-- Mathlib/Data/Ordmap/Ordset.lean
theorem dual_balanceL (l : Ordnode α) (x : α) (r : Ordnode α) :
    dual (balanceL l x r) = balanceR (dual r) x (dual l)

Proof state:
case nil.nil
α : Type u_1
x : α
⊢ (casesOn (id nil)
        (casesOn (id nil) (Ordnode.singleton x) fun ls ll lx lr =>
          casesOn (id ll)
            (casesOn lr (node 2 nil x nil) fun size l lrx r => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
            fun lls l x_1 r =>
            casesOn (id lr) (node 3 ll lx (Ordnode.singleton x)) fun lrs lrl lrx lrr =>
              if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
              else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
        fun rs l x_1 r =>
        casesOn (id nil) (node (rs + 1) nil x nil) fun ls ll lx lr =>
          if ls > delta * rs then
            casesOn (id ll) nil fun lls l x_2 r =>
              casesOn (id lr) nil fun lrs lrl lrx lrr =>
                if lrs < ratio * lls then node (ls + rs + 1) ll lx (node (rs + lrs + 1) lr x nil)
                else node (ls + rs + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + rs + 1) lrr x nil)
          else node (ls + rs + 1) nil x nil).dual =
    casesOn (id nil.dual)
      (casesOn (id nil.dual) (Ordnode.singleton x) fun rs rl rx rr =>
        casesOn (id rr)
          (casesOn rl (node 2 nil x nil.dual) fun size l rlx r =>
            node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
          fun rrs l x_1 r =>
          casesOn (id rl) (node 3 (Ordnode.singleton x) rx rr) fun rls rll rlx rlr =>
            if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
            else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
      fun ls l x_1 r =>
      casesOn (id nil.dual) (node (ls + 1) nil.dual x nil) fun rs rl rx rr =>
        if rs > delta * ls then
          casesOn (id rr) nil fun rrs l x_2 r =>
            casesOn (id rl) nil fun rls rll rlx rlr =>
              if rls < ratio * rrs then node (ls + rs + 1) (node (ls + rls + 1) nil.dual x rl) rx rr
              else
                node (ls + rs + 1) (node (ls + rll.size + 1) nil.dual x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr)
        else node (ls + rs + 1) nil.dual x nil.dual

Tactic:
rfl

Result:
no goals

Proof state:
case pos.node.node
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
ls : ℕ
lx : α
h✝ : ls > delta * rs
lls : ℕ
lll : Ordnode α
llx : α
llr : Ordnode α
lrs : ℕ
lrl : Ordnode α
lrx : α
lrr : Ordnode α
⊢ (if lrs < ratio * lls then
        node (ls + rs + 1) (node lls lll llx llr) lx (node (rs + lrs + 1) (node lrs lrl lrx lrr) x (node rs rl rx rr))
      else
        node (ls + rs + 1) (node (lls + lrl.size + 1) (node lls lll llx llr) lx lrl) lrx
          (node (lrr.size + rs + 1) lrr x (node rs rl rx rr))).dual =
    if lrs < ratio * lls then
      node (rs + ls + 1) (node (rs + lrs + 1) (node rs rr.dual rx rl.dual) x (node lrs lrr.dual lrx lrl.dual)) lx
        (node lls llr.dual llx lll.dual)
    else
      node (rs + ls + 1) (node (rs + lrr.dual.size + 1) (node rs rr.dual rx rl.dual) x lrr.dual) lrx
        (node (lrl.dual.size + lls + 1) lrl.dual lx (node lls llr.dual llx lll.dual))

Tactic:
split_ifs with h <;> simp [h, add_comm]

Result:
no goals

Proof state:
case node.nil
α : Type u_1
x : α
rs : ℕ
rl : Ordnode α
rx : α
rr : Ordnode α
⊢ (casesOn (id (node rs rl rx rr))
        (casesOn (id nil) (Ordnode.singleton x) fun ls ll lx lr =>
          casesOn (id ll)
            (casesOn lr (node 2 nil x nil) fun size l lrx r => node 3 (Ordnode.singleton lx) lrx (Ordnode.singleton x))
            fun lls l x_1 r =>
            casesOn (id lr) (node 3 ll lx (Ordnode.singleton x)) fun lrs lrl lrx lrr =>
              if lrs < ratio * lls then node (ls + 1) ll lx (node (lrs + 1) lr x nil)
              else node (ls + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx (node (lrr.size + 1) lrr x nil))
        fun rs_1 l x_1 r =>
        casesOn (id nil) (node (rs_1 + 1) nil x (node rs rl rx rr)) fun ls ll lx lr =>
          if ls > delta * rs_1 then
            casesOn (id ll) nil fun lls l x_2 r =>
              casesOn (id lr) nil fun lrs lrl lrx lrr =>
                if lrs < ratio * lls then node (ls + rs_1 + 1) ll lx (node (rs_1 + lrs + 1) lr x (node rs rl rx rr))
                else
                  node (ls + rs_1 + 1) (node (lls + lrl.size + 1) ll lx lrl) lrx
                    (node (lrr.size + rs_1 + 1) lrr x (node rs rl rx rr))
          else node (ls + rs_1 + 1) nil x (node rs rl rx rr)).dual =
    casesOn (id (node rs rl rx rr).dual)
      (casesOn (id nil.dual) (Ordnode.singleton x) fun rs rl rx rr =>
        casesOn (id rr)
          (casesOn rl (node 2 nil x nil.dual) fun size l rlx r =>
            node 3 (Ordnode.singleton x) rlx (Ordnode.singleton rx))
          fun rrs l x_1 r =>
          casesOn (id rl) (node 3 (Ordnode.singleton x) rx rr) fun rls rll rlx rlr =>
            if rls < ratio * rrs then node (rs + 1) (node (rls + 1) nil x rl) rx rr
            else node (rs + 1) (node (rll.size + 1) nil x rll) rlx (node (rlr.size + rrs + 1) rlr rx rr))
      fun ls l x_1 r =>
      casesOn (id nil.dual) (node (ls + 1) (node rs rl rx rr).dual x nil) fun rs_1 rl_1 rx_1 rr_1 =>
        if rs_1 > delta * ls then
          casesOn (id rr_1) nil fun rrs l x_2 r =>
            casesOn (id rl_1) nil fun rls rll rlx rlr =>
              if rls < ratio * rrs then
                node (ls + rs_1 + 1) (node (ls + rls + 1) (node rs rl rx rr).dual x rl_1) rx_1 rr_1
              else
                node (ls + rs_1 + 1) (node (ls + rll.size + 1) (node rs rl rx rr).dual x rll) rlx
                  (node (rlr.size + rrs + 1) rlr rx_1 rr_1)
        else node (ls + rs_1 + 1) (node rs rl rx rr).dual x nil.dual

Tactic:
rfl

Result:
no goals