l3lab/ntp-mathlib huggingface dataset:
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · 

Proof state:
case inr.refine_2
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
a b : V
ha : a ∈ {default}ᶜ
hb : b ∈ {default}ᶜ
h' : length (f a) ≤ length (f b)
h : firstDart (f a) (_ : ¬Nil (f a)) = Dart.symm (firstDart (f b) (_ : ¬Nil (f b)))
h1 : (firstDart (f a) (_ : ¬Nil (f a))).toProd.2 = b
h3 : length (f a) = length (f b) + 1
⊢ a = b

Next tactic:
exfalso
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        

Proof state:
case inr.refine_2.h
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
a b : V
ha : a ∈ {default}ᶜ
hb : b ∈ {default}ᶜ
h' : length (f a) ≤ length (f b)
h : firstDart (f a) (_ : ¬Nil (f a)) = Dart.symm (firstDart (f b) (_ : ¬Nil (f b)))
h1 : (firstDart (f a) (_ : ¬Nil (f a))).toProd.2 = b
h3 : length (f a) = length (f b) + 1
⊢ False

Next tactic:
linarith
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · 

Proof state:
case inr.refine_1
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
a b : V
ha : a ∈ {default}ᶜ
hb : b ∈ {default}ᶜ
h' : length (f a) ≤ length (f b)
h : firstDart (f a) (_ : ¬Nil (f a)) = Dart.symm (firstDart (f b) (_ : ¬Nil (f b)))
h1 : (firstDart (f a) (_ : ¬Nil (f a))).toProd.2 = b
⊢ (fun p => IsPath p) (Walk.copy (tail (f a) (_ : ¬Nil (f a))) h1 (_ : default = default))

Next tactic:
simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        

Proof state:
case inr.refine_1
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
a b : V
ha : a ∈ {default}ᶜ
hb : b ∈ {default}ᶜ
h' : length (f a) ≤ length (f b)
h : firstDart (f a) (_ : ¬Nil (f a)) = Dart.symm (firstDart (f b) (_ : ¬Nil (f b)))
h1 : (firstDart (f a) (_ : ¬Nil (f a))).toProd.2 = b
⊢ IsPath (tail (f a) (_ : ¬Nil (f a)))

Next tactic:
exact (hf _).tail _
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  

Proof state:
case surj
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
⊢ ∀ b ∈ edgeFinset G, ∃ a, ∃ (ha : a ∈ {default}ᶜ), (fun w hw => Dart.edge (firstDart (f w) (_ : ¬Nil (f w)))) a ha = b

Next tactic:
case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    intros x y h
    wlog h' : (f x).length ≤ (f y).length generalizing x y
    · rw [Sym2.eq_swap]
      exact this y x h.symm (le_of_not_le h')
    refine ⟨y, ?_, dart_edge_eq_mk'_iff.2 <| Or.inr ?_⟩
    · rintro rfl
      rw [← hf' _ nil IsPath.nil, length_nil,
          ← hf' _ (.cons h .nil) (IsPath.nil.cons <| by simpa using h.ne),
          length_cons, length_nil] at h'
      simp [le_zero_iff, Nat.one_ne_zero] at h'
    rw [← hf' _ (.cons h.symm (f x)) ((cons_isPath_iff _ _).2 ⟨hf _, fun hy => ?contra⟩)]
    rfl
    case contra =>
      suffices : (f x).takeUntil y hy = .cons h .nil
      · rw [← take_spec _ hy] at h'
        simp [this, hf' _ _ ((hf _).dropUntil hy)] at h'
      refine (hG.existsUnique_path _ _).unique ((hf _).takeUntil _) ?_
      simp [h.ne]
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  

Proof state:
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
⊢ ∀ b ∈ edgeFinset G, ∃ a, ∃ (ha : a ∈ {default}ᶜ), (fun w hw => Dart.edge (firstDart (f w) (_ : ¬Nil (f w)))) a ha = b

Next tactic:
case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    intros x y h
    wlog h' : (f x).length ≤ (f y).length generalizing x y
    · rw [Sym2.eq_swap]
      exact this y x h.symm (le_of_not_le h')
    refine ⟨y, ?_, dart_edge_eq_mk'_iff.2 <| Or.inr ?_⟩
    · rintro rfl
      rw [← hf' _ nil IsPath.nil, length_nil,
          ← hf' _ (.cons h .nil) (IsPath.nil.cons <| by simpa using h.ne),
          length_cons, length_nil] at h'
      simp [le_zero_iff, Nat.one_ne_zero] at h'
    rw [← hf' _ (.cons h.symm (f x)) ((cons_isPath_iff _ _).2 ⟨hf _, fun hy => ?contra⟩)]
    rfl
    case contra =>
      suffices : (f x).takeUntil y hy = .cons h .nil
      · rw [← take_spec _ hy] at h'
        simp [this, hf' _ _ ((hf _).dropUntil hy)] at h'
      refine (hG.existsUnique_path _ _).unique ((hf _).takeUntil _) ?_
      simp [h.ne]
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  case surj =>
    

Proof state:
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
⊢ ∀ b ∈ edgeFinset G, ∃ a, ∃ (ha : a ∈ {default}ᶜ), (fun w hw => Dart.edge (firstDart (f w) (_ : ¬Nil (f w)))) a ha = b

Next tactic:
simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    

Proof state:
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
⊢ ∀ (x y : V), Adj G x y → ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(x, y)⟧

Next tactic:
intros x y h
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    intros x y h
    

Proof state:
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝ : Nonempty V
inhabited_h : Inhabited V
this : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
x y : V
h : Adj G x y
⊢ ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(x, y)⟧

Next tactic:
wlog h' : (f x).length ≤ (f y).length generalizing x y
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    intros x y h
    wlog h' : (f x).length ≤ (f y).length generalizing x y
    · 

Proof state:
case inr
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝¹ : Nonempty V
inhabited_h : Inhabited V
this✝ : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
x y : V
h : Adj G x y
this :
  ∀ (x y : V),
    Adj G x y →
      length (f x) ≤ length (f y) → ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(x, y)⟧
h' : ¬length (f x) ≤ length (f y)
⊢ ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(x, y)⟧

Next tactic:
rw [Sym2.eq_swap]
lemma IsTree.card_edgeFinset [Fintype V] [Fintype G.edgeSet] (hG : G.IsTree) :
    Finset.card G.edgeFinset + 1 = Fintype.card V := by
  have := hG.isConnected.nonempty
  inhabit V
  classical
  have : Finset.card ({default} : Finset V)ᶜ + 1 = Fintype.card V := by
    rw [Finset.card_compl, Finset.card_singleton, Nat.sub_add_cancel Fintype.card_pos]
  rw [← this, add_left_inj]
  choose f hf hf' using (hG.existsUnique_path · default)
  refine Eq.symm <| Finset.card_congr
          (fun w hw => ((f w).firstDart <| ?notNil).edge)
          (fun a ha => ?memEdges) ?inj ?surj
  case notNil => exact not_nil_of_ne (by simpa using hw)
  case memEdges => simp
  case inj =>
    intros a b ha hb h
    wlog h' : (f a).length ≤ (f b).length generalizing a b
    · exact Eq.symm (this _ _ hb ha h.symm (le_of_not_le h'))
    rw [dart_edge_eq_iff] at h
    obtain (h | h) := h
    · exact (congrArg (·.fst) h)
    · have h1 : ((f a).firstDart <| not_nil_of_ne (by simpa using ha)).snd = b :=
        congrArg (·.snd) h
      have h3 := congrArg length (hf' _ (((f _).tail _).copy h1 rfl) ?_)
      rw [length_copy, ← add_left_inj 1, length_tail_add_one] at h3
      · exfalso
        linarith
      · simp only [ne_eq, eq_mp_eq_cast, id_eq, isPath_copy]
        exact (hf _).tail _
  case surj =>
    simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet]
    intros x y h
    wlog h' : (f x).length ≤ (f y).length generalizing x y
    · rw [Sym2.eq_swap]
      

Proof state:
case inr
V : Type u
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑(edgeSet G)
hG : IsTree G
this✝¹ : Nonempty V
inhabited_h : Inhabited V
this✝ : Finset.card {default}ᶜ + 1 = Fintype.card V
f : (x : V) → Walk G x default
hf : ∀ (x : V), (fun p => IsPath p) (f x)
hf' : ∀ (x : V) (y : Walk G x default), (fun p => IsPath p) y → y = f x
x y : V
h : Adj G x y
this :
  ∀ (x y : V),
    Adj G x y →
      length (f x) ≤ length (f y) → ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(x, y)⟧
h' : ¬length (f x) ≤ length (f y)
⊢ ∃ a, ∃ (h : ¬a = default), Dart.edge (firstDart (f a) (_ : ¬Nil (f a))) = ⟦(y, x)⟧

Next tactic:
exact this y x h.symm (le_of_not_le h')