Foundation Primitive Recognition Calculus Integer Order Mul Recip Cancel Right A

Cancellation in the signed orbit

In ordinary arithmetic, multiplying a number by its reciprocal gives 1, and multiplying by 1 changes nothing. The declaration mul_recip_cancel_right_assoc_crossEq_self_of_right_not_crossEq_zero states a precise analogue inside Recognition Science's discrete arithmetic of signed orbits (a signed orbit is a signed integer with a sign flag, used to record recognition events). It says: if a right operand is not cross-equal to zero, then multiplying a product by that operand's reciprocal, on the right, yields a result cross-equal to the original left operand.

In symbols, for signed orbits a and b, if b is not cross-equal to zero, then (a * b) * recip(b) is cross-equal to a. Here cross-equal means the two orbits represent the same signed integer value, ignoring any difference in sign flags. The condition that b is not cross-equal to zero is essential: if b were zero, its reciprocal would be undefined in this system, and the cancellation would fail.

The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning its statement is verified by a computer proof checker. It is a structural law about the arithmetic of signed orbits, not a claim about physical measurements or empirical data. It establishes that, under the stated nonzero condition, right multiplication by a reciprocal acts as a right inverse for multiplication, up to cross-equality.

What the declaration does not claim is broader. It does not claim that cancellation works without the nonzero condition, nor that it works for left multiplication in the same way. It does not assert anything about the sign flags of the resulting orbit, only its cross-equality. It says nothing about the meaning of recognition events or the physical interpretation of signed orbits; it is a purely formal algebraic statement within the framework's calculus.

theorem cmp_add_right (a b c : SignedOrbit) :
    SignedOrbit.cmp (SignedOrbit.add a c) (SignedOrbit.add b c) =
      SignedOrbit.cmp a b := by
  cases hcmp : SignedOrbit.cmp a b with
  | lt =>
      have hlt : SignedOrbit.lt a b :=
        (SignedOrbit.cmp_eq_lt_iff a b).mp hcmp
      exact SignedOrbit.cmp_eq_lt_of_lt
        ((SignedOrbit.lt_add_right_iff a b c).mpr hlt)
  | eq =>
      have hbal : SignedOrbit.balanced a b :=
        (SignedOrbit.cmp_eq_eq_iff a b).mp hcmp
      exact SignedOrbit.cmp_eq_eq_of_balanced
        ((SignedOrbit.balanced_add_right_iff a b c).mpr hbal)
  | gt =>
      have hgt : SignedOrbit.lt b a :=
        (SignedOrbit.cmp_eq_gt_iff a b).mp hcmp
      exact SignedOrbit.cmp_eq_gt_of_gt
        ((SignedOrbit.lt_add_right_iff b a c).mpr hgt)

What this page does not claim

The declaration does not claim that cancellation holds without the nonzero condition. It does not claim that left multiplication by a reciprocal cancels in the same way. It does not make any claim about the physical meaning of recognition events.

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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