Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Mul Nonneg Le Q

Ordered multiplication

In mathematics, an ordered set is a collection where any two elements can be compared, and a monotone operation is one that respects that comparison: if a is at most b, then applying the operation to both keeps the inequality. The theorem leQ_mul_nonneg_right (the name is machine shorthand) states a precise monotonicity fact for multiplication on the right in a specific ordered structure called the ratio orbit. Plainly: if r is nonnegative relative to the orbit's zero, and p is at most q, then p multiplied by r is at most q multiplied by r.

The ratio orbit is a discrete ledger, a record of growth events the framework uses to model recognition costs. Its elements are ratios, and its order compares them. The theorem is proved in the machine-checked library of formal theorems: it is a THEOREM, meaning the proof is verified by a computer kernel with no unproved assumptions beyond the standard logical axioms. The declaration is one line in a larger chain that builds ordered arithmetic from primitive recognition events, and it matters because it licenses replacing a ratio by a larger one inside a multiplication without breaking the order, a step later growth arguments rely on.

What the theorem does not claim is equally precise. It does not assert that multiplication by a negative r preserves order; the premise r nonnegative is essential. It does not claim the analogous left-multiplication fact, which would require a separate proof. It does not say anything about the magnitude of the difference, only that the inequality persists. And it does not establish that the ratio orbit's order is total, that every pair of elements is comparable, a stronger property the framework treats separately.

theorem leQ_mul_nonneg_right : ∀ p q r, leQ RatioOrbit.zero r → leQ p q → leQ (RatioOrbit.mul p r) (RatioOrbit.mul q r) := by
  intro p q r hr0 hpq
  unfold leQ at hr0 hpq ⊢
  unfold RatioOrbit.mul at ⊢
  rw [SignedOrbit.le_iff_toInt_le] at hr0 hpq ⊢
  simp only [SignedOrbit.mul_toInt, SignedOrbit.ofOrbit_toInt, DistinctionNat.toNat_mul] at hr0 hpq ⊢
  have h0num : RatioOrbit.zero.num.toInt = 0 := by
    unfold RatioOrbit.zero
    simp only [SignedOrbit.zero_toInt]
  have h0den : RatioOrbit.zero.den.toNat = 1 := by
    unfold RatioOrbit.zero
    rfl
  rw [h0num, h0den] at hr0
  push_cast at hr0 hpq ⊢
  have h_rn : 0 ≤ r.num.toInt := by omega
  have h_rd : 0 ≤ (r.den.toNat : ℤ) := by omega
  have h_coeff : 0 ≤ r.num.toInt * (r.den.toNat : ℤ) := Int.mul_nonneg h_rn h_rd
  have h_diff : 0 ≤ q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ) := by omega
  have h_prod : 0 ≤ r.num.toInt * (r.den.toNat : ℤ) * (q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ)) :=
    Int.mul_nonneg h_coeff h_diff
  have h_identity : q.num.toInt * r.num.toInt * ((p.den.toNat : ℤ) * (r.den.toNat : ℤ))
                    - p.num.toInt * r.num.toInt * ((q.den.toNat : ℤ) * (r.den.toNat : ℤ))
      = r.num.toInt * (r.den.toNat : ℤ) * (q.num.toInt * (p.den.toNat : ℤ) - p.num.toInt * (q.den.toNat : ℤ)) := by
    ring
  rw [← h_identity] at h_prod
  omega

What this page does not claim

Negative r preserves the inequality. Left multiplication is monotone. The ratio orbit's order is total.

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/RatioOrbitOrderMulNonneg.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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