ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.frequently_imp_distrib_right
theorem frequently_imp_distrib_right {f : Filter α} [NeBot f] {p : α → Prop} {q : Prop} :
    (∃ᶠ x in f, p x → q) ↔ (∀ᶠ x in f, p x) → q 
Arguments: {α : Type u} {f : Filter α} [f.NeBot] {p : α → Prop} {q : Prop}
Goal: (∃ᶠ (x : α) in f, p x → q) ↔ (∀ᶠ (x : α) in f, p x) → q
-- Filter.eventually_imp_distrib_right
theorem eventually_imp_distrib_right {f : Filter α} {p : α → Prop} {q : Prop} :
    (∀ᶠ x in f, p x → q) ↔ (∃ᶠ x in f, p x) → q 
Arguments: {α : Type u} {f : Filter α} {p : α → Prop} {q : Prop}
Goal: (∀ᶠ (x : α) in f, p x → q) ↔ (∃ᶠ (x : α) in f, p x) → q
-- Filter.frequently_and_distrib_left
@[simp]
theorem frequently_and_distrib_left {f : Filter α} {p : Prop} {q : α → Prop} :
    (∃ᶠ x in f, p ∧ q x) ↔ p ∧ ∃ᶠ x in f, q x 
Arguments: {α : Type u} {f : Filter α} {p : Prop} {q : α → Prop}
Goal: (∃ᶠ (x : α) in f, p ∧ q x) ↔ p ∧ ∃ᶠ (x : α) in f, q x
-- Filter.frequently_and_distrib_right
@[simp]
theorem frequently_and_distrib_right {f : Filter α} {p : α → Prop} {q : Prop} :
    (∃ᶠ x in f, p x ∧ q) ↔ (∃ᶠ x in f, p x) ∧ q 
Arguments: {α : Type u} {f : Filter α} {p : α → Prop} {q : Prop}
Goal: (∃ᶠ (x : α) in f, p x ∧ q) ↔ (∃ᶠ (x : α) in f, p x) ∧ q
-- Filter.frequently_bot
@[simp]
theorem frequently_bot {p : α → Prop} : ¬∃ᶠ x in ⊥, p x 
Arguments: {α : Type u} {p : α → Prop}
Goal: ¬∃ᶠ (x : α) in ⊥, p x
-- Filter.frequently_top
@[simp]
theorem frequently_top {p : α → Prop} : (∃ᶠ x in ⊤, p x) ↔ ∃ x, p x 
Arguments: {α : Type u} {p : α → Prop}
Goal: (∃ᶠ (x : α) in ⊤, p x) ↔ ∃ x, p x
-- Filter.frequently_principal
@[simp]
theorem frequently_principal {a : Set α} {p : α → Prop} : (∃ᶠ x in 𝓟 a, p x) ↔ ∃ x ∈ a, p x 
Arguments: {α : Type u} {a : Set α} {p : α → Prop}
Goal: (∃ᶠ (x : α) in 𝓟 a, p x) ↔ ∃ x ∈ a, p x
-- Filter.frequently_inf_principal
theorem frequently_inf_principal {f : Filter α} {s : Set α} {p : α → Prop} :
    (∃ᶠ x in f ⊓ 𝓟 s, p x) ↔ ∃ᶠ x in f, x ∈ s ∧ p x 
Arguments: {α : Type u} {f : Filter α} {s : Set α} {p : α → Prop}
Goal: (∃ᶠ (x : α) in f ⊓ 𝓟 s, p x) ↔ ∃ᶠ (x : α) in f, x ∈ s ∧ p x
-- Filter.frequently_sup
theorem frequently_sup {p : α → Prop} {f g : Filter α} :
    (∃ᶠ x in f ⊔ g, p x) ↔ (∃ᶠ x in f, p x) ∨ ∃ᶠ x in g, p x 
Arguments: {α : Type u} {p : α → Prop} {f g : Filter α}
Goal: (∃ᶠ (x : α) in f ⊔ g, p x) ↔ (∃ᶠ (x : α) in f, p x) ∨ ∃ᶠ (x : α) in g, p x
-- Filter.frequently_sSup
@[simp]
theorem frequently_sSup {p : α → Prop} {fs : Set (Filter α)} :
    (∃ᶠ x in sSup fs, p x) ↔ ∃ f ∈ fs, ∃ᶠ x in f, p x 
Arguments: {α : Type u} {p : α → Prop} {fs : Set (Filter α)}
Goal: (∃ᶠ (x : α) in sSup fs, p x) ↔ ∃ f ∈ fs, ∃ᶠ (x : α) in f, p x
-- Filter.frequently_iSup
@[simp]
theorem frequently_iSup {p : α → Prop} {fs : β → Filter α} :
    (∃ᶠ x in ⨆ b, fs b, p x) ↔ ∃ b, ∃ᶠ x in fs b, p x 
Arguments: {α : Type u} {β : Type v} {p : α → Prop} {fs : β → Filter α}
Goal: (∃ᶠ (x : α) in ⨆ b, fs b, p x) ↔ ∃ b, ∃ᶠ (x : α) in fs b, p x
-- Filter.Eventually.choice
theorem Eventually.choice {r : α → β → Prop} {l : Filter α} [l.NeBot] (h : ∀ᶠ x in l, ∃ y, r x y) :
    ∃ f : α → β, ∀ᶠ x in l, r x (f x) 
Arguments: {α : Type u} {β : Type v} {r : α → β → Prop} {l : Filter α} [l.NeBot] (h : ∀ᶠ (x : α) in l, ∃ y, r x y)
Goal: ∃ f, ∀ᶠ (x : α) in l, r x (f x)
-- Filter.EventuallyEq.eventually
theorem EventuallyEq.eventually {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) :
    ∀ᶠ x in l, f x = g x 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g)
Goal: ∀ᶠ (x : α) in l, f x = g x
-- Filter.EventuallyEq.rw
theorem EventuallyEq.rw {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop)
    (hf : ∀ᶠ x in l, p x (f x)) : ∀ᶠ x in l, p x (g x) 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) (p : α → β → Prop) (hf : ∀ᶠ (x : α) in l, p x (f x))
Goal: ∀ᶠ (x : α) in l, p x (g x)
-- Filter.eventuallyEq_set
theorem eventuallyEq_set {s t : Set α} {l : Filter α} : s =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ s ↔ x ∈ t 
Arguments: {α : Type u} {s t : Set α} {l : Filter α}
Goal: s =ᶠ[l] t ↔ ∀ᶠ (x : α) in l, x ∈ s ↔ x ∈ t
-- Filter.eventuallyEq_univ
@[simp]
theorem eventuallyEq_univ {s : Set α} {l : Filter α} : s =ᶠ[l] univ ↔ s ∈ l 
Arguments: {α : Type u} {s : Set α} {l : Filter α}
Goal: s =ᶠ[l] Set.univ ↔ s ∈ l
-- Filter.EventuallyEq.exists_mem
theorem EventuallyEq.exists_mem {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) :
    ∃ s ∈ l, EqOn f g s 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g)
Goal: ∃ s ∈ l, Set.EqOn f g s
-- Filter.eventuallyEq_of_mem
theorem eventuallyEq_of_mem {l : Filter α} {f g : α → β} {s : Set α} (hs : s ∈ l) (h : EqOn f g s) :
    f =ᶠ[l] g 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} {s : Set α} (hs : s ∈ l) (h : Set.EqOn f g s)
Goal: f =ᶠ[l] g
-- Filter.eventuallyEq_iff_exists_mem
theorem eventuallyEq_iff_exists_mem {l : Filter α} {f g : α → β} :
    f =ᶠ[l] g ↔ ∃ s ∈ l, EqOn f g s 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β}
Goal: f =ᶠ[l] g ↔ ∃ s ∈ l, Set.EqOn f g s
-- Filter.EventuallyEq.filter_mono
theorem EventuallyEq.filter_mono {l l' : Filter α} {f g : α → β} (h₁ : f =ᶠ[l] g) (h₂ : l' ≤ l) :
    f =ᶠ[l'] g 
Arguments: {α : Type u} {β : Type v} {l l' : Filter α} {f g : α → β} (h₁ : f =ᶠ[l] g) (h₂ : l' ≤ l)
Goal: f =ᶠ[l'] g
-- Filter.EventuallyEq.refl
@[refl, simp]
theorem EventuallyEq.refl (l : Filter α) (f : α → β) : f =ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} (l : Filter α) (f : α → β)
Goal: f =ᶠ[l] f
-- Filter.EventuallyEq.rfl
protected theorem EventuallyEq.rfl {l : Filter α} {f : α → β} : f =ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f : α → β}
Goal: f =ᶠ[l] f
-- Filter.EventuallyEq.symm
@[symm]
theorem EventuallyEq.symm {f g : α → β} {l : Filter α} (H : f =ᶠ[l] g) : g =ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} {f g : α → β} {l : Filter α} (H : f =ᶠ[l] g)
Goal: g =ᶠ[l] f
-- Filter.EventuallyEq.trans
@[trans]
theorem EventuallyEq.trans {l : Filter α} {f g h : α → β} (H₁ : f =ᶠ[l] g) (H₂ : g =ᶠ[l] h) :
    f =ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g h : α → β} (H₁ : f =ᶠ[l] g) (H₂ : g =ᶠ[l] h)
Goal: f =ᶠ[l] h
-- Filter.EventuallyEq.congr_left
theorem EventuallyEq.congr_left {l : Filter α} {f g h : α → β} (H : f =ᶠ[l] g) :
    f =ᶠ[l] h ↔ g =ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g h : α → β} (H : f =ᶠ[l] g)
Goal: f =ᶠ[l] h ↔ g =ᶠ[l] h
-- Filter.EventuallyEq.congr_right
theorem EventuallyEq.congr_right {l : Filter α} {f g h : α → β} (H : g =ᶠ[l] h) :
    f =ᶠ[l] g ↔ f =ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g h : α → β} (H : g =ᶠ[l] h)
Goal: f =ᶠ[l] g ↔ f =ᶠ[l] h
-- Filter.EventuallyEq.prod_mk
theorem EventuallyEq.prod_mk {l} {f f' : α → β} (hf : f =ᶠ[l] f') {g g' : α → γ} (hg : g =ᶠ[l] g') :
    (fun x => (f x, g x)) =ᶠ[l] fun x => (f' x, g' x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {l : Filter α} {f f' : α → β} (hf : f =ᶠ[l] f') {g g' : α → γ} (hg : g =ᶠ[l] g')
Goal: (fun x => (f x, g x)) =ᶠ[l] fun x => (f' x, g' x)
-- Filter.EventuallyEq.fun_comp
theorem EventuallyEq.fun_comp {f g : α → β} {l : Filter α} (H : f =ᶠ[l] g) (h : β → γ) :
    h ∘ f =ᶠ[l] h ∘ g 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {f g : α → β} {l : Filter α} (H : f =ᶠ[l] g) (h : β → γ)
Goal: h ∘ f =ᶠ[l] h ∘ g
-- Filter.EventuallyEq.comp₂
theorem EventuallyEq.comp₂ {δ} {f f' : α → β} {g g' : α → γ} {l} (Hf : f =ᶠ[l] f') (h : β → γ → δ)
    (Hg : g =ᶠ[l] g') : (fun x => h (f x) (g x)) =ᶠ[l] fun x => h (f' x) (g' x) 
Arguments: {α : Type u} {β : Type v} {γ : Type w} {δ : Type u_2} {f f' : α → β} {g g' : α → γ} {l : Filter α} (Hf : f =ᶠ[l] f') (h : β → γ → δ) (Hg : g =ᶠ[l] g')
Goal: (fun x => h (f x) (g x)) =ᶠ[l] fun x => h (f' x) (g' x)
-- Filter.EventuallyEq.add
@[to_additive]
theorem EventuallyEq.mul [Mul β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g)
    (h' : f' =ᶠ[l] g') : (fun x => f x * f' x) =ᶠ[l] fun x => g x * g' x 
Arguments: {α : Type u} {β : Type v} [Add β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g')
Goal: (fun x => f x + f' x) =ᶠ[l] fun x => g x + g' x
-- Filter.EventuallyEq.mul
@[to_additive]
theorem EventuallyEq.mul [Mul β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g)
    (h' : f' =ᶠ[l] g') : (fun x => f x * f' x) =ᶠ[l] fun x => g x * g' x 
Arguments: {α : Type u} {β : Type v} [Mul β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g')
Goal: (fun x => f x * f' x) =ᶠ[l] fun x => g x * g' x
-- Filter.EventuallyEq.pow_const
@[to_additive const_smul]
theorem EventuallyEq.pow_const {γ} [Pow β γ] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ) :
    (fun x => f x ^ c) =ᶠ[l] fun x => g x ^ c 
Arguments: {α : Type u} {β : Type v} {γ : Type u_2} [Pow β γ] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ)
Goal: (fun x => f x ^ c) =ᶠ[l] fun x => g x ^ c
-- Filter.EventuallyEq.const_smul
@[to_additive const_smul]
theorem EventuallyEq.pow_const {γ} [Pow β γ] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ) :
    (fun x => f x ^ c) =ᶠ[l] fun x => g x ^ c 
Arguments: {α : Type u} {β : Type v} {γ : Type u_2} [SMul γ β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) (c : γ)
Goal: (fun x => c • f x) =ᶠ[l] fun x => c • g x
-- Filter.EventuallyEq.neg
@[to_additive]
theorem EventuallyEq.inv [Inv β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) :
    (fun x => (f x)⁻¹) =ᶠ[l] fun x => (g x)⁻¹ 
Arguments: {α : Type u} {β : Type v} [Neg β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g)
Goal: (fun x => -f x) =ᶠ[l] fun x => -g x
-- Filter.EventuallyEq.inv
@[to_additive]
theorem EventuallyEq.inv [Inv β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) :
    (fun x => (f x)⁻¹) =ᶠ[l] fun x => (g x)⁻¹ 
Arguments: {α : Type u} {β : Type v} [Inv β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g)
Goal: (fun x => (f x)⁻¹) =ᶠ[l] fun x => (g x)⁻¹
-- Filter.EventuallyEq.sub
@[to_additive]
theorem EventuallyEq.div [Div β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g)
    (h' : f' =ᶠ[l] g') : (fun x => f x / f' x) =ᶠ[l] fun x => g x / g' x 
Arguments: {α : Type u} {β : Type v} [Sub β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g')
Goal: (fun x => f x - f' x) =ᶠ[l] fun x => g x - g' x
-- Filter.EventuallyEq.div
@[to_additive]
theorem EventuallyEq.div [Div β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g)
    (h' : f' =ᶠ[l] g') : (fun x => f x / f' x) =ᶠ[l] fun x => g x / g' x 
Arguments: {α : Type u} {β : Type v} [Div β] {f f' g g' : α → β} {l : Filter α} (h : f =ᶠ[l] g) (h' : f' =ᶠ[l] g')
Goal: (fun x => f x / f' x) =ᶠ[l] fun x => g x / g' x
-- Filter.EventuallyEq.vadd
@[to_additive]
theorem EventuallyEq.smul {𝕜} [SMul 𝕜 β] {l : Filter α} {f f' : α → 𝕜} {g g' : α → β}
    (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun x => f x • g x) =ᶠ[l] fun x => f' x • g' x 
Arguments: {α : Type u} {β : Type v} {𝕜 : Type u_2} [VAdd 𝕜 β] {l : Filter α} {f f' : α → 𝕜} {g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: (fun x => f x +ᵥ g x) =ᶠ[l] fun x => f' x +ᵥ g' x
-- Filter.EventuallyEq.smul
@[to_additive]
theorem EventuallyEq.smul {𝕜} [SMul 𝕜 β] {l : Filter α} {f f' : α → 𝕜} {g g' : α → β}
    (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') : (fun x => f x • g x) =ᶠ[l] fun x => f' x • g' x 
Arguments: {α : Type u} {β : Type v} {𝕜 : Type u_2} [SMul 𝕜 β] {l : Filter α} {f f' : α → 𝕜} {g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: (fun x => f x • g x) =ᶠ[l] fun x => f' x • g' x
-- Filter.EventuallyEq.sup
theorem EventuallyEq.sup [Sup β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f')
    (hg : g =ᶠ[l] g') : (fun x => f x ⊔ g x) =ᶠ[l] fun x => f' x ⊔ g' x 
Arguments: {α : Type u} {β : Type v} [Sup β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: (fun x => f x ⊔ g x) =ᶠ[l] fun x => f' x ⊔ g' x
-- Filter.EventuallyEq.inf
theorem EventuallyEq.inf [Inf β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f')
    (hg : g =ᶠ[l] g') : (fun x => f x ⊓ g x) =ᶠ[l] fun x => f' x ⊓ g' x 
Arguments: {α : Type u} {β : Type v} [Inf β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: (fun x => f x ⊓ g x) =ᶠ[l] fun x => f' x ⊓ g' x
-- Filter.EventuallyEq.preimage
theorem EventuallyEq.preimage {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) (s : Set β) :
    f ⁻¹' s =ᶠ[l] g ⁻¹' s 
Arguments: {α : Type u} {β : Type v} {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g) (s : Set β)
Goal: f ⁻¹' s =ᶠ[l] g ⁻¹' s
-- Filter.EventuallyEq.inter
theorem EventuallyEq.inter {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
    (s ∩ s' : Set α) =ᶠ[l] (t ∩ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t')
Goal: s ∩ s' =ᶠ[l] t ∩ t'
-- Filter.EventuallyEq.union
theorem EventuallyEq.union {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
    (s ∪ s' : Set α) =ᶠ[l] (t ∪ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t')
Goal: s ∪ s' =ᶠ[l] t ∪ t'
-- Filter.EventuallyEq.compl
theorem EventuallyEq.compl {s t : Set α} {l : Filter α} (h : s =ᶠ[l] t) :
    (sᶜ : Set α) =ᶠ[l] (tᶜ : Set α) 
Arguments: {α : Type u} {s t : Set α} {l : Filter α} (h : s =ᶠ[l] t)
Goal: sᶜ =ᶠ[l] tᶜ
-- Filter.EventuallyEq.diff
theorem EventuallyEq.diff {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') :
    (s \ s' : Set α) =ᶠ[l] (t \ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t')
Goal: s \ s' =ᶠ[l] t \ t'
-- Filter.EventuallyEq.symmDiff
protected theorem EventuallyEq.symmDiff {s t s' t' : Set α} {l : Filter α}
    (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t') : (s ∆ s' : Set α) =ᶠ[l] (t ∆ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s =ᶠ[l] t) (h' : s' =ᶠ[l] t')
Goal: s ∆ s' =ᶠ[l] t ∆ t'
-- Filter.eventuallyEq_empty
theorem eventuallyEq_empty {s : Set α} {l : Filter α} : s =ᶠ[l] (∅ : Set α) ↔ ∀ᶠ x in l, x ∉ s 
Arguments: {α : Type u} {s : Set α} {l : Filter α}
Goal: s =ᶠ[l] ∅ ↔ ∀ᶠ (x : α) in l, x ∉ s
-- Filter.inter_eventuallyEq_left
theorem inter_eventuallyEq_left {s t : Set α} {l : Filter α} :
    (s ∩ t : Set α) =ᶠ[l] s ↔ ∀ᶠ x in l, x ∈ s → x ∈ t 
Arguments: {α : Type u} {s t : Set α} {l : Filter α}
Goal: s ∩ t =ᶠ[l] s ↔ ∀ᶠ (x : α) in l, x ∈ s → x ∈ t
-- Filter.inter_eventuallyEq_right
theorem inter_eventuallyEq_right {s t : Set α} {l : Filter α} :
    (s ∩ t : Set α) =ᶠ[l] t ↔ ∀ᶠ x in l, x ∈ t → x ∈ s 
Arguments: {α : Type u} {s t : Set α} {l : Filter α}
Goal: s ∩ t =ᶠ[l] t ↔ ∀ᶠ (x : α) in l, x ∈ t → x ∈ s
-- Filter.eventuallyEq_principal
@[simp]
theorem eventuallyEq_principal {s : Set α} {f g : α → β} : f =ᶠ[𝓟 s] g ↔ EqOn f g s 
Arguments: {α : Type u} {β : Type v} {s : Set α} {f g : α → β}
Goal: f =ᶠ[𝓟 s] g ↔ Set.EqOn f g s
-- Filter.eventuallyEq_inf_principal_iff
theorem eventuallyEq_inf_principal_iff {F : Filter α} {s : Set α} {f g : α → β} :
    f =ᶠ[F ⊓ 𝓟 s] g ↔ ∀ᶠ x in F, x ∈ s → f x = g x 
Arguments: {α : Type u} {β : Type v} {F : Filter α} {s : Set α} {f g : α → β}
Goal: f =ᶠ[F ⊓ 𝓟 s] g ↔ ∀ᶠ (x : α) in F, x ∈ s → f x = g x
-- Filter.EventuallyEq.sub_eq
theorem EventuallyEq.sub_eq [AddGroup β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g) :
    f - g =ᶠ[l] 0 
Arguments: {α : Type u} {β : Type v} [AddGroup β] {f g : α → β} {l : Filter α} (h : f =ᶠ[l] g)
Goal: f - g =ᶠ[l] 0
-- Filter.eventuallyEq_iff_sub
theorem eventuallyEq_iff_sub [AddGroup β] {f g : α → β} {l : Filter α} :
    f =ᶠ[l] g ↔ f - g =ᶠ[l] 0 
Arguments: {α : Type u} {β : Type v} [AddGroup β] {f g : α → β} {l : Filter α}
Goal: f =ᶠ[l] g ↔ f - g =ᶠ[l] 0
-- Filter.EventuallyLE.congr
theorem EventuallyLE.congr {f f' g g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
    f' ≤ᶠ[l] g' 
Arguments: {α : Type u} {β : Type v} [LE β] {l : Filter α} {f f' g g' : α → β} (H : f ≤ᶠ[l] g) (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: f' ≤ᶠ[l] g'
-- Filter.eventuallyLE_congr
theorem eventuallyLE_congr {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g') :
    f ≤ᶠ[l] g ↔ f' ≤ᶠ[l] g' 
Arguments: {α : Type u} {β : Type v} [LE β] {l : Filter α} {f f' g g' : α → β} (hf : f =ᶠ[l] f') (hg : g =ᶠ[l] g')
Goal: f ≤ᶠ[l] g ↔ f' ≤ᶠ[l] g'
-- Filter.EventuallyEq.le
theorem EventuallyEq.le (h : f =ᶠ[l] g) : f ≤ᶠ[l] g 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f g : α → β} (h : f =ᶠ[l] g)
Goal: f ≤ᶠ[l] g
-- Filter.EventuallyLE.refl
@[refl]
theorem EventuallyLE.refl (l : Filter α) (f : α → β) : f ≤ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} [Preorder β] (l : Filter α) (f : α → β)
Goal: f ≤ᶠ[l] f
-- Filter.EventuallyLE.rfl
theorem EventuallyLE.rfl : f ≤ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f : α → β}
Goal: f ≤ᶠ[l] f
-- Filter.EventuallyLE.trans
@[trans]
theorem EventuallyLE.trans (H₁ : f ≤ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f g h : α → β} (H₁ : f ≤ᶠ[l] g) (H₂ : g ≤ᶠ[l] h)
Goal: f ≤ᶠ[l] h
-- Filter.EventuallyEq.trans_le
@[trans]
theorem EventuallyEq.trans_le (H₁ : f =ᶠ[l] g) (H₂ : g ≤ᶠ[l] h) : f ≤ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f g h : α → β} (H₁ : f =ᶠ[l] g) (H₂ : g ≤ᶠ[l] h)
Goal: f ≤ᶠ[l] h
-- Filter.EventuallyLE.trans_eq
@[trans]
theorem EventuallyLE.trans_eq (H₁ : f ≤ᶠ[l] g) (H₂ : g =ᶠ[l] h) : f ≤ᶠ[l] h 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f g h : α → β} (H₁ : f ≤ᶠ[l] g) (H₂ : g =ᶠ[l] h)
Goal: f ≤ᶠ[l] h
-- Filter.EventuallyLE.antisymm
theorem EventuallyLE.antisymm [PartialOrder β] {l : Filter α} {f g : α → β} (h₁ : f ≤ᶠ[l] g)
    (h₂ : g ≤ᶠ[l] f) : f =ᶠ[l] g 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f g : α → β} (h₁ : f ≤ᶠ[l] g) (h₂ : g ≤ᶠ[l] f)
Goal: f =ᶠ[l] g
-- Filter.eventuallyLE_antisymm_iff
theorem eventuallyLE_antisymm_iff [PartialOrder β] {l : Filter α} {f g : α → β} :
    f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f g : α → β}
Goal: f =ᶠ[l] g ↔ f ≤ᶠ[l] g ∧ g ≤ᶠ[l] f
-- Filter.EventuallyLE.le_iff_eq
theorem EventuallyLE.le_iff_eq [PartialOrder β] {l : Filter α} {f g : α → β} (h : f ≤ᶠ[l] g) :
    g ≤ᶠ[l] f ↔ g =ᶠ[l] f 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] {l : Filter α} {f g : α → β} (h : f ≤ᶠ[l] g)
Goal: g ≤ᶠ[l] f ↔ g =ᶠ[l] f
-- Filter.Eventually.ne_of_lt
theorem Eventually.ne_of_lt [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ᶠ x in l, f x < g x) :
    ∀ᶠ x in l, f x ≠ g x 
Arguments: {α : Type u} {β : Type v} [Preorder β] {l : Filter α} {f g : α → β} (h : ∀ᶠ (x : α) in l, f x < g x)
Goal: ∀ᶠ (x : α) in l, f x ≠ g x
-- Filter.Eventually.ne_top_of_lt
theorem Eventually.ne_top_of_lt [PartialOrder β] [OrderTop β] {l : Filter α} {f g : α → β}
    (h : ∀ᶠ x in l, f x < g x) : ∀ᶠ x in l, f x ≠ ⊤ 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f g : α → β} (h : ∀ᶠ (x : α) in l, f x < g x)
Goal: ∀ᶠ (x : α) in l, f x ≠ ⊤
-- Filter.Eventually.lt_top_of_ne
theorem Eventually.lt_top_of_ne [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β}
    (h : ∀ᶠ x in l, f x ≠ ⊤) : ∀ᶠ x in l, f x < ⊤ 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β} (h : ∀ᶠ (x : α) in l, f x ≠ ⊤)
Goal: ∀ᶠ (x : α) in l, f x < ⊤
-- Filter.Eventually.lt_top_iff_ne_top
theorem Eventually.lt_top_iff_ne_top [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β} :
    (∀ᶠ x in l, f x < ⊤) ↔ ∀ᶠ x in l, f x ≠ ⊤ 
Arguments: {α : Type u} {β : Type v} [PartialOrder β] [OrderTop β] {l : Filter α} {f : α → β}
Goal: (∀ᶠ (x : α) in l, f x < ⊤) ↔ ∀ᶠ (x : α) in l, f x ≠ ⊤
-- Filter.EventuallyLE.inter
@[mono]
theorem EventuallyLE.inter {s t s' t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t') :
    (s ∩ s' : Set α) ≤ᶠ[l] (t ∩ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t')
Goal: s ∩ s' ≤ᶠ[l] t ∩ t'
-- Filter.EventuallyLE.union
@[mono]
theorem EventuallyLE.union {s t s' t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t') :
    (s ∪ s' : Set α) ≤ᶠ[l] (t ∪ t' : Set α) 
Arguments: {α : Type u} {s t s' t' : Set α} {l : Filter α} (h : s ≤ᶠ[l] t) (h' : s' ≤ᶠ[l] t')
Goal: s ∪ s' ≤ᶠ[l] t ∪ t'
-- Filter.EventuallyLE.iUnion
protected lemma EventuallyLE.iUnion [Finite ι] {s t : ι → Set α}
    (h : ∀ i, s i ≤ᶠ[l] t i) : (⋃ i, s i) ≤ᶠ[l] ⋃ i, t i 
Arguments: {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i)
Goal: ⋃ i, s i ≤ᶠ[l] ⋃ i, t i
-- Filter.EventuallyEq.iUnion
protected lemma EventuallyEq.iUnion [Finite ι] {s t : ι → Set α}
    (h : ∀ i, s i =ᶠ[l] t i) : (⋃ i, s i) =ᶠ[l] ⋃ i, t i 
Arguments: {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i)
Goal: ⋃ i, s i =ᶠ[l] ⋃ i, t i
-- Filter.EventuallyLE.iInter
protected lemma EventuallyLE.iInter [Finite ι] {s t : ι → Set α}
    (h : ∀ i, s i ≤ᶠ[l] t i) : (⋂ i, s i) ≤ᶠ[l] ⋂ i, t i 
Arguments: {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s t : ι → Set α} (h : ∀ (i : ι), s i ≤ᶠ[l] t i)
Goal: ⋂ i, s i ≤ᶠ[l] ⋂ i, t i
-- Filter.EventuallyEq.iInter
protected lemma EventuallyEq.iInter [Finite ι] {s t : ι → Set α}
    (h : ∀ i, s i =ᶠ[l] t i) : (⋂ i, s i) =ᶠ[l] ⋂ i, t i 
Arguments: {α : Type u} {ι : Sort x} {l : Filter α} [Finite ι] {s t : ι → Set α} (h : ∀ (i : ι), s i =ᶠ[l] t i)
Goal: ⋂ i, s i =ᶠ[l] ⋂ i, t i
-- Set.Finite.eventuallyLE_iUnion
lemma _root_.Set.Finite.eventuallyLE_iUnion {ι : Type*} {s : Set ι} (hs : s.Finite)
    {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : (⋃ i ∈ s, f i) ≤ᶠ[l] (⋃ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i)
Goal: ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i
-- Set.Finite.eventuallyEq_iUnion
lemma _root_.Set.Finite.eventuallyEq_iUnion {ι : Type*} {s : Set ι} (hs : s.Finite)
    {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : (⋃ i ∈ s, f i) =ᶠ[l] (⋃ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i
-- Set.Finite.eventuallyLE_iInter
lemma _root_.Set.Finite.eventuallyLE_iInter {ι : Type*} {s : Set ι} (hs : s.Finite)
    {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : (⋂ i ∈ s, f i) ≤ᶠ[l] (⋂ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i)
Goal: ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i
-- Set.Finite.eventuallyEq_iInter
lemma _root_.Set.Finite.eventuallyEq_iInter {ι : Type*} {s : Set ι} (hs : s.Finite)
    {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : (⋂ i ∈ s, f i) =ᶠ[l] (⋂ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} {s : Set ι} (hs : s.Finite) {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ⋂ i ∈ s, f i =ᶠ[l] ⋂ i ∈ s, g i
-- Finset.eventuallyLE_iUnion
lemma _root_.Finset.eventuallyLE_iUnion {ι : Type*} (s : Finset ι) {f g : ι → Set α}
    (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : (⋃ i ∈ s, f i) ≤ᶠ[l] (⋃ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i)
Goal: ⋃ i ∈ s, f i ≤ᶠ[l] ⋃ i ∈ s, g i
-- Finset.eventuallyEq_iUnion
lemma _root_.Finset.eventuallyEq_iUnion {ι : Type*} (s : Finset ι) {f g : ι → Set α}
    (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : (⋃ i ∈ s, f i) =ᶠ[l] (⋃ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i
-- Finset.eventuallyLE_iInter
lemma _root_.Finset.eventuallyLE_iInter {ι : Type*} (s : Finset ι) {f g : ι → Set α}
    (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i) : (⋂ i ∈ s, f i) ≤ᶠ[l] (⋂ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f g : ι → Set α} (hle : ∀ i ∈ s, f i ≤ᶠ[l] g i)
Goal: ⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i
-- Finset.eventuallyEq_iInter
lemma _root_.Finset.eventuallyEq_iInter {ι : Type*} (s : Finset ι) {f g : ι → Set α}
    (heq : ∀ i ∈ s, f i =ᶠ[l] g i) : (⋂ i ∈ s, f i) =ᶠ[l] (⋂ i ∈ s, g i) 
Arguments: {α : Type u} {l : Filter α} {ι : Type u_2} (s : Finset ι) {f g : ι → Set α} (heq : ∀ i ∈ s, f i =ᶠ[l] g i)
Goal: ⋂ i ∈ s, f i =ᶠ[l] ⋂ i ∈ s, g i