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
Finset in Mathlib.Data.Finset.Defs
Finset.range in Mathlib.Data.Finset.Range
FinEnum in Mathlib.Data.FinEnum
FinEnum.toList in Mathlib.Data.FinEnum
BEq in Init.Prelude
DecidableEq in Init.Prelude
Hashable in Init.Prelude
Repr in Init.Data.Repr
Std.HashMap in Std.Data.HashMap.Basic
Std.HashSet in Std.Data.HashSet.Basic
LawfulBEq in Init.Core
List in Init.Prelude
LawfulHashable in Init.Data.LawfulHashable
Inhabited in Init.Prelude
</used_lib_defs>

<lib_lemmas>
Std.HashSet.mem_insert in Std.Data.HashSet.Lemmas
and_true in Init.SimpLemmas
beq_iff_eq in Init.Core
true_and in Init.SimpLemmas
Std.HashMap.getD_eq_getD_getElem? in Std.Data.HashMap.Lemmas
Finset.range_add in Mathlib.Algebra.Group.Nat.Range
FinEnum.mem_toList in Mathlib.Data.FinEnum
</lib_lemmas>

<repo_lemmas>
@[aesop 50% unsafe]
theorem Std.HashMap.mem_of_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] {m : Std.HashMap K V} {k : K} :
    m[k]? = some v → k ∈ m

theorem Std.HashMap.mem_iff_getElem? [BEq K] [LawfulBEq K] [Hashable K] [LawfulHashable K] [Inhabited V] {m : Std.HashMap K V} {k : K} :
     k ∈ m ↔ ∃ v, m[k]? = some v
</repo_lemmas>

<local_ctx>
import Mathlib.Algebra.Group.Nat.Range

import Blase.AutoStructs.BundledNfa

import Blase.FinEnum

open SetRel

abbrev State := Nat

structure RawCNFA (A : Type 0) [BEq A] [Hashable A] [DecidableEq A] [FinEnum A] where
  stateMax : State
  initials : Std.HashSet State
  finals : Std.HashSet State
  trans : Std.HashMap (State × A) (Std.HashSet State)
deriving Repr

section sim

variable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]

@[inline]
def RawCNFA.tr (m : RawCNFA A) s a := m.trans.getD (s, a) ∅

def RawCNFA.states (m : RawCNFA A) : Finset State := Finset.range m.stateMax

instance RawCNFA.statesFinset (m : RawCNFA A) : Fintype m.states := (Finset.range m.stateMax).fintypeCoeSort

end sim

section basics

variable {A : Type} [BEq A] [Hashable A] [DecidableEq A] [FinEnum A]

def RawCNFA.newState (m : RawCNFA A) : State × RawCNFA A :=
  let old := m.stateMax
  let m := { m with stateMax := old + 1 }
  (old, m)

def RawCNFA.addTrans (m : RawCNFA A) (a : A) (s s' : State) : RawCNFA A :=
  let ns := m.trans.getD (s, a) ∅
  let ns := ns.insert s'
  { m with trans :=  m.trans.insert (s, a) ns }

def RawCNFA.addInitial (m : RawCNFA A) (s : State) : RawCNFA A :=
  { m with initials := m.initials.insert s }

def RawCNFA.createSink (m : RawCNFA A) : State × RawCNFA A :=
  let (s, m) := m.newState
  let m := m.addInitial s
  let m := FinEnum.toList (α := A).foldl (init := m) fun m a =>
    m.addTrans a s s
  (s, m)

structure RawCNFA.WF (m : RawCNFA A) where
  initials_lt : ∀ {s}, s ∈ m.initials → s ∈ m.states
  finals_lt : ∀ {s}, s ∈ m.finals → s ∈ m.states
  trans_src_lt : ∀ s_a ∈ m.trans, s_a.1 ∈ m.states
  trans_tgt_lt : s' ∈ m.tr s a → s' ∈ m.states
</local_ctx>

<target_theorem>
@[grind =, simp]
lemma createSink_trans [LawfulBEq A] {m : RawCNFA A} (hwf : m.WF) :
    s₂ ∈ m.createSink.2.tr s₁ a ↔
      (s₁ = m.stateMax ∧ s₂ = m.stateMax) ∨
      (s₁ ∈ m.states ∧ s₂ ∈ m.states ∧ s₂ ∈ m.tr s₁ a) :=
</target_theorem>

