l3lab/ntp-mathlib huggingface dataset:
theorem reachable_iff_exists_finsetWalkLength_nonempty (u v : V) :
    G.Reachable u v ↔ ∃ n : Fin (Fintype.card V), (G.finsetWalkLength n u v).Nonempty := by
  constructor
  · intro r
    refine r.elim_path fun p => ?_
    

Proof state:
case mp
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
u v : V
r : Reachable G u v
p : Path G u v
⊢ ∃ n, Finset.Nonempty (finsetWalkLength G (↑n) u v)

Next tactic:
refine ⟨⟨_, p.isPath.length_lt⟩, p, ?_⟩
theorem reachable_iff_exists_finsetWalkLength_nonempty (u v : V) :
    G.Reachable u v ↔ ∃ n : Fin (Fintype.card V), (G.finsetWalkLength n u v).Nonempty := by
  constructor
  · intro r
    refine r.elim_path fun p => ?_
    refine ⟨⟨_, p.isPath.length_lt⟩, p, ?_⟩
    

Proof state:
case mp
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
u v : V
r : Reachable G u v
p : Path G u v
⊢ ↑p ∈ finsetWalkLength G (↑{ val := Walk.length ↑p, isLt := (_ : Walk.length ↑p < Fintype.card V) }) u v

Next tactic:
simp [Walk.mem_finsetWalkLength_iff_length_eq]
theorem reachable_iff_exists_finsetWalkLength_nonempty (u v : V) :
    G.Reachable u v ↔ ∃ n : Fin (Fintype.card V), (G.finsetWalkLength n u v).Nonempty := by
  constructor
  · intro r
    refine r.elim_path fun p => ?_
    refine ⟨⟨_, p.isPath.length_lt⟩, p, ?_⟩
    simp [Walk.mem_finsetWalkLength_iff_length_eq]
  · 

Proof state:
case mpr
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
u v : V
⊢ (∃ n, Finset.Nonempty (finsetWalkLength G (↑n) u v)) → Reachable G u v

Next tactic:
rintro ⟨_, p, _⟩
theorem reachable_iff_exists_finsetWalkLength_nonempty (u v : V) :
    G.Reachable u v ↔ ∃ n : Fin (Fintype.card V), (G.finsetWalkLength n u v).Nonempty := by
  constructor
  · intro r
    refine r.elim_path fun p => ?_
    refine ⟨⟨_, p.isPath.length_lt⟩, p, ?_⟩
    simp [Walk.mem_finsetWalkLength_iff_length_eq]
  · rintro ⟨_, p, _⟩
    

Proof state:
case mpr.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
u v : V
w✝ : Fin (Fintype.card V)
p : Walk G u v
h✝ : p ∈ finsetWalkLength G (↑w✝) u v
⊢ Reachable G u v

Next tactic:
exact ⟨p⟩
instance : Decidable G.Connected := by
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ Decidable (Connected G)

Next tactic:
rw [connected_iff, ← Finset.univ_nonempty_iff]
instance : Decidable G.Connected := by
  rw [connected_iff, ← Finset.univ_nonempty_iff]
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
⊢ Decidable (Preconnected G ∧ Finset.Nonempty Finset.univ)

Next tactic:
infer_instance
/-- An edge of a graph is a *bridge* if, after removing it, its incident vertices
are no longer reachable from one another. -/
def IsBridge (G : SimpleGraph V) (e : Sym2 V) : Prop :=
  e ∈ G.edgeSet ∧
    Sym2.lift ⟨fun v w => ¬(G \ fromEdgeSet {e}).Reachable v w, by 

Proof state:
V : Type u
V' : Type v
V'' : Type w
G✝ : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
G : SimpleGraph V
e : Sym2 V
⊢ ∀ (a₁ a₂ : V),
    (fun v w => ¬Reachable (G \ fromEdgeSet {e}) v w) a₁ a₂ = (fun v w => ¬Reachable (G \ fromEdgeSet {e}) v w) a₂ a₁

Next tactic:
simp [reachable_comm]
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
⊢ Reachable (G \ fromEdgeSet {⟦(v, w)⟧}) v w ↔ ∃ p, ⟦(v, w)⟧ ∉ Walk.edges p

Next tactic:
constructor
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · 

Proof state:
case mp
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
⊢ Reachable (G \ fromEdgeSet {⟦(v, w)⟧}) v w → ∃ p, ⟦(v, w)⟧ ∉ Walk.edges p

Next tactic:
rintro ⟨p⟩
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    

Proof state:
case mp.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk (G \ fromEdgeSet {⟦(v, w)⟧}) v w
⊢ ∃ p, ⟦(v, w)⟧ ∉ Walk.edges p

Next tactic:
use p.map (Hom.mapSpanningSubgraphs (by simp))
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by 

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk (G \ fromEdgeSet {⟦(v, w)⟧}) v w
⊢ G \ fromEdgeSet {⟦(v, w)⟧} ≤ G

Next tactic:
simp
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    

Proof state:
case h
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk (G \ fromEdgeSet {⟦(v, w)⟧}) v w
⊢ ⟦(v, w)⟧ ∉ Walk.edges (Walk.map (Hom.mapSpanningSubgraphs (_ : G \ fromEdgeSet {⟦(v, w)⟧} ≤ G)) p)

Next tactic:
simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    

Proof state:
case h
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk (G \ fromEdgeSet {⟦(v, w)⟧}) v w
⊢ ¬∃ a ∈ Walk.edges p, a = ⟦(v, w)⟧

Next tactic:
rintro ⟨e, h, rfl⟩
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    rintro ⟨e, h, rfl⟩
    

Proof state:
case h.intro.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk (G \ fromEdgeSet {⟦(v, w)⟧}) v w
h : ⟦(v, w)⟧ ∈ Walk.edges p
⊢ False

Next tactic:
simpa using p.edges_subset_edgeSet h
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    rintro ⟨e, h, rfl⟩
    simpa using p.edges_subset_edgeSet h
  · 

Proof state:
case mpr
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
⊢ (∃ p, ⟦(v, w)⟧ ∉ Walk.edges p) → Reachable (G \ fromEdgeSet {⟦(v, w)⟧}) v w

Next tactic:
rintro ⟨p, h⟩
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    rintro ⟨e, h, rfl⟩
    simpa using p.edges_subset_edgeSet h
  · rintro ⟨p, h⟩
    

Proof state:
case mpr.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk G v w
h : ⟦(v, w)⟧ ∉ Walk.edges p
⊢ Reachable (G \ fromEdgeSet {⟦(v, w)⟧}) v w

Next tactic:
refine ⟨p.transfer _ fun e ep => ?_⟩
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    rintro ⟨e, h, rfl⟩
    simpa using p.edges_subset_edgeSet h
  · rintro ⟨p, h⟩
    refine ⟨p.transfer _ fun e ep => ?_⟩
    

Proof state:
case mpr.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk G v w
h : ⟦(v, w)⟧ ∉ Walk.edges p
e : Sym2 V
ep : e ∈ Walk.edges p
⊢ e ∈ edgeSet (G \ fromEdgeSet {⟦(v, w)⟧})

Next tactic:
simp only [edgeSet_sdiff, edgeSet_fromEdgeSet, edgeSet_sdiff_sdiff_isDiag, Set.mem_diff,
      Set.mem_singleton_iff]
theorem reachable_delete_edges_iff_exists_walk {v w : V} :
    (G \ fromEdgeSet {⟦(v, w)⟧}).Reachable v w ↔ ∃ p : G.Walk v w, ¬⟦(v, w)⟧ ∈ p.edges := by
  constructor
  · rintro ⟨p⟩
    use p.map (Hom.mapSpanningSubgraphs (by simp))
    simp_rw [Walk.edges_map, List.mem_map, Hom.mapSpanningSubgraphs_apply, Sym2.map_id', id.def]
    rintro ⟨e, h, rfl⟩
    simpa using p.edges_subset_edgeSet h
  · rintro ⟨p, h⟩
    refine ⟨p.transfer _ fun e ep => ?_⟩
    simp only [edgeSet_sdiff, edgeSet_fromEdgeSet, edgeSet_sdiff_sdiff_isDiag, Set.mem_diff,
      Set.mem_singleton_iff]
    

Proof state:
case mpr.intro
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
p : Walk G v w
h : ⟦(v, w)⟧ ∉ Walk.edges p
e : Sym2 V
ep : e ∈ Walk.edges p
⊢ e ∈ edgeSet G ∧ ¬e = ⟦(v, w)⟧

Next tactic:
exact ⟨p.edges_subset_edgeSet ep, fun h' => h (h' ▸ ep)⟩
theorem isBridge_iff_adj_and_forall_walk_mem_edges {v w : V} :
    G.IsBridge ⟦(v, w)⟧ ↔ G.Adj v w ∧ ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges := by
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
⊢ IsBridge G ⟦(v, w)⟧ ↔ Adj G v w ∧ ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p

Next tactic:
rw [isBridge_iff, and_congr_right']
theorem isBridge_iff_adj_and_forall_walk_mem_edges {v w : V} :
    G.IsBridge ⟦(v, w)⟧ ↔ G.Adj v w ∧ ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges := by
  rw [isBridge_iff, and_congr_right']
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
v w : V
⊢ ¬Reachable (G \ fromEdgeSet {⟦(v, w)⟧}) v w ↔ ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p

Next tactic:
rw [reachable_delete_edges_iff_exists_walk, not_exists_not]
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
⊢ False

Next tactic:
have hv := c.fst_mem_support_of_mem_edges he
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
⊢ False

Next tactic:
let puw := (c.takeUntil v hv).takeUntil w hw
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
⊢ False

Next tactic:
let pwv := (c.takeUntil v hv).dropUntil w hw
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
⊢ False

Next tactic:
let pvu := c.dropUntil v hv
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
⊢ False

Next tactic:
have : c = (puw.append pwv).append pvu := by simp
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  have : c = (puw.append pwv).append pvu := by 

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
⊢ c = Walk.append (Walk.append puw pwv) pvu

Next tactic:
simp
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  have : c = (puw.append pwv).append pvu := by simp
  -- We have two walks from v to w
  --      pvu     puw
  --   v ----> u ----> w
  --   |               ^
  --    `-------------'
  --      pwv.reverse
  -- so they both contain the edge ⟦(v, w)⟧, but that's a contradiction since c is a trail.
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
this : c = Walk.append (Walk.append puw pwv) pvu
⊢ False

Next tactic:
have hbq := hb (pvu.append puw)
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  have : c = (puw.append pwv).append pvu := by simp
  -- We have two walks from v to w
  --      pvu     puw
  --   v ----> u ----> w
  --   |               ^
  --    `-------------'
  --      pwv.reverse
  -- so they both contain the edge ⟦(v, w)⟧, but that's a contradiction since c is a trail.
  have hbq := hb (pvu.append puw)
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
this : c = Walk.append (Walk.append puw pwv) pvu
hbq : ⟦(v, w)⟧ ∈ Walk.edges (Walk.append pvu puw)
⊢ False

Next tactic:
have hpq' := hb pwv.reverse
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  have : c = (puw.append pwv).append pvu := by simp
  -- We have two walks from v to w
  --      pvu     puw
  --   v ----> u ----> w
  --   |               ^
  --    `-------------'
  --      pwv.reverse
  -- so they both contain the edge ⟦(v, w)⟧, but that's a contradiction since c is a trail.
  have hbq := hb (pvu.append puw)
  have hpq' := hb pwv.reverse
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
this : c = Walk.append (Walk.append puw pwv) pvu
hbq : ⟦(v, w)⟧ ∈ Walk.edges (Walk.append pvu puw)
hpq' : ⟦(v, w)⟧ ∈ Walk.edges (Walk.reverse pwv)
⊢ False

Next tactic:
rw [Walk.edges_reverse, List.mem_reverse] at hpq'
theorem reachable_deleteEdges_iff_exists_cycle.aux [DecidableEq V] {u v w : V}
    (hb : ∀ p : G.Walk v w, ⟦(v, w)⟧ ∈ p.edges) (c : G.Walk u u) (hc : c.IsTrail)
    (he : ⟦(v, w)⟧ ∈ c.edges)
    (hw : w ∈ (c.takeUntil v (c.fst_mem_support_of_mem_edges he)).support) : False := by
  have hv := c.fst_mem_support_of_mem_edges he
  -- decompose c into
  --      puw     pwv     pvu
  --   u ----> w ----> v ----> u
  let puw := (c.takeUntil v hv).takeUntil w hw
  let pwv := (c.takeUntil v hv).dropUntil w hw
  let pvu := c.dropUntil v hv
  have : c = (puw.append pwv).append pvu := by simp
  -- We have two walks from v to w
  --      pvu     puw
  --   v ----> u ----> w
  --   |               ^
  --    `-------------'
  --      pwv.reverse
  -- so they both contain the edge ⟦(v, w)⟧, but that's a contradiction since c is a trail.
  have hbq := hb (pvu.append puw)
  have hpq' := hb pwv.reverse
  rw [Walk.edges_reverse, List.mem_reverse] at hpq'
  

Proof state:
V : Type u
V' : Type v
V'' : Type w
G : SimpleGraph V
G' : SimpleGraph V'
G'' : SimpleGraph V''
inst✝ : DecidableEq V
u v w : V
hb : ∀ (p : Walk G v w), ⟦(v, w)⟧ ∈ Walk.edges p
c : Walk G u u
hc : Walk.IsTrail c
he : ⟦(v, w)⟧ ∈ Walk.edges c
hw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))
hv : v ∈ Walk.support c
puw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw
pwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw
pvu : Walk G v u := Walk.dropUntil c v hv
this : c = Walk.append (Walk.append puw pwv) pvu
hbq : ⟦(v, w)⟧ ∈ Walk.edges (Walk.append pvu puw)
hpq' : ⟦(v, w)⟧ ∈ Walk.edges pwv
⊢ False

Next tactic:
rw [Walk.isTrail_def, this, Walk.edges_append, Walk.edges_append, List.nodup_append_comm,
    ← List.append_assoc, ← Walk.edges_append] at hc