Foundation Primitive Recognition Calculus Grow Signed Orbit Le Congr Right Of Ba

The right-congruence lemma

A signed orbit is, in plain terms, a walk that records each step as a gain or a loss, like a ledger with positive and negative entries. The framework's primitive recognition calculus orders these walks by comparing their cumulative totals. The declaration le_congr_right_of_balanced_cf proves a structural property: if two signed orbits have the same total count when reduced to a canonical form, then they are interchangeable on the right side of the order comparison. Formally, for any signed orbit a and any balanced pair b and b', the statement a.le b holds if and only if a.le b' holds.

This is a congruence lemma: it says that the order relation respects the equivalence of balanced orbits. The proof is short and mechanical, using the fact that the order reduces to a comparison of natural-number totals, and then closing the goal by linear arithmetic. The lemma is a theorem in the machine-checked library of formal theorems, with no unproved assumptions beyond the standard logical axioms.

What the lemma does not claim is just as important. It does not say that balanced orbits are equal as objects, only that they are equivalent for the purpose of the order. It does not extend to the left side: the declaration is specifically about the right argument, and the symmetric statement for the left argument is not part of this theorem. It also does not say anything about the structure of the order beyond this one congruence; it is a local lemma, not a global characterization of the order.

theorem le_congr_right_of_balanced_cf {a b b' : SignedOrbit}
    (h : b.balanced b') :
    a.le b ↔ a.le b' := by
  rw [SignedOrbit.balanced_iff_toNat_eq] at h
  constructor
  · intro hle
    rw [le_iff_toNat_cf] at hle ⊢
    omega
  · intro hle
    rw [le_iff_toNat_cf] at hle ⊢
    omega

What this page does not claim

Balanced orbits are equal as objects, only equivalent for the order. The lemma applies to the left argument of the order. The lemma characterizes the entire order relation.

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

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