Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Tri

The cross-difference identity

In mathematics, a rational number is a fraction of two whole numbers, like 3/4 or 17/5. When working with such fractions, a useful quantity is the cross-difference of two of them: for a/b and c/d, it is the whole number a*d - c*b. This quantity is zero exactly when the two fractions are equal. For instance, 1/2 and 2/4 have cross-difference 1*4 - 2*2 = 0, confirming they represent the same number.

The declaration crossDiff_triangle_id proves a basic algebraic fact about this cross-difference. For any three fractions a, b, and c, the following identity holds: crossDiff(a,c) * den(b) = crossDiff(a,b) * den(c) + crossDiff(b,c) * den(a), where den(x) is the denominator of fraction x. This is a purely computational statement, verified by expanding the definitions and simplifying with ring arithmetic. It is a theorem in the framework's machine-checked library of formal theorems, meaning its proof has been verified step by step by a computer.

In Recognition Science, this identity is part of the technical machinery for building real numbers from sequences of rational numbers. The framework models real numbers not as infinite decimals but as equivalence classes of sequences of fractions that get closer together in a specific, regular way. The cross-difference provides a measure of how close two fractions are, and the triangle identity ensures this measure behaves sensibly: it relates the distance from a to c to the distances from a to b and b to c, weighted by denominators. This is a necessary consistency condition for the whole construction to work.

The identity itself is a narrow algebraic lemma. It does not, by itself, establish that the framework's construction of real numbers is correct or complete. It does not prove that any particular sequence converges, nor does it define what a real number is. Those are separate theorems and definitions in the library. The triangle identity is one small, verified step in a larger chain, ensuring that the cross-difference, a basic tool, satisfies a fundamental algebraic relation.

/-- The three-point cross-difference identity, the algebraic backbone of the
triangle inequality. As rationals `(a-c) = (a-b) + (b-c)`; clearing the three
denominators gives this pure integer identity:
`crossDiff a c * den b = crossDiff a b * den c + crossDiff b c * den a`. -/
theorem crossDiff_triangle_id (a b c : RatioOrbit) :
    crossDiff a c * (b.den.toNat : ℤ) =
      crossDiff a b * (c.den.toNat : ℤ) + crossDiff b c * (a.den.toNat : ℤ) := by
  unfold crossDiff
  ring

/-- The three-point cross-difference identity, the algebraic backbone of the
triangle inequality. As rationals `(a-c) = (a-b) + (b-c)`; clearing the three
denominators gives this pure integer identity:
`crossDiff a c * den b = crossDiff a b * den c + crossDiff b c * den a`. -/
theorem crossDiff_triangle_id (a b c : RatioOrbit) :
    crossDiff a c * (b.den.toNat : ℤ) =
      crossDiff a b * (c.den.toNat : ℤ) + crossDiff b c * (a.den.toNat : ℤ) := by
  unfold crossDiff
  ring

What this page does not claim

The identity does not prove that the framework's construction of real numbers is correct or complete. It does not establish that any particular sequence converges to a real number. It does not define what a real number is within the framework.

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:

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

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