Delta Kernel Semantics Sat Step Succ

The successor step

In formal logic, a de Bruijn index is a way of naming variables by how many binders stand between the variable and its binder, rather than by a name. The theorem sat_stepSucc concerns a formula whose free variable has been shifted one step outward, so that it points past one more binder. The theorem states that this shifted formula is satisfied in an environment where the variable holds the value n exactly when the original formula is satisfied in an environment where the variable holds the value n + 1. In plain words: shifting a variable's index by one is the same as adding one to the value it receives.

This is a recognition event, a discrete record of a formal step, in the framework's kernel. The kernel is the machine-checked library of formal theorems that the Recognition Science framework builds on. The theorem is proved constructively, meaning it does not rely on the law of excluded middle or any choice principle. It is one of the lemmas that the framework's soundness proof needs when it handles the induction rule, the rule that lets a proof step from a base case to the next case.

The theorem itself is narrow. It does not say anything about what the formula means, only how its satisfaction changes when a variable's index and value both shift by one. It is a plumbing fact, the kind of detail that a soundness proof must check so that the whole system can be trusted. Without it, the framework could not be sure that its rules for quantifiers and substitution behave correctly under the simplest kind of variable renaming.

What the theorem does not claim is broader. It does not establish that the framework's soundness theorem holds, nor that the kernel is consistent. It does not say anything about the physical world, about costs, about the golden ratio, or about the forcing chain. It is a lemma, not a conclusion. Its role is to be one of the many small, verified steps that a larger proof can stand on.

/-- The induction-step body means exactly "φ one distinction step up". -/
theorem sat_stepSucc (φ : DFormula) (n : Nat) (ρ : Env) :
    sat (Env.cons n ρ) φ.stepSucc ↔ sat (Env.cons (n + 1) ρ) φ := by
  unfold stepSucc
  rw [sat_subst0]
  have e : (DTerm.succ (DTerm.var 0)).eval (Env.cons n ρ) = n + 1 := rfl
  rw [e, sat_lift]
  refine sat_ext φ (fun m => ?_)
  cases m with
  | zero => simp [liftVar]
  | succ j =>
      have h' : ¬ (j + 1 < 1) := by omega
      have e2 : j + 1 + 1 = (j + 1) + 1 := rfl
      simp [liftVar, h']

/-- The induction-step body means exactly "φ one distinction step up". -/
theorem sat_stepSucc (φ : DFormula) (n : Nat) (ρ : Env) :
    sat (Env.cons n ρ) φ.stepSucc ↔ sat (Env.cons (n + 1) ρ) φ := by
  unfold stepSucc
  rw [sat_subst0]
  have e : (DTerm.succ (DTerm.var 0)).eval (Env.cons n ρ) = n + 1 := rfl
  rw [e, sat_lift]
  refine sat_ext φ (fun m => ?_)
  cases m with
  | zero => simp [liftVar]
  | succ j =>
      have h' : ¬ (j + 1 < 1) := by omega
      have e2 : j + 1 + 1 = (j + 1) + 1 := rfl
      simp [liftVar, h']

What this page does not claim

The theorem does not prove the soundness theorem for the framework's kernel. The theorem does not say anything about the physical world or the forcing chain. The theorem does not establish the consistency of the kernel.

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/Semantics.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: