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>
BEq in Init.Prelude
List in Init.Prelude
Repr in Init.Data.Repr
Vector in Init.Data.Vector.Basic
Bool in Init.Prelude
Int in Init.Data.Int.Basic
String in Init.Prelude
bool in Init.Control.Basic
ite in Init.Prelude
Nat in Init.Prelude
Computation in Mathlib.Data.Seq.Computation
Option in Init.Prelude
Max in Init.Prelude
Max.max in Init.Prelude
</used_lib_defs>

<used_repo_defs>
scoped syntax num : embedded

def instantiateComp (what : Value) (comp : Computation) : Computation :=
  instantiateComputationLvl what 0 comp

def instantiateComputationLvl (what : Value) (level : Nat) : Computation → Computation
| .ret v => .ret <| instantiateValueLvl what level v
| .opCall op v c => .opCall op (instantiateValueLvl what level v) (instantiateComputationLvl what (level+1) c)
| .bind c₁ c₂ => .bind (instantiateComputationLvl what level c₁) (instantiateComputationLvl what (level+1) c₂)
| .ite v c₁ c₂ => .ite (instantiateValueLvl what level v) (instantiateComputationLvl what level c₁) (instantiateComputationLvl what level c₂)
| .app v₁ v₂ => .app (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .handle v c => .handle (instantiateValueLvl what level v) (instantiateComputationLvl what level c)
| .join v₁ v₂ => .join (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .fst v => .fst (instantiateValueLvl what level v)
| .snd v => .snd (instantiateValueLvl what level v)
| .add v₁ v₂ => .add (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .sub v₁ v₂ => .sub (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .max v₁ v₂ => .max (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .lt v₁ v₂ => .lt (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .mul v₁ v₂ => .mul (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .eq v₁ v₂ => .eq (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
termination_by c => sizeOf c
decreasing_by
  all_goals simp
  all_goals grind

def instantiateValueLvl (what : Value) (level : Nat) : Value → Value
| var@(.var (.bvar bvarLevel)) => if bvarLevel = level then what else var
| .lam c => .lam <| instantiateComputationLvl what (level + 1) c
| .hdl h => .hdl <| instantiateHandlerLvl what level h
| .pair v₁ v₂ => .pair (instantiateValueLvl what level v₁) (instantiateValueLvl what level v₂)
| .recfun c => .recfun <| instantiateComputationLvl what (level + 2) c
| .string s => .string s
| .bool b => .bool b
| .unit => .unit
| .var v => .var v
| .num n => .num n
termination_by v => sizeOf v
decreasing_by
  all_goals simp
  all_goals grind

def instantiateHandlerLvl (what : Value) (level : Nat) : Handler → Handler
| ⟨ret, ops'⟩ =>
    ⟨instantiateComputationLvl what (level+1) ret, instantiateOps what level ops'⟩
termination_by h => sizeOf h
decreasing_by
  all_goals simp
  all_goals grind

def instantiateOps (what : Value) (level : Nat) : List OpClause → List OpClause
| [] => []
| ⟨op, body⟩ :: ls =>
  ⟨op, instantiateComputationLvl what (level+2) body⟩ :: instantiateOps what level ls
termination_by l => sizeOf l

structure OpClause where
  op : Name
   
  body : Computation
deriving Repr, BEq

inductive Computation where
| ret : Value → Computation
 
| opCall : Name → Value → Computation → Computation
 
| bind : Computation → Computation → Computation
| ite (val : Value) (trueBranch falseBranch : Computation)
| app : Value → Value → Computation
 
| handle : Value → Computation → Computation
| join : Value → Value → Computation
| fst : Value → Computation
| snd : Value → Computation
| add : Value → Value → Computation
| sub : Value → Value → Computation
| mul : Value → Value → Computation
| max : Value → Value → Computation
| lt : Value → Value → Computation
| eq : Value → Value → Computation
deriving BEq

inductive Value where
| var : Var → Value
| bool : Bool → Value
| string : String → Value
| num : Int → Value
| unit : Value
| pair : Value → Value → Value
 
| lam : Computation → Value
 
| recfun : Computation → Value
| hdl : Handler → Value
deriving BEq

structure Handler where
   
  ret : Computation
  ops : List OpClause
deriving Repr, BEq

inductive Var where
| fvar : Name → Var
| bvar : Nat → Var
deriving Repr, DecidableEq

abbrev Name := String

@[grind cases]
inductive Step : Computation → Computation → Prop
 
| beta v body : Step (.app (.lam body) v) (instantiateComp v body)
 
| recBeta v body : Step (.app (.recfun body) v) (instantiate2 (.recfun body) v body)
 
| iteTrue c₁ c₂ : Step (.ite (.bool true) c₁ c₂) c₁
 
| iteFalse c₁ c₂ : Step (.ite (.bool false) c₁ c₂) c₂
 
| bindStep c₁ c₁' c₂ (h : Step c₁ c₁') : Step (.bind c₁ c₂) (.bind c₁' c₂)
 
| bindReturn v c : Step (.bind (.ret v) c) (instantiateComp v c)
 
| bindOp op v body c : Step (.bind (.opCall op v body) c) (.opCall op v (.bind body c))
 
| handleInner h c₁ c₂ (h₁ : Step c₁ c₂) : Step (.handle (.hdl h) c₁) (.handle (.hdl h) c₂)
 
| handleReturn h v :
  Step (.handle (.hdl h) (.ret v)) (instantiateComp v h.ret)
 
| handleOpHit h op v body c (hop : h.lookup op = some ⟨op, c⟩) :
  Step (.handle (.hdl h) (.opCall op v body))
    (instantiate2 v (.lam (.handle (.hdl h) body)) c)
 
| handleOpMiss h op v body (hop : h.lookup op = none) :
  Step (.handle (.hdl h) (.opCall op v body)) (.opCall op v (.handle (.hdl h) body))
 
| join s₁ s₂ : Step (.join (.string s₁) (.string s₂)) (.ret (.string (strAppend s₁ s₂)))
 
| fstStep v₁ v₂ : Step (.fst (.pair v₁ v₂)) (.ret v₁)
 
| sndStep v₁ v₂ : Step (.snd (.pair v₁ v₂)) (.ret v₂)
| add v₁ v₂ : Step (.add (.num v₁) (.num v₂)) (.ret (.num (v₁ + v₂)))
| sub v₁ v₂ : Step (.sub (.num v₁) (.num v₂)) (.ret (.num (v₁ - v₂)))
| max v₁ v₂ : Step (.max (.num v₁) (.num v₂)) (.ret (.num (Max.max v₁ v₂)))
| lt v₁ v₂ : Step (.lt (.num v₁) (.num v₂)) (.ret (.bool (v₁ < v₂)))
| mul v₁ v₂ : Step (.mul (.num v₁) (.num v₂)) (.ret (.num (v₁ * v₂)))
| eq v₁ v₂ : Step (.eq v₁ v₂) (.ret (.bool (v₁ == v₂)))

def instantiate2 (arg cont : Value) (body : Computation) : Computation :=
  instantiateComputationLvl arg 1 (instantiateComputationLvl cont 0 body)

def strAppend : String → String → String
| "", s => s
| s, "" => s
| s₁, s₂ => s₁ ++ " " ++ s₂

@[grind cases]
inductive StepStar : Computation → Computation → Prop
| refl (c) : StepStar c c
| trans : Step c₁ c₂ → StepStar c₂ c₃ → StepStar c₁ c₃

def Handler.lookup (hdl : Handler) (name : Name) :=
  hdl.ops.find? (·.op == name)

infix:50 " ⤳ " => Step

infix:50 " ⤳⋆ " => StepStar
</used_repo_defs>

<lib_lemmas>
List.find?_some in Init.Data.List.Find
</lib_lemmas>

<local_ctx>
import IntroEffects.SmallStep

def evalSingleStep : Computation → Option Computation
 
| .app (.lam body) v => some <| instantiateComp v body
 
| .app (.recfun body) v => some <| instantiate2 (.recfun body) v body
| .ite (.bool true) c₁ _ => some c₁
| .ite (.bool false) _ c₂ => some c₂
 
| .bind (.ret v) c => some <| instantiateComp v c
 
| .bind (.opCall op v body) c => some <| .opCall op v (.bind body c)
| .bind c₁ c₂ => (evalSingleStep c₁).map (fun c₁' => .bind c₁' c₂)
   
| .handle (.hdl h) (.ret v) => some <| instantiateComp v h.ret
| .handle (.hdl h) (.opCall op v body) =>
  match h.lookup op with
   
  | some ⟨_, c⟩ =>
    let cont := .lam (.handle (.hdl h) body)
    some <| instantiate2 v cont c
   
  | none => some <| .opCall op v (.handle (.hdl h) body)
| .handle (.hdl h) c =>
  (evalSingleStep c).map (fun c' => .handle (.hdl h) c')
| .join (.string s₁) (.string s₂) => some <| .ret (.string (strAppend s₁ s₂))
| .fst (.pair v₁ _) => some <| .ret v₁
| .snd (.pair _ v₂) => some <| .ret v₂
| .add (.num v₁) (.num v₂) => some <| .ret (.num (v₁ + v₂))
| .sub (.num v₁) (.num v₂) => some <| .ret (.num (v₁ - v₂))
| .max (.num v₁) (.num v₂) => some <| .ret (.num (Max.max v₁ v₂))
| .lt (.num v₁) (.num v₂) => some <| .ret (.bool (v₁ < v₂))
| .mul (.num v₁) (.num v₂) => some <| .ret (.num (v₁ * v₂))
| .eq v₁ v₂ => some <| .ret (.bool (v₁ == v₂))
| _ => none

def evalFuel : Nat → Computation → Option Computation
| 0, _ => none
| _+1, .ret v => some <| .ret v
| _+1, .opCall n v c => some <| .opCall n v c
| n+1, c =>
  match evalSingleStep c with
  | some c' => evalFuel n c'
  | none => none
</local_ctx>

<target_theorem>
theorem evalFuel_sound  : evalFuel n c = some v → c ⤳⋆ v :=
</target_theorem>

