ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- mem_lowerBounds_image2
theorem mem_lowerBounds_image2 (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t) :
    f a b ∈ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t)
Goal: f a b ∈ lowerBounds (Set.image2 f s t)
-- image2_upperBounds_upperBounds_subset
theorem image2_upperBounds_upperBounds_subset :
    image2 f (upperBounds s) (upperBounds t) ⊆ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: Set.image2 f (upperBounds s) (upperBounds t) ⊆ upperBounds (Set.image2 f s t)
-- image2_lowerBounds_lowerBounds_subset
theorem image2_lowerBounds_lowerBounds_subset :
    image2 f (lowerBounds s) (lowerBounds t) ⊆ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: Set.image2 f (lowerBounds s) (lowerBounds t) ⊆ lowerBounds (Set.image2 f s t)
-- BddAbove.image2
protected theorem BddAbove.image2 (h₀ : ∀ b, Monotone (swap f b)) (h₁ : ∀ a, Monotone (f a)) :
    BddAbove s → BddAbove t → BddAbove (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: BddAbove s → BddAbove t → BddAbove (Set.image2 f s t)
-- BddBelow.image2
protected theorem BddBelow.image2 (h₀ : ∀ b, Monotone (swap f b)) (h₁ : ∀ a, Monotone (f a)) :
    BddBelow s → BddBelow t → BddBelow (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: BddBelow s → BddBelow t → BddBelow (Set.image2 f s t)
-- IsGreatest.image2
protected theorem IsGreatest.image2 (ha : IsGreatest s a) (hb : IsGreatest t b) :
    IsGreatest (image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : IsGreatest s a) (hb : IsGreatest t b)
Goal: IsGreatest (Set.image2 f s t) (f a b)
-- IsLeast.image2
protected theorem IsLeast.image2 (ha : IsLeast s a) (hb : IsLeast t b) :
    IsLeast (image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : IsLeast s a) (hb : IsLeast t b)
Goal: IsLeast (Set.image2 f s t) (f a b)
-- mem_upperBounds_image2_of_mem_upperBounds_of_mem_lowerBounds
theorem mem_upperBounds_image2_of_mem_upperBounds_of_mem_lowerBounds (ha : a ∈ upperBounds s)
    (hb : b ∈ lowerBounds t) : f a b ∈ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : a ∈ upperBounds s) (hb : b ∈ lowerBounds t)
Goal: f a b ∈ upperBounds (Set.image2 f s t)
-- mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds
theorem mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_upperBounds (ha : a ∈ lowerBounds s)
    (hb : b ∈ upperBounds t) : f a b ∈ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : a ∈ lowerBounds s) (hb : b ∈ upperBounds t)
Goal: f a b ∈ lowerBounds (Set.image2 f s t)
-- image2_upperBounds_lowerBounds_subset_upperBounds_image2
theorem image2_upperBounds_lowerBounds_subset_upperBounds_image2 :
    image2 f (upperBounds s) (lowerBounds t) ⊆ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: Set.image2 f (upperBounds s) (lowerBounds t) ⊆ upperBounds (Set.image2 f s t)
-- image2_lowerBounds_upperBounds_subset_lowerBounds_image2
theorem image2_lowerBounds_upperBounds_subset_lowerBounds_image2 :
    image2 f (lowerBounds s) (upperBounds t) ⊆ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: Set.image2 f (lowerBounds s) (upperBounds t) ⊆ lowerBounds (Set.image2 f s t)
-- BddAbove.bddAbove_image2_of_bddBelow
theorem BddAbove.bddAbove_image2_of_bddBelow :
    BddAbove s → BddBelow t → BddAbove (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: BddAbove s → BddBelow t → BddAbove (Set.image2 f s t)
-- BddBelow.bddBelow_image2_of_bddAbove
theorem BddBelow.bddBelow_image2_of_bddAbove :
    BddBelow s → BddAbove t → BddBelow (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: BddBelow s → BddAbove t → BddBelow (Set.image2 f s t)
-- IsGreatest.isGreatest_image2_of_isLeast
theorem IsGreatest.isGreatest_image2_of_isLeast (ha : IsGreatest s a) (hb : IsLeast t b) :
    IsGreatest (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : IsGreatest s a) (hb : IsLeast t b)
Goal: IsGreatest (Set.image2 f s t) (f a b)
-- IsLeast.isLeast_image2_of_isGreatest
theorem IsLeast.isLeast_image2_of_isGreatest (ha : IsLeast s a) (hb : IsGreatest t b) :
    IsLeast (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Monotone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : IsLeast s a) (hb : IsGreatest t b)
Goal: IsLeast (Set.image2 f s t) (f a b)
-- mem_upperBounds_image2_of_mem_lowerBounds
theorem mem_upperBounds_image2_of_mem_lowerBounds (ha : a ∈ lowerBounds s)
    (hb : b ∈ lowerBounds t) : f a b ∈ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : a ∈ lowerBounds s) (hb : b ∈ lowerBounds t)
Goal: f a b ∈ upperBounds (Set.image2 f s t)
-- mem_lowerBounds_image2_of_mem_upperBounds
theorem mem_lowerBounds_image2_of_mem_upperBounds (ha : a ∈ upperBounds s)
    (hb : b ∈ upperBounds t) : f a b ∈ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : a ∈ upperBounds s) (hb : b ∈ upperBounds t)
Goal: f a b ∈ lowerBounds (Set.image2 f s t)
-- image2_upperBounds_upperBounds_subset_upperBounds_image2
theorem image2_upperBounds_upperBounds_subset_upperBounds_image2 :
    image2 f (lowerBounds s) (lowerBounds t) ⊆ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: Set.image2 f (lowerBounds s) (lowerBounds t) ⊆ upperBounds (Set.image2 f s t)
-- image2_lowerBounds_lowerBounds_subset_lowerBounds_image2
theorem image2_lowerBounds_lowerBounds_subset_lowerBounds_image2 :
    image2 f (upperBounds s) (upperBounds t) ⊆ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: Set.image2 f (upperBounds s) (upperBounds t) ⊆ lowerBounds (Set.image2 f s t)
-- BddBelow.image2_bddAbove
theorem BddBelow.image2_bddAbove : BddBelow s → BddBelow t → BddAbove (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: BddBelow s → BddBelow t → BddAbove (Set.image2 f s t)
-- BddAbove.image2_bddBelow
theorem BddAbove.image2_bddBelow : BddAbove s → BddAbove t → BddBelow (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a))
Goal: BddAbove s → BddAbove t → BddBelow (Set.image2 f s t)
-- IsLeast.isGreatest_image2
theorem IsLeast.isGreatest_image2 (ha : IsLeast s a) (hb : IsLeast t b) :
    IsGreatest (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : IsLeast s a) (hb : IsLeast t b)
Goal: IsGreatest (Set.image2 f s t) (f a b)
-- IsGreatest.isLeast_image2
theorem IsGreatest.isLeast_image2 (ha : IsGreatest s a) (hb : IsGreatest t b) :
    IsLeast (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Antitone (f a)) (ha : IsGreatest s a) (hb : IsGreatest t b)
Goal: IsLeast (Set.image2 f s t) (f a b)
-- mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds
theorem mem_upperBounds_image2_of_mem_upperBounds_of_mem_upperBounds (ha : a ∈ lowerBounds s)
    (hb : b ∈ upperBounds t) : f a b ∈ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : a ∈ lowerBounds s) (hb : b ∈ upperBounds t)
Goal: f a b ∈ upperBounds (Set.image2 f s t)
-- mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_lowerBounds
theorem mem_lowerBounds_image2_of_mem_lowerBounds_of_mem_lowerBounds (ha : a ∈ upperBounds s)
    (hb : b ∈ lowerBounds t) : f a b ∈ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : a ∈ upperBounds s) (hb : b ∈ lowerBounds t)
Goal: f a b ∈ lowerBounds (Set.image2 f s t)
-- image2_lowerBounds_upperBounds_subset_upperBounds_image2
theorem image2_lowerBounds_upperBounds_subset_upperBounds_image2 :
    image2 f (lowerBounds s) (upperBounds t) ⊆ upperBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: Set.image2 f (lowerBounds s) (upperBounds t) ⊆ upperBounds (Set.image2 f s t)
-- image2_upperBounds_lowerBounds_subset_lowerBounds_image2
theorem image2_upperBounds_lowerBounds_subset_lowerBounds_image2 :
    image2 f (upperBounds s) (lowerBounds t) ⊆ lowerBounds (image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: Set.image2 f (upperBounds s) (lowerBounds t) ⊆ lowerBounds (Set.image2 f s t)
-- BddBelow.bddAbove_image2_of_bddAbove
theorem BddBelow.bddAbove_image2_of_bddAbove :
    BddBelow s → BddAbove t → BddAbove (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: BddBelow s → BddAbove t → BddAbove (Set.image2 f s t)
-- BddAbove.bddBelow_image2_of_bddAbove
theorem BddAbove.bddBelow_image2_of_bddAbove :
    BddAbove s → BddBelow t → BddBelow (Set.image2 f s t) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a))
Goal: BddAbove s → BddBelow t → BddBelow (Set.image2 f s t)
-- IsLeast.isGreatest_image2_of_isGreatest
theorem IsLeast.isGreatest_image2_of_isGreatest (ha : IsLeast s a) (hb : IsGreatest t b) :
    IsGreatest (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : IsLeast s a) (hb : IsGreatest t b)
Goal: IsGreatest (Set.image2 f s t) (f a b)
-- IsGreatest.isLeast_image2_of_isLeast
theorem IsGreatest.isLeast_image2_of_isLeast (ha : IsGreatest s a) (hb : IsLeast t b) :
    IsLeast (Set.image2 f s t) (f a b) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} [Preorder α] [Preorder β] [Preorder γ] {f : α → β → γ} {s : Set α} {t : Set β} {a : α} {b : β} (h₀ : ∀ (b : β), Antitone (Function.swap f b)) (h₁ : ∀ (a : α), Monotone (f a)) (ha : IsGreatest s a) (hb : IsLeast t b)
Goal: IsLeast (Set.image2 f s t) (f a b)
-- bddAbove_prod
lemma bddAbove_prod {s : Set (α × β)} :
    BddAbove s ↔ BddAbove (Prod.fst '' s) ∧ BddAbove (Prod.snd '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set (α × β)}
Goal: BddAbove s ↔ BddAbove (Prod.fst '' s) ∧ BddAbove (Prod.snd '' s)
-- bddBelow_prod
lemma bddBelow_prod {s : Set (α × β)} :
    BddBelow s ↔ BddBelow (Prod.fst '' s) ∧ BddBelow (Prod.snd '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set (α × β)}
Goal: BddBelow s ↔ BddBelow (Prod.fst '' s) ∧ BddBelow (Prod.snd '' s)
-- bddAbove_range_prod
lemma bddAbove_range_prod {F : ι → α × β} :
    BddAbove (range F) ↔ BddAbove (range <| Prod.fst ∘ F) ∧ BddAbove (range <| Prod.snd ∘ F) 
Arguments: {ι : Sort x} {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {F : ι → α × β}
Goal: BddAbove (Set.range F) ↔ BddAbove (Set.range (Prod.fst ∘ F)) ∧ BddAbove (Set.range (Prod.snd ∘ F))
-- bddBelow_range_prod
lemma bddBelow_range_prod {F : ι → α × β} :
    BddBelow (range F) ↔ BddBelow (range <| Prod.fst ∘ F) ∧ BddBelow (range <| Prod.snd ∘ F) 
Arguments: {ι : Sort x} {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {F : ι → α × β}
Goal: BddBelow (Set.range F) ↔ BddBelow (Set.range (Prod.fst ∘ F)) ∧ BddBelow (Set.range (Prod.snd ∘ F))
-- isLUB_prod
theorem isLUB_prod {s : Set (α × β)} (p : α × β) :
    IsLUB s p ↔ IsLUB (Prod.fst '' s) p.1 ∧ IsLUB (Prod.snd '' s) p.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set (α × β)} (p : α × β)
Goal: IsLUB s p ↔ IsLUB (Prod.fst '' s) p.1 ∧ IsLUB (Prod.snd '' s) p.2
-- isGLB_prod
theorem isGLB_prod {s : Set (α × β)} (p : α × β) :
    IsGLB s p ↔ IsGLB (Prod.fst '' s) p.1 ∧ IsGLB (Prod.snd '' s) p.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set (α × β)} (p : α × β)
Goal: IsGLB s p ↔ IsGLB (Prod.fst '' s) p.1 ∧ IsGLB (Prod.snd '' s) p.2
-- bddAbove_pi
lemma bddAbove_pi {s : Set (∀ a, π a)} :
    BddAbove s ↔ ∀ a, BddAbove (Function.eval a '' s) 
Arguments: {α : Type u} {π : α → Type u_1} [(a : α) → Preorder (π a)] {s : Set ((a : α) → π a)}
Goal: BddAbove s ↔ ∀ (a : α), BddAbove (Function.eval a '' s)
-- bddBelow_pi
lemma bddBelow_pi {s : Set (∀ a, π a)} :
    BddBelow s ↔ ∀ a, BddBelow (Function.eval a '' s) 
Arguments: {α : Type u} {π : α → Type u_1} [(a : α) → Preorder (π a)] {s : Set ((a : α) → π a)}
Goal: BddBelow s ↔ ∀ (a : α), BddBelow (Function.eval a '' s)
-- bddAbove_range_pi
lemma bddAbove_range_pi {F : ι → ∀ a, π a} :
    BddAbove (range F) ↔ ∀ a, BddAbove (range fun i ↦ F i a) 
Arguments: {α : Type u} {ι : Sort x} {π : α → Type u_1} [(a : α) → Preorder (π a)] {F : ι → (a : α) → π a}
Goal: BddAbove (Set.range F) ↔ ∀ (a : α), BddAbove (Set.range fun i => F i a)
-- bddBelow_range_pi
lemma bddBelow_range_pi {F : ι → ∀ a, π a} :
    BddBelow (range F) ↔ ∀ a, BddBelow (range fun i ↦ F i a) 
Arguments: {α : Type u} {ι : Sort x} {π : α → Type u_1} [(a : α) → Preorder (π a)] {F : ι → (a : α) → π a}
Goal: BddBelow (Set.range F) ↔ ∀ (a : α), BddBelow (Set.range fun i => F i a)
-- isLUB_pi
theorem isLUB_pi {s : Set (∀ a, π a)} {f : ∀ a, π a} :
    IsLUB s f ↔ ∀ a, IsLUB (Function.eval a '' s) (f a) 
Arguments: {α : Type u} {π : α → Type u_1} [(a : α) → Preorder (π a)] {s : Set ((a : α) → π a)} {f : (a : α) → π a}
Goal: IsLUB s f ↔ ∀ (a : α), IsLUB (Function.eval a '' s) (f a)
-- isGLB_pi
theorem isGLB_pi {s : Set (∀ a, π a)} {f : ∀ a, π a} :
    IsGLB s f ↔ ∀ a, IsGLB (Function.eval a '' s) (f a) 
Arguments: {α : Type u} {π : α → Type u_1} [(a : α) → Preorder (π a)] {s : Set ((a : α) → π a)} {f : (a : α) → π a}
Goal: IsGLB s f ↔ ∀ (a : α), IsGLB (Function.eval a '' s) (f a)
-- IsGLB.of_image
theorem IsGLB.of_image [Preorder α] [Preorder β] {f : α → β} (hf : ∀ {x y}, f x ≤ f y ↔ x ≤ y)
    {s : Set α} {x : α} (hx : IsGLB (f '' s) (f x)) : IsGLB s x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y) {s : Set α} {x : α} (hx : IsGLB (f '' s) (f x))
Goal: IsGLB s x
-- IsLUB.of_image
theorem IsLUB.of_image [Preorder α] [Preorder β] {f : α → β} (hf : ∀ {x y}, f x ≤ f y ↔ x ≤ y)
    {s : Set α} {x : α} (hx : IsLUB (f '' s) (f x)) : IsLUB s x 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (hf : ∀ {x y : α}, f x ≤ f y ↔ x ≤ y) {s : Set α} {x : α} (hx : IsLUB (f '' s) (f x))
Goal: IsLUB s x
-- BddAbove.range_mono
lemma BddAbove.range_mono [Preorder β] {f : α → β} (g : α → β) (h : ∀ a, f a ≤ g a)
    (hbdd : BddAbove (range g)) : BddAbove (range f) 
Arguments: {α : Type u} {β : Type v} [Preorder β] {f : α → β} (g : α → β) (h : ∀ (a : α), f a ≤ g a) (hbdd : BddAbove (Set.range g))
Goal: BddAbove (Set.range f)
-- BddBelow.range_mono
lemma BddBelow.range_mono [Preorder β] (f : α → β) {g : α → β} (h : ∀ a, f a ≤ g a)
    (hbdd : BddBelow (range f)) : BddBelow (range g) 
Arguments: {α : Type u} {β : Type v} [Preorder β] (f : α → β) {g : α → β} (h : ∀ (a : α), f a ≤ g a) (hbdd : BddBelow (Set.range f))
Goal: BddBelow (Set.range g)
-- BddAbove.range_comp
lemma BddAbove.range_comp {γ : Type*} [Preorder β] [Preorder γ] {f : α → β} {g : β → γ}
    (hf : BddAbove (range f)) (hg : Monotone g) : BddAbove (range (fun x => g (f x))) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} [Preorder β] [Preorder γ] {f : α → β} {g : β → γ} (hf : BddAbove (Set.range f)) (hg : Monotone g)
Goal: BddAbove (Set.range fun x => g (f x))
-- BddBelow.range_comp
lemma BddBelow.range_comp {γ : Type*} [Preorder β] [Preorder γ] {f : α → β} {g : β → γ}
    (hf : BddBelow (range f)) (hg : Monotone g) : BddBelow (range (fun x => g (f x))) 
Arguments: {α : Type u} {β : Type v} {γ : Type u_1} [Preorder β] [Preorder γ] {f : α → β} {g : β → γ} (hf : BddBelow (Set.range f)) (hg : Monotone g)
Goal: BddBelow (Set.range fun x => g (f x))
-- ScottContinuous.monotone
protected theorem ScottContinuous.monotone (h : ScottContinuous f) : Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} (h : ScottContinuous f)
Goal: Monotone f
-- antitone_Ici
theorem antitone_Ici : Antitone (Ici : α → Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Antitone Set.Ici
-- monotone_Iic
theorem monotone_Iic : Monotone (Iic : α → Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Monotone Set.Iic
-- antitone_Ioi
theorem antitone_Ioi : Antitone (Ioi : α → Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Antitone Set.Ioi
-- monotone_Iio
theorem monotone_Iio : Monotone (Iio : α → Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Monotone Set.Iio
-- Monotone.Ici
protected theorem Monotone.Ici (hf : Monotone f) : Antitone fun x => Ici (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Antitone fun x => Set.Ici (f x)
-- MonotoneOn.Ici
protected theorem MonotoneOn.Ici (hf : MonotoneOn f s) : AntitoneOn (fun x => Ici (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : MonotoneOn f s)
Goal: AntitoneOn (fun x => Set.Ici (f x)) s
-- Antitone.Ici
protected theorem Antitone.Ici (hf : Antitone f) : Monotone fun x => Ici (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f)
Goal: Monotone fun x => Set.Ici (f x)
-- AntitoneOn.Ici
protected theorem AntitoneOn.Ici (hf : AntitoneOn f s) : MonotoneOn (fun x => Ici (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : AntitoneOn f s)
Goal: MonotoneOn (fun x => Set.Ici (f x)) s
-- Monotone.Iic
protected theorem Monotone.Iic (hf : Monotone f) : Monotone fun x => Iic (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Monotone fun x => Set.Iic (f x)
-- MonotoneOn.Iic
protected theorem MonotoneOn.Iic (hf : MonotoneOn f s) : MonotoneOn (fun x => Iic (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : MonotoneOn f s)
Goal: MonotoneOn (fun x => Set.Iic (f x)) s
-- Antitone.Iic
protected theorem Antitone.Iic (hf : Antitone f) : Antitone fun x => Iic (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f)
Goal: Antitone fun x => Set.Iic (f x)
-- AntitoneOn.Iic
protected theorem AntitoneOn.Iic (hf : AntitoneOn f s) : AntitoneOn (fun x => Iic (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : AntitoneOn f s)
Goal: AntitoneOn (fun x => Set.Iic (f x)) s
-- Monotone.Ioi
protected theorem Monotone.Ioi (hf : Monotone f) : Antitone fun x => Ioi (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Antitone fun x => Set.Ioi (f x)
-- MonotoneOn.Ioi
protected theorem MonotoneOn.Ioi (hf : MonotoneOn f s) : AntitoneOn (fun x => Ioi (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : MonotoneOn f s)
Goal: AntitoneOn (fun x => Set.Ioi (f x)) s
-- Antitone.Ioi
protected theorem Antitone.Ioi (hf : Antitone f) : Monotone fun x => Ioi (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Antitone f)
Goal: Monotone fun x => Set.Ioi (f x)
-- AntitoneOn.Ioi
protected theorem AntitoneOn.Ioi (hf : AntitoneOn f s) : MonotoneOn (fun x => Ioi (f x)) s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} {s : Set α} (hf : AntitoneOn f s)
Goal: MonotoneOn (fun x => Set.Ioi (f x)) s
-- Monotone.Iio
protected theorem Monotone.Iio (hf : Monotone f) : Monotone fun x => Iio (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α → β} (hf : Monotone f)
Goal: Monotone fun x => Set.Iio (f x)