ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.Icc_prod_Icc
@[simp]
theorem Icc_prod_Icc (a₁ a₂ : α) (b₁ b₂ : β) : Icc a₁ a₂ ×ˢ Icc b₁ b₂ = Icc (a₁, b₁) (a₂, b₂) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: Set.Icc a₁ a₂ ×ˢ Set.Icc b₁ b₂ = Set.Icc (a₁, b₁) (a₂, b₂)
-- Set.Icc_prod_eq
theorem Icc_prod_eq (a b : α × β) : Icc a b = Icc a.1 b.1 ×ˢ Icc a.2 b.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a b : α × β)
Goal: Set.Icc a b = Set.Icc a.1 b.1 ×ˢ Set.Icc a.2 b.2
-- ex_of_psig
theorem ex_of_psig {α : Sort u} {p : α → Prop} : (Σ' x, p x) → ∃ x, p x
  
Arguments: {α : Sort u} {p : α → Prop}
Goal: (x : α) ×' p x → ∃ x, p x
-- Sigma.eq
protected theorem Sigma.eq {α : Type u} {β : α → Type v} : ∀ {p₁ p₂ : Σ a, β a} (h₁ : p₁.1 = p₂.1),
    (Eq.recOn h₁ p₁.2 : β p₂.1) = p₂.2 → p₁ = p₂
  
Arguments: {α : Type u} {β : α → Type v} {p₁ p₂ : (a : α) × β a} (h₁ : p₁.fst = p₂.fst)
Goal: Eq.recOn h₁ p₁.snd = p₂.snd → p₁ = p₂
-- PSigma.eq
protected theorem PSigma.eq {α : Sort u} {β : α → Sort v} : ∀ {p₁ p₂ : Σ' a, β a} (h₁ : p₁.1 = p₂.1),
    (Eq.recOn h₁ p₁.2 : β p₂.1) = p₂.2 → p₁ = p₂
  
Arguments: {α : Sort u} {β : α → Sort v} {p₁ p₂ : (a : α) ×' β a} (h₁ : p₁.fst = p₂.fst)
Goal: Eq.recOn h₁ p₁.snd = p₂.snd → p₁ = p₂
-- List.mem_cons_eq
theorem mem_cons_eq (a y : α) (l : List α) : (a ∈ y :: l) = (a = y ∨ a ∈ l) 
Arguments: {α : Type u} (a y : α) (l : List α)
Goal: (a ∈ y :: l) = (a = y ∨ a ∈ l)
-- List.not_exists_mem_nil
theorem not_exists_mem_nil (p : α → Prop) : ¬∃ x ∈ @nil α, p x 
Arguments: {α : Type u} (p : α → Prop)
Goal: ¬∃ x, x ∈ [] ∧ p x
-- List.length_mapAccumr
@[simp]
theorem length_mapAccumr :
    ∀ (f : α → σ → σ × β) (x : List α) (s : σ), length (mapAccumr f x s).2 = length x
  
Arguments: {α : Type u} {β : Type v} {σ : Type w₂} (f : α → σ → σ × β) (x : List α) (s : σ)
Goal: (List.mapAccumr f x s).snd.length = x.length
-- List.length_mapAccumr₂
@[simp]
theorem length_mapAccumr₂ :
    ∀ (f : α → β → σ → σ × φ) (x y c), length (mapAccumr₂ f x y c).2 = min (length x) (length y)
  
Arguments: {α : Type u} {β : Type v} {φ : Type w₁} {σ : Type w₂} (f : α → β → σ → σ × φ) (x : List α) (y : List β) (c : σ)
Goal: (List.mapAccumr₂ f x y c).snd.length = min x.length y.length
-- Set.ext
theorem ext {a b : Set α} (h : ∀ (x : α), x ∈ a ↔ x ∈ b) : a = b 
Arguments: {α : Type u} {a b : Set α} (h : ∀ (x : α), x ∈ a ↔ x ∈ b)
Goal: a = b
-- Fin.eq_of_veq
@[deprecated eq_of_val_eq (since := "2024-02-15")]
theorem eq_of_veq : i.val = j.val → i = j 
Arguments: {n : ℕ} {i j : Fin n}
Goal: ↑i = ↑j → i = j
-- Fin.veq_of_eq
@[deprecated val_eq_of_eq (since := "2024-02-15")]
theorem veq_of_eq : i = j → i.val = j.val 
Arguments: {n : ℕ} {i j : Fin n}
Goal: i = j → ↑i = ↑j
-- Fin.ne_of_vne
theorem ne_of_vne (h : i.val ≠ j.val) : i ≠ j 
Arguments: {n : ℕ} {i j : Fin n} (h : ↑i ≠ ↑j)
Goal: i ≠ j
-- Fin.vne_of_ne
theorem vne_of_ne (h : i ≠ j) : i.val ≠ j.val 
Arguments: {n : ℕ} {i j : Fin n} (h : i ≠ j)
Goal: ↑i ≠ ↑j
-- Nat.gcd_def
theorem gcd_def (x y : ℕ) : gcd x y = if x = 0 then y else gcd (y % x) x 
Arguments: (x y : ℕ)
Goal: x.gcd y = if x = 0 then y else (y % x).gcd x
-- UpperSet.infIrred_Ici
@[simp] lemma infIrred_Ici (a : α) : InfIrred (Ici a) 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: InfIrred (UpperSet.Ici a)
-- UpperSet.infIrred_iff_of_finite
@[simp] lemma infIrred_iff_of_finite : InfIrred s ↔ ∃ a, Ici a = s 
Arguments: {α : Type u_1} [PartialOrder α] {s : UpperSet α} [Finite α]
Goal: InfIrred s ↔ ∃ a, UpperSet.Ici a = s
-- LowerSet.supIrred_Iic
@[simp] lemma supIrred_Iic (a : α) : SupIrred (Iic a) 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: SupIrred (LowerSet.Iic a)
-- LowerSet.supIrred_iff_of_finite
@[simp] lemma supIrred_iff_of_finite : SupIrred s ↔ ∃ a, Iic a = s 
Arguments: {α : Type u_1} [PartialOrder α] {s : LowerSet α} [Finite α]
Goal: SupIrred s ↔ ∃ a, LowerSet.Iic a = s
-- OrderEmbedding.supIrredLowerSet_apply
@[simp] lemma supIrredLowerSet_apply (a : α) : supIrredLowerSet a = ⟨Iic a, supIrred_Iic _⟩ 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: OrderEmbedding.supIrredLowerSet a = ⟨LowerSet.Iic a, ⋯⟩
-- OrderEmbedding.infIrredUpperSet_apply
@[simp] lemma infIrredUpperSet_apply (a : α) : infIrredUpperSet a = ⟨Ici a, infIrred_Ici _⟩ 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: OrderEmbedding.infIrredUpperSet a = ⟨UpperSet.Ici a, ⋯⟩
-- OrderEmbedding.supIrredLowerSet_surjective
lemma supIrredLowerSet_surjective : Surjective (supIrredLowerSet (α := α)) 
Arguments: {α : Type u_1} [PartialOrder α] [Finite α]
Goal: Function.Surjective ⇑OrderEmbedding.supIrredLowerSet
-- OrderEmbedding.infIrredUpperSet_surjective
lemma infIrredUpperSet_surjective : Surjective (infIrredUpperSet (α := α)) 
Arguments: {α : Type u_1} [PartialOrder α] [Finite α]
Goal: Function.Surjective ⇑OrderEmbedding.infIrredUpperSet
-- OrderIso.supIrredLowerSet_apply
@[simp] lemma supIrredLowerSet_apply (a : α) : supIrredLowerSet a = ⟨Iic a, supIrred_Iic _⟩ 
Arguments: {α : Type u_1} [PartialOrder α] [Finite α] (a : α)
Goal: OrderIso.supIrredLowerSet a = ⟨LowerSet.Iic a, ⋯⟩
-- OrderIso.infIrredUpperSet_apply
@[simp] lemma infIrredUpperSet_apply (a : α) : infIrredUpperSet a = ⟨Ici a, infIrred_Ici _⟩ 
Arguments: {α : Type u_1} [PartialOrder α] [Finite α] (a : α)
Goal: OrderIso.infIrredUpperSet a = ⟨UpperSet.Ici a, ⋯⟩
-- OrderIso.supIrredLowerSet_symm_apply
@[simp] lemma supIrredLowerSet_symm_apply (s : {s : LowerSet α // SupIrred s}) [Fintype s] :
    supIrredLowerSet.symm s = (s.1 : Set α).toFinset.sup id 
Arguments: {α : Type u_1} [SemilatticeSup α] [OrderBot α] [Finite α] (s : { s // SupIrred s }) [Fintype ↥↑s]
Goal: OrderIso.supIrredLowerSet.symm s = (↑↑s).toFinset.sup id
-- OrderIso.infIrredUpperSet_symm_apply
@[simp] lemma infIrredUpperSet_symm_apply (s : {s : UpperSet α // InfIrred s}) [Fintype s] :
    infIrredUpperSet.symm s = (s.1 : Set α).toFinset.inf id 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderTop α] [Finite α] (s : { s // InfIrred s }) [Fintype ↥↑s]
Goal: OrderIso.infIrredUpperSet.symm s = (↑↑s).toFinset.inf id
-- OrderEmbedding.coe_birkhoffFinset
@[simp] lemma coe_birkhoffFinset (a : α) : birkhoffFinset a = birkhoffSet a 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred] (a : α)
Goal: ↑(OrderEmbedding.birkhoffFinset a) = OrderEmbedding.birkhoffSet a
-- OrderEmbedding.birkhoffSet_sup
@[simp] lemma birkhoffSet_sup (a b : α) : birkhoffSet (a ⊔ b) = birkhoffSet a ∪ birkhoffSet b 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred] (a b : α)
Goal: OrderEmbedding.birkhoffSet (a ⊔ b) = OrderEmbedding.birkhoffSet a ∪ OrderEmbedding.birkhoffSet b
-- OrderEmbedding.birkhoffSet_inf
@[simp] lemma birkhoffSet_inf (a b : α) : birkhoffSet (a ⊓ b) = birkhoffSet a ∩ birkhoffSet b 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred] (a b : α)
Goal: OrderEmbedding.birkhoffSet (a ⊓ b) = OrderEmbedding.birkhoffSet a ∩ OrderEmbedding.birkhoffSet b
-- OrderEmbedding.birkhoffFinset_sup
@[simp] lemma birkhoffFinset_sup (a b : α) :
    birkhoffFinset (a ⊔ b) = birkhoffFinset a ∪ birkhoffFinset b 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred] [DecidableEq α] (a b : α)
Goal: OrderEmbedding.birkhoffFinset (a ⊔ b) = OrderEmbedding.birkhoffFinset a ∪ OrderEmbedding.birkhoffFinset b
-- OrderEmbedding.birkhoffFinset_inf
@[simp] lemma birkhoffFinset_inf (a b : α) :
    birkhoffFinset (a ⊓ b) = birkhoffFinset a ∩ birkhoffFinset b 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred] [DecidableEq α] (a b : α)
Goal: OrderEmbedding.birkhoffFinset (a ⊓ b) = OrderEmbedding.birkhoffFinset a ∩ OrderEmbedding.birkhoffFinset b
-- OrderEmbedding.birkhoffSet_apply
@[simp] lemma birkhoffSet_apply (a : α) :
    birkhoffSet a = OrderIso.lowerSetSupIrred a 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] [Fintype α] [DecidablePred SupIrred] (a : α)
Goal: OrderEmbedding.birkhoffSet a = ↑(OrderIso.lowerSetSupIrred a)
-- LatticeHom.birkhoffFinset_injective
lemma birkhoffFinset_injective : Injective (birkhoffFinset (α := α)) 
Arguments: {α : Type u_1} [DistribLattice α] [Fintype α] [DecidablePred SupIrred]
Goal: Function.Injective ⇑LatticeHom.birkhoffFinset
-- exists_birkhoff_representation
lemma exists_birkhoff_representation.{u} (α : Type u) [Finite α] [DistribLattice α] :
    ∃ (β : Type u) (_ : DecidableEq β) (_ : Fintype β) (f : LatticeHom α (Finset β)),
      Injective f 
Arguments: (α : Type u) [Finite α] [DistribLattice α]
Goal: ∃ β x x_1 f, Function.Injective ⇑f
-- Finset.nonempty_Icc
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Finset.Icc a b).Nonempty ↔ a ≤ b
-- Finset.nonempty_Ico
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Finset.Ico a b).Nonempty ↔ a < b
-- Finset.nonempty_Ioc
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Finset.Ioc a b).Nonempty ↔ a < b
-- Finset.nonempty_Ioo
@[simp]
theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DenselyOrdered α]
Goal: (Finset.Ioo a b).Nonempty ↔ a < b
-- Finset.Icc_eq_empty_iff
@[simp]
theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Icc a b = ∅ ↔ ¬a ≤ b
-- Finset.Ico_eq_empty_iff
@[simp]
theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ico a b = ∅ ↔ ¬a < b
-- Finset.Ioc_eq_empty_iff
@[simp]
theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ioc a b = ∅ ↔ ¬a < b
-- Finset.Ioo_eq_empty_iff
@[simp]
theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DenselyOrdered α]
Goal: Finset.Ioo a b = ∅ ↔ ¬a < b
-- Finset.Ioo_eq_empty
@[simp]
theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : ¬a < b)
Goal: Finset.Ioo a b = ∅
-- Finset.Icc_eq_empty_of_lt
@[simp]
theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b < a)
Goal: Finset.Icc a b = ∅
-- Finset.Ico_eq_empty_of_le
@[simp]
theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Finset.Ico a b = ∅
-- Finset.Ioc_eq_empty_of_le
@[simp]
theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Finset.Ioc a b = ∅
-- Finset.Ioo_eq_empty_of_le
@[simp]
theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Finset.Ioo a b = ∅
-- Finset.left_mem_Icc
theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∈ Finset.Icc a b ↔ a ≤ b
-- Finset.left_mem_Ico
theorem left_mem_Ico : a ∈ Ico a b ↔ a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∈ Finset.Ico a b ↔ a < b
-- Finset.right_mem_Icc
theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∈ Finset.Icc a b ↔ a ≤ b
-- Finset.right_mem_Ioc
theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∈ Finset.Ioc a b ↔ a < b
-- Finset.left_not_mem_Ioc
theorem left_not_mem_Ioc : a ∉ Ioc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∉ Finset.Ioc a b
-- Finset.left_not_mem_Ioo
theorem left_not_mem_Ioo : a ∉ Ioo a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∉ Finset.Ioo a b
-- Finset.right_not_mem_Ico
theorem right_not_mem_Ico : b ∉ Ico a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∉ Finset.Ico a b
-- Finset.right_not_mem_Ioo
theorem right_not_mem_Ioo : b ∉ Ioo a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∉ Finset.Ioo a b
-- Finset.Icc_subset_Icc
theorem Icc_subset_Icc (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂)
Goal: Finset.Icc a₁ b₁ ⊆ Finset.Icc a₂ b₂
-- Finset.Ico_subset_Ico
theorem Ico_subset_Ico (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂)
Goal: Finset.Ico a₁ b₁ ⊆ Finset.Ico a₂ b₂
-- Finset.Ioc_subset_Ioc
theorem Ioc_subset_Ioc (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂)
Goal: Finset.Ioc a₁ b₁ ⊆ Finset.Ioc a₂ b₂
-- Finset.Ioo_subset_Ioo
theorem Ioo_subset_Ioo (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (ha : a₂ ≤ a₁) (hb : b₁ ≤ b₂)
Goal: Finset.Ioo a₁ b₁ ⊆ Finset.Ioo a₂ b₂
-- Finset.Icc_subset_Icc_left
theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Finset.Icc a₂ b ⊆ Finset.Icc a₁ b
-- Finset.Ico_subset_Ico_left
theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Finset.Ico a₂ b ⊆ Finset.Ico a₁ b
-- Finset.Ioc_subset_Ioc_left
theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Finset.Ioc a₂ b ⊆ Finset.Ioc a₁ b
-- Finset.Ioo_subset_Ioo_left
theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Finset.Ioo a₂ b ⊆ Finset.Ioo a₁ b
-- Finset.Icc_subset_Icc_right
theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Finset.Icc a b₁ ⊆ Finset.Icc a b₂
-- Finset.Ico_subset_Ico_right
theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Finset.Ico a b₁ ⊆ Finset.Ico a b₂
-- Finset.Ioc_subset_Ioc_right
theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Finset.Ioc a b₁ ⊆ Finset.Ioc a b₂
-- Finset.Ioo_subset_Ioo_right
theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Finset.Ioo a b₁ ⊆ Finset.Ioo a b₂
-- Finset.Ico_subset_Ioo_left
theorem Ico_subset_Ioo_left (h : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b : α} (h : a₁ < a₂)
Goal: Finset.Ico a₂ b ⊆ Finset.Ioo a₁ b
-- Finset.Ioc_subset_Ioo_right
theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ < b₂)
Goal: Finset.Ioc a b₁ ⊆ Finset.Ioo a b₂
-- Finset.Icc_subset_Ico_right
theorem Icc_subset_Ico_right (h : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b₁ b₂ : α} (h : b₁ < b₂)
Goal: Finset.Icc a b₁ ⊆ Finset.Ico a b₂
-- Finset.Ioo_subset_Ico_self
theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ioo a b ⊆ Finset.Ico a b
-- Finset.Ioo_subset_Ioc_self
theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ioo a b ⊆ Finset.Ioc a b
-- Finset.Ico_subset_Icc_self
theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ico a b ⊆ Finset.Icc a b
-- Finset.Ioc_subset_Icc_self
theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ioc a b ⊆ Finset.Icc a b
-- Finset.Ioo_subset_Icc_self
theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Ioo a b ⊆ Finset.Icc a b
-- Finset.Icc_subset_Icc_iff
theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Finset.Icc a₁ b₁ ⊆ Finset.Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Finset.Icc_subset_Ioo_iff
theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Finset.Icc a₁ b₁ ⊆ Finset.Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂
-- Finset.Icc_subset_Ico_iff
theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Finset.Icc a₁ b₁ ⊆ Finset.Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂
-- Finset.Icc_subset_Ioc_iff
theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Finset.Icc a₁ b₁ ⊆ Finset.Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂
-- Finset.Icc_ssubset_Icc_left
theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) :
    Icc a₁ b₁ ⊂ Icc a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂)
Goal: Finset.Icc a₁ b₁ ⊂ Finset.Icc a₂ b₂
-- Finset.Icc_ssubset_Icc_right
theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) :
    Icc a₁ b₁ ⊂ Icc a₂ b₂ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂)
Goal: Finset.Icc a₁ b₁ ⊂ Finset.Icc a₂ b₂
-- Finset.Ico_self
theorem Ico_self : Ico a a = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ico a a = ∅
-- Finset.Ioc_self
theorem Ioc_self : Ioc a a = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ioc a a = ∅
-- Finset.Ioo_self
theorem Ioo_self : Ioo a a = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ioo a a = ∅
-- Finset.Ico_filter_lt_of_le_left
theorem Ico_filter_lt_of_le_left [DecidablePred (· < c)] (hca : c ≤ a) :
    (Ico a b).filter (· < c) = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hca : c ≤ a)
Goal: Finset.filter (fun x => x < c) (Finset.Ico a b) = ∅
-- Finset.Ico_filter_lt_of_right_le
theorem Ico_filter_lt_of_right_le [DecidablePred (· < c)] (hbc : b ≤ c) :
    (Ico a b).filter (· < c) = Ico a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hbc : b ≤ c)
Goal: Finset.filter (fun x => x < c) (Finset.Ico a b) = Finset.Ico a b
-- Finset.Ico_filter_lt_of_le_right
theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) :
    (Ico a b).filter (· < c) = Ico a c 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hcb : c ≤ b)
Goal: Finset.filter (fun x => x < c) (Finset.Ico a b) = Finset.Ico a c
-- Finset.Ico_filter_le_of_le_left
theorem Ico_filter_le_of_le_left {a b c : α} [DecidablePred (c ≤ ·)] (hca : c ≤ a) :
    (Ico a b).filter (c ≤ ·) = Ico a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => c ≤ x] (hca : c ≤ a)
Goal: Finset.filter (fun x => c ≤ x) (Finset.Ico a b) = Finset.Ico a b
-- Finset.Ico_filter_le_of_right_le
theorem Ico_filter_le_of_right_le {a b : α} [DecidablePred (b ≤ ·)] :
    (Ico a b).filter (b ≤ ·) = ∅ 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun x => b ≤ x]
Goal: Finset.filter (fun x => b ≤ x) (Finset.Ico a b) = ∅
-- Finset.Ico_filter_le_of_left_le
theorem Ico_filter_le_of_left_le {a b c : α} [DecidablePred (c ≤ ·)] (hac : a ≤ c) :
    (Ico a b).filter (c ≤ ·) = Ico c b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => c ≤ x] (hac : a ≤ c)
Goal: Finset.filter (fun x => c ≤ x) (Finset.Ico a b) = Finset.Ico c b
-- Finset.Icc_filter_lt_of_lt_right
theorem Icc_filter_lt_of_lt_right {a b c : α} [DecidablePred (· < c)] (h : b < c) :
    (Icc a b).filter (· < c) = Icc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (h : b < c)
Goal: Finset.filter (fun x => x < c) (Finset.Icc a b) = Finset.Icc a b
-- Finset.Ioc_filter_lt_of_lt_right
theorem Ioc_filter_lt_of_lt_right {a b c : α} [DecidablePred (· < c)] (h : b < c) :
    (Ioc a b).filter (· < c) = Ioc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (h : b < c)
Goal: Finset.filter (fun x => x < c) (Finset.Ioc a b) = Finset.Ioc a b
-- Finset.Iic_filter_lt_of_lt_right
theorem Iic_filter_lt_of_lt_right {α} [Preorder α] [LocallyFiniteOrderBot α] {a c : α}
    [DecidablePred (· < c)] (h : a < c) : (Iic a).filter (· < c) = Iic a 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrderBot α] {a c : α} [DecidablePred fun x => x < c] (h : a < c)
Goal: Finset.filter (fun x => x < c) (Finset.Iic a) = Finset.Iic a
-- Finset.filter_lt_lt_eq_Ioo
theorem filter_lt_lt_eq_Ioo [DecidablePred fun j => a < j ∧ j < b] :
    (univ.filter fun j => a < j ∧ j < b) = Ioo a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype α] [DecidablePred fun j => a < j ∧ j < b]
Goal: Finset.filter (fun j => a < j ∧ j < b) Finset.univ = Finset.Ioo a b
-- Finset.filter_lt_le_eq_Ioc
theorem filter_lt_le_eq_Ioc [DecidablePred fun j => a < j ∧ j ≤ b] :
    (univ.filter fun j => a < j ∧ j ≤ b) = Ioc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype α] [DecidablePred fun j => a < j ∧ j ≤ b]
Goal: Finset.filter (fun j => a < j ∧ j ≤ b) Finset.univ = Finset.Ioc a b
-- Finset.filter_le_lt_eq_Ico
theorem filter_le_lt_eq_Ico [DecidablePred fun j => a ≤ j ∧ j < b] :
    (univ.filter fun j => a ≤ j ∧ j < b) = Ico a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype α] [DecidablePred fun j => a ≤ j ∧ j < b]
Goal: Finset.filter (fun j => a ≤ j ∧ j < b) Finset.univ = Finset.Ico a b
-- Finset.filter_le_le_eq_Icc
theorem filter_le_le_eq_Icc [DecidablePred fun j => a ≤ j ∧ j ≤ b] :
    (univ.filter fun j => a ≤ j ∧ j ≤ b) = Icc a b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype α] [DecidablePred fun j => a ≤ j ∧ j ≤ b]
Goal: Finset.filter (fun j => a ≤ j ∧ j ≤ b) Finset.univ = Finset.Icc a b
-- Finset.nonempty_Ici
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
lemma nonempty_Ici : (Ici a).Nonempty 
Arguments: {α : Type u_2} [Preorder α] {a : α} [LocallyFiniteOrderTop α]
Goal: (Finset.Ici a).Nonempty
-- Finset.nonempty_Ioi
@[simp, aesop safe apply (rule_sets := [finsetNonempty])]
lemma nonempty_Ioi : (Ioi a).Nonempty ↔ ¬ IsMax a 
Arguments: {α : Type u_2} [Preorder α] {a : α} [LocallyFiniteOrderTop α]
Goal: (Finset.Ioi a).Nonempty ↔ ¬IsMax a
-- Finset.Icc_subset_Ici_self
theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Icc a b ⊆ Finset.Ici a