Foundation Primitive Recognition Calculus Orbit Euclidean Quotient Mul Divisor T

The division identity

In ordinary arithmetic, division and multiplication undo each other: if 12 divided by 4 is 3, then 3 times 4 is 12 again. The declaration quotient_mul_divisor_toNat_of_divides proves the same round-trip inside the Recognition Science framework, where numbers are represented as distinction counts, discrete records of recognition events. The formal statement says that when a divisor d divides a number n exactly, the quotient of n by d, multiplied by d, equals n. In symbols, (n / d) * d = n, with the condition that d is not zero and that d divides n without remainder.

The proof is short because the framework's quotient is defined by repeated subtraction, not by a built-in division table. The framework first shows that its quotient, when converted to ordinary natural numbers, matches standard division: the quotient's value is n.toNat / d.toNat. Then it uses the classical fact that for natural numbers, if d divides n, then (n / d) * d = n. The framework's theorem simply translates that classical identity into its own number system, where every operation is defined from the ground up as a ledger of discrete steps.

This identity is a workhorse. It appears in the framework's ratio normalization, the process of reducing a fraction to lowest terms. When the framework divides both numerator and denominator by their greatest common divisor, this theorem guarantees that the reduced denominator, multiplied back by the gcd, recovers the original denominator. That is what lets the framework assert that normalization preserves the ratio's value while making the parts coprime.

The theorem does not claim that division always works, only when the divisor divides the dividend exactly. It says nothing about remainders, which the framework handles separately with a companion theorem: quotient times divisor plus remainder equals the original number. It also does not claim that the quotient operation is efficient or that it matches any particular algorithm beyond the repeated-subtraction definition. What it establishes is a precise, machine-checked guarantee: in the framework's arithmetic, exact division and multiplication are inverses.

theorem quotient_mul_divisor_toNat_of_divides {n d : DistinctionNat}
    (hd : d ≠ zero) (hdiv : divides d n) :
    (quotient n d hd).toNat * d.toNat = n.toNat := by
  rw [quotient_toNat]
  exact Nat.div_mul_cancel ((divides_iff_toNat_dvd d n).mp hdiv)

theorem quotient_toNat (n d : DistinctionNat) (hd : d ≠ zero) :
    (quotient n d hd).toNat = n.toNat / d.toNat := by
  have h := divMod_toNat n d hd
  exact h.1

theorem normalizeRatio_den_mul_gcd_toNat (q : RatioOrbit) :
    (normalizeRatio q).den.toNat *
      (gcd q.num.abs q.den).toNat = q.den.toNat := by
  unfold normalizeRatio
  exact quotient_mul_divisor_toNat_of_divides
    (n := q.den) (d := gcd q.num.abs q.den)
    (gcd_ne_zero_of_right_ne_zero q.num.abs q.den q.den_ne_zero)
    (gcd_divides_right q.num.abs q.den)

What this page does not claim

The theorem does not claim that division works without the exact-divisibility condition. It does not claim anything about the efficiency or computational cost of the repeated-subtraction division algorithm. It does not claim that the framework's quotient operation is defined for zero divisors.

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

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