Theorem: partitionImpl.loop_invariant
Proof state before:
case inv₁
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
  LoopInvariant (arr.swap i j) first i.prev j.prev last →
    ∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
      partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
        LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
⊢ ∀ (k : Fin n), i.prev < k → k < j.prev → swapped[k] < swapped[first]

case inv₂
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
  LoopInvariant (arr.swap i j) first i.prev j.prev last →
    ∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
      partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
        LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
⊢ ∀ (k : Fin n), j.prev < k → k ≤ last → ¬swapped[k] < swapped[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
  LoopInvariant (arr.swap i j) first i.prev j.prev last →
    ∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
      partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
        LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
⊢ i.prev < j.prev → swapped[j.prev] < swapped[first]
Tactic:
. intro k ik kj
  have ik : i.val ≤ k.val := by
    rw [←Nat.sub_add_cancel (Nat.zero_lt_of_lt fi)]
    exact ik
  cases Nat.eq_or_lt_of_le ik with
  | inl ik =>
    have : swapped[k] = arr[j] := by
      simp [Fin.eq_of_val_eq ik.symm]
      apply Vec.get_swap_left
    rw [this, sf]
    exact inv.3 (Nat.lt_of_lt_of_le (Fin.eq_of_val_eq ik.symm ▸ kj) (Nat.sub_le ..))
  | inr ik =>
    have kj : k < j := (Nat.lt_of_lt_of_le kj (Nat.sub_le ..))
    have : swapped[k] = arr[k] := by
      apply Vec.get_swap_neq
      . apply Fin.ne_of_val_ne
        exact Nat.ne_of_gt ik
      . apply Fin.ne_of_val_ne
        exact Nat.ne_of_lt kj
    rw [this, sf]
    exact inv.1 k ik kj
Proof state after:
case inv₂
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
  LoopInvariant (arr.swap i j) first i.prev j.prev last →
    ∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
      partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
        LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
⊢ ∀ (k : Fin n), j.prev < k → k ≤ last → ¬swapped[k] < swapped[first]

case inv₃
α : Type
inst✝ : Ord α
n : Nat
first✝ i✝ j✝ last : Fin n
arr : Vec α n
first i j : Fin n
fi✝ : first ≤ i
ij : i ≤ j
fi : first < i
x✝² : ↑first ≤ ↑i - 1
x✝¹ : ¬arr[i] < arr[first]
x✝ : ↑i - 1 ≤ ↑j - 1
ih :
  LoopInvariant (arr.swap i j) first i.prev j.prev last →
    ∀ (result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n),
      partitionImpl (arr.swap i j) first i.prev j.prev x✝² x✝ = result →
        LoopInvariant result.snd first first ⟨↑result.fst.val, ⋯⟩ last
inv : LoopInvariant arr first i j last
swapped : Vec α n := arr.swap i j
sf : swapped[first] = arr[first]
result : { mid // first ≤ mid ∧ mid ≤ j.prev } × Vec α n := partitionImpl swapped first i.prev j.prev x✝² x✝
⊢ i.prev < j.prev → swapped[j.prev] < swapped[first]