Delta Kernel Sound Mem Of Get Elem

A small structural fact

In the ordinary mathematics of lists, a simple fact holds: if you look up an index in a list and get a value, that value is actually in the list. The declaration mem_of_getElem is the formal statement of this fact, proved by structural recursion. It is a small, unglamorous lemma, the kind that appears in the background of larger proofs.

Its importance here is not the fact itself, but what it represents. This lemma is part of the ledger, a discrete record of every derivation the checker accepts. The kernel, the core that verifies each proof step, certifies itself: every derivation it accepts is true in its canonical model. The ledger records which metatheoretic principles a proof actually used, such as excluded middle or Markov's principle.

In Recognition Science, this lemma is a building block in the proof of conditional soundness. The central theorem sound_cond states that if the checker accepts a derivation and the ledger posts the principles it used, then the conclusion holds in the canonical model. The lemma helps establish that a context hypothesis, an assumption drawn from the list of hypotheses, is indeed satisfied.

What mem_of_getElem does not claim is anything about the forcing chain, the cost function, or the golden ratio. It is not a theorem about physics or the structure of space. It is a structural fact about lists, proved constructively, that supports the kernel's self-certification. Its significance is that it is choice-free, meaning its proof does not rely on classical excluded middle, so the forced fragment of the framework stays maximally trustworthy.

private theorem mem_of_getElem? {α : Type _} :
    ∀ {l : List α} {i : Nat} {a : α}, l[i]? = some a → a ∈ l
  | [], _, _, h => by simp at h
  | x :: xs, 0, a, h => by
      simp only [List.getElem?_cons_zero, Option.some.injEq] at h
      subst h
      exact List.mem_cons_self ..
  | _ :: xs, i + 1, a, h => by
      simp only [List.getElem?_cons_succ] at h
      exact List.mem_cons_of_mem _ (mem_of_getElem? h)

What this page does not claim

This lemma establishes nothing about the forcing chain, the cost function, or the golden ratio. It does not claim that any derivation is true without the principles its ledger posts.

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/Sound.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.

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