ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.Subgraph.sInf_adj
@[simp]
theorem sInf_adj {s : Set G.Subgraph} : (sInf s).Adj a b ↔ (∀ G' ∈ s, Adj G' a b) ∧ G.Adj a b 
Arguments: {V : Type u} {G : SimpleGraph V} {a b : V} {s : Set G.Subgraph}
Goal: (sInf s).Adj a b ↔ (∀ G' ∈ s, G'.Adj a b) ∧ G.Adj a b
-- SimpleGraph.Subgraph.iSup_adj
@[simp]
theorem iSup_adj {f : ι → G.Subgraph} : (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} {a b : V} {f : ι → G.Subgraph}
Goal: (⨆ i, f i).Adj a b ↔ ∃ i, (f i).Adj a b
-- SimpleGraph.Subgraph.iInf_adj
@[simp]
theorem iInf_adj {f : ι → G.Subgraph} : (⨅ i, f i).Adj a b ↔ (∀ i, (f i).Adj a b) ∧ G.Adj a b 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} {a b : V} {f : ι → G.Subgraph}
Goal: (⨅ i, f i).Adj a b ↔ (∀ (i : ι), (f i).Adj a b) ∧ G.Adj a b
-- SimpleGraph.Subgraph.sInf_adj_of_nonempty
theorem sInf_adj_of_nonempty {s : Set G.Subgraph} (hs : s.Nonempty) :
    (sInf s).Adj a b ↔ ∀ G' ∈ s, Adj G' a b 
Arguments: {V : Type u} {G : SimpleGraph V} {a b : V} {s : Set G.Subgraph} (hs : s.Nonempty)
Goal: (sInf s).Adj a b ↔ ∀ G' ∈ s, G'.Adj a b
-- SimpleGraph.Subgraph.iInf_adj_of_nonempty
theorem iInf_adj_of_nonempty [Nonempty ι] {f : ι → G.Subgraph} :
    (⨅ i, f i).Adj a b ↔ ∀ i, (f i).Adj a b 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} {a b : V} [Nonempty ι] {f : ι → G.Subgraph}
Goal: (⨅ i, f i).Adj a b ↔ ∀ (i : ι), (f i).Adj a b
-- SimpleGraph.Subgraph.verts_sSup
@[simp]
theorem verts_sSup (s : Set G.Subgraph) : (sSup s).verts = ⋃ G' ∈ s, verts G' 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph)
Goal: (sSup s).verts = ⋃ G' ∈ s, G'.verts
-- SimpleGraph.Subgraph.verts_sInf
@[simp]
theorem verts_sInf (s : Set G.Subgraph) : (sInf s).verts = ⋂ G' ∈ s, verts G' 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph)
Goal: (sInf s).verts = ⋂ G' ∈ s, G'.verts
-- SimpleGraph.Subgraph.verts_iSup
@[simp]
theorem verts_iSup {f : ι → G.Subgraph} : (⨆ i, f i).verts = ⋃ i, (f i).verts 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} {f : ι → G.Subgraph}
Goal: (⨆ i, f i).verts = ⋃ i, (f i).verts
-- SimpleGraph.Subgraph.verts_iInf
@[simp]
theorem verts_iInf {f : ι → G.Subgraph} : (⨅ i, f i).verts = ⋂ i, (f i).verts 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} {f : ι → G.Subgraph}
Goal: (⨅ i, f i).verts = ⋂ i, (f i).verts
-- SimpleGraph.Subgraph.verts_spanningCoe_injective
theorem verts_spanningCoe_injective :
    (fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: Function.Injective fun G' => (G'.verts, G'.spanningCoe)
-- SimpleGraph.Subgraph.verts_mono
@[gcongr] lemma verts_mono {H H' : G.Subgraph} (h : H ≤ H') : H.verts ⊆ H'.verts 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (h : H ≤ H')
Goal: H.verts ⊆ H'.verts
-- SimpleGraph.Subgraph.verts_monotone
lemma verts_monotone : Monotone (verts : G.Subgraph → Set V) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: Monotone SimpleGraph.Subgraph.verts
-- SimpleGraph.Subgraph.subgraphInhabited_default
@[simps]
instance subgraphInhabited : Inhabited (Subgraph G) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: default = ⊥
-- SimpleGraph.Subgraph.neighborSet_sup
@[simp]
theorem neighborSet_sup {H H' : G.Subgraph} (v : V) :
    (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (v : V)
Goal: (H ⊔ H').neighborSet v = H.neighborSet v ∪ H'.neighborSet v
-- SimpleGraph.Subgraph.neighborSet_inf
@[simp]
theorem neighborSet_inf {H H' : G.Subgraph} (v : V) :
    (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (v : V)
Goal: (H ⊓ H').neighborSet v = H.neighborSet v ∩ H'.neighborSet v
-- SimpleGraph.Subgraph.neighborSet_top
@[simp]
theorem neighborSet_top (v : V) : (⊤ : G.Subgraph).neighborSet v = G.neighborSet v 
Arguments: {V : Type u} {G : SimpleGraph V} (v : V)
Goal: ⊤.neighborSet v = G.neighborSet v
-- SimpleGraph.Subgraph.neighborSet_bot
@[simp]
theorem neighborSet_bot (v : V) : (⊥ : G.Subgraph).neighborSet v = ∅ 
Arguments: {V : Type u} {G : SimpleGraph V} (v : V)
Goal: ⊥.neighborSet v = ∅
-- SimpleGraph.Subgraph.neighborSet_sSup
@[simp]
theorem neighborSet_sSup (s : Set G.Subgraph) (v : V) :
    (sSup s).neighborSet v = ⋃ G' ∈ s, neighborSet G' v 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph) (v : V)
Goal: (sSup s).neighborSet v = ⋃ G' ∈ s, G'.neighborSet v
-- SimpleGraph.Subgraph.neighborSet_sInf
@[simp]
theorem neighborSet_sInf (s : Set G.Subgraph) (v : V) :
    (sInf s).neighborSet v = (⋂ G' ∈ s, neighborSet G' v) ∩ G.neighborSet v 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph) (v : V)
Goal: (sInf s).neighborSet v = (⋂ G' ∈ s, G'.neighborSet v) ∩ G.neighborSet v
-- SimpleGraph.Subgraph.neighborSet_iSup
@[simp]
theorem neighborSet_iSup (f : ι → G.Subgraph) (v : V) :
    (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} (f : ι → G.Subgraph) (v : V)
Goal: (⨆ i, f i).neighborSet v = ⋃ i, (f i).neighborSet v
-- SimpleGraph.Subgraph.neighborSet_iInf
@[simp]
theorem neighborSet_iInf (f : ι → G.Subgraph) (v : V) :
    (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} (f : ι → G.Subgraph) (v : V)
Goal: (⨅ i, f i).neighborSet v = (⋂ i, (f i).neighborSet v) ∩ G.neighborSet v
-- SimpleGraph.Subgraph.edgeSet_top
@[simp]
theorem edgeSet_top : (⊤ : Subgraph G).edgeSet = G.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: ⊤.edgeSet = G.edgeSet
-- SimpleGraph.Subgraph.edgeSet_bot
@[simp]
theorem edgeSet_bot : (⊥ : Subgraph G).edgeSet = ∅ 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: ⊥.edgeSet = ∅
-- SimpleGraph.Subgraph.edgeSet_inf
@[simp]
theorem edgeSet_inf {H₁ H₂ : Subgraph G} : (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V} {H₁ H₂ : G.Subgraph}
Goal: (H₁ ⊓ H₂).edgeSet = H₁.edgeSet ∩ H₂.edgeSet
-- SimpleGraph.Subgraph.edgeSet_sup
@[simp]
theorem edgeSet_sup {H₁ H₂ : Subgraph G} : (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V} {H₁ H₂ : G.Subgraph}
Goal: (H₁ ⊔ H₂).edgeSet = H₁.edgeSet ∪ H₂.edgeSet
-- SimpleGraph.Subgraph.edgeSet_sSup
@[simp]
theorem edgeSet_sSup (s : Set G.Subgraph) : (sSup s).edgeSet = ⋃ G' ∈ s, edgeSet G' 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph)
Goal: (sSup s).edgeSet = ⋃ G' ∈ s, G'.edgeSet
-- SimpleGraph.Subgraph.edgeSet_sInf
@[simp]
theorem edgeSet_sInf (s : Set G.Subgraph) :
    (sInf s).edgeSet = (⋂ G' ∈ s, edgeSet G') ∩ G.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set G.Subgraph)
Goal: (sInf s).edgeSet = (⋂ G' ∈ s, G'.edgeSet) ∩ G.edgeSet
-- SimpleGraph.Subgraph.edgeSet_iSup
@[simp]
theorem edgeSet_iSup (f : ι → G.Subgraph) :
    (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} (f : ι → G.Subgraph)
Goal: (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet
-- SimpleGraph.Subgraph.edgeSet_iInf
@[simp]
theorem edgeSet_iInf (f : ι → G.Subgraph) :
    (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet 
Arguments: {ι : Sort u_1} {V : Type u} {G : SimpleGraph V} (f : ι → G.Subgraph)
Goal: (⨅ i, f i).edgeSet = (⋂ i, (f i).edgeSet) ∩ G.edgeSet
-- SimpleGraph.Subgraph.spanningCoe_top
@[simp]
theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: ⊤.spanningCoe = G
-- SimpleGraph.Subgraph.spanningCoe_bot
@[simp]
theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: ⊥.spanningCoe = ⊥
-- SimpleGraph.toSubgraph_verts
@[simps]
def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph 
Arguments: {V : Type u} {G : SimpleGraph V} (H : SimpleGraph V) (h : H ≤ G)
Goal: (SimpleGraph.toSubgraph H h).verts = Set.univ
-- SimpleGraph.toSubgraph_adj
@[simps]
def _root_.SimpleGraph.toSubgraph (H : SimpleGraph V) (h : H ≤ G) : G.Subgraph 
Arguments: {V : Type u} {G : SimpleGraph V} (H : SimpleGraph V) (h : H ≤ G)
Goal: ∀ (a a_1 : V), (SimpleGraph.toSubgraph H h).Adj a a_1 = H.Adj a a_1
-- SimpleGraph.Subgraph.support_mono
theorem support_mono {H H' : Subgraph G} (h : H ≤ H') : H.support ⊆ H'.support 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (h : H ≤ H')
Goal: H.support ⊆ H'.support
-- SimpleGraph.toSubgraph.isSpanning
theorem _root_.SimpleGraph.toSubgraph.isSpanning (H : SimpleGraph V) (h : H ≤ G) :
    (toSubgraph H h).IsSpanning 
Arguments: {V : Type u} {G : SimpleGraph V} (H : SimpleGraph V) (h : H ≤ G)
Goal: (SimpleGraph.toSubgraph H h).IsSpanning
-- SimpleGraph.Subgraph.spanningCoe_le_of_le
theorem spanningCoe_le_of_le {H H' : Subgraph G} (h : H ≤ H') : H.spanningCoe ≤ H'.spanningCoe 
Arguments: {V : Type u} {G : SimpleGraph V} {H H' : G.Subgraph} (h : H ≤ H')
Goal: H.spanningCoe ≤ H'.spanningCoe
-- SimpleGraph.Subgraph.edgeSet_mono
theorem edgeSet_mono {H₁ H₂ : Subgraph G} (h : H₁ ≤ H₂) : H₁.edgeSet ≤ H₂.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V} {H₁ H₂ : G.Subgraph} (h : H₁ ≤ H₂)
Goal: H₁.edgeSet ≤ H₂.edgeSet
-- Disjoint.edgeSet
theorem _root_.Disjoint.edgeSet {H₁ H₂ : Subgraph G} (h : Disjoint H₁ H₂) :
    Disjoint H₁.edgeSet H₂.edgeSet 
Arguments: {V : Type u} {G : SimpleGraph V} {H₁ H₂ : G.Subgraph} (h : Disjoint H₁ H₂)
Goal: Disjoint H₁.edgeSet H₂.edgeSet
-- SimpleGraph.Subgraph.map_adj
@[simps]
protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph)
Goal: ∀ (a a_1 : W), (SimpleGraph.Subgraph.map f H).Adj a a_1 = Relation.Map H.Adj (⇑f) (⇑f) a a_1
-- SimpleGraph.Subgraph.map_verts
@[simps]
protected def map {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) : G'.Subgraph 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph)
Goal: (SimpleGraph.Subgraph.map f H).verts = ⇑f '' H.verts
-- SimpleGraph.Subgraph.map_monotone
theorem map_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.map f) 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G')
Goal: Monotone (SimpleGraph.Subgraph.map f)
-- SimpleGraph.Subgraph.map_sup
theorem map_sup {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph} :
    (H ⊔ H').map f = H.map f ⊔ H'.map f 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {H H' : G.Subgraph}
Goal: SimpleGraph.Subgraph.map f (H ⊔ H') = SimpleGraph.Subgraph.map f H ⊔ SimpleGraph.Subgraph.map f H'
-- SimpleGraph.Subgraph.comap_adj
@[simps]
protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) (u v : V)
Goal: (SimpleGraph.Subgraph.comap f H).Adj u v = (G.Adj u v ∧ H.Adj (f u) (f v))
-- SimpleGraph.Subgraph.comap_verts
@[simps]
protected def comap {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph) : G.Subgraph 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (H : G'.Subgraph)
Goal: (SimpleGraph.Subgraph.comap f H).verts = ⇑f ⁻¹' H.verts
-- SimpleGraph.Subgraph.comap_monotone
theorem comap_monotone {G' : SimpleGraph W} (f : G →g G') : Monotone (Subgraph.comap f) 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G')
Goal: Monotone (SimpleGraph.Subgraph.comap f)
-- SimpleGraph.Subgraph.map_le_iff_le_comap
theorem map_le_iff_le_comap {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph) :
    H.map f ≤ H' ↔ H ≤ H'.comap f 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (H : G.Subgraph) (H' : G'.Subgraph)
Goal: SimpleGraph.Subgraph.map f H ≤ H' ↔ H ≤ SimpleGraph.Subgraph.comap f H'
-- SimpleGraph.Subgraph.inclusion_apply_coe
@[simps]
def inclusion {x y : Subgraph G} (h : x ≤ y) : x.coe →g y.coe 
Arguments: {V : Type u} {G : SimpleGraph V} {x y : G.Subgraph} (h : x ≤ y) (v : ↑x.verts)
Goal: ↑((SimpleGraph.Subgraph.inclusion h) v) = ↑v
-- SimpleGraph.Subgraph.inclusion.injective
theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) 
Arguments: {V : Type u} {G : SimpleGraph V} {x y : G.Subgraph} (h : x ≤ y)
Goal: Function.Injective ⇑(SimpleGraph.Subgraph.inclusion h)
-- SimpleGraph.Subgraph.hom_apply
@[simps]
protected def hom (x : Subgraph G) : x.coe →g G 
Arguments: {V : Type u} {G : SimpleGraph V} (x : G.Subgraph) (v : ↑x.verts)
Goal: x.hom v = ↑v
-- SimpleGraph.Subgraph.coe_hom
@[simp] lemma coe_hom (x : Subgraph G) :
    (x.hom : x.verts → V) = (fun (v : x.verts) => (v : V)) 
Arguments: {V : Type u} {G : SimpleGraph V} (x : G.Subgraph)
Goal: ⇑x.hom = fun v => ↑v
-- SimpleGraph.Subgraph.hom.injective
theorem hom.injective {x : Subgraph G} : Function.Injective x.hom 
Arguments: {V : Type u} {G : SimpleGraph V} {x : G.Subgraph}
Goal: Function.Injective ⇑x.hom
-- SimpleGraph.Subgraph.spanningHom_apply
@[simps]
def spanningHom (x : Subgraph G) : x.spanningCoe →g G 
Arguments: {V : Type u} {G : SimpleGraph V} (x : G.Subgraph) (a : V)
Goal: x.spanningHom a = id a
-- SimpleGraph.Subgraph.spanningHom.injective
theorem spanningHom.injective {x : Subgraph G} : Function.Injective x.spanningHom 
Arguments: {V : Type u} {G : SimpleGraph V} {x : G.Subgraph}
Goal: Function.Injective ⇑x.spanningHom
-- SimpleGraph.Subgraph.neighborSet_subset_of_subgraph
theorem neighborSet_subset_of_subgraph {x y : Subgraph G} (h : x ≤ y) (v : V) :
    x.neighborSet v ⊆ y.neighborSet v 
Arguments: {V : Type u} {G : SimpleGraph V} {x y : G.Subgraph} (h : x ≤ y) (v : V)
Goal: x.neighborSet v ⊆ y.neighborSet v
-- SimpleGraph.Subgraph.IsSpanning.card_verts
theorem IsSpanning.card_verts [Fintype V] {G' : Subgraph G} [Fintype G'.verts] (h : G'.IsSpanning) :
    G'.verts.toFinset.card = Fintype.card V 
Arguments: {V : Type u} {G : SimpleGraph V} [Fintype V] {G' : G.Subgraph} [Fintype ↑G'.verts] (h : G'.IsSpanning)
Goal: G'.verts.toFinset.card = Fintype.card V
-- SimpleGraph.Subgraph.finset_card_neighborSet_eq_degree
theorem finset_card_neighborSet_eq_degree {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
    (G'.neighborSet v).toFinset.card = G'.degree v 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} {v : V} [Fintype ↑(G'.neighborSet v)]
Goal: (G'.neighborSet v).toFinset.card = G'.degree v
-- SimpleGraph.Subgraph.degree_le
theorem degree_le (G' : Subgraph G) (v : V) [Fintype (G'.neighborSet v)]
    [Fintype (G.neighborSet v)] : G'.degree v ≤ G.degree v 
Arguments: {V : Type u} {G : SimpleGraph V} (G' : G.Subgraph) (v : V) [Fintype ↑(G'.neighborSet v)] [Fintype ↑(G.neighborSet v)]
Goal: G'.degree v ≤ G.degree v
-- SimpleGraph.Subgraph.degree_le'
theorem degree_le' (G' G'' : Subgraph G) (h : G' ≤ G'') (v : V) [Fintype (G'.neighborSet v)]
    [Fintype (G''.neighborSet v)] : G'.degree v ≤ G''.degree v 
Arguments: {V : Type u} {G : SimpleGraph V} (G' G'' : G.Subgraph) (h : G' ≤ G'') (v : V) [Fintype ↑(G'.neighborSet v)] [Fintype ↑(G''.neighborSet v)]
Goal: G'.degree v ≤ G''.degree v
-- SimpleGraph.Subgraph.coe_degree
@[simp]
theorem coe_degree (G' : Subgraph G) (v : G'.verts) [Fintype (G'.coe.neighborSet v)]
    [Fintype (G'.neighborSet v)] : G'.coe.degree v = G'.degree v 
Arguments: {V : Type u} {G : SimpleGraph V} (G' : G.Subgraph) (v : ↑G'.verts) [Fintype ↑(G'.coe.neighborSet v)] [Fintype ↑(G'.neighborSet ↑v)]
Goal: G'.coe.degree v = G'.degree ↑v
-- SimpleGraph.Subgraph.degree_spanningCoe
@[simp]
theorem degree_spanningCoe {G' : G.Subgraph} (v : V) [Fintype (G'.neighborSet v)]
    [Fintype (G'.spanningCoe.neighborSet v)] : G'.spanningCoe.degree v = G'.degree v 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (v : V) [Fintype ↑(G'.neighborSet v)] [Fintype ↑(G'.spanningCoe.neighborSet v)]
Goal: G'.spanningCoe.degree v = G'.degree v
-- SimpleGraph.Subgraph.degree_eq_one_iff_unique_adj
theorem degree_eq_one_iff_unique_adj {G' : Subgraph G} {v : V} [Fintype (G'.neighborSet v)] :
    G'.degree v = 1 ↔ ∃! w : V, G'.Adj v w 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} {v : V} [Fintype ↑(G'.neighborSet v)]
Goal: G'.degree v = 1 ↔ ∃! w, G'.Adj v w
-- SimpleGraph.singletonSubgraph_le_iff
@[simp]
theorem singletonSubgraph_le_iff (v : V) (H : G.Subgraph) :
    G.singletonSubgraph v ≤ H ↔ v ∈ H.verts 
Arguments: {V : Type u} {G : SimpleGraph V} (v : V) (H : G.Subgraph)
Goal: G.singletonSubgraph v ≤ H ↔ v ∈ H.verts
-- SimpleGraph.map_singletonSubgraph
@[simp]
theorem map_singletonSubgraph (f : G →g G') {v : V} :
    Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v) 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {v : V}
Goal: SimpleGraph.Subgraph.map f (G.singletonSubgraph v) = G'.singletonSubgraph (f v)
-- SimpleGraph.neighborSet_singletonSubgraph
@[simp]
theorem neighborSet_singletonSubgraph (v w : V) : (G.singletonSubgraph v).neighborSet w = ∅ 
Arguments: {V : Type u} {G : SimpleGraph V} (v w : V)
Goal: (G.singletonSubgraph v).neighborSet w = ∅
-- SimpleGraph.edgeSet_singletonSubgraph
@[simp]
theorem edgeSet_singletonSubgraph (v : V) : (G.singletonSubgraph v).edgeSet = ∅ 
Arguments: {V : Type u} {G : SimpleGraph V} (v : V)
Goal: (G.singletonSubgraph v).edgeSet = ∅
-- SimpleGraph.eq_singletonSubgraph_iff_verts_eq
theorem eq_singletonSubgraph_iff_verts_eq (H : G.Subgraph) {v : V} :
    H = G.singletonSubgraph v ↔ H.verts = {v} 
Arguments: {V : Type u} {G : SimpleGraph V} (H : G.Subgraph) {v : V}
Goal: H = G.singletonSubgraph v ↔ H.verts = {v}
-- SimpleGraph.edgeSet_subgraphOfAdj
@[simp]
theorem edgeSet_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
    (G.subgraphOfAdj hvw).edgeSet = {s(v, w)} 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).edgeSet = {s(v, w)}
-- SimpleGraph.subgraphOfAdj_le_of_adj
lemma subgraphOfAdj_le_of_adj {v w : V} (H : G.Subgraph) (h : H.Adj v w) :
    G.subgraphOfAdj (H.adj_sub h) ≤ H 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (H : G.Subgraph) (h : H.Adj v w)
Goal: G.subgraphOfAdj ⋯ ≤ H
-- SimpleGraph.subgraphOfAdj_symm
theorem subgraphOfAdj_symm {v w : V} (hvw : G.Adj v w) :
    G.subgraphOfAdj hvw.symm = G.subgraphOfAdj hvw 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (hvw : G.Adj v w)
Goal: G.subgraphOfAdj ⋯ = G.subgraphOfAdj hvw
-- SimpleGraph.map_subgraphOfAdj
@[simp]
theorem map_subgraphOfAdj (f : G →g G') {v w : V} (hvw : G.Adj v w) :
    Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj (f.map_adj hvw) 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {v w : V} (hvw : G.Adj v w)
Goal: SimpleGraph.Subgraph.map f (G.subgraphOfAdj hvw) = G'.subgraphOfAdj ⋯
-- SimpleGraph.neighborSet_subgraphOfAdj_subset
theorem neighborSet_subgraphOfAdj_subset {u v w : V} (hvw : G.Adj v w) :
    (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w} 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).neighborSet u ⊆ {v, w}
-- SimpleGraph.neighborSet_fst_subgraphOfAdj
@[simp]
theorem neighborSet_fst_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
    (G.subgraphOfAdj hvw).neighborSet v = {w} 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).neighborSet v = {w}
-- SimpleGraph.neighborSet_snd_subgraphOfAdj
@[simp]
theorem neighborSet_snd_subgraphOfAdj {v w : V} (hvw : G.Adj v w) :
    (G.subgraphOfAdj hvw).neighborSet w = {v} 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).neighborSet w = {v}
-- SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne
@[simp]
theorem neighborSet_subgraphOfAdj_of_ne_of_ne {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v)
    (hw : u ≠ w) : (G.subgraphOfAdj hvw).neighborSet u = ∅ 
Arguments: {V : Type u} {G : SimpleGraph V} {u v w : V} (hvw : G.Adj v w) (hv : u ≠ v) (hw : u ≠ w)
Goal: (G.subgraphOfAdj hvw).neighborSet u = ∅
-- SimpleGraph.neighborSet_subgraphOfAdj
theorem neighborSet_subgraphOfAdj [DecidableEq V] {u v w : V} (hvw : G.Adj v w) :
    (G.subgraphOfAdj hvw).neighborSet u =
    (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅ 
Arguments: {V : Type u} {G : SimpleGraph V} [DecidableEq V] {u v w : V} (hvw : G.Adj v w)
Goal: (G.subgraphOfAdj hvw).neighborSet u = (if u = v then {w} else ∅) ∪ if u = w then {v} else ∅
-- SimpleGraph.singletonSubgraph_fst_le_subgraphOfAdj
theorem singletonSubgraph_fst_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
    G.singletonSubgraph u ≤ G.subgraphOfAdj h 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} {h : G.Adj u v}
Goal: G.singletonSubgraph u ≤ G.subgraphOfAdj h
-- SimpleGraph.singletonSubgraph_snd_le_subgraphOfAdj
theorem singletonSubgraph_snd_le_subgraphOfAdj {u v : V} {h : G.Adj u v} :
    G.singletonSubgraph v ≤ G.subgraphOfAdj h 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} {h : G.Adj u v}
Goal: G.singletonSubgraph v ≤ G.subgraphOfAdj h
-- SimpleGraph.support_subgraphOfAdj
@[simp]
lemma support_subgraphOfAdj {u v : V} (h : G.Adj u v) :
    (G.subgraphOfAdj h).support = {u , v} 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} (h : G.Adj u v)
Goal: (G.subgraphOfAdj h).support = {u, v}
-- SimpleGraph.Subgraph.verts_coeSubgraph
@[simp]
lemma verts_coeSubgraph {G' : Subgraph G} (G'' : Subgraph G'.coe) :
    G''.coeSubgraph.verts = (G''.verts : Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (G'' : G'.coe.Subgraph)
Goal: (SimpleGraph.Subgraph.coeSubgraph G'').verts = Subtype.val '' G''.verts
-- SimpleGraph.Subgraph.coeSubgraph_adj
lemma coeSubgraph_adj {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V) :
    (G'.coeSubgraph G'').Adj v w ↔
      ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩ 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (G'' : G'.coe.Subgraph) (v w : V)
Goal: (SimpleGraph.Subgraph.coeSubgraph G'').Adj v w ↔ ∃ (hv : v ∈ G'.verts) (hw : w ∈ G'.verts), G''.Adj ⟨v, hv⟩ ⟨w, hw⟩
-- SimpleGraph.Subgraph.restrict_adj
lemma restrict_adj {G' G'' : G.Subgraph} (v w : G'.verts) :
    (G'.restrict G'').Adj v w ↔ G'.Adj v w ∧ G''.Adj v w 
Arguments: {V : Type u} {G : SimpleGraph V} {G' G'' : G.Subgraph} (v w : ↑G'.verts)
Goal: (SimpleGraph.Subgraph.restrict G'').Adj v w ↔ G'.Adj ↑v ↑w ∧ G''.Adj ↑v ↑w
-- SimpleGraph.Subgraph.restrict_coeSubgraph
theorem restrict_coeSubgraph {G' : G.Subgraph} (G'' : G'.coe.Subgraph) :
    Subgraph.restrict (Subgraph.coeSubgraph G'') = G'' 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (G'' : G'.coe.Subgraph)
Goal: SimpleGraph.Subgraph.restrict (SimpleGraph.Subgraph.coeSubgraph G'') = G''
-- SimpleGraph.Subgraph.coeSubgraph_injective
theorem coeSubgraph_injective (G' : G.Subgraph) :
    Function.Injective (Subgraph.coeSubgraph : G'.coe.Subgraph → G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} (G' : G.Subgraph)
Goal: Function.Injective SimpleGraph.Subgraph.coeSubgraph
-- SimpleGraph.Subgraph.coeSubgraph_le
lemma coeSubgraph_le {H : G.Subgraph} (H' : H.coe.Subgraph) :
    Subgraph.coeSubgraph H' ≤ H 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (H' : H.coe.Subgraph)
Goal: SimpleGraph.Subgraph.coeSubgraph H' ≤ H
-- SimpleGraph.Subgraph.coeSubgraph_restrict_eq
lemma coeSubgraph_restrict_eq {H : G.Subgraph} (H' : G.Subgraph) :
    Subgraph.coeSubgraph (H.restrict H') = H ⊓ H' 
Arguments: {V : Type u} {G : SimpleGraph V} {H : G.Subgraph} (H' : G.Subgraph)
Goal: SimpleGraph.Subgraph.coeSubgraph (SimpleGraph.Subgraph.restrict H') = H ⊓ H'
-- SimpleGraph.Subgraph.deleteEdges_verts
@[simp]
theorem deleteEdges_verts : (G'.deleteEdges s).verts = G'.verts 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (s : Set (Sym2 V))
Goal: (G'.deleteEdges s).verts = G'.verts
-- SimpleGraph.Subgraph.deleteEdges_adj
@[simp]
theorem deleteEdges_adj (v w : V) : (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ ¬s(v, w) ∈ s 
Arguments: {V : Type u} {G : SimpleGraph V} {G' : G.Subgraph} (s : Set (Sym2 V)) (v w : V)
Goal: (G'.deleteEdges s).Adj v w ↔ G'.Adj v w ∧ s(v, w) ∉ s