ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Rel.swap_mem_interedges_iff
@[simp]
theorem swap_mem_interedges_iff (hr : Symmetric r) {x : α × α} :
    x.swap ∈ interedges r s t ↔ x ∈ interedges r t s 
Arguments: {α : Type u_4} {r : α → α → Prop} [DecidableRel r] {s t : Finset α} (hr : Symmetric r) {x : α × α}
Goal: x.swap ∈ Rel.interedges r s t ↔ x ∈ Rel.interedges r t s
-- Rel.mk_mem_interedges_comm
theorem mk_mem_interedges_comm : (a, b) ∈ interedges r s t ↔ (b, a) ∈ interedges r t s 
Arguments: {α : Type u_4} {r : α → α → Prop} [DecidableRel r] {s t : Finset α} {a b : α} (hr : Symmetric r)
Goal: (a, b) ∈ Rel.interedges r s t ↔ (b, a) ∈ Rel.interedges r t s
-- Rel.card_interedges_comm
theorem card_interedges_comm (s t : Finset α) : (interedges r s t).card = (interedges r t s).card 
Arguments: {α : Type u_4} {r : α → α → Prop} [DecidableRel r] (hr : Symmetric r) (s t : Finset α)
Goal: (Rel.interedges r s t).card = (Rel.interedges r t s).card
-- Rel.edgeDensity_comm
theorem edgeDensity_comm (hr : Symmetric r) (s t : Finset α) :
    edgeDensity r s t = edgeDensity r t s 
Arguments: {α : Type u_4} {r : α → α → Prop} [DecidableRel r] (hr : Symmetric r) (s t : Finset α)
Goal: Rel.edgeDensity r s t = Rel.edgeDensity r t s
-- SimpleGraph.interedges_def
theorem interedges_def (s t : Finset α) :
    G.interedges s t = (s ×ˢ t).filter fun e ↦ G.Adj e.1 e.2 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: G.interedges s t = Finset.filter (fun e => G.Adj e.1 e.2) (s ×ˢ t)
-- SimpleGraph.edgeDensity_def
theorem edgeDensity_def (s t : Finset α) :
    G.edgeDensity s t = (G.interedges s t).card / (s.card * t.card) 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: G.edgeDensity s t = ↑(G.interedges s t).card / (↑s.card * ↑t.card)
-- SimpleGraph.card_interedges_div_card
theorem card_interedges_div_card (s t : Finset α) :
    ((G.interedges s t).card : ℚ) / (s.card * t.card) = G.edgeDensity s t 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: ↑(G.interedges s t).card / (↑s.card * ↑t.card) = G.edgeDensity s t
-- SimpleGraph.mem_interedges_iff
theorem mem_interedges_iff {x : α × α} : x ∈ G.interedges s t ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ G.Adj x.1 x.2 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} {x : α × α}
Goal: x ∈ G.interedges s t ↔ x.1 ∈ s ∧ x.2 ∈ t ∧ G.Adj x.1 x.2
-- SimpleGraph.mk_mem_interedges_iff
theorem mk_mem_interedges_iff : (a, b) ∈ G.interedges s t ↔ a ∈ s ∧ b ∈ t ∧ G.Adj a b 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} {a b : α}
Goal: (a, b) ∈ G.interedges s t ↔ a ∈ s ∧ b ∈ t ∧ G.Adj a b
-- SimpleGraph.interedges_empty_left
@[simp]
theorem interedges_empty_left (t : Finset α) : G.interedges ∅ t = ∅ 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (t : Finset α)
Goal: G.interedges ∅ t = ∅
-- SimpleGraph.interedges_mono
theorem interedges_mono : s₂ ⊆ s₁ → t₂ ⊆ t₁ → G.interedges s₂ t₂ ⊆ G.interedges s₁ t₁ 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s₁ s₂ t₁ t₂ : Finset α}
Goal: s₂ ⊆ s₁ → t₂ ⊆ t₁ → G.interedges s₂ t₂ ⊆ G.interedges s₁ t₁
-- SimpleGraph.interedges_disjoint_left
theorem interedges_disjoint_left (hs : Disjoint s₁ s₂) (t : Finset α) :
    Disjoint (G.interedges s₁ t) (G.interedges s₂ t) 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s₁ s₂ : Finset α} (hs : Disjoint s₁ s₂) (t : Finset α)
Goal: Disjoint (G.interedges s₁ t) (G.interedges s₂ t)
-- SimpleGraph.interedges_disjoint_right
theorem interedges_disjoint_right (s : Finset α) (ht : Disjoint t₁ t₂) :
    Disjoint (G.interedges s t₁) (G.interedges s t₂) 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {t₁ t₂ : Finset α} (s : Finset α) (ht : Disjoint t₁ t₂)
Goal: Disjoint (G.interedges s t₁) (G.interedges s t₂)
-- SimpleGraph.interedges_biUnion_left
theorem interedges_biUnion_left (s : Finset ι) (t : Finset α) (f : ι → Finset α) :
    G.interedges (s.biUnion f) t = s.biUnion fun a ↦ G.interedges (f a) t 
Arguments: {ι : Type u_2} {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] [DecidableEq α] (s : Finset ι) (t : Finset α) (f : ι → Finset α)
Goal: G.interedges (s.biUnion f) t = s.biUnion fun a => G.interedges (f a) t
-- SimpleGraph.interedges_biUnion_right
theorem interedges_biUnion_right (s : Finset α) (t : Finset ι) (f : ι → Finset α) :
    G.interedges s (t.biUnion f) = t.biUnion fun b ↦ G.interedges s (f b) 
Arguments: {ι : Type u_2} {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] [DecidableEq α] (s : Finset α) (t : Finset ι) (f : ι → Finset α)
Goal: G.interedges s (t.biUnion f) = t.biUnion fun b => G.interedges s (f b)
-- SimpleGraph.interedges_biUnion
theorem interedges_biUnion (s : Finset ι) (t : Finset κ) (f : ι → Finset α) (g : κ → Finset α) :
    G.interedges (s.biUnion f) (t.biUnion g) =
      (s ×ˢ t).biUnion fun ab ↦ G.interedges (f ab.1) (g ab.2) 
Arguments: {ι : Type u_2} {κ : Type u_3} {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] [DecidableEq α] (s : Finset ι) (t : Finset κ) (f : ι → Finset α) (g : κ → Finset α)
Goal: G.interedges (s.biUnion f) (t.biUnion g) = (s ×ˢ t).biUnion fun ab => G.interedges (f ab.1) (g ab.2)
-- SimpleGraph.card_interedges_add_card_interedges_compl
theorem card_interedges_add_card_interedges_compl (h : Disjoint s t) :
    (G.interedges s t).card + (Gᶜ.interedges s t).card = s.card * t.card 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} [DecidableEq α] (h : Disjoint s t)
Goal: (G.interedges s t).card + (Gᶜ.interedges s t).card = s.card * t.card
-- SimpleGraph.edgeDensity_add_edgeDensity_compl
theorem edgeDensity_add_edgeDensity_compl (hs : s.Nonempty) (ht : t.Nonempty) (h : Disjoint s t) :
    G.edgeDensity s t + Gᶜ.edgeDensity s t = 1 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} [DecidableEq α] (hs : s.Nonempty) (ht : t.Nonempty) (h : Disjoint s t)
Goal: G.edgeDensity s t + Gᶜ.edgeDensity s t = 1
-- SimpleGraph.card_interedges_le_mul
theorem card_interedges_le_mul (s t : Finset α) : (G.interedges s t).card ≤ s.card * t.card 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: (G.interedges s t).card ≤ s.card * t.card
-- SimpleGraph.edgeDensity_nonneg
theorem edgeDensity_nonneg (s t : Finset α) : 0 ≤ G.edgeDensity s t 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: 0 ≤ G.edgeDensity s t
-- SimpleGraph.edgeDensity_le_one
theorem edgeDensity_le_one (s t : Finset α) : G.edgeDensity s t ≤ 1 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: G.edgeDensity s t ≤ 1
-- SimpleGraph.edgeDensity_empty_left
@[simp]
theorem edgeDensity_empty_left (t : Finset α) : G.edgeDensity ∅ t = 0 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (t : Finset α)
Goal: G.edgeDensity ∅ t = 0
-- SimpleGraph.edgeDensity_empty_right
@[simp]
theorem edgeDensity_empty_right (s : Finset α) : G.edgeDensity s ∅ = 0 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s : Finset α)
Goal: G.edgeDensity s ∅ = 0
-- SimpleGraph.swap_mem_interedges_iff
@[simp]
theorem swap_mem_interedges_iff {x : α × α} : x.swap ∈ G.interedges s t ↔ x ∈ G.interedges t s 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} {x : α × α}
Goal: x.swap ∈ G.interedges s t ↔ x ∈ G.interedges t s
-- SimpleGraph.mk_mem_interedges_comm
theorem mk_mem_interedges_comm : (a, b) ∈ G.interedges s t ↔ (b, a) ∈ G.interedges t s 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α} {a b : α}
Goal: (a, b) ∈ G.interedges s t ↔ (b, a) ∈ G.interedges t s
-- SimpleGraph.edgeDensity_comm
theorem edgeDensity_comm (s t : Finset α) : G.edgeDensity s t = G.edgeDensity t s 
Arguments: {α : Type u_4} (G : SimpleGraph α) [DecidableRel G.Adj] (s t : Finset α)
Goal: G.edgeDensity s t = G.edgeDensity t s
-- SimpleGraph.IsUniform.mono
theorem IsUniform.mono {ε' : 𝕜} (h : ε ≤ ε') (hε : IsUniform G ε s t) : IsUniform G ε' s t 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} {ε' : 𝕜} (h : ε ≤ ε') (hε : G.IsUniform ε s t)
Goal: G.IsUniform ε' s t
-- SimpleGraph.IsUniform.symm
theorem IsUniform.symm : Symmetric (IsUniform G ε) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜}
Goal: Symmetric (G.IsUniform ε)
-- SimpleGraph.isUniform_comm
theorem isUniform_comm : IsUniform G ε s t ↔ IsUniform G ε t s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α}
Goal: G.IsUniform ε s t ↔ G.IsUniform ε t s
-- SimpleGraph.isUniform_one
lemma isUniform_one : G.IsUniform (1 : 𝕜) s t 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {s t : Finset α}
Goal: G.IsUniform 1 s t
-- SimpleGraph.IsUniform.pos
lemma IsUniform.pos (hG : G.IsUniform ε s t) : 0 < ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (hG : G.IsUniform ε s t)
Goal: 0 < ε
-- SimpleGraph.isUniform_singleton
@[simp] lemma isUniform_singleton : G.IsUniform ε {a} {b} ↔ 0 < ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} {a b : α}
Goal: G.IsUniform ε {a} {b} ↔ 0 < ε
-- SimpleGraph.not_isUniform_zero
theorem not_isUniform_zero : ¬G.IsUniform (0 : 𝕜) s t 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {s t : Finset α}
Goal: ¬G.IsUniform 0 s t
-- SimpleGraph.not_isUniform_iff
theorem not_isUniform_iff :
    ¬G.IsUniform ε s t ↔ ∃ s', s' ⊆ s ∧ ∃ t', t' ⊆ t ∧ ↑s.card * ε ≤ s'.card ∧
      ↑t.card * ε ≤ t'.card ∧ ε ≤ |G.edgeDensity s' t' - G.edgeDensity s t| 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α}
Goal: ¬G.IsUniform ε s t ↔
  ∃ s' ⊆ s, ∃ t' ⊆ t, ↑s.card * ε ≤ ↑s'.card ∧ ↑t.card * ε ≤ ↑t'.card ∧ ε ≤ ↑|G.edgeDensity s' t' - G.edgeDensity s t|
-- SimpleGraph.left_nonuniformWitnesses_subset
theorem left_nonuniformWitnesses_subset (h : ¬G.IsUniform ε s t) :
    (G.nonuniformWitnesses ε s t).1 ⊆ s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: (G.nonuniformWitnesses ε s t).1 ⊆ s
-- SimpleGraph.left_nonuniformWitnesses_card
theorem left_nonuniformWitnesses_card (h : ¬G.IsUniform ε s t) :
    (s.card : 𝕜) * ε ≤ (G.nonuniformWitnesses ε s t).1.card 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: ↑s.card * ε ≤ ↑(G.nonuniformWitnesses ε s t).1.card
-- SimpleGraph.right_nonuniformWitnesses_subset
theorem right_nonuniformWitnesses_subset (h : ¬G.IsUniform ε s t) :
    (G.nonuniformWitnesses ε s t).2 ⊆ t 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: (G.nonuniformWitnesses ε s t).2 ⊆ t
-- SimpleGraph.right_nonuniformWitnesses_card
theorem right_nonuniformWitnesses_card (h : ¬G.IsUniform ε s t) :
    (t.card : 𝕜) * ε ≤ (G.nonuniformWitnesses ε s t).2.card 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: ↑t.card * ε ≤ ↑(G.nonuniformWitnesses ε s t).2.card
-- SimpleGraph.nonuniformWitnesses_spec
theorem nonuniformWitnesses_spec (h : ¬G.IsUniform ε s t) :
    ε ≤
      |G.edgeDensity (G.nonuniformWitnesses ε s t).1 (G.nonuniformWitnesses ε s t).2 -
          G.edgeDensity s t| 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: ε ≤ ↑|G.edgeDensity (G.nonuniformWitnesses ε s t).1 (G.nonuniformWitnesses ε s t).2 - G.edgeDensity s t|
-- SimpleGraph.nonuniformWitness_subset
theorem nonuniformWitness_subset (h : ¬G.IsUniform ε s t) : G.nonuniformWitness ε s t ⊆ s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: G.nonuniformWitness ε s t ⊆ s
-- SimpleGraph.le_card_nonuniformWitness
theorem le_card_nonuniformWitness (h : ¬G.IsUniform ε s t) :
    (s.card : 𝕜) * ε ≤ (G.nonuniformWitness ε s t).card 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t)
Goal: ↑s.card * ε ≤ ↑(G.nonuniformWitness ε s t).card
-- SimpleGraph.nonuniformWitness_spec
theorem nonuniformWitness_spec (h₁ : s ≠ t) (h₂ : ¬G.IsUniform ε s t) : ε ≤ |G.edgeDensity
    (G.nonuniformWitness ε s t) (G.nonuniformWitness ε t s) - G.edgeDensity s t| 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h₁ : s ≠ t) (h₂ : ¬G.IsUniform ε s t)
Goal: ε ≤ ↑|G.edgeDensity (G.nonuniformWitness ε s t) (G.nonuniformWitness ε t s) - G.edgeDensity s t|
-- Finpartition.mk_mem_sparsePairs
@[simp]
lemma mk_mem_sparsePairs (u v : Finset α) (ε : 𝕜) :
    (u, v) ∈ P.sparsePairs G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ G.edgeDensity u v < ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj] (u v : Finset α) (ε : 𝕜)
Goal: (u, v) ∈ P.sparsePairs G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ ↑(G.edgeDensity u v) < ε
-- Finpartition.sparsePairs_mono
lemma sparsePairs_mono {ε ε' : 𝕜} (h : ε ≤ ε') : P.sparsePairs G ε ⊆ P.sparsePairs G ε' 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj] {ε ε' : 𝕜} (h : ε ≤ ε')
Goal: P.sparsePairs G ε ⊆ P.sparsePairs G ε'
-- Finpartition.mk_mem_nonUniforms
@[simp] lemma mk_mem_nonUniforms :
    (u, v) ∈ P.nonUniforms G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ ¬G.IsUniform ε u v 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} {u v : Finset α}
Goal: (u, v) ∈ P.nonUniforms G ε ↔ u ∈ P.parts ∧ v ∈ P.parts ∧ u ≠ v ∧ ¬G.IsUniform ε u v
-- Finpartition.nonUniforms_mono
theorem nonUniforms_mono {ε ε' : 𝕜} (h : ε ≤ ε') : P.nonUniforms G ε' ⊆ P.nonUniforms G ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj] {ε ε' : 𝕜} (h : ε ≤ ε')
Goal: P.nonUniforms G ε' ⊆ P.nonUniforms G ε
-- Finpartition.nonUniforms_bot
theorem nonUniforms_bot (hε : 0 < ε) : (⊥ : Finpartition A).nonUniforms G ε = ∅ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} (hε : 0 < ε)
Goal: ⊥.nonUniforms G ε = ∅
-- Finpartition.bot_isUniform
lemma bot_isUniform (hε : 0 < ε) : (⊥ : Finpartition A).IsUniform G ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (G : SimpleGraph α) [DecidableRel G.Adj] {ε : 𝕜} (hε : 0 < ε)
Goal: ⊥.IsUniform G ε
-- Finpartition.isUniform_one
lemma isUniform_one : P.IsUniform G (1 : 𝕜) 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj]
Goal: P.IsUniform G 1
-- Finpartition.IsUniform.mono
theorem IsUniform.mono {ε ε' : 𝕜} (hP : P.IsUniform G ε) (h : ε ≤ ε') : P.IsUniform G ε' 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε ε' : 𝕜} (hP : P.IsUniform G ε) (h : ε ≤ ε')
Goal: P.IsUniform G ε'
-- Finpartition.isUniformOfEmpty
theorem isUniformOfEmpty (hP : P.parts = ∅) : P.IsUniform G ε 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (hP : P.parts = ∅)
Goal: P.IsUniform G ε
-- Finpartition.nonempty_of_not_uniform
theorem nonempty_of_not_uniform (h : ¬P.IsUniform G ε) : P.parts.Nonempty 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (h : ¬P.IsUniform G ε)
Goal: P.parts.Nonempty
-- Finpartition.nonuniformWitness_mem_nonuniformWitnesses
theorem nonuniformWitness_mem_nonuniformWitnesses (h : ¬G.IsUniform ε s t) (ht : t ∈ P.parts)
    (hst : s ≠ t) : G.nonuniformWitness ε s t ∈ P.nonuniformWitnesses G ε s 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} {s t : Finset α} (h : ¬G.IsUniform ε s t) (ht : t ∈ P.parts) (hst : s ≠ t)
Goal: G.nonuniformWitness ε s t ∈ P.nonuniformWitnesses G ε s
-- Finpartition.IsEquipartition.card_interedges_sparsePairs_le'
lemma IsEquipartition.card_interedges_sparsePairs_le' (hP : P.IsEquipartition)
    (hε : 0 ≤ ε) :
    ((P.sparsePairs G ε).biUnion fun (U, V) ↦ G.interedges U V).card ≤
      ε * (A.card + P.parts.card) ^ 2 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (hP : P.IsEquipartition) (hε : 0 ≤ ε)
Goal: ↑((P.sparsePairs G ε).biUnion fun x =>
        match x with
        | (U, V) => G.interedges U V).card ≤
  ε * (↑A.card + ↑P.parts.card) ^ 2
-- Finpartition.IsEquipartition.card_interedges_sparsePairs_le
lemma IsEquipartition.card_interedges_sparsePairs_le (hP : P.IsEquipartition) (hε : 0 ≤ ε) :
    ((P.sparsePairs G ε).biUnion fun (U, V) ↦ G.interedges U V).card ≤ 4 * ε * A.card ^ 2 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (hP : P.IsEquipartition) (hε : 0 ≤ ε)
Goal: ↑((P.sparsePairs G ε).biUnion fun x =>
        match x with
        | (U, V) => G.interedges U V).card ≤
  4 * ε * ↑A.card ^ 2
-- _private.Mathlib.Combinatorics.SimpleGraph.Regularity.Uniform.0.Finpartition.aux
private lemma aux {i j : ℕ} (hj : 0 < j) : j * (j - 1) * (i / j + 1) ^ 2 < (i + j) ^ 2 
Arguments: {i j : ℕ} (hj : 0 < j)
Goal: j * (j - 1) * (i / j + 1) ^ 2 < (i + j) ^ 2
-- Finpartition.IsEquipartition.card_biUnion_offDiag_le'
lemma IsEquipartition.card_biUnion_offDiag_le' (hP : P.IsEquipartition) :
    ((P.parts.biUnion offDiag).card : 𝕜) ≤ A.card * (A.card + P.parts.card) / P.parts.card 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} (hP : P.IsEquipartition)
Goal: ↑(P.parts.biUnion Finset.offDiag).card ≤ ↑A.card * (↑A.card + ↑P.parts.card) / ↑P.parts.card
-- Finpartition.IsEquipartition.card_biUnion_offDiag_le
lemma IsEquipartition.card_biUnion_offDiag_le (hε : 0 < ε) (hP : P.IsEquipartition)
    (hP' : 4 / ε ≤ P.parts.card) : (P.parts.biUnion offDiag).card ≤ ε / 2 * A.card ^ 2 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {ε : 𝕜} (hε : 0 < ε) (hP : P.IsEquipartition) (hP' : 4 / ε ≤ ↑P.parts.card)
Goal: ↑(P.parts.biUnion Finset.offDiag).card ≤ ε / 2 * ↑A.card ^ 2
-- Finpartition.IsEquipartition.sum_nonUniforms_lt'
lemma IsEquipartition.sum_nonUniforms_lt' (hA : A.Nonempty) (hε : 0 < ε) (hP : P.IsEquipartition)
    (hG : P.IsUniform G ε) :
    ∑ i ∈ P.nonUniforms G ε, (i.1.card * i.2.card : 𝕜) < ε * (A.card + P.parts.card) ^ 2 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (hA : A.Nonempty) (hε : 0 < ε) (hP : P.IsEquipartition) (hG : P.IsUniform G ε)
Goal: ∑ i ∈ P.nonUniforms G ε, ↑i.1.card * ↑i.2.card < ε * (↑A.card + ↑P.parts.card) ^ 2
-- Finpartition.IsEquipartition.sum_nonUniforms_lt
lemma IsEquipartition.sum_nonUniforms_lt (hA : A.Nonempty) (hε : 0 < ε) (hP : P.IsEquipartition)
    (hG : P.IsUniform G ε) :
    ((P.nonUniforms G ε).biUnion fun (U, V) ↦ U ×ˢ V).card < 4 * ε * A.card ^ 2 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜} (hA : A.Nonempty) (hε : 0 < ε) (hP : P.IsEquipartition) (hG : P.IsUniform G ε)
Goal: ↑((P.nonUniforms G ε).biUnion fun x =>
        match x with
        | (U, V) => U ×ˢ V).card <
  4 * ε * ↑A.card ^ 2
-- SimpleGraph.regularityReduced_adj
@[simps] def regularityReduced (ε δ : 𝕜) : SimpleGraph α 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} (P : Finpartition A) (G : SimpleGraph α) [DecidableRel G.Adj] (ε δ : 𝕜) (a b : α)
Goal: (SimpleGraph.regularityReduced P G ε δ).Adj a b =
  (G.Adj a b ∧ ∃ U ∈ P.parts, ∃ V ∈ P.parts, a ∈ U ∧ b ∈ V ∧ U ≠ V ∧ G.IsUniform ε U V ∧ δ ≤ ↑(G.edgeDensity U V))
-- SimpleGraph.regularityReduced_le
lemma regularityReduced_le : G.regularityReduced P ε δ ≤ G 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε δ : 𝕜}
Goal: SimpleGraph.regularityReduced P G ε δ ≤ G
-- SimpleGraph.regularityReduced_mono
lemma regularityReduced_mono {ε₁ ε₂ : 𝕜} (hε : ε₁ ≤ ε₂) :
    G.regularityReduced P ε₁ δ ≤ G.regularityReduced P ε₂ δ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {δ ε₁ ε₂ : 𝕜} (hε : ε₁ ≤ ε₂)
Goal: SimpleGraph.regularityReduced P G ε₁ δ ≤ SimpleGraph.regularityReduced P G ε₂ δ
-- SimpleGraph.regularityReduced_anti
lemma regularityReduced_anti {δ₁ δ₂ : 𝕜} (hδ : δ₁ ≤ δ₂) :
    G.regularityReduced P ε δ₂ ≤ G.regularityReduced P ε δ₁ 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε δ₁ δ₂ : 𝕜} (hδ : δ₁ ≤ δ₂)
Goal: SimpleGraph.regularityReduced P G ε δ₂ ≤ SimpleGraph.regularityReduced P G ε δ₁
-- SimpleGraph.unreduced_edges_subset
lemma unreduced_edges_subset :
    (A ×ˢ A).filter (fun (x, y) ↦ G.Adj x y ∧ ¬ (G.regularityReduced P (ε/8) (ε/4)).Adj x y) ⊆
      (P.nonUniforms G (ε/8)).biUnion (fun (U, V) ↦ U ×ˢ V) ∪ P.parts.biUnion offDiag ∪
        (P.sparsePairs G (ε/4)).biUnion fun (U, V) ↦ G.interedges U V 
Arguments: {α : Type u_1} {𝕜 : Type u_2} [LinearOrderedField 𝕜] [DecidableEq α] {A : Finset α} {P : Finpartition A} {G : SimpleGraph α} [DecidableRel G.Adj] {ε : 𝕜}
Goal: Finset.filter
    (fun x =>
      match x with
      | (x, y) => G.Adj x y ∧ ¬(SimpleGraph.regularityReduced P G (ε / 8) (ε / 4)).Adj x y)
    (A ×ˢ A) ⊆
  ((P.nonUniforms G (ε / 8)).biUnion fun x =>
        match x with
        | (U, V) => U ×ˢ V) ∪
      P.parts.biUnion Finset.offDiag ∪
    (P.sparsePairs G (ε / 4)).biUnion fun x =>
      match x with
      | (U, V) => G.interedges U V
-- SimpleGraph.ComponentCompl.supp_injective
@[ext]
theorem ComponentCompl.supp_injective :
    Function.Injective (ComponentCompl.supp : G.ComponentCompl K → Set V) 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V}
Goal: Function.Injective SimpleGraph.ComponentCompl.supp
-- SimpleGraph.ComponentCompl.supp_inj
theorem ComponentCompl.supp_inj {C D : G.ComponentCompl K} : C.supp = D.supp ↔ C = D 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {C D : G.ComponentCompl K}
Goal: C.supp = D.supp ↔ C = D
-- SimpleGraph.ComponentCompl.mem_supp_iff
@[simp]
theorem ComponentCompl.mem_supp_iff {v : V} {C : ComponentCompl G K} :
    v ∈ C ↔ ∃ vK : v ∉ K, G.componentComplMk vK = C 
Arguments: {V : Type u} {G : SimpleGraph V} {K : Set V} {v : V} {C : G.ComponentCompl K}
Goal: v ∈ C ↔ ∃ (vK : v ∉ K), G.componentComplMk vK = C