Foundation Primitive Recognition Calculus Real Cauchy Zero Lt Of Positive

A bridge from positive to ordered

The declaration zero_lt_of_positive is a theorem in a machine-checked library of formal mathematics. It states a simple fact about the framework's rational numbers: if a number is positive, then it is greater than zero. In symbols, if positive q holds, then lt 0 q follows. The proof is short and direct, relying on the library's definitions of positivity and the less-than relation, which are themselves tied to the usual ordering of rational numbers.

This fact is a bridge inside a larger construction. The framework builds its real numbers from Cauchy sequences of rationals, where a sequence is Cauchy if its terms get close to one another in a specific sense. The closeness is measured by a cost function, a way of assigning a nonnegative number to a pair of terms. The theorem zero_lt_of_positive is used to show that a sequence is Cauchy: it converts the assumption that a gap is positive into the needed inequality that the gap is greater than zero. It is a small but necessary step in the framework's account of real numbers.

The theorem does not claim that the framework's definition of positivity is the only one, nor does it assert that the real numbers so constructed are the same as the classical real numbers. It only establishes a logical implication within the framework's own definitions. The framework's real numbers are a quotient of Cauchy sequences, and this theorem is part of the groundwork for that construction, not a claim about the nature of mathematical truth.

theorem zero_lt_of_positive {q : PRCRat}
    (h : positive q) : lt 0 q := by
  rw [lt_iff_toRat_lt]
  simpa using (positive_iff_toRat_pos q).mp h

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

/-- A PRC Cauchy sequence is a completed orbit-indexed rational ledger whose
J-cost distance eventually falls below every positive PRC rational tolerance. -/
structure PRCCauchySeq where
  term : Nat → PRCRat
  cauchy :
    ∀ eps : PRCRat, PRCRat.positive eps →
      ∃ N : Nat, ∀ m n : Nat, N ≤ m → N ≤ n →
        PRCRat.lt (PRCJCostDistance (term m) (term n)) eps

What this page does not claim

The theorem does not claim that the framework's positivity is the only possible definition. The theorem does not assert that the framework's real numbers are identical to classical real numbers. The theorem does not establish any property of the real numbers beyond the logical implication it states.

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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