case nil =>
  intro xs hxsys haxs hpxs _ hpa
  dsimp [insertWithAux, insertWithAux.loop]
  exact ⟨⟨hpa, hpxs⟩, True.intro⟩
α : Type u
r : α → α → Prop
inst✝¹ : StrictLinearOrder r
inst✝ : DecidableRel r
β : α → Type v
f : (a : α) → β a → β a → Option (β a)
ys : KVChain r β
a : α
b : β a
p : α → Prop
⊢ ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs KVChain.nil) (haxs : boundLeft a xs),
    List.forAll (p ∘ Sigma.fst) xs.toList →
      List.forAll (p ∘ Sigma.fst) KVChain.nil.toList →
        p a →
          List.forAll (p ∘ Sigma.fst) (insertWithAux f xs KVChain.nil hxsys a b haxs).val.fst.toList ∧
            List.forAll (p ∘ Sigma.fst) (insertWithAux f xs KVChain.nil hxsys a b haxs).val.snd.toList
no goals