Foundation Primitive Recognition Calculus Real Completeness Prcjcost Distance Th

A continuity guarantee

A distance function on rational numbers is continuous when small changes in the inputs produce small changes in the output. The theorem PRCJCostDistanceThreeLegModulusTarget_proved establishes this property for a particular distance function called PRCJCostDistance, which measures the cost of recognition between two rational numbers. The theorem states that for any positive tolerance epsilon, there exists a positive delta such that if the distances from a to b, b to c, and c to d are all less than delta, then the distance from a to d is less than epsilon. This is a formalized version of the triangle inequality, a fundamental property that any reasonable notion of distance should satisfy.

The theorem is part of a larger effort to construct real numbers from a ledger of recognition events. In this framework, a ledger is a discrete record of such events, and rational numbers are represented as sequences of these records. The distance function PRCJCostDistance measures how different two such sequences are. The theorem is proved in a machine-checked library of formal theorems, meaning that a computer program has verified every step of the proof. This provides a high degree of confidence in the result, as it rules out the possibility of human error in the reasoning.

The theorem does not claim that PRCJCostDistance is the only possible distance function, nor does it claim that the construction of real numbers from ledgers is complete. It only establishes the continuity property for this specific distance function. Other properties, such as the completeness of the resulting real number system, are proved separately in other theorems within the same library. The theorem also does not make any claims about the physical world; it is purely a mathematical statement about a formal construction.

In plain terms, the theorem guarantees that the distance function behaves well: it does not have any sudden jumps or discontinuities. This is a necessary condition for the distance function to be useful in defining limits and convergence, which are essential for building a complete number system. The proof of this theorem is a small but important step in the larger project of showing that the framework's construction of real numbers is mathematically sound.

theorem PRCJCostDistanceThreeLegModulusTarget_proved :
    PRCJCostDistanceThreeLegModulusTarget := by
  intro eps heps
  rcases PRCJCostDistanceTriangleModulusTarget_proved eps heps with
    ⟨eta, heta_pos, heta_tri⟩
  rcases PRCJCostDistanceTriangleModulusTarget_proved eta heta_pos with
    ⟨theta, htheta_pos, htheta_tri⟩
  rcases PRCUnitFraction_eventually_lt heta_pos with ⟨Neta, hNeta⟩
  rcases PRCUnitFraction_eventually_lt htheta_pos with ⟨Ntheta, hNtheta⟩
  let delta := PRCUnitFraction (max Neta Ntheta)
  have hdelta_pos : PRCRat.positive delta := PRCUnitFraction_positive _
  have hdelta_lt_eta : PRCRat.lt delta eta := by
    exact hNeta (max Neta Ntheta) (Nat.le_max_left Neta Ntheta)
  have hdelta_lt_theta : PRCRat.lt delta theta := by
    exact hNtheta (max Neta Ntheta) (Nat.le_max_right Neta Ntheta)
  refine ⟨delta, hdelta_pos, ?_⟩
  intro a b c d hab hbc hcd
  have hab_eta : PRCRat.lt (PRCJCostDistance a b) eta :=
    PRCRat.lt_trans hab hdelta_lt_eta
  have hbc_theta : PRCRat.lt (PRCJCostDistance b c) theta :=
    PRCRat.lt_trans hbc hdelta_lt_theta
  have hcd_theta : PRCRat.lt (PRCJCostDistance c d) theta :=
    PRCRat.lt_trans hcd hdelta_lt_theta
  have hbd_eta : PRCRat.lt (PRCJCostDistance b d) eta :=
    htheta_tri b c d hbc_theta hcd_theta
  exact heta_tri a b d hab_eta hbd_eta

theorem PRCJCostDistanceThreeLegModulusTarget_proved :
    PRCJCostDistanceThreeLegModulusTarget := by
  intro eps heps
  rcases PRCJCostDistanceTriangleModulusTarget_proved eps heps with
    ⟨eta, heta_pos, heta_tri⟩
  rcases PRCJCostDistanceTriangleModulusTarget_proved eta heta_pos with
    ⟨theta, htheta_pos, htheta_tri⟩
  rcases PRCUnitFraction_eventually_lt heta_pos with ⟨Neta, hNeta⟩
  rcases PRCUnitFraction_eventually_lt htheta_pos with ⟨Ntheta, hNtheta⟩
  let delta := PRCUnitFraction (max Neta Ntheta)
  have hdelta_pos : PRCRat.positive delta := PRCUnitFraction_positive _
  have hdelta_lt_eta : PRCRat.lt delta eta := by
    exact hNeta (max Neta Ntheta) (Nat.le_max_left Neta Ntheta)
  have hdelta_lt_theta : PRCRat.lt delta theta := by
    exact hNtheta (max Neta Ntheta) (Nat.le_max_right Neta Ntheta)
  refine ⟨delta, hdelta_pos, ?_⟩
  intro a b c d hab hbc hcd
  have hab_eta : PRCRat.lt (PRCJCostDistance a b) eta :=
    PRCRat.lt_trans hab hdelta_lt_eta
  have hbc_theta : PRCRat.lt (PRCJCostDistance b c) theta :=
    PRCRat.lt_trans hbc hdelta_lt_theta
  have hcd_theta : PRCRat.lt (PRCJCostDistance c d) theta :=
    PRCRat.lt_trans hcd hdelta_lt_theta
  have hbd_eta : PRCRat.lt (PRCJCostDistance b d) eta :=
    htheta_tri b c d hbc_theta hcd_theta
  exact heta_tri a b d hab_eta hbd_eta

What this page does not claim

The theorem does not claim that PRCJCostDistance is the only possible distance function. The theorem does not claim that the construction of real numbers from ledgers is complete. The theorem does not make any claims about the physical world.

Verify this page

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$ lake env lean IndisputableMonolith/Foundation/PrimitiveRecognitionCalculus/RealCompleteness.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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