File: Mathlib/Topology/Order/LeftRight.lean

full_name: frequently_lt_nhds
code: lemma frequently_lt_nhds (a : α) [NeBot (𝓝[<] a)] : ∃ᶠ x in 𝓝 a, x < a 
goal: ∃ᶠ (x : α) in 𝓝 a, x < a

full_name: frequently_gt_nhds
code: lemma frequently_gt_nhds (a : α) [NeBot (𝓝[>] a)] : ∃ᶠ x in 𝓝 a, a < x 
goal: ∃ᶠ (x : α) in 𝓝 a, a < x

full_name: Filter.Eventually.exists_lt
code: theorem Filter.Eventually.exists_lt {a : α} [NeBot (𝓝[<] a)] {p : α → Prop}
    (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b < a, p b 
goal: ∃ b < a, p b

full_name: Filter.Eventually.exists_gt
code: theorem Filter.Eventually.exists_gt {a : α} [NeBot (𝓝[>] a)] {p : α → Prop}
    (h : ∀ᶠ x in 𝓝 a, p x) : ∃ b > a, p b 
goal: ∃ b > a, p b

full_name: nhdsWithin_Ici_neBot
code: theorem nhdsWithin_Ici_neBot {a b : α} (H₂ : a ≤ b) : NeBot (𝓝[Ici a] b) 
goal: (𝓝[Set.Ici a] b).NeBot

full_name: nhdsWithin_Ici_self_neBot
code: instance nhdsWithin_Ici_self_neBot (a : α) : NeBot (𝓝[≥] a) 
goal: (𝓝[≥] a).NeBot

full_name: nhdsWithin_Iic_neBot
code: theorem nhdsWithin_Iic_neBot {a b : α} (H : a ≤ b) : NeBot (𝓝[Iic b] a) 
goal: (𝓝[Set.Iic b] a).NeBot

full_name: nhdsWithin_Iic_self_neBot
code: instance nhdsWithin_Iic_self_neBot (a : α) : NeBot (𝓝[≤] a) 
goal: (𝓝[≤] a).NeBot

full_name: nhds_left'_le_nhds_ne
code: theorem nhds_left'_le_nhds_ne (a : α) : 𝓝[<] a ≤ 𝓝[≠] a 
goal: 𝓝[<] a ≤ 𝓝[≠] a

full_name: nhds_right'_le_nhds_ne
code: theorem nhds_right'_le_nhds_ne (a : α) : 𝓝[>] a ≤ 𝓝[≠] a 
goal: 𝓝[>] a ≤ 𝓝[≠] a

full_name: IsAntichain.interior_eq_empty
code: lemma IsAntichain.interior_eq_empty [∀ x : α, (𝓝[<] x).NeBot] {s : Set α}
    (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ 
goal: interior s = ∅

full_name: IsAntichain.interior_eq_empty'
code: lemma IsAntichain.interior_eq_empty' [∀ x : α, (𝓝[>] x).NeBot] {s : Set α}
    (hs : IsAntichain (· ≤ ·) s) : interior s = ∅ 
goal: interior s = ∅

full_name: continuousWithinAt_Ioi_iff_Ici
code: theorem continuousWithinAt_Ioi_iff_Ici {a : α} {f : α → β} :
    ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a 
goal: ContinuousWithinAt f (Set.Ioi a) a ↔ ContinuousWithinAt f (Set.Ici a) a

full_name: continuousWithinAt_Iio_iff_Iic
code: theorem continuousWithinAt_Iio_iff_Iic {a : α} {f : α → β} :
    ContinuousWithinAt f (Iio a) a ↔ ContinuousWithinAt f (Iic a) a 
goal: ContinuousWithinAt f (Set.Iio a) a ↔ ContinuousWithinAt f (Set.Iic a) a

full_name: nhds_left_sup_nhds_right
code: theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a 
goal: 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a

full_name: nhds_left'_sup_nhds_right
code: theorem nhds_left'_sup_nhds_right (a : α) : 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a 
goal: 𝓝[<] a ⊔ 𝓝[≥] a = 𝓝 a

full_name: nhds_left_sup_nhds_right'
code: theorem nhds_left_sup_nhds_right' (a : α) : 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a 
goal: 𝓝[≤] a ⊔ 𝓝[>] a = 𝓝 a

full_name: nhds_left'_sup_nhds_right'
code: theorem nhds_left'_sup_nhds_right' (a : α) : 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a 
goal: 𝓝[<] a ⊔ 𝓝[>] a = 𝓝[≠] a

full_name: nhdsWithin_right_sup_nhds_singleton
code: lemma nhdsWithin_right_sup_nhds_singleton (a : α) :
    𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a 
goal: 𝓝[>] a ⊔ 𝓝[{a}] a = 𝓝[≥] a

full_name: continuousAt_iff_continuous_left_right
code: theorem continuousAt_iff_continuous_left_right {a : α} {f : α → β} :
    ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a 
goal: ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iic a) a ∧ ContinuousWithinAt f (Set.Ici a) a

full_name: continuousAt_iff_continuous_left'_right'
code: theorem continuousAt_iff_continuous_left'_right' {a : α} {f : α → β} :
    ContinuousAt f a ↔ ContinuousWithinAt f (Iio a) a ∧ ContinuousWithinAt f (Ioi a) a 
goal: ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iio a) a ∧ ContinuousWithinAt f (Set.Ioi a) a