ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.Path.cons_isCycle
theorem cons_isCycle {u v : V} (p : G.Path v u) (h : G.Adj u v)
    (he : ¬s(u, v) ∈ (p : G.Walk v u).edges) : (Walk.cons h ↑p).IsCycle 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Path v u) (h : G.Adj u v) (he : s(u, v) ∉ (↑p).edges)
Goal: (SimpleGraph.Walk.cons h ↑p).IsCycle
-- SimpleGraph.Walk.bypass_copy
@[simp]
theorem bypass_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :
    (p.copy hu hv).bypass = p.bypass.copy hu hv 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v u' v' : V} (p : G.Walk u v) (hu : u = u') (hv : v = v')
Goal: (p.copy hu hv).bypass = p.bypass.copy hu hv
-- SimpleGraph.Walk.bypass_isPath
theorem bypass_isPath {u v : V} (p : G.Walk u v) : p.bypass.IsPath 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: p.bypass.IsPath
-- SimpleGraph.Walk.length_bypass_le
theorem length_bypass_le {u v : V} (p : G.Walk u v) : p.bypass.length ≤ p.length 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: p.bypass.length ≤ p.length
-- SimpleGraph.Walk.bypass_eq_self_of_length_le
lemma bypass_eq_self_of_length_le {u v : V} (p : G.Walk u v) (h : p.length ≤ p.bypass.length) :
    p.bypass = p 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v) (h : p.length ≤ p.bypass.length)
Goal: p.bypass = p
-- SimpleGraph.Walk.support_bypass_subset
theorem support_bypass_subset {u v : V} (p : G.Walk u v) : p.bypass.support ⊆ p.support 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: p.bypass.support ⊆ p.support
-- SimpleGraph.Walk.support_toPath_subset
theorem support_toPath_subset {u v : V} (p : G.Walk u v) :
    (p.toPath : G.Walk u v).support ⊆ p.support 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: (↑p.toPath).support ⊆ p.support
-- SimpleGraph.Walk.darts_bypass_subset
theorem darts_bypass_subset {u v : V} (p : G.Walk u v) : p.bypass.darts ⊆ p.darts 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: p.bypass.darts ⊆ p.darts
-- SimpleGraph.Walk.edges_bypass_subset
theorem edges_bypass_subset {u v : V} (p : G.Walk u v) : p.bypass.edges ⊆ p.edges 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: p.bypass.edges ⊆ p.edges
-- SimpleGraph.Walk.darts_toPath_subset
theorem darts_toPath_subset {u v : V} (p : G.Walk u v) : (p.toPath : G.Walk u v).darts ⊆ p.darts 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: (↑p.toPath).darts ⊆ p.darts
-- SimpleGraph.Walk.edges_toPath_subset
theorem edges_toPath_subset {u v : V} (p : G.Walk u v) : (p.toPath : G.Walk u v).edges ⊆ p.edges 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v : V} (p : G.Walk u v)
Goal: (↑p.toPath).edges ⊆ p.edges
-- SimpleGraph.Walk.map_isPath_of_injective
theorem map_isPath_of_injective (hinj : Function.Injective f) (hp : p.IsPath) :
    (p.map f).IsPath 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {f : G →g G'} {u v : V} {p : G.Walk u v} (hinj : Function.Injective ⇑f) (hp : p.IsPath)
Goal: (SimpleGraph.Walk.map f p).IsPath
-- SimpleGraph.Walk.IsPath.of_map
protected theorem IsPath.of_map {f : G →g G'} (hp : (p.map f).IsPath) : p.IsPath 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {u v : V} {p : G.Walk u v} {f : G →g G'} (hp : (SimpleGraph.Walk.map f p).IsPath)
Goal: p.IsPath
-- SimpleGraph.Walk.map_isPath_iff_of_injective
theorem map_isPath_iff_of_injective (hinj : Function.Injective f) : (p.map f).IsPath ↔ p.IsPath 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {f : G →g G'} {u v : V} {p : G.Walk u v} (hinj : Function.Injective ⇑f)
Goal: (SimpleGraph.Walk.map f p).IsPath ↔ p.IsPath
-- SimpleGraph.Walk.map_isTrail_iff_of_injective
theorem map_isTrail_iff_of_injective (hinj : Function.Injective f) :
    (p.map f).IsTrail ↔ p.IsTrail 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {f : G →g G'} {u v : V} {p : G.Walk u v} (hinj : Function.Injective ⇑f)
Goal: (SimpleGraph.Walk.map f p).IsTrail ↔ p.IsTrail
-- SimpleGraph.Walk.map_isCycle_iff_of_injective
theorem map_isCycle_iff_of_injective {p : G.Walk u u} (hinj : Function.Injective f) :
    (p.map f).IsCycle ↔ p.IsCycle 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {f : G →g G'} {u : V} {p : G.Walk u u} (hinj : Function.Injective ⇑f)
Goal: (SimpleGraph.Walk.map f p).IsCycle ↔ p.IsCycle
-- SimpleGraph.Walk.mapLe_isTrail
@[simp]
theorem mapLe_isTrail {G G' : SimpleGraph V} (h : G ≤ G') {u v : V} {p : G.Walk u v} :
    (p.mapLe h).IsTrail ↔ p.IsTrail 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') {u v : V} {p : G.Walk u v}
Goal: (SimpleGraph.Walk.mapLe h p).IsTrail ↔ p.IsTrail
-- SimpleGraph.Walk.mapLe_isPath
@[simp]
theorem mapLe_isPath {G G' : SimpleGraph V} (h : G ≤ G') {u v : V} {p : G.Walk u v} :
    (p.mapLe h).IsPath ↔ p.IsPath 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') {u v : V} {p : G.Walk u v}
Goal: (SimpleGraph.Walk.mapLe h p).IsPath ↔ p.IsPath
-- SimpleGraph.Walk.mapLe_isCycle
@[simp]
theorem mapLe_isCycle {G G' : SimpleGraph V} (h : G ≤ G') {u : V} {p : G.Walk u u} :
    (p.mapLe h).IsCycle ↔ p.IsCycle 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') {u : V} {p : G.Walk u u}
Goal: (SimpleGraph.Walk.mapLe h p).IsCycle ↔ p.IsCycle
-- SimpleGraph.Path.map_coe
@[simps]
protected def map (f : G →g G') (hinj : Function.Injective f) {u v : V} (p : G.Path u v) :
    G'.Path (f u) (f v) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (f : G →g G') (hinj : Function.Injective ⇑f) {u v : V} (p : G.Path u v)
Goal: ↑(SimpleGraph.Path.map f hinj p) = SimpleGraph.Walk.map f ↑p
-- SimpleGraph.Path.map_injective
theorem map_injective {f : G →g G'} (hinj : Function.Injective f) (u v : V) :
    Function.Injective (Path.map f hinj : G.Path u v → G'.Path (f u) (f v)) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {f : G →g G'} (hinj : Function.Injective ⇑f) (u v : V)
Goal: Function.Injective (SimpleGraph.Path.map f hinj)
-- SimpleGraph.Path.mapEmbedding_coe
@[simps!]
protected def mapEmbedding (f : G ↪g G') {u v : V} (p : G.Path u v) : G'.Path (f u) (f v) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (f : G ↪g G') {u v : V} (p : G.Path u v)
Goal: ↑(SimpleGraph.Path.mapEmbedding f p) = SimpleGraph.Walk.map f.toHom ↑p
-- SimpleGraph.Path.mapEmbedding_injective
theorem mapEmbedding_injective (f : G ↪g G') (u v : V) :
    Function.Injective (Path.mapEmbedding f : G.Path u v → G'.Path (f u) (f v)) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (f : G ↪g G') (u v : V)
Goal: Function.Injective (SimpleGraph.Path.mapEmbedding f)
-- SimpleGraph.Walk.IsPath.transfer
protected theorem IsPath.transfer (hp) (pp : p.IsPath) :
    (p.transfer H hp).IsPath 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} {H : SimpleGraph V} (p : G.Walk u v) (hp : ∀ e ∈ p.edges, e ∈ H.edgeSet) (pp : p.IsPath)
Goal: (p.transfer H hp).IsPath
-- SimpleGraph.Walk.IsCycle.transfer
protected theorem IsCycle.transfer {q : G.Walk u u} (qc : q.IsCycle) (hq) :
    (q.transfer H hq).IsCycle 
Arguments: {V : Type u} {G : SimpleGraph V} {u : V} {H : SimpleGraph V} {q : G.Walk u u} (qc : q.IsCycle) (hq : ∀ e ∈ q.edges, e ∈ H.edgeSet)
Goal: (q.transfer H hq).IsCycle
-- SimpleGraph.Walk.IsPath.toDeleteEdges
protected theorem IsPath.toDeleteEdges (s : Set (Sym2 V))
    {p : G.Walk v w} (h : p.IsPath) (hp) : (p.toDeleteEdges s hp).IsPath 
Arguments: {V : Type u} (G : SimpleGraph V) {v w : V} (s : Set (Sym2 V)) {p : G.Walk v w} (h : p.IsPath) (hp : ∀ e ∈ p.edges, e ∉ s)
Goal: (SimpleGraph.Walk.toDeleteEdges s p hp).IsPath
-- SimpleGraph.Walk.IsCycle.toDeleteEdges
protected theorem IsCycle.toDeleteEdges (s : Set (Sym2 V))
    {p : G.Walk v v} (h : p.IsCycle) (hp) : (p.toDeleteEdges s hp).IsCycle 
Arguments: {V : Type u} (G : SimpleGraph V) {v : V} (s : Set (Sym2 V)) {p : G.Walk v v} (h : p.IsCycle) (hp : ∀ e ∈ p.edges, e ∉ s)
Goal: (SimpleGraph.Walk.toDeleteEdges s p hp).IsCycle
-- SimpleGraph.Walk.toDeleteEdges_copy
@[simp]
theorem toDeleteEdges_copy {v u u' v' : V} (s : Set (Sym2 V))
    (p : G.Walk u v) (hu : u = u') (hv : v = v') (h) :
    (p.copy hu hv).toDeleteEdges s h =
      (p.toDeleteEdges s (by subst_vars; exact h)).copy hu hv 
Arguments: {V : Type u} (G : SimpleGraph V) {v u u' v' : V} (s : Set (Sym2 V)) (p : G.Walk u v) (hu : u = u') (hv : v = v') (h : ∀ e ∈ (p.copy hu hv).edges, e ∉ s)
Goal: SimpleGraph.Walk.toDeleteEdges s (p.copy hu hv) h = (SimpleGraph.Walk.toDeleteEdges s p ⋯).copy hu hv
-- SimpleGraph.reachable_iff_nonempty_univ
theorem reachable_iff_nonempty_univ {u v : V} :
    G.Reachable u v ↔ (Set.univ : Set (G.Walk u v)).Nonempty 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V}
Goal: G.Reachable u v ↔ Set.univ.Nonempty
-- SimpleGraph.not_reachable_iff_isEmpty_walk
lemma not_reachable_iff_isEmpty_walk {u v : V} : ¬G.Reachable u v ↔ IsEmpty (G.Walk u v) 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V}
Goal: ¬G.Reachable u v ↔ IsEmpty (G.Walk u v)
-- SimpleGraph.Reachable.elim
protected theorem Reachable.elim {p : Prop} {u v : V} (h : G.Reachable u v)
    (hp : G.Walk u v → p) : p 
Arguments: {V : Type u} {G : SimpleGraph V} {p : Prop} {u v : V} (h : G.Reachable u v) (hp : G.Walk u v → p)
Goal: p
-- SimpleGraph.Reachable.elim_path
protected theorem Reachable.elim_path {p : Prop} {u v : V} (h : G.Reachable u v)
    (hp : G.Path u v → p) : p 
Arguments: {V : Type u} {G : SimpleGraph V} {p : Prop} {u v : V} (h : G.Reachable u v) (hp : G.Path u v → p)
Goal: p
-- SimpleGraph.Walk.reachable
protected theorem Walk.reachable {G : SimpleGraph V} {u v : V} (p : G.Walk u v) : G.Reachable u v 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: G.Reachable u v
-- SimpleGraph.Adj.reachable
protected theorem Adj.reachable {u v : V} (h : G.Adj u v) : G.Reachable u v 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (h : G.Adj u v)
Goal: G.Reachable u v
-- SimpleGraph.Reachable.refl
@[refl]
protected theorem Reachable.refl (u : V) : G.Reachable u u 
Arguments: {V : Type u} {G : SimpleGraph V} (u : V)
Goal: G.Reachable u u
-- SimpleGraph.Reachable.rfl
protected theorem Reachable.rfl {u : V} : G.Reachable u u 
Arguments: {V : Type u} {G : SimpleGraph V} {u : V}
Goal: G.Reachable u u
-- SimpleGraph.Reachable.symm
@[symm]
protected theorem Reachable.symm {u v : V} (huv : G.Reachable u v) : G.Reachable v u 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (huv : G.Reachable u v)
Goal: G.Reachable v u
-- SimpleGraph.reachable_comm
theorem reachable_comm {u v : V} : G.Reachable u v ↔ G.Reachable v u 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V}
Goal: G.Reachable u v ↔ G.Reachable v u
-- SimpleGraph.Reachable.trans
@[trans]
protected theorem Reachable.trans {u v w : V} (huv : G.Reachable u v) (hvw : G.Reachable v w) :
    G.Reachable u w 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (huv : G.Reachable u v) (hvw : G.Reachable v w)
Goal: G.Reachable u w
-- SimpleGraph.reachable_iff_reflTransGen
theorem reachable_iff_reflTransGen (u v : V) :
    G.Reachable u v ↔ Relation.ReflTransGen G.Adj u v 
Arguments: {V : Type u} {G : SimpleGraph V} (u v : V)
Goal: G.Reachable u v ↔ Relation.ReflTransGen G.Adj u v
-- SimpleGraph.Reachable.map
protected theorem Reachable.map {u v : V} {G : SimpleGraph V} {G' : SimpleGraph V'} (f : G →g G')
    (h : G.Reachable u v) : G'.Reachable (f u) (f v) 
Arguments: {V : Type u} {V' : Type v} {u v : V} {G : SimpleGraph V} {G' : SimpleGraph V'} (f : G →g G') (h : G.Reachable u v)
Goal: G'.Reachable (f u) (f v)
-- SimpleGraph.Reachable.mono
@[mono]
protected lemma Reachable.mono {u v : V} {G G' : SimpleGraph V}
    (h : G ≤ G') (Guv : G.Reachable u v) : G'.Reachable u v 
Arguments: {V : Type u} {u v : V} {G G' : SimpleGraph V} (h : G ≤ G') (Guv : G.Reachable u v)
Goal: G'.Reachable u v
-- SimpleGraph.Iso.reachable_iff
theorem Iso.reachable_iff {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {u v : V} :
    G'.Reachable (φ u) (φ v) ↔ G.Reachable u v 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {u v : V}
Goal: G'.Reachable (φ u) (φ v) ↔ G.Reachable u v
-- SimpleGraph.Iso.symm_apply_reachable
theorem Iso.symm_apply_reachable {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {u : V}
    {v : V'} : G.Reachable (φ.symm v) u ↔ G'.Reachable v (φ u) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {u : V} {v : V'}
Goal: G.Reachable (φ.symm v) u ↔ G'.Reachable v (φ u)
-- SimpleGraph.reachable_is_equivalence
theorem reachable_is_equivalence : Equivalence G.Reachable 
Arguments: {V : Type u} (G : SimpleGraph V)
Goal: Equivalence G.Reachable
-- SimpleGraph.Preconnected.map
theorem Preconnected.map {G : SimpleGraph V} {H : SimpleGraph V'} (f : G →g H) (hf : Surjective f)
    (hG : G.Preconnected) : H.Preconnected 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {H : SimpleGraph V'} (f : G →g H) (hf : Function.Surjective ⇑f) (hG : G.Preconnected)
Goal: H.Preconnected
-- SimpleGraph.Preconnected.mono
@[mono]
protected lemma Preconnected.mono  {G G' : SimpleGraph V} (h : G ≤ G') (hG : G.Preconnected) :
    G'.Preconnected 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') (hG : G.Preconnected)
Goal: G'.Preconnected
-- SimpleGraph.top_preconnected
lemma top_preconnected : (⊤ : SimpleGraph V).Preconnected 
Arguments: {V : Type u}
Goal: ⊤.Preconnected
-- SimpleGraph.Iso.preconnected_iff
theorem Iso.preconnected_iff {G : SimpleGraph V} {H : SimpleGraph V'} (e : G ≃g H) :
    G.Preconnected ↔ H.Preconnected 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {H : SimpleGraph V'} (e : G ≃g H)
Goal: G.Preconnected ↔ H.Preconnected
-- SimpleGraph.Connected.preconnected
@[mk_iff]
structure Connected : Prop where
  protected preconnected : G.Preconnected
  protected [nonempty : Nonempty V]
Arguments: {V : Type u} {G : SimpleGraph V} (self : G.Connected)
Goal: G.Preconnected
-- SimpleGraph.connected_iff
@[mk_iff]
structure Connected : Prop where
  protected preconnected : G.Preconnected
  protected [nonempty : Nonempty V]
Arguments: {V : Type u} (G : SimpleGraph V)
Goal: G.Connected ↔ G.Preconnected ∧ Nonempty V
-- SimpleGraph.Connected.nonempty
@[mk_iff]
structure Connected : Prop where
  protected preconnected : G.Preconnected
  protected [nonempty : Nonempty V]
Arguments: {V : Type u} {G : SimpleGraph V} (self : G.Connected)
Goal: Nonempty V
-- SimpleGraph.connected_iff_exists_forall_reachable
lemma connected_iff_exists_forall_reachable : G.Connected ↔ ∃ v, ∀ w, G.Reachable v w 
Arguments: {V : Type u} (G : SimpleGraph V)
Goal: G.Connected ↔ ∃ v, ∀ (w : V), G.Reachable v w
-- SimpleGraph.Connected.map
theorem Connected.map {G : SimpleGraph V} {H : SimpleGraph V'} (f : G →g H) (hf : Surjective f)
    (hG : G.Connected) : H.Connected 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {H : SimpleGraph V'} (f : G →g H) (hf : Function.Surjective ⇑f) (hG : G.Connected)
Goal: H.Connected
-- SimpleGraph.Connected.mono
@[mono]
protected lemma Connected.mono {G G' : SimpleGraph V} (h : G ≤ G')
    (hG : G.Connected) : G'.Connected 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') (hG : G.Connected)
Goal: G'.Connected
-- SimpleGraph.top_connected
lemma top_connected [Nonempty V] : (⊤ : SimpleGraph V).Connected 
Arguments: {V : Type u} [Nonempty V]
Goal: ⊤.Connected
-- SimpleGraph.Iso.connected_iff
theorem Iso.connected_iff {G : SimpleGraph V} {H : SimpleGraph V'} (e : G ≃g H) :
    G.Connected ↔ H.Connected 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {H : SimpleGraph V'} (e : G ≃g H)
Goal: G.Connected ↔ H.Connected
-- SimpleGraph.ConnectedComponent.inhabited_default
@[simps]
instance inhabited [Inhabited V] : Inhabited G.ConnectedComponent 
Arguments: {V : Type u} {G : SimpleGraph V} [Inhabited V]
Goal: default = G.connectedComponentMk default
-- SimpleGraph.ConnectedComponent.ind
@[elab_as_elim]
protected theorem ind {β : G.ConnectedComponent → Prop}
    (h : ∀ v : V, β (G.connectedComponentMk v)) (c : G.ConnectedComponent) : β c 
Arguments: {V : Type u} {G : SimpleGraph V} {β : G.ConnectedComponent → Prop} (h : ∀ (v : V), β (G.connectedComponentMk v)) (c : G.ConnectedComponent)
Goal: β c
-- SimpleGraph.ConnectedComponent.ind₂
@[elab_as_elim]
protected theorem ind₂ {β : G.ConnectedComponent → G.ConnectedComponent → Prop}
    (h : ∀ v w : V, β (G.connectedComponentMk v) (G.connectedComponentMk w))
    (c d : G.ConnectedComponent) : β c d 
Arguments: {V : Type u} {G : SimpleGraph V} {β : G.ConnectedComponent → G.ConnectedComponent → Prop} (h : ∀ (v w : V), β (G.connectedComponentMk v) (G.connectedComponentMk w)) (c d : G.ConnectedComponent)
Goal: β c d
-- SimpleGraph.ConnectedComponent.sound
protected theorem sound {v w : V} :
    G.Reachable v w → G.connectedComponentMk v = G.connectedComponentMk w 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V}
Goal: G.Reachable v w → G.connectedComponentMk v = G.connectedComponentMk w
-- SimpleGraph.ConnectedComponent.exact
protected theorem exact {v w : V} :
    G.connectedComponentMk v = G.connectedComponentMk w → G.Reachable v w 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V}
Goal: G.connectedComponentMk v = G.connectedComponentMk w → G.Reachable v w
-- SimpleGraph.ConnectedComponent.eq
@[simp]
protected theorem eq {v w : V} :
    G.connectedComponentMk v = G.connectedComponentMk w ↔ G.Reachable v w 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V}
Goal: G.connectedComponentMk v = G.connectedComponentMk w ↔ G.Reachable v w
-- SimpleGraph.ConnectedComponent.connectedComponentMk_eq_of_adj
theorem connectedComponentMk_eq_of_adj {v w : V} (a : G.Adj v w) :
    G.connectedComponentMk v = G.connectedComponentMk w 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (a : G.Adj v w)
Goal: G.connectedComponentMk v = G.connectedComponentMk w
-- SimpleGraph.ConnectedComponent.lift_mk
@[simp]
protected theorem lift_mk {β : Sort*} {f : V → β}
    {h : ∀ (v w : V) (p : G.Walk v w), p.IsPath → f v = f w} {v : V} :
    ConnectedComponent.lift f h (G.connectedComponentMk v) = f v 
Arguments: {V : Type u} {G : SimpleGraph V} {β : Sort u_1} {f : V → β} {h : ∀ (v w : V) (p : G.Walk v w), p.IsPath → f v = f w} {v : V}
Goal: SimpleGraph.ConnectedComponent.lift f h (G.connectedComponentMk v) = f v
-- SimpleGraph.ConnectedComponent.exists
protected theorem «exists» {p : G.ConnectedComponent → Prop} :
    (∃ c : G.ConnectedComponent, p c) ↔ ∃ v, p (G.connectedComponentMk v) 
Arguments: {V : Type u} {G : SimpleGraph V} {p : G.ConnectedComponent → Prop}
Goal: (∃ c, p c) ↔ ∃ v, p (G.connectedComponentMk v)
-- SimpleGraph.ConnectedComponent.forall
protected theorem «forall» {p : G.ConnectedComponent → Prop} :
    (∀ c : G.ConnectedComponent, p c) ↔ ∀ v, p (G.connectedComponentMk v) 
Arguments: {V : Type u} {G : SimpleGraph V} {p : G.ConnectedComponent → Prop}
Goal: (∀ (c : G.ConnectedComponent), p c) ↔ ∀ (v : V), p (G.connectedComponentMk v)
-- SimpleGraph.Preconnected.subsingleton_connectedComponent
theorem _root_.SimpleGraph.Preconnected.subsingleton_connectedComponent (h : G.Preconnected) :
    Subsingleton G.ConnectedComponent 
Arguments: {V : Type u} {G : SimpleGraph V} (h : G.Preconnected)
Goal: Subsingleton G.ConnectedComponent
-- SimpleGraph.ConnectedComponent.map_mk
@[simp]
theorem map_mk (φ : G →g G') (v : V) :
    (G.connectedComponentMk v).map φ = G'.connectedComponentMk (φ v) 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (φ : G →g G') (v : V)
Goal: SimpleGraph.ConnectedComponent.map φ (G.connectedComponentMk v) = G'.connectedComponentMk (φ v)
-- SimpleGraph.ConnectedComponent.map_id
@[simp]
theorem map_id (C : ConnectedComponent G) : C.map Hom.id = C 
Arguments: {V : Type u} {G : SimpleGraph V} (C : G.ConnectedComponent)
Goal: SimpleGraph.ConnectedComponent.map SimpleGraph.Hom.id C = C
-- SimpleGraph.ConnectedComponent.map_comp
@[simp]
theorem map_comp (C : G.ConnectedComponent) (φ : G →g G') (ψ : G' →g G'') :
    (C.map φ).map ψ = C.map (ψ.comp φ) 
Arguments: {V : Type u} {V' : Type v} {V'' : Type w} {G : SimpleGraph V} {G' : SimpleGraph V'} {G'' : SimpleGraph V''} (C : G.ConnectedComponent) (φ : G →g G') (ψ : G' →g G'')
Goal: SimpleGraph.ConnectedComponent.map ψ (SimpleGraph.ConnectedComponent.map φ C) =
  SimpleGraph.ConnectedComponent.map (ψ.comp φ) C
-- SimpleGraph.ConnectedComponent.iso_image_comp_eq_map_iff_eq_comp
@[simp]
theorem iso_image_comp_eq_map_iff_eq_comp {C : G.ConnectedComponent} :
    G'.connectedComponentMk (φ v) = C.map ↑(↑φ : G ↪g G') ↔ G.connectedComponentMk v = C 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {v : V} {C : G.ConnectedComponent}
Goal: G'.connectedComponentMk (φ v) = SimpleGraph.ConnectedComponent.map (RelIso.toRelEmbedding φ).toRelHom C ↔
  G.connectedComponentMk v = C
-- SimpleGraph.ConnectedComponent.iso_inv_image_comp_eq_iff_eq_map
@[simp]
theorem iso_inv_image_comp_eq_iff_eq_map {C : G.ConnectedComponent} :
    G.connectedComponentMk (φ.symm v') = C ↔ G'.connectedComponentMk v' = C.map φ 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} {φ : G ≃g G'} {v' : V'} {C : G.ConnectedComponent}
Goal: G.connectedComponentMk (φ.symm v') = C ↔
  G'.connectedComponentMk v' = SimpleGraph.ConnectedComponent.map (RelIso.toRelEmbedding φ).toRelHom C
-- SimpleGraph.Iso.connectedComponentEquiv_symm_apply
@[simps]
def connectedComponentEquiv (φ : G ≃g G') : G.ConnectedComponent ≃ G'.ConnectedComponent 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (φ : G ≃g G') (C : G'.ConnectedComponent)
Goal: φ.connectedComponentEquiv.symm C = SimpleGraph.ConnectedComponent.map (RelIso.toRelEmbedding φ.symm).toRelHom C
-- SimpleGraph.Iso.connectedComponentEquiv_apply
@[simps]
def connectedComponentEquiv (φ : G ≃g G') : G.ConnectedComponent ≃ G'.ConnectedComponent 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (φ : G ≃g G') (C : G.ConnectedComponent)
Goal: φ.connectedComponentEquiv C = SimpleGraph.ConnectedComponent.map (RelIso.toRelEmbedding φ).toRelHom C
-- SimpleGraph.Iso.connectedComponentEquiv_refl
@[simp]
theorem connectedComponentEquiv_refl :
    (Iso.refl : G ≃g G).connectedComponentEquiv = Equiv.refl _ 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: SimpleGraph.Iso.refl.connectedComponentEquiv = Equiv.refl G.ConnectedComponent
-- SimpleGraph.Iso.connectedComponentEquiv_symm
@[simp]
theorem connectedComponentEquiv_symm (φ : G ≃g G') :
    φ.symm.connectedComponentEquiv = φ.connectedComponentEquiv.symm 
Arguments: {V : Type u} {V' : Type v} {G : SimpleGraph V} {G' : SimpleGraph V'} (φ : G ≃g G')
Goal: φ.symm.connectedComponentEquiv = φ.connectedComponentEquiv.symm
-- SimpleGraph.Iso.connectedComponentEquiv_trans
@[simp]
theorem connectedComponentEquiv_trans (φ : G ≃g G') (φ' : G' ≃g G'') :
    connectedComponentEquiv (φ.trans φ') =
    φ.connectedComponentEquiv.trans φ'.connectedComponentEquiv 
Arguments: {V : Type u} {V' : Type v} {V'' : Type w} {G : SimpleGraph V} {G' : SimpleGraph V'} {G'' : SimpleGraph V''} (φ : G ≃g G') (φ' : G' ≃g G'')
Goal: SimpleGraph.Iso.connectedComponentEquiv (RelIso.trans φ φ') = φ.connectedComponentEquiv.trans φ'.connectedComponentEquiv
-- SimpleGraph.ConnectedComponent.supp_injective
@[ext]
theorem supp_injective :
    Function.Injective (ConnectedComponent.supp : G.ConnectedComponent → Set V) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: Function.Injective SimpleGraph.ConnectedComponent.supp
-- SimpleGraph.ConnectedComponent.supp_inj
@[simp]
theorem supp_inj {C D : G.ConnectedComponent} : C.supp = D.supp ↔ C = D 
Arguments: {V : Type u} {G : SimpleGraph V} {C D : G.ConnectedComponent}
Goal: C.supp = D.supp ↔ C = D