l3lab/ntp-mathlib huggingface dataset:
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
          

Proof state:
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
this : size (node lrs lrl lrx lrr) ≤ 1
⊢ size lrl = 0 ∧ size lrr = 0

Next tactic:
rwa [size, sl.2.2.1, Nat.succ_le_succ_iff, le_zero_iff, add_eq_zero_iff] at this
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
        

Proof state:
case node.nil.nil.node
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
hl : Balanced (node' nil lx (node lrs lrl lrx lrr))
sl : Sized (node ls nil lx (node lrs lrl lrx lrr))
this : size lrl = 0 ∧ size lrr = 0
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
    if 0 + lrs + 1 ≤ 1 then node' (node' nil lx (node lrs lrl lrx lrr)) x nil
    else
      if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs lrl lrx lrr)) x nil
      else
        if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs lrl lrx lrr)) x nil
        else node' (node' nil lx (node lrs lrl lrx lrr)) x nil

Next tactic:
cases sl.2.2.2.1.size_eq_zero.1 this.1
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
        

Proof state:
case node.nil.nil.node.refl
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrs : ℕ
lrx : α
lrr : Ordnode α
hl : Balanced (node' nil lx (node lrs nil lrx lrr))
sl : Sized (node ls nil lx (node lrs nil lrx lrr))
this : size nil = 0 ∧ size lrr = 0
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
    if 0 + lrs + 1 ≤ 1 then node' (node' nil lx (node lrs nil lrx lrr)) x nil
    else
      if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs nil lrx lrr)) x nil
      else
        if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs nil lrx lrr)) x nil
        else node' (node' nil lx (node lrs nil lrx lrr)) x nil

Next tactic:
cases sl.2.2.2.2.size_eq_zero.1 this.2
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
        

Proof state:
case node.nil.nil.node.refl.refl
α : Type u_1
x : α
ls : ℕ
lx : α
hr : Balanced nil
sr : Sized nil
lrs : ℕ
lrx : α
hl : Balanced (node' nil lx (node lrs nil lrx nil))
sl : Sized (node ls nil lx (node lrs nil lrx nil))
this : size nil = 0 ∧ size nil = 0
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
    if 0 + lrs + 1 ≤ 1 then node' (node' nil lx (node lrs nil lrx nil)) x nil
    else
      if 0 > delta * (0 + lrs + 1) then rotateL (node' nil lx (node lrs nil lrx nil)) x nil
      else
        if 0 + lrs + 1 > 0 then rotateR (node' nil lx (node lrs nil lrx nil)) x nil
        else node' (node' nil lx (node lrs nil lrx nil)) x nil

Next tactic:
obtain rfl : lrs = 1 := sl.2.2.1
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
        

Proof state:
case node.nil.nil.node.refl.refl
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))
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) =
    if 0 + 1 + 1 ≤ 1 then node' (node' nil lx (node 1 nil lrx nil)) x nil
    else
      if 0 > delta * (0 + 1 + 1) then rotateL (node' nil lx (node 1 nil lrx nil)) x nil
      else
        if 0 + 1 + 1 > 0 then rotateR (node' nil lx (node 1 nil lrx nil)) x nil
        else node' (node' nil lx (node 1 nil lrx nil)) x nil

Next tactic:
rw [if_neg, if_neg, if_pos, rotateR, if_neg]
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]; · 

Proof state:
case node.nil.nil.node.refl.refl
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))
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) = node4R nil lx (node 1 nil lrx nil) x nil

Next tactic:
rfl
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
        all_goals 

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

Next tactic:
dsimp only [size]
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
        all_goals dsimp only [size]; 

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))
⊢ ¬1 < ratio * 0

Next tactic:
decide
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
        all_goals 

Proof state:
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

Next tactic:
dsimp only [size]