ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- PiLp.nnnorm_equiv
@[simp] theorem nnnorm_equiv (f : PiLp ∞ β) : ‖WithLp.equiv ⊤ _ f‖₊ = ‖f‖₊ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (f : PiLp ⊤ β)
Goal: ‖(WithLp.equiv ⊤ ((i : ι) → β i)) f‖₊ = ‖f‖₊
-- PiLp.nnnorm_equiv_symm
@[simp] theorem nnnorm_equiv_symm (f : ∀ i, β i) : ‖(WithLp.equiv ⊤ _).symm f‖₊ = ‖f‖₊ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (f : (i : ι) → β i)
Goal: ‖(WithLp.equiv ⊤ ((i : ι) → β i)).symm f‖₊ = ‖f‖₊
-- PiLp.norm_equiv
@[simp] theorem norm_equiv (f : PiLp ∞ β) : ‖WithLp.equiv ⊤ _ f‖ = ‖f‖ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (f : PiLp ⊤ β)
Goal: ‖(WithLp.equiv ⊤ ((i : ι) → β i)) f‖ = ‖f‖
-- PiLp.norm_equiv_symm
@[simp] theorem norm_equiv_symm (f : ∀ i, β i) : ‖(WithLp.equiv ⊤ _).symm f‖ = ‖f‖ 
Arguments: {ι : Type u_2} {β : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] (f : (i : ι) → β i)
Goal: ‖(WithLp.equiv ⊤ ((i : ι) → β i)).symm f‖ = ‖f‖
-- PiLp.norm_eq_of_nat
theorem norm_eq_of_nat {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type*}
    [∀ i, SeminormedAddCommGroup (β i)] (n : ℕ) (h : p = n) (f : PiLp p β) :
    ‖f‖ = (∑ i, ‖f i‖ ^ n) ^ (1 / (n : ℝ)) 
Arguments: {ι : Type u_2} [Fintype ι] {p : ℝ≥0∞} [Fact (1 ≤ p)] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (n : ℕ) (h : p = ↑n) (f : PiLp p β)
Goal: ‖f‖ = (∑ i : ι, ‖f i‖ ^ n) ^ (1 / ↑n)
-- PiLp.norm_eq_of_L2
theorem norm_eq_of_L2 {β : ι → Type*} [∀ i, SeminormedAddCommGroup (β i)] (x : PiLp 2 β) :
    ‖x‖ = √(∑ i : ι, ‖x i‖ ^ 2) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (x : PiLp 2 β)
Goal: ‖x‖ = √(∑ i : ι, ‖x i‖ ^ 2)
-- PiLp.nnnorm_eq_of_L2
theorem nnnorm_eq_of_L2 {β : ι → Type*} [∀ i, SeminormedAddCommGroup (β i)] (x : PiLp 2 β) :
    ‖x‖₊ = NNReal.sqrt (∑ i : ι, ‖x i‖₊ ^ 2) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (x : PiLp 2 β)
Goal: ‖x‖₊ = NNReal.sqrt (∑ i : ι, ‖x i‖₊ ^ 2)
-- PiLp.norm_sq_eq_of_L2
theorem norm_sq_eq_of_L2 (β : ι → Type*) [∀ i, SeminormedAddCommGroup (β i)] (x : PiLp 2 β) :
    ‖x‖ ^ 2 = ∑ i : ι, ‖x i‖ ^ 2 
Arguments: {ι : Type u_2} [Fintype ι] (β : ι → Type u_5) [(i : ι) → SeminormedAddCommGroup (β i)] (x : PiLp 2 β)
Goal: ‖x‖ ^ 2 = ∑ i : ι, ‖x i‖ ^ 2
-- PiLp.dist_eq_of_L2
theorem dist_eq_of_L2 {β : ι → Type*} [∀ i, SeminormedAddCommGroup (β i)] (x y : PiLp 2 β) :
    dist x y = √(∑ i, dist (x i) (y i) ^ 2) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp 2 β)
Goal: dist x y = √(∑ i : ι, dist (x i) (y i) ^ 2)
-- PiLp.nndist_eq_of_L2
theorem nndist_eq_of_L2 {β : ι → Type*} [∀ i, SeminormedAddCommGroup (β i)] (x y : PiLp 2 β) :
    nndist x y = NNReal.sqrt (∑ i, nndist (x i) (y i) ^ 2) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp 2 β)
Goal: nndist x y = NNReal.sqrt (∑ i : ι, nndist (x i) (y i) ^ 2)
-- PiLp.edist_eq_of_L2
theorem edist_eq_of_L2 {β : ι → Type*} [∀ i, SeminormedAddCommGroup (β i)] (x y : PiLp 2 β) :
    edist x y = (∑ i, edist (x i) (y i) ^ 2) ^ (1 / 2 : ℝ) 
Arguments: {ι : Type u_2} [Fintype ι] {β : ι → Type u_5} [(i : ι) → SeminormedAddCommGroup (β i)] (x y : PiLp 2 β)
Goal: edist x y = (∑ i : ι, edist (x i) (y i) ^ 2) ^ (1 / 2)
-- LinearIsometryEquiv.piLpCongrLeft_apply
@[simp]
theorem _root_.LinearIsometryEquiv.piLpCongrLeft_apply (e : ι ≃ ι') (v : PiLp p fun _ : ι => E) :
    LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e v = Equiv.piCongrLeft' (fun _ : ι => E) e v 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] {ι' : Type u_5} [Fintype ι'] {E : Type u_6} [SeminormedAddCommGroup E] [Module 𝕜 E] (e : ι ≃ ι') (v : PiLp p fun x => E)
Goal: (LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e) v = (Equiv.piCongrLeft' (fun x => E) e) v
-- LinearIsometryEquiv.piLpCongrLeft_symm
@[simp]
theorem _root_.LinearIsometryEquiv.piLpCongrLeft_symm (e : ι ≃ ι') :
    (LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e).symm =
      LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e.symm 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] {ι' : Type u_5} [Fintype ι'] {E : Type u_6} [SeminormedAddCommGroup E] [Module 𝕜 E] (e : ι ≃ ι')
Goal: (LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e).symm = LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e.symm
-- LinearIsometryEquiv.piLpCongrLeft_single
@[simp high]
theorem _root_.LinearIsometryEquiv.piLpCongrLeft_single [DecidableEq ι] [DecidableEq ι']
    (e : ι ≃ ι') (i : ι) (v : E) :
    LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e ((WithLp.equiv p (_ → E)).symm <| Pi.single i v) =
      (WithLp.equiv p (_ → E)).symm (Pi.single (e i) v) 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] {ι' : Type u_5} [Fintype ι'] {E : Type u_6} [SeminormedAddCommGroup E] [Module 𝕜 E] [DecidableEq ι] [DecidableEq ι'] (e : ι ≃ ι') (i : ι) (v : E)
Goal: (LinearIsometryEquiv.piLpCongrLeft p 𝕜 E e) ((WithLp.equiv p (ι → E)).symm (Pi.single i v)) =
  (WithLp.equiv p (ι' → E)).symm (Pi.single (e i) v)
-- LinearIsometryEquiv.piLpCongrRight_apply
@[simp]
theorem _root_.LinearIsometryEquiv.piLpCongrRight_apply (e : ∀ i, α i ≃ₗᵢ[𝕜] β i) (x : PiLp p α) :
    LinearIsometryEquiv.piLpCongrRight p e x =
      (WithLp.equiv p _).symm (fun i => e i (x i)) 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} {α : ι → Type u_3} {β : ι → Type u_4} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (α i)] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (α i)] [(i : ι) → Module 𝕜 (β i)] (e : (i : ι) → α i ≃ₗᵢ[𝕜] β i) (x : PiLp p α)
Goal: (LinearIsometryEquiv.piLpCongrRight p e) x = (WithLp.equiv p ((i : ι) → β i)).symm fun i => (e i) (x i)
-- LinearIsometryEquiv.piLpCongrRight_refl
@[simp]
theorem _root_.LinearIsometryEquiv.piLpCongrRight_refl :
    LinearIsometryEquiv.piLpCongrRight p (fun i => .refl 𝕜 (α i)) = .refl _ _ 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} {α : ι → Type u_3} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (α i)] [(i : ι) → Module 𝕜 (α i)]
Goal: (LinearIsometryEquiv.piLpCongrRight p fun i => LinearIsometryEquiv.refl 𝕜 (α i)) = LinearIsometryEquiv.refl 𝕜 (PiLp p α)
-- LinearIsometryEquiv.piLpCongrRight_symm
@[simp]
theorem _root_.LinearIsometryEquiv.piLpCongrRight_symm (e : ∀ i, α i ≃ₗᵢ[𝕜] β i) :
    (LinearIsometryEquiv.piLpCongrRight p e).symm =
      LinearIsometryEquiv.piLpCongrRight p (fun i => (e i).symm) 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} {α : ι → Type u_3} {β : ι → Type u_4} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (α i)] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (α i)] [(i : ι) → Module 𝕜 (β i)] (e : (i : ι) → α i ≃ₗᵢ[𝕜] β i)
Goal: (LinearIsometryEquiv.piLpCongrRight p e).symm = LinearIsometryEquiv.piLpCongrRight p fun i => (e i).symm
-- LinearIsometryEquiv.piLpCongrRight_single
@[simp high]
theorem _root_.LinearIsometryEquiv.piLpCongrRight_single (e : ∀ i, α i ≃ₗᵢ[𝕜] β i) [DecidableEq ι]
    (i : ι) (v : α i) :
    LinearIsometryEquiv.piLpCongrRight p e ((WithLp.equiv p (∀ i, α i)).symm <| Pi.single i v) =
      (WithLp.equiv p (∀ i, β i)).symm (Pi.single i (e _ v)) 
Arguments: {p : ℝ≥0∞} {𝕜 : Type u_1} {ι : Type u_2} {α : ι → Type u_3} {β : ι → Type u_4} [Fact (1 ≤ p)] [Fintype ι] [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (α i)] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (α i)] [(i : ι) → Module 𝕜 (β i)] (e : (i : ι) → α i ≃ₗᵢ[𝕜] β i) [DecidableEq ι] (i : ι) (v : α i)
Goal: (LinearIsometryEquiv.piLpCongrRight p e) ((WithLp.equiv p ((i : ι) → α i)).symm (Pi.single i v)) =
  (WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i ((e i) v))
-- LinearIsometryEquiv.piLpCurry_apply
@[simp] theorem _root_.LinearIsometryEquiv.piLpCurry_apply
    (f : PiLp p (fun i : Sigma κ => α i.1 i.2)) :
    _root_.LinearIsometryEquiv.piLpCurry 𝕜 p α f =
      (WithLp.equiv _ _).symm (fun i => (WithLp.equiv _ _).symm <|
        Sigma.curry (WithLp.equiv _ _ f) i) 
Arguments: {𝕜 : Type u_1} [Semiring 𝕜] {ι : Type u_5} {κ : ι → Type u_6} (p : ℝ≥0∞) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → Fintype (κ i)] (α : (i : ι) → κ i → Type u_7) [(i : ι) → (k : κ i) → SeminormedAddCommGroup (α i k)] [(i : ι) → (k : κ i) → Module 𝕜 (α i k)] (f : PiLp p fun i => α i.fst i.snd)
Goal: (LinearIsometryEquiv.piLpCurry 𝕜 p α) f =
  (WithLp.equiv p ((i : ι) → WithLp p ((y : κ i) → α i y))).symm fun i =>
    (WithLp.equiv p ((y : κ i) → α i y)).symm (Sigma.curry ((WithLp.equiv p ((x : Sigma κ) → α x.fst x.snd)) f) i)
-- LinearIsometryEquiv.piLpCurry_symm_apply
@[simp] theorem _root_.LinearIsometryEquiv.piLpCurry_symm_apply
    (f : PiLp p (fun i => PiLp p (α i))) :
    (_root_.LinearIsometryEquiv.piLpCurry 𝕜 p α).symm f =
      (WithLp.equiv _ _).symm (Sigma.uncurry fun i j => f i j) 
Arguments: {𝕜 : Type u_1} [Semiring 𝕜] {ι : Type u_5} {κ : ι → Type u_6} (p : ℝ≥0∞) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → Fintype (κ i)] (α : (i : ι) → κ i → Type u_7) [(i : ι) → (k : κ i) → SeminormedAddCommGroup (α i k)] [(i : ι) → (k : κ i) → Module 𝕜 (α i k)] (f : PiLp p fun i => PiLp p (α i))
Goal: (LinearIsometryEquiv.piLpCurry 𝕜 p α).symm f =
  (WithLp.equiv p ((x : Sigma κ) → α x.fst x.snd)).symm (Sigma.uncurry fun i j => f i j)
-- PiLp.nnnorm_equiv_symm_single
@[simp]
theorem nnnorm_equiv_symm_single [hp : Fact (1 ≤ p)] (i : ι) (b : β i) :
    ‖(WithLp.equiv p (∀ i, β i)).symm (Pi.single i b)‖₊ = ‖b‖₊ 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] [hp : Fact (1 ≤ p)] (i : ι) (b : β i)
Goal: ‖(WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b)‖₊ = ‖b‖₊
-- PiLp.norm_equiv_symm_single
@[simp]
theorem norm_equiv_symm_single (i : ι) (b : β i) :
    ‖(WithLp.equiv p (∀ i, β i)).symm (Pi.single i b)‖ = ‖b‖ 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b : β i)
Goal: ‖(WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b)‖ = ‖b‖
-- PiLp.nndist_equiv_symm_single_same
@[simp]
theorem nndist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
    nndist
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₁))
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₂)) =
      nndist b₁ b₂ 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b₁ b₂ : β i)
Goal: nndist ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₁))
    ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₂)) =
  nndist b₁ b₂
-- PiLp.dist_equiv_symm_single_same
@[simp]
theorem dist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
    dist
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₁))
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₂)) =
      dist b₁ b₂ 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b₁ b₂ : β i)
Goal: dist ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₁)) ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₂)) =
  dist b₁ b₂
-- PiLp.edist_equiv_symm_single_same
@[simp]
theorem edist_equiv_symm_single_same (i : ι) (b₁ b₂ : β i) :
    edist
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₁))
        ((WithLp.equiv p (∀ i, β i)).symm (Pi.single i b₂)) =
      edist b₁ b₂ 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} (β : ι → Type u_4) [Fact (1 ≤ p)] [Fintype ι] [(i : ι) → SeminormedAddCommGroup (β i)] [DecidableEq ι] (i : ι) (b₁ b₂ : β i)
Goal: edist ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₁))
    ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₂)) =
  edist b₁ b₂
-- PiLp.nnnorm_equiv_symm_const
theorem nnnorm_equiv_symm_const {β} [SeminormedAddCommGroup β] (hp : p ≠ ∞) (b : β) :
    ‖(WithLp.equiv p (ι → β)).symm (Function.const _ b)‖₊ =
      (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖b‖₊ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] (hp : p ≠ ⊤) (b : β)
Goal: ‖(WithLp.equiv p (ι → β)).symm (Function.const ι b)‖₊ = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊
-- PiLp.nnnorm_equiv_symm_const'
theorem nnnorm_equiv_symm_const' {β} [SeminormedAddCommGroup β] [Nonempty ι] (b : β) :
    ‖(WithLp.equiv p (ι → β)).symm (Function.const _ b)‖₊ =
      (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖b‖₊ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] [Nonempty ι] (b : β)
Goal: ‖(WithLp.equiv p (ι → β)).symm (Function.const ι b)‖₊ = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊
-- PiLp.norm_equiv_symm_const
theorem norm_equiv_symm_const {β} [SeminormedAddCommGroup β] (hp : p ≠ ∞) (b : β) :
    ‖(WithLp.equiv p (ι → β)).symm (Function.const _ b)‖ =
      (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖b‖ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] (hp : p ≠ ⊤) (b : β)
Goal: ‖(WithLp.equiv p (ι → β)).symm (Function.const ι b)‖ = ↑↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖
-- PiLp.norm_equiv_symm_const'
theorem norm_equiv_symm_const' {β} [SeminormedAddCommGroup β] [Nonempty ι] (b : β) :
    ‖(WithLp.equiv p (ι → β)).symm (Function.const _ b)‖ =
      (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖b‖ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] [Nonempty ι] (b : β)
Goal: ‖(WithLp.equiv p (ι → β)).symm (Function.const ι b)‖ = ↑↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖
-- PiLp.nnnorm_equiv_symm_one
theorem nnnorm_equiv_symm_one {β} [SeminormedAddCommGroup β] (hp : p ≠ ∞) [One β] :
    ‖(WithLp.equiv p (ι → β)).symm 1‖₊ =
      (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖(1 : β)‖₊ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] (hp : p ≠ ⊤) [One β]
Goal: ‖(WithLp.equiv p (ι → β)).symm 1‖₊ = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖1‖₊
-- PiLp.norm_equiv_symm_one
theorem norm_equiv_symm_one {β} [SeminormedAddCommGroup β] (hp : p ≠ ∞) [One β] :
    ‖(WithLp.equiv p (ι → β)).symm 1‖ = (Fintype.card ι : ℝ≥0) ^ (1 / p).toReal * ‖(1 : β)‖ 
Arguments: {p : ℝ≥0∞} {ι : Type u_2} [Fact (1 ≤ p)] [Fintype ι] {β : Type u_5} [SeminormedAddCommGroup β] (hp : p ≠ ⊤) [One β]
Goal: ‖(WithLp.equiv p (ι → β)).symm 1‖ = ↑↑(Fintype.card ι) ^ (1 / p).toReal * ‖1‖
-- PiLp.continuousLinearEquiv_symm_apply
@[simps! (config := .asFn) apply symm_apply]
protected def continuousLinearEquiv : PiLp p β ≃L[𝕜] ∀ i, β i 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) {ι : Type u_2} (β : ι → Type u_4) [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (β i)]
Goal: ⇑(PiLp.continuousLinearEquiv p 𝕜 β).symm = ⇑(WithLp.equiv p ((i : ι) → β i)).symm
-- PiLp.continuousLinearEquiv_apply
@[simps! (config := .asFn) apply symm_apply]
protected def continuousLinearEquiv : PiLp p β ≃L[𝕜] ∀ i, β i 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) {ι : Type u_2} (β : ι → Type u_4) [Semiring 𝕜] [(i : ι) → SeminormedAddCommGroup (β i)] [(i : ι) → Module 𝕜 (β i)]
Goal: ⇑(PiLp.continuousLinearEquiv p 𝕜 β) = ⇑(WithLp.equiv p ((i : ι) → β i))
-- PiLp.basisFun_apply
@[simp]
theorem basisFun_apply [DecidableEq ι] (i) :
    basisFun p 𝕜 ι i = (WithLp.equiv p _).symm (Pi.single i 1) 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (ι : Type u_2) [Finite ι] [Ring 𝕜] [DecidableEq ι] (i : ι)
Goal: (PiLp.basisFun p 𝕜 ι) i = (WithLp.equiv p (ι → 𝕜)).symm (Pi.single i 1)
-- PiLp.basisFun_repr
@[simp]
theorem basisFun_repr (x : PiLp p fun _ : ι => 𝕜) (i : ι) : (basisFun p 𝕜 ι).repr x i = x i 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (ι : Type u_2) [Finite ι] [Ring 𝕜] (x : PiLp p fun x => 𝕜) (i : ι)
Goal: ((PiLp.basisFun p 𝕜 ι).repr x) i = x i
-- PiLp.basisFun_equivFun
@[simp]
theorem basisFun_equivFun : (basisFun p 𝕜 ι).equivFun = WithLp.linearEquiv p 𝕜 (ι → 𝕜) 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (ι : Type u_2) [Finite ι] [Ring 𝕜]
Goal: (PiLp.basisFun p 𝕜 ι).equivFun = WithLp.linearEquiv p 𝕜 (ι → 𝕜)
-- PiLp.basisFun_eq_pi_basisFun
theorem basisFun_eq_pi_basisFun :
    basisFun p 𝕜 ι = (Pi.basisFun 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (ι : Type u_2) [Finite ι] [Ring 𝕜]
Goal: PiLp.basisFun p 𝕜 ι = (Pi.basisFun 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm
-- PiLp.basisFun_map
@[simp]
theorem basisFun_map :
    (basisFun p 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜)) = Pi.basisFun 𝕜 ι 
Arguments: (p : ℝ≥0∞) (𝕜 : Type u_1) (ι : Type u_2) [Finite ι] [Ring 𝕜]
Goal: (PiLp.basisFun p 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜)) = Pi.basisFun 𝕜 ι
-- PiLp.basis_toMatrix_basisFun_mul
nonrec theorem basis_toMatrix_basisFun_mul [Fintype ι]
    {𝕜} [SeminormedCommRing 𝕜] (b : Basis ι 𝕜 (PiLp p fun _ : ι => 𝕜))
    (A : Matrix ι ι 𝕜) :
    b.toMatrix (PiLp.basisFun _ _ _) * A =
      Matrix.of fun i j => b.repr ((WithLp.equiv _ _).symm (Aᵀ j)) i 
Arguments: (p : ℝ≥0∞) {ι : Type u_2} [Fintype ι] {𝕜 : Type u_5} [SeminormedCommRing 𝕜] (b : Basis ι 𝕜 (PiLp p fun x => 𝕜)) (A : Matrix ι ι 𝕜)
Goal: b.toMatrix ⇑(PiLp.basisFun p 𝕜 ι) * A =
  Matrix.of fun i j => (b.repr ((WithLp.equiv p ((i : ι) → (fun x => 𝕜) i)).symm (A.transpose j))) i
-- AddCircle.norm_coe_mul
@[simp]
theorem norm_coe_mul (x : ℝ) (t : ℝ) :
    ‖(↑(t * x) : AddCircle (t * p))‖ = |t| * ‖(x : AddCircle p)‖ 
Arguments: (p x t : ℝ)
Goal: ‖↑(t * x)‖ = |t| * ‖↑x‖
-- AddCircle.norm_neg_period
theorem norm_neg_period (x : ℝ) : ‖(x : AddCircle (-p))‖ = ‖(x : AddCircle p)‖ 
Arguments: (p x : ℝ)
Goal: ‖↑x‖ = ‖↑x‖
-- AddCircle.norm_eq_of_zero
@[simp]
theorem norm_eq_of_zero {x : ℝ} : ‖(x : AddCircle (0 : ℝ))‖ = |x| 
Arguments: {x : ℝ}
Goal: ‖↑x‖ = |x|
-- AddCircle.norm_eq
theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| 
Arguments: (p : ℝ) {x : ℝ}
Goal: ‖↑x‖ = |x - ↑(round (p⁻¹ * x)) * p|
-- AddCircle.norm_eq'
theorem norm_eq' (hp : 0 < p) {x : ℝ} : ‖(x : AddCircle p)‖ = p * |p⁻¹ * x - round (p⁻¹ * x)| 
Arguments: (p : ℝ) (hp : 0 < p) {x : ℝ}
Goal: ‖↑x‖ = p * |p⁻¹ * x - ↑(round (p⁻¹ * x))|
-- AddCircle.norm_le_half_period
theorem norm_le_half_period {x : AddCircle p} (hp : p ≠ 0) : ‖x‖ ≤ |p| / 2 
Arguments: (p : ℝ) {x : AddCircle p} (hp : p ≠ 0)
Goal: ‖x‖ ≤ |p| / 2
-- AddCircle.norm_half_period_eq
@[simp]
theorem norm_half_period_eq : ‖(↑(p / 2) : AddCircle p)‖ = |p| / 2 
Arguments: (p : ℝ)
Goal: ‖↑(p / 2)‖ = |p| / 2
-- AddCircle.norm_coe_eq_abs_iff
theorem norm_coe_eq_abs_iff {x : ℝ} (hp : p ≠ 0) : ‖(x : AddCircle p)‖ = |x| ↔ |x| ≤ |p| / 2 
Arguments: (p : ℝ) {x : ℝ} (hp : p ≠ 0)
Goal: ‖↑x‖ = |x| ↔ |x| ≤ |p| / 2
-- AddCircle.closedBall_eq_univ_of_half_period_le
theorem closedBall_eq_univ_of_half_period_le (hp : p ≠ 0) (x : AddCircle p) {ε : ℝ}
    (hε : |p| / 2 ≤ ε) : closedBall x ε = univ 
Arguments: (p : ℝ) (hp : p ≠ 0) (x : AddCircle p) {ε : ℝ} (hε : |p| / 2 ≤ ε)
Goal: Metric.closedBall x ε = Set.univ
-- AddCircle.coe_real_preimage_closedBall_period_zero
@[simp]
theorem coe_real_preimage_closedBall_period_zero (x ε : ℝ) :
    (↑) ⁻¹' closedBall (x : AddCircle (0 : ℝ)) ε = closedBall x ε 
Arguments: (x ε : ℝ)
Goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε = Metric.closedBall x ε
-- AddCircle.coe_real_preimage_closedBall_eq_iUnion
theorem coe_real_preimage_closedBall_eq_iUnion (x ε : ℝ) :
    (↑) ⁻¹' closedBall (x : AddCircle p) ε = ⋃ z : ℤ, closedBall (x + z • p) ε 
Arguments: (p x ε : ℝ)
Goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε = ⋃ z, Metric.closedBall (x + z • p) ε
-- AddCircle.coe_real_preimage_closedBall_inter_eq
theorem coe_real_preimage_closedBall_inter_eq {x ε : ℝ} (s : Set ℝ)
    (hs : s ⊆ closedBall x (|p| / 2)) :
    (↑) ⁻¹' closedBall (x : AddCircle p) ε ∩ s = if ε < |p| / 2 then closedBall x ε ∩ s else s 
Arguments: (p : ℝ) {x ε : ℝ} (s : Set ℝ) (hs : s ⊆ Metric.closedBall x (|p| / 2))
Goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε ∩ s = if ε < |p| / 2 then Metric.closedBall x ε ∩ s else s
-- AddCircle.norm_div_natCast
theorem norm_div_natCast {m n : ℕ} :
    ‖(↑(↑m / ↑n * p) : AddCircle p)‖ = p * (↑(min (m % n) (n - m % n)) / n) 
Arguments: {p : ℝ} [hp : Fact (0 < p)] {m n : ℕ}
Goal: ‖↑(↑m / ↑n * p)‖ = p * (↑(min (m % n) (n - m % n)) / ↑n)
-- AddCircle.exists_norm_eq_of_isOfFinAddOrder
theorem exists_norm_eq_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u) :
    ∃ k : ℕ, ‖u‖ = p * (k / addOrderOf u) 
Arguments: {p : ℝ} [hp : Fact (0 < p)] {u : AddCircle p} (hu : IsOfFinAddOrder u)
Goal: ∃ k, ‖u‖ = p * (↑k / ↑(addOrderOf u))
-- AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder
theorem le_add_order_smul_norm_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u)
    (hu' : u ≠ 0) : p ≤ addOrderOf u • ‖u‖ 
Arguments: {p : ℝ} [hp : Fact (0 < p)] {u : AddCircle p} (hu : IsOfFinAddOrder u) (hu' : u ≠ 0)
Goal: p ≤ addOrderOf u • ‖u‖
-- UnitAddCircle.norm_eq
theorem norm_eq {x : ℝ} : ‖(x : UnitAddCircle)‖ = |x - round x| 
Arguments: {x : ℝ}
Goal: ‖↑x‖ = |x - ↑(round x)|
-- ZeroAtInftyContinuousMapClass.norm_le
theorem ZeroAtInftyContinuousMapClass.norm_le (f : 𝓕) (ε : ℝ) (hε : 0 < ε) :
    ∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε 
Arguments: {E : Type u_1} {F : Type u_2} {𝓕 : Type u_3} [SeminormedAddGroup E] [SeminormedAddCommGroup F] [FunLike 𝓕 E F] [ZeroAtInftyContinuousMapClass 𝓕 E F] (f : 𝓕) (ε : ℝ) (hε : 0 < ε)
Goal: ∃ r, ∀ (x : E), r < ‖x‖ → ‖f x‖ < ε
-- zero_at_infty_of_norm_le
theorem zero_at_infty_of_norm_le (f : E → F)
    (h : ∀ (ε : ℝ) (_hε : 0 < ε), ∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε) :
    Tendsto f (cocompact E) (𝓝 0) 
Arguments: {E : Type u_1} {F : Type u_2} [SeminormedAddGroup E] [SeminormedAddCommGroup F] [ProperSpace E] (f : E → F) (h : ∀ (ε : ℝ), 0 < ε → ∃ r, ∀ (x : E), r < ‖x‖ → ‖f x‖ < ε)
Goal: Filter.Tendsto f (Filter.cocompact E) (𝓝 0)
-- AddGroupSeminorm.map_zero'
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
  -- Porting note: can't extend `ZeroHom G ℝ` because otherwise `to_additive` won't work since
  -- we aren't using old structures
    protected toFun : G → ℝ
    protected map_zero' : toFun 0 = 0
    protected add_le' : ∀ r s, toFun (r + s) ≤ toFun r + toFun s
    protected neg' : ∀ r, toFun (-r) = toFun r
Arguments: {G : Type u_7} [AddGroup G] (self : AddGroupSeminorm G)
Goal: self.toFun 0 = 0
-- AddGroupSeminorm.neg'
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
  -- Porting note: can't extend `ZeroHom G ℝ` because otherwise `to_additive` won't work since
  -- we aren't using old structures
    protected toFun : G → ℝ
    protected map_zero' : toFun 0 = 0
    protected add_le' : ∀ r s, toFun (r + s) ≤ toFun r + toFun s
    protected neg' : ∀ r, toFun (-r) = toFun r
Arguments: {G : Type u_7} [AddGroup G] (self : AddGroupSeminorm G) (r : G)
Goal: self.toFun (-r) = self.toFun r
-- AddGroupSeminorm.add_le'
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
  -- Porting note: can't extend `ZeroHom G ℝ` because otherwise `to_additive` won't work since
  -- we aren't using old structures
    protected toFun : G → ℝ
    protected map_zero' : toFun 0 = 0
    protected add_le' : ∀ r s, toFun (r + s) ≤ toFun r + toFun s
    protected neg' : ∀ r, toFun (-r) = toFun r
Arguments: {G : Type u_7} [AddGroup G] (self : AddGroupSeminorm G) (r s : G)
Goal: self.toFun (r + s) ≤ self.toFun r + self.toFun s
-- GroupSeminorm.inv'
@[to_additive]
structure GroupSeminorm (G : Type*) [Group G] where
    protected toFun : G → ℝ
    protected map_one' : toFun 1 = 0
    protected mul_le' : ∀ x y, toFun (x * y) ≤ toFun x + toFun y
    protected inv' : ∀ x, toFun x⁻¹ = toFun x
Arguments: {G : Type u_7} [Group G] (self : GroupSeminorm G) (x : G)
Goal: self.toFun x⁻¹ = self.toFun x
-- GroupSeminorm.mul_le'
@[to_additive]
structure GroupSeminorm (G : Type*) [Group G] where
    protected toFun : G → ℝ
    protected map_one' : toFun 1 = 0
    protected mul_le' : ∀ x y, toFun (x * y) ≤ toFun x + toFun y
    protected inv' : ∀ x, toFun x⁻¹ = toFun x
Arguments: {G : Type u_7} [Group G] (self : GroupSeminorm G) (x y : G)
Goal: self.toFun (x * y) ≤ self.toFun x + self.toFun y