l3lab/ntp-mathlib huggingface dataset:
/-- A fully faithful functor reflects limits. -/
instance fullyFaithfulReflectsLimits [Full F] [Faithful F] : ReflectsLimitsOfSize.{w, w'} F where
  reflectsLimitsOfShape {J} 𝒥₁ :=
    { reflectsLimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsLimit.mkConeMorphism fun s =>
                (Cones.functoriality K F).preimage (t.liftConeMorphism _)) <| by
              apply fun s m => (Cones.functoriality K F).map_injective _
              intro s m
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cone K
t : IsLimit (F.mapCone c)
s : Cone K
m : s ⟶ c
⊢ (Cones.functoriality K F).map m =
    (Cones.functoriality K F).map
      ((Cones.functoriality K F).preimage (IsLimit.liftConeMorphism t ((Cones.functoriality K F).obj s)))

Next tactic:
rw [Functor.image_preimage]
/-- A fully faithful functor reflects limits. -/
instance fullyFaithfulReflectsLimits [Full F] [Faithful F] : ReflectsLimitsOfSize.{w, w'} F where
  reflectsLimitsOfShape {J} 𝒥₁ :=
    { reflectsLimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsLimit.mkConeMorphism fun s =>
                (Cones.functoriality K F).preimage (t.liftConeMorphism _)) <| by
              apply fun s m => (Cones.functoriality K F).map_injective _
              intro s m
              rw [Functor.image_preimage]
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cone K
t : IsLimit (F.mapCone c)
s : Cone K
m : s ⟶ c
⊢ (Cones.functoriality K F).map m = IsLimit.liftConeMorphism t ((Cones.functoriality K F).obj s)

Next tactic:
apply t.uniq_cone_morphism
/-- A fully faithful functor reflects colimits. -/
instance fullyFaithfulReflectsColimits [Full F] [Faithful F] :
    ReflectsColimitsOfSize.{w, w'} F where
  reflectsColimitsOfShape {J} 𝒥₁ :=
    { reflectsColimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsColimit.mkCoconeMorphism fun s =>
                (Cocones.functoriality K F).preimage (t.descCoconeMorphism _)) <| by
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cocone K
t : IsColimit (F.mapCocone c)
⊢ ∀ (s : Cocone K) (m : c ⟶ s),
    m = (Cocones.functoriality K F).preimage (IsColimit.descCoconeMorphism t ((Cocones.functoriality K F).obj s))

Next tactic:
apply fun s m => (Cocones.functoriality K F).map_injective _
/-- A fully faithful functor reflects colimits. -/
instance fullyFaithfulReflectsColimits [Full F] [Faithful F] :
    ReflectsColimitsOfSize.{w, w'} F where
  reflectsColimitsOfShape {J} 𝒥₁ :=
    { reflectsColimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsColimit.mkCoconeMorphism fun s =>
                (Cocones.functoriality K F).preimage (t.descCoconeMorphism _)) <| by
              apply fun s m => (Cocones.functoriality K F).map_injective _
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cocone K
t : IsColimit (F.mapCocone c)
⊢ ∀ (s : Cocone K) (m : c ⟶ s),
    (Cocones.functoriality K F).map m =
      (Cocones.functoriality K F).map
        ((Cocones.functoriality K F).preimage (IsColimit.descCoconeMorphism t ((Cocones.functoriality K F).obj s)))

Next tactic:
intro s m
/-- A fully faithful functor reflects colimits. -/
instance fullyFaithfulReflectsColimits [Full F] [Faithful F] :
    ReflectsColimitsOfSize.{w, w'} F where
  reflectsColimitsOfShape {J} 𝒥₁ :=
    { reflectsColimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsColimit.mkCoconeMorphism fun s =>
                (Cocones.functoriality K F).preimage (t.descCoconeMorphism _)) <| by
              apply fun s m => (Cocones.functoriality K F).map_injective _
              intro s m
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cocone K
t : IsColimit (F.mapCocone c)
s : Cocone K
m : c ⟶ s
⊢ (Cocones.functoriality K F).map m =
    (Cocones.functoriality K F).map
      ((Cocones.functoriality K F).preimage (IsColimit.descCoconeMorphism t ((Cocones.functoriality K F).obj s)))

Next tactic:
rw [Functor.image_preimage]
/-- A fully faithful functor reflects colimits. -/
instance fullyFaithfulReflectsColimits [Full F] [Faithful F] :
    ReflectsColimitsOfSize.{w, w'} F where
  reflectsColimitsOfShape {J} 𝒥₁ :=
    { reflectsColimit := fun {K} =>
        { reflects := fun {c} t =>
            (IsColimit.mkCoconeMorphism fun s =>
                (Cocones.functoriality K F).preimage (t.descCoconeMorphism _)) <| by
              apply fun s m => (Cocones.functoriality K F).map_injective _
              intro s m
              rw [Functor.image_preimage]
              

Proof state:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J✝ : Type w
inst✝² : Category.{w', w} J✝
K✝ : J✝ ⥤ C
F : C ⥤ D
inst✝¹ : Full F
inst✝ : Faithful F
J : Type w'
𝒥₁ : Category.{w, w'} J
K : J ⥤ C
c : Cocone K
t : IsColimit (F.mapCocone c)
s : Cocone K
m : c ⟶ s
⊢ (Cocones.functoriality K F).map m = IsColimit.descCoconeMorphism t ((Cocones.functoriality K F).obj s)

Next tactic:
apply t.uniq_cocone_morphism
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, cons.injEq, true_and]
    · rw [Thunk.get, ih]
  left_inv := by
    

Proof state:
α : Type u_1
⊢ LeftInverse toList ofList

Next tactic:
intro xs
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, cons.injEq, true_and]
    · rw [Thunk.get, ih]
  left_inv := by
    intro xs
    

Proof state:
α : Type u_1
xs : List α
⊢ toList (ofList xs) = xs

Next tactic:
induction xs
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, cons.injEq, true_and]
    · rw [Thunk.get, ih]
  left_inv := by
    intro xs
    induction xs
    · 

Proof state:
case nil
α : Type u_1
⊢ toList (ofList []) = []

Next tactic:
rfl
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, cons.injEq, true_and]
    · rw [Thunk.get, ih]
  left_inv := by
    intro xs
    induction xs
    · rfl
    · 

Proof state:
case cons
α : Type u_1
head✝ : α
tail✝ : List α
tail_ih✝ : toList (ofList tail✝) = tail✝
⊢ toList (ofList (head✝ :: tail✝)) = head✝ :: tail✝

Next tactic:
simpa [ofList, toList]
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    

Proof state:
α : Type u_1
⊢ Function.RightInverse toList ofList

Next tactic:
intro xs
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    

Proof state:
α : Type u_1
xs : LazyList α
⊢ ofList (toList xs) = xs

Next tactic:
induction' xs using LazyList.rec with _ _ _ _ ih
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · 

Proof state:
case nil
α : Type u_1
⊢ ofList (toList nil) = nil

Next tactic:
rfl
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · 

Proof state:
case cons
α : Type u_1
xs : LazyList α
hd✝ : α
tl✝ : Thunk (LazyList α)
tl_ih✝ : ?m.972 tl✝
⊢ ofList (toList (cons hd✝ tl✝)) = cons hd✝ tl✝

Next tactic:
simpa only [toList, ofList, cons.injEq, true_and]
/-- Isomorphism between strict and lazy lists. -/
def listEquivLazyList (α : Type*) : List α ≃ LazyList α where
  toFun := LazyList.ofList
  invFun := LazyList.toList
  right_inv := by
    intro xs
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, cons.injEq, true_and]
    · 

Proof state:
case mk
α : Type u_1
xs : LazyList α
fn✝ : Unit → LazyList α
ih : ∀ (a : Unit), ofList (toList (fn✝ a)) = fn✝ a
⊢ { fn := fun x => ofList (toList (Thunk.get { fn := fn✝ })) } = { fn := fn✝ }

Next tactic:
rw [Thunk.get, ih]
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        

Proof state:
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : ¬Thunk.get xs = Thunk.get ys
⊢ Decidable (cons x xs = cons y ys)

Next tactic:
apply isFalse
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : ¬Thunk.get xs = Thunk.get ys
⊢ ¬cons x xs = cons y ys

Next tactic:
simp only [cons.injEq, not_and]
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : ¬Thunk.get xs = Thunk.get ys
⊢ x = y → ¬xs = ys

Next tactic:
intro _ xs_ys
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : ¬Thunk.get xs = Thunk.get ys
a✝ : x = y
xs_ys : xs = ys
⊢ False

Next tactic:
apply h2
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : ¬Thunk.get xs = Thunk.get ys
a✝ : x = y
xs_ys : xs = ys
⊢ Thunk.get xs = Thunk.get ys

Next tactic:
rw [xs_ys]
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by 

Proof state:
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : Thunk.get xs = Thunk.get ys
⊢ Decidable (cons x xs = cons y ys)

Next tactic:
apply isTrue
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : Thunk.get xs = Thunk.get ys
⊢ cons x xs = cons y ys

Next tactic:
congr
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; 

Proof state:
case h.e_tl
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : Thunk.get xs = Thunk.get ys
⊢ xs = ys

Next tactic:
ext
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; 

Proof state:
case h.e_tl.eq
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : x = y
h2 : Thunk.get xs = Thunk.get ys
⊢ Thunk.get xs = Thunk.get ys

Next tactic:
exact h2
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by 

Proof state:
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : ¬x = y
⊢ Decidable (cons x xs = cons y ys)

Next tactic:
apply isFalse
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : ¬x = y
⊢ ¬cons x xs = cons y ys

Next tactic:
simp only [cons.injEq, not_and]
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : ¬x = y
⊢ x = y → ¬xs = ys

Next tactic:
intro
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; intro; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
x : α
xs : Thunk (LazyList α)
y : α
ys : Thunk (LazyList α)
h : ¬x = y
a✝ : x = y
⊢ ¬xs = ys

Next tactic:
contradiction
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; intro; contradiction
  | nil, cons _ _ => by 

Proof state:
α : Type u
inst✝ : DecidableEq α
hd✝ : α
tl✝ : Thunk (LazyList α)
⊢ Decidable (nil = cons hd✝ tl✝)

Next tactic:
apply isFalse
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; intro; contradiction
  | nil, cons _ _ => by apply isFalse; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
hd✝ : α
tl✝ : Thunk (LazyList α)
⊢ ¬nil = cons hd✝ tl✝

Next tactic:
simp
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; intro; contradiction
  | nil, cons _ _ => by apply isFalse; simp
  | cons _ _, nil => by 

Proof state:
α : Type u
inst✝ : DecidableEq α
hd✝ : α
tl✝ : Thunk (LazyList α)
⊢ Decidable (cons hd✝ tl✝ = nil)

Next tactic:
apply isFalse
instance decidableEq {α : Type u} [DecidableEq α] : DecidableEq (LazyList α)
  | nil, nil => isTrue rfl
  | cons x xs, cons y ys =>
    if h : x = y then
      match decidableEq xs.get ys.get with
      | isFalse h2 => by
        apply isFalse; simp only [cons.injEq, not_and]; intro _ xs_ys; apply h2; rw [xs_ys]
      | isTrue h2 => by apply isTrue; congr; ext; exact h2
    else by apply isFalse; simp only [cons.injEq, not_and]; intro; contradiction
  | nil, cons _ _ => by apply isFalse; simp
  | cons _ _, nil => by apply isFalse; 

Proof state:
case h
α : Type u
inst✝ : DecidableEq α
hd✝ : α
tl✝ : Thunk (LazyList α)
⊢ ¬cons hd✝ tl✝ = nil

Next tactic:
simp
instance : LawfulTraversable LazyList := by
  

Proof state:
⊢ LawfulTraversable LazyList

Next tactic:
apply Equiv.isLawfulTraversable' listEquivLazyList
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> 

Proof state:
case h₀
⊢ ∀ {α β : Type ?u.27717} (f : α → β), Functor.map f = Equiv.map listEquivLazyList f

Next tactic:
intros
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> 

Proof state:
case h₁
⊢ ∀ {α β : Type ?u.27717} (f : β), Functor.mapConst f = (Equiv.map listEquivLazyList ∘ const α) f

Next tactic:
intros
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> 

Proof state:
case h₂
⊢ ∀ {F : Type ?u.27717 → Type ?u.27717} [inst : Applicative F] [inst_1 : LawfulApplicative F] {α β : Type ?u.27717}
    (f : α → F β), traverse f = Equiv.traverse listEquivLazyList f

Next tactic:
intros
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> 

Proof state:
case h₀
α✝ β✝ : Type ?u.27717
f✝ : α✝ → β✝
⊢ Functor.map f✝ = Equiv.map listEquivLazyList f✝

Next tactic:
ext
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> 

Proof state:
case h₁
α✝ β✝ : Type ?u.27717
f✝ : β✝
⊢ Functor.mapConst f✝ = (Equiv.map listEquivLazyList ∘ const α✝) f✝

Next tactic:
ext
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> 

Proof state:
case h₂
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f✝ : α✝ → F✝ β✝
⊢ traverse f✝ = Equiv.traverse listEquivLazyList f✝

Next tactic:
ext
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> 

Proof state:
case h₀.h
α✝ β✝ : Type ?u.27717
f✝ : α✝ → β✝
x✝ : LazyList α✝
⊢ f✝ <$> x✝ = Equiv.map listEquivLazyList f✝ x✝

Next tactic:
rename_i f xs
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> 

Proof state:
case h₁.h
α✝ β✝ : Type ?u.27717
f✝ : β✝
x✝ : LazyList α✝
⊢ Functor.mapConst f✝ x✝ = (Equiv.map listEquivLazyList ∘ const α✝) f✝ x✝

Next tactic:
rename_i f xs
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> 

Proof state:
case h₂.h
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f✝ : α✝ → F✝ β✝
x✝ : LazyList α✝
⊢ traverse f✝ x✝ = Equiv.traverse listEquivLazyList f✝ x✝

Next tactic:
rename_i f xs
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
  · 

Proof state:
case h₀.h
α✝ β✝ : Type ?u.27717
f : α✝ → β✝
xs : LazyList α✝
⊢ f <$> xs = Equiv.map listEquivLazyList f xs

Next tactic:
induction' xs using LazyList.rec with _ _ _ _ ih
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
  · induction' xs using LazyList.rec with _ _ _ _ ih
    · 

Proof state:
case h₀.h.nil
α✝ β✝ : Type ?u.27717
f : α✝ → β✝
⊢ f <$> nil = Equiv.map listEquivLazyList f nil

Next tactic:
rfl
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
  · induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · 

Proof state:
case h₀.h.cons
α✝ β✝ : Type ?u.27717
f : α✝ → β✝
xs : LazyList α✝
hd✝ : α✝
tl✝ : Thunk (LazyList α✝)
tl_ih✝ : ?m.27945 tl✝
⊢ f <$> cons hd✝ tl✝ = Equiv.map listEquivLazyList f (cons hd✝ tl✝)

Next tactic:
simpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk,
        LazyList.traverse, Seq.seq, toList, ofList, cons.injEq, true_and]
instance : LawfulTraversable LazyList := by
  apply Equiv.isLawfulTraversable' listEquivLazyList <;> intros <;> ext <;> rename_i f xs
  · induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [Equiv.map, Functor.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk,
        LazyList.traverse, Seq.seq, toList, ofList, cons.injEq, true_and]
    · 

Proof state:
case h₀.h.mk
α✝ β✝ : Type ?u.27717
f : α✝ → β✝
xs : LazyList α✝
fn✝ : Unit → LazyList α✝
ih : ∀ (a : Unit), f <$> fn✝ a = Equiv.map listEquivLazyList f (fn✝ a)
⊢ Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn✝ })) =
    { fn := fun x => ofList (List.map f (toList (Thunk.get { fn := fn✝ }))) }

Next tactic:
ext