ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Metric.isBounded_range_of_tendsto_cofinite
theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
    IsBounded (range f) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} {a : α} (hf : Filter.Tendsto f Filter.cofinite (𝓝 a))
Goal: Bornology.IsBounded (Set.range f)
-- Metric.isBounded_of_compactSpace
theorem isBounded_of_compactSpace [CompactSpace α] : IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [CompactSpace α]
Goal: Bornology.IsBounded s
-- Metric.isBounded_range_of_tendsto
theorem isBounded_range_of_tendsto (u : ℕ → α) {x : α} (hu : Tendsto u atTop (𝓝 x)) :
    IsBounded (range u) 
Arguments: {α : Type u} [PseudoMetricSpace α] (u : ℕ → α) {x : α} (hu : Filter.Tendsto u Filter.atTop (𝓝 x))
Goal: Bornology.IsBounded (Set.range u)
-- Metric.disjoint_nhds_cobounded
theorem disjoint_nhds_cobounded (x : α) : Disjoint (𝓝 x) (cobounded α) 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: Disjoint (𝓝 x) (Bornology.cobounded α)
-- Metric.disjoint_cobounded_nhds
theorem disjoint_cobounded_nhds (x : α) : Disjoint (cobounded α) (𝓝 x) 
Arguments: {α : Type u} [PseudoMetricSpace α] (x : α)
Goal: Disjoint (Bornology.cobounded α) (𝓝 x)
-- Metric.disjoint_nhdsSet_cobounded
theorem disjoint_nhdsSet_cobounded {s : Set α} (hs : IsCompact s) : Disjoint (𝓝ˢ s) (cobounded α) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s)
Goal: Disjoint (𝓝ˢ s) (Bornology.cobounded α)
-- Metric.disjoint_cobounded_nhdsSet
theorem disjoint_cobounded_nhdsSet {s : Set α} (hs : IsCompact s) : Disjoint (cobounded α) (𝓝ˢ s) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : IsCompact s)
Goal: Disjoint (Bornology.cobounded α) (𝓝ˢ s)
-- Metric.exists_isBounded_image_of_tendsto
theorem exists_isBounded_image_of_tendsto {α β : Type*} [PseudoMetricSpace β]
    {l : Filter α} {f : α → β} {x : β} (hf : Tendsto f l (𝓝 x)) :
    ∃ s ∈ l, IsBounded (f '' s) 
Arguments: {α : Type u_3} {β : Type u_4} [PseudoMetricSpace β] {l : Filter α} {f : α → β} {x : β} (hf : Filter.Tendsto f l (𝓝 x))
Goal: ∃ s ∈ l, Bornology.IsBounded (f '' s)
-- Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_forall_continuousWithinAt
    [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k)
    (hf : ∀ x ∈ k, ContinuousWithinAt f s x) :
    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousWithinAt f s x)
Goal: ∃ t, k ⊆ t ∧ IsOpen t ∧ Bornology.IsBounded (f '' (t ∩ s))
-- Metric.exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt
theorem exists_isOpen_isBounded_image_of_isCompact_of_forall_continuousAt [TopologicalSpace β]
    {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x) :
    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {k : Set β} {f : β → α} (hk : IsCompact k) (hf : ∀ x ∈ k, ContinuousAt f x)
Goal: ∃ t, k ⊆ t ∧ IsOpen t ∧ Bornology.IsBounded (f '' t)
-- Metric.exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn
theorem exists_isOpen_isBounded_image_inter_of_isCompact_of_continuousOn [TopologicalSpace β]
    {k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s) :
    ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' (t ∩ s)) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k) (hks : k ⊆ s) (hf : ContinuousOn f s)
Goal: ∃ t, k ⊆ t ∧ IsOpen t ∧ Bornology.IsBounded (f '' (t ∩ s))
-- Metric.exists_isOpen_isBounded_image_of_isCompact_of_continuousOn
theorem exists_isOpen_isBounded_image_of_isCompact_of_continuousOn [TopologicalSpace β]
    {k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s)
    (hf : ContinuousOn f s) : ∃ t, k ⊆ t ∧ IsOpen t ∧ IsBounded (f '' t) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] [TopologicalSpace β] {k s : Set β} {f : β → α} (hk : IsCompact k) (hs : IsOpen s) (hks : k ⊆ s) (hf : ContinuousOn f s)
Goal: ∃ t, k ⊆ t ∧ IsOpen t ∧ Bornology.IsBounded (f '' t)
-- Metric.isCompact_of_isClosed_isBounded
theorem isCompact_of_isClosed_isBounded [ProperSpace α] (hc : IsClosed s) (hb : IsBounded s) :
    IsCompact s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [ProperSpace α] (hc : IsClosed s) (hb : Bornology.IsBounded s)
Goal: IsCompact s
-- Bornology.IsBounded.isCompact_closure
theorem _root_.Bornology.IsBounded.isCompact_closure [ProperSpace α] (h : IsBounded s) :
    IsCompact (closure s) 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [ProperSpace α] (h : Bornology.IsBounded s)
Goal: IsCompact (closure s)
-- Metric.isCompact_iff_isClosed_bounded
theorem isCompact_iff_isClosed_bounded [T2Space α] [ProperSpace α] :
    IsCompact s ↔ IsClosed s ∧ IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} [T2Space α] [ProperSpace α]
Goal: IsCompact s ↔ IsClosed s ∧ Bornology.IsBounded s
-- Metric.compactSpace_iff_isBounded_univ
theorem compactSpace_iff_isBounded_univ [ProperSpace α] :
    CompactSpace α ↔ IsBounded (univ : Set α) 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α]
Goal: CompactSpace α ↔ Bornology.IsBounded Set.univ
-- totallyBounded_Icc
theorem _root_.totallyBounded_Icc (a b : α) : TotallyBounded (Icc a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: TotallyBounded (Set.Icc a b)
-- totallyBounded_Ico
theorem _root_.totallyBounded_Ico (a b : α) : TotallyBounded (Ico a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: TotallyBounded (Set.Ico a b)
-- totallyBounded_Ioc
theorem _root_.totallyBounded_Ioc (a b : α) : TotallyBounded (Ioc a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: TotallyBounded (Set.Ioc a b)
-- totallyBounded_Ioo
theorem _root_.totallyBounded_Ioo (a b : α) : TotallyBounded (Ioo a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: TotallyBounded (Set.Ioo a b)
-- Metric.isBounded_Icc
theorem isBounded_Icc (a b : α) : IsBounded (Icc a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: Bornology.IsBounded (Set.Icc a b)
-- Metric.isBounded_Ico
theorem isBounded_Ico (a b : α) : IsBounded (Ico a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: Bornology.IsBounded (Set.Ico a b)
-- Metric.isBounded_Ioc
theorem isBounded_Ioc (a b : α) : IsBounded (Ioc a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: Bornology.IsBounded (Set.Ioc a b)
-- Metric.isBounded_Ioo
theorem isBounded_Ioo (a b : α) : IsBounded (Ioo a b) 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] (a b : α)
Goal: Bornology.IsBounded (Set.Ioo a b)
-- Metric.isBounded_of_bddAbove_of_bddBelow
theorem isBounded_of_bddAbove_of_bddBelow {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s) :
    IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] [Preorder α] [CompactIccSpace α] {s : Set α} (h₁ : BddAbove s) (h₂ : BddBelow s)
Goal: Bornology.IsBounded s
-- Metric.diam_nonneg
theorem diam_nonneg : 0 ≤ diam s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: 0 ≤ Metric.diam s
-- Metric.diam_subsingleton
theorem diam_subsingleton (hs : s.Subsingleton) : diam s = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : s.Subsingleton)
Goal: Metric.diam s = 0
-- Metric.diam_empty
@[simp]
theorem diam_empty : diam (∅ : Set α) = 0 
Arguments: {α : Type u} [PseudoMetricSpace α]
Goal: Metric.diam ∅ = 0
-- Metric.diam_singleton
@[simp]
theorem diam_singleton : diam ({x} : Set α) = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α}
Goal: Metric.diam {x} = 0
-- Metric.diam_zero
@[to_additive (attr := simp)]
theorem diam_one [One α] : diam (1 : Set α) = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] [Zero α]
Goal: Metric.diam 0 = 0
-- Metric.diam_one
@[to_additive (attr := simp)]
theorem diam_one [One α] : diam (1 : Set α) = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] [One α]
Goal: Metric.diam 1 = 0
-- Metric.diam_pair
theorem diam_pair : diam ({x, y} : Set α) = dist x y 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y : α}
Goal: Metric.diam {x, y} = dist x y
-- Metric.diam_triple
theorem diam_triple :
    Metric.diam ({x, y, z} : Set α) = max (max (dist x y) (dist x z)) (dist y z) 
Arguments: {α : Type u} [PseudoMetricSpace α] {x y z : α}
Goal: Metric.diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z)
-- Metric.ediam_le_of_forall_dist_le
theorem ediam_le_of_forall_dist_le {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
    EMetric.diam s ≤ ENNReal.ofReal C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C)
Goal: EMetric.diam s ≤ ENNReal.ofReal C
-- Metric.diam_le_of_forall_dist_le
theorem diam_le_of_forall_dist_le {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) :
    diam s ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {C : ℝ} (h₀ : 0 ≤ C) (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C)
Goal: Metric.diam s ≤ C
-- Metric.diam_le_of_forall_dist_le_of_nonempty
theorem diam_le_of_forall_dist_le_of_nonempty (hs : s.Nonempty) {C : ℝ}
    (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C) : diam s ≤ C 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (hs : s.Nonempty) {C : ℝ} (h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C)
Goal: Metric.diam s ≤ C
-- Metric.dist_le_diam_of_mem'
theorem dist_le_diam_of_mem' (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s) :
    dist x y ≤ diam s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {x y : α} (h : EMetric.diam s ≠ ⊤) (hx : x ∈ s) (hy : y ∈ s)
Goal: dist x y ≤ Metric.diam s
-- Metric.isBounded_iff_ediam_ne_top
theorem isBounded_iff_ediam_ne_top : IsBounded s ↔ EMetric.diam s ≠ ⊤ 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: Bornology.IsBounded s ↔ EMetric.diam s ≠ ⊤
-- Metric.ediam_eq_top_iff_unbounded
theorem ediam_eq_top_iff_unbounded : EMetric.diam s = ⊤ ↔ ¬IsBounded s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α}
Goal: EMetric.diam s = ⊤ ↔ ¬Bornology.IsBounded s
-- Metric.ediam_univ_eq_top_iff_noncompact
theorem ediam_univ_eq_top_iff_noncompact [ProperSpace α] :
    EMetric.diam (univ : Set α) = ∞ ↔ NoncompactSpace α 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α]
Goal: EMetric.diam Set.univ = ⊤ ↔ NoncompactSpace α
-- Metric.ediam_univ_of_noncompact
@[simp]
theorem ediam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] :
    EMetric.diam (univ : Set α) = ∞ 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] [NoncompactSpace α]
Goal: EMetric.diam Set.univ = ⊤
-- Metric.diam_univ_of_noncompact
@[simp]
theorem diam_univ_of_noncompact [ProperSpace α] [NoncompactSpace α] : diam (univ : Set α) = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] [NoncompactSpace α]
Goal: Metric.diam Set.univ = 0
-- Metric.dist_le_diam_of_mem
theorem dist_le_diam_of_mem (h : IsBounded s) (hx : x ∈ s) (hy : y ∈ s) : dist x y ≤ diam s 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {x y : α} (h : Bornology.IsBounded s) (hx : x ∈ s) (hy : y ∈ s)
Goal: dist x y ≤ Metric.diam s
-- Metric.ediam_of_unbounded
theorem ediam_of_unbounded (h : ¬IsBounded s) : EMetric.diam s = ∞ 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : ¬Bornology.IsBounded s)
Goal: EMetric.diam s = ⊤
-- Metric.diam_eq_zero_of_unbounded
theorem diam_eq_zero_of_unbounded (h : ¬IsBounded s) : diam s = 0 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} (h : ¬Bornology.IsBounded s)
Goal: Metric.diam s = 0
-- Metric.diam_mono
theorem diam_mono {s t : Set α} (h : s ⊆ t) (ht : IsBounded t) : diam s ≤ diam t 
Arguments: {α : Type u} [PseudoMetricSpace α] {s t : Set α} (h : s ⊆ t) (ht : Bornology.IsBounded t)
Goal: Metric.diam s ≤ Metric.diam t
-- Metric.diam_union
theorem diam_union {t : Set α} (xs : x ∈ s) (yt : y ∈ t) :
    diam (s ∪ t) ≤ diam s + dist x y + diam t 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {x y : α} {t : Set α} (xs : x ∈ s) (yt : y ∈ t)
Goal: Metric.diam (s ∪ t) ≤ Metric.diam s + dist x y + Metric.diam t
-- Metric.diam_union'
theorem diam_union' {t : Set α} (h : (s ∩ t).Nonempty) : diam (s ∪ t) ≤ diam s + diam t 
Arguments: {α : Type u} [PseudoMetricSpace α] {s t : Set α} (h : (s ∩ t).Nonempty)
Goal: Metric.diam (s ∪ t) ≤ Metric.diam s + Metric.diam t
-- Metric.diam_le_of_subset_closedBall
theorem diam_le_of_subset_closedBall {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ closedBall x r) :
    diam s ≤ 2 * r 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : Set α} {x : α} {r : ℝ} (hr : 0 ≤ r) (h : s ⊆ Metric.closedBall x r)
Goal: Metric.diam s ≤ 2 * r
-- Metric.diam_closedBall
theorem diam_closedBall {r : ℝ} (h : 0 ≤ r) : diam (closedBall x r) ≤ 2 * r 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h : 0 ≤ r)
Goal: Metric.diam (Metric.closedBall x r) ≤ 2 * r
-- Metric.diam_ball
theorem diam_ball {r : ℝ} (h : 0 ≤ r) : diam (ball x r) ≤ 2 * r 
Arguments: {α : Type u} [PseudoMetricSpace α] {x : α} {r : ℝ} (h : 0 ≤ r)
Goal: Metric.diam (Metric.ball x r) ≤ 2 * r
-- IsComplete.nonempty_iInter_of_nonempty_biInter
theorem _root_.IsComplete.nonempty_iInter_of_nonempty_biInter {s : ℕ → Set α}
    (h0 : IsComplete (s 0)) (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n))
    (h : ∀ N, (⋂ n ≤ N, s n).Nonempty) (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) :
    (⋂ n, s n).Nonempty 
Arguments: {α : Type u} [PseudoMetricSpace α] {s : ℕ → Set α} (h0 : IsComplete (s 0)) (hs : ∀ (n : ℕ), IsClosed (s n)) (h's : ∀ (n : ℕ), Bornology.IsBounded (s n)) (h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty) (h' : Filter.Tendsto (fun n => Metric.diam (s n)) Filter.atTop (𝓝 0))
Goal: (⋂ n, s n).Nonempty
-- Metric.nonempty_iInter_of_nonempty_biInter
theorem nonempty_iInter_of_nonempty_biInter [CompleteSpace α] {s : ℕ → Set α}
    (hs : ∀ n, IsClosed (s n)) (h's : ∀ n, IsBounded (s n)) (h : ∀ N, (⋂ n ≤ N, s n).Nonempty)
    (h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)) : (⋂ n, s n).Nonempty 
Arguments: {α : Type u} [PseudoMetricSpace α] [CompleteSpace α] {s : ℕ → Set α} (hs : ∀ (n : ℕ), IsClosed (s n)) (h's : ∀ (n : ℕ), Bornology.IsBounded (s n)) (h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty) (h' : Filter.Tendsto (fun n => Metric.diam (s n)) Filter.atTop (𝓝 0))
Goal: (⋂ n, s n).Nonempty
-- Metric.cobounded_eq_cocompact
theorem Metric.cobounded_eq_cocompact [ProperSpace α] : cobounded α = cocompact α 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α]
Goal: Bornology.cobounded α = Filter.cocompact α
-- tendsto_dist_right_cocompact_atTop
theorem tendsto_dist_right_cocompact_atTop [ProperSpace α] (x : α) :
    Tendsto (dist · x) (cocompact α) atTop 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] (x : α)
Goal: Filter.Tendsto (fun x_1 => dist x_1 x) (Filter.cocompact α) Filter.atTop
-- tendsto_dist_left_cocompact_atTop
theorem tendsto_dist_left_cocompact_atTop [ProperSpace α] (x : α) :
    Tendsto (dist x) (cocompact α) atTop 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] (x : α)
Goal: Filter.Tendsto (dist x) (Filter.cocompact α) Filter.atTop
-- comap_dist_left_atTop_eq_cocompact
theorem comap_dist_left_atTop_eq_cocompact [ProperSpace α] (x : α) :
    comap (dist x) atTop = cocompact α 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] (x : α)
Goal: Filter.comap (dist x) Filter.atTop = Filter.cocompact α
-- tendsto_cocompact_of_tendsto_dist_comp_atTop
theorem tendsto_cocompact_of_tendsto_dist_comp_atTop {f : β → α} {l : Filter β} (x : α)
    (h : Tendsto (fun y => dist (f y) x) l atTop) : Tendsto f l (cocompact α) 
Arguments: {α : Type u} {β : Type v} [PseudoMetricSpace α] {f : β → α} {l : Filter β} (x : α) (h : Filter.Tendsto (fun y => dist (f y) x) l Filter.atTop)
Goal: Filter.Tendsto f l (Filter.cocompact α)
-- Metric.finite_isBounded_inter_isClosed
theorem Metric.finite_isBounded_inter_isClosed [ProperSpace α] {K s : Set α} [DiscreteTopology s]
    (hK : IsBounded K) (hs : IsClosed s) : Set.Finite (K ∩ s) 
Arguments: {α : Type u} [PseudoMetricSpace α] [ProperSpace α] {K s : Set α} [DiscreteTopology ↑s] (hK : Bornology.IsBounded K) (hs : IsClosed s)
Goal: (K ∩ s).Finite
-- Bundle.Trivial.trivialization.coordChangeL
theorem trivialization.coordChangeL (b : B) :
    (trivialization B F).coordChangeL 𝕜 (trivialization B F) b =
      ContinuousLinearEquiv.refl 𝕜 F 
Arguments: {𝕜 : Type u_1} (B : Type u_2) (F : Type u_3) [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [TopologicalSpace B] (b : B)
Goal: Trivialization.coordChangeL 𝕜 (Bundle.Trivial.trivialization B F) (Bundle.Trivial.trivialization B F) b =
  ContinuousLinearEquiv.refl 𝕜 F
-- Trivialization.coordChangeL_prod
@[simp]
theorem coordChangeL_prod [e₁.IsLinear 𝕜] [e₁'.IsLinear 𝕜] [e₂.IsLinear 𝕜] [e₂'.IsLinear 𝕜] ⦃b⦄
    (hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet) :
    ((e₁.prod e₂).coordChangeL 𝕜 (e₁'.prod e₂') b : F₁ × F₂ →L[𝕜] F₁ × F₂) =
      (e₁.coordChangeL 𝕜 e₁' b : F₁ →L[𝕜] F₁).prodMap (e₂.coordChangeL 𝕜 e₂' b) 
Arguments: (𝕜 : Type u_1) {B : Type u_2} [NontriviallyNormedField 𝕜] [TopologicalSpace B] {F₁ : Type u_3} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {E₁ : B → Type u_4} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_5} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {E₂ : B → Type u_6} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → AddCommMonoid (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [(x : B) → AddCommMonoid (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] (e₁ e₁' : Trivialization F₁ Bundle.TotalSpace.proj) (e₂ e₂' : Trivialization F₂ Bundle.TotalSpace.proj) [Trivialization.IsLinear 𝕜 e₁] [Trivialization.IsLinear 𝕜 e₁'] [Trivialization.IsLinear 𝕜 e₂] [Trivialization.IsLinear 𝕜 e₂'] ⦃b : B⦄ (hb : b ∈ (e₁.prod e₂).baseSet ∩ (e₁'.prod e₂').baseSet)
Goal: ↑(Trivialization.coordChangeL 𝕜 (e₁.prod e₂) (e₁'.prod e₂') b) =
  (↑(Trivialization.coordChangeL 𝕜 e₁ e₁' b)).prodMap ↑(Trivialization.coordChangeL 𝕜 e₂ e₂' b)
-- Trivialization.prod_apply
theorem prod_apply [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B} (hx₁ : x ∈ e₁.baseSet)
    (hx₂ : x ∈ e₂.baseSet) (v₁ : E₁ x) (v₂ : E₂ x) :
    prod e₁ e₂ ⟨x, (v₁, v₂)⟩ =
      ⟨x, e₁.continuousLinearEquivAt 𝕜 x hx₁ v₁, e₂.continuousLinearEquivAt 𝕜 x hx₂ v₂⟩ 
Arguments: (𝕜 : Type u_1) {B : Type u_2} [NontriviallyNormedField 𝕜] [TopologicalSpace B] {F₁ : Type u_3} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {E₁ : B → Type u_4} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_5} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {E₂ : B → Type u_6} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → AddCommMonoid (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [(x : B) → AddCommMonoid (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₁ E₁] [FiberBundle F₂ E₂] [Trivialization.IsLinear 𝕜 e₁] [Trivialization.IsLinear 𝕜 e₂] {x : B} (hx₁ : x ∈ e₁.baseSet) (hx₂ : x ∈ e₂.baseSet) (v₁ : E₁ x) (v₂ : E₂ x)
Goal: ↑(e₁.prod e₂) { proj := x, snd := (v₁, v₂) } =
  (x, (Trivialization.continuousLinearEquivAt 𝕜 e₁ x hx₁) v₁, (Trivialization.continuousLinearEquivAt 𝕜 e₂ x hx₂) v₂)
-- Trivialization.continuousLinearEquivAt_prod
@[simp] -- Porting note: changed arguments to make `simpNF` happy: merged `hx₁` and `hx₂` into `hx`
theorem Trivialization.continuousLinearEquivAt_prod {e₁ : Trivialization F₁ (π F₁ E₁)}
    {e₂ : Trivialization F₂ (π F₂ E₂)} [e₁.IsLinear 𝕜] [e₂.IsLinear 𝕜] {x : B}
    (hx : x ∈ (e₁.prod e₂).baseSet) :
    (e₁.prod e₂).continuousLinearEquivAt 𝕜 x hx =
      (e₁.continuousLinearEquivAt 𝕜 x hx.1).prod (e₂.continuousLinearEquivAt 𝕜 x hx.2) 
Arguments: {𝕜 : Type u_1} {B : Type u_2} [NontriviallyNormedField 𝕜] [TopologicalSpace B] {F₁ : Type u_3} [NormedAddCommGroup F₁] [NormedSpace 𝕜 F₁] {E₁ : B → Type u_4} [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_5} [NormedAddCommGroup F₂] [NormedSpace 𝕜 F₂] {E₂ : B → Type u_6} [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [(x : B) → AddCommMonoid (E₁ x)] [(x : B) → Module 𝕜 (E₁ x)] [(x : B) → AddCommMonoid (E₂ x)] [(x : B) → Module 𝕜 (E₂ x)] [(x : B) → TopologicalSpace (E₁ x)] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₁ E₁] [FiberBundle F₂ E₂] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [Trivialization.IsLinear 𝕜 e₁] [Trivialization.IsLinear 𝕜 e₂] {x : B} (hx : x ∈ (e₁.prod e₂).baseSet)
Goal: Trivialization.continuousLinearEquivAt 𝕜 (e₁.prod e₂) x hx =
  (Trivialization.continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (Trivialization.continuousLinearEquivAt 𝕜 e₂ x ⋯)
-- Subtype.dist_eq
lemma Subtype.dist_eq {p : α → Prop} (x y : Subtype p) : dist x y = dist (x : α) y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {p : α → Prop} (x y : Subtype p)
Goal: dist x y = dist ↑x ↑y
-- Subtype.nndist_eq
lemma Subtype.nndist_eq {p : α → Prop} (x y : Subtype p) : nndist x y = nndist (x : α) y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] {p : α → Prop} (x y : Subtype p)
Goal: nndist x y = nndist ↑x ↑y
-- AddOpposite.dist_unop
@[to_additive (attr := simp)]
lemma dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : αᵃᵒᵖ)
Goal: dist (AddOpposite.unop x) (AddOpposite.unop y) = dist x y
-- MulOpposite.dist_unop
@[to_additive (attr := simp)]
lemma dist_unop (x y : αᵐᵒᵖ) : dist (unop x) (unop y) = dist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : αᵐᵒᵖ)
Goal: dist (MulOpposite.unop x) (MulOpposite.unop y) = dist x y
-- AddOpposite.dist_op
@[to_additive (attr := simp)]
lemma dist_op (x y : α) : dist (op x) (op y) = dist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : α)
Goal: dist (AddOpposite.op x) (AddOpposite.op y) = dist x y
-- MulOpposite.dist_op
@[to_additive (attr := simp)]
lemma dist_op (x y : α) : dist (op x) (op y) = dist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : α)
Goal: dist (MulOpposite.op x) (MulOpposite.op y) = dist x y
-- AddOpposite.nndist_unop
@[to_additive (attr := simp)]
lemma nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : αᵃᵒᵖ)
Goal: nndist (AddOpposite.unop x) (AddOpposite.unop y) = nndist x y
-- MulOpposite.nndist_unop
@[to_additive (attr := simp)]
lemma nndist_unop (x y : αᵐᵒᵖ) : nndist (unop x) (unop y) = nndist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : αᵐᵒᵖ)
Goal: nndist (MulOpposite.unop x) (MulOpposite.unop y) = nndist x y
-- AddOpposite.nndist_op
@[to_additive (attr := simp)]
lemma nndist_op (x y : α) : nndist (op x) (op y) = nndist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : α)
Goal: nndist (AddOpposite.op x) (AddOpposite.op y) = nndist x y
-- MulOpposite.nndist_op
@[to_additive (attr := simp)]
lemma nndist_op (x y : α) : nndist (op x) (op y) = nndist x y 
Arguments: {α : Type u_1} [PseudoMetricSpace α] (x y : α)
Goal: nndist (MulOpposite.op x) (MulOpposite.op y) = nndist x y
-- NNReal.dist_eq
lemma NNReal.dist_eq (a b : ℝ≥0) : dist a b = |(a : ℝ) - b| 
Arguments: (a b : ℝ≥0)
Goal: dist a b = |↑a - ↑b|
-- NNReal.nndist_eq
lemma NNReal.nndist_eq (a b : ℝ≥0) : nndist a b = max (a - b) (b - a) 
Arguments: (a b : ℝ≥0)
Goal: nndist a b = max (a - b) (b - a)
-- NNReal.nndist_zero_eq_val
@[simp]
lemma NNReal.nndist_zero_eq_val (z : ℝ≥0) : nndist 0 z = z 
Arguments: (z : ℝ≥0)
Goal: nndist 0 z = z
-- NNReal.nndist_zero_eq_val'
@[simp]
lemma NNReal.nndist_zero_eq_val' (z : ℝ≥0) : nndist z 0 = z 
Arguments: (z : ℝ≥0)
Goal: nndist z 0 = z
-- NNReal.le_add_nndist
lemma NNReal.le_add_nndist (a b : ℝ≥0) : a ≤ b + nndist a b 
Arguments: (a b : ℝ≥0)
Goal: a ≤ b + nndist a b
-- NNReal.ball_zero_eq_Ico'
lemma NNReal.ball_zero_eq_Ico' (c : ℝ≥0) :
    Metric.ball (0 : ℝ≥0) c.toReal = Set.Ico 0 c 
Arguments: (c : ℝ≥0)
Goal: Metric.ball 0 ↑c = Set.Ico 0 c