Foundation Primitive Recognition Calculus Orbit Arithmetic Add Succ Eq

The successor step

The declaration add_succ_eq is a theorem about a formal system of counting positions. It states that for any two positions a and b, adding a to the successor of b equals the successor of adding a to b. In symbols: a + succ b = succ (a + b). This is the recursive clause that defines addition for the system's natural numbers, mirroring the standard Peano arithmetic definition. The theorem is proved by reflexivity, meaning it holds directly from the definition of the add function.

The system in question is the δ-orbit, a discrete sequence of positions generated by repeated application of a successor operation. Within the Recognition Science framework, this orbit models the primitive act of distinguishing one event from the next. The declaration is part of a larger machine-checked library of formal theorems that builds arithmetic on this basis. Its role is foundational: it ensures that the addition operation behaves as expected when the second argument is a successor, which is the step that allows induction proofs to proceed.

The theorem does not claim that this definition of addition is the only possible one, nor does it assert anything about the physical or metaphysical nature of the orbit. It is a purely formal statement about a recursively defined operation. The framework's broader claims about recognition and cost are not established by this declaration alone; they depend on a chain of further theorems. The declaration is a building block, not a conclusion.

theorem add_succ_eq (a b : DistinctionNat) :
    a + succ b = succ (a + b) := rfl

theorem add_succ_eq (a b : DistinctionNat) :
    a + succ b = succ (a + b) := rfl

What this page does not claim

This declaration does not prove the commutativity or associativity of addition. It does not establish any property of the δ-orbit beyond the formal behavior of the successor operation. It does not provide a physical interpretation of the orbit positions.

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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