Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcreal Ne

The negation congruence

A real number is often built as a limit of rational numbers. In the Recognition Science framework, the raw material is a ledger, a discrete record of events, here an infinite sequence of rational values. Two such sequences are considered equivalent when they get arbitrarily close to each other in a special distance called the recognition cost, which measures how hard it is to tell the two sequences apart. The declaration PRCRealNegCongruenceTarget_proved establishes that if two sequences are equivalent in this sense, then their pointwise negations are also equivalent. In plain words: negating every entry in each sequence preserves the property of being indistinguishable at the level of the cost distance.

The proof is a short argument that relies on a symmetry of the cost distance. The theorem PRCJCostDistance_neg_neg shows that the distance between two negated values equals the distance between the original values. The proof of the congruence target applies this fact at each index of the sequences, so the same bound that shows the original sequences are close also shows their negations are close. This is a formal result in the framework's machine-checked library of formal theorems, meaning the logic has been verified by a computer.

This congruence property is a necessary technical step for constructing a complete ordered field from the ledgers. It ensures that the operation of negation is well-defined on equivalence classes of sequences, which is part of building the real numbers inside the framework. The declaration is one of several closure and congruence targets that together support the construction. It does not, by itself, prove that the real numbers exist or that the full field structure is complete; those are separate targets that rely on additional assumptions.

/-- Negation respects null equivalence. -/
theorem PRCRealNegCongruenceTarget_proved :
    PRCRealNegCongruenceTarget := by
  intro u v huv eps heps
  rcases huv eps heps with ⟨N, hN⟩
  refine ⟨N, ?_⟩
  intro n hn
  change PRCRat.lt (PRCJCostDistance (-(u.term n)) (-(v.term n))) eps
  rw [PRCJCostDistance_neg_neg]
  exact hN n hn

/-- J-cost distance is invariant under negating both endpoints. -/
theorem PRCJCostDistance_neg_neg (a b : PRCRat) :
    PRCJCostDistance (-a) (-b) = PRCJCostDistance a b := by
  apply PRCRat.toRat_injective
  rw [PRCJCostDistance_toRat, PRCJCostDistance_toRat]
  simp [PRCJCostDistanceRatDisplay]
  ring_nf

What this page does not claim

This proof does not establish that the real numbers exist as a complete ordered field within the framework. This proof does not show that negation is the only operation that preserves equivalence. This proof does not claim that the cost distance itself is a metric in the usual sense.

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

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