Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Of

The cross-difference lemma

In the framework's recognition calculus, a discrete record of events and their costs, rational numbers appear as ratios of integers. The declaration crossDiff_of_crossEq states a simple algebraic fact: if two such ratios are equal, then a certain integer expression built from their numerators and denominators is zero. The expression, called the cross-difference, is a.num * b.den - b.num * a.den for ratios a and b. The lemma proves that equality of ratios implies the cross-difference vanishes.

This is not a deep theorem; it is a computational convenience. The framework's library of machine-checked formal theorems uses the cross-difference to avoid working with rational numbers directly, since integer arithmetic is simpler to verify. The lemma itself is proved by rewriting the definition of ratio equality and simplifying with the omega tactic, which handles linear integer arithmetic. It is a stepping stone, not a destination.

What the lemma does not claim is important. It does not say that a zero cross-difference implies the ratios are equal; that converse is a separate statement, and the library does not derive it from this lemma. It also does not establish anything about the cost function, the framework's measure of recognition effort, or about the physical constants that the framework derives. The lemma is purely about rational arithmetic, and its role is to support the construction of a completion of the rationals, not to prove any physical law.

/-- Cross-equal rationals have zero cross-difference. -/
theorem crossDiff_of_crossEq (a b : RatioOrbit) (h : RatioOrbit.crossEq a b) :
    crossDiff a b = 0 := by
  rw [RatioOrbit.crossEq_iff_toIntCross] at h
  unfold crossDiff
  omega

What this page does not claim

The lemma does not prove the converse, that a zero cross-difference implies the ratios are equal. The lemma does not establish any property of the cost function or the framework's physical constants. The lemma does not itself construct the completion of the rationals; it only supports that construction.

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/EtaCompletionM0a.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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