internlm/Lean-Github huggingface dataset:
Theorem: KVChain.forAll_fst_insertWithAux
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
ha₁xs : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a₁ b₁ ys ha₁ys).toList
hpa : p a
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
⊢ List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.snd.toList
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
ha₁xs : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a₁ b₁ ys ha₁ys).toList
hpa : p a
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
this : a = a₁
⊢ List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.snd.toList
Theorem: KVChain.forAll_fst_insertWithAux
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
ha₁xs : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a₁ b₁ ys ha₁ys).toList
hpa : p a
ha₁a : ¬r a₁ a
haa₁ : ¬r a a₁
this : a = a₁
⊢ List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b hzxs).val.snd.toList
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
⊢ List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) ha₁xs a b hzxs).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) ha₁xs a b hzxs).val.snd.toList
Theorem: KVChain.forAll_fst_insertWithAux
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
⊢ List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) ha₁xs a b hzxs).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst) (insertWithAux f xs (KVChain.cons a b₁ ys ha₁ys) ha₁xs a b hzxs).val.snd.toList
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
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
⊢ List.forAll (p ∘ Sigma.fst)
      (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 hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst)
      (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 hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).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)
ys✝ : KVChain r β
a : α
b : β a
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
⊢ List.forAll (p ∘ Sigma.fst)
      (Option.casesOn none { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst)
      (Option.casesOn none { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.snd.toList
Tactic:
case none => exact ⟨hpxs, hpys.right⟩
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)
ys✝ : KVChain r β
a : α
b : β a
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
bf : β a
⊢ List.forAll (p ∘ Sigma.fst)
      (Option.casesOn (some bf) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst)
      (Option.casesOn (some bf) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.snd.toList
Tactic:
case some bf => exact ⟨⟨hpys.left, hpxs⟩, hpys.right⟩
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)
ys✝ : KVChain r β
a : α
b : β a
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
⊢ List.forAll (p ∘ Sigma.fst)
      (Option.casesOn none { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst)
      (Option.casesOn none { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.snd.toList
Tactic:
exact ⟨hpxs, hpys.right⟩
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)
ys✝ : KVChain r β
a : α
b : β a
p : α → Prop
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),
    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
xs : KVChain (flip r) fun a => β a
hzxs : boundLeft a xs
hpxs : List.forAll (p ∘ Sigma.fst) xs.toList
hpa : p a
b₁ : β a
ha₁ys : boundLeft a ys
ha₁xs : hingeable xs (KVChain.cons a b₁ ys ha₁ys)
hpys : List.forAll (p ∘ Sigma.fst) (KVChain.cons a b₁ ys ha₁ys).toList
ha₁a haa₁ : ¬r a a
bf : β a
⊢ List.forAll (p ∘ Sigma.fst)
      (Option.casesOn (some bf) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.fst.toList ∧
    List.forAll (p ∘ Sigma.fst)
      (Option.casesOn (some bf) { val := (xs, ys), property := (_ : hingeable (xs, ys).fst (xs, ys).snd) } fun bf =>
              { val := (KVChain.cons a bf xs hzxs, ys),
                property :=
                  (_ : hingeable (KVChain.cons a bf xs hzxs) (KVChain.cons a bf xs hzxs, ys).snd) }).val.snd.toList
Tactic:
exact ⟨⟨hpys.left, hpxs⟩, hpys.right⟩
Proof state after:
no goals
Theorem: KVChain.boundLeft_insertWithAux_fst
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
⊢ ∀ {c : α}, r a c → boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
Tactic:
intro c hac
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
c : α
hac : r a c
⊢ boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
Theorem: KVChain.boundLeft_insertWithAux_fst
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
c : α
hac : r a c
⊢ boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
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
c : α
hac : r a c
⊢ ∀ {xs : KVChain (flip r) β} {hxsys : hingeable xs ys} {haxs : boundLeft a xs},
    boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
⊢ ∀ {xs : KVChain (flip r) β} {hxsys : hingeable xs ys} {haxs : boundLeft a xs},
    boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
Tactic:
apply ys.recOnList (motive:=λ ys=>∀ xs hxsys haxs, (insertWithAux f xs ys hxsys a b haxs).val.fst.boundLeft c)
Proof state after:
case nil
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
c : α
hac : r a c
⊢ ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs KVChain.nil) (haxs : boundLeft a xs),
    boundLeft c (insertWithAux f xs KVChain.nil hxsys a b haxs).val.fst

case cons
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
c : α
hac : r a c
⊢ ∀ (a_1 : α) (b_1 : β a_1) (xs : KVChain r fun a => β a) (haxs : boundLeft a_1 xs),
    (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        xs →
      (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        (KVChain.cons a_1 b_1 xs haxs)
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
⊢ ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs KVChain.nil) (haxs : boundLeft a xs),
    boundLeft c (insertWithAux f xs KVChain.nil hxsys a b haxs).val.fst
Tactic:
case nil =>
  intros; dsimp; exact hac
Proof state after:
no goals
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
⊢ ∀ (a_1 : α) (b_1 : β a_1) (xs : KVChain r fun a => β a) (haxs : boundLeft a_1 xs),
    (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        xs →
      (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        (KVChain.cons a_1 b_1 xs haxs)
Tactic:
case cons =>
  intro a₁ b₁ ys ha₁ys h_ind xs ha₁xs haxs
  apply dite (r a₁ a) <;> intro ha₁a
  . rw [insertWithAux_cons_rel ha₁a]
    exact h_ind _ (cons_hingeable.symm ▸ ha₁ys) (by exact ha₁a)
  . apply dite (r a a₁) <;> intro haa₁
    . rw [insertWithAux_cons_flip haa₁]
      exact hac
    . have : a = a₁ := Trichotomous.eq_of_incomp ⟨haa₁, ha₁a⟩
      cases this
      rw [insertWithAux_cons_rfl]
      cases f a b₁ b
      case none =>
        apply boundLeft_descend _ haxs
        exact λ _ => flip (trans (r:=r) (s:=r) (t:=r)) hac
      case some bf => exact hac
Proof state after:
no goals
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
⊢ ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs KVChain.nil) (haxs : boundLeft a xs),
    boundLeft c (insertWithAux f xs KVChain.nil hxsys a b haxs).val.fst
Tactic:
intros
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
c : α
hac : r a c
xs✝ : KVChain (flip r) fun a => β a
hxsys✝ : hingeable xs✝ KVChain.nil
haxs✝ : boundLeft a xs✝
⊢ boundLeft c (insertWithAux f xs✝ KVChain.nil hxsys✝ a b haxs✝).val.fst
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
xs✝ : KVChain (flip r) fun a => β a
hxsys✝ : hingeable xs✝ KVChain.nil
haxs✝ : boundLeft a xs✝
⊢ boundLeft c (insertWithAux f xs✝ KVChain.nil hxsys✝ a b haxs✝).val.fst
Tactic:
dsimp
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
c : α
hac : r a c
xs✝ : KVChain (flip r) fun a => β a
hxsys✝ : hingeable xs✝ KVChain.nil
haxs✝ : boundLeft a xs✝
⊢ boundLeft c (KVChain.cons a b xs✝ haxs✝)
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
xs✝ : KVChain (flip r) fun a => β a
hxsys✝ : hingeable xs✝ KVChain.nil
haxs✝ : boundLeft a xs✝
⊢ boundLeft c (KVChain.cons a b xs✝ haxs✝)
Tactic:
exact hac
Proof state after:
no goals
Theorem: KVChain.boundLeft_insertWithAux_fst
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)
ys : KVChain r β
a : α
b : β a
c : α
hac : r a c
⊢ ∀ (a_1 : α) (b_1 : β a_1) (xs : KVChain r fun a => β a) (haxs : boundLeft a_1 xs),
    (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        xs →
      (fun ys =>
          ∀ (xs : KVChain (flip r) fun a => β a) (hxsys : hingeable xs ys) (haxs : boundLeft a xs),
            boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst)
        (KVChain.cons a_1 b_1 xs haxs)
Tactic:
intro a₁ b₁ ys ha₁ys h_ind xs ha₁xs haxs
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
c : α
hac : r a c
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),
    boundLeft c (insertWithAux f xs ys hxsys a b haxs).val.fst
xs : KVChain (flip r) fun a => β a
ha₁xs : hingeable xs (KVChain.cons a₁ b₁ ys ha₁ys)
haxs : boundLeft a xs
⊢ boundLeft c (insertWithAux f xs (KVChain.cons a₁ b₁ ys ha₁ys) ha₁xs a b haxs).val.fst