-- Mathlib/Data/Ordmap/Ordset.lean
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

Proof state:
case pos.node.nil
α : Type u_1
x : α
ls : ℕ
lx : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
hr : (node rs rl rx rr).Balanced
sr : (node rs rl rx rr).Sized
h : ¬delta * ls < rs
h_1 : delta * rs < ls
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
hl : (node ls (node lls lll llx llr) lx nil).Balanced
sl : (node ls (node lls lll llx llr) lx nil).Sized
ld : delta ≤ (node lls lll llx llr).size + nil.size
⊢ (node ls (node lls lll llx llr) lx nil).rotateR x (node rs rl rx rr) =
    rec nil
      (fun size l x_1 r l_ih r_ih =>
        rec nil
          (fun size_1 l x_2 r l_ih r_ih =>
            if size_1 < ratio * size then
              node (ls + rs + 1) (node lls lll llx llr) lx (node (size_1 + rs + 1) nil x (node rs rl rx rr))
            else
              node (ls + rs + 1) (node (size + l.size + 1) (node lls lll llx llr) lx l) x_2
                (node (r.size + rs + 1) r x (node rs rl rx rr)))
          nil)
      (node lls lll llx llr)

Tactic:
exact absurd (le_trans ld (balancedSz_zero.1 hl.1)) (by decide)

Result:
no goals

Proof state:
α : Type u_1
l : Ordnode α
x : α
r : Ordnode α
hl : l.Balanced
hr : r.Balanced
sl : l.Sized
sr : r.Sized
⊢ l.balance x r = l.balance' x r

Tactic:
obtain - | ⟨ls, ll, lx, lr⟩ := l

Result:
case nil
α : Type u_1
x : α
r : Ordnode α
hr : r.Balanced
sr : r.Sized
hl : nil.Balanced
sl : nil.Sized
⊢ nil.balance x r = nil.balance' x r

case node
α : Type u_1
x : α
r : Ordnode α
hr : r.Balanced
sr : r.Sized
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
⊢ (node ls ll lx lr).balance x r = (node ls ll lx lr).balance' x r

Proof state:
α : Type u_1
x : α
hl : nil.Balanced
sl : nil.Sized
rs : ℕ
rx : α
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
hr : ((node rls rll rlx rlr).node' rx nil).Balanced
sr : (node rs (node rls rll rlx rlr) rx nil).Sized
this : (node rls rll rlx rlr).size ≤ 1
⊢ rll.size = 0 ∧ rlr.size = 0

Tactic:
rwa [size, sr.2.1.1, Nat.succ_le_succ_iff, Nat.le_zero, add_eq_zero] at this

Result:
no goals

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

Tactic:
obtain rfl : lrs = 1 := sl.2.2.1

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

Proof state:
case node.nil.node.nil.refl.refl.hc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
llx : α
this : nil.size = 0 ∧ nil.size = 0
hl : ((node 1 nil llx nil).node' lx nil).Balanced
sl : (node ls (node 1 nil llx nil) lx nil).Sized
⊢ nil.size < ratio * (node 1 nil llx nil).size

Tactic:
dsimp only [size]

Result:
case node.nil.node.nil.refl.refl.hc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
llx : α
this : nil.size = 0 ∧ nil.size = 0
hl : ((node 1 nil llx nil).node' lx nil).Balanced
sl : (node ls (node 1 nil llx nil) lx nil).Sized
⊢ 0 < ratio * 1

Proof state:
case nil.node.nil.node.refl.refl
α : Type u_1
x : α
hl : nil.Balanced
sl : nil.Sized
rs : ℕ
rx rrx : α
this : nil.size = 0 ∧ nil.size = 0
hr : (nil.node' rx (node 1 nil rrx nil)).Balanced
sr : (node rs nil rx (node 1 nil rrx nil)).Sized
⊢ node 3 (Ordnode.singleton x) rx (node 1 nil rrx nil) = nil.node3L x nil rx (node 1 nil rrx nil)

Tactic:
rfl

Result:
no goals

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

Tactic:
rw [if_neg, if_pos, rotateR_node, if_neg]

Result:
case node.nil.nil.node.refl.refl
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lrx : α
this : nil.size = 0 ∧ nil.size = 0
hl : (nil.node' lx (node 1 nil lrx nil)).Balanced
sl : (node ls nil lx (node 1 nil lrx nil)).Sized
⊢ node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x) = nil.node4R lx (node 1 nil lrx nil) x nil

case node.nil.nil.node.refl.refl.hnc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lrx : α
this : nil.size = 0 ∧ nil.size = 0
hl : (nil.node' lx (node 1 nil lrx nil)).Balanced
sl : (node ls nil lx (node 1 nil lrx nil)).Sized
⊢ ¬(node 1 nil lrx nil).size < ratio * nil.size

case node.nil.nil.node.refl.refl.hc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lrx : α
this : nil.size = 0 ∧ nil.size = 0
hl : (nil.node' lx (node 1 nil lrx nil)).Balanced
sl : (node ls nil lx (node 1 nil lrx nil)).Sized
⊢ 0 + 1 + 1 > 0

case node.nil.nil.node.refl.refl.hnc
α : Type u_1
x : α
ls : ℕ
lx : α
hr : nil.Balanced
sr : nil.Sized
lrx : α
this : nil.size = 0 ∧ nil.size = 0
hl : (nil.node' lx (node 1 nil lrx nil)).Balanced
sl : (node ls nil lx (node 1 nil lrx nil)).Sized
⊢ ¬0 + 1 + 1 ≤ 1

Proof state:
case neg
α : Type u_1
x : α
ls : ℕ
ll : Ordnode α
lx : α
lr : Ordnode α
hl : (node ls ll lx lr).Balanced
sl : (node ls ll lx lr).Sized
rs : ℕ
rx : α
h : delta * ls < rs
rls : ℕ
rll : Ordnode α
rlx : α
rlr : Ordnode α
rrs : ℕ
rrl : Ordnode α
rrx : α
rrr : Ordnode α
hr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Balanced
sr : (node rs (node rls rll rlx rlr) rx (node rrs rrl rrx rrr)).Sized
rd : delta ≤ (node rls rll rlx rlr).size + (node rrs rrl rrx rrr).size
h✝ : ¬rls < ratio * rrs
⊢ ls + rll.size + 1 + (rlr.size + rrs + 1) = ls + (rll.size + rlr.size + 1 + rrs + 1)

Tactic:
abel

Result:
no goals