l3lab/ntp-mathlib huggingface dataset:
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]
    · ext; 

Proof state:
case h₀.h.mk.eq
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.get (Thunk.pure (LazyList.traverse f (Thunk.get { fn := fn✝ }))) =
    Thunk.get { fn := fun x => ofList (List.map f (toList (Thunk.get { fn := fn✝ }))) }

Next tactic:
apply ih
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]
    · ext; apply ih
  · 

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

Next tactic:
simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    

Proof state:
case h₁.h
α✝ β✝ : Type ?u.27717
f : β✝
xs : LazyList α✝
⊢ LazyList.traverse (const α✝ f) xs = ofList (const α✝ f <$> toList 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
    · 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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · 

Proof state:
case h₁.h.nil
α✝ β✝ : Type ?u.27717
f : β✝
⊢ LazyList.traverse (const α✝ f) nil = ofList (const α✝ f <$> toList 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
    · 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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · 

Proof state:
case h₁.h.cons
α✝ β✝ : Type ?u.27717
f : β✝
xs : LazyList α✝
hd✝ : α✝
tl✝ : Thunk (LazyList α✝)
tl_ih✝ : ?m.29738 tl✝
⊢ LazyList.traverse (const α✝ f) (cons hd✝ tl✝) = ofList (const α✝ f <$> toList (cons hd✝ tl✝))

Next tactic:
simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, 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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · 

Proof state:
case h₁.h.mk
α✝ β✝ : Type ?u.27717
f : β✝
xs : LazyList α✝
fn✝ : Unit → LazyList α✝
ih : ∀ (a : Unit), LazyList.traverse (const α✝ f) (fn✝ a) = ofList (const α✝ f <$> toList (fn✝ a))
⊢ Thunk.pure (LazyList.traverse (const α✝ f) (Thunk.get { fn := fn✝ })) =
    { fn := fun x => ofList (List.map (const α✝ f) (toList (Thunk.get { fn := fn✝ }))) }

Next tactic:
congr
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; 

Proof state:
case h₁.h.mk.e_a
α✝ β✝ : Type ?u.27717
f : β✝
xs : LazyList α✝
fn✝ : Unit → LazyList α✝
ih : ∀ (a : Unit), LazyList.traverse (const α✝ f) (fn✝ a) = ofList (const α✝ f <$> toList (fn✝ a))
⊢ LazyList.traverse (const α✝ f) (Thunk.get { fn := fn✝ }) =
    ofList (List.map (const α✝ f) (toList (Thunk.get { fn := fn✝ })))

Next tactic:
apply ih
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · 

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✝
xs : LazyList α✝
⊢ traverse f xs = Equiv.traverse listEquivLazyList f xs

Next tactic:
simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    

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✝
xs : LazyList α✝
⊢ LazyList.traverse f xs = ofList <$> List.traverse f (toList xs)

Next tactic:
induction' xs using LazyList.rec with _ tl ih _ ih
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    induction' xs using LazyList.rec with _ tl ih _ ih
    · 

Proof state:
case h₂.h.nil
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
⊢ LazyList.traverse f nil = ofList <$> List.traverse f (toList nil)

Next tactic:
simp only [List.traverse, map_pure]
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    induction' xs using LazyList.rec with _ tl ih _ ih
    · simp only [List.traverse, map_pure]; 

Proof state:
case h₂.h.nil
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
⊢ LazyList.traverse f nil = pure (ofList [])

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
    · 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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    induction' xs using LazyList.rec with _ tl ih _ ih
    · simp only [List.traverse, map_pure]; rfl
    · 

Proof state:
case h₂.h.cons
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
xs : LazyList α✝
hd✝ : α✝
tl : Thunk (LazyList α✝)
ih : ?m.31326 tl
⊢ LazyList.traverse f (cons hd✝ tl) = ofList <$> List.traverse f (toList (cons hd✝ tl))

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

Proof state:
case h₂.h.cons
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
xs : LazyList α✝
hd✝ : α✝
tl : Thunk (LazyList α✝)
ih : LazyList.traverse f (Thunk.get tl) = ofList <$> List.traverse f (toList (Thunk.get tl))
⊢ LazyList.traverse f (cons hd✝ tl) = ofList <$> List.traverse f (toList (cons hd✝ tl))

Next tactic:
simp only [traverse._eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq]
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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    induction' xs using LazyList.rec with _ tl ih _ ih
    · simp only [List.traverse, map_pure]; rfl
    · replace ih : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih
      simp only [traverse._eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq]
      · 

Proof state:
case h₂.h.cons
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
xs : LazyList α✝
hd✝ : α✝
tl : Thunk (LazyList α✝)
ih : LazyList.traverse f (Thunk.get tl) = ofList <$> List.traverse f (toList (Thunk.get tl))
⊢ (Seq.seq (((fun x => x ∘ Thunk.pure ∘ ofList) ∘ cons) <$> f hd✝) fun x => List.traverse f (toList (Thunk.get tl))) =
    Seq.seq (((fun x => ofList ∘ x) ∘ List.cons) <$> f hd✝) fun x => List.traverse f (toList (Thunk.get tl))

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
    · 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]
    · ext; apply ih
  · simp only [Equiv.map, listEquivLazyList, Equiv.coe_fn_symm_mk, Equiv.coe_fn_mk, comp,
      Functor.mapConst]
    induction' xs using LazyList.rec with _ _ _ _ ih
    · rfl
    · simpa only [toList, ofList, LazyList.traverse, Seq.seq, Functor.map, cons.injEq, true_and]
    · congr; apply ih
  · simp only [traverse, Equiv.traverse, listEquivLazyList, Equiv.coe_fn_mk, Equiv.coe_fn_symm_mk]
    induction' xs using LazyList.rec with _ tl ih _ ih
    · simp only [List.traverse, map_pure]; rfl
    · replace ih : tl.get.traverse f = ofList <$> tl.get.toList.traverse f := ih
      simp only [traverse._eq_2, ih, Functor.map_map, seq_map_assoc, toList, List.traverse, map_seq]
      · rfl
    · 

Proof state:
case h₂.h.mk
F✝ : Type ?u.27717 → Type ?u.27717
inst✝¹ : Applicative F✝
inst✝ : LawfulApplicative F✝
α✝ β✝ : Type ?u.27717
f : α✝ → F✝ β✝
xs : LazyList α✝
fn✝ : Unit → LazyList α✝
ih : ∀ (a : Unit), LazyList.traverse f (fn✝ a) = ofList <$> List.traverse f (toList (fn✝ a))
⊢ LazyList.traverse f (Thunk.get { fn := fn✝ }) = ofList <$> List.traverse f (toList (Thunk.get { fn := fn✝ }))

Next tactic:
apply ih
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
  

Proof state:
α : Type u_1
xs : LazyList α
⊢ append xs (Thunk.pure nil) = xs

Next tactic:
induction' xs using LazyList.rec with _ _ _ _ ih
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · 

Proof state:
case nil
α : Type u_1
⊢ append nil (Thunk.pure nil) = nil

Next tactic:
rfl
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
  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.80128 tl✝
⊢ append (cons hd✝ tl✝) (Thunk.pure nil) = cons hd✝ tl✝

Next tactic:
simpa only [append, cons.injEq, true_and]
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · rfl
  · simpa only [append, cons.injEq, true_and]
  · 

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

Next tactic:
ext
theorem append_nil {α} (xs : LazyList α) : xs.append (Thunk.pure LazyList.nil) = xs := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · rfl
  · simpa only [append, cons.injEq, true_and]
  · ext; 

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

Next tactic:
apply ih
theorem append_assoc {α} (xs ys zs : LazyList α) :
    (xs.append ys).append zs = xs.append (ys.append zs) := by
  

Proof state:
α : Type u_1
xs ys zs : LazyList α
⊢ append (append xs { fn := fun x => ys }) { fn := fun x => zs } =
    append xs { fn := fun x => append ys { fn := fun x => zs } }

Next tactic:
induction' xs using LazyList.rec with _ _ _ _ ih
theorem append_assoc {α} (xs ys zs : LazyList α) :
    (xs.append ys).append zs = xs.append (ys.append zs) := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · 

Proof state:
case nil
α : Type u_1
ys zs : LazyList α
⊢ append (append nil { fn := fun x => ys }) { fn := fun x => zs } =
    append nil { fn := fun x => append ys { fn := fun x => zs } }

Next tactic:
rfl
theorem append_assoc {α} (xs ys zs : LazyList α) :
    (xs.append ys).append zs = xs.append (ys.append zs) := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · rfl
  · 

Proof state:
case cons
α : Type u_1
xs ys zs : LazyList α
hd✝ : α
tl✝ : Thunk (LazyList α)
tl_ih✝ : ?m.83206 tl✝
⊢ append (append (cons hd✝ tl✝) { fn := fun x => ys }) { fn := fun x => zs } =
    append (cons hd✝ tl✝) { fn := fun x => append ys { fn := fun x => zs } }

Next tactic:
simpa only [append, cons.injEq, true_and]
theorem append_assoc {α} (xs ys zs : LazyList α) :
    (xs.append ys).append zs = xs.append (ys.append zs) := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · rfl
  · simpa only [append, cons.injEq, true_and]
  · 

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

Next tactic:
ext
theorem append_assoc {α} (xs ys zs : LazyList α) :
    (xs.append ys).append zs = xs.append (ys.append zs) := by
  induction' xs using LazyList.rec with _ _ _ _ ih
  · rfl
  · simpa only [append, cons.injEq, true_and]
  · ext; 

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

Next tactic:
apply ih
theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
    (xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by
  

Proof state:
α : Type u_1
β : Type u_2
xs : LazyList α
ys : Thunk (LazyList α)
f : α → LazyList β
⊢ LazyList.bind (append xs ys) f = append (LazyList.bind xs f) { fn := fun x => LazyList.bind (Thunk.get ys) f }

Next tactic:
match xs with
  | LazyList.nil => rfl
  | LazyList.cons x xs =>
    simp only [append, Thunk.get, LazyList.bind]
    have := append_bind xs.get ys f
    simp only [Thunk.get] at this
    rw [this, append_assoc]
theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
    (xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by
  match xs with
  | LazyList.nil => 

Proof state:
α : Type u_1
β : Type u_2
xs : LazyList α
ys : Thunk (LazyList α)
f : α → LazyList β
⊢ LazyList.bind (append nil ys) f = append (LazyList.bind nil f) { fn := fun x => LazyList.bind (Thunk.get ys) f }

Next tactic:
rfl
theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
    (xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by
  match xs with
  | LazyList.nil => rfl
  | LazyList.cons x xs =>
    

Proof state:
α : Type u_1
β : Type u_2
xs✝ : LazyList α
ys : Thunk (LazyList α)
f : α → LazyList β
x : α
xs : Thunk (LazyList α)
⊢ LazyList.bind (append (cons x xs) ys) f =
    append (LazyList.bind (cons x xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }

Next tactic:
simp only [append, Thunk.get, LazyList.bind]
theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
    (xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by
  match xs with
  | LazyList.nil => rfl
  | LazyList.cons x xs =>
    simp only [append, Thunk.get, LazyList.bind]
    

Proof state:
α : Type u_1
β : Type u_2
xs✝ : LazyList α
ys : Thunk (LazyList α)
f : α → LazyList β
x : α
xs : Thunk (LazyList α)
⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =
    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })
      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }

Next tactic:
have := append_bind xs.get ys f
theorem append_bind {α β} (xs : LazyList α) (ys : Thunk (LazyList α)) (f : α → LazyList β) :
    (xs.append ys).bind f = (xs.bind f).append (ys.get.bind f) := by
  match xs with
  | LazyList.nil => rfl
  | LazyList.cons x xs =>
    simp only [append, Thunk.get, LazyList.bind]
    have := append_bind xs.get ys f
    

Proof state:
α : Type u_1
β : Type u_2
xs✝ : LazyList α
ys : Thunk (LazyList α)
f : α → LazyList β
x : α
xs : Thunk (LazyList α)
this :
  LazyList.bind (append (Thunk.get xs) ys) f =
    append (LazyList.bind (Thunk.get xs) f) { fn := fun x => LazyList.bind (Thunk.get ys) f }
⊢ append (f x) { fn := fun x => LazyList.bind (append (Thunk.fn xs ()) ys) f } =
    append (append (f x) { fn := fun x => LazyList.bind (Thunk.fn xs ()) f })
      { fn := fun x => LazyList.bind (Thunk.fn ys ()) f }

Next tactic:
simp only [Thunk.get] at this