ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CompositionAsSet.boundary_zero
@[simp]
theorem boundary_zero : (c.boundary ⟨0, c.card_boundaries_pos⟩ : Fin (n + 1)) = 0 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.boundary ⟨0, ⋯⟩ = 0
-- CompositionAsSet.boundary_length
@[simp]
theorem boundary_length : c.boundary ⟨c.length, c.length_lt_card_boundaries⟩ = Fin.last n 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.boundary ⟨c.length, ⋯⟩ = Fin.last n
-- CompositionAsSet.blocksFun_pos
theorem blocksFun_pos (i : Fin c.length) : 0 < c.blocksFun i 
Arguments: {n : ℕ} (c : CompositionAsSet n) (i : Fin c.length)
Goal: 0 < c.blocksFun i
-- CompositionAsSet.blocks_length
@[simp]
theorem blocks_length : c.blocks.length = c.length 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.blocks.length = c.length
-- CompositionAsSet.blocks_partial_sum
theorem blocks_partial_sum {i : ℕ} (h : i < c.boundaries.card) :
    (c.blocks.take i).sum = c.boundary ⟨i, h⟩ 
Arguments: {n : ℕ} (c : CompositionAsSet n) {i : ℕ} (h : i < c.boundaries.card)
Goal: (List.take i c.blocks).sum = ↑(c.boundary ⟨i, h⟩)
-- CompositionAsSet.mem_boundaries_iff_exists_blocks_sum_take_eq
theorem mem_boundaries_iff_exists_blocks_sum_take_eq {j : Fin (n + 1)} :
    j ∈ c.boundaries ↔ ∃ i < c.boundaries.card, (c.blocks.take i).sum = j 
Arguments: {n : ℕ} (c : CompositionAsSet n) {j : Fin (n + 1)}
Goal: j ∈ c.boundaries ↔ ∃ i < c.boundaries.card, (List.take i c.blocks).sum = ↑j
-- CompositionAsSet.blocks_sum
theorem blocks_sum : c.blocks.sum = n 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.blocks.sum = n
-- Composition.toCompositionAsSet_length
@[simp]
theorem Composition.toCompositionAsSet_length (c : Composition n) :
    c.toCompositionAsSet.length = c.length 
Arguments: {n : ℕ} (c : Composition n)
Goal: c.toCompositionAsSet.length = c.length
-- CompositionAsSet.toComposition_length
@[simp]
theorem CompositionAsSet.toComposition_length (c : CompositionAsSet n) :
    c.toComposition.length = c.length 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.toComposition.length = c.length
-- Composition.toCompositionAsSet_blocks
@[simp]
theorem Composition.toCompositionAsSet_blocks (c : Composition n) :
    c.toCompositionAsSet.blocks = c.blocks 
Arguments: {n : ℕ} (c : Composition n)
Goal: c.toCompositionAsSet.blocks = c.blocks
-- CompositionAsSet.toComposition_blocks
@[simp]
theorem CompositionAsSet.toComposition_blocks (c : CompositionAsSet n) :
    c.toComposition.blocks = c.blocks 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.toComposition.blocks = c.blocks
-- CompositionAsSet.toComposition_boundaries
@[simp]
theorem CompositionAsSet.toComposition_boundaries (c : CompositionAsSet n) :
    c.toComposition.boundaries = c.boundaries 
Arguments: {n : ℕ} (c : CompositionAsSet n)
Goal: c.toComposition.boundaries = c.boundaries
-- Composition.toCompositionAsSet_boundaries
@[simp]
theorem Composition.toCompositionAsSet_boundaries (c : Composition n) :
    c.toCompositionAsSet.boundaries = c.boundaries 
Arguments: {n : ℕ} (c : Composition n)
Goal: c.toCompositionAsSet.boundaries = c.boundaries
-- composition_card
theorem composition_card (n : ℕ) : Fintype.card (Composition n) = 2 ^ (n - 1) 
Arguments: (n : ℕ)
Goal: Fintype.card (Composition n) = 2 ^ (n - 1)
-- sup_sdiff_injOn
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
    { x | Disjoint u x ∧ v ≤ x }.InjOn fun x => (x ⊔ u) \ v 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (u v : α)
Goal: Set.InjOn (fun x => (x ⊔ u) \ v) {x | Disjoint u x ∧ v ≤ x}
-- UV.compress_of_disjoint_of_le
theorem compress_of_disjoint_of_le (hua : Disjoint u a) (hva : v ≤ a) :
    compress u v a = (a ⊔ u) \ v 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {u v a : α} (hua : Disjoint u a) (hva : v ≤ a)
Goal: UV.compress u v a = (a ⊔ u) \ v
-- UV.compress_of_disjoint_of_le'
theorem compress_of_disjoint_of_le' (hva : Disjoint v a) (hua : u ≤ a) :
    compress u v ((a ⊔ v) \ u) = a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {u v a : α} (hva : Disjoint v a) (hua : u ≤ a)
Goal: UV.compress u v ((a ⊔ v) \ u) = a
-- UV.compress_self
@[simp]
theorem compress_self (u a : α) : compress u u a = a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] (u a : α)
Goal: UV.compress u u a = a
-- UV.compress_sdiff_sdiff
@[simp]
theorem compress_sdiff_sdiff (a b : α) : compress (a \ b) (b \ a) b = a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] (a b : α)
Goal: UV.compress (a \ b) (b \ a) b = a
-- UV.compress_idem
@[simp]
theorem compress_idem (u v a : α) : compress u v (compress u v a) = compress u v a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] (u v a : α)
Goal: UV.compress u v (UV.compress u v a) = UV.compress u v a
-- UV.compress_injOn
theorem compress_injOn : Set.InjOn (compress u v) ↑(s.filter (compress u v · ∉ s)) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v : α} [DecidableEq α]
Goal: Set.InjOn (UV.compress u v) ↑(Finset.filter (fun x => UV.compress u v x ∉ s) s)
-- UV.mem_compression
theorem mem_compression :
    a ∈ 𝓒 u v s ↔ a ∈ s ∧ compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, compress u v b = a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α]
Goal: a ∈ UV.compression u v s ↔ a ∈ s ∧ UV.compress u v a ∈ s ∨ a ∉ s ∧ ∃ b ∈ s, UV.compress u v b = a
-- UV.IsCompressed.eq
protected theorem IsCompressed.eq (h : IsCompressed u v s) : 𝓒 u v s = s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v : α} [DecidableEq α] (h : UV.IsCompressed u v s)
Goal: UV.compression u v s = s
-- UV.compression_self
@[simp]
theorem compression_self (u : α) (s : Finset α) : 𝓒 u u s = s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] [DecidableEq α] (u : α) (s : Finset α)
Goal: UV.compression u u s = s
-- UV.isCompressed_self
theorem isCompressed_self (u : α) (s : Finset α) : IsCompressed u u s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] [DecidableEq α] (u : α) (s : Finset α)
Goal: UV.IsCompressed u u s
-- UV.compress_disjoint
theorem compress_disjoint :
    Disjoint (s.filter (compress u v · ∈ s)) ((s.image <| compress u v).filter (· ∉ s)) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v : α} [DecidableEq α]
Goal: Disjoint (Finset.filter (fun x => UV.compress u v x ∈ s) s)
  (Finset.filter (fun x => x ∉ s) (Finset.image (UV.compress u v) s))
-- UV.compress_mem_compression
theorem compress_mem_compression (ha : a ∈ s) : compress u v a ∈ 𝓒 u v s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (ha : a ∈ s)
Goal: UV.compress u v a ∈ UV.compression u v s
-- UV.compress_mem_compression_of_mem_compression
theorem compress_mem_compression_of_mem_compression (ha : a ∈ 𝓒 u v s) :
    compress u v a ∈ 𝓒 u v s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (ha : a ∈ UV.compression u v s)
Goal: UV.compress u v a ∈ UV.compression u v s
-- UV.compression_idem
@[simp]
theorem compression_idem (u v : α) (s : Finset α) : 𝓒 u v (𝓒 u v s) = 𝓒 u v s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] [DecidableEq α] (u v : α) (s : Finset α)
Goal: UV.compression u v (UV.compression u v s) = UV.compression u v s
-- UV.card_compression
@[simp]
theorem card_compression (u v : α) (s : Finset α) : (𝓒 u v s).card = s.card 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] [DecidableEq α] (u v : α) (s : Finset α)
Goal: (UV.compression u v s).card = s.card
-- UV.le_of_mem_compression_of_not_mem
theorem le_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : u ≤ a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (h : a ∈ UV.compression u v s) (ha : a ∉ s)
Goal: u ≤ a
-- UV.disjoint_of_mem_compression_of_not_mem
theorem disjoint_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) : Disjoint v a 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (h : a ∈ UV.compression u v s) (ha : a ∉ s)
Goal: Disjoint v a
-- UV.sup_sdiff_mem_of_mem_compression_of_not_mem
theorem sup_sdiff_mem_of_mem_compression_of_not_mem (h : a ∈ 𝓒 u v s) (ha : a ∉ s) :
    (a ⊔ v) \ u ∈ s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (h : a ∈ UV.compression u v s) (ha : a ∉ s)
Goal: (a ⊔ v) \ u ∈ s
-- UV.sup_sdiff_mem_of_mem_compression
theorem sup_sdiff_mem_of_mem_compression (ha : a ∈ 𝓒 u v s) (hva : v ≤ a) (hua : Disjoint u a) :
    (a ⊔ u) \ v ∈ s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (ha : a ∈ UV.compression u v s) (hva : v ≤ a) (hua : Disjoint u a)
Goal: (a ⊔ u) \ v ∈ s
-- UV.mem_of_mem_compression
theorem mem_of_mem_compression (ha : a ∈ 𝓒 u v s) (hva : v ≤ a) (hvu : v = ⊥ → u = ⊥) :
    a ∈ s 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] [DecidableRel Disjoint] [DecidableRel fun x x_1 => x ≤ x_1] {s : Finset α} {u v a : α} [DecidableEq α] (ha : a ∈ UV.compression u v s) (hva : v ≤ a) (hvu : v = ⊥ → u = ⊥)
Goal: a ∈ s
-- UV.card_compress
theorem card_compress (huv : u.card = v.card) (a : Finset α) : (compress u v a).card = a.card 
Arguments: {α : Type u_1} [DecidableEq α] {u v : Finset α} (huv : u.card = v.card) (a : Finset α)
Goal: (UV.compress u v a).card = a.card
-- Set.Sized.uvCompression
lemma _root_.Set.Sized.uvCompression (huv : u.card = v.card) (h𝒜 : (𝒜 : Set (Finset α)).Sized r) :
    (𝓒 u v 𝒜 : Set (Finset α)).Sized r 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {u v : Finset α} {r : ℕ} (huv : u.card = v.card) (h𝒜 : Set.Sized r ↑𝒜)
Goal: Set.Sized r ↑(UV.compression u v 𝒜)
-- _private.Mathlib.Combinatorics.SetFamily.Compression.UV.0.UV.aux
private theorem aux (huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
    v = ∅ → u = ∅ 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {u v : Finset α} (huv : ∀ x ∈ u, ∃ y ∈ v, UV.IsCompressed (u.erase x) (v.erase y) 𝒜)
Goal: v = ∅ → u = ∅
-- UV.shadow_compression_subset_compression_shadow
theorem shadow_compression_subset_compression_shadow (u v : Finset α)
    (huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
    ∂ (𝓒 u v 𝒜) ⊆ 𝓒 u v (∂ 𝒜) 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} (u v : Finset α) (huv : ∀ x ∈ u, ∃ y ∈ v, UV.IsCompressed (u.erase x) (v.erase y) 𝒜)
Goal: (UV.compression u v 𝒜).shadow ⊆ UV.compression u v 𝒜.shadow
-- UV.card_shadow_compression_le
theorem card_shadow_compression_le (u v : Finset α)
    (huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜) :
    (∂ (𝓒 u v 𝒜)).card ≤ (∂ 𝒜).card 
Arguments: {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} (u v : Finset α) (huv : ∀ x ∈ u, ∃ y ∈ v, UV.IsCompressed (u.erase x) (v.erase y) 𝒜)
Goal: (UV.compression u v 𝒜).shadow.card ≤ 𝒜.shadow.card
-- SimpleGraph.map_adj
@[simp]
theorem map_adj (f : V ↪ W) (G : SimpleGraph V) (u v : W) :
    (G.map f).Adj u v ↔ ∃ u' v' : V, G.Adj u' v' ∧ f u' = u ∧ f v' = v 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph V) (u v : W)
Goal: (SimpleGraph.map f G).Adj u v ↔ ∃ u' v', G.Adj u' v' ∧ f u' = u ∧ f v' = v
-- SimpleGraph.map_adj_apply
lemma map_adj_apply {G : SimpleGraph V} {f : V ↪ W} {a b : V} :
    (G.map f).Adj (f a) (f b) ↔ G.Adj a b 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {f : V ↪ W} {a b : V}
Goal: (SimpleGraph.map f G).Adj (f a) (f b) ↔ G.Adj a b
-- SimpleGraph.map_monotone
theorem map_monotone (f : V ↪ W) : Monotone (SimpleGraph.map f) 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W)
Goal: Monotone (SimpleGraph.map f)
-- SimpleGraph.map_id
@[simp] lemma map_id : G.map (Function.Embedding.refl _) = G 
Arguments: {V : Type u_1} (G : SimpleGraph V)
Goal: SimpleGraph.map (Function.Embedding.refl V) G = G
-- SimpleGraph.map_map
@[simp] lemma map_map (f : V ↪ W) (g : W ↪ X) : (G.map f).map g = G.map (f.trans g) 
Arguments: {V : Type u_1} {W : Type u_2} {X : Type u_3} (G : SimpleGraph V) (f : V ↪ W) (g : W ↪ X)
Goal: SimpleGraph.map g (SimpleGraph.map f G) = SimpleGraph.map (f.trans g) G
-- SimpleGraph.comap_adj
@[simp] lemma comap_adj {G : SimpleGraph W} {f : V → W} :
    (G.comap f).Adj u v ↔ G.Adj (f u) (f v) 
Arguments: {V : Type u_1} {W : Type u_2} {u v : V} {G : SimpleGraph W} {f : V → W}
Goal: (SimpleGraph.comap f G).Adj u v ↔ G.Adj (f u) (f v)
-- SimpleGraph.comap_id
@[simp] lemma comap_id {G : SimpleGraph V} : G.comap id = G 
Arguments: {V : Type u_1} {G : SimpleGraph V}
Goal: SimpleGraph.comap id G = G
-- SimpleGraph.comap_comap
@[simp] lemma comap_comap {G : SimpleGraph X} (f : V → W) (g : W → X) :
  (G.comap g).comap f = G.comap (g ∘ f) 
Arguments: {V : Type u_1} {W : Type u_2} {X : Type u_3} {G : SimpleGraph X} (f : V → W) (g : W → X)
Goal: SimpleGraph.comap f (SimpleGraph.comap g G) = SimpleGraph.comap (g ∘ f) G
-- SimpleGraph.comap_symm
lemma comap_symm (G : SimpleGraph V) (e : V ≃ W) :
    G.comap e.symm.toEmbedding = G.map e.toEmbedding 
Arguments: {V : Type u_1} {W : Type u_2} (G : SimpleGraph V) (e : V ≃ W)
Goal: SimpleGraph.comap (⇑e.symm.toEmbedding) G = SimpleGraph.map e.toEmbedding G
-- SimpleGraph.map_symm
lemma map_symm (G : SimpleGraph W) (e : V ≃ W) :
    G.map e.symm.toEmbedding = G.comap e.toEmbedding 
Arguments: {V : Type u_1} {W : Type u_2} (G : SimpleGraph W) (e : V ≃ W)
Goal: SimpleGraph.map e.symm.toEmbedding G = SimpleGraph.comap (⇑e.toEmbedding) G
-- SimpleGraph.comap_monotone
theorem comap_monotone (f : V ↪ W) : Monotone (SimpleGraph.comap f) 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W)
Goal: Monotone (SimpleGraph.comap ⇑f)
-- SimpleGraph.comap_map_eq
@[simp]
theorem comap_map_eq (f : V ↪ W) (G : SimpleGraph V) : (G.map f).comap f = G 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph V)
Goal: SimpleGraph.comap (⇑f) (SimpleGraph.map f G) = G
-- SimpleGraph.leftInverse_comap_map
theorem leftInverse_comap_map (f : V ↪ W) :
    Function.LeftInverse (SimpleGraph.comap f) (SimpleGraph.map f) 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W)
Goal: Function.LeftInverse (SimpleGraph.comap ⇑f) (SimpleGraph.map f)
-- SimpleGraph.map_injective
theorem map_injective (f : V ↪ W) : Function.Injective (SimpleGraph.map f) 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W)
Goal: Function.Injective (SimpleGraph.map f)
-- SimpleGraph.comap_surjective
theorem comap_surjective (f : V ↪ W) : Function.Surjective (SimpleGraph.comap f) 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W)
Goal: Function.Surjective (SimpleGraph.comap ⇑f)
-- SimpleGraph.map_le_iff_le_comap
theorem map_le_iff_le_comap (f : V ↪ W) (G : SimpleGraph V) (G' : SimpleGraph W) :
    G.map f ≤ G' ↔ G ≤ G'.comap f 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph V) (G' : SimpleGraph W)
Goal: SimpleGraph.map f G ≤ G' ↔ G ≤ SimpleGraph.comap (⇑f) G'
-- SimpleGraph.map_comap_le
theorem map_comap_le (f : V ↪ W) (G : SimpleGraph W) : (G.comap f).map f ≤ G 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph W)
Goal: SimpleGraph.map f (SimpleGraph.comap (⇑f) G) ≤ G
-- SimpleGraph.le_comap_of_subsingleton
lemma le_comap_of_subsingleton (f : V → W) [Subsingleton V] : G ≤ G'.comap f 
Arguments: {V : Type u_1} {W : Type u_2} (G : SimpleGraph V) (G' : SimpleGraph W) (f : V → W) [Subsingleton V]
Goal: G ≤ SimpleGraph.comap f G'
-- SimpleGraph.map_le_of_subsingleton
lemma map_le_of_subsingleton (f : V ↪ W) [Subsingleton V] : G.map f ≤ G' 
Arguments: {V : Type u_1} {W : Type u_2} (G : SimpleGraph V) (G' : SimpleGraph W) (f : V ↪ W) [Subsingleton V]
Goal: SimpleGraph.map f G ≤ G'
-- Equiv.simpleGraph_apply
@[simps apply]
protected def _root_.Equiv.simpleGraph (e : V ≃ W) : SimpleGraph V ≃ SimpleGraph W 
Arguments: {V : Type u_1} {W : Type u_2} (e : V ≃ W) (G : SimpleGraph V)
Goal: e.simpleGraph G = SimpleGraph.comap (⇑e.symm) G
-- Equiv.simpleGraph_refl
@[simp] lemma _root_.Equiv.simpleGraph_refl : (Equiv.refl V).simpleGraph = Equiv.refl _ 
Arguments: {V : Type u_1}
Goal: (Equiv.refl V).simpleGraph = Equiv.refl (SimpleGraph V)
-- Equiv.simpleGraph_trans
@[simp] lemma _root_.Equiv.simpleGraph_trans (e₁ : V ≃ W) (e₂ : W ≃ X) :
  (e₁.trans e₂).simpleGraph = e₁.simpleGraph.trans e₂.simpleGraph 
Arguments: {V : Type u_1} {W : Type u_2} {X : Type u_3} (e₁ : V ≃ W) (e₂ : W ≃ X)
Goal: (e₁.trans e₂).simpleGraph = e₁.simpleGraph.trans e₂.simpleGraph
-- Equiv.symm_simpleGraph
@[simp]
lemma _root_.Equiv.symm_simpleGraph (e : V ≃ W) : e.simpleGraph.symm = e.symm.simpleGraph 
Arguments: {V : Type u_1} {W : Type u_2} (e : V ≃ W)
Goal: e.simpleGraph.symm = e.symm.simpleGraph
-- SimpleGraph.induce_singleton_eq_top
@[simp] lemma induce_singleton_eq_top (v : V) : G.induce {v} = ⊤ 
Arguments: {V : Type u_1} (G : SimpleGraph V) (v : V)
Goal: SimpleGraph.induce {v} G = ⊤
-- SimpleGraph.induce_spanningCoe
theorem induce_spanningCoe {s : Set V} {G : SimpleGraph s} : G.spanningCoe.induce s = G 
Arguments: {V : Type u_1} {s : Set V} {G : SimpleGraph ↑s}
Goal: SimpleGraph.induce s G.spanningCoe = G
-- SimpleGraph.spanningCoe_induce_le
theorem spanningCoe_induce_le (s : Set V) : (G.induce s).spanningCoe ≤ G 
Arguments: {V : Type u_1} (G : SimpleGraph V) (s : Set V)
Goal: (SimpleGraph.induce s G).spanningCoe ≤ G
-- SimpleGraph.Hom.coe_id
@[simp, norm_cast] lemma coe_id : ⇑(Hom.id : G →g G) = id 
Arguments: {V : Type u_1} {G : SimpleGraph V}
Goal: ⇑SimpleGraph.Hom.id = id
-- SimpleGraph.Hom.map_adj
theorem map_adj {v w : V} (h : G.Adj v w) : G'.Adj (f v) (f w) 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {v w : V} (h : G.Adj v w)
Goal: G'.Adj (f v) (f w)
-- SimpleGraph.Hom.map_mem_edgeSet
theorem map_mem_edgeSet {e : Sym2 V} (h : e ∈ G.edgeSet) : e.map f ∈ G'.edgeSet 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {e : Sym2 V} (h : e ∈ G.edgeSet)
Goal: Sym2.map (⇑f) e ∈ G'.edgeSet
-- SimpleGraph.Hom.apply_mem_neighborSet
theorem apply_mem_neighborSet {v w : V} (h : w ∈ G.neighborSet v) : f w ∈ G'.neighborSet (f v) 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') {v w : V} (h : w ∈ G.neighborSet v)
Goal: f w ∈ G'.neighborSet (f v)
-- SimpleGraph.Hom.mapEdgeSet_coe
@[simps]
def mapEdgeSet (e : G.edgeSet) : G'.edgeSet 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (e : ↑G.edgeSet)
Goal: ↑(f.mapEdgeSet e) = Sym2.map ⇑f ↑e
-- SimpleGraph.Hom.mapNeighborSet_coe
@[simps]
def mapNeighborSet (v : V) (w : G.neighborSet v) : G'.neighborSet (f v) 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (v : V) (w : ↑(G.neighborSet v))
Goal: ↑(f.mapNeighborSet v w) = f ↑w
-- SimpleGraph.Hom.mapDart_apply
@[simp]
theorem mapDart_apply (d : G.Dart) : f.mapDart d = ⟨d.1.map f f, f.map_adj d.2⟩ 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (d : G.Dart)
Goal: f.mapDart d = { toProd := Prod.map (⇑f) (⇑f) d.toProd, adj := ⋯ }
-- SimpleGraph.Hom.mapSpanningSubgraphs_apply
@[simps]
def mapSpanningSubgraphs {G G' : SimpleGraph V} (h : G ≤ G') : G →g G' 
Arguments: {V : Type u_1} {G G' : SimpleGraph V} (h : G ≤ G') (x : V)
Goal: (SimpleGraph.Hom.mapSpanningSubgraphs h) x = x
-- SimpleGraph.Hom.mapEdgeSet.injective
theorem mapEdgeSet.injective (hinj : Function.Injective f) : Function.Injective f.mapEdgeSet 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G →g G') (hinj : Function.Injective ⇑f)
Goal: Function.Injective f.mapEdgeSet
-- SimpleGraph.Hom.injective_of_top_hom
theorem injective_of_top_hom (f : (⊤ : SimpleGraph V) →g G') : Function.Injective f 
Arguments: {V : Type u_1} {W : Type u_2} {G' : SimpleGraph W} (f : ⊤ →g G')
Goal: Function.Injective ⇑f
-- SimpleGraph.Hom.comap_apply
@[simps]
protected def comap (f : V → W) (G : SimpleGraph W) : G.comap f →g G 
Arguments: {V : Type u_1} {W : Type u_2} (f : V → W) (G : SimpleGraph W)
Goal: ∀ (a : V), (SimpleGraph.Hom.comap f G) a = f a
-- SimpleGraph.Hom.coe_comp
@[simp]
theorem coe_comp (f' : G' →g G'') (f : G →g G') : ⇑(f'.comp f) = f' ∘ f 
Arguments: {V : Type u_1} {W : Type u_2} {X : Type u_3} {G : SimpleGraph V} {G' : SimpleGraph W} {G'' : SimpleGraph X} (f' : G' →g G'') (f : G →g G')
Goal: ⇑(f'.comp f) = ⇑f' ∘ ⇑f
-- SimpleGraph.Hom.coe_ofLE
@[simp, norm_cast] lemma coe_ofLE (h : G₁ ≤ G₂) : ⇑(ofLE h) = id 
Arguments: {V : Type u_1} {G₁ G₂ : SimpleGraph V} (h : G₁ ≤ G₂)
Goal: ⇑(SimpleGraph.Hom.ofLE h) = id
-- SimpleGraph.Embedding.coe_toHom
@[simp] lemma coe_toHom (f : G ↪g H) : ⇑f.toHom = f 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {H : SimpleGraph W} (f : G ↪g H)
Goal: ⇑f.toHom = ⇑f
-- SimpleGraph.Embedding.map_adj_iff
@[simp] theorem map_adj_iff {v w : V} : G'.Adj (f v) (f w) ↔ G.Adj v w 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G ↪g G') {v w : V}
Goal: G'.Adj (f v) (f w) ↔ G.Adj v w
-- SimpleGraph.Embedding.map_mem_edgeSet_iff
theorem map_mem_edgeSet_iff {e : Sym2 V} : e.map f ∈ G'.edgeSet ↔ e ∈ G.edgeSet 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G ↪g G') {e : Sym2 V}
Goal: Sym2.map (⇑f) e ∈ G'.edgeSet ↔ e ∈ G.edgeSet
-- SimpleGraph.Embedding.apply_mem_neighborSet_iff
theorem apply_mem_neighborSet_iff {v w : V} : f w ∈ G'.neighborSet (f v) ↔ w ∈ G.neighborSet v 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G ↪g G') {v w : V}
Goal: f w ∈ G'.neighborSet (f v) ↔ w ∈ G.neighborSet v
-- SimpleGraph.Embedding.mapEdgeSet_apply
@[simps]
def mapEdgeSet : G.edgeSet ↪ G'.edgeSet 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G ↪g G') (e : ↑G.edgeSet)
Goal: f.mapEdgeSet e = SimpleGraph.Hom.mapEdgeSet (RelEmbedding.toRelHom f) e
-- SimpleGraph.Embedding.mapNeighborSet_apply_coe
@[simps]
def mapNeighborSet (v : V) : G.neighborSet v ↪ G'.neighborSet (f v) 
Arguments: {V : Type u_1} {W : Type u_2} {G : SimpleGraph V} {G' : SimpleGraph W} (f : G ↪g G') (v : V) (w : ↑(G.neighborSet v))
Goal: ↑((f.mapNeighborSet v) w) = f ↑w
-- SimpleGraph.Embedding.comap_apply
@[simp]
theorem comap_apply (f : V ↪ W) (G : SimpleGraph W) (v : V) :
    SimpleGraph.Embedding.comap f G v = f v 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph W) (v : V)
Goal: (SimpleGraph.Embedding.comap f G) v = f v
-- SimpleGraph.Embedding.map_apply
@[simp]
theorem map_apply (f : V ↪ W) (G : SimpleGraph V) (v : V) :
    SimpleGraph.Embedding.map f G v = f v 
Arguments: {V : Type u_1} {W : Type u_2} (f : V ↪ W) (G : SimpleGraph V) (v : V)
Goal: (SimpleGraph.Embedding.map f G) v = f v
-- SimpleGraph.Embedding.coe_completeGraph
@[simp] lemma coe_completeGraph {α β : Type*} (f : α ↪ β) : ⇑(Embedding.completeGraph f) = f 
Arguments: {α : Type u_4} {β : Type u_5} (f : α ↪ β)
Goal: ⇑(SimpleGraph.Embedding.completeGraph f) = ⇑f