ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AntisymmRel.image
theorem AntisymmRel.image {a b : α} (h : AntisymmRel (· ≤ ·) a b) {f : α → β} (hf : Monotone f) :
    AntisymmRel (· ≤ ·) (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : α} (h : AntisymmRel (fun x x_1 => x ≤ x_1) a b) {f : α → β} (hf : Monotone f)
Goal: AntisymmRel (fun x x_1 => x ≤ x_1) (f a) (f b)
-- antisymmetrization_fibration
theorem antisymmetrization_fibration :
    Relation.Fibration (· < ·) (· < ·) (@toAntisymmetrization α (· ≤ ·) _) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Relation.Fibration (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (toAntisymmetrization fun x x_1 => x ≤ x_1)
-- acc_antisymmetrization_iff
theorem acc_antisymmetrization_iff : Acc (· < ·)
    (@toAntisymmetrization α (· ≤ ·) _ a) ↔ Acc (· < ·) a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Acc (fun x x_1 => x < x_1) (toAntisymmetrization (fun x x_1 => x ≤ x_1) a) ↔ Acc (fun x x_1 => x < x_1) a
-- wellFounded_antisymmetrization_iff
theorem wellFounded_antisymmetrization_iff :
    WellFounded (@LT.lt (Antisymmetrization α (· ≤ ·)) _) ↔ WellFounded (@LT.lt α _) 
Arguments: {α : Type u_1} [Preorder α]
Goal: WellFounded LT.lt ↔ WellFounded LT.lt
-- toAntisymmetrization_le_toAntisymmetrization_iff
@[simp]
theorem toAntisymmetrization_le_toAntisymmetrization_iff :
    @toAntisymmetrization α (· ≤ ·) _ a ≤ @toAntisymmetrization α (· ≤ ·) _ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: toAntisymmetrization (fun x x_1 => x ≤ x_1) a ≤ toAntisymmetrization (fun x x_1 => x ≤ x_1) b ↔ a ≤ b
-- toAntisymmetrization_lt_toAntisymmetrization_iff
@[simp]
theorem toAntisymmetrization_lt_toAntisymmetrization_iff :
    @toAntisymmetrization α (· ≤ ·) _ a < @toAntisymmetrization α (· ≤ ·) _ b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: toAntisymmetrization (fun x x_1 => x ≤ x_1) a < toAntisymmetrization (fun x x_1 => x ≤ x_1) b ↔ a < b
-- ofAntisymmetrization_le_ofAntisymmetrization_iff
@[simp]
theorem ofAntisymmetrization_le_ofAntisymmetrization_iff {a b : Antisymmetrization α (· ≤ ·)} :
    ofAntisymmetrization (· ≤ ·) a ≤ ofAntisymmetrization (· ≤ ·) b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : Antisymmetrization α fun x x_1 => x ≤ x_1}
Goal: ofAntisymmetrization (fun x x_1 => x ≤ x_1) a ≤ ofAntisymmetrization (fun x x_1 => x ≤ x_1) b ↔ a ≤ b
-- ofAntisymmetrization_lt_ofAntisymmetrization_iff
@[simp]
theorem ofAntisymmetrization_lt_ofAntisymmetrization_iff {a b : Antisymmetrization α (· ≤ ·)} :
    ofAntisymmetrization (· ≤ ·) a < ofAntisymmetrization (· ≤ ·) b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : Antisymmetrization α fun x x_1 => x ≤ x_1}
Goal: ofAntisymmetrization (fun x x_1 => x ≤ x_1) a < ofAntisymmetrization (fun x x_1 => x ≤ x_1) b ↔ a < b
-- toAntisymmetrization_mono
@[mono]
theorem toAntisymmetrization_mono : Monotone (@toAntisymmetrization α (· ≤ ·) _) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Monotone (toAntisymmetrization fun x x_1 => x ≤ x_1)
-- _private.Mathlib.Order.Antisymmetrization.0.liftFun_antisymmRel
private theorem liftFun_antisymmRel (f : α →o β) :
    ((AntisymmRel.setoid α (· ≤ ·)).r ⇒ (AntisymmRel.setoid β (· ≤ ·)).r) f f 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α →o β)
Goal: (Setoid.r ⇒ Setoid.r) ⇑f ⇑f
-- OrderHom.coe_antisymmetrization
@[simp]
theorem OrderHom.coe_antisymmetrization (f : α →o β) :
    ⇑f.antisymmetrization = Quotient.map' f (liftFun_antisymmRel f) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α →o β)
Goal: ⇑f.antisymmetrization = Quotient.map' ⇑f ⋯
-- OrderHom.antisymmetrization_apply
theorem OrderHom.antisymmetrization_apply (f : α →o β) (a : Antisymmetrization α (· ≤ ·)) :
    f.antisymmetrization a = Quotient.map' f (liftFun_antisymmRel f) a 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α →o β) (a : Antisymmetrization α fun x x_1 => x ≤ x_1)
Goal: f.antisymmetrization a = Quotient.map' ⇑f ⋯ a
-- OrderHom.antisymmetrization_apply_mk
@[simp]
theorem OrderHom.antisymmetrization_apply_mk (f : α →o β) (a : α) :
    f.antisymmetrization (toAntisymmetrization _ a) = toAntisymmetrization _ (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α →o β) (a : α)
Goal: f.antisymmetrization (toAntisymmetrization (fun x x_1 => x ≤ x_1) a) = toAntisymmetrization (fun x x_1 => x ≤ x_1) (f a)
-- OrderEmbedding.ofAntisymmetrization_apply
@[simps]
noncomputable def OrderEmbedding.ofAntisymmetrization : Antisymmetrization α (· ≤ ·) ↪o α 
Arguments: (α : Type u_1) [Preorder α]
Goal: ∀ (a : Antisymmetrization α fun x x_1 => x ≤ x_1),
  (OrderEmbedding.ofAntisymmetrization α) a = ofAntisymmetrization (fun x x_1 => x ≤ x_1) a
-- OrderIso.dualAntisymmetrization_apply
@[simp]
theorem OrderIso.dualAntisymmetrization_apply (a : α) :
    OrderIso.dualAntisymmetrization _ (toDual <| toAntisymmetrization _ a) =
      toAntisymmetrization _ (toDual a) 
Arguments: (α : Type u_1) [Preorder α] (a : α)
Goal: (OrderIso.dualAntisymmetrization α) (OrderDual.toDual (toAntisymmetrization (fun x x_1 => x ≤ x_1) a)) =
  toAntisymmetrization (fun x x_1 => x ≤ x_1) (OrderDual.toDual a)
-- OrderIso.dualAntisymmetrization_symm_apply
@[simp]
theorem OrderIso.dualAntisymmetrization_symm_apply (a : α) :
    (OrderIso.dualAntisymmetrization _).symm (toAntisymmetrization _ <| toDual a) =
      toDual (toAntisymmetrization _ a) 
Arguments: (α : Type u_1) [Preorder α] (a : α)
Goal: (OrderIso.dualAntisymmetrization α).symm (toAntisymmetrization (fun x x_1 => x ≤ x_1) (OrderDual.toDual a)) =
  OrderDual.toDual (toAntisymmetrization (fun x x_1 => x ≤ x_1) a)
-- Set.dual_uIcc
@[simp] lemma dual_uIcc (a b : α) : [[toDual a, toDual b]] = ofDual ⁻¹' [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] (a b : α)
Goal: Set.uIcc (OrderDual.toDual a) (OrderDual.toDual b) = ⇑OrderDual.ofDual ⁻¹' Set.uIcc a b
-- Set.uIcc_of_le
@[simp]
lemma uIcc_of_le (h : a ≤ b) : [[a, b]] = Icc a b 
Arguments: {α : Type u_1} [Lattice α] {a b : α} (h : a ≤ b)
Goal: Set.uIcc a b = Set.Icc a b
-- Set.uIcc_of_ge
@[simp]
lemma uIcc_of_ge (h : b ≤ a) : [[a, b]] = Icc b a 
Arguments: {α : Type u_1} [Lattice α] {a b : α} (h : b ≤ a)
Goal: Set.uIcc a b = Set.Icc b a
-- Set.uIcc_comm
lemma uIcc_comm (a b : α) : [[a, b]] = [[b, a]] 
Arguments: {α : Type u_1} [Lattice α] (a b : α)
Goal: Set.uIcc a b = Set.uIcc b a
-- Set.uIcc_of_lt
lemma uIcc_of_lt (h : a < b) : [[a, b]] = Icc a b 
Arguments: {α : Type u_1} [Lattice α] {a b : α} (h : a < b)
Goal: Set.uIcc a b = Set.Icc a b
-- Set.uIcc_of_gt
lemma uIcc_of_gt (h : b < a) : [[a, b]] = Icc b a 
Arguments: {α : Type u_1} [Lattice α] {a b : α} (h : b < a)
Goal: Set.uIcc a b = Set.Icc b a
-- Set.uIcc_self
lemma uIcc_self : [[a, a]] = {a} 
Arguments: {α : Type u_1} [Lattice α] {a : α}
Goal: Set.uIcc a a = {a}
-- Set.nonempty_uIcc
@[simp] lemma nonempty_uIcc : [[a, b]].Nonempty 
Arguments: {α : Type u_1} [Lattice α] {a b : α}
Goal: (Set.uIcc a b).Nonempty
-- Set.Icc_subset_uIcc
lemma Icc_subset_uIcc : Icc a b ⊆ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b : α}
Goal: Set.Icc a b ⊆ Set.uIcc a b
-- Set.Icc_subset_uIcc'
lemma Icc_subset_uIcc' : Icc b a ⊆ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b : α}
Goal: Set.Icc b a ⊆ Set.uIcc a b
-- Set.left_mem_uIcc
@[simp] lemma left_mem_uIcc : a ∈ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b : α}
Goal: a ∈ Set.uIcc a b
-- Set.right_mem_uIcc
@[simp] lemma right_mem_uIcc : b ∈ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b : α}
Goal: b ∈ Set.uIcc a b
-- Set.mem_uIcc_of_le
lemma mem_uIcc_of_le (ha : a ≤ x) (hb : x ≤ b) : x ∈ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b x : α} (ha : a ≤ x) (hb : x ≤ b)
Goal: x ∈ Set.uIcc a b
-- Set.mem_uIcc_of_ge
lemma mem_uIcc_of_ge (hb : b ≤ x) (ha : x ≤ a) : x ∈ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b x : α} (hb : b ≤ x) (ha : x ≤ a)
Goal: x ∈ Set.uIcc a b
-- Set.uIcc_subset_uIcc
lemma uIcc_subset_uIcc (h₁ : a₁ ∈ [[a₂, b₂]]) (h₂ : b₁ ∈ [[a₂, b₂]]) :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] 
Arguments: {α : Type u_1} [Lattice α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ∈ Set.uIcc a₂ b₂) (h₂ : b₁ ∈ Set.uIcc a₂ b₂)
Goal: Set.uIcc a₁ b₁ ⊆ Set.uIcc a₂ b₂
-- Set.uIcc_subset_Icc
lemma uIcc_subset_Icc (ha : a₁ ∈ Icc a₂ b₂) (hb : b₁ ∈ Icc a₂ b₂) :
    [[a₁, b₁]] ⊆ Icc a₂ b₂ 
Arguments: {α : Type u_1} [Lattice α] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ∈ Set.Icc a₂ b₂) (hb : b₁ ∈ Set.Icc a₂ b₂)
Goal: Set.uIcc a₁ b₁ ⊆ Set.Icc a₂ b₂
-- Set.uIcc_subset_uIcc_iff_mem
lemma uIcc_subset_uIcc_iff_mem :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₁ ∈ [[a₂, b₂]] ∧ b₁ ∈ [[a₂, b₂]] 
Arguments: {α : Type u_1} [Lattice α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.uIcc a₁ b₁ ⊆ Set.uIcc a₂ b₂ ↔ a₁ ∈ Set.uIcc a₂ b₂ ∧ b₁ ∈ Set.uIcc a₂ b₂
-- Set.uIcc_subset_uIcc_iff_le'
lemma uIcc_subset_uIcc_iff_le' :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₂ ⊓ b₂ ≤ a₁ ⊓ b₁ ∧ a₁ ⊔ b₁ ≤ a₂ ⊔ b₂ 
Arguments: {α : Type u_1} [Lattice α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.uIcc a₁ b₁ ⊆ Set.uIcc a₂ b₂ ↔ a₂ ⊓ b₂ ≤ a₁ ⊓ b₁ ∧ a₁ ⊔ b₁ ≤ a₂ ⊔ b₂
-- Set.uIcc_subset_uIcc_right
lemma uIcc_subset_uIcc_right (h : x ∈ [[a, b]]) : [[x, b]] ⊆ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b x : α} (h : x ∈ Set.uIcc a b)
Goal: Set.uIcc x b ⊆ Set.uIcc a b
-- Set.uIcc_subset_uIcc_left
lemma uIcc_subset_uIcc_left (h : x ∈ [[a, b]]) : [[a, x]] ⊆ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] {a b x : α} (h : x ∈ Set.uIcc a b)
Goal: Set.uIcc a x ⊆ Set.uIcc a b
-- Set.bdd_below_bdd_above_iff_subset_uIcc
lemma bdd_below_bdd_above_iff_subset_uIcc (s : Set α) :
    BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ [[a, b]] 
Arguments: {α : Type u_1} [Lattice α] (s : Set α)
Goal: BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Set.uIcc a b
-- Set.uIcc_prod_uIcc
@[simp]
theorem uIcc_prod_uIcc (a₁ a₂ : α) (b₁ b₂ : β) :
    [[a₁, a₂]] ×ˢ [[b₁, b₂]] = [[(a₁, b₁), (a₂, b₂)]] 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: Set.uIcc a₁ a₂ ×ˢ Set.uIcc b₁ b₂ = Set.uIcc (a₁, b₁) (a₂, b₂)
-- Set.uIcc_prod_eq
theorem uIcc_prod_eq (a b : α × β) : [[a, b]] = [[a.1, b.1]] ×ˢ [[a.2, b.2]] 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] (a b : α × β)
Goal: Set.uIcc a b = Set.uIcc a.1 b.1 ×ˢ Set.uIcc a.2 b.2
-- Set.eq_of_mem_uIcc_of_mem_uIcc
lemma eq_of_mem_uIcc_of_mem_uIcc (ha : a ∈ [[b, c]]) (hb : b ∈ [[a, c]]) : a = b 
Arguments: {α : Type u_1} [DistribLattice α] {a b c : α} (ha : a ∈ Set.uIcc b c) (hb : b ∈ Set.uIcc a c)
Goal: a = b
-- Set.eq_of_mem_uIcc_of_mem_uIcc'
lemma eq_of_mem_uIcc_of_mem_uIcc' : b ∈ [[a, c]] → c ∈ [[a, b]] → b = c 
Arguments: {α : Type u_1} [DistribLattice α] {a b c : α}
Goal: b ∈ Set.uIcc a c → c ∈ Set.uIcc a b → b = c
-- Set.uIcc_injective_right
lemma uIcc_injective_right (a : α) : Injective fun b => uIcc b a 
Arguments: {α : Type u_1} [DistribLattice α] (a : α)
Goal: Function.Injective fun b => Set.uIcc b a
-- Set.uIcc_injective_left
lemma uIcc_injective_left (a : α) : Injective (uIcc a) 
Arguments: {α : Type u_1} [DistribLattice α] (a : α)
Goal: Function.Injective (Set.uIcc a)
-- MonotoneOn.mapsTo_uIcc
lemma _root_.MonotoneOn.mapsTo_uIcc (hf : MonotoneOn f (uIcc a b)) :
    MapsTo f (uIcc a b) (uIcc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : MonotoneOn f (Set.uIcc a b))
Goal: Set.MapsTo f (Set.uIcc a b) (Set.uIcc (f a) (f b))
-- AntitoneOn.mapsTo_uIcc
lemma _root_.AntitoneOn.mapsTo_uIcc (hf : AntitoneOn f (uIcc a b)) :
    MapsTo f (uIcc a b) (uIcc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : AntitoneOn f (Set.uIcc a b))
Goal: Set.MapsTo f (Set.uIcc a b) (Set.uIcc (f a) (f b))
-- Monotone.mapsTo_uIcc
lemma _root_.Monotone.mapsTo_uIcc (hf : Monotone f) : MapsTo f (uIcc a b) (uIcc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : Monotone f)
Goal: Set.MapsTo f (Set.uIcc a b) (Set.uIcc (f a) (f b))
-- Antitone.mapsTo_uIcc
lemma _root_.Antitone.mapsTo_uIcc (hf : Antitone f) : MapsTo f (uIcc a b) (uIcc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : Antitone f)
Goal: Set.MapsTo f (Set.uIcc a b) (Set.uIcc (f a) (f b))
-- MonotoneOn.image_uIcc_subset
lemma _root_.MonotoneOn.image_uIcc_subset (hf : MonotoneOn f (uIcc a b)) :
    f '' uIcc a b ⊆ uIcc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : MonotoneOn f (Set.uIcc a b))
Goal: f '' Set.uIcc a b ⊆ Set.uIcc (f a) (f b)
-- AntitoneOn.image_uIcc_subset
lemma _root_.AntitoneOn.image_uIcc_subset (hf : AntitoneOn f (uIcc a b)) :
    f '' uIcc a b ⊆ uIcc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : AntitoneOn f (Set.uIcc a b))
Goal: f '' Set.uIcc a b ⊆ Set.uIcc (f a) (f b)
-- Monotone.image_uIcc_subset
lemma _root_.Monotone.image_uIcc_subset (hf : Monotone f) : f '' uIcc a b ⊆ uIcc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : Monotone f)
Goal: f '' Set.uIcc a b ⊆ Set.uIcc (f a) (f b)
-- Antitone.image_uIcc_subset
lemma _root_.Antitone.image_uIcc_subset (hf : Antitone f) : f '' uIcc a b ⊆ uIcc (f a) (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] {f : α → β} {a b : α} (hf : Antitone f)
Goal: f '' Set.uIcc a b ⊆ Set.uIcc (f a) (f b)
-- Set.Icc_min_max
theorem Icc_min_max : Icc (min a b) (max a b) = [[a, b]] 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Icc (min a b) (max a b) = Set.uIcc a b
-- Set.uIcc_of_not_le
lemma uIcc_of_not_le (h : ¬a ≤ b) : [[a, b]] = Icc b a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : ¬a ≤ b)
Goal: Set.uIcc a b = Set.Icc b a
-- Set.uIcc_of_not_ge
lemma uIcc_of_not_ge (h : ¬b ≤ a) : [[a, b]] = Icc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : ¬b ≤ a)
Goal: Set.uIcc a b = Set.Icc a b
-- Set.uIcc_eq_union
lemma uIcc_eq_union : [[a, b]] = Icc a b ∪ Icc b a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.uIcc a b = Set.Icc a b ∪ Set.Icc b a
-- Set.mem_uIcc
lemma mem_uIcc : a ∈ [[b, c]] ↔ b ≤ a ∧ a ≤ c ∨ c ≤ a ∧ a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: a ∈ Set.uIcc b c ↔ b ≤ a ∧ a ≤ c ∨ c ≤ a ∧ a ≤ b
-- Set.not_mem_uIcc_of_lt
lemma not_mem_uIcc_of_lt (ha : c < a) (hb : c < b) : c ∉ [[a, b]] 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (ha : c < a) (hb : c < b)
Goal: c ∉ Set.uIcc a b
-- Set.not_mem_uIcc_of_gt
lemma not_mem_uIcc_of_gt (ha : a < c) (hb : b < c) : c ∉ [[a, b]] 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (ha : a < c) (hb : b < c)
Goal: c ∉ Set.uIcc a b
-- Set.uIcc_subset_uIcc_iff_le
lemma uIcc_subset_uIcc_iff_le :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ min a₂ b₂ ≤ min a₁ b₁ ∧ max a₁ b₁ ≤ max a₂ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.uIcc a₁ b₁ ⊆ Set.uIcc a₂ b₂ ↔ min a₂ b₂ ≤ min a₁ b₁ ∧ max a₁ b₁ ≤ max a₂ b₂
-- Set.uIcc_subset_uIcc_union_uIcc
lemma uIcc_subset_uIcc_union_uIcc : [[a, c]] ⊆ [[a, b]] ∪ [[b, c]] 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.uIcc a c ⊆ Set.uIcc a b ∪ Set.uIcc b c
-- Set.monotone_or_antitone_iff_uIcc
lemma monotone_or_antitone_iff_uIcc :
    Monotone f ∨ Antitone f ↔ ∀ a b c, c ∈ [[a, b]] → f c ∈ [[f a, f b]] 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] {f : α → β}
Goal: Monotone f ∨ Antitone f ↔ ∀ (a b c : α), c ∈ Set.uIcc a b → f c ∈ Set.uIcc (f a) (f b)
-- Set.monotoneOn_or_antitoneOn_iff_uIcc
lemma monotoneOn_or_antitoneOn_iff_uIcc :
    MonotoneOn f s ∨ AntitoneOn f s ↔
      ∀ᵉ (a ∈ s) (b ∈ s) (c ∈ s), c ∈ [[a, b]] → f c ∈ [[f a, f b]] 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Set α}
Goal: MonotoneOn f s ∨ AntitoneOn f s ↔ ∀ a ∈ s, ∀ b ∈ s, ∀ c ∈ s, c ∈ Set.uIcc a b → f c ∈ Set.uIcc (f a) (f b)
-- Set.uIoc_of_le
@[simp] lemma uIoc_of_le (h : a ≤ b) : Ι a b = Ioc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Ι a b = Set.Ioc a b
-- Set.uIoc_of_ge
@[simp] lemma uIoc_of_ge (h : b ≤ a) : Ι a b = Ioc b a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : b ≤ a)
Goal: Ι a b = Set.Ioc b a
-- Set.uIoc_eq_union
lemma uIoc_eq_union : Ι a b = Ioc a b ∪ Ioc b a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Ι a b = Set.Ioc a b ∪ Set.Ioc b a
-- Set.mem_uIoc
lemma mem_uIoc : a ∈ Ι b c ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: a ∈ Ι b c ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b
-- Set.not_mem_uIoc
lemma not_mem_uIoc : a ∉ Ι b c ↔ a ≤ b ∧ a ≤ c ∨ c < a ∧ b < a 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: a ∉ Ι b c ↔ a ≤ b ∧ a ≤ c ∨ c < a ∧ b < a
-- Set.left_mem_uIoc
@[simp] lemma left_mem_uIoc : a ∈ Ι a b ↔ b < a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: a ∈ Ι a b ↔ b < a
-- Set.right_mem_uIoc
@[simp] lemma right_mem_uIoc : b ∈ Ι a b ↔ a < b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: b ∈ Ι a b ↔ a < b
-- Set.forall_uIoc_iff
lemma forall_uIoc_iff {P : α → Prop} :
    (∀ x ∈ Ι a b, P x) ↔ (∀ x ∈ Ioc a b, P x) ∧ ∀ x ∈ Ioc b a, P x 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} {P : α → Prop}
Goal: (∀ x ∈ Ι a b, P x) ↔ (∀ x ∈ Set.Ioc a b, P x) ∧ ∀ x ∈ Set.Ioc b a, P x
-- Set.uIoc_subset_uIoc_of_uIcc_subset_uIcc
lemma uIoc_subset_uIoc_of_uIcc_subset_uIcc {a b c d : α}
    (h : [[a, b]] ⊆ [[c, d]]) : Ι a b ⊆ Ι c d 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h : Set.uIcc a b ⊆ Set.uIcc c d)
Goal: Ι a b ⊆ Ι c d
-- Set.uIoc_comm
lemma uIoc_comm (a b : α) : Ι a b = Ι b a 
Arguments: {α : Type u_1} [LinearOrder α] (a b : α)
Goal: Ι a b = Ι b a
-- Set.Ioc_subset_uIoc
lemma Ioc_subset_uIoc : Ioc a b ⊆ Ι a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioc a b ⊆ Ι a b
-- Set.Ioc_subset_uIoc'
lemma Ioc_subset_uIoc' : Ioc a b ⊆ Ι b a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioc a b ⊆ Ι b a
-- Set.uIoc_subset_uIcc
lemma uIoc_subset_uIcc : Ι a b ⊆ uIcc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Ι a b ⊆ Set.uIcc a b
-- Set.eq_of_mem_uIoc_of_mem_uIoc
lemma eq_of_mem_uIoc_of_mem_uIoc : a ∈ Ι b c → b ∈ Ι a c → a = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: a ∈ Ι b c → b ∈ Ι a c → a = b
-- Set.eq_of_mem_uIoc_of_mem_uIoc'
lemma eq_of_mem_uIoc_of_mem_uIoc' : b ∈ Ι a c → c ∈ Ι a b → b = c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: b ∈ Ι a c → c ∈ Ι a b → b = c
-- Set.eq_of_not_mem_uIoc_of_not_mem_uIoc
lemma eq_of_not_mem_uIoc_of_not_mem_uIoc (ha : a ≤ c) (hb : b ≤ c) :
    a ∉ Ι b c → b ∉ Ι a c → a = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (ha : a ≤ c) (hb : b ≤ c)
Goal: a ∉ Ι b c → b ∉ Ι a c → a = b
-- Set.uIoc_injective_right
lemma uIoc_injective_right (a : α) : Injective fun b => Ι b a 
Arguments: {α : Type u_1} [LinearOrder α] (a : α)
Goal: Function.Injective fun b => Ι b a
-- Set.uIoc_injective_left
lemma uIoc_injective_left (a : α) : Injective (Ι a) 
Arguments: {α : Type u_1} [LinearOrder α] (a : α)
Goal: Function.Injective (Ι a)
-- OrderDual.toDual_symm_eq
@[simp]
theorem toDual_symm_eq : (@toDual α).symm = ofDual 
Arguments: {α : Type u_1}
Goal: OrderDual.toDual.symm = OrderDual.ofDual
-- OrderDual.ofDual_symm_eq
@[simp]
theorem ofDual_symm_eq : (@ofDual α).symm = toDual 
Arguments: {α : Type u_1}
Goal: OrderDual.ofDual.symm = OrderDual.toDual
-- OrderDual.toDual_ofDual
@[simp]
theorem toDual_ofDual (a : αᵒᵈ) : toDual (ofDual a) = a 
Arguments: {α : Type u_1} (a : αᵒᵈ)
Goal: OrderDual.toDual (OrderDual.ofDual a) = a
-- OrderDual.ofDual_toDual
@[simp]
theorem ofDual_toDual (a : α) : ofDual (toDual a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: OrderDual.ofDual (OrderDual.toDual a) = a
-- OrderDual.toDual_inj
theorem toDual_inj {a b : α} : toDual a = toDual b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: OrderDual.toDual a = OrderDual.toDual b ↔ a = b
-- OrderDual.ofDual_inj
theorem ofDual_inj {a b : αᵒᵈ} : ofDual a = ofDual b ↔ a = b 
Arguments: {α : Type u_1} {a b : αᵒᵈ}
Goal: OrderDual.ofDual a = OrderDual.ofDual b ↔ a = b
-- OrderDual.toDual_le_toDual
@[simp]
theorem toDual_le_toDual [LE α] {a b : α} : toDual a ≤ toDual b ↔ b ≤ a 
Arguments: {α : Type u_1} [LE α] {a b : α}
Goal: OrderDual.toDual a ≤ OrderDual.toDual b ↔ b ≤ a
-- OrderDual.toDual_lt_toDual
@[simp]
theorem toDual_lt_toDual [LT α] {a b : α} : toDual a < toDual b ↔ b < a 
Arguments: {α : Type u_1} [LT α] {a b : α}
Goal: OrderDual.toDual a < OrderDual.toDual b ↔ b < a
-- OrderDual.ofDual_le_ofDual
@[simp]
theorem ofDual_le_ofDual [LE α] {a b : αᵒᵈ} : ofDual a ≤ ofDual b ↔ b ≤ a 
Arguments: {α : Type u_1} [LE α] {a b : αᵒᵈ}
Goal: OrderDual.ofDual a ≤ OrderDual.ofDual b ↔ b ≤ a
-- OrderDual.ofDual_lt_ofDual
@[simp]
theorem ofDual_lt_ofDual [LT α] {a b : αᵒᵈ} : ofDual a < ofDual b ↔ b < a 
Arguments: {α : Type u_1} [LT α] {a b : αᵒᵈ}
Goal: OrderDual.ofDual a < OrderDual.ofDual b ↔ b < a
-- OrderDual.le_toDual
theorem le_toDual [LE α] {a : αᵒᵈ} {b : α} : a ≤ toDual b ↔ b ≤ ofDual a 
Arguments: {α : Type u_1} [LE α] {a : αᵒᵈ} {b : α}
Goal: a ≤ OrderDual.toDual b ↔ b ≤ OrderDual.ofDual a
-- OrderDual.lt_toDual
theorem lt_toDual [LT α] {a : αᵒᵈ} {b : α} : a < toDual b ↔ b < ofDual a 
Arguments: {α : Type u_1} [LT α] {a : αᵒᵈ} {b : α}
Goal: a < OrderDual.toDual b ↔ b < OrderDual.ofDual a
-- OrderDual.toDual_le
theorem toDual_le [LE α] {a : α} {b : αᵒᵈ} : toDual a ≤ b ↔ ofDual b ≤ a 
Arguments: {α : Type u_1} [LE α] {a : α} {b : αᵒᵈ}
Goal: OrderDual.toDual a ≤ b ↔ OrderDual.ofDual b ≤ a
-- OrderDual.toDual_lt
theorem toDual_lt [LT α] {a : α} {b : αᵒᵈ} : toDual a < b ↔ ofDual b < a 
Arguments: {α : Type u_1} [LT α] {a : α} {b : αᵒᵈ}
Goal: OrderDual.toDual a < b ↔ OrderDual.ofDual b < a
-- OrderDual.forall
@[simp]
protected theorem «forall» {p : αᵒᵈ → Prop} : (∀ a, p a) ↔ ∀ a, p (toDual a) 
Arguments: {α : Type u_1} {p : αᵒᵈ → Prop}
Goal: (∀ (a : αᵒᵈ), p a) ↔ ∀ (a : α), p (OrderDual.toDual a)
-- OrderDual.exists
@[simp]
protected theorem «exists» {p : αᵒᵈ → Prop} : (∃ a, p a) ↔ ∃ a, p (toDual a) 
Arguments: {α : Type u_1} {p : αᵒᵈ → Prop}
Goal: (∃ a, p a) ↔ ∃ a, p (OrderDual.toDual a)
-- toLex_symm_eq
@[simp]
theorem toLex_symm_eq : (@toLex α).symm = ofLex 
Arguments: {α : Type u_1}
Goal: toLex.symm = ofLex
-- ofLex_symm_eq
@[simp]
theorem ofLex_symm_eq : (@ofLex α).symm = toLex 
Arguments: {α : Type u_1}
Goal: ofLex.symm = toLex
-- toLex_ofLex
@[simp]
theorem toLex_ofLex (a : Lex α) : toLex (ofLex a) = a 
Arguments: {α : Type u_1} (a : Lex α)
Goal: toLex (ofLex a) = a
-- ofLex_toLex
@[simp]
theorem ofLex_toLex (a : α) : ofLex (toLex a) = a 
Arguments: {α : Type u_1} (a : α)
Goal: ofLex (toLex a) = a
-- toLex_inj
theorem toLex_inj {a b : α} : toLex a = toLex b ↔ a = b 
Arguments: {α : Type u_1} {a b : α}
Goal: toLex a = toLex b ↔ a = b
-- ofLex_inj
theorem ofLex_inj {a b : Lex α} : ofLex a = ofLex b ↔ a = b 
Arguments: {α : Type u_1} {a b : Lex α}
Goal: ofLex a = ofLex b ↔ a = b
-- Lex.forall
@[simp] lemma Lex.forall {p : Lex α → Prop} : (∀ a, p a) ↔ ∀ a, p (toLex a) 
Arguments: {α : Type u_1} {p : Lex α → Prop}
Goal: (∀ (a : Lex α), p a) ↔ ∀ (a : α), p (toLex a)
-- Lex.exists
@[simp] lemma Lex.exists {p : Lex α → Prop} : (∃ a, p a) ↔ ∃ a, p (toLex a) 
Arguments: {α : Type u_1} {p : Lex α → Prop}
Goal: (∃ a, p a) ↔ ∃ a, p (toLex a)
-- Multiset.mem_Icc
@[simp] lemma mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Multiset.Icc a b ↔ a ≤ x ∧ x ≤ b
-- Multiset.mem_Ico
@[simp] lemma mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Multiset.Ico a b ↔ a ≤ x ∧ x < b
-- Multiset.mem_Ioc
@[simp] lemma mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Multiset.Ioc a b ↔ a < x ∧ x ≤ b
-- Multiset.mem_Ioo
@[simp] lemma mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Multiset.Ioo a b ↔ a < x ∧ x < b
-- Multiset.mem_Ici
@[simp] lemma mem_Ici : x ∈ Ici a ↔ a ≤ x 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α}
Goal: x ∈ Multiset.Ici a ↔ a ≤ x