ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- isUpperSet_compl
@[simp]
theorem isUpperSet_compl : IsUpperSet sᶜ ↔ IsLowerSet s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: IsUpperSet sᶜ ↔ IsLowerSet s
-- isLowerSet_compl
@[simp]
theorem isLowerSet_compl : IsLowerSet sᶜ ↔ IsUpperSet s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: IsLowerSet sᶜ ↔ IsUpperSet s
-- IsUpperSet.union
theorem IsUpperSet.union (hs : IsUpperSet s) (ht : IsUpperSet t) : IsUpperSet (s ∪ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsUpperSet s) (ht : IsUpperSet t)
Goal: IsUpperSet (s ∪ t)
-- IsLowerSet.union
theorem IsLowerSet.union (hs : IsLowerSet s) (ht : IsLowerSet t) : IsLowerSet (s ∪ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsLowerSet s) (ht : IsLowerSet t)
Goal: IsLowerSet (s ∪ t)
-- IsUpperSet.inter
theorem IsUpperSet.inter (hs : IsUpperSet s) (ht : IsUpperSet t) : IsUpperSet (s ∩ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsUpperSet s) (ht : IsUpperSet t)
Goal: IsUpperSet (s ∩ t)
-- IsLowerSet.inter
theorem IsLowerSet.inter (hs : IsLowerSet s) (ht : IsLowerSet t) : IsLowerSet (s ∩ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsLowerSet s) (ht : IsLowerSet t)
Goal: IsLowerSet (s ∩ t)
-- isUpperSet_sUnion
theorem isUpperSet_sUnion {S : Set (Set α)} (hf : ∀ s ∈ S, IsUpperSet s) : IsUpperSet (⋃₀ S) 
Arguments: {α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsUpperSet s)
Goal: IsUpperSet (⋃₀ S)
-- isLowerSet_sUnion
theorem isLowerSet_sUnion {S : Set (Set α)} (hf : ∀ s ∈ S, IsLowerSet s) : IsLowerSet (⋃₀ S) 
Arguments: {α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsLowerSet s)
Goal: IsLowerSet (⋃₀ S)
-- isUpperSet_iUnion
theorem isUpperSet_iUnion {f : ι → Set α} (hf : ∀ i, IsUpperSet (f i)) : IsUpperSet (⋃ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {f : ι → Set α} (hf : ∀ (i : ι), IsUpperSet (f i))
Goal: IsUpperSet (⋃ i, f i)
-- isLowerSet_iUnion
theorem isLowerSet_iUnion {f : ι → Set α} (hf : ∀ i, IsLowerSet (f i)) : IsLowerSet (⋃ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {f : ι → Set α} (hf : ∀ (i : ι), IsLowerSet (f i))
Goal: IsLowerSet (⋃ i, f i)
-- isUpperSet_iUnion₂
theorem isUpperSet_iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsUpperSet (f i j)) :
    IsUpperSet (⋃ (i) (j), f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {f : (i : ι) → κ i → Set α} (hf : ∀ (i : ι) (j : κ i), IsUpperSet (f i j))
Goal: IsUpperSet (⋃ i, ⋃ j, f i j)
-- isLowerSet_iUnion₂
theorem isLowerSet_iUnion₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsLowerSet (f i j)) :
    IsLowerSet (⋃ (i) (j), f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {f : (i : ι) → κ i → Set α} (hf : ∀ (i : ι) (j : κ i), IsLowerSet (f i j))
Goal: IsLowerSet (⋃ i, ⋃ j, f i j)
-- isUpperSet_sInter
theorem isUpperSet_sInter {S : Set (Set α)} (hf : ∀ s ∈ S, IsUpperSet s) : IsUpperSet (⋂₀ S) 
Arguments: {α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsUpperSet s)
Goal: IsUpperSet (⋂₀ S)
-- isLowerSet_sInter
theorem isLowerSet_sInter {S : Set (Set α)} (hf : ∀ s ∈ S, IsLowerSet s) : IsLowerSet (⋂₀ S) 
Arguments: {α : Type u_1} [LE α] {S : Set (Set α)} (hf : ∀ s ∈ S, IsLowerSet s)
Goal: IsLowerSet (⋂₀ S)
-- isUpperSet_iInter
theorem isUpperSet_iInter {f : ι → Set α} (hf : ∀ i, IsUpperSet (f i)) : IsUpperSet (⋂ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {f : ι → Set α} (hf : ∀ (i : ι), IsUpperSet (f i))
Goal: IsUpperSet (⋂ i, f i)
-- isLowerSet_iInter
theorem isLowerSet_iInter {f : ι → Set α} (hf : ∀ i, IsLowerSet (f i)) : IsLowerSet (⋂ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {f : ι → Set α} (hf : ∀ (i : ι), IsLowerSet (f i))
Goal: IsLowerSet (⋂ i, f i)
-- isUpperSet_iInter₂
theorem isUpperSet_iInter₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsUpperSet (f i j)) :
    IsUpperSet (⋂ (i) (j), f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {f : (i : ι) → κ i → Set α} (hf : ∀ (i : ι) (j : κ i), IsUpperSet (f i j))
Goal: IsUpperSet (⋂ i, ⋂ j, f i j)
-- isLowerSet_iInter₂
theorem isLowerSet_iInter₂ {f : ∀ i, κ i → Set α} (hf : ∀ i j, IsLowerSet (f i j)) :
    IsLowerSet (⋂ (i) (j), f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {f : (i : ι) → κ i → Set α} (hf : ∀ (i : ι) (j : κ i), IsLowerSet (f i j))
Goal: IsLowerSet (⋂ i, ⋂ j, f i j)
-- isLowerSet_preimage_ofDual_iff
@[simp]
theorem isLowerSet_preimage_ofDual_iff : IsLowerSet (ofDual ⁻¹' s) ↔ IsUpperSet s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: IsLowerSet (⇑OrderDual.ofDual ⁻¹' s) ↔ IsUpperSet s
-- isUpperSet_preimage_ofDual_iff
@[simp]
theorem isUpperSet_preimage_ofDual_iff : IsUpperSet (ofDual ⁻¹' s) ↔ IsLowerSet s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: IsUpperSet (⇑OrderDual.ofDual ⁻¹' s) ↔ IsLowerSet s
-- isLowerSet_preimage_toDual_iff
@[simp]
theorem isLowerSet_preimage_toDual_iff {s : Set αᵒᵈ} : IsLowerSet (toDual ⁻¹' s) ↔ IsUpperSet s 
Arguments: {α : Type u_1} [LE α] {s : Set αᵒᵈ}
Goal: IsLowerSet (⇑OrderDual.toDual ⁻¹' s) ↔ IsUpperSet s
-- isUpperSet_preimage_toDual_iff
@[simp]
theorem isUpperSet_preimage_toDual_iff {s : Set αᵒᵈ} : IsUpperSet (toDual ⁻¹' s) ↔ IsLowerSet s 
Arguments: {α : Type u_1} [LE α] {s : Set αᵒᵈ}
Goal: IsUpperSet (⇑OrderDual.toDual ⁻¹' s) ↔ IsLowerSet s
-- IsUpperSet.isLowerSet_preimage_coe
lemma IsUpperSet.isLowerSet_preimage_coe (hs : IsUpperSet s) :
    IsLowerSet ((↑) ⁻¹' t : Set s) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsUpperSet s)
Goal: IsLowerSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t
-- IsLowerSet.isUpperSet_preimage_coe
lemma IsLowerSet.isUpperSet_preimage_coe (hs : IsLowerSet s) :
    IsUpperSet ((↑) ⁻¹' t : Set s) ↔ ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsLowerSet s)
Goal: IsUpperSet (Subtype.val ⁻¹' t) ↔ ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t
-- IsUpperSet.sdiff
lemma IsUpperSet.sdiff (hs : IsUpperSet s) (ht : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t) :
    IsUpperSet (s \ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsUpperSet s) (ht : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t)
Goal: IsUpperSet (s \ t)
-- IsLowerSet.sdiff
lemma IsLowerSet.sdiff (hs : IsLowerSet s) (ht : ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t) :
    IsLowerSet (s \ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsLowerSet s) (ht : ∀ b ∈ s, ∀ c ∈ t, c ≤ b → b ∈ t)
Goal: IsLowerSet (s \ t)
-- IsUpperSet.sdiff_of_isLowerSet
lemma IsUpperSet.sdiff_of_isLowerSet (hs : IsUpperSet s) (ht : IsLowerSet t) : IsUpperSet (s \ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsUpperSet s) (ht : IsLowerSet t)
Goal: IsUpperSet (s \ t)
-- IsLowerSet.sdiff_of_isUpperSet
lemma IsLowerSet.sdiff_of_isUpperSet (hs : IsLowerSet s) (ht : IsUpperSet t) : IsLowerSet (s \ t) 
Arguments: {α : Type u_1} [LE α] {s t : Set α} (hs : IsLowerSet s) (ht : IsUpperSet t)
Goal: IsLowerSet (s \ t)
-- IsUpperSet.erase
lemma IsUpperSet.erase (hs : IsUpperSet s) (has : ∀ b ∈ s, b ≤ a → b = a) : IsUpperSet (s \ {a}) 
Arguments: {α : Type u_1} [LE α] {s : Set α} {a : α} (hs : IsUpperSet s) (has : ∀ b ∈ s, b ≤ a → b = a)
Goal: IsUpperSet (s \ {a})
-- IsLowerSet.erase
lemma IsLowerSet.erase (hs : IsLowerSet s) (has : ∀ b ∈ s, a ≤ b → b = a) : IsLowerSet (s \ {a}) 
Arguments: {α : Type u_1} [LE α] {s : Set α} {a : α} (hs : IsLowerSet s) (has : ∀ b ∈ s, a ≤ b → b = a)
Goal: IsLowerSet (s \ {a})
-- isUpperSet_Ici
theorem isUpperSet_Ici : IsUpperSet (Ici a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: IsUpperSet (Set.Ici a)
-- isLowerSet_Iic
theorem isLowerSet_Iic : IsLowerSet (Iic a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: IsLowerSet (Set.Iic a)
-- isUpperSet_Ioi
theorem isUpperSet_Ioi : IsUpperSet (Ioi a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: IsUpperSet (Set.Ioi a)
-- isLowerSet_Iio
theorem isLowerSet_Iio : IsLowerSet (Iio a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: IsLowerSet (Set.Iio a)
-- isUpperSet_iff_Ici_subset
theorem isUpperSet_iff_Ici_subset : IsUpperSet s ↔ ∀ ⦃a⦄, a ∈ s → Ici a ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsUpperSet s ↔ ∀ ⦃a : α⦄, a ∈ s → Set.Ici a ⊆ s
-- isLowerSet_iff_Iic_subset
theorem isLowerSet_iff_Iic_subset : IsLowerSet s ↔ ∀ ⦃a⦄, a ∈ s → Iic a ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: IsLowerSet s ↔ ∀ ⦃a : α⦄, a ∈ s → Set.Iic a ⊆ s
-- IsUpperSet.Ioi_subset
theorem IsUpperSet.Ioi_subset (h : IsUpperSet s) ⦃a⦄ (ha : a ∈ s) : Ioi a ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : IsUpperSet s) ⦃a : α⦄ (ha : a ∈ s)
Goal: Set.Ioi a ⊆ s
-- IsLowerSet.Iio_subset
theorem IsLowerSet.Iio_subset (h : IsLowerSet s) ⦃a⦄ (ha : a ∈ s) : Iio a ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : IsLowerSet s) ⦃a : α⦄ (ha : a ∈ s)
Goal: Set.Iio a ⊆ s
-- IsUpperSet.ordConnected
theorem IsUpperSet.ordConnected (h : IsUpperSet s) : s.OrdConnected 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : IsUpperSet s)
Goal: s.OrdConnected
-- IsLowerSet.ordConnected
theorem IsLowerSet.ordConnected (h : IsLowerSet s) : s.OrdConnected 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : IsLowerSet s)
Goal: s.OrdConnected
-- IsUpperSet.preimage
theorem IsUpperSet.preimage (hs : IsUpperSet s) {f : β → α} (hf : Monotone f) :
    IsUpperSet (f ⁻¹' s : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} (hs : IsUpperSet s) {f : β → α} (hf : Monotone f)
Goal: IsUpperSet (f ⁻¹' s)
-- IsLowerSet.preimage
theorem IsLowerSet.preimage (hs : IsLowerSet s) {f : β → α} (hf : Monotone f) :
    IsLowerSet (f ⁻¹' s : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} (hs : IsLowerSet s) {f : β → α} (hf : Monotone f)
Goal: IsLowerSet (f ⁻¹' s)
-- IsUpperSet.image
theorem IsUpperSet.image (hs : IsUpperSet s) (f : α ≃o β) : IsUpperSet (f '' s : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} (hs : IsUpperSet s) (f : α ≃o β)
Goal: IsUpperSet (⇑f '' s)
-- IsLowerSet.image
theorem IsLowerSet.image (hs : IsLowerSet s) (f : α ≃o β) : IsLowerSet (f '' s : Set β) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} (hs : IsLowerSet s) (f : α ≃o β)
Goal: IsLowerSet (⇑f '' s)
-- OrderEmbedding.image_Ici
theorem OrderEmbedding.image_Ici (e : α ↪o β) (he : IsUpperSet (range e)) (a : α) :
    e '' Ici a = Ici (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : IsUpperSet (Set.range ⇑e)) (a : α)
Goal: ⇑e '' Set.Ici a = Set.Ici (e a)
-- OrderEmbedding.image_Iic
theorem OrderEmbedding.image_Iic (e : α ↪o β) (he : IsLowerSet (range e)) (a : α) :
    e '' Iic a = Iic (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : IsLowerSet (Set.range ⇑e)) (a : α)
Goal: ⇑e '' Set.Iic a = Set.Iic (e a)
-- OrderEmbedding.image_Ioi
theorem OrderEmbedding.image_Ioi (e : α ↪o β) (he : IsUpperSet (range e)) (a : α) :
    e '' Ioi a = Ioi (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : IsUpperSet (Set.range ⇑e)) (a : α)
Goal: ⇑e '' Set.Ioi a = Set.Ioi (e a)
-- OrderEmbedding.image_Iio
theorem OrderEmbedding.image_Iio (e : α ↪o β) (he : IsLowerSet (range e)) (a : α) :
    e '' Iio a = Iio (e a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : IsLowerSet (Set.range ⇑e)) (a : α)
Goal: ⇑e '' Set.Iio a = Set.Iio (e a)
-- Set.monotone_mem
@[simp]
theorem Set.monotone_mem : Monotone (· ∈ s) ↔ IsUpperSet s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: (Monotone fun x => x ∈ s) ↔ IsUpperSet s
-- Set.antitone_mem
@[simp]
theorem Set.antitone_mem : Antitone (· ∈ s) ↔ IsLowerSet s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: (Antitone fun x => x ∈ s) ↔ IsLowerSet s
-- isUpperSet_setOf
@[simp]
theorem isUpperSet_setOf : IsUpperSet { a | p a } ↔ Monotone p 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop}
Goal: IsUpperSet {a | p a} ↔ Monotone p
-- isLowerSet_setOf
@[simp]
theorem isLowerSet_setOf : IsLowerSet { a | p a } ↔ Antitone p 
Arguments: {α : Type u_1} [Preorder α] {p : α → Prop}
Goal: IsLowerSet {a | p a} ↔ Antitone p
-- IsUpperSet.upperBounds_subset
lemma IsUpperSet.upperBounds_subset (hs : IsUpperSet s) : s.Nonempty → upperBounds s ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (hs : IsUpperSet s)
Goal: s.Nonempty → upperBounds s ⊆ s
-- IsLowerSet.lowerBounds_subset
lemma IsLowerSet.lowerBounds_subset (hs : IsLowerSet s) : s.Nonempty → lowerBounds s ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (hs : IsLowerSet s)
Goal: s.Nonempty → lowerBounds s ⊆ s
-- IsLowerSet.top_mem
theorem IsLowerSet.top_mem (hs : IsLowerSet s) : ⊤ ∈ s ↔ s = univ 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderTop α] (hs : IsLowerSet s)
Goal: ⊤ ∈ s ↔ s = Set.univ
-- IsUpperSet.top_mem
theorem IsUpperSet.top_mem (hs : IsUpperSet s) : ⊤ ∈ s ↔ s.Nonempty 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderTop α] (hs : IsUpperSet s)
Goal: ⊤ ∈ s ↔ s.Nonempty
-- IsUpperSet.not_top_mem
theorem IsUpperSet.not_top_mem (hs : IsUpperSet s) : ⊤ ∉ s ↔ s = ∅ 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderTop α] (hs : IsUpperSet s)
Goal: ⊤ ∉ s ↔ s = ∅
-- IsUpperSet.bot_mem
theorem IsUpperSet.bot_mem (hs : IsUpperSet s) : ⊥ ∈ s ↔ s = univ 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderBot α] (hs : IsUpperSet s)
Goal: ⊥ ∈ s ↔ s = Set.univ
-- IsLowerSet.bot_mem
theorem IsLowerSet.bot_mem (hs : IsLowerSet s) : ⊥ ∈ s ↔ s.Nonempty 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderBot α] (hs : IsLowerSet s)
Goal: ⊥ ∈ s ↔ s.Nonempty
-- IsLowerSet.not_bot_mem
theorem IsLowerSet.not_bot_mem (hs : IsLowerSet s) : ⊥ ∉ s ↔ s = ∅ 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [OrderBot α] (hs : IsLowerSet s)
Goal: ⊥ ∉ s ↔ s = ∅
-- IsUpperSet.not_bddAbove
theorem IsUpperSet.not_bddAbove (hs : IsUpperSet s) : s.Nonempty → ¬BddAbove s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [NoMaxOrder α] (hs : IsUpperSet s)
Goal: s.Nonempty → ¬BddAbove s
-- not_bddAbove_Ici
theorem not_bddAbove_Ici : ¬BddAbove (Ici a) 
Arguments: {α : Type u_1} [Preorder α] (a : α) [NoMaxOrder α]
Goal: ¬BddAbove (Set.Ici a)
-- not_bddAbove_Ioi
theorem not_bddAbove_Ioi : ¬BddAbove (Ioi a) 
Arguments: {α : Type u_1} [Preorder α] (a : α) [NoMaxOrder α]
Goal: ¬BddAbove (Set.Ioi a)
-- IsLowerSet.not_bddBelow
theorem IsLowerSet.not_bddBelow (hs : IsLowerSet s) : s.Nonempty → ¬BddBelow s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [NoMinOrder α] (hs : IsLowerSet s)
Goal: s.Nonempty → ¬BddBelow s
-- not_bddBelow_Iic
theorem not_bddBelow_Iic : ¬BddBelow (Iic a) 
Arguments: {α : Type u_1} [Preorder α] (a : α) [NoMinOrder α]
Goal: ¬BddBelow (Set.Iic a)
-- not_bddBelow_Iio
theorem not_bddBelow_Iio : ¬BddBelow (Iio a) 
Arguments: {α : Type u_1} [Preorder α] (a : α) [NoMinOrder α]
Goal: ¬BddBelow (Set.Iio a)
-- isUpperSet_iff_forall_lt
theorem isUpperSet_iff_forall_lt : IsUpperSet s ↔ ∀ ⦃a b : α⦄, a < b → a ∈ s → b ∈ s 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α}
Goal: IsUpperSet s ↔ ∀ ⦃a b : α⦄, a < b → a ∈ s → b ∈ s
-- isLowerSet_iff_forall_lt
theorem isLowerSet_iff_forall_lt : IsLowerSet s ↔ ∀ ⦃a b : α⦄, b < a → a ∈ s → b ∈ s 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α}
Goal: IsLowerSet s ↔ ∀ ⦃a b : α⦄, b < a → a ∈ s → b ∈ s
-- isUpperSet_iff_Ioi_subset
theorem isUpperSet_iff_Ioi_subset : IsUpperSet s ↔ ∀ ⦃a⦄, a ∈ s → Ioi a ⊆ s 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α}
Goal: IsUpperSet s ↔ ∀ ⦃a : α⦄, a ∈ s → Set.Ioi a ⊆ s
-- isLowerSet_iff_Iio_subset
theorem isLowerSet_iff_Iio_subset : IsLowerSet s ↔ ∀ ⦃a⦄, a ∈ s → Iio a ⊆ s 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α}
Goal: IsLowerSet s ↔ ∀ ⦃a : α⦄, a ∈ s → Set.Iio a ⊆ s
-- Relation.Fibration.isLowerSet_image
lemma Fibration.isLowerSet_image [LE α] [LE β] (hf : Fibration (· ≤ ·) (· ≤ ·) f)
    {s : Set α} (hs : IsLowerSet s) : IsLowerSet (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [LE α] [LE β] (hf : Relation.Fibration (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) f) {s : Set α} (hs : IsLowerSet s)
Goal: IsLowerSet (f '' s)
-- Relation.fibration_iff_isLowerSet_image_Iic
lemma fibration_iff_isLowerSet_image_Iic [Preorder α] [LE β] :
    Fibration (· ≤ ·) (· ≤ ·) f ↔ ∀ x, IsLowerSet (f '' Iic x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [LE β]
Goal: Relation.Fibration (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) f ↔ ∀ (x : α), IsLowerSet (f '' Set.Iic x)
-- Relation.fibration_iff_isLowerSet_image
lemma fibration_iff_isLowerSet_image [Preorder α] [LE β] :
    Fibration (· ≤ ·) (· ≤ ·) f ↔ ∀ s, IsLowerSet s → IsLowerSet (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [LE β]
Goal: Relation.Fibration (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) f ↔ ∀ (s : Set α), IsLowerSet s → IsLowerSet (f '' s)
-- Relation.fibration_iff_image_Iic
lemma fibration_iff_image_Iic [Preorder α] [Preorder β] (hf : Monotone f) :
    Fibration (· ≤ ·) (· ≤ ·) f ↔ ∀ x, f '' Iic x = Iic (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] (hf : Monotone f)
Goal: Relation.Fibration (fun x x_1 => x ≤ x_1) (fun x x_1 => x ≤ x_1) f ↔ ∀ (x : α), f '' Set.Iic x = Set.Iic (f x)
-- Relation.Fibration.isUpperSet_image
lemma Fibration.isUpperSet_image [LE α] [LE β] (hf : Fibration (· ≥ ·) (· ≥ ·) f)
    {s : Set α} (hs : IsUpperSet s) : IsUpperSet (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [LE α] [LE β] (hf : Relation.Fibration (fun x x_1 => x ≥ x_1) (fun x x_1 => x ≥ x_1) f) {s : Set α} (hs : IsUpperSet s)
Goal: IsUpperSet (f '' s)
-- Relation.fibration_iff_isUpperSet_image_Ici
lemma fibration_iff_isUpperSet_image_Ici [Preorder α] [LE β] :
    Fibration (· ≥ ·) (· ≥ ·) f ↔ ∀ x, IsUpperSet (f '' Ici x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [LE β]
Goal: Relation.Fibration (fun x x_1 => x ≥ x_1) (fun x x_1 => x ≥ x_1) f ↔ ∀ (x : α), IsUpperSet (f '' Set.Ici x)
-- Relation.fibration_iff_isUpperSet_image
lemma fibration_iff_isUpperSet_image [Preorder α] [LE β] :
    Fibration (· ≥ ·) (· ≥ ·) f ↔ ∀ s, IsUpperSet s → IsUpperSet (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [LE β]
Goal: Relation.Fibration (fun x x_1 => x ≥ x_1) (fun x x_1 => x ≥ x_1) f ↔ ∀ (s : Set α), IsUpperSet s → IsUpperSet (f '' s)
-- Relation.fibration_iff_image_Ici
lemma fibration_iff_image_Ici [Preorder α] [Preorder β] (hf : Monotone f) :
    Fibration (· ≥ ·) (· ≥ ·) f ↔ ∀ x, f '' Ici x = Ici (f x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} [Preorder α] [Preorder β] (hf : Monotone f)
Goal: Relation.Fibration (fun x x_1 => x ≥ x_1) (fun x x_1 => x ≥ x_1) f ↔ ∀ (x : α), f '' Set.Ici x = Set.Ici (f x)
-- IsUpperSet.total
theorem IsUpperSet.total (hs : IsUpperSet s) (ht : IsUpperSet t) : s ⊆ t ∨ t ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α} (hs : IsUpperSet s) (ht : IsUpperSet t)
Goal: s ⊆ t ∨ t ⊆ s
-- IsLowerSet.total
theorem IsLowerSet.total (hs : IsLowerSet s) (ht : IsLowerSet t) : s ⊆ t ∨ t ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α} (hs : IsLowerSet s) (ht : IsLowerSet t)
Goal: s ⊆ t ∨ t ⊆ s
-- UpperSet.upper'
@[inherit_doc IsUpperSet]
structure UpperSet (α : Type*) [LE α] where
    carrier : Set α
    upper' : IsUpperSet carrier
Arguments: {α : Type u_6} [LE α] (self : UpperSet α)
Goal: IsUpperSet self.carrier
-- LowerSet.lower'
@[inherit_doc IsLowerSet]
structure LowerSet (α : Type*) [LE α] where
    carrier : Set α
    lower' : IsLowerSet carrier
Arguments: {α : Type u_6} [LE α] (self : LowerSet α)
Goal: IsLowerSet self.carrier
-- UpperSet.ext
@[ext]
theorem ext {s t : UpperSet α} : (s : Set α) = t → s = t 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α}
Goal: ↑s = ↑t → s = t
-- UpperSet.carrier_eq_coe
@[simp]
theorem carrier_eq_coe (s : UpperSet α) : s.carrier = s 
Arguments: {α : Type u_1} [LE α] (s : UpperSet α)
Goal: s.carrier = ↑s
-- UpperSet.upper
@[simp] protected lemma upper (s : UpperSet α) : IsUpperSet (s : Set α) 
Arguments: {α : Type u_1} [LE α] (s : UpperSet α)
Goal: IsUpperSet ↑s
-- UpperSet.coe_mk
@[simp, norm_cast] lemma coe_mk (s : Set α) (hs) : mk s hs = s 
Arguments: {α : Type u_1} [LE α] (s : Set α) (hs : IsUpperSet s)
Goal: ↑{ carrier := s, upper' := hs } = s
-- UpperSet.mem_mk
@[simp] lemma mem_mk {s : Set α} (hs) {a : α} : a ∈ mk s hs ↔ a ∈ s 
Arguments: {α : Type u_1} [LE α] {s : Set α} (hs : IsUpperSet s) {a : α}
Goal: a ∈ { carrier := s, upper' := hs } ↔ a ∈ s
-- LowerSet.ext
@[ext]
theorem ext {s t : LowerSet α} : (s : Set α) = t → s = t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α}
Goal: ↑s = ↑t → s = t
-- LowerSet.carrier_eq_coe
@[simp]
theorem carrier_eq_coe (s : LowerSet α) : s.carrier = s 
Arguments: {α : Type u_1} [LE α] (s : LowerSet α)
Goal: s.carrier = ↑s
-- LowerSet.lower
@[simp] protected lemma lower (s : LowerSet α) : IsLowerSet (s : Set α) 
Arguments: {α : Type u_1} [LE α] (s : LowerSet α)
Goal: IsLowerSet ↑s
-- LowerSet.coe_mk
@[simp, norm_cast] lemma coe_mk (s : Set α) (hs) : mk s hs = s 
Arguments: {α : Type u_1} [LE α] (s : Set α) (hs : IsLowerSet s)
Goal: ↑{ carrier := s, lower' := hs } = s
-- LowerSet.mem_mk
@[simp] lemma mem_mk {s : Set α} (hs) {a : α} : a ∈ mk s hs ↔ a ∈ s 
Arguments: {α : Type u_1} [LE α] {s : Set α} (hs : IsLowerSet s) {a : α}
Goal: a ∈ { carrier := s, lower' := hs } ↔ a ∈ s
-- UpperSet.coe_subset_coe
@[simp 1100, norm_cast]
theorem coe_subset_coe : (s : Set α) ⊆ t ↔ t ≤ s 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α}
Goal: ↑s ⊆ ↑t ↔ t ≤ s
-- UpperSet.coe_ssubset_coe
@[simp 1100, norm_cast] lemma coe_ssubset_coe : (s : Set α) ⊂ t ↔ t < s 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α}
Goal: ↑s ⊂ ↑t ↔ t < s
-- UpperSet.coe_top
@[simp, norm_cast]
theorem coe_top : ((⊤ : UpperSet α) : Set α) = ∅ 
Arguments: {α : Type u_1} [LE α]
Goal: ↑⊤ = ∅
-- UpperSet.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : UpperSet α) : Set α) = univ 
Arguments: {α : Type u_1} [LE α]
Goal: ↑⊥ = Set.univ
-- UpperSet.coe_eq_univ
@[simp, norm_cast]
theorem coe_eq_univ : (s : Set α) = univ ↔ s = ⊥ 
Arguments: {α : Type u_1} [LE α] {s : UpperSet α}
Goal: ↑s = Set.univ ↔ s = ⊥
-- UpperSet.coe_eq_empty
@[simp, norm_cast]
theorem coe_eq_empty : (s : Set α) = ∅ ↔ s = ⊤ 
Arguments: {α : Type u_1} [LE α] {s : UpperSet α}
Goal: ↑s = ∅ ↔ s = ⊤
-- UpperSet.coe_nonempty
@[simp, norm_cast] lemma coe_nonempty : (s : Set α).Nonempty ↔ s ≠ ⊤ 
Arguments: {α : Type u_1} [LE α] {s : UpperSet α}
Goal: (↑s).Nonempty ↔ s ≠ ⊤
-- UpperSet.coe_sup
@[simp, norm_cast]
theorem coe_sup (s t : UpperSet α) : (↑(s ⊔ t) : Set α) = (s : Set α) ∩ t 
Arguments: {α : Type u_1} [LE α] (s t : UpperSet α)
Goal: ↑(s ⊔ t) = ↑s ∩ ↑t
-- UpperSet.coe_inf
@[simp, norm_cast]
theorem coe_inf (s t : UpperSet α) : (↑(s ⊓ t) : Set α) = (s : Set α) ∪ t 
Arguments: {α : Type u_1} [LE α] (s t : UpperSet α)
Goal: ↑(s ⊓ t) = ↑s ∪ ↑t
-- UpperSet.coe_sSup
@[simp, norm_cast]
theorem coe_sSup (S : Set (UpperSet α)) : (↑(sSup S) : Set α) = ⋂ s ∈ S, ↑s 
Arguments: {α : Type u_1} [LE α] (S : Set (UpperSet α))
Goal: ↑(sSup S) = ⋂ s ∈ S, ↑s
-- UpperSet.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (UpperSet α)) : (↑(sInf S) : Set α) = ⋃ s ∈ S, ↑s 
Arguments: {α : Type u_1} [LE α] (S : Set (UpperSet α))
Goal: ↑(sInf S) = ⋃ s ∈ S, ↑s
-- UpperSet.coe_iSup
@[simp, norm_cast]
theorem coe_iSup (f : ι → UpperSet α) : (↑(⨆ i, f i) : Set α) = ⋂ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → UpperSet α)
Goal: ↑(⨆ i, f i) = ⋂ i, ↑(f i)
-- UpperSet.coe_iInf
@[simp, norm_cast]
theorem coe_iInf (f : ι → UpperSet α) : (↑(⨅ i, f i) : Set α) = ⋃ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → UpperSet α)
Goal: ↑(⨅ i, f i) = ⋃ i, ↑(f i)
-- UpperSet.coe_iSup₂
@[norm_cast] -- Porting note: no longer a `simp`
theorem coe_iSup₂ (f : ∀ i, κ i → UpperSet α) :
    (↑(⨆ (i) (j), f i j) : Set α) = ⋂ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → UpperSet α)
Goal: ↑(⨆ i, ⨆ j, f i j) = ⋂ i, ⋂ j, ↑(f i j)
-- UpperSet.coe_iInf₂
@[norm_cast] -- Porting note: no longer a `simp`
theorem coe_iInf₂ (f : ∀ i, κ i → UpperSet α) :
    (↑(⨅ (i) (j), f i j) : Set α) = ⋃ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → UpperSet α)
Goal: ↑(⨅ i, ⨅ j, f i j) = ⋃ i, ⋃ j, ↑(f i j)