l3lab/ntp-mathlib huggingface dataset:
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; 

Proof state:
case neg.intro.inl
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬0 + 0 ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr = 0
ml0 : size ml = 0
⊢ BalancedSz (size l) (size ml) ∧
    BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1)

Next tactic:
cases mm (by decide)
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬0 + 0 ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr = 0
ml0 : size ml = 0
⊢ 0 + 0 ≤ 1

Next tactic:
decide
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    

Proof state:
case neg.intro.inr
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
⊢ BalancedSz (size l) (size ml) ∧
    BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1)

Next tactic:
have : 2 * size l ≤ size ml + size mr + 1 := by
      have := Nat.mul_le_mul_left ratio lr₁
      rw [mul_left_comm, mul_add] at this
      have := le_trans this (add_le_add_left mr₁ _)
      rw [← Nat.succ_mul] at this
      exact (mul_le_mul_left (by decide)).1 this
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    have : 2 * size l ≤ size ml + size mr + 1 := by
      

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
⊢ 2 * size l ≤ size ml + size mr + 1

Next tactic:
have := Nat.mul_le_mul_left ratio lr₁
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    have : 2 * size l ≤ size ml + size mr + 1 := by
      have := Nat.mul_le_mul_left ratio lr₁
      

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
this : ratio * (delta * size l) ≤ ratio * (size ml + size mr + 1 + size r)
⊢ 2 * size l ≤ size ml + size mr + 1

Next tactic:
rw [mul_left_comm, mul_add] at this
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    have : 2 * size l ≤ size ml + size mr + 1 := by
      have := Nat.mul_le_mul_left ratio lr₁
      rw [mul_left_comm, mul_add] at this
      

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
this : delta * (ratio * size l) ≤ ratio * (size ml + size mr + 1) + ratio * size r
⊢ 2 * size l ≤ size ml + size mr + 1

Next tactic:
have := le_trans this (add_le_add_left mr₁ _)
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    have : 2 * size l ≤ size ml + size mr + 1 := by
      have := Nat.mul_le_mul_left ratio lr₁
      rw [mul_left_comm, mul_add] at this
      have := le_trans this (add_le_add_left mr₁ _)
      

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
this✝ : delta * (ratio * size l) ≤ ratio * (size ml + size mr + 1) + ratio * size r
this : delta * (ratio * size l) ≤ ratio * (size ml + size mr + 1) + (size ml + size mr + 1)
⊢ 2 * size l ≤ size ml + size mr + 1

Next tactic:
rw [← Nat.succ_mul] at this
theorem Valid'.node4L {l} {x : α} {m} {y : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hm : Valid' x m y)
    (hr : Valid' (↑y) r o₂) (Hm : 0 < size m)
    (H : size l = 0 ∧ size m = 1 ∧ size r ≤ 1 ∨
        0 < size l ∧
          ratio * size r ≤ size m ∧
            delta * size l ≤ size m + size r ∧
              3 * (size m + size r) ≤ 16 * size l + 9 ∧ size m ≤ delta * size r) :
    Valid' o₁ (@node4L α l x m y r) o₂ := by
  cases' m with s ml z mr; · cases Hm
  suffices
    BalancedSz (size l) (size ml) ∧
      BalancedSz (size mr) (size r) ∧ BalancedSz (size l + size ml + 1) (size mr + size r + 1) from
    Valid'.node' (hl.node' hm.left this.1) (hm.right.node' hr this.2.1) this.2.2
  rcases H with (⟨l0, m1, r0⟩ | ⟨l0, mr₁, lr₁, lr₂, mr₂⟩)
  · rw [hm.2.size_eq, Nat.succ_inj', add_eq_zero_iff] at m1
    rw [l0, m1.1, m1.2]; revert r0; rcases size r with (_ | _ | _) <;>
      [decide; decide; (intro r0; unfold BalancedSz delta; linarith)]
  · rcases Nat.eq_zero_or_pos (size r) with r0 | r0
    · rw [r0] at mr₂; cases not_le_of_lt Hm mr₂
    rw [hm.2.size_eq] at lr₁ lr₂ mr₁ mr₂
    by_cases mm : size ml + size mr ≤ 1
    · have r1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans mr₁ (Nat.succ_le_succ mm) : _ ≤ ratio * 1)) r0
      rw [r1, add_assoc] at lr₁
      have l1 :=
        le_antisymm
          ((mul_le_mul_left (by decide)).1 (le_trans lr₁ (add_le_add_right mm 2) : _ ≤ delta * 1))
          l0
      rw [l1, r1]
      revert mm; cases size ml <;> cases size mr <;> intro mm
      · decide
      · rw [Nat.zero_eq, zero_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
        decide
      · rcases mm with (_ | ⟨⟨⟩⟩); decide
      · rw [Nat.succ_add] at mm; rcases mm with (_ | ⟨⟨⟩⟩)
    rcases hm.3.1.resolve_left mm with ⟨mm₁, mm₂⟩
    rcases Nat.eq_zero_or_pos (size ml) with ml0 | ml0
    · rw [ml0, mul_zero, le_zero_iff] at mm₂
      rw [ml0, mm₂] at mm; cases mm (by decide)
    have : 2 * size l ≤ size ml + size mr + 1 := by
      have := Nat.mul_le_mul_left ratio lr₁
      rw [mul_left_comm, mul_add] at this
      have := le_trans this (add_le_add_left mr₁ _)
      rw [← Nat.succ_mul] at this
      

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x y : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
hr : Valid' (↑y) r o₂
s : ℕ
ml : Ordnode α
z : α
mr : Ordnode α
hm : Valid' (↑x) (Ordnode.node s ml z mr) ↑y
Hm : 0 < size (Ordnode.node s ml z mr)
l0 : 0 < size l
mr₁ : ratio * size r ≤ size ml + size mr + 1
lr₁ : delta * size l ≤ size ml + size mr + 1 + size r
lr₂ : 3 * (size ml + size mr + 1 + size r) ≤ 16 * size l + 9
mr₂ : size ml + size mr + 1 ≤ delta * size r
r0 : size r > 0
mm : ¬size ml + size mr ≤ 1
mm₁ : size ml ≤ delta * size mr
mm₂ : size mr ≤ delta * size ml
ml0 : size ml > 0
this✝ : delta * (ratio * size l) ≤ ratio * (size ml + size mr + 1) + ratio * size r
this : delta * (ratio * size l) ≤ Nat.succ ratio * (size ml + size mr + 1)
⊢ 2 * size l ≤ size ml + size mr + 1

Next tactic:
exact (mul_le_mul_left (by decide)).1 this