internlm/Lean-Github huggingface dataset:
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : r a₁ a
⊢ (insertWithAux f (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys
          (_ : hingeable (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys) a b ha₁a).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
rw [insertWithAux_cons_rel ha₁a]
Proof state after:
case t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : r a₁ a
⊢ (insertWithAux f (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys
          (_ : hingeable (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys) a b ha₁a).val.snd =
    (insertWithAux f (KVChain.cons a₁ b₁ KVChain.nil (_ : boundLeft a₁ KVChain.nil)) ys
          (_ : hingeable (KVChain.cons a₁ b₁ KVChain.nil (_ : boundLeft a₁ KVChain.nil)) ys) a b ha₁a).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : r a₁ a
⊢ (insertWithAux f (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys
          (_ : hingeable (KVChain.cons a₁ b₁ xs (_ : boundLeft a₁ xs)) ys) a b ha₁a).val.snd =
    (insertWithAux f (KVChain.cons a₁ b₁ KVChain.nil (_ : boundLeft a₁ KVChain.nil)) ys
          (_ : hingeable (KVChain.cons a₁ b₁ KVChain.nil (_ : boundLeft a₁ KVChain.nil)) ys) a b ha₁a).val.snd
Tactic:
rw [h_ind, h_ind ⟨[⟨a₁,b₁⟩], _⟩]
Proof state after:
no goals
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
apply dite (r a a₁) <;> intro haa₁
Proof state after:
case e.t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd

case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd

case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
. rw [insertWithAux_cons_flip haa₁]
  rw [insertWithAux_cons_flip haa₁]
Proof state after:
case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
. have : a = a₁ := Trichotomous.eq_of_incomp ⟨haa₁, ha₁a⟩
  cases this
  rw [insertWithAux_cons_rfl]
  rw [insertWithAux_cons_rfl]
  cases f a b₁ b <;> rfl
Proof state after:
no goals
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
rw [insertWithAux_cons_flip haa₁]
Proof state after:
case e.t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : r a a₁
⊢ { val := (KVChain.cons a b xs haxs, KVChain.cons a₁ b₁ ys ha₁ys), property := haa₁ }.val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.t
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : r a a₁
⊢ { val := (KVChain.cons a b xs haxs, KVChain.cons a₁ b₁ ys ha₁ys), property := haa₁ }.val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
rw [insertWithAux_cons_flip haa₁]
Proof state after:
no goals
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
have : a = a₁ := Trichotomous.eq_of_incomp ⟨haa₁, ha₁a⟩
Proof state after:
case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
this : a = a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e
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
a₁ : α
b₁ : β a₁
ys : KVChain r fun a => β a
ha₁ys : boundLeft a₁ ys
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
hxsys : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
this : a = a₁
⊢ (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a₁ b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
cases this
Proof state after:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) hxsys a b haxs).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
rw [insertWithAux_cons_rfl]
Proof state after:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (Option.casesOn (f a b₁ b) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
          { val := (KVChain.cons a bf xs haxs, ys),
            property := (_ : hingeable (KVChain.cons a bf xs haxs) (KVChain.cons a bf xs haxs, ys).snd) }).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (Option.casesOn (f a b₁ b) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
          { val := (KVChain.cons a bf xs haxs, ys),
            property := (_ : hingeable (KVChain.cons a bf xs haxs) (KVChain.cons a bf xs haxs, ys).snd) }).val.snd =
    (insertWithAux f KVChain.nil (KVChain.cons a b₁ ys ha₁ys) True.intro a b True.intro).val.snd
Tactic:
rw [insertWithAux_cons_rfl]
Proof state after:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (Option.casesOn (f a b₁ b) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
          { val := (KVChain.cons a bf xs haxs, ys),
            property := (_ : hingeable (KVChain.cons a bf xs haxs) (KVChain.cons a bf xs haxs, ys).snd) }).val.snd =
    (Option.casesOn (f a b₁ b)
          { val := (KVChain.nil, ys), property := (_ : hingeable (KVChain.nil, ys).fst (KVChain.nil, ys).snd) }
          fun bf =>
          { val := (KVChain.cons a bf KVChain.nil True.intro, ys),
            property :=
              (_ :
                hingeable (KVChain.cons a bf KVChain.nil True.intro)
                  (KVChain.cons a bf KVChain.nil True.intro, ys).snd) }).val.snd
Theorem: KVChain.toList_snd_insertWithAux_inv
Proof state before:
case e.e.refl
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
ys : KVChain r fun a => β a
h_ind :
  ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
    (insertWithAux f xs ys hxsys a b haxs).val.snd = (insertWithAux f KVChain.nil ys True.intro a b True.intro).val.snd
xs : KVChain (flip r) fun a => β a
haxs : boundLeft a xs
b₁ : β a
ha₁ys : boundLeft a ys
hxsys : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
ha₁a haa₁ : ¬r a a
⊢ (Option.casesOn (f a b₁ b) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
          { val := (KVChain.cons a bf xs haxs, ys),
            property := (_ : hingeable (KVChain.cons a bf xs haxs) (KVChain.cons a bf xs haxs, ys).snd) }).val.snd =
    (Option.casesOn (f a b₁ b)
          { val := (KVChain.nil, ys), property := (_ : hingeable (KVChain.nil, ys).fst (KVChain.nil, ys).snd) }
          fun bf =>
          { val := (KVChain.cons a bf KVChain.nil True.intro, ys),
            property :=
              (_ :
                hingeable (KVChain.cons a bf KVChain.nil True.intro)
                  (KVChain.cons a bf KVChain.nil True.intro, ys).snd) }).val.snd
Tactic:
cases f a b₁ b <;> rfl
Proof state after:
no goals
Theorem: KVChain.forAll_fst_insertWithAux
Proof state before:
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 (flip r) β
ys : KVChain r β
hxsys : hingeable xs ys
a : α
b : β a
haxs : boundLeft a xs
⊢ ∀ {p : α → Prop},
    List.forAll (p ∘ Sigma.fst) xs.toList →
      List.forAll (p ∘ Sigma.fst) ys.toList →
        p a →
          List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.fst.toList ∧
            List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.snd.toList
Tactic:
intro p
Proof state after:
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 (flip r) β
ys : KVChain r β
hxsys : hingeable xs ys
a : α
b : β a
haxs : boundLeft a xs
p : α → Prop
⊢ List.forAll (p ∘ Sigma.fst) xs.toList →
    List.forAll (p ∘ Sigma.fst) ys.toList →
      p a →
        List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.fst.toList ∧
          List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.snd.toList
Theorem: KVChain.forAll_fst_insertWithAux
Proof state before:
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 (flip r) β
ys : KVChain r β
hxsys : hingeable xs ys
a : α
b : β a
haxs : boundLeft a xs
p : α → Prop
⊢ List.forAll (p ∘ Sigma.fst) xs.toList →
    List.forAll (p ∘ Sigma.fst) ys.toList →
      p a →
        List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.fst.toList ∧
          List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.snd.toList
Tactic:
revert xs
Proof state after:
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) β} {hxsys : hingeable xs ys} {haxs : boundLeft a xs},
    List.forAll (p ∘ Sigma.fst) xs.toList →
      List.forAll (p ∘ Sigma.fst) ys.toList →
        p a →
          List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.fst.toList ∧
            List.forAll (p ∘ Sigma.fst) (insertWithAux f xs ys hxsys a b haxs).val.snd.toList