Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcreal Ad

Addition and the ledger

The real numbers are usually built from rationals by filling gaps: a real number is an infinite sequence of rationals that gets arbitrarily close to itself. The framework of Recognition Science builds the same object from a different starting point. Its raw material is the ledger, a discrete record of events indexed by natural numbers, where each entry is a rational number. A real number is then an equivalence class of ledgers that settle down: a Cauchy sequence under a particular cost-based distance.

The declaration PRCRealAddCongruenceTarget_proved establishes a property that addition must have for this construction to work. It proves that if two ledgers represent the same real number, and two other ledgers represent the same real number, then adding the first pair of ledgers entry by entry gives a ledger equivalent to adding the second pair. In plain terms: adding real numbers is well-defined. The result of adding two reals does not depend on which ledger you chose to represent them. This is the congruence property for addition, and it is proved in the machine-checked library of formal theorems.

The proof itself is short and rests on a symmetry in the cost distance. The distance function J(x) = (x + 1/x)/2 - 1, which measures recognition cost, satisfies J(c+a, c+b) = J(a,b) and J(-a, -b) = J(a,b). These two identities mean the cost distance is unchanged by translating both arguments or by negating both. The congruence proof uses these to show that adding a fixed ledger to two equivalent ledgers keeps them equivalent, and that negating preserves equivalence. The formal theorem PRCJCostDistance_add_left and PRCJCostDistance_neg_neg are the lemmas that carry this.

What the declaration does not claim is broader. It proves only the congruence property for addition. It does not prove that addition is closed: that adding two Cauchy ledgers always yields another Cauchy ledger. That is a separate target, PRCRealAddClosureTarget, which the pack does not show as proved. Nor does it prove the same congruence for multiplication or order. Those remain separate targets in the library. The complete ordered field structure, with all operations and completeness, is stated as a conditional certificate in the framework, not as a finished theorem.

What the reader can now see is the shape of the framework's construction: it is not a shortcut past the classical theory of real numbers, but a re-derivation of it from a cost-based ledger, one operation at a time. Addition is the first operation to pass the well-definedness test. The framework proves this one step, and the remaining steps are targets, not achievements.

/-- Addition respects null equivalence. -/
theorem PRCRealAddCongruenceTarget_proved :
    PRCRealAddCongruenceTarget := by
  intro u u' v v' huu hvv eps heps
  rcases PRCJCostDistanceTriangleModulusTarget_proved eps heps with
    ⟨delta, hdelta_pos, hdelta⟩
  rcases huu delta hdelta_pos with ⟨Nu, hNu⟩
  rcases hvv delta hdelta_pos with ⟨Nv, hNv⟩
  refine ⟨max Nu Nv, ?_⟩
  intro n hn
  have hnu : Nu ≤ n := le_trans (Nat.le_max_left Nu Nv) hn
  have hnv : Nv ≤ n := le_trans (Nat.le_max_right Nu Nv) hn
  exact hdelta
    ((u.term n) + (v.term n))
    ((u'.term n) + (v.term n))
    ((u'.term n) + (v'.term n))
    (by
      rw [PRCJCostDistance_add_right]
      exact hNu n hnu)
    (by
      rw [PRCJCostDistance_add_left]
      exact hNv n hnv)

/-- J-cost distance is invariant under translating both endpoints on the left. -/
theorem PRCJCostDistance_add_left (a b c : PRCRat) :
    PRCJCostDistance (c + a) (c + b) = PRCJCostDistance a b := by
  apply PRCRat.toRat_injective
  rw [PRCJCostDistance_toRat, PRCJCostDistance_toRat]
  simp [PRCJCostDistanceRatDisplay]

/-- 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

The declaration does not prove that addition is closed on Cauchy ledgers. The declaration does not prove congruence for multiplication or order. The declaration does not establish the full complete ordered field structure.

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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