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"):
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>
List.length in Init.Prelude
List.lengthTR in Init.Prelude
List.get in Init.Prelude
List.get!Internal in Init.GetElem
List.head in Init.Data.List.Basic
List.head! in Init.Data.List.BasicAux
List.tail in Init.Data.List.Basic
List.tail! in Init.Data.List.BasicAux
List.cons in Init.Prelude
List.consFixedLengthDigits in Mathlib.Data.Nat.Digits.Lemmas
List.append in Init.Prelude
List.appendTR in Init.Data.List.Basic
List.reverse in Init.Data.List.Basic
List.reverseAux in Init.Data.List.Basic
List.map in Init.Prelude
List.mapA in Init.Data.List.Control
List.filter in Init.Data.List.Basic
List.filterAuxM in Init.Data.List.Control
List.foldl in Init.Prelude
List.foldlIdx in Batteries.Data.List.Basic
Nat.succ in Init.Prelude
Nat.succPNat in Mathlib.Data.PNat.Defs
Nat.pred in Init.Prelude
Nat.add in Init.Prelude
Nat.sub in Init.Prelude
Nat.subDigitChar in Init.Data.Repr
Nat.mul in Init.Prelude
Nat.multichoose in Mathlib.Data.Nat.Choose.Basic
Nat.div in Init.Prelude
Nat.div.go in Init.Prelude
Nat.mod in Init.Prelude
Nat.mod.inductionOn in Init.Data.Nat.Div.Basic
Nat.min in Init.Data.Nat.Basic
Nat.minFac in Mathlib.Data.Nat.Prime.Defs
Nat.max in Init.Data.Nat.Basic
Nat.maxPowDiv in Mathlib.Data.Nat.MaxPowDiv
Nat.pow in Init.Prelude
Nat.lt in Init.Prelude
Nat.lt_iff_le_and_not_ge in Init.Data.Nat.Basic
Nat.le in Init.Prelude
</used_lib_defs>

<lib_lemmas>
List.length_nil in Init.Data.List.Basic
List.length_cons in Init.Data.List.Basic
List.length_append in Init.Data.List.Basic
List.length_reverse in Init.Data.List.Lemmas
List.length_map in Init.Data.List.Lemmas
List.length_mapAccumr in Mathlib.Data.List.Defs
List.length_filterMap_eq_countP in Init.Data.List.Count
List.length_filterMap_le in Init.Data.List.Lemmas
List.length_take in Init.Data.List.Nat.TakeDrop
List.length_take_le in Init.Data.List.Nat.TakeDrop
List.length_drop in Init.Data.List.TakeDrop
List.length_dropLast in Init.Data.List.Lemmas
List.get_cons_succ in Init.Data.List.Lemmas
List.get_cons_succ' in Init.Data.List.Lemmas
List.get_cons_zero in Init.Data.List.Lemmas
List.mem_cons in Init.Data.List.Lemmas
List.mem_cons_of_mem in Init.Data.List.Lemmas
List.mem_cons_self in Init.Data.List.Lemmas
List.mem_append in Init.Data.List.Lemmas
List.mem_append_cons_self in Init.Data.List.Lemmas
List.mem_map in Init.Data.List.Lemmas
List.mem_mapFinIdx in Init.Data.List.MapIdx
List.mem_filter in Init.Data.List.Lemmas
List.mem_filterMap in Init.Data.List.Lemmas
List.filter_cons in Init.Data.List.Lemmas
List.filter_cons_of_neg in Init.Data.List.Lemmas
List.map_cons in Init.Data.List.Basic
List.map_const in Init.Data.List.Lemmas
List.reverse_cons in Init.Data.List.Basic
List.reverse_cons' in Mathlib.Data.List.Basic
Nat.add_comm in Init.Data.Nat.Basic
Nat.add_assoc in Init.Data.Nat.Basic
Nat.add_zero in Init.Core
Nat.zero_add in Init.Data.Nat.Basic
Nat.sub_self in Init.Data.Nat.Basic
Nat.sub_self_add in Init.Data.Nat.Basic
Nat.sub_zero in Init.Data.Nat.Basic
Nat.mul_comm in Init.Data.Nat.Basic
Nat.div_self in Init.Data.Nat.Div.Basic
Nat.mod_self in Init.Data.Nat.Div.Basic
Nat.lt_succ in Init.Data.Nat.Basic
Nat.lt_succ_floor in Mathlib.Algebra.Order.Floor.Semiring
Nat.lt_succ_iff in Init.Data.Nat.Basic
Nat.succ_lt_of_lt_pred in Init.Data.Nat.Basic
Nat.succ_lt_succ in Init.Data.Nat.Basic
Nat.le_succ in Init.Prelude
Nat.le_succ_iff in Init.Data.Nat.Lemmas
Nat.lt_of_lt_floor in Mathlib.Algebra.Order.Floor.Semiring
Nat.lt_of_lt_of_eq in Init.Data.Nat.Basic
Nat.lt_of_le_of_lt in Init.Prelude
Nat.lt_of_le_of_ne in Init.Prelude
Nat.le_of_lt in Init.Data.Nat.Basic
Nat.le_of_lt_add_of_dvd in Init.Data.Nat.Lemmas
Nat.le_trans in Init.Prelude
Nat.lt_trans in Init.Prelude
Nat.not_lt in Init.Data.Nat.Basic
Nat.not_lt_eq in Init.Data.Nat.Basic
Nat.lt_irrefl in Init.Prelude
Nat.le_refl in Init.Prelude
decide_eq_true in Init.Prelude
decide_eq_false in Init.Prelude
of_decide_eq_true in Init.Prelude
of_decide_eq_false in Init.Prelude
eq_comm in Init.Core
ne_comm in Init.Core
Iff.intro in Init.Core
Iff.mp in Init.Core
Iff.mpr in Init.Core
And.intro in Init.Prelude
And.left in Init.Prelude
And.right in Init.Prelude
Or.inl in Init.Prelude
Or.inr in Init.Prelude
Classical.em in Init.Classical
</lib_lemmas>

<local_ctx>
import Mathlib

@[reducible, simp]
def trapRainWater_precond (height : List Nat) : Prop :=
  True

def trapRainWater (height : List Nat) (h_precond : trapRainWater_precond (height)) : Nat :=
  Id.run do
    let mut left := 0
    let mut right := height.length - 1
    let mut leftMax := 0
    let mut rightMax := 0
    let mut water := 0

    while left < right do
      let hLeft := height[left]!
      let hRight := height[right]!

      if hLeft < hRight then
        if hLeft >= leftMax then
          leftMax := hLeft
        else
          water := water + (leftMax - hLeft)
        left := left + 1
      else
        if hRight >= rightMax then
          rightMax := hRight
        else
          water := water + (rightMax - hRight)
        right := right - 1

    return water

@[reducible, simp]
def trapRainWater_postcond (height : List Nat) (result: Nat) (h_precond : trapRainWater_precond (height)) : Prop :=
  let waterAt := List.range height.length |>.map (fun i =>
    let lmax := List.take (i+1) height |>.foldl Nat.max 0
    let rmax := List.drop i height |>.foldl Nat.max 0
    Nat.min lmax rmax - height[i]!)

  result - (waterAt.foldl (· + ·) 0) = 0 ∧ (waterAt.foldl (· + ·) 0) ≤ result
</local_ctx>

<target_theorem>
theorem trapRainWater_spec_satisfied (height: List Nat) (h_precond : trapRainWater_precond (height)) :
    trapRainWater_postcond (height) (trapRainWater (height) h_precond) h_precond
</target_theorem>