Foundation Primitive Recognition Calculus Integer Order Recip Num Mul Num Balanc

The reciprocal numerator balance

An ordinary fraction like 3/4 has a reciprocal, 4/3. The numerator of the reciprocal is the denominator of the original, and the denominator of the reciprocal is the numerator of the original. The declaration recip_num_mul_num_balanced_ofOrbit_den_mul_abs_of_not_balanced_zero is a machine-checked theorem that states a version of this fact for the framework's own arithmetic objects, called ratio orbits (a ratio orbit is a pair of signed integers that represent a fraction, with equality up to a common scale).

In plain language, the theorem says: take a ratio orbit a. If its numerator is not balanced with zero (meaning the numerator is not zero), then the absolute value of the numerator of the reciprocal of a is equal to the denominator of a. The proof relies on a companion fact: under the same condition, the denominator of the reciprocal of a is equal to the absolute value of the numerator of a. Together, these two statements are exactly the familiar swap of numerator and denominator that holds for ordinary fractions.

The theorem is not about any particular number. It is a general law about the framework's arithmetic: it holds for every ratio orbit whose numerator is not zero. The condition matters. If the numerator is zero, the reciprocal is not defined in the usual way, and the theorem does not apply. The declaration also does not claim anything about the sign of the numbers involved; it only concerns their absolute values, which are always nonnegative.

This result is a small but necessary piece of the framework's foundation. It ensures that the operation of taking a reciprocal behaves consistently with the framework's notion of equality, which is based on balance rather than on the usual equality of integers. Without this theorem, the framework could not cleanly reason about fractions and their reciprocals, which are central to its later developments involving ratios and scaling.

theorem recipNonzero_num_abs_eq_den (a : RatioOrbit)
    (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero) :
    (RatioOrbit.recipNonzero a h).num.abs = a.den := by
  by_cases hnonneg : a.num.nonnegFlag = true
  · rw [RatioOrbit.recipNonzero_num_eq_of_nonnegFlag (a := a) (h := h) hnonneg,
      SignedOrbit.abs_ofOrbit]
  · have hneg : a.num.negativeFlag = true := by
      rw [SignedOrbit.negativeFlag_eq_true_iff_nonnegFlag_eq_false]
      cases hflag : a.num.nonnegFlag with
      | false => rfl
      | true =>
          exfalso
          exact hnonneg hflag
    rw [RatioOrbit.recipNonzero_num_eq_of_negativeFlag (a := a) (h := h) hneg,
      SignedOrbit.abs_negate_ofOrbit]

theorem recipNonzero_den_eq_abs (a : RatioOrbit)
    (h : ¬ SignedOrbit.balanced a.num SignedOrbit.zero) :
    (RatioOrbit.recipNonzero a h).den = a.num.abs := by
  rfl

What this page does not claim

The theorem does not claim that the reciprocal of a ratio orbit with zero numerator is defined or behaves in any particular way. The theorem does not claim anything about the sign of the numerator or denominator, only their absolute values. The theorem does not establish any property about the reciprocal of a ratio orbit when the numerator is balanced with zero.

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

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