-- Mathlib/Data/Ordmap/Ordset.lean
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

Proof state:
case neg.inr.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : ¬rl.size < ratio * rr.size
rl0 : rl.size > 0
l0 : l.size = 0
⊢ l.size = 0 ∧ rl.size = 1 ∧ rr.size ≤ 1 ∨
    0 < l.size ∧
      ratio * rr.size ≤ rl.size ∧
        delta * l.size ≤ rl.size + rr.size ∧ 3 * (rl.size + rr.size) ≤ 16 * l.size + 9 ∧ rl.size ≤ delta * rr.size

Tactic:
replace H3 := H3_0 l0

Result:
case neg.inr.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : ¬rl.size < ratio * rr.size
rl0 : rl.size > 0
l0 : l.size = 0
H3 : rl.size + rr.size ≤ 2
⊢ l.size = 0 ∧ rl.size = 1 ∧ rr.size ≤ 1 ∨
    0 < l.size ∧
      ratio * rr.size ≤ rl.size ∧
        delta * l.size ≤ rl.size + rr.size ∧ 3 * (rl.size + rr.size) ≤ 16 * l.size + 9 ∧ rl.size ≤ delta * rr.size

Proof state:
case neg.inr.inl.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : ¬rl.size < ratio * rr.size
rl0 : rl.size > 0
l0 : l.size = 0
H3 : rl.size + rr.size ≤ 2
rr0 : rr.size = 0
⊢ l.size = 0 ∧ rl.size = 1 ∧ rr.size ≤ 1 ∨
    0 < l.size ∧
      ratio * rr.size ≤ rl.size ∧
        delta * l.size ≤ rl.size + rr.size ∧ 3 * (rl.size + rr.size) ≤ 16 * l.size + 9 ∧ rl.size ≤ delta * rr.size

Tactic:
have := hr.3.1

Result:
case neg.inr.inl.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : ¬rl.size < ratio * rr.size
rl0 : rl.size > 0
l0 : l.size = 0
H3 : rl.size + rr.size ≤ 2
rr0 : rr.size = 0
this : BalancedSz rl.size rr.size
⊢ l.size = 0 ∧ rl.size = 1 ∧ rr.size ≤ 1 ∨
    0 < l.size ∧
      ratio * rr.size ≤ rl.size ∧
        delta * l.size ≤ rl.size + rr.size ∧ 3 * (rl.size + rr.size) ≤ 16 * l.size + 9 ∧ rl.size ≤ delta * rr.size

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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
⊢ l.size = 0 → rl.size + rr.size ≤ 2

Tactic:
intro l0

Result:
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
l0 : l.size = 0
⊢ rl.size + rr.size ≤ 2

Proof state:
case pos.inr.intro.refine_4
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : rl.size < ratio * rr.size
rr0 : rr.size > 0
l0 : l.size > 0
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
left✝ : rl.size ≤ delta * rr.size
hb₂ : rr.size ≤ delta * rl.size
⊢ rr.size ≤ delta * (l.size + rl.size + 1)

Tactic:
exact
  le_trans hb₂
    (Nat.mul_le_mul_left _ <| le_trans (Nat.le_add_left _ _) (Nat.le_add_right _ _))

Result:
no goals

Proof state:
case pos.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : rl.size < ratio * rr.size
rr0 : rr.size > 0
l0 : l.size = 0
⊢ BalancedSz 0 rl.size ∧ BalancedSz (0 + rl.size + 1) rr.size

Tactic:
replace H3 := H3_0 l0

Result:
case pos.inl
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : rl.size < ratio * rr.size
rr0 : rr.size > 0
l0 : l.size = 0
H3 : rl.size + rr.size ≤ 2
⊢ BalancedSz 0 rl.size ∧ BalancedSz (0 + rl.size + 1) rr.size

Proof state:
case neg.inr.inr
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 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1
H2 : delta * l.size ≤ rl.size + rr.size
H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2
H3_0 : l.size = 0 → rl.size + rr.size ≤ 2
H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3
ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1
hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1
h : ¬rl.size < ratio * rr.size
rl0 : rl.size > 0
l0 : l.size > 0
⊢ l.size = 0 ∧ rl.size = 1 ∧ rr.size ≤ 1 ∨
    0 < l.size ∧
      ratio * rr.size ≤ rl.size ∧
        delta * l.size ≤ rl.size + rr.size ∧ 3 * (rl.size + rr.size) ≤ 16 * l.size + 9 ∧ rl.size ≤ delta * rr.size

Tactic:
exact
  Or.inr ⟨l0, not_lt.1 h, H2, Valid'.rotateL_lemma₄ (H3p l0), (hr.3.1.resolve_left (hlp l0)).1⟩

Result:
no goals