small but load-bearing step in the eta completion, the process of adjoining limits to the ratio orbits so that the framework's arithmetic has access to the full real continuum. The library's machine-checked proof shows that the identification is coherent: equivalent sequences cannot be split apart by the completion map. This is a prerequisite for any later claim that a particular sequence represents a particular real number, such as the golden ratio or the framework's constants.

The theorem does not, by itself, establish that the completion contains all real numbers, nor that the equivalence relation is the usual one of Cauchy sequences. It is a local coherence fact about one construction step. The broader claims about what the completed reals contain, and how they relate to the classical reals, are separate theorems in the same file, each with its own proof and its own scope.

/-- Equivalent regular sequences determine equal constructive reals. -/
theorem mk_eq_mk_of_equiv {s t : RegularSeq} (h : equiv s t) : mk s = mk t :=
  Quot.sound h

What this page does not claim

The theorem does not claim that every real number is represented by some sequence in the completion. The theorem does not claim that the equivalence relation is the usual Cauchy-sequence equivalence. The theorem does not claim that the completion map is injective or surjective.

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/Grow/EtaCompletionM0a.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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