Foundation Primitive Recognition Calculus Grow Signed Orbit Le Mul Right Iff Of

When multiplying preserves order

An ordering inequality, like 2 ≤ 3, stays true when you multiply both sides by a positive number: 2×5 ≤ 3×5. The framework's ledger, a discrete record of recognition events, builds a number system where the same question is subtle. Its elements, called signed orbits, can be multiplied, and one asks whether multiplying both sides of an order comparison by the same element always preserves the comparison. The answer is not automatic, because the ledger's arithmetic is not the familiar arithmetic of integers.

The framework proves a precise condition. If the multiplier is nonnegative (its nonnegFlag is true) and is not balanced against zero, then multiplying preserves order: for any two elements z and w, z ≤ w holds exactly when z times the multiplier is ≤ w times the multiplier. The proof is machine-checked in the framework's library of formal theorems. The condition matters: the multiplier must be genuinely positive in the ledger's sense, not merely nonzero, and it must not be balanced against zero, a special degenerate case the framework excludes explicitly.

The theorem is a structural lemma, not a statement about any particular numeric values. It says that the ledger's order and multiplication interact cleanly for a whole class of multipliers. A reader should not take it as a claim that every nonzero element preserves order, nor as a statement about ordinary integer arithmetic. The framework's signed orbits are a different object, and the lemma is one step in building a larger theory of growth and comparison in the ledger.

theorem le_mul_right_iff_of_nonnegFlag_of_not_balanced_zero_cf :
    ∀ (a z w : SignedOrbit), a.nonnegFlag = true →
    ¬ a.balanced SignedOrbit.zero →
    ((z.mul a).le (w.mul a) ↔ z.le w) := by
  intro a z w hanonneg ha
  have han : a.neg.toNat ≤ a.pos.toNat := by
    have h := hanonneg
    unfold SignedOrbit.nonnegFlag at h
    rwa [leq_eq_true_iff_cf] at h
  have hlt : a.neg.toNat < a.pos.toNat := by
    rcases Nat.lt_or_ge a.neg.toNat a.pos.toNat with h | h
    · exact h
    · exfalso; apply ha
      rw [SignedOrbit.balanced_iff_toNat_eq]
      rw [show SignedOrbit.zero.neg.toNat = 0 from rfl,
          show SignedOrbit.zero.pos.toNat = 0 from rfl]
      omega
  rw [le_iff_toNat_cf, le_iff_toNat_cf]
  have eqL : (w.mul a).neg.toNat + (z.mul a).pos.toNat =
      a.neg.toNat * (w.pos.toNat + z.neg.toNat) + a.pos.toNat * (w.neg.toNat + z.pos.toNat) := by
    simp only [SignedOrbit.mul_pos, SignedOrbit.mul_neg,
              DistinctionNat.toNat_add, DistinctionNat.toNat_mul]
    ring
  have eqR : (w.mul a).pos.toNat + (z.mul a).neg.toNat =
      a.pos.toNat * (w.pos.toNat + z.neg.toNat) + a.neg.toNat * (w.neg.toNat + z.pos.toNat) := by
    simp only [SignedOrbit.mul_pos, SignedOrbit.mul_neg,
              DistinctionNat.toNat_add, DistinctionNat.toNat_mul]
    ring
  rw [eqL, eqR]
  exact cross_le_iff (w.neg.toNat + z.pos.toNat) (w.pos.toNat + z.neg.toNat)
      a.pos.toNat a.neg.toNat hlt

What this page does not claim

The theorem does not claim that every nonzero multiplier preserves order. The theorem does not apply to ordinary integer arithmetic. The theorem does not claim anything about the values of specific signed orbits.

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

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