Foundation Primitive Recognition Calculus Real Cauchy Lt Iff To Rat Lt

The rational order bridge

The declaration lt_iff_toRat_lt is a theorem about how to compare two rational numbers in the Recognition Science framework. It states that the framework's own strict order on rationals, written a < b, holds exactly when the ordinary rational comparison a.toRat < b.toRat holds. In plain terms, the framework's order is not a new exotic ordering; it agrees perfectly with the standard one. The proof unfolds the framework's definition of lt, which says a < b means the gap b - a is positive, and then uses the fact that a rational is positive precisely when its standard representation is positive.

This agreement matters because the framework builds its real numbers from Cauchy sequences of these rationals. The theorem ensures that when the framework talks about one rational being less than another, it is talking about the same relation any mathematician would recognize. It is a bridge between the framework's constructed world and the classical one, verified by a machine-checked library of formal theorems. The declaration is a small but foundational piece: it lets later constructions, such as defining Cauchy sequences and their equivalence, rely on a familiar and trustworthy comparison.

The theorem does not claim to define the real numbers. It only establishes the order on rationals. The framework's real numbers are a separate construction, built as a quotient of Cauchy sequences under a relation of being eventually close. That construction, and the theorems about it, are distinct from this order theorem. The declaration also does not claim that the framework's order is the only possible one, nor does it address how the order extends to the real numbers themselves. Those are separate questions, answered by other parts of the framework.

theorem lt_iff_toRat_lt (a b : PRCRat) :
    lt a b ↔ a.toRat < b.toRat := by
  unfold lt
  rw [positive_iff_toRat_pos]
  rw [PRCRat.sub_eq, PRCRat.toRat_sub]
  constructor
  · intro h
    linarith
  · intro h
    linarith

/-- PRC-native strict order on rationals: `a < b` means the positive gap
`b - a` has a positive ratio-orbit representative. -/
def lt (a b : PRCRat) : Prop :=
  positive (b - a)

What this page does not claim

The theorem does not define the real numbers. The theorem does not claim the framework's order is the only possible order on rationals. The theorem does not address how the order extends to the framework's real numbers.

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

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