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>
inline in Init.Core
Finset in Mathlib.Data.Finset.Defs
OptionT in Init.Control.Option
OptionT.lift in Init.Control.Option
PMF in Mathlib.Probability.ProbabilityMassFunction.Basic
PMF.uniformOfFinset in Mathlib.Probability.Distributions.Uniform
ENNReal in Mathlib.Data.ENNReal.Basic
Option in Init.Prelude
OptionT.mk in Init.Control.Option
reduceDIte in Lean.Meta.Tactic.Simp.BuiltinSimprocs.Core
PEmpty in Init.Prelude
PUnit in Init.Prelude
IsEmpty in Mathlib.Logic.IsEmpty
List in Init.Prelude
List.count in Init.Data.List.Basic
Fin in Init.Prelude
Unit in Init.Prelude
Fintype in Mathlib.Data.Fintype.Defs
Vector in Init.Data.Vector.Basic
Vector.insertIdx in Init.Data.Vector.Basic
Bool in Init.Prelude
Pure in Init.Prelude
</used_lib_defs>

<used_repo_defs>
notation "[⊥" "|" oa "]" => probFailure oa

notation "[]ₒ" => emptySpec

notation "$[0.." n "]" => uniformFin n

notation:50 "$[" n "⋯" m "]" => uniformFin' n m

example : OracleComp probSpec ℕ := do
  let x ← $[314⋯31415]; let y ← $[0⋯x]
  return x + 2 * y

notation "[=" x "|" oa "]" => probOutput oa x

class HasEvalDist (m : Type u → Type v) [Monad m] where
  evalDist {α : Type u} (mx : m α) : SPMF α
  evalDist_pure {α : Type u} (x : α) : evalDist (pure x : m α) = pure x
  evalDist_bind {α β : Type u} (mx : m α) (my : α → m β) :
    evalDist (mx >>= my) = evalDist mx >>= fun x => evalDist (my x)

protected class DecidableEq (spec : OracleSpec ι) where
  domain_decidableEq' (i : ι) : DecidableEq (spec.domain i)
  range_decidableEq' (i : ι) : DecidableEq (spec.range i)

def OracleSpec (ι : Type u) : Type (max u (v + 1)) :=
  (i : ι) → Type v × Type v

@[inline, reducible]
protected def domain (spec : OracleSpec ι) (i : ι) : Type v := (spec i).1

@[inline, reducible]
protected def range (spec : OracleSpec ι) (i : ι) : Type w := (spec i).2

noncomputable def probFailure (oa : OracleComp spec α) : ℝ≥0∞ :=
  (evalDist oa).run none

def OracleComp {ι : Type u} (spec : OracleSpec.{u,v} ι) :
    Type w → Type (max u (v + 1) w) :=
  OptionT (FreeMonad (OracleQuery.{u,v} spec))

inductive OracleQuery {ι : Type u} (spec : OracleSpec.{u,v} ι) : Type v → Type (max u v)
  | query (i : ι) (t : spec.domain i) : OracleQuery spec (spec.range i)

inductive FreeMonad (f : Type u → Type v) (α : Type w) : Type (max (u + 1) v w)
  | protected pure (x : α) : FreeMonad f α
  | roll {β : Type u} (x : f β) (r : β → FreeMonad f α) : FreeMonad f α

noncomputable def probOutput (oa : OracleComp spec α) (x : α) : ℝ≥0∞ :=
  (evalDist oa).run (some x)

instance isEmpty : IsEmpty (OracleQuery []ₒ α) where false | query i t => i.elim

@[inline, reducible]
def emptySpec : OracleSpec PEmpty := λ _ ↦ (PUnit, PUnit)

@[reducible, inline] def uniformFin (n : ℕ) : ProbComp (Fin (n + 1)) :=
  unifSpec.query n ()

abbrev ProbComp : Type z → Type (max z 1) := OracleComp unifSpec

@[inline, reducible] def unifSpec : OracleSpec.{0,0} ℕ :=
  λ n ↦ (Unit, Fin (n + 1))

@[reducible, inline] def uniformFin' (n m : ℕ) : OracleComp probSpec (Fin (m + 1)) :=
  probSpec.query m n

def probSpec : OracleSpec.{0,0} ℕ :=
  fun n => (ℕ, Fin (n + 1))

@[elab_as_elim]
def induction {α} {motive : {n : ℕ} → Vector α n → Sort*} (v_empty : motive #v[])
  (v_insert : {n : ℕ} → (hd : α) → (tl : Vector α n) → motive tl → motive (tl.insertIdx 0 hd))
    {m : ℕ} : (v : Vector α m) → motive v :=

notation "Pr[=" x "|" mx "]" => probOutput mx x

notation "Pr[⊥" "|" mx "]" => probFailure mx

notation "[=" x "|" oa "]" => probOutput oa x

notation "[⊥" "|" oa "]" => probFailure oa

notation "$[0.." n "]" => uniformFin n

notation:50 "$[" n "⋯" m "]" => uniformFin' n m

example : OracleComp probSpec ℕ := do
  let x ← $[314⋯31415]; let y ← $[0⋯x]
  return x + 2 * y

notation "[]ₒ" => emptySpec
</used_repo_defs>

<lib_lemmas>
Finset.nonempty_iff_ne_empty in Mathlib.Data.Finset.Empty
div_eq_mul_inv in Mathlib.Algebra.Group.Defs
eq_comm in Init.Core
Finset.mem_toList in Mathlib.Data.Finset.Dedup
List.isEmpty_eq_false_iff_exists_mem in Init.Data.List.Lemmas
ENNReal.inv_eq_zero in Mathlib.Data.ENNReal.Inv
ENNReal.natCast_ne_top in Mathlib.Data.ENNReal.Basic
Finset.not_nonempty_iff_eq_empty in Mathlib.Data.Finset.Empty
PMF.bind_apply in Mathlib.Probability.ProbabilityMassFunction.Monad
PMF.pure_apply in Mathlib.Probability.ProbabilityMassFunction.Monad
PMF.uniformOfFinset_apply in Mathlib.Probability.Distributions.Uniform
imp_false in Init.Core
ite_eq_ite in Init.PropLemmas
ite_eq_right_iff in Init.PropLemmas
mul_ite in Mathlib.Algebra.Notation.Defs
mul_one in Mathlib.Algebra.Group.Defs
ne_eq in Init.SimpLemmas
symm in Mathlib.Order.Defs.Unbundled
tsum_eq_single in Mathlib.Topology.Algebra.InfiniteSum.Basic
</lib_lemmas>

<repo_lemmas>
lemma probOutput_map_eq_sum_fintype_ite [Fintype α] [DecidableEq β] (y : β) :
    [= y | f <$> oa] = ∑ x : α, if y = f x then [= x | oa] else 0

lemma probOutput_map_eq_tsum_ite [DecidableEq β] (y : β) :
    [= y | f <$> oa] = ∑' x : α, if y = f x then [= x | oa] else 0

lemma probOutput_bind_eq_tsum (y : β) :
    [= y | oa >>= ob] = ∑' x : α, [= x | oa] * [= y | ob x]

@[simp]
lemma probOutput_pure [DecidableEq α] (y : α) :
    [= y | (pure x : OracleComp spec α)] = if y = x then 1 else 0

lemma List.countP_eq_sum_fin_ite {α : Type*} (xs : List α) (p : α → Bool) :
    (∑ i : Fin xs.length, if p xs[i] then 1 else 0) = xs.countP p

@[simp]
lemma PMF.monad_pure_eq_pure {α : Type u} (x : α) :
    (Pure.pure x : PMF α) = PMF.pure x

@[simp]
lemma PMF.monad_bind_eq_bind {α β : Type u}
      (p : PMF α) (q : α → PMF β) : p >>= q = p.bind q
</repo_lemmas>

<local_ctx>
import VCVio.OracleComp.DistSemantics.Prod

import Batteries.Control.OptionT

open OracleSpec BigOperators ENNReal

namespace OracleComp

section uniformSelect

prefix : 50 "$" => uniformSelect _

variable {cont β : Type} [h : HasUniformSelect cont β]

end uniformSelect

section uniformSelectList

variable {α : Type}

end uniformSelectList

section uniformSelectVector

end uniformSelectVector

section uniformSelectFinset

variable {α : Type}
</local_ctx>

<target_theorem>
@[simp] lemma evalDist_uniformSelectFinset [DecidableEq α] (s : Finset α) :
    evalDist ($ s) = if hs : s.Nonempty then
      OptionT.lift (PMF.uniformOfFinset s hs) else failure :=
</target_theorem>

