ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.bounded_ge_Ioo
theorem bounded_ge_Ioo [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioo a b)
-- Set.bounded_ge_Ioc
theorem bounded_ge_Ioc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioc a b)
-- Set.bounded_ge_Ico
theorem bounded_ge_Ico [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ico a b)
-- Set.bounded_ge_Icc
theorem bounded_ge_Icc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Icc a b)
-- Set.unbounded_le_Ioi
theorem unbounded_le_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· ≤ ·) (Ioi a) 
Arguments: {α : Type u_1} [SemilatticeSup α] [NoMaxOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (Set.Ioi a)
-- Set.unbounded_le_Ici
theorem unbounded_le_Ici [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· ≤ ·) (Ici a) 
Arguments: {α : Type u_1} [SemilatticeSup α] [NoMaxOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (Set.Ici a)
-- Set.unbounded_lt_Ioi
theorem unbounded_lt_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
    Unbounded (· < ·) (Ioi a) 
Arguments: {α : Type u_1} [SemilatticeSup α] [NoMaxOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x < x_1) (Set.Ioi a)
-- Set.unbounded_lt_Ici
theorem unbounded_lt_Ici [SemilatticeSup α] (a : α) : Unbounded (· < ·) (Ici a) 
Arguments: {α : Type u_1} [SemilatticeSup α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x < x_1) (Set.Ici a)
-- Set.bounded_inter_not
theorem bounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
    Bounded r (s ∩ { b | ¬r b a }) ↔ Bounded r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m) (a : α)
Goal: Set.Bounded r (s ∩ {b | ¬r b a}) ↔ Set.Bounded r s
-- Set.unbounded_inter_not
theorem unbounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
    Unbounded r (s ∩ { b | ¬r b a }) ↔ Unbounded r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (H : ∀ (a b : α), ∃ m, ∀ (c : α), r c a ∨ r c b → r c m) (a : α)
Goal: Set.Unbounded r (s ∩ {b | ¬r b a}) ↔ Set.Unbounded r s
-- Set.bounded_le_inter_not_le
theorem bounded_le_inter_not_le [SemilatticeSup α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Bounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeSup α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s
-- Set.unbounded_le_inter_not_le
theorem unbounded_le_inter_not_le [SemilatticeSup α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Unbounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeSup α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | ¬b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s
-- Set.bounded_le_inter_lt
theorem bounded_le_inter_lt [LinearOrder α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Bounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s
-- Set.unbounded_le_inter_lt
theorem unbounded_le_inter_lt [LinearOrder α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Unbounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s
-- Set.bounded_le_inter_le
theorem bounded_le_inter_le [LinearOrder α] (a : α) :
    Bounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Set.Bounded (fun x x_1 => x ≤ x_1) s
-- Set.unbounded_le_inter_le
theorem unbounded_le_inter_le [LinearOrder α] (a : α) :
    Unbounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s
-- Set.bounded_lt_inter_not_lt
theorem bounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
    Bounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Bounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeSup α] (a : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | ¬b < a}) ↔ Set.Bounded (fun x x_1 => x < x_1) s
-- Set.unbounded_lt_inter_not_lt
theorem unbounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Unbounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeSup α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | ¬b < a}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s
-- Set.bounded_lt_inter_le
theorem bounded_lt_inter_le [LinearOrder α] (a : α) :
    Bounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | a ≤ b}) ↔ Set.Bounded (fun x x_1 => x < x_1) s
-- Set.unbounded_lt_inter_le
theorem unbounded_lt_inter_le [LinearOrder α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s
-- Set.bounded_lt_inter_lt
theorem bounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
    Bounded (· < ·) (s ∩ { b | a < b }) ↔ Bounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] [NoMaxOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Set.Bounded (fun x x_1 => x < x_1) s
-- Set.unbounded_lt_inter_lt
theorem unbounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
    Unbounded (· < ·) (s ∩ { b | a < b }) ↔ Unbounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] [NoMaxOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x < x_1) (s ∩ {b | a < b}) ↔ Set.Unbounded (fun x x_1 => x < x_1) s
-- Set.bounded_ge_inter_not_ge
theorem bounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Bounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeInf α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | ¬a ≤ b}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_ge_inter_not_ge
theorem unbounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Unbounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeInf α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | ¬a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s
-- Set.bounded_ge_inter_gt
theorem bounded_ge_inter_gt [LinearOrder α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Bounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b < a}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_ge_inter_gt
theorem unbounded_ge_inter_gt [LinearOrder α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Unbounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b < a}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s
-- Set.bounded_ge_inter_ge
theorem bounded_ge_inter_ge [LinearOrder α] (a : α) :
    Bounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b ≤ a}) ↔ Set.Bounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_ge_iff_unbounded_inter_ge
theorem unbounded_ge_iff_unbounded_inter_ge [LinearOrder α] (a : α) :
    Unbounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) (s ∩ {b | b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s
-- Set.bounded_gt_inter_not_gt
theorem bounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
    Bounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Bounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeInf α] (a : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | ¬a < b}) ↔ Set.Bounded (fun x x_1 => x > x_1) s
-- Set.unbounded_gt_inter_not_gt
theorem unbounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Unbounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [SemilatticeInf α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | ¬a < b}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s
-- Set.bounded_gt_inter_ge
theorem bounded_gt_inter_ge [LinearOrder α] (a : α) :
    Bounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | b ≤ a}) ↔ Set.Bounded (fun x x_1 => x > x_1) s
-- Set.unbounded_inter_ge
theorem unbounded_inter_ge [LinearOrder α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | b ≤ a}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s
-- Set.bounded_gt_inter_gt
theorem bounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
    Bounded (· > ·) (s ∩ { b | b < a }) ↔ Bounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] [NoMinOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (s ∩ {b | b < a}) ↔ Set.Bounded (fun x x_1 => x > x_1) s
-- Set.unbounded_gt_inter_gt
theorem unbounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
    Unbounded (· > ·) (s ∩ { b | b < a }) ↔ Unbounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α] [NoMinOrder α] (a : α)
Goal: Set.Unbounded (fun x x_1 => x > x_1) (s ∩ {b | b < a}) ↔ Set.Unbounded (fun x x_1 => x > x_1) s
-- FilterBasis.inter_sets
structure FilterBasis (α : Type*) where
    sets : Set (Set α)
    nonempty : sets.Nonempty
    inter_sets {x y} : x ∈ sets → y ∈ sets → ∃ z ∈ sets, z ⊆ x ∩ y
Arguments: {α : Type u_6} (self : FilterBasis α) {x y : Set α}
Goal: x ∈ self.sets → y ∈ self.sets → ∃ z ∈ self.sets, z ⊆ x ∩ y
-- FilterBasis.nonempty
structure FilterBasis (α : Type*) where
    sets : Set (Set α)
    nonempty : sets.Nonempty
    inter_sets {x y} : x ∈ sets → y ∈ sets → ∃ z ∈ sets, z ⊆ x ∩ y
Arguments: {α : Type u_6} (self : FilterBasis α)
Goal: self.sets.Nonempty
-- FilterBasis.mem_sets
@[simp] theorem FilterBasis.mem_sets {s : Set α} {B : FilterBasis α} : s ∈ B.sets ↔ s ∈ B 
Arguments: {α : Type u_1} {s : Set α} {B : FilterBasis α}
Goal: s ∈ B.sets ↔ s ∈ B
-- Filter.IsBasis.nonempty
structure Filter.IsBasis (p : ι → Prop) (s : ι → Set α) : Prop where
    nonempty : ∃ i, p i
    inter : ∀ {i j}, p i → p j → ∃ k, p k ∧ s k ⊆ s i ∩ s j
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (self : Filter.IsBasis p s)
Goal: ∃ i, p i
-- Filter.IsBasis.inter
structure Filter.IsBasis (p : ι → Prop) (s : ι → Set α) : Prop where
    nonempty : ∃ i, p i
    inter : ∀ {i j}, p i → p j → ∃ k, p k ∧ s k ⊆ s i ∩ s j
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (self : Filter.IsBasis p s) {i j : ι}
Goal: p i → p j → ∃ k, p k ∧ s k ⊆ s i ∩ s j
-- Filter.IsBasis.mem_filterBasis_iff
theorem mem_filterBasis_iff {U : Set α} : U ∈ h.filterBasis ↔ ∃ i, p i ∧ s i = U 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) {U : Set α}
Goal: U ∈ h.filterBasis ↔ ∃ i, p i ∧ s i = U
-- FilterBasis.mem_filter_iff
theorem mem_filter_iff (B : FilterBasis α) {U : Set α} : U ∈ B.filter ↔ ∃ s ∈ B, s ⊆ U 
Arguments: {α : Type u_1} (B : FilterBasis α) {U : Set α}
Goal: U ∈ B.filter ↔ ∃ s ∈ B, s ⊆ U
-- FilterBasis.mem_filter_of_mem
theorem mem_filter_of_mem (B : FilterBasis α) {U : Set α} : U ∈ B → U ∈ B.filter 
Arguments: {α : Type u_1} (B : FilterBasis α) {U : Set α}
Goal: U ∈ B → U ∈ B.filter
-- FilterBasis.eq_iInf_principal
theorem eq_iInf_principal (B : FilterBasis α) : B.filter = ⨅ s : B.sets, 𝓟 s 
Arguments: {α : Type u_1} (B : FilterBasis α)
Goal: B.filter = ⨅ s, 𝓟 ↑s
-- FilterBasis.generate
protected theorem generate (B : FilterBasis α) : generate B.sets = B.filter 
Arguments: {α : Type u_1} (B : FilterBasis α)
Goal: Filter.generate B.sets = B.filter
-- Filter.IsBasis.mem_filter_iff
protected theorem mem_filter_iff (h : IsBasis p s) {U : Set α} :
    U ∈ h.filter ↔ ∃ i, p i ∧ s i ⊆ U 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s) {U : Set α}
Goal: U ∈ h.filter ↔ ∃ i, p i ∧ s i ⊆ U
-- Filter.IsBasis.filter_eq_generate
theorem filter_eq_generate (h : IsBasis p s) : h.filter = generate { U | ∃ i, p i ∧ s i = U } 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s)
Goal: h.filter = Filter.generate {U | ∃ i, p i ∧ s i = U}
-- Filter.HasBasis.mem_iff'
structure HasBasis (l : Filter α) (p : ι → Prop) (s : ι → Set α) : Prop where
    mem_iff' : ∀ t : Set α, t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (self : l.HasBasis p s) (t : Set α)
Goal: t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t
-- Filter.hasBasis_generate
theorem hasBasis_generate (s : Set (Set α)) :
    (generate s).HasBasis (fun t => Set.Finite t ∧ t ⊆ s) fun t => ⋂₀ t 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: (Filter.generate s).HasBasis (fun t => t.Finite ∧ t ⊆ s) fun t => ⋂₀ t
-- Filter.FilterBasis.ofSets_sets
lemma FilterBasis.ofSets_sets (s : Set (Set α)) :
    (FilterBasis.ofSets s).sets = sInter '' { t | Set.Finite t ∧ t ⊆ s } 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: (Filter.FilterBasis.ofSets s).sets = Set.sInter '' {t | t.Finite ∧ t ⊆ s}
-- Filter.HasBasis.mem_iff
theorem HasBasis.mem_iff (hl : l.HasBasis p s) : t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (hl : l.HasBasis p s)
Goal: t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t
-- Filter.HasBasis.eq_of_same_basis
theorem HasBasis.eq_of_same_basis (hl : l.HasBasis p s) (hl' : l'.HasBasis p s) : l = l' 
Arguments: {α : Type u_1} {ι : Sort u_4} {l l' : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) (hl' : l'.HasBasis p s)
Goal: l = l'
-- Filter.hasBasis_iff
theorem hasBasis_iff : l.HasBasis p s ↔ ∀ t, t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α}
Goal: l.HasBasis p s ↔ ∀ (t : Set α), t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t
-- Filter.HasBasis.ex_mem
theorem HasBasis.ex_mem (h : l.HasBasis p s) : ∃ i, p i 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: ∃ i, p i
-- Filter.HasBasis.nonempty
protected theorem HasBasis.nonempty (h : l.HasBasis p s) : Nonempty ι 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: Nonempty ι
-- Filter.IsBasis.hasBasis
protected theorem IsBasis.hasBasis (h : IsBasis p s) : HasBasis h.filter p s 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {s : ι → Set α} (h : Filter.IsBasis p s)
Goal: h.filter.HasBasis p s
-- Filter.HasBasis.mem_of_superset
protected theorem HasBasis.mem_of_superset (hl : l.HasBasis p s) (hi : p i) (ht : s i ⊆ t) :
    t ∈ l 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} {i : ι} (hl : l.HasBasis p s) (hi : p i) (ht : s i ⊆ t)
Goal: t ∈ l
-- Filter.HasBasis.mem_of_mem
theorem HasBasis.mem_of_mem (hl : l.HasBasis p s) (hi : p i) : s i ∈ l 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {i : ι} (hl : l.HasBasis p s) (hi : p i)
Goal: s i ∈ l
-- Filter.HasBasis.property_index
theorem HasBasis.property_index (h : l.HasBasis p s) (ht : t ∈ l) : p (h.index t ht) 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l)
Goal: p ↑(h.index t ht)
-- Filter.HasBasis.set_index_mem
theorem HasBasis.set_index_mem (h : l.HasBasis p s) (ht : t ∈ l) : s (h.index t ht) ∈ l 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l)
Goal: s ↑(h.index t ht) ∈ l
-- Filter.HasBasis.set_index_subset
theorem HasBasis.set_index_subset (h : l.HasBasis p s) (ht : t ∈ l) : s (h.index t ht) ⊆ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {t : Set α} (h : l.HasBasis p s) (ht : t ∈ l)
Goal: s ↑(h.index t ht) ⊆ t
-- Filter.HasBasis.isBasis
theorem HasBasis.isBasis (h : l.HasBasis p s) : IsBasis p s 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: Filter.IsBasis p s
-- Filter.HasBasis.filter_eq
theorem HasBasis.filter_eq (h : l.HasBasis p s) : h.isBasis.filter = l 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: ⋯.filter = l
-- Filter.HasBasis.eq_generate
theorem HasBasis.eq_generate (h : l.HasBasis p s) : l = generate { U | ∃ i, p i ∧ s i = U } 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: l = Filter.generate {U | ∃ i, p i ∧ s i = U}
-- Filter.generate_eq_generate_inter
theorem generate_eq_generate_inter (s : Set (Set α)) :
    generate s = generate (sInter '' { t | Set.Finite t ∧ t ⊆ s }) 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: Filter.generate s = Filter.generate (Set.sInter '' {t | t.Finite ∧ t ⊆ s})
-- Filter.ofSets_filter_eq_generate
theorem ofSets_filter_eq_generate (s : Set (Set α)) :
    (FilterBasis.ofSets s).filter = generate s 
Arguments: {α : Type u_1} (s : Set (Set α))
Goal: (Filter.FilterBasis.ofSets s).filter = Filter.generate s
-- FilterBasis.hasBasis
protected theorem _root_.FilterBasis.hasBasis (B : FilterBasis α) :
    HasBasis B.filter (fun s : Set α => s ∈ B) id 
Arguments: {α : Type u_1} (B : FilterBasis α)
Goal: B.filter.HasBasis (fun s => s ∈ B) id
-- Filter.HasBasis.to_hasBasis'
theorem HasBasis.to_hasBasis' (hl : l.HasBasis p s) (h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
    (h' : ∀ i', p' i' → s' i' ∈ l) : l.HasBasis p' s' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (h : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i) (h' : ∀ (i' : ι'), p' i' → s' i' ∈ l)
Goal: l.HasBasis p' s'
-- Filter.HasBasis.to_hasBasis
theorem HasBasis.to_hasBasis (hl : l.HasBasis p s) (h : ∀ i, p i → ∃ i', p' i' ∧ s' i' ⊆ s i)
    (h' : ∀ i', p' i' → ∃ i, p i ∧ s i ⊆ s' i') : l.HasBasis p' s' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} {p' : ι' → Prop} {s' : ι' → Set α} (hl : l.HasBasis p s) (h : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i) (h' : ∀ (i' : ι'), p' i' → ∃ i, p i ∧ s i ⊆ s' i')
Goal: l.HasBasis p' s'
-- Filter.HasBasis.congr
protected lemma HasBasis.congr (hl : l.HasBasis p s) {p' s'} (hp : ∀ i, p i ↔ p' i)
    (hs : ∀ i, p i → s i = s' i) : l.HasBasis p' s' 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {p' : ι → Prop} {s' : ι → Set α} (hp : ∀ (i : ι), p i ↔ p' i) (hs : ∀ (i : ι), p i → s i = s' i)
Goal: l.HasBasis p' s'
-- Filter.HasBasis.to_subset
theorem HasBasis.to_subset (hl : l.HasBasis p s) {t : ι → Set α} (h : ∀ i, p i → t i ⊆ s i)
    (ht : ∀ i, p i → t i ∈ l) : l.HasBasis p t 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {t : ι → Set α} (h : ∀ (i : ι), p i → t i ⊆ s i) (ht : ∀ (i : ι), p i → t i ∈ l)
Goal: l.HasBasis p t
-- Filter.HasBasis.eventually_iff
theorem HasBasis.eventually_iff (hl : l.HasBasis p s) {q : α → Prop} :
    (∀ᶠ x in l, q x) ↔ ∃ i, p i ∧ ∀ ⦃x⦄, x ∈ s i → q x 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {q : α → Prop}
Goal: (∀ᶠ (x : α) in l, q x) ↔ ∃ i, p i ∧ ∀ ⦃x : α⦄, x ∈ s i → q x
-- Filter.HasBasis.frequently_iff
theorem HasBasis.frequently_iff (hl : l.HasBasis p s) {q : α → Prop} :
    (∃ᶠ x in l, q x) ↔ ∀ i, p i → ∃ x ∈ s i, q x 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {q : α → Prop}
Goal: (∃ᶠ (x : α) in l, q x) ↔ ∀ (i : ι), p i → ∃ x ∈ s i, q x
-- Filter.HasBasis.exists_iff
theorem HasBasis.exists_iff (hl : l.HasBasis p s) {P : Set α → Prop}
    (mono : ∀ ⦃s t⦄, s ⊆ t → P t → P s) : (∃ s ∈ l, P s) ↔ ∃ i, p i ∧ P (s i) 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {P : Set α → Prop} (mono : ∀ ⦃s t : Set α⦄, s ⊆ t → P t → P s)
Goal: (∃ s ∈ l, P s) ↔ ∃ i, p i ∧ P (s i)
-- Filter.HasBasis.forall_iff
theorem HasBasis.forall_iff (hl : l.HasBasis p s) {P : Set α → Prop}
    (mono : ∀ ⦃s t⦄, s ⊆ t → P s → P t) : (∀ s ∈ l, P s) ↔ ∀ i, p i → P (s i) 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s) {P : Set α → Prop} (mono : ∀ ⦃s t : Set α⦄, s ⊆ t → P s → P t)
Goal: (∀ s ∈ l, P s) ↔ ∀ (i : ι), p i → P (s i)
-- Filter.HasBasis.neBot_iff
protected theorem HasBasis.neBot_iff (hl : l.HasBasis p s) :
    NeBot l ↔ ∀ {i}, p i → (s i).Nonempty 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s)
Goal: l.NeBot ↔ ∀ {i : ι}, p i → (s i).Nonempty
-- Filter.HasBasis.eq_bot_iff
theorem HasBasis.eq_bot_iff (hl : l.HasBasis p s) : l = ⊥ ↔ ∃ i, p i ∧ s i = ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (hl : l.HasBasis p s)
Goal: l = ⊥ ↔ ∃ i, p i ∧ s i = ∅
-- Filter.generate_neBot_iff
theorem generate_neBot_iff {s : Set (Set α)} :
    NeBot (generate s) ↔ ∀ t, t ⊆ s → t.Finite → (⋂₀ t).Nonempty 
Arguments: {α : Type u_1} {s : Set (Set α)}
Goal: (Filter.generate s).NeBot ↔ ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty
-- Filter.basis_sets
theorem basis_sets (l : Filter α) : l.HasBasis (fun s : Set α => s ∈ l) id 
Arguments: {α : Type u_1} (l : Filter α)
Goal: l.HasBasis (fun s => s ∈ l) id
-- Filter.asBasis_filter
theorem asBasis_filter (f : Filter α) : f.asBasis.filter = f 
Arguments: {α : Type u_1} (f : Filter α)
Goal: f.asBasis.filter = f
-- Filter.hasBasis_self
theorem hasBasis_self {l : Filter α} {P : Set α → Prop} :
    HasBasis l (fun s => s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t 
Arguments: {α : Type u_1} {l : Filter α} {P : Set α → Prop}
Goal: l.HasBasis (fun s => s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t
-- Filter.HasBasis.comp_surjective
theorem HasBasis.comp_surjective (h : l.HasBasis p s) {g : ι' → ι} (hg : Function.Surjective g) :
    l.HasBasis (p ∘ g) (s ∘ g) 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) {g : ι' → ι} (hg : Function.Surjective g)
Goal: l.HasBasis (p ∘ g) (s ∘ g)
-- Filter.HasBasis.comp_equiv
theorem HasBasis.comp_equiv (h : l.HasBasis p s) (e : ι' ≃ ι) : l.HasBasis (p ∘ e) (s ∘ e) 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) (e : ι' ≃ ι)
Goal: l.HasBasis (p ∘ ⇑e) (s ∘ ⇑e)
-- Filter.HasBasis.to_image_id'
theorem HasBasis.to_image_id' (h : l.HasBasis p s) : l.HasBasis (fun t ↦ ∃ i, p i ∧ s i = t) id 
Arguments: {α : Type u_1} {ι : Sort u_4} {l : Filter α} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: l.HasBasis (fun t => ∃ i, p i ∧ s i = t) id
-- Filter.HasBasis.to_image_id
theorem HasBasis.to_image_id {ι : Type*} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s) :
    l.HasBasis (· ∈ s '' {i | p i}) id 
Arguments: {α : Type u_1} {l : Filter α} {ι : Type u_6} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p s)
Goal: l.HasBasis (fun x => x ∈ s '' {i | p i}) id