ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint.inj₁
class ExplicitDisjoint (t : Finset (α × β × γ)) : Prop where
  inj₀ : ∀ ⦃a b c a'⦄, (a, b, c) ∈ t → (a', b, c) ∈ t → a = a'
  inj₁ : ∀ ⦃a b c b'⦄, (a, b, c) ∈ t → (a, b', c) ∈ t → b = b'
  inj₂ : ∀ ⦃a b c c'⦄, (a, b, c) ∈ t → (a, b, c') ∈ t → c = c'
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [self : SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint t] ⦃a : α⦄ ⦃b : β⦄ ⦃c : γ⦄ ⦃b' : β⦄
Goal: (a, b, c) ∈ t → (a, b', c) ∈ t → b = b'
-- SimpleGraph.TripartiteFromTriangles.NoAccidental.eq_or_eq_or_eq
class NoAccidental (t : Finset (α × β × γ)) : Prop where
  eq_or_eq_or_eq : ∀ ⦃a a' b b' c c'⦄, (a', b, c) ∈ t → (a, b', c) ∈ t → (a, b, c') ∈ t →
    a = a' ∨ b = b' ∨ c = c'
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [self : SimpleGraph.TripartiteFromTriangles.NoAccidental t] ⦃a a' : α⦄ ⦃b b' : β⦄ ⦃c c' : γ⦄
Goal: (a', b, c) ∈ t → (a, b', c) ∈ t → (a, b, c') ∈ t → a = a' ∨ b = b' ∨ c = c'
-- SimpleGraph.TripartiteFromTriangles.graph_triple
 lemma graph_triple ⦃x y z⦄ :
  (graph t).Adj x y → (graph t).Adj x z → (graph t).Adj y z → ∃ a b c,
    ({in₀ a, in₁ b, in₂ c} : Finset (α ⊕ β ⊕ γ)) = {x, y, z} ∧ (graph t).Adj (in₀ a) (in₁ b) ∧
      (graph t).Adj (in₀ a) (in₂ c) ∧ (graph t).Adj (in₁ b) (in₂ c) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [DecidableEq α] [DecidableEq β] [DecidableEq γ] ⦃x y z : α ⊕ β ⊕ γ⦄
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).Adj x y →
  (SimpleGraph.TripartiteFromTriangles.graph t).Adj x z →
    (SimpleGraph.TripartiteFromTriangles.graph t).Adj y z →
      ∃ a b c,
        {Sum3.in₀ a, Sum3.in₁ b, Sum3.in₂ c} = {x, y, z} ∧
          (SimpleGraph.TripartiteFromTriangles.graph t).Adj (Sum3.in₀ a) (Sum3.in₁ b) ∧
            (SimpleGraph.TripartiteFromTriangles.graph t).Adj (Sum3.in₀ a) (Sum3.in₂ c) ∧
              (SimpleGraph.TripartiteFromTriangles.graph t).Adj (Sum3.in₁ b) (Sum3.in₂ c)
-- SimpleGraph.TripartiteFromTriangles.toTriangle_apply
@[simps] def toTriangle : α × β × γ ↪ Finset (α ⊕ β ⊕ γ) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [DecidableEq α] [DecidableEq β] [DecidableEq γ] (x : α × β × γ)
Goal: SimpleGraph.TripartiteFromTriangles.toTriangle x = {Sum3.in₀ x.1, Sum3.in₁ x.2.1, Sum3.in₂ x.2.2}
-- SimpleGraph.TripartiteFromTriangles.toTriangle_is3Clique
lemma toTriangle_is3Clique (hx : x ∈ t) : (graph t).IsNClique 3 (toTriangle x) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} {x : α × β × γ} [DecidableEq α] [DecidableEq β] [DecidableEq γ] (hx : x ∈ t)
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).IsNClique 3 (SimpleGraph.TripartiteFromTriangles.toTriangle x)
-- SimpleGraph.TripartiteFromTriangles.exists_mem_toTriangle
lemma exists_mem_toTriangle {x y : α ⊕ β ⊕ γ} (hxy : (graph t).Adj x y) :
    ∃ z ∈ t, x ∈ toTriangle z ∧ y ∈ toTriangle z 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [DecidableEq α] [DecidableEq β] [DecidableEq γ] {x y : α ⊕ β ⊕ γ} (hxy : (SimpleGraph.TripartiteFromTriangles.graph t).Adj x y)
Goal: ∃ z ∈ t, x ∈ SimpleGraph.TripartiteFromTriangles.toTriangle z ∧ y ∈ SimpleGraph.TripartiteFromTriangles.toTriangle z
-- SimpleGraph.TripartiteFromTriangles.is3Clique_iff
nonrec lemma is3Clique_iff [NoAccidental t] {s : Finset (α ⊕ β ⊕ γ)} :
    (graph t).IsNClique 3 s ↔ ∃ x, x ∈ t ∧ toTriangle x = s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [DecidableEq α] [DecidableEq β] [DecidableEq γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t] {s : Finset (α ⊕ β ⊕ γ)}
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).IsNClique 3 s ↔
  ∃ x ∈ t, SimpleGraph.TripartiteFromTriangles.toTriangle x = s
-- SimpleGraph.TripartiteFromTriangles.toTriangle_surjOn
lemma toTriangle_surjOn [NoAccidental t] :
    (t : Set (α × β × γ)).SurjOn toTriangle ((graph t).cliqueSet 3) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {t : Finset (α × β × γ)} [DecidableEq α] [DecidableEq β] [DecidableEq γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: Set.SurjOn (⇑SimpleGraph.TripartiteFromTriangles.toTriangle) (↑t)
  ((SimpleGraph.TripartiteFromTriangles.graph t).cliqueSet 3)
-- SimpleGraph.TripartiteFromTriangles.map_toTriangle_disjoint
lemma map_toTriangle_disjoint [ExplicitDisjoint t] :
    (t.map toTriangle : Set (Finset (α ⊕ β ⊕ γ))).Pairwise
      fun x y ↦ (x ∩ y : Set (α ⊕ β ⊕ γ)).Subsingleton 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint t]
Goal: (↑(Finset.map SimpleGraph.TripartiteFromTriangles.toTriangle t)).Pairwise fun x y => (↑x ∩ ↑y).Subsingleton
-- SimpleGraph.TripartiteFromTriangles.cliqueSet_eq_image
lemma cliqueSet_eq_image [NoAccidental t] : (graph t).cliqueSet 3 = toTriangle '' t 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).cliqueSet 3 = ⇑SimpleGraph.TripartiteFromTriangles.toTriangle '' ↑t
-- SimpleGraph.TripartiteFromTriangles.cliqueFinset_eq_image
lemma cliqueFinset_eq_image [NoAccidental t] : (graph t).cliqueFinset 3 = t.image toTriangle 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [Fintype α] [Fintype β] [Fintype γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).cliqueFinset 3 =
  Finset.image (⇑SimpleGraph.TripartiteFromTriangles.toTriangle) t
-- SimpleGraph.TripartiteFromTriangles.cliqueFinset_eq_map
lemma cliqueFinset_eq_map [NoAccidental t] : (graph t).cliqueFinset 3 = t.map toTriangle 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [Fintype α] [Fintype β] [Fintype γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).cliqueFinset 3 =
  Finset.map SimpleGraph.TripartiteFromTriangles.toTriangle t
-- SimpleGraph.TripartiteFromTriangles.card_triangles
@[simp] lemma card_triangles [NoAccidental t] : ((graph t).cliqueFinset 3).card = t.card 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [Fintype α] [Fintype β] [Fintype γ] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: ((SimpleGraph.TripartiteFromTriangles.graph t).cliqueFinset 3).card = t.card
-- SimpleGraph.TripartiteFromTriangles.farFromTriangleFree
lemma farFromTriangleFree [ExplicitDisjoint t] {ε : 𝕜}
    (ht : ε * ((Fintype.card α + Fintype.card β + Fintype.card γ) ^ 2 : ℕ) ≤ t.card) :
    (graph t).FarFromTriangleFree ε 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {𝕜 : Type u_4} [LinearOrderedField 𝕜] (t : Finset (α × β × γ)) [DecidableEq α] [DecidableEq β] [DecidableEq γ] [Fintype α] [Fintype β] [Fintype γ] [SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint t] {ε : 𝕜} (ht : ε * ↑((Fintype.card α + Fintype.card β + Fintype.card γ) ^ 2) ≤ ↑t.card)
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).FarFromTriangleFree ε
-- SimpleGraph.TripartiteFromTriangles.locallyLinear
lemma locallyLinear [ExplicitDisjoint t] [NoAccidental t] : (graph t).LocallyLinear 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} (t : Finset (α × β × γ)) [SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint t] [SimpleGraph.TripartiteFromTriangles.NoAccidental t]
Goal: (SimpleGraph.TripartiteFromTriangles.graph t).LocallyLinear
-- mem_derangements_iff_fixedPoints_eq_empty
theorem mem_derangements_iff_fixedPoints_eq_empty {f : Perm α} :
    f ∈ derangements α ↔ fixedPoints f = ∅ 
Arguments: {α : Type u_1} {f : Equiv.Perm α}
Goal: f ∈ derangements α ↔ Function.fixedPoints ⇑f = ∅
-- derangements.Equiv.RemoveNone.mem_fiber
theorem RemoveNone.mem_fiber (a : Option α) (f : Perm α) :
    f ∈ RemoveNone.fiber a ↔
      ∃ F : Perm (Option α), F ∈ derangements (Option α) ∧ F none = a ∧ removeNone F = f 
Arguments: {α : Type u_1} [DecidableEq α] (a : Option α) (f : Equiv.Perm α)
Goal: f ∈ derangements.Equiv.RemoveNone.fiber a ↔ ∃ F ∈ derangements (Option α), F none = a ∧ Equiv.removeNone F = f
-- derangements.Equiv.RemoveNone.fiber_none
theorem RemoveNone.fiber_none : RemoveNone.fiber (@none α) = ∅ 
Arguments: {α : Type u_1} [DecidableEq α]
Goal: derangements.Equiv.RemoveNone.fiber none = ∅
-- derangements.Equiv.RemoveNone.fiber_some
theorem RemoveNone.fiber_some (a : α) :
    RemoveNone.fiber (some a) = { f : Perm α | fixedPoints f ⊆ {a} } 
Arguments: {α : Type u_1} [DecidableEq α] (a : α)
Goal: derangements.Equiv.RemoveNone.fiber (some a) = {f | Function.fixedPoints ⇑f ⊆ {a}}
-- numDerangements_tendsto_inv_e
theorem numDerangements_tendsto_inv_e :
    Tendsto (fun n => (numDerangements n : ℝ) / n.factorial) atTop (𝓝 (Real.exp (-1))) 
Goal: Filter.Tendsto (fun n => ↑(numDerangements n) / ↑n.factorial) Filter.atTop (𝓝 (Real.exp (-1)))
-- card_derangements_invariant
theorem card_derangements_invariant {α β : Type*} [Fintype α] [DecidableEq α] [Fintype β]
    [DecidableEq β] (h : card α = card β) : card (derangements α) = card (derangements β) 
Arguments: {α : Type u_2} {β : Type u_3} [Fintype α] [DecidableEq α] [Fintype β] [DecidableEq β] (h : Fintype.card α = Fintype.card β)
Goal: Fintype.card ↑(derangements α) = Fintype.card ↑(derangements β)
-- card_derangements_fin_add_two
theorem card_derangements_fin_add_two (n : ℕ) :
    card (derangements (Fin (n + 2))) =
      (n + 1) * card (derangements (Fin n)) + (n + 1) * card (derangements (Fin (n + 1))) 
Arguments: (n : ℕ)
Goal: Fintype.card ↑(derangements (Fin (n + 2))) =
  (n + 1) * Fintype.card ↑(derangements (Fin n)) + (n + 1) * Fintype.card ↑(derangements (Fin (n + 1)))
-- numDerangements_zero
@[simp]
theorem numDerangements_zero : numDerangements 0 = 1 
Goal: numDerangements 0 = 1
-- numDerangements_one
@[simp]
theorem numDerangements_one : numDerangements 1 = 0 
Goal: numDerangements 1 = 0
-- numDerangements_add_two
theorem numDerangements_add_two (n : ℕ) :
    numDerangements (n + 2) = (n + 1) * (numDerangements n + numDerangements (n + 1)) 
Arguments: (n : ℕ)
Goal: numDerangements (n + 2) = (n + 1) * (numDerangements n + numDerangements (n + 1))
-- numDerangements_succ
theorem numDerangements_succ (n : ℕ) :
    (numDerangements (n + 1) : ℤ) = (n + 1) * (numDerangements n : ℤ) - (-1) ^ n 
Arguments: (n : ℕ)
Goal: ↑(numDerangements (n + 1)) = (↑n + 1) * ↑(numDerangements n) - (-1) ^ n
-- card_derangements_fin_eq_numDerangements
theorem card_derangements_fin_eq_numDerangements {n : ℕ} :
    card (derangements (Fin n)) = numDerangements n 
Arguments: {n : ℕ}
Goal: Fintype.card ↑(derangements (Fin n)) = numDerangements n
-- card_derangements_eq_numDerangements
theorem card_derangements_eq_numDerangements (α : Type*) [Fintype α] [DecidableEq α] :
    card (derangements α) = numDerangements (card α) 
Arguments: (α : Type u_2) [Fintype α] [DecidableEq α]
Goal: Fintype.card ↑(derangements α) = numDerangements (Fintype.card α)
-- numDerangements_sum
theorem numDerangements_sum (n : ℕ) :
    (numDerangements n : ℤ) =
      ∑ k ∈ Finset.range (n + 1), (-1 : ℤ) ^ k * Nat.ascFactorial (k + 1) (n - k) 
Arguments: (n : ℕ)
Goal: ↑(numDerangements n) = ∑ k ∈ Finset.range (n + 1), (-1) ^ k * ↑((k + 1).ascFactorial (n - k))
-- Finset.exists_subset_add_sub
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t 
Arguments: {α : Type u_1} [DecidableEq α] [AddCommGroup α] (s : Finset α) {t : Finset α} (ht : t.Nonempty)
Goal: ∃ u, u.card * t.card ≤ (s + t).card ∧ s ⊆ u + t - t
-- Finset.exists_subset_mul_div
@[to_additive "**Ruzsa's covering lemma**"]
theorem exists_subset_mul_div (ht : t.Nonempty) :
    ∃ u : Finset α, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t 
Arguments: {α : Type u_1} [DecidableEq α] [CommGroup α] (s : Finset α) {t : Finset α} (ht : t.Nonempty)
Goal: ∃ u, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t
-- Set.exists_subset_add_sub
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite 
Arguments: {α : Type u_1} [AddCommGroup α] {s t : Set α} (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty)
Goal: ∃ u, Nat.card ↑u * Nat.card ↑t ≤ Nat.card ↑(s + t) ∧ s ⊆ u + t - t ∧ u.Finite
-- Set.exists_subset_mul_div
@[to_additive "**Ruzsa's covering lemma**. Version for sets. For finsets,
see `Finset.exists_subset_add_sub`."]
lemma exists_subset_mul_div (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty) :
    ∃ u : Set α, Nat.card u * Nat.card t ≤ Nat.card (s * t) ∧ s ⊆ u * t / t ∧ u.Finite 
Arguments: {α : Type u_1} [CommGroup α] {s t : Set α} (hs : s.Finite) (ht' : t.Finite) (ht : t.Nonempty)
Goal: ∃ u, Nat.card ↑u * Nat.card ↑t ≤ Nat.card ↑(s * t) ∧ s ⊆ u * t / t ∧ u.Finite
-- SimpleGraph.LocallyLinear.edgeDisjointTriangles
protected lemma LocallyLinear.edgeDisjointTriangles : G.LocallyLinear → G.EdgeDisjointTriangles 
Arguments: {α : Type u_1} {G : SimpleGraph α}
Goal: G.LocallyLinear → G.EdgeDisjointTriangles
-- SimpleGraph.EdgeDisjointTriangles.mono
nonrec lemma EdgeDisjointTriangles.mono (h : G ≤ H) (hH : H.EdgeDisjointTriangles) :
    G.EdgeDisjointTriangles 
Arguments: {α : Type u_1} {G H : SimpleGraph α} (h : G ≤ H) (hH : H.EdgeDisjointTriangles)
Goal: G.EdgeDisjointTriangles
-- SimpleGraph.edgeDisjointTriangles_bot
@[simp] lemma edgeDisjointTriangles_bot : (⊥ : SimpleGraph α).EdgeDisjointTriangles 
Arguments: {α : Type u_1}
Goal: ⊥.EdgeDisjointTriangles
-- SimpleGraph.locallyLinear_bot
@[simp] lemma locallyLinear_bot : (⊥ : SimpleGraph α).LocallyLinear 
Arguments: {α : Type u_1}
Goal: ⊥.LocallyLinear
-- SimpleGraph.EdgeDisjointTriangles.map
lemma EdgeDisjointTriangles.map (f : α ↪ β) (hG : G.EdgeDisjointTriangles) :
    (G.map f).EdgeDisjointTriangles 
Arguments: {α : Type u_1} {β : Type u_2} {G : SimpleGraph α} (f : α ↪ β) (hG : G.EdgeDisjointTriangles)
Goal: (SimpleGraph.map f G).EdgeDisjointTriangles
-- SimpleGraph.LocallyLinear.map
lemma LocallyLinear.map (f : α ↪ β) (hG : G.LocallyLinear) : (G.map f).LocallyLinear 
Arguments: {α : Type u_1} {β : Type u_2} {G : SimpleGraph α} (f : α ↪ β) (hG : G.LocallyLinear)
Goal: (SimpleGraph.map f G).LocallyLinear
-- SimpleGraph.locallyLinear_comap
@[simp] lemma locallyLinear_comap {G : SimpleGraph β} {e : α ≃ β} :
    (G.comap e).LocallyLinear ↔ G.LocallyLinear 
Arguments: {α : Type u_1} {β : Type u_2} {G : SimpleGraph β} {e : α ≃ β}
Goal: (SimpleGraph.comap (⇑e) G).LocallyLinear ↔ G.LocallyLinear
-- SimpleGraph.edgeDisjointTriangles_iff_mem_sym2_subsingleton
lemma edgeDisjointTriangles_iff_mem_sym2_subsingleton :
    G.EdgeDisjointTriangles ↔
      ∀ ⦃e : Sym2 α⦄, ¬ e.IsDiag → {s ∈ G.cliqueSet 3 | e ∈ (s : Finset α).sym2}.Subsingleton 
Arguments: {α : Type u_1} {G : SimpleGraph α}
Goal: G.EdgeDisjointTriangles ↔ ∀ ⦃e : Sym2 α⦄, ¬e.IsDiag → {s | s ∈ G.cliqueSet 3 ∧ e ∈ s.sym2}.Subsingleton
-- SimpleGraph.EdgeDisjointTriangles.card_edgeFinset_le
lemma EdgeDisjointTriangles.card_edgeFinset_le (hG : G.EdgeDisjointTriangles) :
    3 * (G.cliqueFinset 3).card ≤ G.edgeFinset.card 
Arguments: {α : Type u_1} {G : SimpleGraph α} [DecidableEq α] [Fintype α] [DecidableRel G.Adj] (hG : G.EdgeDisjointTriangles)
Goal: 3 * (G.cliqueFinset 3).card ≤ G.edgeFinset.card
-- SimpleGraph.LocallyLinear.card_edgeFinset
lemma LocallyLinear.card_edgeFinset (hG : G.LocallyLinear) :
    G.edgeFinset.card = 3 * (G.cliqueFinset 3).card 
Arguments: {α : Type u_1} {G : SimpleGraph α} [DecidableEq α] [Fintype α] [DecidableRel G.Adj] (hG : G.LocallyLinear)
Goal: G.edgeFinset.card = 3 * (G.cliqueFinset 3).card
-- SimpleGraph.farFromTriangleFree_iff
theorem farFromTriangleFree_iff :
    G.FarFromTriangleFree ε ↔ ∀ ⦃H : SimpleGraph α⦄, [DecidableRel H.Adj] → H ≤ G → H.CliqueFree 3 →
      ε * (card α ^ 2 : ℕ) ≤ G.edgeFinset.card - H.edgeFinset.card 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj]
Goal: G.FarFromTriangleFree ε ↔
  ∀ ⦃H : SimpleGraph α⦄ [inst : DecidableRel H.Adj],
    H ≤ G → H.CliqueFree 3 → ε * ↑(Fintype.card α ^ 2) ≤ ↑G.edgeFinset.card - ↑H.edgeFinset.card
-- SimpleGraph.FarFromTriangleFree.mono
nonrec theorem FarFromTriangleFree.mono (hε : G.FarFromTriangleFree ε) (h : δ ≤ ε) :
    G.FarFromTriangleFree δ 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε δ : 𝕜} [Fintype α] [DecidableRel G.Adj] (hε : G.FarFromTriangleFree ε) (h : δ ≤ ε)
Goal: G.FarFromTriangleFree δ
-- SimpleGraph.FarFromTriangleFree.cliqueFinset_nonempty'
theorem FarFromTriangleFree.cliqueFinset_nonempty' (hH : H ≤ G) (hG : G.FarFromTriangleFree ε)
    (hcard : G.edgeFinset.card - H.edgeFinset.card < ε * (card α ^ 2 : ℕ)) :
    (H.cliqueFinset 3).Nonempty 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G H : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [DecidableRel H.Adj] [DecidableEq α] (hH : H ≤ G) (hG : G.FarFromTriangleFree ε) (hcard : ↑G.edgeFinset.card - ↑H.edgeFinset.card < ε * ↑(Fintype.card α ^ 2))
Goal: (H.cliqueFinset 3).Nonempty
-- _private.Mathlib.Combinatorics.SimpleGraph.Triangle.Basic.0.SimpleGraph.farFromTriangleFree_of_disjoint_triangles_aux
private lemma farFromTriangleFree_of_disjoint_triangles_aux {tris : Finset (Finset α)}
    (htris : tris ⊆ G.cliqueFinset 3)
    (pd : (tris : Set (Finset α)).Pairwise fun x y ↦ (x ∩ y : Set α).Subsingleton) (hHG : H ≤ G)
    (hH : H.CliqueFree 3) : tris.card ≤ G.edgeFinset.card - H.edgeFinset.card 
Arguments: {α : Type u_1} {G H : SimpleGraph α} [Fintype α] [DecidableRel G.Adj] [DecidableRel H.Adj] [DecidableEq α] {tris : Finset (Finset α)} (htris : tris ⊆ G.cliqueFinset 3) (pd : (↑tris).Pairwise fun x y => (↑x ∩ ↑y).Subsingleton) (hHG : H ≤ G) (hH : H.CliqueFree 3)
Goal: tris.card ≤ G.edgeFinset.card - H.edgeFinset.card
-- SimpleGraph.farFromTriangleFree_of_disjoint_triangles
lemma farFromTriangleFree_of_disjoint_triangles (tris : Finset (Finset α))
    (htris : tris ⊆ G.cliqueFinset 3)
    (pd : (tris : Set (Finset α)).Pairwise fun x y ↦ (x ∩ y : Set α).Subsingleton)
    (tris_big : ε * (card α ^ 2 : ℕ) ≤ tris.card) :
    G.FarFromTriangleFree ε 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [DecidableEq α] (tris : Finset (Finset α)) (htris : tris ⊆ G.cliqueFinset 3) (pd : (↑tris).Pairwise fun x y => (↑x ∩ ↑y).Subsingleton) (tris_big : ε * ↑(Fintype.card α ^ 2) ≤ ↑tris.card)
Goal: G.FarFromTriangleFree ε
-- SimpleGraph.EdgeDisjointTriangles.farFromTriangleFree
protected lemma EdgeDisjointTriangles.farFromTriangleFree (hG : G.EdgeDisjointTriangles)
    (tris_big : ε * (card α ^ 2 : ℕ) ≤ (G.cliqueFinset 3).card) :
    G.FarFromTriangleFree ε 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [DecidableEq α] (hG : G.EdgeDisjointTriangles) (tris_big : ε * ↑(Fintype.card α ^ 2) ≤ ↑(G.cliqueFinset 3).card)
Goal: G.FarFromTriangleFree ε
-- SimpleGraph.FarFromTriangleFree.lt_half
lemma FarFromTriangleFree.lt_half (hG : G.FarFromTriangleFree ε) : ε < 2⁻¹ 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] (hG : G.FarFromTriangleFree ε)
Goal: ε < 2⁻¹
-- SimpleGraph.FarFromTriangleFree.lt_one
lemma FarFromTriangleFree.lt_one (hG : G.FarFromTriangleFree ε) : ε < 1 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] (hG : G.FarFromTriangleFree ε)
Goal: ε < 1
-- SimpleGraph.FarFromTriangleFree.nonpos
theorem FarFromTriangleFree.nonpos (h₀ : G.FarFromTriangleFree ε) (h₁ : G.CliqueFree 3) :
    ε ≤ 0 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] (h₀ : G.FarFromTriangleFree ε) (h₁ : G.CliqueFree 3)
Goal: ε ≤ 0
-- SimpleGraph.CliqueFree.not_farFromTriangleFree
theorem CliqueFree.not_farFromTriangleFree (hG : G.CliqueFree 3) (hε : 0 < ε) :
    ¬G.FarFromTriangleFree ε 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] (hG : G.CliqueFree 3) (hε : 0 < ε)
Goal: ¬G.FarFromTriangleFree ε
-- SimpleGraph.FarFromTriangleFree.not_cliqueFree
theorem FarFromTriangleFree.not_cliqueFree (hG : G.FarFromTriangleFree ε) (hε : 0 < ε) :
    ¬G.CliqueFree 3 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] (hG : G.FarFromTriangleFree ε) (hε : 0 < ε)
Goal: ¬G.CliqueFree 3
-- SimpleGraph.FarFromTriangleFree.cliqueFinset_nonempty
theorem FarFromTriangleFree.cliqueFinset_nonempty [DecidableEq α]
    (hG : G.FarFromTriangleFree ε) (hε : 0 < ε) : (G.cliqueFinset 3).Nonempty 
Arguments: {α : Type u_1} {𝕜 : Type u_3} [LinearOrderedField 𝕜] {G : SimpleGraph α} {ε : 𝕜} [Fintype α] [DecidableRel G.Adj] [Nonempty α] [DecidableEq α] (hG : G.FarFromTriangleFree ε) (hε : 0 < ε)
Goal: (G.cliqueFinset 3).Nonempty
-- Combinatorics.Subspace.proper
@[ext]
structure Subspace (η α ι : Type*) where
    idxFun : ι → α ⊕ η
    proper : ∀ e, ∃ i, idxFun i = Sum.inr e
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} (self : Combinatorics.Subspace η α ι) (e : η)
Goal: ∃ i, self.idxFun i = Sum.inr e
-- Combinatorics.Subspace.coe_apply
lemma coe_apply (l : Subspace η α ι) (x : η → α) (i : ι) : l x i = (l.idxFun i).elim id x 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} (l : Combinatorics.Subspace η α ι) (x : η → α) (i : ι)
Goal: ↑l x i = Sum.elim id x (l.idxFun i)
-- Combinatorics.Subspace.coe_injective
lemma coe_injective [Nontrivial α] : Injective ((⇑) : Subspace η α ι → (η → α) → ι → α) 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} [Nontrivial α]
Goal: Function.Injective Combinatorics.Subspace.toFun
-- Combinatorics.Subspace.apply_def
lemma apply_def (l : Subspace η α ι) (x : η → α) (i : ι) : l x i = (l.idxFun i).elim id x 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} (l : Combinatorics.Subspace η α ι) (x : η → α) (i : ι)
Goal: ↑l x i = Sum.elim id x (l.idxFun i)
-- Combinatorics.Subspace.apply_inl
lemma apply_inl (h : l.idxFun i = Sum.inl a) : l x i = a 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} {l : Combinatorics.Subspace η α ι} {x : η → α} {i : ι} {a : α} (h : l.idxFun i = Sum.inl a)
Goal: ↑l x i = a
-- Combinatorics.Subspace.apply_inr
lemma apply_inr (h : l.idxFun i = Sum.inr e) : l x i = x e 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} {l : Combinatorics.Subspace η α ι} {x : η → α} {i : ι} {e : η} (h : l.idxFun i = Sum.inr e)
Goal: ↑l x i = x e
-- Combinatorics.Subspace.reindex_apply
@[simp] lemma reindex_apply (l : Subspace η α ι) (eη : η ≃ η') (eα : α ≃ α') (eι : ι ≃ ι') (x i) :
    l.reindex eη eα eι x i = eα (l (eα.symm ∘ x ∘ eη) <| eι.symm i) 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} {η' : Type u_9} {α' : Type u_10} {ι' : Type u_11} (l : Combinatorics.Subspace η α ι) (eη : η ≃ η') (eα : α ≃ α') (eι : ι ≃ ι') (x : η' → α') (i : ι')
Goal: ↑(l.reindex eη eα eι) x i = eα (↑l (⇑eα.symm ∘ x ∘ ⇑eη) (eι.symm i))
-- Combinatorics.Subspace.reindex_isMono
@[simp] lemma reindex_isMono {eη : η ≃ η'} {eα : α ≃ α'} {eι : ι ≃ ι'} {C : (ι' → α') → κ} :
    (l.reindex eη eα eι).IsMono C ↔ l.IsMono fun x ↦ C <| eα ∘ x ∘ eι.symm 
Arguments: {η : Type u_5} {α : Type u_6} {ι : Type u_7} {κ : Type u_8} {l : Combinatorics.Subspace η α ι} {η' : Type u_9} {α' : Type u_10} {ι' : Type u_11} {eη : η ≃ η'} {eα : α ≃ α'} {eι : ι ≃ ι'} {C : (ι' → α') → κ}
Goal: Combinatorics.Subspace.IsMono C (l.reindex eη eα eι) ↔ Combinatorics.Subspace.IsMono (fun x => C (⇑eα ∘ x ∘ ⇑eι.symm)) l