Research proof index

Generalized free-ization. For any β : _root_.LO.FirstOrder.ArithmeticSemiformula ℕ m, substituting the free-variable atoms &0 … &(m-1) for its m bound slots equals ⌜β ⇜ (&·)⌝. This is the forward recognizer's tool: once IsSemiformula.sound yields a β with ⌜β⌝ = b, this computes subst (fvarVec m) b. (Specializes to `subst_fvarVec_quot...

termBShift shifts the bound-variable depth up by exactly one (on well-formed terms): so t is a level-m term iff termBShift t is level-(m+1). The ←-direction recovers the lowered arity, which is how the bounded-∀ bound (a termBShift-image) is recognized as a bShift of a real term of the outer arity.

mem_iff math (C = Hierarchy 𝚺 1). Mirrors chUniv_mem_iff, threading the IsSigma1 K side condition through isSigma1_iff_hierarchy.

Closure inversion (forward keystone). A freevar-free level-m formula β whose internal bv is m and which substitutes back to succInd γ is exactly the fixitr-image, so its m-fold closure is (succInd γ).univCl'. Mirror of bv_quote_fixitr's ≥-direction inversion; the genuine remaining math.

(⟹) A 𝚺₁-recognized code is the code of a 𝚺₁ formula. Meta-induction on the formula: atoms are 𝚺₁ unconditionally; ∧/∨/∃ recurse; the ^∀ case is forced into the bounded shape by the recognizer (IsSigma1.of_all), and the bound is a bShift-image (positivity via termBV_termBShift_le), so Hierarchy.ball applies.

(⟸) Every 𝚺₁ formula has a 𝚺₁-recognized code. By sigma₁_induction'.