l3lab/ntp-mathlib huggingface dataset:
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * size (Ordnode.node rs rl rx rr) ≤ 9 * size l + 5 ∨ size (Ordnode.node rs rl rx rr) ≤ 3
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
rw [hr.2.size_eq] at H3
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr + 1) ≤ 9 * size l + 5 ∨ size rl + size rr + 1 ≤ 3
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
⊢ size l = 0 → size rl + size rr ≤ 2

Next tactic:
intro l0
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
l0 : size l = 0
⊢ size rl + size rr ≤ 2

Next tactic:
rw [l0] at H3
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2
l0 : size l = 0
⊢ size rl + size rr ≤ 2

Next tactic:
exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2
l0 : size l = 0
h : 2 * (size rl + size rr) ≤ 9 * 0 + 3
⊢ 0 < 2

Next tactic:
decide
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * 0 + 3 ∨ size rl + size rr ≤ 2
l0 : size l = 0
h : 2 * (size rl + size rr) ≤ 9 * 0 + 3
⊢ 9 * 0 + 3 ≤ 2 * 2

Next tactic:
decide
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
l0 : 1 ≤ size l
⊢ size rl + size rr ≤ 2 → 2 * (size rl + size rr) ≤ 9 * size l + 3

Next tactic:
intro
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
l0 : 1 ≤ size l
a✝ : size rl + size rr ≤ 2
⊢ 2 * (size rl + size rr) ≤ 9 * size l + 3

Next tactic:
linarith
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; linarith
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
⊢ ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1

Next tactic:
intros
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
a✝² b✝ : ℕ
a✝¹ : 1 ≤ a✝²
a✝ : a✝² + b✝ ≤ 2
⊢ b✝ ≤ 1

Next tactic:
linarith
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; linarith
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
have hlp : size l > 0 → ¬size rl + size rr ≤ 1 := fun l0 hb =>
    absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by decide)
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; linarith
  have hlp : size l > 0 → ¬size rl + size rr ≤ 1 := fun l0 hb =>
    absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by 

Proof state:
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
l0 : size l > 0
hb : size rl + size rr ≤ 1
⊢ ¬delta * Nat.succ 0 ≤ 1

Next tactic:
decide
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; linarith
  have hlp : size l > 0 → ¬size rl + size rr ≤ 1 := fun l0 hb =>
    absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by decide)
  

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : size l > 0 → ¬size rl + size rr ≤ 1
⊢ Valid' o₁ (Ordnode.rotateL l x (Ordnode.node rs rl rx rr)) o₂

Next tactic:
rw [Ordnode.rotateL]
theorem Valid'.rotateL {l} {x : α} {r o₁ o₂} (hl : Valid' o₁ l x) (hr : Valid' x r o₂)
    (H1 : ¬size l + size r ≤ 1) (H2 : delta * size l < size r)
    (H3 : 2 * size r ≤ 9 * size l + 5 ∨ size r ≤ 3) : Valid' o₁ (@rotateL α l x r) o₂ := by
  cases' r with rs rl rx rr; · cases H2
  rw [hr.2.size_eq, Nat.lt_succ_iff] at H2
  rw [hr.2.size_eq] at H3
  replace H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2 :=
    H3.imp (@Nat.le_of_add_le_add_right _ 2 _) Nat.le_of_succ_le_succ
  have H3_0 : size l = 0 → size rl + size rr ≤ 2 := by
    intro l0; rw [l0] at H3
    exact
      (or_iff_right_of_imp fun h => (mul_le_mul_left (by decide)).1 (le_trans h (by decide))).1 H3
  have H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3 := fun l0 : 1 ≤ size l =>
    (or_iff_left_of_imp <| by intro; linarith).1 H3
  have ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1 := by intros; linarith
  have hlp : size l > 0 → ¬size rl + size rr ≤ 1 := fun l0 hb =>
    absurd (le_trans (le_trans (Nat.mul_le_mul_left _ l0) H2) hb) (by decide)
  rw [Ordnode.rotateL]; 

Proof state:
case node
α : Type u_1
inst✝ : Preorder α
l : Ordnode α
x : α
o₁ : WithBot α
o₂ : WithTop α
hl : Valid' o₁ l ↑x
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂
H1 : ¬size l + size (Ordnode.node rs rl rx rr) ≤ 1
H2 : delta * size l ≤ size rl + size rr
H3 : 2 * (size rl + size rr) ≤ 9 * size l + 3 ∨ size rl + size rr ≤ 2
H3_0 : size l = 0 → size rl + size rr ≤ 2
H3p : size l > 0 → 2 * (size rl + size rr) ≤ 9 * size l + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : size l > 0 → ¬size rl + size rr ≤ 1
⊢ Valid' o₁ (if size rl < ratio * size rr then Ordnode.node3L l x rl rx rr else Ordnode.node4L l x rl rx rr) o₂

Next tactic:
split_ifs with h