l3lab/ntp-mathlib huggingface dataset:
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        

Proof state:
case pos.inr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v x u : V
huv : Adj G u x
p : Walk G x x
ht : IsTrail p ∧ ⟦(u, x)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ x ≠ x → x ≠ x ∧ x ≠ x
⊢ ¬u = x

Next tactic:
exact huv.ne
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · 

Proof state:
case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ v ≠ w → x ≠ v ∧ x ≠ w
h : ¬decide (x = u ∨ x = v) = true
⊢ Even (List.countP (fun e => decide (x ∈ e)) (edges p) + 0) ↔ ¬u = w → ¬x = u ∧ ¬x = w

Next tactic:
rw [decide_eq_true_eq, not_or] at h
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      

Proof state:
case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ v ≠ w → x ≠ v ∧ x ≠ w
h : ¬x = u ∧ ¬x = v
⊢ Even (List.countP (fun e => decide (x ∈ e)) (edges p) + 0) ↔ ¬u = w → ¬x = u ∧ ¬x = w

Next tactic:
simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      

Proof state:
case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
h : ¬x = u ∧ ¬x = v
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = w → ¬x = w
⊢ Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬u = w → ¬x = w

Next tactic:
rw [ih]
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      

Proof state:
case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
h : ¬x = u ∧ ¬x = v
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = w → ¬x = w
⊢ ¬v = w → ¬x = w ↔ ¬u = w → ¬x = w

Next tactic:
constructor
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · 

Proof state:
case neg.mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
h : ¬x = u ∧ ¬x = v
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = w → ¬x = w
⊢ (¬v = w → ¬x = w) → ¬u = w → ¬x = w

Next tactic:
rintro h' h'' rfl
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          

Proof state:
case neg.mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v : V
huv : Adj G u v
h : ¬x = u ∧ ¬x = v
p : Walk G v x
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = x → ¬x = x
h' : ¬v = x → ¬x = x
h'' : ¬u = x
⊢ False

Next tactic:
simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
          

Proof state:
case neg.mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v : V
huv : Adj G u v
h : ¬x = u ∧ ¬x = v
p : Walk G v x
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = x → ¬x = x
h'' : ¬u = x
h' : v = x
⊢ False

Next tactic:
cases h'
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
          cases h'
          

Proof state:
case neg.mp.refl
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v x u : V
h'' : ¬u = x
huv : Adj G u x
h : ¬x = u ∧ ¬x = x
p : Walk G x x
ht : IsTrail p ∧ ⟦(u, x)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬x = x → ¬x = x
⊢ False

Next tactic:
simp only [not_true, and_false, false_and] at h
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · 

Proof state:
case neg.mpr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v w : V
huv : Adj G u v
p : Walk G v w
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
h : ¬x = u ∧ ¬x = v
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = w → ¬x = w
⊢ (¬u = w → ¬x = w) → ¬v = w → ¬x = w

Next tactic:
rintro h' h'' rfl
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          

Proof state:
case neg.mpr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v : V
huv : Adj G u v
h : ¬x = u ∧ ¬x = v
p : Walk G v x
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = x → ¬x = x
h' : ¬u = x → ¬x = x
h'' : ¬v = x
⊢ False

Next tactic:
simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
          

Proof state:
case neg.mpr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u✝ v✝ x u v : V
huv : Adj G u v
h : ¬x = u ∧ ¬x = v
p : Walk G v x
ht : IsTrail p ∧ ⟦(u, v)⟧ ∉ edges p
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = x → ¬x = x
h'' : ¬v = x
h' : u = x
⊢ False

Next tactic:
cases h'
theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
    Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by
  induction' p with u u v w huv p ih
  · simp
  · rw [cons_isTrail_iff] at ht
    specialize ih ht.1
    simp only [List.countP_cons, Ne.def, edges_cons, Sym2.mem_iff]
    split_ifs with h
    · rw [decide_eq_true_eq] at h
      obtain (rfl | rfl) := h
      · rw [Nat.even_add_one, ih]
        simp only [huv.ne, imp_false, Ne.def, not_false_iff, true_and_iff, not_forall,
          Classical.not_not, exists_prop, eq_self_iff_true, not_true, false_and_iff,
          and_iff_right_iff_imp]
        rintro rfl rfl
        exact G.loopless _ huv
      · rw [Nat.even_add_one, ih, ← not_iff_not]
        simp only [huv.ne.symm, Ne.def, eq_self_iff_true, not_true, false_and_iff, not_forall,
          not_false_iff, exists_prop, and_true_iff, Classical.not_not, true_and_iff, iff_and_self]
        rintro rfl
        exact huv.ne
    · rw [decide_eq_true_eq, not_or] at h
      simp only [h.1, h.2, not_false_iff, true_and_iff, add_zero, Ne.def] at ih ⊢
      rw [ih]
      constructor <;>
        · rintro h' h'' rfl
          simp only [imp_false, eq_self_iff_true, not_true, Classical.not_not] at h'
          cases h'
          

Proof state:
case neg.mpr.refl
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v✝ x v : V
p : Walk G v x
ih : Even (List.countP (fun e => decide (x ∈ e)) (edges p)) ↔ ¬v = x → ¬x = x
h'' : ¬v = x
huv : Adj G x v
h : ¬x = x ∧ ¬x = v
ht : IsTrail p ∧ ⟦(x, v)⟧ ∉ edges p
⊢ False

Next tactic:
simp only [not_true, and_false, false_and] at h
theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by
  

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
⊢ IsTrail p

Next tactic:
rw [isTrail_def, List.nodup_iff_count_le_one]
theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by
  rw [isTrail_def, List.nodup_iff_count_le_one]
  

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
⊢ ∀ (a : Sym2 V), List.count a (edges p) ≤ 1

Next tactic:
intro e
theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by
  rw [isTrail_def, List.nodup_iff_count_le_one]
  intro e
  

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
⊢ List.count e (edges p) ≤ 1

Next tactic:
by_cases he : e ∈ p.edges
theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by
  rw [isTrail_def, List.nodup_iff_count_le_one]
  intro e
  by_cases he : e ∈ p.edges
  · 

Proof state:
case pos
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
he : e ∈ edges p
⊢ List.count e (edges p) ≤ 1

Next tactic:
exact (h e (edges_subset_edgeSet _ he)).le
theorem IsEulerian.isTrail {u v : V} {p : G.Walk u v} (h : p.IsEulerian) : p.IsTrail := by
  rw [isTrail_def, List.nodup_iff_count_le_one]
  intro e
  by_cases he : e ∈ p.edges
  · exact (h e (edges_subset_edgeSet _ he)).le
  · 

Proof state:
case neg
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
he : e ∉ edges p
⊢ List.count e (edges p) ≤ 1

Next tactic:
simp [he]
theorem IsEulerian.mem_edges_iff {u v : V} {p : G.Walk u v} (h : p.IsEulerian) {e : Sym2 V} :
    e ∈ p.edges ↔ e ∈ G.edgeSet :=
  ⟨ fun h => p.edges_subset_edgeSet h
  , fun he => by 

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
he : e ∈ edgeSet G
⊢ e ∈ edges p

Next tactic:
apply List.count_pos_iff_mem.mp
theorem IsEulerian.mem_edges_iff {u v : V} {p : G.Walk u v} (h : p.IsEulerian) {e : Sym2 V} :
    e ∈ p.edges ↔ e ∈ G.edgeSet :=
  ⟨ fun h => p.edges_subset_edgeSet h
  , fun he => by apply List.count_pos_iff_mem.mp; 

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
he : e ∈ edgeSet G
⊢ 0 < List.count e (edges p)

Next tactic:
simpa using (h e he).ge
/-- The edge set of an Eulerian graph is finite. -/
def IsEulerian.fintypeEdgeSet {u v : V} {p : G.Walk u v} (h : p.IsEulerian) :
    Fintype G.edgeSet :=
  Fintype.ofFinset h.isTrail.edgesFinset fun e => by
    

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
e : Sym2 V
⊢ e ∈ IsTrail.edgesFinset (_ : IsTrail p) ↔ e ∈ edgeSet G

Next tactic:
simp only [Finset.mem_mk, Multiset.mem_coe, h.mem_edges_iff]
theorem isEulerian_iff {u v : V} (p : G.Walk u v) :
    p.IsEulerian ↔ p.IsTrail ∧ ∀ e, e ∈ G.edgeSet → e ∈ p.edges := by
  

Proof state:
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
⊢ IsEulerian p ↔ IsTrail p ∧ ∀ e ∈ edgeSet G, e ∈ edges p

Next tactic:
constructor
theorem isEulerian_iff {u v : V} (p : G.Walk u v) :
    p.IsEulerian ↔ p.IsTrail ∧ ∀ e, e ∈ G.edgeSet → e ∈ p.edges := by
  constructor
  · 

Proof state:
case mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
⊢ IsEulerian p → IsTrail p ∧ ∀ e ∈ edgeSet G, e ∈ edges p

Next tactic:
intro h
theorem isEulerian_iff {u v : V} (p : G.Walk u v) :
    p.IsEulerian ↔ p.IsTrail ∧ ∀ e, e ∈ G.edgeSet → e ∈ p.edges := by
  constructor
  · intro h
    

Proof state:
case mp
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
h : IsEulerian p
⊢ IsTrail p ∧ ∀ e ∈ edgeSet G, e ∈ edges p

Next tactic:
exact ⟨h.isTrail, fun _ => h.mem_edges_iff.mpr⟩
theorem isEulerian_iff {u v : V} (p : G.Walk u v) :
    p.IsEulerian ↔ p.IsTrail ∧ ∀ e, e ∈ G.edgeSet → e ∈ p.edges := by
  constructor
  · intro h
    exact ⟨h.isTrail, fun _ => h.mem_edges_iff.mpr⟩
  · 

Proof state:
case mpr
V : Type u_1
G : SimpleGraph V
inst✝ : DecidableEq V
u v : V
p : Walk G u v
⊢ (IsTrail p ∧ ∀ e ∈ edgeSet G, e ∈ edges p) → IsEulerian p

Next tactic:
rintro ⟨h, hl⟩