Theorem: Ordnode.Valid'.eraseMax_aux
File: Mathlib/Data/Ordmap/Ordset.lean

Proof steps:
state_before: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (Ordnode.node s l x r) o₂
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
state_after: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (Ordnode.node s l x r) o₂
this : Ordnode.node s l x r = l.node' x r
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
tactic: have := H.2.eq_node'
state_before: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (Ordnode.node s l x r) o₂
this : Ordnode.node s l x r = l.node' x r
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
state_after: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (l.node' x r) o₂
this : Ordnode.node s l x r = l.node' x r
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
tactic: rw [this] at H
state_before: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (l.node' x r) o₂
this : Ordnode.node s l x r = l.node' x r
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
state_after: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (l.node' x r) o₂
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
tactic: clear this
state_before: α : Type u_1
inst✝ : Preorder α
s : ℕ
l : Ordnode α
x : α
r : Ordnode α
o₁ : WithBot α
o₂ : WithTop α
H : Valid' o₁ (l.node' x r) o₂
⊢ Valid' o₁ (l.node' x r).eraseMax ↑(findMax' x r) ∧ (l.node' x r).size = (l.node' x r).eraseMax.size + 1
state_after: case nil
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x nil) o₂
⊢ Valid' o₁ (l.node' x nil).eraseMax ↑(findMax' x nil) ∧ (l.node' x nil).size = (l.node' x nil).eraseMax.size + 1

case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)) o₂
⊢ Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)).eraseMax ↑(findMax' x (Ordnode.node rs rl rx rr)) ∧
    (l.node' x (Ordnode.node rs rl rx rr)).size = (l.node' x (Ordnode.node rs rl rx rr)).eraseMax.size + 1
tactic: induction' r with rs rl rx rr _ IHrr generalizing l x o₁
state_before: case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)) o₂
⊢ Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)).eraseMax ↑(findMax' x (Ordnode.node rs rl rx rr)) ∧
    (l.node' x (Ordnode.node rs rl rx rr)).size = (l.node' x (Ordnode.node rs rl rx rr)).eraseMax.size + 1
state_after: case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
⊢ Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)).eraseMax ↑(findMax' x (Ordnode.node rs rl rx rr)) ∧
    (l.node' x (Ordnode.node rs rl rx rr)).size = (l.node' x (Ordnode.node rs rl rx rr)).eraseMax.size + 1
tactic: have := H.2.2.2.eq_node'
state_before: case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
⊢ Valid' o₁ (l.node' x (Ordnode.node rs rl rx rr)).eraseMax ↑(findMax' x (Ordnode.node rs rl rx rr)) ∧
    (l.node' x (Ordnode.node rs rl rx rr)).size = (l.node' x (Ordnode.node rs rl rx rr)).eraseMax.size + 1
state_after: case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
⊢ Valid' o₁ (l.node' x (rl.node' rx rr)).eraseMax ↑(findMax' x (rl.node' rx rr)) ∧
    (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
tactic: rw [this] at H ⊢
state_before: case node
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
⊢ Valid' o₁ (l.node' x (rl.node' rx rr)).eraseMax ↑(findMax' x (rl.node' rx rr)) ∧
    (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
state_after: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ Valid' o₁ (l.node' x (rl.node' rx rr)).eraseMax ↑(findMax' x (rl.node' rx rr)) ∧
    (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
tactic: rcases IHrr H.right with ⟨h, e⟩
state_before: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ Valid' o₁ (l.node' x (rl.node' rx rr)).eraseMax ↑(findMax' x (rl.node' rx rr)) ∧
    (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
state_after: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
tactic: refine ⟨Valid'.balanceL H.left h (Or.inr ⟨_, Or.inr e, H.3.1⟩), ?_⟩
state_before: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ (l.node' x (rl.node' rx rr)).size = (l.node' x (rl.node' rx rr)).eraseMax.size + 1
state_after: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ (l.node' x (rl.node' rx rr)).size = l.size + (rl.node' rx rr).eraseMax.size + 1 + 1
tactic: rw [eraseMax, size_balanceL H.3.2.1 h.3 H.2.2.1 h.2 (Or.inr ⟨_, Or.inr e, H.3.1⟩)]
state_before: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ (l.node' x (rl.node' rx rr)).size = l.size + (rl.node' rx rr).eraseMax.size + 1 + 1
state_after: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ l.size + ((rl.node' rx rr).eraseMax.size + 1) + 1 = l.size + (rl.node' rx rr).eraseMax.size + 1 + 1
tactic: rw [size_node, e]
state_before: case node.intro
α : Type u_1
inst✝ : Preorder α
s : ℕ
o₂ : WithTop α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
l_ih✝ :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rl) o₂ →
      Valid' o₁ (l.node' x rl).eraseMax ↑(findMax' x rl) ∧ (l.node' x rl).size = (l.node' x rl).eraseMax.size + 1
IHrr :
  ∀ {l : Ordnode α} {x : α} {o₁ : WithBot α},
    Valid' o₁ (l.node' x rr) o₂ →
      Valid' o₁ (l.node' x rr).eraseMax ↑(findMax' x rr) ∧ (l.node' x rr).size = (l.node' x rr).eraseMax.size + 1
l : Ordnode α
x : α
o₁ : WithBot α
H : Valid' o₁ (l.node' x (rl.node' rx rr)) o₂
this : Ordnode.node rs rl rx rr = rl.node' rx rr
h : Valid' (↑x) (rl.node' rx rr).eraseMax ↑(findMax' rx rr)
e : (rl.node' rx rr).size = (rl.node' rx rr).eraseMax.size + 1
⊢ l.size + ((rl.node' rx rr).eraseMax.size + 1) + 1 = l.size + (rl.node' rx rr).eraseMax.size + 1 + 1
state_after: no goals
tactic: rfl