Delta Kernel Syntax Step Succ

The successor step

In formal logic, a formula like "x is even" contains a variable that can be replaced by a specific number. The declaration stepSucc is a syntactic operation that performs exactly one such replacement: it takes a formula and substitutes the successor function S for its first free variable. If the formula reads "x is even", applying stepSucc produces "S(x) is even", meaning "the next number after x is even". The operation leaves all other free variables untouched, and it shifts the bound variables correctly to avoid capture. This is a purely mechanical, structural transformation on the formula's syntax.

The operation belongs to the δ-kernel, a machine-checked library of formal theorems that builds arithmetic from a single primitive distinction: the ability to tell one object from the next. The signature contains only zero, successor, addition, and multiplication. Successor is the primitive step, the raw act of moving from one natural number to the next. stepSucc lifts that primitive step from the level of numbers to the level of formulas: it applies the successor step inside a formula's body, at the position of its bound variable. The definition is two lines long and uses only the standard operations of variable shifting and substitution.

The declaration establishes a precise technical fact about the syntax of the object logic. It defines what it means to advance a formula's induction step by one. It does not prove any arithmetic theorem, does not assign meaning to the successor function, and does not claim that the step corresponds to any physical process. It is a definition, not a result. The framework's larger claims about forcing and recognition cost are built elsewhere, on top of this syntax; stepSucc itself only provides the mechanical operation that later proofs can use.

What stepSucc does not claim is as important as what it does. It does not assert that the successor function is the only possible way to count. It does not prove that induction is valid, or that the natural numbers exist in any metaphysical sense. It does not say that the step from n to n+1 has any particular cost or significance. The declaration is deliberately minimal: it defines a syntactic transformation and nothing more. Any interpretation of what the successor step means, or why it matters, belongs to other parts of the framework, not to this declaration.

/-- The induction-step body: `φ` with its bound variable advanced by one
distinction step, i.e. `φ[x ↦ S x]` for the de Bruijn variable 0,
leaving all other free variables fixed. -/
def stepSucc (φ : DFormula) : DFormula :=
  (φ.lift 1 1).subst 0 (DTerm.succ (DTerm.var 0))

/-- The induction-step body: `φ` with its bound variable advanced by one
distinction step, i.e. `φ[x ↦ S x]` for the de Bruijn variable 0,
leaving all other free variables fixed. -/
def stepSucc (φ : DFormula) : DFormula :=
  (φ.lift 1 1).subst 0 (DTerm.succ (DTerm.var 0))

What this page does not claim

stepSucc does not prove any arithmetic theorem or assign meaning to the successor function. The declaration does not claim that the natural numbers exist or that induction is valid. stepSucc does not assert that counting by successor is the only possible way to enumerate objects.

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

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