Theorem: partition.partition
Proof state before:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
  { inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
    inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
    inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
  partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
    (Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
    result.snd[result.fst.val] = arr[first] ∧
      ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]
Tactic:
have p₃ (k : Fin n) (mk : mid < k.val) (kl : k ≤ last) : ¬swapped[k] <o arr[first] := by
  have : swapped[k] = arr'[k] := by
    apply Vec.get_swap_neq
    . apply Fin.ne_of_val_ne
      exact Nat.ne_of_gt (Nat.lt_of_le_of_lt mid.property.1 mk)
    . apply Fin.ne_of_val_ne
      exact Nat.ne_of_gt mk
  rw [this, ←first_eq]
  exact inv.2 k mk kl
Proof state after:
α : Type
inst✝ : Ord α
n : Nat
arr : Vec α n
first last : Fin n
fl : first ≤ last
result : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n
eq : _root_.partition arr first last fl = result
afterLoop : { mid // first ≤ mid ∧ mid ≤ last } × Vec α n := partitionImpl arr first last last fl ⋯
mid : { mid // first ≤ mid ∧ mid ≤ last } := afterLoop.fst
arr' : Vec α n := afterLoop.snd
this✝¹ : ↑mid.val < n
swapped : Vec α n := arr'.swap first ⟨↑mid.val, this✝¹⟩
this✝ : result.fst = mid
this : result.snd = swapped
first_eq : arr'[first] = arr[first]
inv₀ : partitionImpl.LoopInvariant arr first last last last :=
  { inv₁ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_trans kl lk)),
    inv₂ := fun k lk kl => False.elim (Nat.lt_irrefl (↑k) (Nat.lt_of_le_of_lt kl lk)),
    inv₃ := fun ll => False.elim (Nat.lt_irrefl (↑last) ll) }
inv : partitionImpl.LoopInvariant afterLoop.snd first first ⟨↑afterLoop.fst.val, ⋯⟩ last :=
  partitionImpl.loop_invariant arr first last last last fl (Nat.le_refl ↑last) inv₀ afterLoop
    (Eq.refl (partitionImpl arr first last last fl (Nat.le_refl ↑last)))
p₁ : ∀ (k : Fin n), first ≤ k → ↑k < ↑mid.val → swapped[k] < arr[first]
p₂ : swapped[mid.val] = arr[first]
p₃ : ∀ (k : Fin n), ↑mid.val < ↑k → k ≤ last → ¬swapped[k] < arr[first]
⊢ (∀ (k : Fin n), first ≤ k → ↑k < ↑result.fst.val → result.snd[k] < arr[first]) ∧
    result.snd[result.fst.val] = arr[first] ∧
      ∀ (k : Fin n), ↑result.fst.val < ↑k → k ≤ last → ¬result.snd[k] < arr[first]