ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.incidence_other_neighbor_edge
theorem incidence_other_neighbor_edge {v w : V} (h : w ∈ G.neighborSet v) :
    G.otherVertexOfIncident (G.mem_incidence_iff_neighbor.mpr h) = w 
Arguments: {V : Type u} (G : SimpleGraph V) [DecidableEq V] {v w : V} (h : w ∈ G.neighborSet v)
Goal: G.otherVertexOfIncident ⋯ = w
-- SimpleGraph.incidenceSetEquivNeighborSet_symm_apply_coe
@[simps]
def incidenceSetEquivNeighborSet (v : V) : G.incidenceSet v ≃ G.neighborSet v 
Arguments: {V : Type u} (G : SimpleGraph V) [DecidableEq V] (v : V) (w : ↑(G.neighborSet v))
Goal: ↑((G.incidenceSetEquivNeighborSet v).symm w) = s(v, ↑w)
-- SimpleGraph.incidenceSetEquivNeighborSet_apply_coe
@[simps]
def incidenceSetEquivNeighborSet (v : V) : G.incidenceSet v ≃ G.neighborSet v 
Arguments: {V : Type u} (G : SimpleGraph V) [DecidableEq V] (v : V) (e : ↑(G.incidenceSet v))
Goal: ↑((G.incidenceSetEquivNeighborSet v) e) = G.otherVertexOfIncident ⋯
-- SimpleGraph.deleteEdges_adj
@[simp] lemma deleteEdges_adj : (G.deleteEdges s).Adj v w ↔ G.Adj v w ∧ ¬s(v, w) ∈ s 
Arguments: {V : Type u} {G : SimpleGraph V} {v w : V} {s : Set (Sym2 V)}
Goal: (G.deleteEdges s).Adj v w ↔ G.Adj v w ∧ s(v, w) ∉ s
-- SimpleGraph.deleteEdges_edgeSet
@[simp] lemma deleteEdges_edgeSet (G G' : SimpleGraph V) : G.deleteEdges G'.edgeSet = G \ G' 
Arguments: {V : Type u} (G G' : SimpleGraph V)
Goal: G.deleteEdges G'.edgeSet = G \ G'
-- SimpleGraph.deleteEdges_deleteEdges
@[simp]
theorem deleteEdges_deleteEdges (s s' : Set (Sym2 V)) :
    (G.deleteEdges s).deleteEdges s' = G.deleteEdges (s ∪ s') 
Arguments: {V : Type u} {G : SimpleGraph V} (s s' : Set (Sym2 V))
Goal: (G.deleteEdges s).deleteEdges s' = G.deleteEdges (s ∪ s')
-- SimpleGraph.deleteEdges_empty
@[simp] lemma deleteEdges_empty : G.deleteEdges ∅ = G 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: G.deleteEdges ∅ = G
-- SimpleGraph.deleteEdges_univ
@[simp] lemma deleteEdges_univ : G.deleteEdges Set.univ = ⊥ 
Arguments: {V : Type u} {G : SimpleGraph V}
Goal: G.deleteEdges Set.univ = ⊥
-- SimpleGraph.deleteEdges_le
lemma deleteEdges_le (s : Set (Sym2 V)) : G.deleteEdges s ≤ G 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set (Sym2 V))
Goal: G.deleteEdges s ≤ G
-- SimpleGraph.deleteEdges_anti
lemma deleteEdges_anti (h : s₁ ⊆ s₂) : G.deleteEdges s₂ ≤ G.deleteEdges s₁ 
Arguments: {V : Type u} {G : SimpleGraph V} {s₁ s₂ : Set (Sym2 V)} (h : s₁ ⊆ s₂)
Goal: G.deleteEdges s₂ ≤ G.deleteEdges s₁
-- SimpleGraph.deleteEdges_mono
lemma deleteEdges_mono (h : G ≤ H) : G.deleteEdges s ≤ H.deleteEdges s 
Arguments: {V : Type u} {G H : SimpleGraph V} {s : Set (Sym2 V)} (h : G ≤ H)
Goal: G.deleteEdges s ≤ H.deleteEdges s
-- SimpleGraph.deleteEdges_eq_self
@[simp] lemma deleteEdges_eq_self : G.deleteEdges s = G ↔ Disjoint G.edgeSet s 
Arguments: {V : Type u} {G : SimpleGraph V} {s : Set (Sym2 V)}
Goal: G.deleteEdges s = G ↔ Disjoint G.edgeSet s
-- SimpleGraph.deleteEdges_eq_inter_edgeSet
theorem deleteEdges_eq_inter_edgeSet (s : Set (Sym2 V)) :
    G.deleteEdges s = G.deleteEdges (s ∩ G.edgeSet) 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set (Sym2 V))
Goal: G.deleteEdges s = G.deleteEdges (s ∩ G.edgeSet)
-- SimpleGraph.deleteEdges_sdiff_eq_of_le
theorem deleteEdges_sdiff_eq_of_le {H : SimpleGraph V} (h : H ≤ G) :
    G.deleteEdges (G.edgeSet \ H.edgeSet) = H 
Arguments: {V : Type u} {G H : SimpleGraph V} (h : H ≤ G)
Goal: G.deleteEdges (G.edgeSet \ H.edgeSet) = H
-- SimpleGraph.edgeSet_deleteEdges
theorem edgeSet_deleteEdges (s : Set (Sym2 V)) : (G.deleteEdges s).edgeSet = G.edgeSet \ s 
Arguments: {V : Type u} {G : SimpleGraph V} (s : Set (Sym2 V))
Goal: (G.deleteEdges s).edgeSet = G.edgeSet \ s
-- SimpleGraph.Subgraph.IsMatching.toEdge_eq_of_adj
theorem IsMatching.toEdge_eq_of_adj (h : M.IsMatching) (hv : v ∈ M.verts) (hvw : M.Adj v w) :
    h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} {v w : V} (h : M.IsMatching) (hv : v ∈ M.verts) (hvw : M.Adj v w)
Goal: h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩
-- SimpleGraph.Subgraph.IsMatching.toEdge.surjective
theorem IsMatching.toEdge.surjective (h : M.IsMatching) : Surjective h.toEdge 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} (h : M.IsMatching)
Goal: Function.Surjective h.toEdge
-- SimpleGraph.Subgraph.IsMatching.toEdge_eq_toEdge_of_adj
theorem IsMatching.toEdge_eq_toEdge_of_adj (h : M.IsMatching)
    (hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.Adj v w) :
    h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩ 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} {v w : V} (h : M.IsMatching) (hv : v ∈ M.verts) (hw : w ∈ M.verts) (ha : M.Adj v w)
Goal: h.toEdge ⟨v, hv⟩ = h.toEdge ⟨w, hw⟩
-- SimpleGraph.Subgraph.IsMatching.map_ofLE
lemma IsMatching.map_ofLE (h : M.IsMatching) (hGG' : G ≤ G') :
    (M.map (Hom.ofLE hGG')).IsMatching 
Arguments: {V : Type u_1} {G G' : SimpleGraph V} {M : G.Subgraph} (h : M.IsMatching) (hGG' : G ≤ G')
Goal: (SimpleGraph.Subgraph.map (SimpleGraph.Hom.ofLE hGG') M).IsMatching
-- SimpleGraph.Subgraph.IsMatching.sup
lemma IsMatching.sup (hM : M.IsMatching) (hM' : M'.IsMatching)
    (hd : Disjoint M.support M'.support) : (M ⊔ M').IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M M' : G.Subgraph} (hM : M.IsMatching) (hM' : M'.IsMatching) (hd : Disjoint M.support M'.support)
Goal: (M ⊔ M').IsMatching
-- SimpleGraph.Subgraph.IsMatching.iSup
lemma IsMatching.iSup {ι : Sort _} {f : ι → Subgraph G} (hM : (i : ι) → (f i).IsMatching)
    (hd : Pairwise fun i j ↦ Disjoint (f i).support (f j).support) :
    (⨆ i, f i).IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {ι : Type u_2} {f : ι → G.Subgraph} (hM : ∀ (i : ι), (f i).IsMatching) (hd : Pairwise fun i j => Disjoint (f i).support (f j).support)
Goal: (⨆ i, f i).IsMatching
-- SimpleGraph.Subgraph.IsMatching.subgraphOfAdj
lemma IsMatching.subgraphOfAdj (h : G.Adj v w) : (G.subgraphOfAdj h).IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {v w : V} (h : G.Adj v w)
Goal: (G.subgraphOfAdj h).IsMatching
-- SimpleGraph.Subgraph.IsMatching.coeSubgraph
lemma IsMatching.coeSubgraph {G' : Subgraph G} {M : Subgraph G'.coe} (hM : M.IsMatching) :
    M.coeSubgraph.IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {G' : G.Subgraph} {M : G'.coe.Subgraph} (hM : M.IsMatching)
Goal: (SimpleGraph.Subgraph.coeSubgraph M).IsMatching
-- SimpleGraph.Subgraph.IsMatching.support_eq_verts
theorem IsMatching.support_eq_verts (h : M.IsMatching) : M.support = M.verts 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} (h : M.IsMatching)
Goal: M.support = M.verts
-- SimpleGraph.Subgraph.isMatching_iff_forall_degree
theorem isMatching_iff_forall_degree [∀ v, Fintype (M.neighborSet v)] :
    M.IsMatching ↔ ∀ v : V, v ∈ M.verts → M.degree v = 1 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [(v : V) → Fintype ↑(M.neighborSet v)]
Goal: M.IsMatching ↔ ∀ v ∈ M.verts, M.degree v = 1
-- SimpleGraph.Subgraph.IsMatching.even_card
theorem IsMatching.even_card [Fintype M.verts] (h : M.IsMatching) : Even M.verts.toFinset.card 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [Fintype ↑M.verts] (h : M.IsMatching)
Goal: Even M.verts.toFinset.card
-- SimpleGraph.Subgraph.isPerfectMatching_iff
theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph}
Goal: M.IsPerfectMatching ↔ ∀ (v : V), ∃! w, M.Adj v w
-- SimpleGraph.Subgraph.isPerfectMatching_iff_forall_degree
theorem isPerfectMatching_iff_forall_degree [∀ v, Fintype (M.neighborSet v)] :
    M.IsPerfectMatching ↔ ∀ v, M.degree v = 1 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [(v : V) → Fintype ↑(M.neighborSet v)]
Goal: M.IsPerfectMatching ↔ ∀ (v : V), M.degree v = 1
-- SimpleGraph.Subgraph.IsPerfectMatching.even_card
theorem IsPerfectMatching.even_card [Fintype V] (h : M.IsPerfectMatching) :
    Even (Fintype.card V) 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [Fintype V] (h : M.IsPerfectMatching)
Goal: Even (Fintype.card V)
-- SimpleGraph.Subgraph.IsMatching.induce_connectedComponent
lemma IsMatching.induce_connectedComponent (h : M.IsMatching) (c : ConnectedComponent G) :
    (M.induce (M.verts ∩ c.supp)).IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} (h : M.IsMatching) (c : G.ConnectedComponent)
Goal: (M.induce (M.verts ∩ c.supp)).IsMatching
-- SimpleGraph.Subgraph.IsPerfectMatching.induce_connectedComponent_isMatching
lemma IsPerfectMatching.induce_connectedComponent_isMatching (h : M.IsPerfectMatching)
    (c : ConnectedComponent G) : (M.induce c.supp).IsMatching 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} (h : M.IsPerfectMatching) (c : G.ConnectedComponent)
Goal: (M.induce c.supp).IsMatching
-- SimpleGraph.ConnectedComponent.even_card_of_isPerfectMatching
lemma even_card_of_isPerfectMatching [DecidableEq V]
    (c : ConnectedComponent G) (hM : M.IsPerfectMatching) :
    Even (Fintype.card c.supp) 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [Fintype V] [DecidableRel G.Adj] [DecidableEq V] (c : G.ConnectedComponent) (hM : M.IsPerfectMatching)
Goal: Even (Fintype.card ↑c.supp)
-- SimpleGraph.ConnectedComponent.odd_matches_node_outside
lemma odd_matches_node_outside {u : Set V} {c : ConnectedComponent (Subgraph.deleteVerts ⊤ u).coe}
    (hM : M.IsPerfectMatching) (codd : Odd (Nat.card c.supp)) :
    ∃ᵉ (w ∈ u) (v : ((⊤ : G.Subgraph).deleteVerts u).verts), M.Adj v w ∧ v ∈ c.supp 
Arguments: {V : Type u_1} {G : SimpleGraph V} {M : G.Subgraph} [Fintype V] {u : Set V} {c : (⊤.deleteVerts u).coe.ConnectedComponent} (hM : M.IsPerfectMatching) (codd : Odd (Nat.card ↑c.supp))
Goal: ∃ w ∈ u, ∃ v, M.Adj (↑v) w ∧ v ∈ c.supp
-- CategoryTheory.instCategorySimplicialObject_comp_app
@[simps!]
instance : Category (SimplicialObject C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y Z : SimplexCategoryᵒᵖ ⥤ C} (α : X ⟶ Y) (β : Y ⟶ Z) (X_1 : SimplexCategoryᵒᵖ),
  (α ≫ β).app X_1 = α.app X_1 ≫ β.app X_1
-- CategoryTheory.instCategorySimplicialObject_id_app
@[simps!]
instance : Category (SimplicialObject C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : SimplexCategoryᵒᵖ ⥤ C) (X : SimplexCategoryᵒᵖ)
Goal: (𝟙 F).app X = 𝟙 (F.obj X)
-- CategoryTheory.SimplicialObject.hom_ext
@[ext]
lemma hom_ext {X Y : SimplicialObject C} (f g : X ⟶ Y)
    (h : ∀ (n : SimplexCategoryᵒᵖ), f.app n = g.app n) : f = g 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : CategoryTheory.SimplicialObject C} (f g : X ⟶ Y) (h : ∀ (n : SimplexCategoryᵒᵖ), f.app n = g.app n)
Goal: f = g
-- CategoryTheory.SimplicialObject.eqToIso_refl
@[simp]
theorem eqToIso_refl {n : ℕ} (h : n = n) : X.eqToIso h = Iso.refl _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} (h : n = n)
Goal: X.eqToIso h = CategoryTheory.Iso.refl (X.obj (Opposite.op (SimplexCategory.mk n)))
-- CategoryTheory.SimplicialObject.δ_comp_δ
@[reassoc]
theorem δ_comp_δ {n} {i j : Fin (n + 2)} (H : i ≤ j) :
    X.δ j.succ ≫ X.δ i = X.δ (Fin.castSucc i) ≫ X.δ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i j : Fin (n + 2)} (H : i ≤ j)
Goal: X.δ j.succ ≫ X.δ i = X.δ i.castSucc ≫ X.δ j
-- CategoryTheory.SimplicialObject.δ_comp_δ'
@[reassoc]
theorem δ_comp_δ' {n} {i : Fin (n + 2)} {j : Fin (n + 3)} (H : Fin.castSucc i < j) :
    X.δ j ≫ X.δ i =
      X.δ (Fin.castSucc i) ≫
        X.δ (j.pred fun (hj : j = 0) => by simp [hj, Fin.not_lt_zero] at H) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 3)} (H : i.castSucc < j)
Goal: X.δ j ≫ X.δ i = X.δ i.castSucc ≫ X.δ (j.pred ⋯)
-- CategoryTheory.SimplicialObject.δ_comp_δ''
@[reassoc]
theorem δ_comp_δ'' {n} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : i ≤ Fin.castSucc j) :
    X.δ j.succ ≫ X.δ (i.castLT (Nat.lt_of_le_of_lt (Fin.le_iff_val_le_val.mp H) j.is_lt)) =
      X.δ i ≫ X.δ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : i ≤ j.castSucc)
Goal: X.δ j.succ ≫ X.δ (i.castLT ⋯) = X.δ i ≫ X.δ j
-- CategoryTheory.SimplicialObject.δ_comp_δ_self
@[reassoc]
theorem δ_comp_δ_self {n} {i : Fin (n + 2)} :
    X.δ (Fin.castSucc i) ≫ X.δ i = X.δ i.succ ≫ X.δ i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 2)}
Goal: X.δ i.castSucc ≫ X.δ i = X.δ i.succ ≫ X.δ i
-- CategoryTheory.SimplicialObject.δ_comp_δ_self'
@[reassoc]
theorem δ_comp_δ_self' {n} {j : Fin (n + 3)} {i : Fin (n + 2)} (H : j = Fin.castSucc i) :
    X.δ j ≫ X.δ i = X.δ i.succ ≫ X.δ i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {j : Fin (n + 3)} {i : Fin (n + 2)} (H : j = i.castSucc)
Goal: X.δ j ≫ X.δ i = X.δ i.succ ≫ X.δ i
-- CategoryTheory.SimplicialObject.δ_comp_σ_of_le
@[reassoc]
theorem δ_comp_σ_of_le {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ Fin.castSucc j) :
    X.σ j.succ ≫ X.δ (Fin.castSucc i) = X.δ i ≫ X.σ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : i ≤ j.castSucc)
Goal: X.σ j.succ ≫ X.δ i.castSucc = X.δ i ≫ X.σ j
-- CategoryTheory.SimplicialObject.δ_comp_σ_self
@[reassoc]
theorem δ_comp_σ_self {n} {i : Fin (n + 1)} : X.σ i ≫ X.δ (Fin.castSucc i) = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 1)}
Goal: X.σ i ≫ X.δ i.castSucc = 𝟙 (X.obj (Opposite.op (SimplexCategory.mk n)))
-- CategoryTheory.SimplicialObject.δ_comp_σ_self'
@[reassoc]
theorem δ_comp_σ_self' {n} {j : Fin (n + 2)} {i : Fin (n + 1)} (H : j = Fin.castSucc i) :
    X.σ i ≫ X.δ j = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {j : Fin (n + 2)} {i : Fin (n + 1)} (H : j = i.castSucc)
Goal: X.σ i ≫ X.δ j = 𝟙 (X.obj (Opposite.op (SimplexCategory.mk n)))
-- CategoryTheory.SimplicialObject.δ_comp_σ_succ
@[reassoc]
theorem δ_comp_σ_succ {n} {i : Fin (n + 1)} : X.σ i ≫ X.δ i.succ = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 1)}
Goal: X.σ i ≫ X.δ i.succ = 𝟙 (X.obj (Opposite.op (SimplexCategory.mk n)))
-- CategoryTheory.SimplicialObject.δ_comp_σ_succ'
@[reassoc]
theorem δ_comp_σ_succ' {n} {j : Fin (n + 2)} {i : Fin (n + 1)} (H : j = i.succ) :
    X.σ i ≫ X.δ j = 𝟙 _ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {j : Fin (n + 2)} {i : Fin (n + 1)} (H : j = i.succ)
Goal: X.σ i ≫ X.δ j = 𝟙 (X.obj (Opposite.op (SimplexCategory.mk n)))
-- CategoryTheory.SimplicialObject.δ_comp_σ_of_gt
@[reassoc]
theorem δ_comp_σ_of_gt {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : Fin.castSucc j < i) :
    X.σ (Fin.castSucc j) ≫ X.δ i.succ = X.δ i ≫ X.σ j 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : j.castSucc < i)
Goal: X.σ j.castSucc ≫ X.δ i.succ = X.δ i ≫ X.σ j
-- CategoryTheory.SimplicialObject.δ_comp_σ_of_gt'
@[reassoc]
theorem δ_comp_σ_of_gt' {n} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : j.succ < i) :
    X.σ j ≫ X.δ i =
      X.δ (i.pred fun (hi : i = 0) => by simp only [Fin.not_lt_zero, hi] at H) ≫
        X.σ (j.castLT ((add_lt_add_iff_right 1).mp (lt_of_lt_of_le H i.is_le))) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i : Fin (n + 3)} {j : Fin (n + 2)} (H : j.succ < i)
Goal: X.σ j ≫ X.δ i = X.δ (i.pred ⋯) ≫ X.σ (j.castLT ⋯)
-- CategoryTheory.SimplicialObject.σ_comp_σ
@[reassoc]
theorem σ_comp_σ {n} {i j : Fin (n + 1)} (H : i ≤ j) :
    X.σ j ≫ X.σ (Fin.castSucc i) = X.σ i ≫ X.σ j.succ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject C) {n : ℕ} {i j : Fin (n + 1)} (H : i ≤ j)
Goal: X.σ j ≫ X.σ i.castSucc = X.σ i ≫ X.σ j.succ
-- CategoryTheory.SimplicialObject.δ_naturality
@[reassoc (attr := simp)]
theorem δ_naturality {X' X : SimplicialObject C} (f : X ⟶ X') {n : ℕ} (i : Fin (n + 2)) :
    X.δ i ≫ f.app (op [n]) = f.app (op [n + 1]) ≫ X'.δ i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X' X : CategoryTheory.SimplicialObject C} (f : X ⟶ X') {n : ℕ} (i : Fin (n + 2))
Goal: X.δ i ≫ f.app (Opposite.op (SimplexCategory.mk n)) = f.app (Opposite.op (SimplexCategory.mk (n + 1))) ≫ X'.δ i
-- CategoryTheory.SimplicialObject.σ_naturality
@[reassoc (attr := simp)]
theorem σ_naturality {X' X : SimplicialObject C} (f : X ⟶ X') {n : ℕ} (i : Fin (n + 1)) :
    X.σ i ≫ f.app (op [n + 1]) = f.app (op [n]) ≫ X'.σ i 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X' X : CategoryTheory.SimplicialObject C} (f : X ⟶ X') {n : ℕ} (i : Fin (n + 1))
Goal: X.σ i ≫ f.app (Opposite.op (SimplexCategory.mk (n + 1))) = f.app (Opposite.op (SimplexCategory.mk n)) ≫ X'.σ i
-- CategoryTheory.SimplicialObject.whiskering_map_app_app
@[simps!]
def whiskering (D : Type*) [Category D] : (C ⥤ D) ⥤ SimplicialObject C ⥤ SimplicialObject D 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D]
Goal: ∀ {X Y : C ⥤ D} (τ : X ⟶ Y) (F : SimplexCategoryᵒᵖ ⥤ C) (c : SimplexCategoryᵒᵖ),
  (((CategoryTheory.SimplicialObject.whiskering C D).map τ).app F).app c = τ.app (F.obj c)
-- CategoryTheory.SimplicialObject.whiskering_obj_obj_map
@[simps!]
def whiskering (D : Type*) [Category D] : (C ⥤ D) ⥤ SimplicialObject C ⥤ SimplicialObject D 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D) (F : SimplexCategoryᵒᵖ ⥤ C)
Goal: ∀ {X Y : SimplexCategoryᵒᵖ} (f : X ⟶ Y),
  (((CategoryTheory.SimplicialObject.whiskering C D).obj H).obj F).map f = H.map (F.map f)
-- CategoryTheory.SimplicialObject.whiskering_obj_obj_obj
@[simps!]
def whiskering (D : Type*) [Category D] : (C ⥤ D) ⥤ SimplicialObject C ⥤ SimplicialObject D 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D) (F : SimplexCategoryᵒᵖ ⥤ C) (X : SimplexCategoryᵒᵖ)
Goal: (((CategoryTheory.SimplicialObject.whiskering C D).obj H).obj F).obj X = H.obj (F.obj X)
-- CategoryTheory.SimplicialObject.whiskering_obj_map_app
@[simps!]
def whiskering (D : Type*) [Category D] : (C ⥤ D) ⥤ SimplicialObject C ⥤ SimplicialObject D 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D)
Goal: ∀ {X Y : SimplexCategoryᵒᵖ ⥤ C} (α : X ⟶ Y) (X_1 : SimplexCategoryᵒᵖ),
  (((CategoryTheory.SimplicialObject.whiskering C D).obj H).map α).app X_1 = H.map (α.app X_1)
-- CategoryTheory.SimplicialObject.Truncated.whiskering_obj_map_app
@[simps!]
def whiskering {n} (D : Type*) [Category D] : (C ⥤ D) ⥤ Truncated C n ⥤ Truncated D n 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {n : ℕ} (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D)
Goal: ∀ {X Y : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C} (α : X ⟶ Y) (X_1 : (SimplexCategory.Truncated n)ᵒᵖ),
  (((CategoryTheory.SimplicialObject.Truncated.whiskering C D).obj H).map α).app X_1 = H.map (α.app X_1)
-- CategoryTheory.SimplicialObject.Truncated.whiskering_obj_obj_map
@[simps!]
def whiskering {n} (D : Type*) [Category D] : (C ⥤ D) ⥤ Truncated C n ⥤ Truncated D n 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {n : ℕ} (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D) (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C)
Goal: ∀ {X Y : (SimplexCategory.Truncated n)ᵒᵖ} (f : X ⟶ Y),
  (((CategoryTheory.SimplicialObject.Truncated.whiskering C D).obj H).obj F).map f = H.map (F.map f)
-- CategoryTheory.SimplicialObject.Truncated.whiskering_obj_obj_obj
@[simps!]
def whiskering {n} (D : Type*) [Category D] : (C ⥤ D) ⥤ Truncated C n ⥤ Truncated D n 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {n : ℕ} (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D] (H : C ⥤ D) (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C) (X : (SimplexCategory.Truncated n)ᵒᵖ)
Goal: (((CategoryTheory.SimplicialObject.Truncated.whiskering C D).obj H).obj F).obj X = H.obj (F.obj X)
-- CategoryTheory.SimplicialObject.Truncated.whiskering_map_app_app
@[simps!]
def whiskering {n} (D : Type*) [Category D] : (C ⥤ D) ⥤ Truncated C n ⥤ Truncated D n 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] {n : ℕ} (D : Type u_1) [CategoryTheory.Category.{u_2, u_1} D]
Goal: ∀ {X Y : C ⥤ D} (τ : X ⟶ Y) (F : (SimplexCategory.Truncated n)ᵒᵖ ⥤ C) (c : (SimplexCategory.Truncated n)ᵒᵖ),
  (((CategoryTheory.SimplicialObject.Truncated.whiskering C D).map τ).app F).app c = τ.app (F.obj c)
-- CategoryTheory.SimplicialObject.instCategoryAugmented_id_left_app
@[simps!]
instance : Category (Augmented C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C))
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ), (𝟙 X).left.app X_1 = 𝟙 (X.left.obj X_1)
-- CategoryTheory.SimplicialObject.instCategoryAugmented_id_right
@[simps!]
instance : Category (Augmented C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C))
Goal: (𝟙 X).right = 𝟙 X.right
-- CategoryTheory.SimplicialObject.instCategoryAugmented_comp_left_app
@[simps!]
instance : Category (Augmented C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y Z : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C)}
  (f : X ⟶ Y) (g : Y ⟶ Z) (X_1 : SimplexCategoryᵒᵖ), (f ≫ g).left.app X_1 = f.left.app X_1 ≫ g.left.app X_1
-- CategoryTheory.SimplicialObject.instCategoryAugmented_comp_right
@[simps!]
instance : Category (Augmented C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y Z : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C)}
  (f : X ⟶ Y) (g : Y ⟶ Z), (f ≫ g).right = f.right ≫ g.right
-- CategoryTheory.SimplicialObject.Augmented.hom_ext
@[ext]
lemma hom_ext {X Y : Augmented C} (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right) :
    f = g 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X Y : CategoryTheory.SimplicialObject.Augmented C} (f g : X ⟶ Y) (h₁ : f.left = g.left) (h₂ : f.right = g.right)
Goal: f = g
-- CategoryTheory.SimplicialObject.Augmented.drop_obj
@[simps!]
def drop : Augmented C ⥤ SimplicialObject C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C))
Goal: CategoryTheory.SimplicialObject.Augmented.drop.obj X = X.left
-- CategoryTheory.SimplicialObject.Augmented.drop_map
@[simps!]
def drop : Augmented C ⥤ SimplicialObject C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C)}
  (f : X ⟶ Y), CategoryTheory.SimplicialObject.Augmented.drop.map f = f.left
-- CategoryTheory.SimplicialObject.Augmented.point_obj
@[simps!]
def point : Augmented C ⥤ C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C))
Goal: CategoryTheory.SimplicialObject.Augmented.point.obj X = X.right
-- CategoryTheory.SimplicialObject.Augmented.point_map
@[simps!]
def point : Augmented C ⥤ C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C)}
  (f : X ⟶ Y), CategoryTheory.SimplicialObject.Augmented.point.map f = f.right