Foundation Primitive Recognition Calculus Real Null Setoid Prcnull Distance Seto

The triangle modulus

The declaration PRCNullDistanceSetoidTarget_of_triangle_modulus is a formal theorem in the framework's machine-checked library of formal theorems. It establishes a conditional bridge: if a certain local analytic condition holds, then a global structure follows. The condition is called the triangle modulus. It says that for any positive tolerance, there is a smaller positive tolerance such that if two consecutive distances are each below the smaller tolerance, the composed distance is below the original tolerance. This is a precise, quantitative version of the triangle inequality, stated for a distance function derived from the framework's cost function.

The global structure that follows is a setoid: an equivalence relation on Cauchy sequences of rationals, where two sequences are equivalent when their null distance is zero. In plain terms, the theorem proves that the relation "the distance between two sequences is zero" is transitive, given the triangle modulus. With reflexivity and symmetry already in hand, transitivity completes the proof that this relation is an equivalence. That equivalence is what allows the framework to define a quotient carrier, PRCRealNull, which is its intended real number system built from the recognition ledger, a discrete record of events. The theorem itself is a conditional: it does not prove the triangle modulus holds; it proves that if it does, the setoid and quotient follow.

The declaration is part of a build order, step 9, and the framework's audit records it as living under trace closure, meaning the final quotient still depends on an open analytic obligation. The theorem's role is to isolate that obligation: all quotient mechanics, the equivalence relation, the quotient type, and the embedding of rationals, are complete once the triangle modulus is proved. The remaining work is analytic, not structural. The framework's library shows this by providing a conditional certificate that records the precise remaining theorem and proves that this theorem is sufficient to construct the null-distance setoid and quotient carrier.

What the declaration does not claim is equally sharp. It does not claim that the triangle modulus is true; that remains an open analytic target. It does not claim that the quotient is a complete or ordered field, only that it is a setoid. It does not claim that the real number system is finished; the final carrier still lives under trace closure. The theorem is a precise conditional, and its value is in the clarity of the remaining obligation: one local analytic inequality, once proved, completes the construction.

/-- The exact setoid target follows from the sharper triangle-modulus target. -/
theorem PRCNullDistanceSetoidTarget_of_triangle_modulus
    (htri : PRCJCostDistanceTriangleModulusTarget) :
    PRCNullDistanceSetoidTarget :=
  PRCNullDistanceSetoidTarget_of_transitive
    (PRCNullDistanceTransitiveTarget_of_triangle_modulus htri)

/-- The exact setoid target follows from the sharper triangle-modulus target. -/
theorem PRCNullDistanceSetoidTarget_of_triangle_modulus
    (htri : PRCJCostDistanceTriangleModulusTarget) :
    PRCNullDistanceSetoidTarget :=
  PRCNullDistanceSetoidTarget_of_transitive
    (PRCNullDistanceTransitiveTarget_of_triangle_modulus htri)

/-- The analytic triangle modulus is sufficient for null-distance
transitivity. All remaining work here is completed-orbit index bookkeeping. -/
theorem PRCNullDistanceTransitiveTarget_of_triangle_modulus
    (htri : PRCJCostDistanceTriangleModulusTarget) :
    PRCNullDistanceTransitiveTarget := by
  intro u v w huv hvw eps heps
  rcases htri eps heps with ⟨delta, hdelta_pos, hdelta⟩
  rcases huv delta hdelta_pos with ⟨Nuv, hNuv⟩
  rcases hvw delta hdelta_pos with ⟨Nvw, hNvw⟩
  refine ⟨max Nuv Nvw, ?_⟩
  intro n hn
  have hn_uv : Nuv ≤ n := le_trans (Nat.le_max_left Nuv Nvw) hn
  have hn_vw : Nvw ≤ n := le_trans (Nat.le_max_right Nuv Nvw) hn
  exact hdelta (u.term n) (v.term n) (w.term n)
    (hNuv n hn_uv) (hNvw n hn_vw)

/-- Build Order step 9 conditional closure: no quotient mechanics remain once
the local J-cost triangle modulus is proved. -/
theorem real_null_setoid_conditional_certificate :
    PRCRealNullSetoidConditionalCertificate where
  triangle_modulus_target := rfl
  transitive_from_triangle := PRCNullDistanceTransitiveTarget_of_triangle_modulus
  setoid_from_transitive := PRCNullDistanceSetoidTarget_of_transitive
  setoid_from_triangle := PRCNullDistanceSetoidTarget_of_triangle_modulus
  quotient_from_transitive := by
    intro htrans
    exact ⟨PRCRealNull.ofRat htrans 0⟩
  rat_embedding_from_transitive := by
    intro htrans
    exact ⟨PRCRealNull.ofRat htrans⟩
  strength_tag := rfl

What this page does not claim

The theorem does not prove the triangle modulus holds; that remains an open analytic obligation. The theorem does not claim the quotient is a complete or ordered field, only that it is a setoid. The theorem does not claim the real number system is finished, as the final carrier still lives under trace closure.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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