Foundation Primitive Recognition Calculus Orbit Arithmetic Succ Add Eq

What succ_add_eq proves

In ordinary arithmetic, adding one to a number and then adding another number gives the same result as adding the two numbers first and then adding one. For example, (1 + 2) + 1 equals 1 + (2 + 1); both are 4. The declaration succ_add_eq proves the same rule inside a formal system that models counting as a sequence of distinct recognition events, a discrete record of events called a ledger. The theorem states that for any two counts a and b, the expression (a + 1) + b equals (a + b) + 1.

The proof works by induction on b: it checks the base case where b is zero, then shows that if the rule holds for b, it holds for b + 1. The result is one of several basic properties of addition on this counting structure, alongside commutativity (a + b = b + a) and associativity ((a + b) + c = a + (b + c)). These properties are proved in a machine-checked library of formal theorems, meaning the reasoning is verified step by step by a computer.

What the theorem does not claim is anything about the physical world. It does not say that recognition events actually occur in a particular order, nor that the ledger corresponds to any specific experimental observation. It is a purely formal statement about a mathematical structure. The theorem also does not establish that addition is commutative; that is a separate theorem. Finally, it does not claim that the counting structure is the only possible one; it simply proves a property of the structure as defined.

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

theorem succ_add_eq (a b : DistinctionNat) :
    succ a + b = succ (a + b) := by
  induction b with
  | zero => rfl
  | succ n ih =>
      show succ (succ a + n) = succ (succ (a + 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 recognition events occur in any particular physical order. The theorem does not claim that addition is commutative; that is a separate theorem. The theorem does not claim that the counting structure is the only possible one.

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)

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