ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UniformSpace.Completion.continuous_map₂
theorem continuous_map₂ {δ} [TopologicalSpace δ] {f : α → β → γ} {a : δ → Completion α}
    {b : δ → Completion β} (ha : Continuous a) (hb : Continuous b) :
    Continuous fun d : δ => Completion.map₂ f (a d) (b d) 
Arguments: {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] {δ : Type u_4} [TopologicalSpace δ] {f : α → β → γ} {a : δ → UniformSpace.Completion α} {b : δ → UniformSpace.Completion β} (ha : Continuous a) (hb : Continuous b)
Goal: Continuous fun d => UniformSpace.Completion.map₂ f (a d) (b d)
-- UniformSpace.Completion.map₂_coe_coe
theorem map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f) :
    Completion.map₂ f (a : Completion α) (b : Completion β) = f a b 
Arguments: {α : Type u_1} [UniformSpace α] {β : Type u_2} [UniformSpace β] {γ : Type u_3} [UniformSpace γ] (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f)
Goal: UniformSpace.Completion.map₂ f (↑α a) (↑β b) = ↑γ (f a b)
-- exists_subset_iUnion_ball_radius_lt
theorem exists_subset_iUnion_ball_radius_lt {r : ι → ℝ} (hs : IsClosed s)
    (uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
    ∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i 
Arguments: {α : Type u} {ι : Type v} [MetricSpace α] [ProperSpace α] {c : ι → α} {s : Set α} {r : ι → ℝ} (hs : IsClosed s) (uf : ∀ x ∈ s, {i | x ∈ Metric.ball (c i) (r i)}.Finite) (us : s ⊆ ⋃ i, Metric.ball (c i) (r i))
Goal: ∃ r', s ⊆ ⋃ i, Metric.ball (c i) (r' i) ∧ ∀ (i : ι), r' i < r i
-- exists_iUnion_ball_eq_radius_lt
theorem exists_iUnion_ball_eq_radius_lt {r : ι → ℝ} (uf : ∀ x, { i | x ∈ ball (c i) (r i) }.Finite)
    (uU : ⋃ i, ball (c i) (r i) = univ) :
    ∃ r' : ι → ℝ, ⋃ i, ball (c i) (r' i) = univ ∧ ∀ i, r' i < r i 
Arguments: {α : Type u} {ι : Type v} [MetricSpace α] [ProperSpace α] {c : ι → α} {r : ι → ℝ} (uf : ∀ (x : α), {i | x ∈ Metric.ball (c i) (r i)}.Finite) (uU : ⋃ i, Metric.ball (c i) (r i) = Set.univ)
Goal: ∃ r', ⋃ i, Metric.ball (c i) (r' i) = Set.univ ∧ ∀ (i : ι), r' i < r i
-- exists_subset_iUnion_ball_radius_pos_lt
theorem exists_subset_iUnion_ball_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i) (hs : IsClosed s)
    (uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
    ∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i ∈ Ioo 0 (r i) 
Arguments: {α : Type u} {ι : Type v} [MetricSpace α] [ProperSpace α] {c : ι → α} {s : Set α} {r : ι → ℝ} (hr : ∀ (i : ι), 0 < r i) (hs : IsClosed s) (uf : ∀ x ∈ s, {i | x ∈ Metric.ball (c i) (r i)}.Finite) (us : s ⊆ ⋃ i, Metric.ball (c i) (r i))
Goal: ∃ r', s ⊆ ⋃ i, Metric.ball (c i) (r' i) ∧ ∀ (i : ι), r' i ∈ Set.Ioo 0 (r i)
-- exists_iUnion_ball_eq_radius_pos_lt
theorem exists_iUnion_ball_eq_radius_pos_lt {r : ι → ℝ} (hr : ∀ i, 0 < r i)
    (uf : ∀ x, { i | x ∈ ball (c i) (r i) }.Finite) (uU : ⋃ i, ball (c i) (r i) = univ) :
    ∃ r' : ι → ℝ, ⋃ i, ball (c i) (r' i) = univ ∧ ∀ i, r' i ∈ Ioo 0 (r i) 
Arguments: {α : Type u} {ι : Type v} [MetricSpace α] [ProperSpace α] {c : ι → α} {r : ι → ℝ} (hr : ∀ (i : ι), 0 < r i) (uf : ∀ (x : α), {i | x ∈ Metric.ball (c i) (r i)}.Finite) (uU : ⋃ i, Metric.ball (c i) (r i) = Set.univ)
Goal: ∃ r', ⋃ i, Metric.ball (c i) (r' i) = Set.univ ∧ ∀ (i : ι), r' i ∈ Set.Ioo 0 (r i)
-- exists_locallyFinite_subset_iUnion_ball_radius_lt
theorem exists_locallyFinite_subset_iUnion_ball_radius_lt (hs : IsClosed s) {R : α → ℝ}
    (hR : ∀ x ∈ s, 0 < R x) :
    ∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
      (∀ i, c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
        (LocallyFinite fun i => ball (c i) (r' i)) ∧ s ⊆ ⋃ i, ball (c i) (r i) 
Arguments: {α : Type u} [MetricSpace α] [ProperSpace α] {s : Set α} (hs : IsClosed s) {R : α → ℝ} (hR : ∀ x ∈ s, 0 < R x)
Goal: ∃ ι c r r',
  (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
    (LocallyFinite fun i => Metric.ball (c i) (r' i)) ∧ s ⊆ ⋃ i, Metric.ball (c i) (r i)
-- exists_locallyFinite_iUnion_eq_ball_radius_lt
theorem exists_locallyFinite_iUnion_eq_ball_radius_lt {R : α → ℝ} (hR : ∀ x, 0 < R x) :
    ∃ (ι : Type u) (c : ι → α) (r r' : ι → ℝ),
      (∀ i, 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
        (LocallyFinite fun i => ball (c i) (r' i)) ∧ ⋃ i, ball (c i) (r i) = univ 
Arguments: {α : Type u} [MetricSpace α] [ProperSpace α] {R : α → ℝ} (hR : ∀ (x : α), 0 < R x)
Goal: ∃ ι c r r',
  (∀ (i : ι), 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧
    (LocallyFinite fun i => Metric.ball (c i) (r' i)) ∧ ⋃ i, Metric.ball (c i) (r i) = Set.univ
-- tendsto_ite
lemma tendsto_ite {β : Type*} {p : ι → Prop} [DecidablePred p] {q : Prop} [Decidable q]
    {a b : β} {F G : Filter β}
    (haG : {a}ᶜ ∈ G) (hbF : {b}ᶜ ∈ F) (haF : principal {a} ≤ F) (hbG : principal {b} ≤ G) :
    Tendsto (fun i ↦ if p i then a else b) L (if q then F else G) ↔ ∀ᶠ i in L, p i ↔ q 
Arguments: {ι : Type u_3} (L : Filter ι) {β : Type u_4} {p : ι → Prop} [DecidablePred p] {q : Prop} [Decidable q] {a b : β} {F G : Filter β} (haG : {a}ᶜ ∈ G) (hbF : {b}ᶜ ∈ F) (haF : 𝓟 {a} ≤ F) (hbG : 𝓟 {b} ≤ G)
Goal: Filter.Tendsto (fun i => if p i then a else b) L (if q then F else G) ↔ ∀ᶠ (i : ι) in L, p i ↔ q
-- tendsto_indicator_const_apply_iff_eventually'
lemma tendsto_indicator_const_apply_iff_eventually' (b : β)
    (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) (x : α) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b) x) L (𝓝 (A.indicator (fun (_ : α) ↦ b) x))
      ↔ ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) (x : α)
Goal: Filter.Tendsto (fun i => (As i).indicator (fun x => b) x) L (𝓝 (A.indicator (fun x => b) x)) ↔
  ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- tendsto_indicator_const_iff_forall_eventually'
lemma tendsto_indicator_const_iff_forall_eventually'
    (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ ∀ x, ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0)
Goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- tendsto_indicator_const_apply_iff_eventually
@[simp] lemma tendsto_indicator_const_apply_iff_eventually [T1Space β] (b : β) [NeZero b]
    (x : α) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b) x) L (𝓝 (A.indicator (fun (_ : α) ↦ b) x))
      ↔ ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} [T1Space β] (b : β) [NeZero b] (x : α)
Goal: Filter.Tendsto (fun i => (As i).indicator (fun x => b) x) L (𝓝 (A.indicator (fun x => b) x)) ↔
  ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- tendsto_indicator_const_iff_forall_eventually
@[simp] lemma tendsto_indicator_const_iff_forall_eventually [T1Space β] (b : β) [NeZero b] :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ ∀ x, ∀ᶠ i in L, (x ∈ As i ↔ x ∈ A) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} [T1Space β] (b : β) [NeZero b]
Goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
-- tendsto_indicator_const_iff_tendsto_pi_pure'
lemma tendsto_indicator_const_iff_tendsto_pi_pure'
    (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0) :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ (Tendsto As L <| Filter.pi (pure <| · ∈ A)) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} (b : β) (nhd_b : {0}ᶜ ∈ 𝓝 b) (nhd_o : {b}ᶜ ∈ 𝓝 0)
Goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  Filter.Tendsto As L (Filter.pi fun x => pure (x ∈ A))
-- tendsto_indicator_const_iff_tendsto_pi_pure
lemma tendsto_indicator_const_iff_tendsto_pi_pure [T1Space β] (b : β) [NeZero b] :
    Tendsto (fun i ↦ (As i).indicator (fun (_ : α) ↦ b)) L (𝓝 (A.indicator (fun (_ : α) ↦ b)))
      ↔ (Tendsto As L <| Filter.pi (pure <| · ∈ A)) 
Arguments: {α : Type u_1} {A : Set α} {β : Type u_2} [Zero β] [TopologicalSpace β] {ι : Type u_3} (L : Filter ι) {As : ι → Set α} [T1Space β] (b : β) [NeZero b]
Goal: Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔
  Filter.Tendsto As L (Filter.pi fun x => pure (x ∈ A))
-- Pretrivialization.open_baseSet
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
  open_target : IsOpen target
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (self : Pretrivialization F proj)
Goal: IsOpen self.baseSet
-- Pretrivialization.source_eq
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
  open_target : IsOpen target
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (self : Pretrivialization F proj)
Goal: self.source = proj ⁻¹' self.baseSet
-- Pretrivialization.open_target
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
  open_target : IsOpen target
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (self : Pretrivialization F proj)
Goal: IsOpen self.target
-- Pretrivialization.proj_toFun
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
  open_target : IsOpen target
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (self : Pretrivialization F proj) (p : Z)
Goal: p ∈ self.source → (↑self.toPartialEquiv p).1 = proj p
-- Pretrivialization.target_eq
structure Pretrivialization (proj : Z → B) extends PartialEquiv Z (B × F) where
  open_target : IsOpen target
  baseSet : Set B
  open_baseSet : IsOpen baseSet
  source_eq : source = proj ⁻¹' baseSet
  target_eq : target = baseSet ×ˢ univ
  proj_toFun : ∀ p ∈ source, (toFun p).1 = proj p
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (self : Pretrivialization F proj)
Goal: self.target = self.baseSet ×ˢ Set.univ
-- Pretrivialization.ext'
@[ext]
lemma ext' (e e' : Pretrivialization F proj) (h₁ : e.toPartialEquiv = e'.toPartialEquiv)
    (h₂ : e.baseSet = e'.baseSet) : e = e' 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e e' : Pretrivialization F proj) (h₁ : e.toPartialEquiv = e'.toPartialEquiv) (h₂ : e.baseSet = e'.baseSet)
Goal: e = e'
-- Pretrivialization.ext
lemma ext {e e' : Pretrivialization F proj} (h₁ : ∀ x, e x = e' x)
    (h₂ : ∀ x, e.toPartialEquiv.symm x = e'.toPartialEquiv.symm x) (h₃ : e.baseSet = e'.baseSet) :
    e = e' 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} {e e' : Pretrivialization F proj} (h₁ : ∀ (x : Z), ↑e x = ↑e' x) (h₂ : ∀ (x : B × F), ↑e.symm x = ↑e'.symm x) (h₃ : e.baseSet = e'.baseSet)
Goal: e = e'
-- Pretrivialization.toPartialEquiv_injective
lemma toPartialEquiv_injective [Nonempty F] :
    Injective (toPartialEquiv : Pretrivialization F proj → PartialEquiv Z (B × F)) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} [Nonempty F]
Goal: Function.Injective Pretrivialization.toPartialEquiv
-- Pretrivialization.coe_coe
@[simp, mfld_simps]
theorem coe_coe : ⇑e.toPartialEquiv = e 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj)
Goal: ↑e.toPartialEquiv = ↑e
-- Pretrivialization.coe_fst
@[simp, mfld_simps]
theorem coe_fst (ex : x ∈ e.source) : (e x).1 = proj x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: (↑e x).1 = proj x
-- Pretrivialization.mem_source
theorem mem_source : x ∈ e.source ↔ proj x ∈ e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z}
Goal: x ∈ e.source ↔ proj x ∈ e.baseSet
-- Pretrivialization.coe_fst'
theorem coe_fst' (ex : proj x ∈ e.baseSet) : (e x).1 = proj x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (ex : proj x ∈ e.baseSet)
Goal: (↑e x).1 = proj x
-- Pretrivialization.eqOn
protected theorem eqOn : EqOn (Prod.fst ∘ e) proj e.source 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj)
Goal: Set.EqOn (Prod.fst ∘ ↑e) proj e.source
-- Pretrivialization.mk_proj_snd
theorem mk_proj_snd (ex : x ∈ e.source) : (proj x, (e x).2) = e x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: (proj x, (↑e x).2) = ↑e x
-- Pretrivialization.mk_proj_snd'
theorem mk_proj_snd' (ex : proj x ∈ e.baseSet) : (proj x, (e x).2) = e x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (ex : proj x ∈ e.baseSet)
Goal: (proj x, (↑e x).2) = ↑e x
-- Pretrivialization.mem_target
theorem mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : B × F}
Goal: x ∈ e.target ↔ x.1 ∈ e.baseSet
-- Pretrivialization.proj_symm_apply
theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.toPartialEquiv.symm x) = x.1 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : B × F} (hx : x ∈ e.target)
Goal: proj (↑e.symm x) = x.1
-- Pretrivialization.proj_symm_apply'
theorem proj_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
    proj (e.toPartialEquiv.symm (b, x)) = b 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {b : B} {x : F} (hx : b ∈ e.baseSet)
Goal: proj (↑e.symm (b, x)) = b
-- Pretrivialization.proj_surjOn_baseSet
theorem proj_surjOn_baseSet [Nonempty F] : Set.SurjOn proj e.source e.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) [Nonempty F]
Goal: Set.SurjOn proj e.source e.baseSet
-- Pretrivialization.apply_symm_apply
theorem apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.toPartialEquiv.symm x) = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : B × F} (hx : x ∈ e.target)
Goal: ↑e (↑e.symm x) = x
-- Pretrivialization.apply_symm_apply'
theorem apply_symm_apply' {b : B} {x : F} (hx : b ∈ e.baseSet) :
    e (e.toPartialEquiv.symm (b, x)) = (b, x) 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {b : B} {x : F} (hx : b ∈ e.baseSet)
Goal: ↑e (↑e.symm (b, x)) = (b, x)
-- Pretrivialization.symm_apply_apply
theorem symm_apply_apply {x : Z} (hx : x ∈ e.source) : e.toPartialEquiv.symm (e x) = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (hx : x ∈ e.source)
Goal: ↑e.symm (↑e x) = x
-- Pretrivialization.symm_apply_mk_proj
@[simp, mfld_simps]
theorem symm_apply_mk_proj {x : Z} (ex : x ∈ e.source) :
    e.toPartialEquiv.symm (proj x, (e x).2) = x 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) {x : Z} (ex : x ∈ e.source)
Goal: ↑e.symm (proj x, (↑e x).2) = x
-- Pretrivialization.preimage_symm_proj_baseSet
@[simp, mfld_simps]
theorem preimage_symm_proj_baseSet :
    e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj)
Goal: ↑e.symm ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target
-- Pretrivialization.preimage_symm_proj_inter
@[simp, mfld_simps]
theorem preimage_symm_proj_inter (s : Set B) :
    e.toPartialEquiv.symm ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ univ = (s ∩ e.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e : Pretrivialization F proj) (s : Set B)
Goal: ↑e.symm ⁻¹' (proj ⁻¹' s) ∩ e.baseSet ×ˢ Set.univ = (s ∩ e.baseSet) ×ˢ Set.univ
-- Pretrivialization.target_inter_preimage_symm_source_eq
theorem target_inter_preimage_symm_source_eq (e f : Pretrivialization F proj) :
    f.target ∩ f.toPartialEquiv.symm ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e f : Pretrivialization F proj)
Goal: f.target ∩ ↑f.symm ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ Set.univ
-- Pretrivialization.trans_source
theorem trans_source (e f : Pretrivialization F proj) :
    (f.toPartialEquiv.symm.trans e.toPartialEquiv).source = (e.baseSet ∩ f.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e f : Pretrivialization F proj)
Goal: (f.symm.trans e.toPartialEquiv).source = (e.baseSet ∩ f.baseSet) ×ˢ Set.univ
-- Pretrivialization.symm_trans_symm
theorem symm_trans_symm (e e' : Pretrivialization F proj) :
    (e.toPartialEquiv.symm.trans e'.toPartialEquiv).symm
      = e'.toPartialEquiv.symm.trans e.toPartialEquiv 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e e' : Pretrivialization F proj)
Goal: (e.symm.trans e'.toPartialEquiv).symm = e'.symm.trans e.toPartialEquiv
-- Pretrivialization.symm_trans_source_eq
theorem symm_trans_source_eq (e e' : Pretrivialization F proj) :
    (e.toPartialEquiv.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e e' : Pretrivialization F proj)
Goal: (e.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ Set.univ
-- Pretrivialization.symm_trans_target_eq
theorem symm_trans_target_eq (e e' : Pretrivialization F proj) :
    (e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ 
Arguments: {B : Type u_2} {F : Type u_3} {Z : Type u_5} [TopologicalSpace B] [TopologicalSpace F] {proj : Z → B} (e e' : Pretrivialization F proj)
Goal: (e.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ Set.univ
-- Pretrivialization.coe_mem_source
@[simp]
theorem coe_mem_source : ↑y ∈ e'.source ↔ b ∈ e'.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] (e' : Pretrivialization F Bundle.TotalSpace.proj) {b : B} {y : E b}
Goal: { proj := b, snd := y } ∈ e'.source ↔ b ∈ e'.baseSet
-- Pretrivialization.coe_coe_fst
@[simp, mfld_simps]
theorem coe_coe_fst (hb : b ∈ e'.baseSet) : (e' y).1 = b 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] (e' : Pretrivialization F Bundle.TotalSpace.proj) {b : B} {y : E b} (hb : b ∈ e'.baseSet)
Goal: (↑e' { proj := b, snd := y }).1 = b
-- Pretrivialization.mk_mem_target
theorem mk_mem_target {x : B} {y : F} : (x, y) ∈ e'.target ↔ x ∈ e'.baseSet 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] (e' : Pretrivialization F Bundle.TotalSpace.proj) {x : B} {y : F}
Goal: (x, y) ∈ e'.target ↔ x ∈ e'.baseSet
-- Pretrivialization.symm_coe_proj
theorem symm_coe_proj {x : B} {y : F} (e' : Pretrivialization F (π F E)) (h : x ∈ e'.baseSet) :
    (e'.toPartialEquiv.symm (x, y)).1 = x 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] {x : B} {y : F} (e' : Pretrivialization F Bundle.TotalSpace.proj) (h : x ∈ e'.baseSet)
Goal: (↑e'.symm (x, y)).proj = x
-- Pretrivialization.symm_apply
theorem symm_apply (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.toPartialEquiv.symm (b, y)).2 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: e.symm b y = cast ⋯ (↑e.symm (b, y)).snd
-- Pretrivialization.symm_apply_of_not_mem
theorem symm_apply_of_not_mem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet)
    (y : F) : e.symm b y = 0 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∉ e.baseSet) (y : F)
Goal: e.symm b y = 0
-- Pretrivialization.coe_symm_of_not_mem
theorem coe_symm_of_not_mem (e : Pretrivialization F (π F E)) {b : B} (hb : b ∉ e.baseSet) :
    (e.symm b : F → E b) = 0 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∉ e.baseSet)
Goal: e.symm b = 0
-- Pretrivialization.mk_symm
theorem mk_symm (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet) (y : F) :
    TotalSpace.mk b (e.symm b y) = e.toPartialEquiv.symm (b, y) 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : F)
Goal: { proj := b, snd := e.symm b y } = ↑e.symm (b, y)
-- Pretrivialization.symm_proj_apply
theorem symm_proj_apply (e : Pretrivialization F (π F E)) (z : TotalSpace F E)
    (hz : z.proj ∈ e.baseSet) : e.symm z.proj (e z).2 = z.2 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) (z : Bundle.TotalSpace F E) (hz : z.proj ∈ e.baseSet)
Goal: e.symm z.proj (↑e z).2 = z.snd
-- Pretrivialization.symm_apply_apply_mk
theorem symm_apply_apply_mk (e : Pretrivialization F (π F E)) {b : B} (hb : b ∈ e.baseSet)
    (y : E b) : e.symm b (e ⟨b, y⟩).2 = y 
Arguments: {B : Type u_2} {F : Type u_3} {E : B → Type u_4} [TopologicalSpace B] [TopologicalSpace F] [(x : B) → Zero (E x)] (e : Pretrivialization F Bundle.TotalSpace.proj) {b : B} (hb : b ∈ e.baseSet) (y : E b)
Goal: e.symm b (↑e { proj := b, snd := y }).2 = y