have getM (k : Fin n) (eq : mid.val = k.val): sorted[k] = arr[first] := by
  let mid' : Fin n := ⟨mid.val, mn⟩
  have : k = mid' := by
    apply Fin.eq_of_val_eq
    simp [eq]
  subst this
  rw [quickSortImpl.get_lt (k := mid') (lt := Nat.lt_succ_self ..)]
  cases Nat.eq_zero_or_pos mid.val with
  | inl zero =>
    have zero : mid.val = Fin.mk 0 mid.val.pos := by
      apply Fin.eq_of_val_eq
      simp [zero]
    simp [zero] at *
    exact m
  | inr pos =>
    rw [quickSortImpl.get_gt (k := mid') (gt := Nat.sub_lt_self (by decide) (Nat.zero_lt_of_lt pos))]
    exact m
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
  sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
    (↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
        (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
    true
case step
α : Type
inst✝ : Order α
n : Nat
arr : Vec α n
first : Nat
last : Fin n
lt : first < ↑last
parted : { mid // first ≤ ↑mid ∧ mid ≤ last } × Vec α n
eq : partition arr ⟨first, ⋯⟩ last ⋯ = parted
ih₁ : sortedRange (quickSortImpl parted.snd first parted.fst.val.prev) first ↑parted.fst.val.prev
ih₂ :
  sortedRange (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)
    (↑parted.fst.val + 1) ↑last
this : first < n
i j : Fin n
fi : first ≤ ↑i
ij : i ≤ j
jl : ↑j ≤ ↑last
mid : { mid // first ≤ ↑mid ∧ mid ≤ last } := parted.fst
mn : ↑mid.val < n
sorted : Vec α n := quickSortImpl (quickSortImpl parted.snd first mid.val.prev) (↑mid.val + 1) last
l : ∀ (k : Fin n), ⟨first, this⟩ ≤ k → ↑k < ↑parted.fst.val → parted.snd[k] < arr[⟨first, this⟩]
m : parted.snd[parted.fst.val] = arr[⟨first, this⟩]
r : ∀ (k : Fin n), ↑parted.fst.val < ↑k → k ≤ last → ¬parted.snd[k] < arr[⟨first, this⟩]
getL : ∀ (k : Fin n), first ≤ ↑k → k < mid.val → sorted[k] < arr[first]
getM : ∀ (k : Fin n), ↑mid.val = ↑k → sorted[k] = arr[first]
⊢ (compare (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[i]
        (quickSortImpl (quickSortImpl parted.snd first parted.fst.val.prev) (↑parted.fst.val + 1) last)[j]).isLE =
    true