ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.Connected.exists_walk_length_eq_edist
protected theorem Connected.exists_walk_length_eq_edist  (hconn : G.Connected) (u v : V) :
    ∃ p : G.Walk u v, p.length = G.edist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} (hconn : G.Connected) (u v : V)
Goal: ∃ p, ↑p.length = G.edist u v
-- SimpleGraph.edist_le
theorem edist_le (p : G.Walk u v) :
    G.edist u v ≤ p.length 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: G.edist u v ≤ ↑p.length
-- SimpleGraph.edist_eq_zero_iff
@[simp]
theorem edist_eq_zero_iff :
    G.edist u v = 0 ↔ u = v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.edist u v = 0 ↔ u = v
-- SimpleGraph.edist_self
theorem edist_self : edist G v v = 0 
Arguments: {V : Type u_1} {G : SimpleGraph V} {v : V}
Goal: G.edist v v = 0
-- SimpleGraph.edist_pos_of_ne
theorem edist_pos_of_ne (hne : u ≠ v) :
    0 < G.edist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hne : u ≠ v)
Goal: 0 < G.edist u v
-- SimpleGraph.edist_eq_top_of_not_reachable
lemma edist_eq_top_of_not_reachable (h : ¬G.Reachable u v) :
    G.edist u v = ⊤ 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : ¬G.Reachable u v)
Goal: G.edist u v = ⊤
-- SimpleGraph.reachable_of_edist_ne_top
theorem reachable_of_edist_ne_top (h : G.edist u v ≠ ⊤) :
    G.Reachable u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : G.edist u v ≠ ⊤)
Goal: G.Reachable u v
-- SimpleGraph.exists_walk_of_edist_ne_top
lemma exists_walk_of_edist_ne_top (h : G.edist u v ≠ ⊤) :
    ∃ p : G.Walk u v, p.length = G.edist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : G.edist u v ≠ ⊤)
Goal: ∃ p, ↑p.length = G.edist u v
-- SimpleGraph.edist_triangle
protected theorem edist_triangle : G.edist u w ≤ G.edist u v + G.edist v w 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v w : V}
Goal: G.edist u w ≤ G.edist u v + G.edist v w
-- SimpleGraph.edist_comm
theorem edist_comm : G.edist u v = G.edist v u 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.edist u v = G.edist v u
-- SimpleGraph.exists_walk_of_edist_eq_coe
lemma exists_walk_of_edist_eq_coe {k : ℕ} (h : G.edist u v = k) :
    ∃ p : G.Walk u v, p.length = k 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} {k : ℕ} (h : G.edist u v = ↑k)
Goal: ∃ p, p.length = k
-- SimpleGraph.edist_ne_top_iff_reachable
lemma edist_ne_top_iff_reachable : G.edist u v ≠ ⊤ ↔ G.Reachable u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.edist u v ≠ ⊤ ↔ G.Reachable u v
-- SimpleGraph.edist_eq_one_iff_adj
@[simp]
theorem edist_eq_one_iff_adj : G.edist u v = 1 ↔ G.Adj u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.edist u v = 1 ↔ G.Adj u v
-- SimpleGraph.dist_eq_sInf
theorem dist_eq_sInf : G.dist u v = sInf (Set.range (Walk.length : G.Walk u v → ℕ)) 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.dist u v = sInf (Set.range SimpleGraph.Walk.length)
-- SimpleGraph.Reachable.exists_walk_length_eq_dist
protected theorem Reachable.exists_walk_length_eq_dist (hr : G.Reachable u v) :
    ∃ p : G.Walk u v, p.length = G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hr : G.Reachable u v)
Goal: ∃ p, p.length = G.dist u v
-- SimpleGraph.Connected.exists_walk_length_eq_dist
protected theorem Connected.exists_walk_length_eq_dist (hconn : G.Connected) (u v : V) :
    ∃ p : G.Walk u v, p.length = G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} (hconn : G.Connected) (u v : V)
Goal: ∃ p, p.length = G.dist u v
-- SimpleGraph.dist_le
theorem dist_le (p : G.Walk u v) : G.dist u v ≤ p.length 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (p : G.Walk u v)
Goal: G.dist u v ≤ p.length
-- SimpleGraph.dist_eq_zero_iff_eq_or_not_reachable
@[simp]
theorem dist_eq_zero_iff_eq_or_not_reachable :
    G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v
-- SimpleGraph.dist_self
theorem dist_self : dist G v v = 0 
Arguments: {V : Type u_1} {G : SimpleGraph V} {v : V}
Goal: G.dist v v = 0
-- SimpleGraph.Reachable.dist_eq_zero_iff
protected theorem Reachable.dist_eq_zero_iff (hr : G.Reachable u v) :
    G.dist u v = 0 ↔ u = v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hr : G.Reachable u v)
Goal: G.dist u v = 0 ↔ u = v
-- SimpleGraph.Reachable.pos_dist_of_ne
protected theorem Reachable.pos_dist_of_ne (h : G.Reachable u v) (hne : u ≠ v) :
    0 < G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : G.Reachable u v) (hne : u ≠ v)
Goal: 0 < G.dist u v
-- SimpleGraph.Connected.dist_eq_zero_iff
protected theorem Connected.dist_eq_zero_iff (hconn : G.Connected) :
    G.dist u v = 0 ↔ u = v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hconn : G.Connected)
Goal: G.dist u v = 0 ↔ u = v
-- SimpleGraph.Connected.pos_dist_of_ne
protected theorem Connected.pos_dist_of_ne (hconn : G.Connected) (hne : u ≠ v) :
    0 < G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hconn : G.Connected) (hne : u ≠ v)
Goal: 0 < G.dist u v
-- SimpleGraph.dist_eq_zero_of_not_reachable
theorem dist_eq_zero_of_not_reachable (h : ¬G.Reachable u v) : G.dist u v = 0 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : ¬G.Reachable u v)
Goal: G.dist u v = 0
-- SimpleGraph.nonempty_of_pos_dist
theorem nonempty_of_pos_dist (h : 0 < G.dist u v) :
    (Set.univ : Set (G.Walk u v)).Nonempty 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : 0 < G.dist u v)
Goal: Set.univ.Nonempty
-- SimpleGraph.Connected.dist_triangle
protected theorem Connected.dist_triangle (hconn : G.Connected) :
    G.dist u w ≤ G.dist u v + G.dist v w 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v w : V} (hconn : G.Connected)
Goal: G.dist u w ≤ G.dist u v + G.dist v w
-- SimpleGraph.dist_comm
theorem dist_comm : G.dist u v = G.dist v u 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.dist u v = G.dist v u
-- SimpleGraph.dist_ne_zero_iff_ne_and_reachable
lemma dist_ne_zero_iff_ne_and_reachable : G.dist u v ≠ 0 ↔ u ≠ v ∧ G.Reachable u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.dist u v ≠ 0 ↔ u ≠ v ∧ G.Reachable u v
-- SimpleGraph.Reachable.of_dist_ne_zero
lemma Reachable.of_dist_ne_zero (h : G.dist u v ≠ 0) : G.Reachable u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : G.dist u v ≠ 0)
Goal: G.Reachable u v
-- SimpleGraph.exists_walk_of_dist_ne_zero
lemma exists_walk_of_dist_ne_zero (h : G.dist u v ≠ 0) :
    ∃ p : G.Walk u v, p.length = G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (h : G.dist u v ≠ 0)
Goal: ∃ p, p.length = G.dist u v
-- SimpleGraph.dist_eq_one_iff_adj
@[simp]
theorem dist_eq_one_iff_adj : G.dist u v = 1 ↔ G.Adj u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V}
Goal: G.dist u v = 1 ↔ G.Adj u v
-- SimpleGraph.Walk.isPath_of_length_eq_dist
theorem Walk.isPath_of_length_eq_dist (p : G.Walk u v) (hp : p.length = G.dist u v) :
    p.IsPath 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (p : G.Walk u v) (hp : p.length = G.dist u v)
Goal: p.IsPath
-- SimpleGraph.Reachable.exists_path_of_dist
lemma Reachable.exists_path_of_dist (hr : G.Reachable u v) :
    ∃ (p : G.Walk u v), p.IsPath ∧ p.length = G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} {u v : V} (hr : G.Reachable u v)
Goal: ∃ p, p.IsPath ∧ p.length = G.dist u v
-- SimpleGraph.Connected.exists_path_of_dist
lemma Connected.exists_path_of_dist (hconn : G.Connected) (u v : V) :
    ∃ (p : G.Walk u v), p.IsPath ∧ p.length = G.dist u v 
Arguments: {V : Type u_1} {G : SimpleGraph V} (hconn : G.Connected) (u v : V)
Goal: ∃ p, p.IsPath ∧ p.length = G.dist u v
-- SimpleGraph.end_componentCompl_infinite
lemma end_componentCompl_infinite (e : G.end) (K : (Finset V)ᵒᵖ) :
    ((e : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j) K).supp.Infinite 
Arguments: {V : Type} (G : SimpleGraph V) (e : ↑G.end) (K : (Finset V)ᵒᵖ)
Goal: (SimpleGraph.ComponentCompl.supp (↑e K)).Infinite
-- SimpleGraph.nonempty_ends_of_infinite
lemma nonempty_ends_of_infinite [LocallyFinite G] [Fact G.Preconnected] [Infinite V] :
    G.end.Nonempty 
Arguments: {V : Type} (G : SimpleGraph V) [G.LocallyFinite] [Fact G.Preconnected] [Infinite V]
Goal: G.end.Nonempty
-- SimpleGraph.Finsubgraph.coe_bot
@[simp, norm_cast] lemma coe_bot : (⊥ : G.Finsubgraph) = (⊥ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: ↑⊥ = ⊥
-- SimpleGraph.Finsubgraph.coe_sup
@[simp, norm_cast]
lemma coe_sup (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⊔ G₂) = (G₁ ⊔ G₂ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} (G₁ G₂ : G.Finsubgraph)
Goal: ↑(G₁ ⊔ G₂) = ↑G₁ ⊔ ↑G₂
-- SimpleGraph.Finsubgraph.coe_inf
@[simp, norm_cast]
lemma coe_inf (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⊓ G₂) = (G₁ ⊓ G₂ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} (G₁ G₂ : G.Finsubgraph)
Goal: ↑(G₁ ⊓ G₂) = ↑G₁ ⊓ ↑G₂
-- SimpleGraph.Finsubgraph.coe_sdiff
@[simp, norm_cast]
lemma coe_sdiff (G₁ G₂ : G.Finsubgraph) : ↑(G₁ \ G₂) = (G₁ \ G₂ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} (G₁ G₂ : G.Finsubgraph)
Goal: ↑(G₁ \ G₂) = ↑G₁ \ ↑G₂
-- SimpleGraph.Finsubgraph.coe_top
@[simp, norm_cast] lemma coe_top : (⊤ : G.Finsubgraph) = (⊤ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V]
Goal: ↑⊤ = ⊤
-- SimpleGraph.Finsubgraph.coe_compl
@[simp, norm_cast] lemma coe_compl (G' : G.Finsubgraph) : ↑(G'ᶜ) = (G'ᶜ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] (G' : G.Finsubgraph)
Goal: ↑G'ᶜ = (↑G')ᶜ
-- SimpleGraph.Finsubgraph.coe_hnot
@[simp, norm_cast] lemma coe_hnot (G' : G.Finsubgraph) : ↑(￢G') = (￢G' : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] (G' : G.Finsubgraph)
Goal: ↑(￢G') = ￢↑G'
-- SimpleGraph.Finsubgraph.coe_himp
@[simp, norm_cast]
lemma coe_himp (G₁ G₂ : G.Finsubgraph) : ↑(G₁ ⇨ G₂) = (G₁ ⇨ G₂ : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] (G₁ G₂ : G.Finsubgraph)
Goal: ↑(G₁ ⇨ G₂) = ↑G₁ ⇨ ↑G₂
-- SimpleGraph.Finsubgraph.coe_sSup
@[simp, norm_cast]
lemma coe_sSup (s : Set G.Finsubgraph) : sSup s = (⨆ G ∈ s, G : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] (s : Set G.Finsubgraph)
Goal: ↑(sSup s) = ⨆ G_1 ∈ s, ↑G_1
-- SimpleGraph.Finsubgraph.coe_sInf
@[simp, norm_cast]
lemma coe_sInf (s : Set G.Finsubgraph) : sInf s = (⨅ G ∈ s, G : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] (s : Set G.Finsubgraph)
Goal: ↑(sInf s) = ⨅ G_1 ∈ s, ↑G_1
-- SimpleGraph.Finsubgraph.coe_iSup
@[simp, norm_cast]
lemma coe_iSup {ι : Sort*} (f : ι → G.Finsubgraph) : ⨆ i, f i = (⨆ i, f i : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] {ι : Sort u_1} (f : ι → G.Finsubgraph)
Goal: ↑(⨆ i, f i) = ⨆ i, ↑(f i)
-- SimpleGraph.Finsubgraph.coe_iInf
@[simp, norm_cast]
lemma coe_iInf {ι : Sort*} (f : ι → G.Finsubgraph) : ⨅ i, f i = (⨅ i, f i : G.Subgraph) 
Arguments: {V : Type u} {G : SimpleGraph V} [Finite V] {ι : Sort u_1} (f : ι → G.Finsubgraph)
Goal: ↑(⨅ i, f i) = ⨅ i, ↑(f i)
-- SimpleGraph.singletonFinsubgraph_le_adj_left
theorem singletonFinsubgraph_le_adj_left {u v : V} {e : G.Adj u v} :
    singletonFinsubgraph u ≤ finsubgraphOfAdj e 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} {e : G.Adj u v}
Goal: SimpleGraph.singletonFinsubgraph u ≤ SimpleGraph.finsubgraphOfAdj e
-- SimpleGraph.singletonFinsubgraph_le_adj_right
theorem singletonFinsubgraph_le_adj_right {u v : V} {e : G.Adj u v} :
    singletonFinsubgraph v ≤ finsubgraphOfAdj e 
Arguments: {V : Type u} {G : SimpleGraph V} {u v : V} {e : G.Adj u v}
Goal: SimpleGraph.singletonFinsubgraph v ≤ SimpleGraph.finsubgraphOfAdj e
-- SimpleGraph.nonempty_hom_of_forall_finite_subgraph_hom
theorem nonempty_hom_of_forall_finite_subgraph_hom [Finite W]
    (h : ∀ G' : G.Subgraph, G'.verts.Finite → G'.coe →g F) : Nonempty (G →g F) 
Arguments: {V : Type u} {W : Type v} {G : SimpleGraph V} {F : SimpleGraph W} [Finite W] (h : (G' : G.Subgraph) → G'.verts.Finite → G'.coe →g F)
Goal: Nonempty (G →g F)
-- SimpleGraph.Coloring.valid
theorem Coloring.valid {v w : V} (h : G.Adj v w) : C v ≠ C w 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) {v w : V} (h : G.Adj v w)
Goal: C v ≠ C w
-- SimpleGraph.Coloring.mem_colorClass
theorem Coloring.mem_colorClass (v : V) : v ∈ C.colorClass (C v) 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) (v : V)
Goal: v ∈ C.colorClass (C v)
-- SimpleGraph.Coloring.colorClasses_isPartition
theorem Coloring.colorClasses_isPartition : Setoid.IsPartition C.colorClasses 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α)
Goal: Setoid.IsPartition C.colorClasses
-- SimpleGraph.Coloring.mem_colorClasses
theorem Coloring.mem_colorClasses {v : V} : C.colorClass (C v) ∈ C.colorClasses 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) {v : V}
Goal: C.colorClass (C v) ∈ C.colorClasses
-- SimpleGraph.Coloring.colorClasses_finite
theorem Coloring.colorClasses_finite [Finite α] : C.colorClasses.Finite 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) [Finite α]
Goal: C.colorClasses.Finite
-- SimpleGraph.Coloring.card_colorClasses_le
theorem Coloring.card_colorClasses_le [Fintype α] [Fintype C.colorClasses] :
    Fintype.card C.colorClasses ≤ Fintype.card α 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) [Fintype α] [Fintype ↑C.colorClasses]
Goal: Fintype.card ↑C.colorClasses ≤ Fintype.card α
-- SimpleGraph.Coloring.not_adj_of_mem_colorClass
theorem Coloring.not_adj_of_mem_colorClass {c : α} {v w : V} (hv : v ∈ C.colorClass c)
    (hw : w ∈ C.colorClass c) : ¬G.Adj v w 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) {c : α} {v w : V} (hv : v ∈ C.colorClass c) (hw : w ∈ C.colorClass c)
Goal: ¬G.Adj v w
-- SimpleGraph.Coloring.color_classes_independent
theorem Coloring.color_classes_independent (c : α) : IsAntichain G.Adj (C.colorClass c) 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} (C : G.Coloring α) (c : α)
Goal: IsAntichain G.Adj (C.colorClass c)
-- SimpleGraph.colorable_of_isEmpty
theorem colorable_of_isEmpty [IsEmpty V] (n : ℕ) : G.Colorable n 
Arguments: {V : Type u} (G : SimpleGraph V) [IsEmpty V] (n : ℕ)
Goal: G.Colorable n
-- SimpleGraph.isEmpty_of_colorable_zero
theorem isEmpty_of_colorable_zero (h : G.Colorable 0) : IsEmpty V 
Arguments: {V : Type u} (G : SimpleGraph V) (h : G.Colorable 0)
Goal: IsEmpty V
-- SimpleGraph.chromaticNumber_eq_biInf
lemma chromaticNumber_eq_biInf {G : SimpleGraph V} :
    G.chromaticNumber = ⨅ n ∈ setOf G.Colorable, (n : ℕ∞) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: G.chromaticNumber = ⨅ n ∈ setOf G.Colorable, ↑n
-- SimpleGraph.chromaticNumber_eq_iInf
lemma chromaticNumber_eq_iInf {G : SimpleGraph V} :
    G.chromaticNumber = ⨅ n : {m | G.Colorable m}, (n : ℕ∞) 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: G.chromaticNumber = ⨅ n, ↑↑n
-- SimpleGraph.Colorable.chromaticNumber_eq_sInf
lemma Colorable.chromaticNumber_eq_sInf {G : SimpleGraph V} {n} (h : G.Colorable n) :
    G.chromaticNumber = sInf {n' : ℕ | G.Colorable n'} 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ} (h : G.Colorable n)
Goal: G.chromaticNumber = ↑(sInf {n' | G.Colorable n'})
-- SimpleGraph.coe_recolorOfEmbedding
@[simp] lemma coe_recolorOfEmbedding (f : α ↪ β) :
    ⇑(G.recolorOfEmbedding f) = (Embedding.completeGraph f).toHom.comp 
Arguments: {V : Type u} (G : SimpleGraph V) {α : Type u_1} {β : Type u_2} (f : α ↪ β)
Goal: ⇑(G.recolorOfEmbedding f) = (SimpleGraph.Embedding.completeGraph f).toHom.comp
-- SimpleGraph.coe_recolorOfEquiv
@[simp] lemma coe_recolorOfEquiv (f : α ≃ β) :
    ⇑(G.recolorOfEquiv f) = (Embedding.completeGraph f).toHom.comp 
Arguments: {V : Type u} (G : SimpleGraph V) {α : Type u_1} {β : Type u_2} (f : α ≃ β)
Goal: ⇑(G.recolorOfEquiv f) = (SimpleGraph.Embedding.completeGraph f.toEmbedding).toHom.comp
-- SimpleGraph.coe_recolorOfCardLE
@[simp] lemma coe_recolorOfCardLE [Fintype α] [Fintype β] (hαβ : card α ≤ card β) :
    ⇑(G.recolorOfCardLE hαβ) =
      (Embedding.completeGraph (Embedding.nonempty_of_card_le hαβ).some).toHom.comp 
Arguments: {V : Type u} (G : SimpleGraph V) {α : Type u_1} {β : Type u_2} [Fintype α] [Fintype β] (hαβ : Fintype.card α ≤ Fintype.card β)
Goal: ⇑(G.recolorOfCardLE hαβ) = (SimpleGraph.Embedding.completeGraph ⋯.some).toHom.comp
-- SimpleGraph.Colorable.mono
theorem Colorable.mono {n m : ℕ} (h : n ≤ m) (hc : G.Colorable n) : G.Colorable m 
Arguments: {V : Type u} {G : SimpleGraph V} {n m : ℕ} (h : n ≤ m) (hc : G.Colorable n)
Goal: G.Colorable m
-- SimpleGraph.Coloring.colorable
theorem Coloring.colorable [Fintype α] (C : G.Coloring α) : G.Colorable (Fintype.card α) 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} [Fintype α] (C : G.Coloring α)
Goal: G.Colorable (Fintype.card α)
-- SimpleGraph.colorable_of_fintype
theorem colorable_of_fintype (G : SimpleGraph V) [Fintype V] : G.Colorable (Fintype.card V) 
Arguments: {V : Type u} (G : SimpleGraph V) [Fintype V]
Goal: G.Colorable (Fintype.card V)
-- SimpleGraph.Colorable.of_embedding
theorem Colorable.of_embedding {V' : Type*} {G' : SimpleGraph V'} (f : G ↪g G') {n : ℕ}
    (h : G'.Colorable n) : G.Colorable n 
Arguments: {V : Type u} {G : SimpleGraph V} {V' : Type u_3} {G' : SimpleGraph V'} (f : G ↪g G') {n : ℕ} (h : G'.Colorable n)
Goal: G.Colorable n
-- SimpleGraph.colorable_iff_exists_bdd_nat_coloring
theorem colorable_iff_exists_bdd_nat_coloring (n : ℕ) :
    G.Colorable n ↔ ∃ C : G.Coloring ℕ, ∀ v, C v < n 
Arguments: {V : Type u} {G : SimpleGraph V} (n : ℕ)
Goal: G.Colorable n ↔ ∃ C, ∀ (v : V), C v < n
-- SimpleGraph.colorable_set_nonempty_of_colorable
theorem colorable_set_nonempty_of_colorable {n : ℕ} (hc : G.Colorable n) :
    { n : ℕ | G.Colorable n }.Nonempty 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ} (hc : G.Colorable n)
Goal: {n | G.Colorable n}.Nonempty
-- SimpleGraph.chromaticNumber_bddBelow
theorem chromaticNumber_bddBelow : BddBelow { n : ℕ | G.Colorable n } 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: BddBelow {n | G.Colorable n}
-- SimpleGraph.Colorable.chromaticNumber_le
theorem Colorable.chromaticNumber_le {n : ℕ} (hc : G.Colorable n) : G.chromaticNumber ≤ n 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ} (hc : G.Colorable n)
Goal: G.chromaticNumber ≤ ↑n
-- SimpleGraph.chromaticNumber_ne_top_iff_exists
theorem chromaticNumber_ne_top_iff_exists : G.chromaticNumber ≠ ⊤ ↔ ∃ n, G.Colorable n 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: G.chromaticNumber ≠ ⊤ ↔ ∃ n, G.Colorable n
-- SimpleGraph.chromaticNumber_le_iff_colorable
theorem chromaticNumber_le_iff_colorable {n : ℕ} : G.chromaticNumber ≤ n ↔ G.Colorable n 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ}
Goal: G.chromaticNumber ≤ ↑n ↔ G.Colorable n
-- SimpleGraph.chromaticNumber_le_card
@[deprecated Colorable.chromaticNumber_le (since := "2024-03-21")]
theorem chromaticNumber_le_card [Fintype α] (C : G.Coloring α) :
    G.chromaticNumber ≤ Fintype.card α 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} [Fintype α] (C : G.Coloring α)
Goal: G.chromaticNumber ≤ ↑(Fintype.card α)
-- SimpleGraph.colorable_chromaticNumber
theorem colorable_chromaticNumber {m : ℕ} (hc : G.Colorable m) :
    G.Colorable (ENat.toNat G.chromaticNumber) 
Arguments: {V : Type u} {G : SimpleGraph V} {m : ℕ} (hc : G.Colorable m)
Goal: G.Colorable G.chromaticNumber.toNat
-- SimpleGraph.colorable_chromaticNumber_of_fintype
theorem colorable_chromaticNumber_of_fintype (G : SimpleGraph V) [Finite V] :
    G.Colorable (ENat.toNat G.chromaticNumber) 
Arguments: {V : Type u} (G : SimpleGraph V) [Finite V]
Goal: G.Colorable G.chromaticNumber.toNat
-- SimpleGraph.chromaticNumber_le_one_of_subsingleton
theorem chromaticNumber_le_one_of_subsingleton (G : SimpleGraph V) [Subsingleton V] :
    G.chromaticNumber ≤ 1 
Arguments: {V : Type u} (G : SimpleGraph V) [Subsingleton V]
Goal: G.chromaticNumber ≤ 1
-- SimpleGraph.chromaticNumber_eq_zero_of_isempty
theorem chromaticNumber_eq_zero_of_isempty (G : SimpleGraph V) [IsEmpty V] :
    G.chromaticNumber = 0 
Arguments: {V : Type u} (G : SimpleGraph V) [IsEmpty V]
Goal: G.chromaticNumber = 0
-- SimpleGraph.isEmpty_of_chromaticNumber_eq_zero
theorem isEmpty_of_chromaticNumber_eq_zero (G : SimpleGraph V) [Finite V]
    (h : G.chromaticNumber = 0) : IsEmpty V 
Arguments: {V : Type u} (G : SimpleGraph V) [Finite V] (h : G.chromaticNumber = 0)
Goal: IsEmpty V
-- SimpleGraph.chromaticNumber_pos
theorem chromaticNumber_pos [Nonempty V] {n : ℕ} (hc : G.Colorable n) : 0 < G.chromaticNumber 
Arguments: {V : Type u} {G : SimpleGraph V} [Nonempty V] {n : ℕ} (hc : G.Colorable n)
Goal: 0 < G.chromaticNumber
-- SimpleGraph.colorable_of_chromaticNumber_ne_top
theorem colorable_of_chromaticNumber_ne_top (h : G.chromaticNumber ≠ ⊤) :
    G.Colorable (ENat.toNat G.chromaticNumber) 
Arguments: {V : Type u} {G : SimpleGraph V} (h : G.chromaticNumber ≠ ⊤)
Goal: G.Colorable G.chromaticNumber.toNat
-- SimpleGraph.Colorable.mono_left
theorem Colorable.mono_left {G' : SimpleGraph V} (h : G ≤ G') {n : ℕ} (hc : G'.Colorable n) :
    G.Colorable n 
Arguments: {V : Type u} {G G' : SimpleGraph V} (h : G ≤ G') {n : ℕ} (hc : G'.Colorable n)
Goal: G.Colorable n
-- SimpleGraph.chromaticNumber_le_of_forall_imp
theorem chromaticNumber_le_of_forall_imp {V' : Type*} {G' : SimpleGraph V'}
    (h : ∀ n, G'.Colorable n → G.Colorable n) :
    G.chromaticNumber ≤ G'.chromaticNumber 
Arguments: {V : Type u} {G : SimpleGraph V} {V' : Type u_3} {G' : SimpleGraph V'} (h : ∀ (n : ℕ), G'.Colorable n → G.Colorable n)
Goal: G.chromaticNumber ≤ G'.chromaticNumber
-- SimpleGraph.chromaticNumber_mono
theorem chromaticNumber_mono (G' : SimpleGraph V)
    (h : G ≤ G') : G.chromaticNumber ≤ G'.chromaticNumber 
Arguments: {V : Type u} {G : SimpleGraph V} (G' : SimpleGraph V) (h : G ≤ G')
Goal: G.chromaticNumber ≤ G'.chromaticNumber
-- SimpleGraph.chromaticNumber_mono_of_embedding
theorem chromaticNumber_mono_of_embedding {V' : Type*} {G' : SimpleGraph V'}
    (f : G ↪g G') : G.chromaticNumber ≤ G'.chromaticNumber 
Arguments: {V : Type u} {G : SimpleGraph V} {V' : Type u_3} {G' : SimpleGraph V'} (f : G ↪g G')
Goal: G.chromaticNumber ≤ G'.chromaticNumber
-- SimpleGraph.card_le_chromaticNumber_iff_forall_surjective
lemma card_le_chromaticNumber_iff_forall_surjective [Fintype α] :
    card α ≤ G.chromaticNumber ↔ ∀ C : G.Coloring α, Surjective C 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} [Fintype α]
Goal: ↑(Fintype.card α) ≤ G.chromaticNumber ↔ ∀ (C : G.Coloring α), Function.Surjective ⇑C
-- SimpleGraph.le_chromaticNumber_iff_forall_surjective
lemma le_chromaticNumber_iff_forall_surjective :
    n ≤ G.chromaticNumber ↔ ∀ C : G.Coloring (Fin n), Surjective C 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ}
Goal: ↑n ≤ G.chromaticNumber ↔ ∀ (C : G.Coloring (Fin n)), Function.Surjective ⇑C
-- SimpleGraph.chromaticNumber_eq_card_iff_forall_surjective
lemma chromaticNumber_eq_card_iff_forall_surjective [Fintype α] (hG : G.Colorable (card α)) :
    G.chromaticNumber = card α ↔ ∀ C : G.Coloring α, Surjective C 
Arguments: {V : Type u} {G : SimpleGraph V} {α : Type u_1} [Fintype α] (hG : G.Colorable (Fintype.card α))
Goal: G.chromaticNumber = ↑(Fintype.card α) ↔ ∀ (C : G.Coloring α), Function.Surjective ⇑C
-- SimpleGraph.chromaticNumber_eq_iff_forall_surjective
lemma chromaticNumber_eq_iff_forall_surjective (hG : G.Colorable n) :
    G.chromaticNumber = n ↔ ∀ C : G.Coloring (Fin n), Surjective C 
Arguments: {V : Type u} {G : SimpleGraph V} {n : ℕ} (hG : G.Colorable n)
Goal: G.chromaticNumber = ↑n ↔ ∀ (C : G.Coloring (Fin n)), Function.Surjective ⇑C
-- SimpleGraph.chromaticNumber_bot
theorem chromaticNumber_bot [Nonempty V] : (⊥ : SimpleGraph V).chromaticNumber = 1 
Arguments: {V : Type u} [Nonempty V]
Goal: ⊥.chromaticNumber = 1
-- SimpleGraph.chromaticNumber_top
@[simp]
theorem chromaticNumber_top [Fintype V] : (⊤ : SimpleGraph V).chromaticNumber = Fintype.card V 
Arguments: {V : Type u} [Fintype V]
Goal: ⊤.chromaticNumber = ↑(Fintype.card V)
-- SimpleGraph.chromaticNumber_top_eq_top_of_infinite
theorem chromaticNumber_top_eq_top_of_infinite (V : Type*) [Infinite V] :
    (⊤ : SimpleGraph V).chromaticNumber = ⊤ 
Arguments: (V : Type u_3) [Infinite V]
Goal: ⊤.chromaticNumber = ⊤