Foundation Primitive Recognition Calculus Orbit Arithmetic Add Zero Eq

The zero rule

Addition is one of the first operations a child learns: put three apples with two apples, and you have five apples. The rule that adding zero changes nothing is so obvious that it is rarely stated as a discovery. Yet in a formal system, even this fact must be proved or defined, and the statement add_zero_eq is the Recognition Science library's way of recording it.

The framework builds counting from a primitive notion: a ledger, a discrete record of distinct marks. The type DistinctionNat is its version of the natural numbers, built from a zero mark and a successor operation that adds one more mark. Addition is defined by recursion: adding zero to any position returns that position, and adding a successor to a position means taking the successor of the sum. The theorem add_zero_eq states exactly the first clause: for any position a, a + zero = a. Its proof is immediate, marked as rfl, meaning the two sides are definitionally equal; the statement is true because of how addition was defined, not because of a separate argument.

This theorem is the first of a family. The library proves zero + a = a by induction, and then commutativity, associativity, and cancellation laws follow. A transport theorem shows that the orbit's addition matches ordinary natural number addition when mapped to the standard type, and an injectivity theorem shows that equal displays come from equal positions. The zero rule is the anchor for all of these: it is the base case that makes the recursion work.

What add_zero_eq does not claim is just as important. It does not claim that zero is a number in the ordinary sense, nor that the ledger's marks correspond to physical objects. It does not assert that addition is commutative; that is a separate theorem proved later. It does not say anything about the cost of recognition, the golden ratio, or any of the framework's larger results. It is a narrow, precise statement about one operation on one type, and its value lies in being the foundation on which the rest of the arithmetic is built.

In plain terms, the declaration establishes that in the framework's counting system, adding nothing is a no-op. It is a definitional truth, not an empirical discovery, and it is the first step in showing that the framework's arithmetic behaves like the arithmetic everyone already knows.

theorem add_zero_eq (a : DistinctionNat) :
    a + zero = a := rfl

theorem add_zero_eq (a : DistinctionNat) :
    a + zero = a := rfl

/-- K4.5. Addition of orbit positions: concatenation of repetition. -/
def add : DistinctionNat → DistinctionNat → DistinctionNat
  | a, zero => a
  | a, succ b => succ (add a b)

theorem zero_add_eq (a : DistinctionNat) :
    zero + a = a := by
  induction a with
  | zero => rfl
  | succ n ih =>
      show succ (zero + n) = succ n
      rw [ih]

theorem add_comm (a b : DistinctionNat) :
    a + b = b + a := by
  induction a with
  | zero =>
      rw [zero_add_eq, add_zero_eq]
  | succ n ih =>
      rw [succ_add_eq, add_succ_eq, ih]

theorem add_assoc (a b c : DistinctionNat) :
    (a + b) + c = a + (b + c) := by
  induction c with
  | zero => rfl
  | succ n ih =>
      show (a + b) + succ n = a + (b + succ n)
      rw [add_succ_eq, add_succ_eq, add_succ_eq, ih]

What this page does not claim

The theorem does not claim that zero is a number in the ordinary sense. It does not assert that addition is commutative; that is a separate theorem. It does not say anything about the cost of recognition, the golden ratio, or any larger framework results.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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