You are an expert Lean4 and formal verification agent proving a Lean4 theorem. Your output **must** strictly adhere to the specified format.

### Input Information
You will receive the following components, relevant to the target theorem:
* <used_lib_defs>...</used_lib_defs>: Imported library definitions directly used by the theorem.
* <used_repo_defs>...</used_repo_defs>: Definitions from the local project (but outside the current file), directly or transitively used by the theorem.
* <lib_lemmas>...</lib_lemmas>: Imported library lemmas/theorems that may assist the proof.
* <repo_lemmas>...</repo_lemmas>: Lemmas/theorems defined in the local project outside this file.
* <local_ctx>...</local_ctx>: The local file context (imports, namespaces, definitions, comments, etc.) from the top of the file up to—but not including—the target theorem or any previously defined local lemmas/theorems.
* <target_theorem>...</target_theorem>: The target theorem statement, ending with `:=`.

**Notes**
1. Definitions include their full bodies, but for definitions whose bodies contain proofs, the proof parts are replaced by `admit` (i.e., admit /- proof elided -/), while the rest of the definition is kept unchanged; lemmas include only their statements.
2. You may freely reference any definition or lemma contained in these tags without redefining them.
3. The tags for used definitions and helpful lemmas (used_lib_defs, used_repo_defs, lib_lemmas, repo_lemmas) may be omitted if empty.

### Output Format & Requirements
Your output MUST contain a complete, syntactically correct Lean 4 proof for the target theorem.

**Strict Output Structure:**
<reasoning> 
...
</reasoning> 

<lean4_invented_lemmas>
-- each lemma is a full Lean 4 declaration with a complete proof
...
</lean4_invented_lemmas>

<lean4_proof>
-- Lean 4 proof body of the target theorem
...
</lean4_proof>

**Requirements:**
1. <reasoning> and <lean4_invented_lemmas> are OPTIONAL. Include them only if they aid proof construction.
2. <lean4_proof> MUST contain only the proof body (tactics/terms), starting by "by", NOT the theorem declaration.
3. <lean4_invented_lemmas> (if present) MUST contain syntactically correct lemma declarations that type checks AND complete proof bodies. Defining a lemma using `axiom` is NOT allowed.
4. <reasoning> contains high-level proof planning.
5. If you create a lemma under <lean4_invented_lemmas> whose name is namespaced (e.g., X.Y.Z), you MUST always refer to it using its fully qualified name. Do NOT rely on shortened names. For example, write rw [X.Y.Z], not rw [Z] for invented lemmas.
6. All Lean 4 code MUST be syntactically correct. No `sorry`, `admit`, or non-Lean text. Comments allowed using `--` or `/- ... -/`.
7. Follow the tags and structure EXACTLY. No extra text before, between, or after the sections.
8. Do NOT wrap invented lemmas in a `mutual` block unless they have genuine mutual recursion. Declare each lemma independently.

### Examples are provided here (example input are under "User Prompt", and example assistant response are under "Agent Response"):
Example 1 ------------------- 
User Prompt:
<used_lib_defs>
structure BitVec (w : Nat) where
  /-- Construct a `BitVec w` from a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  ofFin ::

  /-- Interpret a bitvector as a number less than `2^w`.
  O(1), because we use `Fin` as the internal representation of a bitvector. -/
  toFin : Fin (hPow 2 w)
</used_lib_defs>

<repo_lemmas>
theorem List.getElem_append_left{α : Type u_1} {l₁ l₂ : List α} {i : Nat} (hn : i < l₁.length) :\n(l₁ ++ l₂)[i] = l₁[i]
</repo_lemmas>

<local_ctx>
import Mathlib.Computability.NFA
import Mathlib.Data.FinEnum
import Mathlib.Data.Rel
import Mathlib.Data.Vector.Basic
import Blase.AutoStructs.ForLean

open Set
open Mathlib
open SetRel

-- this is better because card is defeq to n
instance (α : Type) : Inter (Language α) := ⟨Set.inter⟩
instance (α : Type) : Union (Language α) := ⟨Set.union⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width.
-/
structure BitVecs (n : Nat) where
  w : Nat
  bvs : List.Vector (BitVec w) n

def BitVecs.cast (bvs : BitVecs n) (h : n = n') : BitVecs n' :=
  { w := bvs.w, bvs := h ▸ bvs.bvs }

abbrev BitVecs.empty : BitVecs n := ⟨0, List.Vector.replicate n .nil⟩
abbrev BitVecs.singleton {w : Nat} (bv : BitVec w) : BitVecs 1 := ⟨w, bv ::ᵥ .nil⟩
abbrev BitVecs.pair {w : Nat} (bv1 bv2 : BitVec w) : BitVecs 2 := ⟨w, bv1 ::ᵥ bv2 ::ᵥ .nil⟩

/--
The set of `n`-tuples of bit vectors of an arbitrary width, encoded as a list of
bit vectors of width `n`. The width of the encoded bit vectors is the length of
the list.
-/
abbrev BitVecs' (n : Nat) := List (BitVec n)

@[simps]
def dec (bvs' : BitVecs' n) : BitVecs n where
  w := bvs'.length
  bvs := List.Vector.ofFn fun k => BitVec.ofFn fun i => bvs'[i].getLsbD k

@[simp]
</local_ctx>

<target_theorem>
lemma dec_snoc n (bvs' : BitVecs' n) (a : BitVec n) :
  dec (bvs' ++ [a]) =
    { w := bvs'.length + 1
      bvs := List.Vector.ofFn fun k =>
        BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k) } :=
</target_theorem>

Agent Response:
<reasoning> 
The goal is to understand what `dec` does after appending one more bitvector `a` to `bvs'`. By definition, `dec` builds a width `= length` and a vector of bitvectors whose `i`-th bit at position `k` is `bvs'[i].getLsbD k`. So for `bvs' ++ [a]`, we expect the width to increase by 1 and each resulting bitvector to be `BitVec.cons` of the new bit coming from `a`.

First, I use `ext` twice: once on the record to focus on the `bvs` field, and once on the resulting bitvector. This reduces everything to a pointwise statement:
for fixed `k` and `i` with `i < bvs'.length + 1`,
the bit `(bvs' ++ [a])[i].getLsbD k` should match the bit at index `i` of `BitVec.cons (a.getLsbD k) ((dec bvs').bvs.get k)`.

At this point I need a clean description of indexing into `BitVec.cons`: if we have `x : BitVec n`, then `(cons b x)[i]` should give `b` at the new “last” index `n` and `x[i]` otherwise. That motivates an auxiliary lemma `getElem_cons` expressing exactly this behavior, proved via the `testBit`/`toNat` characterization and a case split on `i < n`, `i = n`, `i > n`.

Using `getElem_cons`, the RHS becomes an `if` on whether `i = bvs'.length`. So I split into cases:
- If `i = bvs'.length`, then `(bvs' ++ [a])[i]` is the appended `a`, so both sides reduce to `a.getLsbD k`.
- If `i ≠ bvs'.length`, the bound `i < bvs'.length + 1` forces `i < bvs'.length`. Then `(bvs' ++ [a])[i] = bvs'[i]` by `List.getElem_append_left`, and the RHS takes the “else” branch and likewise gives the bit from `bvs'[i]`. In both subcases, `simp` (plus the definition of `dec`) finishes.

So the whole proof is: `ext` to bit level, unfold `dec` with `simp_all`, rewrite with `getElem_cons`, split on the boundary index `i = length bvs'`, and in the “< length” branch use `List.getElem_append_left` and bounds to line up both sides.
</reasoning> 

<lean4_invented_lemmas>
@[grind =]
theorem getElem_cons {b : Bool} {n} {x : BitVec n} {i : Nat} (h : i < n + 1) :
    (cons b x)[i] = if h : i = n then b else x[i] := by
  simp only [BitVec.getElem_eq_testBit_toNat, BitVec.toNat_cons, Nat.testBit_or]
  rw [Nat.testBit_shiftLeft]
  rcases Nat.lt_trichotomy i n with i_lt_n | i_eq_n | n_lt_i
  · have p1 : ¬(n ≤ i) := by omega
    have p2 : i ≠ n := by omega
    simp [p1, p2]
  · simp only [i_eq_n, ge_iff_le, Nat.le_refl, decide_true, Nat.sub_self, Nat.testBit_zero,
    Bool.true_and, BitVec.testBit_toNat, BitVec.getLsbD_of_ge, Bool.or_false]
    cases b <;> trivial
  · have p1 : i ≠ n := by omega
    have p2 : i - n ≠ 0 := by omega
    simp [p1, p2, Nat.testBit_bool_toNat]
</lean4_invented_lemmas>

<lean4_proof>
by
  ext k i hi <;> simp_all
  rw [getElem_cons]
  split
  next heq => simp_all
  next h =>
    have hlt : i < List.length bvs' := by simp at hi; omega
    rw [List.getElem_append_left hlt]
    simp
</lean4_proof>

### Your task:
<used_lib_defs>
Nat in Init.Prelude
Fin in Init.Prelude
Subspace in Mathlib.Algebra.Module.Submodule.Basic
Set in Mathlib.Data.Set.Defs
Set.Ico in Mathlib.Order.Interval.Set.Defs
Submodule in Mathlib.Algebra.Module.Submodule.Defs
Submodule.span in Mathlib.LinearAlgebra.Span.Defs
Polynomial.C in Mathlib.Algebra.Polynomial.Basic
Algebra in Mathlib.Algebra.Algebra.Defs
Field in Mathlib.Algebra.Field.Defs
IsLinearMap in Mathlib.Algebra.Module.LinearMap.Defs
toFun in ToMathlib.Control.Monad.Hom
Submodule.map in Mathlib.Algebra.Module.Submodule.Map
Module.Basis in Mathlib.LinearAlgebra.Basis.Defs
Disjoint in Mathlib.Order.Disjoint
Fact in Mathlib.Logic.Basic
LinearEquiv in Mathlib.Algebra.Module.Equiv.Defs
LinearEquiv.ofBijective in Mathlib.Algebra.Module.Submodule.Equiv
LinearIndependent in Mathlib.LinearAlgebra.LinearIndependent.Defs
LinearMap in Mathlib.Algebra.Module.LinearMap.Defs
LinearMap.codRestrict in Mathlib.Algebra.Module.Submodule.LinearMap
LinearMap.ker in Mathlib.Algebra.Module.Submodule.Ker
Module.Basis.span in Mathlib.LinearAlgebra.Basis.Basic
Set.range in Mathlib.Data.Set.Operations
Submodule.subtype in Mathlib.Algebra.Module.Submodule.LinearMap
Subtype in Init.Prelude
Finset in Mathlib.Data.Finset.Defs
Finset.univ in Mathlib.Data.Fintype.Defs
Ne in Init.Core
NeZero in Init.Data.NeZero
Decidable in Init.Prelude
reduceDIte in Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core
Fintype in Mathlib.Data.Fintype.Defs
Fintype.card in Mathlib.Data.Fintype.Card
Subsingleton in Init.Core
Units in Mathlib.Algebra.Group.Units.Defs
Units.mk0 in Mathlib.Algebra.GroupWithZero.Units.Basic
Units.val in Mathlib.Algebra.Group.Units.Defs
</used_lib_defs>

<used_repo_defs>
def getBit (k n : Nat) : Nat := (n >>> k) &&& 1

noncomputable def normalizedW (i : Fin r) : L[X] :=
  C (1 / (W 𝔽q β i).eval (β i)) * W 𝔽q β i

noncomputable def W (i : Fin r) : L[X] :=
  ∏ u : U 𝔽q β i, (X - C u.val)

def U (i : Fin r) : Subspace 𝔽q L := Submodule.span 𝔽q (β '' (Set.Ico 0 i))

noncomputable def polyEvalLinearMap {L 𝔽q : Type*} [Field L] [Field 𝔽q] [Algebra 𝔽q L]
  (p : L[X]) (hp_add : IsLinearMap 𝔽q (fun x : L => p.eval x)) : L →ₗ[𝔽q] L :=
{
  toFun    := fun x => p.eval x,
  map_add' := hp_add.map_add,
  map_smul' := hp_add.map_smul
}

def binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary : ∀ j: Fin n, m j ≤ 1) : Fin (2^n) :=
</used_repo_defs>

<lib_lemmas>
Fintype.card_le_one_iff_subsingleton in Mathlib.Data.Fintype.EquivFin
Fintype.card_units in Mathlib.Data.Fintype.Units
Nat.le_of_eq in Init.Data.Nat.Basic
Subsingleton.elim in Init.Core
congr_arg in Batteries.Logic
Decidable.not_not in Init.PropLemmas
Fin.eta in Init.Data.Fin.Lemmas
Fin.is_lt in Init.Data.Fin.Lemmas
Polynomial.ext in Mathlib.Algebra.Polynomial.Basic
imp_false in Init.Core
ite_eq_left_iff in Init.PropLemmas
ite_eq_right_iff in Init.PropLemmas
ne_eq in Init.SimpLemmas
one_ne_zero in Mathlib.Algebra.NeZero
zero_ne_one in Mathlib.Algebra.NeZero
</lib_lemmas>

<repo_lemmas>
lemma Xⱼ_zero_eq_one (ℓ : ℕ) (h_ℓ : ℓ ≤ r) :
  Xⱼ 𝔽q β ℓ h_ℓ ⟨0, by exact Nat.two_pow_pos ℓ⟩ = 1

lemma getBit_of_binaryFinMapToNat {n : ℕ} (m : Fin n → ℕ) (h_binary: ∀ j: Fin n, m j ≤ 1) :
    ∀ k: ℕ, Nat.getBit k (binaryFinMapToNat m h_binary).val
      = if h_k: k < n then m ⟨k, by omega⟩ else 0

lemma and_two_pow_eq_zero_of_getBit_0 {n i : ℕ} (h_getBit : getBit i n = 0)
    : n &&& (2 ^ i) = 0

lemma and_eq_zero_iff_and_each_getBit_eq_zero {n m : ℕ} :
    n &&& m = 0 ↔ ∀ k, Nat.getBit k n &&& Nat.getBit k m = 0

lemma eq_iff_eq_all_getBits {n m : ℕ} : n = m ↔ ∀ k, getBit k n = getBit k m

lemma shiftRight_and_one_distrib {n m k : ℕ} :
    Nat.getBit k (n &&& m) = Nat.getBit k n &&& Nat.getBit k m

lemma getBit_two_pow {i k : ℕ} : (getBit k (2^i) = if i == k then 1 else 0)

lemma getBit_of_lt_two_pow {n: ℕ} (a: Fin (2^n)) (k: ℕ) :
  getBit k a = if k < n then getBit k a else 0

lemma getBit_eq_testBit (k n : Nat) : getBit k n = if n.testBit k then 1 else 0

lemma getBit_lt_2 {k n : Nat} : getBit k n < 2

lemma getBit_of_xor {n m k: ℕ} : getBit k (n ^^^ m) = getBit k n ^^^ getBit k m

lemma getBit_zero_eq_zero {k : Nat} : getBit k 0 = 0

lemma sum_of_and_eq_zero_is_xor {n m : ℕ} (h_n_AND_m : n &&& m = 0) : n + m = n ^^^ m

lemma sum_eq_xor_plus_twice_and (n : Nat) : ∀ m : ℕ, n + m = (n ^^^ m) + 2 * (n &&& m)

lemma and_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)
  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):
  n &&& m = (n1 &&& m1) * 2 + (bn &&& bm)

lemma div_2_form {nD2 b : ℕ} (h_b : b < 2):
  (nD2 * 2 + b) / 2 = nD2

lemma or_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)
  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):
  n ||

lemma xor_by_split_lowBits {n m n1 m1 bn bm : ℕ} (h_bn : bn < 2) (h_bm : bm < 2)
  (h_n : n = n1 * 2 + bn) (h_m : m = m1 * 2 + bm):
  n ^^^ m = (n1 ^^^ m1) * 2 + (bn ^^^ bm)
</repo_lemmas>

<local_ctx>
import ArkLib.Data.FieldTheory.AdditiveNTT.NovelPolynomialBasis

import Mathlib.Tactic

import Mathlib.Data.Finsupp.Defs

import Mathlib.LinearAlgebra.LinearIndependent.Defs

open Polynomial AdditiveNTT Module

namespace AdditiveNTT

variable {r : ℕ} [NeZero r]

variable {L : Type u} [Field L] [Fintype L] [DecidableEq L]

variable (𝔽q : Type u) [Field 𝔽q] [Fintype 𝔽q] [DecidableEq 𝔽q]
  [h_Fq_char_prime : Fact (Nat.Prime (ringChar 𝔽q))] [hF₂ : Fact (Fintype.card 𝔽q = 2)]

variable [Algebra 𝔽q L]

variable (β : Fin r → L) [hβ_lin_indep : Fact (LinearIndependent 𝔽q β)]
  [h_β₀_eq_1 : Fact (β 0 = 1)]

variable {ℓ R_rate : ℕ} (h_ℓ_add_R_rate : ℓ + R_rate < r)-- ℓ ∈ {1, ..., r-1}

section IntermediateStructures

noncomputable def sDomain (i : Fin r) : Subspace 𝔽q L :=
  let W_i_norm := normalizedW 𝔽q β i
  let h_W_i_norm_is_additive : IsLinearMap 𝔽q (fun x : L => W_i_norm.eval x) :=
    AdditiveNTT.normalizedW_is_additive 𝔽q β i
  Submodule.map (polyEvalLinearMap W_i_norm h_W_i_norm_is_additive)
    (U 𝔽q β ⟨ℓ + R_rate, h_ℓ_add_R_rate⟩)

def sBasis (i : Fin r) (h_i : i < ℓ + R_rate) : Fin (ℓ + R_rate - i) → L :=
  fun k => β ⟨i + k.val, by admit /- proof elided -/
  ⟩

noncomputable def sDomain_basis (i : Fin r) (h_i : i < ℓ + R_rate) :
    Basis (Fin (ℓ + R_rate - i)) 𝔽q (
      sDomain 𝔽q β h_ℓ_add_R_rate i) :=

noncomputable section DomainBijection

def splitPointIntoCoeffs (i : Fin r) (h_i : i < ℓ + R_rate)
  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :
  Fin (ℓ + R_rate - i.val) → ℕ := fun j =>
    if ((sDomain_basis 𝔽q β
    h_ℓ_add_R_rate i h_i).repr x j = 0) then
      0 else 1

noncomputable def sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)
  (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :
  Fin (2^(ℓ + R_rate - i.val)) :=

def finToBinaryCoeffs (i : Fin r) (idx : Fin (2 ^ (ℓ + R_rate - i.val))) :
  Fin (ℓ + R_rate - i.val) → 𝔽q := fun j =>
    if (Nat.getBit (k:=j) (n:=idx)) = 1 then (1 : 𝔽q) else (0 : 𝔽q)
</local_ctx>

<target_theorem>
omit h_β₀_eq_1 in
lemma finToBinaryCoeffs_sDomainToFin (i : Fin r) (h_i : i < ℓ + R_rate)
    (x : sDomain 𝔽q β h_ℓ_add_R_rate i) :
    let pointFinIdx :=
</target_theorem>

