. apply (xs.cons a₁ b₁ ha₁xs).ascend.forAll_of_predHead
  . exact λ _ _ h₁ h₂ => trans (t:=r) h₁ h₂
  . exact hdxs
case a.a
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)
xs✝ : KVChain r β
a : α
b : β a
a₁ : α
b₁ : β a₁
xs : KVChain r fun a => β a
ha₁xs : boundLeft a₁ xs
a✝ : ∀ (d : α), boundLeft d xs → r d a → boundLeft d (insertWith f xs a b)
d : α
hdxs : boundLeft d (KVChain.cons a₁ b₁ xs ha₁xs)
hda : r d a
⊢ List.forAll (r d ∘ Sigma.fst) (KVChain.cons a₁ b₁ xs ha₁xs).toList

case a.a
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)
xs✝ : KVChain r β
a : α
b : β a
a₁ : α
b₁ : β a₁
xs : KVChain r fun a => β a
ha₁xs : boundLeft a₁ xs
a✝ : ∀ (d : α), boundLeft d xs → r d a → boundLeft d (insertWith f xs a b)
d : α
hdxs : boundLeft d (KVChain.cons a₁ b₁ xs ha₁xs)
hda : r d a
⊢ r d a
case a.a
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)
xs✝ : KVChain r β
a : α
b : β a
a₁ : α
b₁ : β a₁
xs : KVChain r fun a => β a
ha₁xs : boundLeft a₁ xs
a✝ : ∀ (d : α), boundLeft d xs → r d a → boundLeft d (insertWith f xs a b)
d : α
hdxs : boundLeft d (KVChain.cons a₁ b₁ xs ha₁xs)
hda : r d a
⊢ r d a