Foundation Primitive Recognition Calculus Grow Ratio Orbit Order Add Mono Le Iff

The order bridge

A ledger, a discrete record of events, must compare its entries: which came first, which is larger, which is later in the growth sequence. In the Recognition Science framework, entries are signed orbits, objects that carry both a positive and a negative count. The lemma le_iff_toInt_le_cf states a simple fact: one signed orbit is at most another in the framework's order exactly when its integer translation, its positive count minus its negative count, is at most the other's. That is, the order on signed orbits and the usual order on integers agree perfectly.

The proof routes through a companion lemma on natural numbers, le_iff_toNat_cf, which compares the positive and negative counts directly. The integer version then unfolds each signed orbit into the difference of its two counts and applies a routine arithmetic check. The declaration is private, meaning it is an internal bridge for later proofs, not a public theorem. Its role is to give the framework a clean, choice-free way to translate order statements into integer arithmetic, avoiding a separate path that would rely on the axiom of choice.

What the lemma does not claim is equally precise. It does not assert that the order itself is total, that every pair of signed orbits is comparable; it only says that when the order holds, the integer comparison follows, and conversely. It does not say that the integer translation is injective or that it preserves addition; it only bridges the order. And it says nothing about the ratio order, the order on growth ratios that appears in a separate theorem about translation invariance. That theorem, leQ_add_right, is a different statement: adding the same ratio to both sides of a ratio order preserves the order.

In plain terms, the lemma is a bookkeeping convenience. The framework's library of machine-checked formal theorems uses it to keep comparisons honest and free of choice assumptions. A reader who wants to know whether the order on signed orbits matches ordinary integer comparison now has a direct answer: yes, exactly, and the proof is fully constructive.

/-- Choice-free Int order bridge, routed through the CF Nat bridge `le_iff_toNat_cf`
    (never through the choice-tainted `SignedOrbit.le_iff_toInt_le`). -/
private lemma le_iff_toInt_le_cf (a b : SignedOrbit) :
    SignedOrbit.le a b ↔ a.toInt ≤ b.toInt := by
  rw [le_iff_toNat_cf, toInt_eq a, toInt_eq b]
  constructor
  · intro hh; omega
  · intro hh; omega

/-- Choice-free Int order bridge, routed through the CF Nat bridge `le_iff_toNat_cf`
    (never through the choice-tainted `SignedOrbit.le_iff_toInt_le`). -/
private lemma le_iff_toInt_le_cf (a b : SignedOrbit) :
    SignedOrbit.le a b ↔ a.toInt ≤ b.toInt := by
  rw [le_iff_toNat_cf, toInt_eq a, toInt_eq b]
  constructor
  · intro hh; omega
  · intro hh; omega

/-- Choice-free Int order bridge, routed through the CF Nat bridge `le_iff_toNat_cf`
    (never through the choice-tainted `SignedOrbit.le_iff_toInt_le`). -/
private lemma le_iff_toInt_le_cf (a b : SignedOrbit) :
    SignedOrbit.le a b ↔ a.toInt ≤ b.toInt := by
  rw [le_iff_toNat_cf, toInt_eq a, toInt_eq b]
  constructor
  · intro hh; omega
  · intro hh; omega

What this page does not claim

The lemma does not assert that the order on signed orbits is total. The integer translation is not claimed to be injective or to preserve addition. The lemma says nothing about the ratio order or its translation invariance theorem.

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

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