theorem balance_eq_balance' {l x r} (hl : Balanced l) (hr : Balanced r) (sl : Sized l)
    (sr : Sized r) : @balance α l x r = balance' l x r := by
  cases' l with ls ll lx lr
  · cases' r with rs rl rx rr
    · rfl
    · rw [sr.eq_node'] at hr ⊢
      cases' rl with rls rll rlx rlr <;> cases' rr with rrs rrl rrx rrr <;>
        dsimp [balance, balance']
      · rfl
      · have : size rrl = 0 ∧ size rrr = 0 := by
          have := balancedSz_zero.1 hr.1.symm
          rwa [size, sr.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
        cases sr.2.2.2.1.size_eq_zero.1 this.1
        cases sr.2.2.2.2.size_eq_zero.1 this.2
        obtain rfl : rrs = 1 := sr.2.2.1
        rw [if_neg, if_pos, rotateL, if_pos]; · rfl
        all_goals dsimp only [size]; decide
      · have : size rll = 0 ∧ size rlr = 0 := by
          have := balancedSz_zero.1 hr.1
          rwa [size, sr.2.1.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
        cases sr.2.1.2.1.size_eq_zero.1 this.1
        cases sr.2.1.2.2.size_eq_zero.1 this.2
        obtain rfl : rls = 1 := sr.2.1.1
        rw [if_neg, if_pos, rotateL, if_neg]; · rfl
        all_goals dsimp only [size]; decide
      · symm; rw [zero_add, if_neg, if_pos, rotateL]
        · dsimp only [size_node]; split_ifs
          · simp [node3L, node']; abel
          · simp [node4L, node', sr.2.1.1]; abel
        · apply Nat.zero_lt_succ
        · exact not_le_of_gt (Nat.succ_lt_succ (add_pos sr.2.1.pos sr.2.2.pos))
  · cases' r with rs rl rx rr
    · rw [sl.eq_node'] at hl ⊢
      cases' ll with lls lll llx llr <;> cases' lr with lrs lrl lrx lrr <;>
        dsimp [balance, balance']
      · rfl
      · have : size lrl = 0 ∧ size lrr = 0 := by
          have := balancedSz_zero.1 hl.1.symm
          rwa [size, sl.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
        cases sl.2.2.2.1.size_eq_zero.1 this.1
        cases sl.2.2.2.2.size_eq_zero.1 this.2
        obtain rfl : lrs = 1 := sl.2.2.1
        rw [if_neg, if_neg, if_pos, rotateR, if_neg]; · rfl
        

Proof state:
case node.nil.nil.node.refl.refl.hnc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrx : α
this : size nil = 0 ∧ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
⊢ ¬size (node 1 nil lrx nil) < ratio * size nil
case node.nil.nil.node.refl.refl.hc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrx : α
this : size nil = 0 ∧ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
⊢ 0 + 1 + 1 > 0
case node.nil.nil.node.refl.refl.hnc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrx : α
this : size nil = 0 ∧ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
⊢ ¬0 > delta * (0 + 1 + 1)
case node.nil.nil.node.refl.refl.hnc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrx : α
this : size nil = 0 ∧ size nil = 0
hl : Balanced (node' nil lx (node 1 nil lrx nil))
sl : Sized (node ls nil lx (node 1 nil lrx nil))
⊢ ¬0 + 1 + 1 ≤ 1

Next tactic:
all_goals dsimp only [size]; decide