ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Trivialization.continuousLinearMap_apply
theorem Trivialization.continuousLinearMap_apply
    (p : TotalSpace (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂)) :
    e₁.continuousLinearMap σ e₂ p =
      ⟨p.1, (e₂.continuousLinearMapAt 𝕜₂ p.1 : _ →L[𝕜₂] _).comp
        (p.2.comp (e₁.symmL 𝕜₁ p.1 : F₁ →L[𝕜₁] E₁ p.1) : F₁ →SL[σ] E₂ p.1)⟩ 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] {e₁ : Trivialization F₁ Bundle.TotalSpace.proj} {e₂ : Trivialization F₂ Bundle.TotalSpace.proj} [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] [he₁ : MemTrivializationAtlas e₁] [he₂ : MemTrivializationAtlas e₂] (p : Bundle.TotalSpace (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂))
Goal: ↑(Trivialization.continuousLinearMap σ e₁ e₂) p =
  (p.proj,
    (Trivialization.continuousLinearMapAt 𝕜₂ e₂ p.proj).comp
      (ContinuousLinearMap.comp p.snd (Trivialization.symmL 𝕜₁ e₁ p.proj)))
-- hom_trivializationAt_apply
theorem hom_trivializationAt_apply (x₀ : B)
    (x : TotalSpace (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂)) :
    trivializationAt (F₁ →SL[σ] F₂) (fun x => E₁ x →SL[σ] E₂ x) x₀ x =
      ⟨x.1, inCoordinates F₁ E₁ F₂ E₂ x₀ x.1 x₀ x.1 x.2⟩ 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] (x₀ : B) (x : Bundle.TotalSpace (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂))
Goal: ↑(trivializationAt (F₁ →SL[σ] F₂) (fun x => E₁ x →SL[σ] E₂ x) x₀) x =
  (x.proj, ContinuousLinearMap.inCoordinates F₁ E₁ F₂ E₂ x₀ x.proj x₀ x.proj x.snd)
-- hom_trivializationAt_source
@[simp, mfld_simps]
theorem hom_trivializationAt_source (x₀ : B) :
    (trivializationAt (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) x₀).source =
      π (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) ⁻¹'
        ((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] (x₀ : B)
Goal: (trivializationAt (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) x₀).source =
  Bundle.TotalSpace.proj ⁻¹' ((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet)
-- hom_trivializationAt_target
@[simp, mfld_simps]
theorem hom_trivializationAt_target (x₀ : B) :
    (trivializationAt (F₁ →SL[σ] F₂) (fun x => E₁ x →SL[σ] E₂ x) x₀).target =
      ((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) ×ˢ Set.univ 
Arguments: {𝕜₁ : Type u_1} [NontriviallyNormedField 𝕜₁] {𝕜₂ : Type u_2} [NontriviallyNormedField 𝕜₂] (σ : 𝕜₁ →+* 𝕜₂) [iσ : RingHomIsometric σ] {B : Type u_3} {F₁ : Type u_4} [NormedAddCommGroup F₁] [NormedSpace 𝕜₁ F₁] {E₁ : B → Type u_5} [(x : B) → AddCommGroup (E₁ x)] [(x : B) → Module 𝕜₁ (E₁ x)] [TopologicalSpace (Bundle.TotalSpace F₁ E₁)] {F₂ : Type u_6} [NormedAddCommGroup F₂] [NormedSpace 𝕜₂ F₂] {E₂ : B → Type u_7} [(x : B) → AddCommGroup (E₂ x)] [(x : B) → Module 𝕜₂ (E₂ x)] [TopologicalSpace (Bundle.TotalSpace F₂ E₂)] [TopologicalSpace B] [(x : B) → TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁] [(x : B) → TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂] [∀ (x : B), TopologicalAddGroup (E₂ x)] [∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)] (x₀ : B)
Goal: (trivializationAt (F₁ →SL[σ] F₂) (fun x => E₁ x →SL[σ] E₂ x) x₀).target =
  ((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet) ×ˢ Set.univ
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_self
private theorem glueDist_self (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) : ∀ x, glueDist Φ Ψ ε x x = 0
  
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X ⊕ Y)
Goal: Metric.glueDist Φ Ψ ε x x = 0
-- Metric.glueDist_glued_points
theorem glueDist_glued_points [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z) :
    glueDist Φ Ψ ε (.inl (Φ p)) (.inr (Ψ p)) = ε 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (p : Z)
Goal: Metric.glueDist Φ Ψ ε (Sum.inl (Φ p)) (Sum.inr (Ψ p)) = ε
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_comm
private theorem glueDist_comm (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :
    ∀ x y, glueDist Φ Ψ ε x y = glueDist Φ Ψ ε y x
  
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y : X ⊕ Y)
Goal: Metric.glueDist Φ Ψ ε x y = Metric.glueDist Φ Ψ ε y x
-- Metric.glueDist_swap
theorem glueDist_swap (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) :
    ∀ x y, glueDist Ψ Φ ε x.swap y.swap = glueDist Φ Ψ ε x y
  
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y : X ⊕ Y)
Goal: Metric.glueDist Ψ Φ ε x.swap y.swap = Metric.glueDist Φ Ψ ε x y
-- Metric.le_glueDist_inl_inr
theorem le_glueDist_inl_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
    ε ≤ glueDist Φ Ψ ε (.inl x) (.inr y) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y : Y)
Goal: ε ≤ Metric.glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y)
-- Metric.le_glueDist_inr_inl
theorem le_glueDist_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x y) :
    ε ≤ glueDist Φ Ψ ε (.inr x) (.inl y) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : Y) (y : X)
Goal: ε ≤ Metric.glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y)
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_triangle_inl_inr_inr
private theorem glueDist_triangle_inl_inr_inr (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y z : Y) :
    glueDist Φ Ψ ε (.inl x) (.inr z) ≤
      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inr z) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (x : X) (y z : Y)
Goal: Metric.glueDist Φ Ψ ε (Sum.inl x) (Sum.inr z) ≤
  Metric.glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) + Metric.glueDist Φ Ψ ε (Sum.inr y) (Sum.inr z)
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_triangle_inl_inr_inl
private theorem glueDist_triangle_inl_inr_inl (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)
    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x : X) (y : Y) (z : X) :
    glueDist Φ Ψ ε (.inl x) (.inl z) ≤
      glueDist Φ Ψ ε (.inl x) (.inr y) + glueDist Φ Ψ ε (.inr y) (.inl z) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x : X) (y : Y) (z : X)
Goal: Metric.glueDist Φ Ψ ε (Sum.inl x) (Sum.inl z) ≤
  Metric.glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) + Metric.glueDist Φ Ψ ε (Sum.inr y) (Sum.inl z)
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.glueDist_triangle
private theorem glueDist_triangle (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ)
    (H : ∀ p q, |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) :
    ∀ x y z, glueDist Φ Ψ ε x z ≤ glueDist Φ Ψ ε x y + glueDist Φ Ψ ε y z
  
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] [Nonempty Z] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε) (x y z : X ⊕ Y)
Goal: Metric.glueDist Φ Ψ ε x z ≤ Metric.glueDist Φ Ψ ε x y + Metric.glueDist Φ Ψ ε y z
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.eq_of_glueDist_eq_zero
private theorem eq_of_glueDist_eq_zero (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) :
    ∀ p q : X ⊕ Y, glueDist Φ Ψ ε p q = 0 → p = q
  
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] (Φ : Z → X) (Ψ : Z → Y) (ε : ℝ) (ε0 : 0 < ε) (p q : X ⊕ Y)
Goal: Metric.glueDist Φ Ψ ε p q = 0 → p = q
-- Metric.Sum.mem_uniformity_iff_glueDist
theorem Sum.mem_uniformity_iff_glueDist (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
    s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ a b, glueDist Φ Ψ ε a b < δ → (a, b) ∈ s 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} {ε : ℝ} (hε : 0 < ε) (s : Set ((X ⊕ Y) × (X ⊕ Y)))
Goal: s ∈ 𝓤 (X ⊕ Y) ↔ ∃ δ > 0, ∀ (a b : X ⊕ Y), Metric.glueDist Φ Ψ ε a b < δ → (a, b) ∈ s
-- Metric.Sum.dist_eq_glueDist
theorem Sum.dist_eq_glueDist {p q : X ⊕ Y} (x : X) (y : Y) :
    Sum.dist p q =
      glueDist (fun _ : Unit => Nonempty.some ⟨x⟩) (fun _ : Unit => Nonempty.some ⟨y⟩) 1 p q 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {p q : X ⊕ Y} (x : X) (y : Y)
Goal: Metric.Sum.dist p q = Metric.glueDist (fun x_1 => ⋯.some) (fun x => ⋯.some) 1 p q
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.Sum.dist_comm
private theorem Sum.dist_comm (x y : X ⊕ Y) : Sum.dist x y = Sum.dist y x 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] (x y : X ⊕ Y)
Goal: Metric.Sum.dist x y = Metric.Sum.dist y x
-- Metric.Sum.one_le_dist_inl_inr
theorem Sum.one_le_dist_inl_inr {x : X} {y : Y} : 1 ≤ Sum.dist (.inl x) (.inr y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {x : X} {y : Y}
Goal: 1 ≤ Metric.Sum.dist (Sum.inl x) (Sum.inr y)
-- Metric.Sum.one_le_dist_inr_inl
theorem Sum.one_le_dist_inr_inl {x : X} {y : Y} : 1 ≤ Sum.dist (.inr y) (.inl x) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {x : X} {y : Y}
Goal: 1 ≤ Metric.Sum.dist (Sum.inr y) (Sum.inl x)
-- _private.Mathlib.Topology.MetricSpace.Gluing.0.Metric.Sum.mem_uniformity
private theorem Sum.mem_uniformity (s : Set ((X ⊕ Y) × (X ⊕ Y))) :
    s ∈ 𝓤 (X ⊕ Y) ↔ ∃ ε > 0, ∀ a b, Sum.dist a b < ε → (a, b) ∈ s 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] (s : Set ((X ⊕ Y) × (X ⊕ Y)))
Goal: s ∈ 𝓤 (X ⊕ Y) ↔ ∃ ε > 0, ∀ (a b : X ⊕ Y), Metric.Sum.dist a b < ε → (a, b) ∈ s
-- Metric.Sum.dist_eq
theorem Sum.dist_eq {x y : X ⊕ Y} : dist x y = Sum.dist x y 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y] {x y : X ⊕ Y}
Goal: dist x y = Metric.Sum.dist x y
-- Metric.isometry_inl
theorem isometry_inl : Isometry (Sum.inl : X → X ⊕ Y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y]
Goal: Isometry Sum.inl
-- Metric.isometry_inr
theorem isometry_inr : Isometry (Sum.inr : Y → X ⊕ Y) 
Arguments: {X : Type u} {Y : Type v} [MetricSpace X] [MetricSpace Y]
Goal: Isometry Sum.inr
-- Metric.Sigma.dist_same
@[simp]
theorem dist_same (i : ι) (x y : E i) : dist (Sigma.mk i x) ⟨i, y⟩ = dist x y 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (i : ι) (x y : E i)
Goal: dist ⟨i, x⟩ ⟨i, y⟩ = dist x y
-- Metric.Sigma.dist_ne
@[simp]
theorem dist_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
    dist (⟨i, x⟩ : Σk, E k) ⟨j, y⟩ = dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨y⟩) y 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] {i j : ι} (h : i ≠ j) (x : E i) (y : E j)
Goal: dist ⟨i, x⟩ ⟨j, y⟩ = dist x ⋯.some + 1 + dist ⋯.some y
-- Metric.Sigma.one_le_dist_of_ne
theorem one_le_dist_of_ne {i j : ι} (h : i ≠ j) (x : E i) (y : E j) :
    1 ≤ dist (⟨i, x⟩ : Σk, E k) ⟨j, y⟩ 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] {i j : ι} (h : i ≠ j) (x : E i) (y : E j)
Goal: 1 ≤ dist ⟨i, x⟩ ⟨j, y⟩
-- Metric.Sigma.fst_eq_of_dist_lt_one
theorem fst_eq_of_dist_lt_one (x y : Σi, E i) (h : dist x y < 1) : x.1 = y.1 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (x y : (i : ι) × E i) (h : dist x y < 1)
Goal: x.fst = y.fst
-- Metric.Sigma.dist_triangle
protected theorem dist_triangle (x y z : Σi, E i) : dist x z ≤ dist x y + dist y z 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (x y z : (i : ι) × E i)
Goal: dist x z ≤ dist x y + dist y z
-- Metric.Sigma.isOpen_iff
protected theorem isOpen_iff (s : Set (Σi, E i)) :
    IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (s : Set ((i : ι) × E i))
Goal: IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s
-- Metric.Sigma.isometry_mk
theorem isometry_mk (i : ι) : Isometry (Sigma.mk i : E i → Σk, E k) 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] (i : ι)
Goal: Isometry (Sigma.mk i)
-- Metric.Sigma.completeSpace
protected theorem completeSpace [∀ i, CompleteSpace (E i)] : CompleteSpace (Σi, E i) 
Arguments: {ι : Type u_1} {E : ι → Type u_2} [(i : ι) → MetricSpace (E i)] [∀ (i : ι), CompleteSpace (E i)]
Goal: CompleteSpace ((i : ι) × E i)
-- Metric.toGlue_commute
theorem toGlue_commute (hΦ : Isometry Φ) (hΨ : Isometry Ψ) :
    toGlueL hΦ hΨ ∘ Φ = toGlueR hΦ hΨ ∘ Ψ 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ)
Goal: Metric.toGlueL hΦ hΨ ∘ Φ = Metric.toGlueR hΦ hΨ ∘ Ψ
-- Metric.toGlueL_isometry
theorem toGlueL_isometry (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (toGlueL hΦ hΨ) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ)
Goal: Isometry (Metric.toGlueL hΦ hΨ)
-- Metric.toGlueR_isometry
theorem toGlueR_isometry (hΦ : Isometry Φ) (hΨ : Isometry Ψ) : Isometry (toGlueR hΦ hΨ) 
Arguments: {X : Type u} {Y : Type v} {Z : Type w} [Nonempty Z] [MetricSpace Z] [MetricSpace X] [MetricSpace Y] {Φ : Z → X} {Ψ : Z → Y} (hΦ : Isometry Φ) (hΨ : Isometry Ψ)
Goal: Isometry (Metric.toGlueR hΦ hΨ)
-- Metric.inductiveLimitDist_eq_dist
theorem inductiveLimitDist_eq_dist (I : ∀ n, Isometry (f n)) (x y : Σn, X n) :
    ∀ m (hx : x.1 ≤ m) (hy : y.1 ≤ m), inductiveLimitDist f x y =
      dist (leRecOn hx (f _) x.2 : X m) (leRecOn hy (f _) y.2 : X m)
  
Arguments: {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (x y : (n : ℕ) × X n) (m : ℕ) (hx : x.fst ≤ m) (hy : y.fst ≤ m)
Goal: Metric.inductiveLimitDist f x y = dist (Nat.leRecOn hx (fun {k} => f k) x.snd) (Nat.leRecOn hy (fun {k} => f k) y.snd)
-- Metric.toInductiveLimit_isometry
theorem toInductiveLimit_isometry (I : ∀ n, Isometry (f n)) (n : ℕ) :
    Isometry (toInductiveLimit I n) 
Arguments: {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (n : ℕ)
Goal: Isometry (Metric.toInductiveLimit I n)
-- Metric.toInductiveLimit_commute
theorem toInductiveLimit_commute (I : ∀ n, Isometry (f n)) (n : ℕ) :
    toInductiveLimit I n.succ ∘ f n = toInductiveLimit I n 
Arguments: {X : ℕ → Type u} [(n : ℕ) → MetricSpace (X n)] {f : (n : ℕ) → X n → X (n + 1)} (I : ∀ (n : ℕ), Isometry (f n)) (n : ℕ)
Goal: Metric.toInductiveLimit I n.succ ∘ f n = Metric.toInductiveLimit I n
-- CocompactMap.cocompact_tendsto'
structure CocompactMap (α : Type u) (β : Type v) [TopologicalSpace α] [TopologicalSpace β] extends
  ContinuousMap α β : Type max u v where
    cocompact_tendsto' : Tendsto toFun (cocompact α) (cocompact β)
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] (self : CocompactMap α β)
Goal: Filter.Tendsto self.toFun (Filter.cocompact α) (Filter.cocompact β)
-- CocompactMapClass.cocompact_tendsto
class CocompactMapClass (F : Type*) (α β : outParam Type*) [TopologicalSpace α]
  [TopologicalSpace β] [FunLike F α β] extends ContinuousMapClass F α β : Prop where
    cocompact_tendsto (f : F) : Tendsto f (cocompact α) (cocompact β)
Arguments: {F : Type u_1} {α : outParam (Type u_2)} {β : outParam (Type u_3)} [TopologicalSpace α] [TopologicalSpace β] [FunLike F α β] [self : CocompactMapClass F α β] (f : F)
Goal: Filter.Tendsto (⇑f) (Filter.cocompact α) (Filter.cocompact β)
-- CocompactMap.coe_toContinuousMap
@[simp]
theorem coe_toContinuousMap {f : CocompactMap α β} : (f.toContinuousMap : α → β) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : CocompactMap α β}
Goal: ⇑f.toContinuousMap = ⇑f
-- CocompactMap.ext
@[ext]
theorem ext {f g : CocompactMap α β} (h : ∀ x, f x = g x) : f = g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f g : CocompactMap α β} (h : ∀ (x : α), f x = g x)
Goal: f = g
-- CocompactMap.coe_copy
@[simp]
theorem coe_copy (f : CocompactMap α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : CocompactMap α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- CocompactMap.copy_eq
theorem copy_eq (f : CocompactMap α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : CocompactMap α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- CocompactMap.coe_mk
@[simp]
theorem coe_mk (f : C(α, β)) (h : Tendsto f (cocompact α) (cocompact β)) :
    ⇑(⟨f, h⟩ : CocompactMap α β) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) (h : Filter.Tendsto (⇑f) (Filter.cocompact α) (Filter.cocompact β))
Goal: ⇑{ toContinuousMap := f, cocompact_tendsto' := h } = ⇑f
-- CocompactMap.coe_id
@[simp]
theorem coe_id : ⇑(CocompactMap.id α) = id 
Arguments: (α : Type u_1) [TopologicalSpace α]
Goal: ⇑(CocompactMap.id α) = id
-- CocompactMap.coe_comp
@[simp]
theorem coe_comp (f : CocompactMap β γ) (g : CocompactMap α β) : ⇑(comp f g) = f ∘ g 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : CocompactMap β γ) (g : CocompactMap α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- CocompactMap.comp_apply
@[simp]
theorem comp_apply (f : CocompactMap β γ) (g : CocompactMap α β) (a : α) : comp f g a = f (g a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : CocompactMap β γ) (g : CocompactMap α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- CocompactMap.comp_assoc
@[simp]
theorem comp_assoc (f : CocompactMap γ δ) (g : CocompactMap β γ) (h : CocompactMap α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] [TopologicalSpace δ] (f : CocompactMap γ δ) (g : CocompactMap β γ) (h : CocompactMap α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- CocompactMap.id_comp
@[simp]
theorem id_comp (f : CocompactMap α β) : (CocompactMap.id _).comp f = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : CocompactMap α β)
Goal: (CocompactMap.id β).comp f = f
-- CocompactMap.comp_id
@[simp]
theorem comp_id (f : CocompactMap α β) : f.comp (CocompactMap.id _) = f 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : CocompactMap α β)
Goal: f.comp (CocompactMap.id α) = f
-- CocompactMap.tendsto_of_forall_preimage
theorem tendsto_of_forall_preimage {f : α → β} (h : ∀ s, IsCompact s → IsCompact (f ⁻¹' s)) :
    Tendsto f (cocompact α) (cocompact β) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (h : ∀ (s : Set β), IsCompact s → IsCompact (f ⁻¹' s))
Goal: Filter.Tendsto f (Filter.cocompact α) (Filter.cocompact β)
-- CocompactMap.isCompact_preimage_of_isClosed
theorem isCompact_preimage_of_isClosed (f : CocompactMap α β)
    ⦃s : Set β⦄ (hs : IsCompact s) (h's : IsClosed s) :
    IsCompact (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s) (h's : IsClosed s)
Goal: IsCompact (⇑f ⁻¹' s)
-- CocompactMap.isCompact_preimage
theorem isCompact_preimage [T2Space β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s) :
    IsCompact (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [T2Space β] (f : CocompactMap α β) ⦃s : Set β⦄ (hs : IsCompact s)
Goal: IsCompact (⇑f ⁻¹' s)
-- Homeomorph.toCocompactMap_toFun
@[simps]
def Homeomorph.toCocompactMap {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    (f : α ≃ₜ β) : CocompactMap α β 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (a : α)
Goal: f.toCocompactMap a = f a
-- ContinuousMap.le_def
theorem le_def [PartialOrder β] {f g : C(α, β)} : f ≤ g ↔ ∀ a, f a ≤ g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [PartialOrder β] {f g : C(α, β)}
Goal: f ≤ g ↔ ∀ (a : α), f a ≤ g a
-- ContinuousMap.lt_def
theorem lt_def [PartialOrder β] {f g : C(α, β)} : f < g ↔ (∀ a, f a ≤ g a) ∧ ∃ a, f a < g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [PartialOrder β] {f g : C(α, β)}
Goal: f < g ↔ (∀ (a : α), f a ≤ g a) ∧ ∃ a, f a < g a
-- ContinuousMap.coe_sup
@[simp, norm_cast] lemma coe_sup (f g : C(α, β)) : ⇑(f ⊔ g) = ⇑f ⊔ g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g : C(α, β))
Goal: ⇑(f ⊔ g) = ⇑f ⊔ ⇑g
-- ContinuousMap.sup_apply
@[simp] lemma sup_apply (f g : C(α, β)) (a : α) : (f ⊔ g) a = f a ⊔ g a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] (f g : C(α, β)) (a : α)
Goal: (f ⊔ g) a = f a ⊔ g a
-- ContinuousMap.sup'_apply
lemma sup'_apply {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) (a : α) :
    s.sup' H f a = s.sup' H fun i ↦ f i a 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] {ι : Type u_4} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) (a : α)
Goal: (s.sup' H f) a = s.sup' H fun i => (f i) a
-- ContinuousMap.coe_sup'
@[simp, norm_cast]
lemma coe_sup' {ι : Type*} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β)) :
    ⇑(s.sup' H f) = s.sup' H fun i ↦ ⇑(f i) 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SemilatticeSup β] [ContinuousSup β] {ι : Type u_4} {s : Finset ι} (H : s.Nonempty) (f : ι → C(α, β))
Goal: ⇑(s.sup' H f) = s.sup' H fun i => ⇑(f i)
-- ContinuousMap.coe_inf
@[simp, norm_cast] lemma coe_inf (f g : C(α, β)) : ⇑(f ⊓ g) = ⇑f ⊓ g 
Arguments: {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] [SemilatticeInf β] [ContinuousInf β] (f g : C(α, β))
Goal: ⇑(f ⊓ g) = ⇑f ⊓ ⇑g