ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AntivaryOn.subset
protected theorem AntivaryOn.subset (hst : s ⊆ t) (h : AntivaryOn f g t) : AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s t : Set ι} (hst : s ⊆ t) (h : AntivaryOn f g t)
Goal: AntivaryOn f g s
-- monovary_const_left
theorem monovary_const_left (g : ι → β) (a : α) : Monovary (const ι a) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (g : ι → β) (a : α)
Goal: Monovary (Function.const ι a) g
-- antivary_const_left
theorem antivary_const_left (g : ι → β) (a : α) : Antivary (const ι a) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (g : ι → β) (a : α)
Goal: Antivary (Function.const ι a) g
-- monovary_const_right
theorem monovary_const_right (f : ι → α) (b : β) : Monovary f (const ι b) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : ι → α) (b : β)
Goal: Monovary f (Function.const ι b)
-- antivary_const_right
theorem antivary_const_right (f : ι → α) (b : β) : Antivary f (const ι b) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : ι → α) (b : β)
Goal: Antivary f (Function.const ι b)
-- monovary_self
theorem monovary_self (f : ι → α) : Monovary f f 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] (f : ι → α)
Goal: Monovary f f
-- monovaryOn_self
theorem monovaryOn_self (f : ι → α) (s : Set ι) : MonovaryOn f f s 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] (f : ι → α) (s : Set ι)
Goal: MonovaryOn f f s
-- Subsingleton.monovary
protected theorem Subsingleton.monovary [Subsingleton ι] (f : ι → α) (g : ι → β) : Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Subsingleton ι] (f : ι → α) (g : ι → β)
Goal: Monovary f g
-- Subsingleton.antivary
protected theorem Subsingleton.antivary [Subsingleton ι] (f : ι → α) (g : ι → β) : Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Subsingleton ι] (f : ι → α) (g : ι → β)
Goal: Antivary f g
-- Subsingleton.monovaryOn
protected theorem Subsingleton.monovaryOn [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι) :
    MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι)
Goal: MonovaryOn f g s
-- Subsingleton.antivaryOn
protected theorem Subsingleton.antivaryOn [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι) :
    AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι)
Goal: AntivaryOn f g s
-- monovaryOn_const_left
theorem monovaryOn_const_left (g : ι → β) (a : α) (s : Set ι) : MonovaryOn (const ι a) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (g : ι → β) (a : α) (s : Set ι)
Goal: MonovaryOn (Function.const ι a) g s
-- antivaryOn_const_left
theorem antivaryOn_const_left (g : ι → β) (a : α) (s : Set ι) : AntivaryOn (const ι a) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (g : ι → β) (a : α) (s : Set ι)
Goal: AntivaryOn (Function.const ι a) g s
-- monovaryOn_const_right
theorem monovaryOn_const_right (f : ι → α) (b : β) (s : Set ι) : MonovaryOn f (const ι b) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : ι → α) (b : β) (s : Set ι)
Goal: MonovaryOn f (Function.const ι b) s
-- antivaryOn_const_right
theorem antivaryOn_const_right (f : ι → α) (b : β) (s : Set ι) : AntivaryOn f (const ι b) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] (f : ι → α) (b : β) (s : Set ι)
Goal: AntivaryOn f (Function.const ι b) s
-- Monovary.comp_right
theorem Monovary.comp_right (h : Monovary f g) (k : ι' → ι) : Monovary (f ∘ k) (g ∘ k) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} (h : Monovary f g) (k : ι' → ι)
Goal: Monovary (f ∘ k) (g ∘ k)
-- Antivary.comp_right
theorem Antivary.comp_right (h : Antivary f g) (k : ι' → ι) : Antivary (f ∘ k) (g ∘ k) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} (h : Antivary f g) (k : ι' → ι)
Goal: Antivary (f ∘ k) (g ∘ k)
-- MonovaryOn.comp_right
theorem MonovaryOn.comp_right (h : MonovaryOn f g s) (k : ι' → ι) :
    MonovaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} (h : MonovaryOn f g s) (k : ι' → ι)
Goal: MonovaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s)
-- AntivaryOn.comp_right
theorem AntivaryOn.comp_right (h : AntivaryOn f g s) (k : ι' → ι) :
    AntivaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} (h : AntivaryOn f g s) (k : ι' → ι)
Goal: AntivaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s)
-- Monovary.comp_monotone_left
theorem Monovary.comp_monotone_left (h : Monovary f g) (hf : Monotone f') : Monovary (f' ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} (h : Monovary f g) (hf : Monotone f')
Goal: Monovary (f' ∘ f) g
-- Monovary.comp_antitone_left
theorem Monovary.comp_antitone_left (h : Monovary f g) (hf : Antitone f') : Antivary (f' ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} (h : Monovary f g) (hf : Antitone f')
Goal: Antivary (f' ∘ f) g
-- Antivary.comp_monotone_left
theorem Antivary.comp_monotone_left (h : Antivary f g) (hf : Monotone f') : Antivary (f' ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} (h : Antivary f g) (hf : Monotone f')
Goal: Antivary (f' ∘ f) g
-- Antivary.comp_antitone_left
theorem Antivary.comp_antitone_left (h : Antivary f g) (hf : Antitone f') : Monovary (f' ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} (h : Antivary f g) (hf : Antitone f')
Goal: Monovary (f' ∘ f) g
-- MonovaryOn.comp_monotone_on_left
theorem MonovaryOn.comp_monotone_on_left (h : MonovaryOn f g s) (hf : Monotone f') :
    MonovaryOn (f' ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {s : Set ι} (h : MonovaryOn f g s) (hf : Monotone f')
Goal: MonovaryOn (f' ∘ f) g s
-- MonovaryOn.comp_antitone_on_left
theorem MonovaryOn.comp_antitone_on_left (h : MonovaryOn f g s) (hf : Antitone f') :
    AntivaryOn (f' ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {s : Set ι} (h : MonovaryOn f g s) (hf : Antitone f')
Goal: AntivaryOn (f' ∘ f) g s
-- AntivaryOn.comp_monotone_on_left
theorem AntivaryOn.comp_monotone_on_left (h : AntivaryOn f g s) (hf : Monotone f') :
    AntivaryOn (f' ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {s : Set ι} (h : AntivaryOn f g s) (hf : Monotone f')
Goal: AntivaryOn (f' ∘ f) g s
-- AntivaryOn.comp_antitone_on_left
theorem AntivaryOn.comp_antitone_on_left (h : AntivaryOn f g s) (hf : Antitone f') :
    MonovaryOn (f' ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [Preorder β] [Preorder γ] {f : ι → α} {f' : α → γ} {g : ι → β} {s : Set ι} (h : AntivaryOn f g s) (hf : Antitone f')
Goal: MonovaryOn (f' ∘ f) g s
-- Monovary.dual
theorem Monovary.dual : Monovary f g → Monovary (toDual ∘ f) (toDual ∘ g) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Monovary f g → Monovary (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g)
-- Antivary.dual
theorem Antivary.dual : Antivary f g → Antivary (toDual ∘ f) (toDual ∘ g) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Antivary f g → Antivary (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g)
-- Monovary.dual_left
theorem Monovary.dual_left : Monovary f g → Antivary (toDual ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Monovary f g → Antivary (⇑OrderDual.toDual ∘ f) g
-- Antivary.dual_left
theorem Antivary.dual_left : Antivary f g → Monovary (toDual ∘ f) g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Antivary f g → Monovary (⇑OrderDual.toDual ∘ f) g
-- Monovary.dual_right
theorem Monovary.dual_right : Monovary f g → Antivary f (toDual ∘ g) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Monovary f g → Antivary f (⇑OrderDual.toDual ∘ g)
-- Antivary.dual_right
theorem Antivary.dual_right : Antivary f g → Monovary f (toDual ∘ g) 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Antivary f g → Monovary f (⇑OrderDual.toDual ∘ g)
-- MonovaryOn.dual
theorem MonovaryOn.dual : MonovaryOn f g s → MonovaryOn (toDual ∘ f) (toDual ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f g s → MonovaryOn (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g) s
-- AntivaryOn.dual
theorem AntivaryOn.dual : AntivaryOn f g s → AntivaryOn (toDual ∘ f) (toDual ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f g s → AntivaryOn (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g) s
-- MonovaryOn.dual_left
theorem MonovaryOn.dual_left : MonovaryOn f g s → AntivaryOn (toDual ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f g s → AntivaryOn (⇑OrderDual.toDual ∘ f) g s
-- AntivaryOn.dual_left
theorem AntivaryOn.dual_left : AntivaryOn f g s → MonovaryOn (toDual ∘ f) g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f g s → MonovaryOn (⇑OrderDual.toDual ∘ f) g s
-- MonovaryOn.dual_right
theorem MonovaryOn.dual_right : MonovaryOn f g s → AntivaryOn f (toDual ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f g s → AntivaryOn f (⇑OrderDual.toDual ∘ g) s
-- AntivaryOn.dual_right
theorem AntivaryOn.dual_right : AntivaryOn f g s → MonovaryOn f (toDual ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f g s → MonovaryOn f (⇑OrderDual.toDual ∘ g) s
-- monovary_toDual_left
@[simp]
theorem monovary_toDual_left : Monovary (toDual ∘ f) g ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Monovary (⇑OrderDual.toDual ∘ f) g ↔ Antivary f g
-- monovary_toDual_right
@[simp]
theorem monovary_toDual_right : Monovary f (toDual ∘ g) ↔ Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Monovary f (⇑OrderDual.toDual ∘ g) ↔ Antivary f g
-- antivary_toDual_left
@[simp]
theorem antivary_toDual_left : Antivary (toDual ∘ f) g ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Antivary (⇑OrderDual.toDual ∘ f) g ↔ Monovary f g
-- antivary_toDual_right
@[simp]
theorem antivary_toDual_right : Antivary f (toDual ∘ g) ↔ Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β}
Goal: Antivary f (⇑OrderDual.toDual ∘ g) ↔ Monovary f g
-- monovaryOn_toDual_left
@[simp]
theorem monovaryOn_toDual_left : MonovaryOn (toDual ∘ f) g s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn (⇑OrderDual.toDual ∘ f) g s ↔ AntivaryOn f g s
-- monovaryOn_toDual_right
@[simp]
theorem monovaryOn_toDual_right : MonovaryOn f (toDual ∘ g) s ↔ AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f (⇑OrderDual.toDual ∘ g) s ↔ AntivaryOn f g s
-- antivaryOn_toDual_left
@[simp]
theorem antivaryOn_toDual_left : AntivaryOn (toDual ∘ f) g s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn (⇑OrderDual.toDual ∘ f) g s ↔ MonovaryOn f g s
-- antivaryOn_toDual_right
@[simp]
theorem antivaryOn_toDual_right : AntivaryOn f (toDual ∘ g) s ↔ MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f (⇑OrderDual.toDual ∘ g) s ↔ MonovaryOn f g s
-- monovary_id_iff
@[simp]
theorem monovary_id_iff : Monovary f id ↔ Monotone f 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] {f : ι → α} [PartialOrder ι]
Goal: Monovary f id ↔ Monotone f
-- antivary_id_iff
@[simp]
theorem antivary_id_iff : Antivary f id ↔ Antitone f 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] {f : ι → α} [PartialOrder ι]
Goal: Antivary f id ↔ Antitone f
-- monovaryOn_id_iff
@[simp]
theorem monovaryOn_id_iff : MonovaryOn f id s ↔ MonotoneOn f s 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] {f : ι → α} {s : Set ι} [PartialOrder ι]
Goal: MonovaryOn f id s ↔ MonotoneOn f s
-- antivaryOn_id_iff
@[simp]
theorem antivaryOn_id_iff : AntivaryOn f id s ↔ AntitoneOn f s 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder α] {f : ι → α} {s : Set ι} [PartialOrder ι]
Goal: AntivaryOn f id s ↔ AntitoneOn f s
-- Monotone.monovary
protected theorem Monotone.monovary (hf : Monotone f) (hg : Monotone g) : Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} [LinearOrder ι] (hf : Monotone f) (hg : Monotone g)
Goal: Monovary f g
-- Monotone.antivary
protected theorem Monotone.antivary (hf : Monotone f) (hg : Antitone g) : Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} [LinearOrder ι] (hf : Monotone f) (hg : Antitone g)
Goal: Antivary f g
-- Antitone.monovary
protected theorem Antitone.monovary (hf : Antitone f) (hg : Antitone g) : Monovary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} [LinearOrder ι] (hf : Antitone f) (hg : Antitone g)
Goal: Monovary f g
-- Antitone.antivary
protected theorem Antitone.antivary (hf : Antitone f) (hg : Monotone g) : Antivary f g 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} [LinearOrder ι] (hf : Antitone f) (hg : Monotone g)
Goal: Antivary f g
-- MonotoneOn.monovaryOn
protected theorem MonotoneOn.monovaryOn (hf : MonotoneOn f s) (hg : MonotoneOn g s) :
    MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} [LinearOrder ι] (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonovaryOn f g s
-- MonotoneOn.antivaryOn
protected theorem MonotoneOn.antivaryOn (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} [LinearOrder ι] (hf : MonotoneOn f s) (hg : AntitoneOn g s)
Goal: AntivaryOn f g s
-- AntitoneOn.monovaryOn
protected theorem AntitoneOn.monovaryOn (hf : AntitoneOn f s) (hg : AntitoneOn g s) :
    MonovaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} [LinearOrder ι] (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: MonovaryOn f g s
-- AntitoneOn.antivaryOn
protected theorem AntitoneOn.antivaryOn (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    AntivaryOn f g s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Preorder β] {f : ι → α} {g : ι → β} {s : Set ι} [LinearOrder ι] (hf : AntitoneOn f s) (hg : MonotoneOn g s)
Goal: AntivaryOn f g s
-- MonovaryOn.comp_monotoneOn_right
theorem MonovaryOn.comp_monotoneOn_right (h : MonovaryOn f g s) (hg : MonotoneOn g' (g '' s)) :
    MonovaryOn f (g' ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {g : ι → β} {g' : β → γ} {s : Set ι} (h : MonovaryOn f g s) (hg : MonotoneOn g' (g '' s))
Goal: MonovaryOn f (g' ∘ g) s
-- MonovaryOn.comp_antitoneOn_right
theorem MonovaryOn.comp_antitoneOn_right (h : MonovaryOn f g s) (hg : AntitoneOn g' (g '' s)) :
    AntivaryOn f (g' ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {g : ι → β} {g' : β → γ} {s : Set ι} (h : MonovaryOn f g s) (hg : AntitoneOn g' (g '' s))
Goal: AntivaryOn f (g' ∘ g) s
-- AntivaryOn.comp_monotoneOn_right
theorem AntivaryOn.comp_monotoneOn_right (h : AntivaryOn f g s) (hg : MonotoneOn g' (g '' s)) :
    AntivaryOn f (g' ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {g : ι → β} {g' : β → γ} {s : Set ι} (h : AntivaryOn f g s) (hg : MonotoneOn g' (g '' s))
Goal: AntivaryOn f (g' ∘ g) s
-- AntivaryOn.comp_antitoneOn_right
theorem AntivaryOn.comp_antitoneOn_right (h : AntivaryOn f g s) (hg : AntitoneOn g' (g '' s)) :
    MonovaryOn f (g' ∘ g) s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [Preorder α] [LinearOrder β] [Preorder γ] {f : ι → α} {g : ι → β} {g' : β → γ} {s : Set ι} (h : AntivaryOn f g s) (hg : AntitoneOn g' (g '' s))
Goal: MonovaryOn f (g' ∘ g) s
-- Monovary.symm
@[symm]
protected theorem Monovary.symm (h : Monovary f g) : Monovary g f 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [LinearOrder β] {f : ι → α} {g : ι → β} (h : Monovary f g)
Goal: Monovary g f
-- Antivary.symm
@[symm]
protected theorem Antivary.symm (h : Antivary f g) : Antivary g f 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [LinearOrder β] {f : ι → α} {g : ι → β} (h : Antivary f g)
Goal: Antivary g f
-- MonovaryOn.symm
@[symm]
protected theorem MonovaryOn.symm (h : MonovaryOn f g s) : MonovaryOn g f s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [LinearOrder β] {f : ι → α} {g : ι → β} {s : Set ι} (h : MonovaryOn f g s)
Goal: MonovaryOn g f s
-- AntivaryOn.symm
@[symm]
protected theorem AntivaryOn.symm (h : AntivaryOn f g s) : AntivaryOn g f s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [LinearOrder β] {f : ι → α} {g : ι → β} {s : Set ι} (h : AntivaryOn f g s)
Goal: AntivaryOn g f s
-- monovary_comm
theorem monovary_comm : Monovary f g ↔ Monovary g f 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [LinearOrder α] [LinearOrder β] {f : ι → α} {g : ι → β}
Goal: Monovary f g ↔ Monovary g f
-- antivary_comm
theorem antivary_comm : Antivary f g ↔ Antivary g f 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [LinearOrder α] [LinearOrder β] {f : ι → α} {g : ι → β}
Goal: Antivary f g ↔ Antivary g f
-- monovaryOn_comm
theorem monovaryOn_comm : MonovaryOn f g s ↔ MonovaryOn g f s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [LinearOrder α] [LinearOrder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: MonovaryOn f g s ↔ MonovaryOn g f s
-- antivaryOn_comm
theorem antivaryOn_comm : AntivaryOn f g s ↔ AntivaryOn g f s 
Arguments: {ι : Type u_1} {α : Type u_3} {β : Type u_4} [LinearOrder α] [LinearOrder β] {f : ι → α} {g : ι → β} {s : Set ι}
Goal: AntivaryOn f g s ↔ AntivaryOn g f s
-- Fintype.IsSimpleOrder.univ
theorem univ : (Finset.univ : Finset α) = {⊤, ⊥} 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]
Goal: Finset.univ = {⊤, ⊥}
-- Fintype.IsSimpleOrder.card
theorem card : Fintype.card α = 2 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]
Goal: Fintype.card α = 2
-- exists_covby_infinite_Ici_of_infinite_Ici
theorem exists_covby_infinite_Ici_of_infinite_Ici [IsStronglyAtomic α]
    (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) :
    ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} [IsStronglyAtomic α] (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite)
Goal: ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite
-- exists_covby_infinite_Iic_of_infinite_Iic
theorem exists_covby_infinite_Iic_of_infinite_Iic [IsStronglyCoatomic α]
    (ha : (Set.Iic a).Infinite) (hfin : {x | x ⋖ a}.Finite) :
    ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} [IsStronglyCoatomic α] (ha : (Set.Iic a).Infinite) (hfin : {x | x ⋖ a}.Finite)
Goal: ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite
-- Nat.Icc_eq_range'
theorem Icc_eq_range' : Icc a b = ⟨List.range' a (b + 1 - a), List.nodup_range' _ _⟩ 
Arguments: (a b : ℕ)
Goal: Finset.Icc a b = { val := ↑(List.range' a (b + 1 - a)), nodup := ⋯ }
-- Nat.Ico_eq_range'
theorem Ico_eq_range' : Ico a b = ⟨List.range' a (b - a), List.nodup_range' _ _⟩ 
Arguments: (a b : ℕ)
Goal: Finset.Ico a b = { val := ↑(List.range' a (b - a)), nodup := ⋯ }
-- Nat.Ioc_eq_range'
theorem Ioc_eq_range' : Ioc a b = ⟨List.range' (a + 1) (b - a), List.nodup_range' _ _⟩ 
Arguments: (a b : ℕ)
Goal: Finset.Ioc a b = { val := ↑(List.range' (a + 1) (b - a)), nodup := ⋯ }
-- Nat.Ioo_eq_range'
theorem Ioo_eq_range' : Ioo a b = ⟨List.range' (a + 1) (b - a - 1), List.nodup_range' _ _⟩ 
Arguments: (a b : ℕ)
Goal: Finset.Ioo a b = { val := ↑(List.range' (a + 1) (b - a - 1)), nodup := ⋯ }
-- Nat.uIcc_eq_range'
theorem uIcc_eq_range' :
    uIcc a b = ⟨List.range' (min a b) (max a b + 1 - min a b), List.nodup_range' _ _⟩ 
Arguments: (a b : ℕ)
Goal: Finset.uIcc a b = { val := ↑(List.range' (min a b) (max a b + 1 - min a b)), nodup := ⋯ }
-- Nat.Iio_eq_range
theorem Iio_eq_range : Iio = range 
Goal: Finset.Iio = Finset.range
-- Nat.Ico_zero_eq_range
@[simp]
theorem Ico_zero_eq_range : Ico 0 = range 
Goal: Finset.Ico 0 = Finset.range
-- Nat.range_eq_Icc_zero_sub_one
lemma range_eq_Icc_zero_sub_one (n : ℕ) (hn : n ≠ 0) : range n = Icc 0 (n - 1) 
Arguments: (n : ℕ) (hn : n ≠ 0)
Goal: Finset.range n = Finset.Icc 0 (n - 1)
-- Finset.range_eq_Ico
theorem _root_.Finset.range_eq_Ico : range = Ico 0 
Goal: Finset.range = Finset.Ico 0
-- Nat.card_Icc
@[simp]
theorem card_Icc : (Icc a b).card = b + 1 - a 
Arguments: (a b : ℕ)
Goal: (Finset.Icc a b).card = b + 1 - a
-- Nat.card_Ico
@[simp]
theorem card_Ico : (Ico a b).card = b - a 
Arguments: (a b : ℕ)
Goal: (Finset.Ico a b).card = b - a
-- Nat.card_Ioc
@[simp]
theorem card_Ioc : (Ioc a b).card = b - a 
Arguments: (a b : ℕ)
Goal: (Finset.Ioc a b).card = b - a
-- Nat.card_Ioo
@[simp]
theorem card_Ioo : (Ioo a b).card = b - a - 1 
Arguments: (a b : ℕ)
Goal: (Finset.Ioo a b).card = b - a - 1
-- Nat.card_uIcc
@[simp]
theorem card_uIcc : (uIcc a b).card = (b - a : ℤ).natAbs + 1 
Arguments: (a b : ℕ)
Goal: (Finset.uIcc a b).card = (↑b - ↑a).natAbs + 1
-- Nat.card_Iic
@[simp]
lemma card_Iic : (Iic b).card = b + 1 
Arguments: (b : ℕ)
Goal: (Finset.Iic b).card = b + 1
-- Nat.card_Iio
@[simp]
theorem card_Iio : (Iio b).card = b 
Arguments: (b : ℕ)
Goal: (Finset.Iio b).card = b
-- Nat.card_fintypeIcc
theorem card_fintypeIcc : Fintype.card (Set.Icc a b) = b + 1 - a 
Arguments: (a b : ℕ)
Goal: Fintype.card ↑(Set.Icc a b) = b + 1 - a
-- Nat.card_fintypeIco
theorem card_fintypeIco : Fintype.card (Set.Ico a b) = b - a 
Arguments: (a b : ℕ)
Goal: Fintype.card ↑(Set.Ico a b) = b - a