ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.l2_opNNNorm_def
lemma l2_opNNNorm_def (A : Matrix m n 𝕜) :
    ‖A‖₊ = ‖(toEuclideanLin (𝕜 := 𝕜) (m := m) (n := n)).trans toContinuousLinearMap A‖₊ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜)
Goal: ‖A‖₊ = ‖(Matrix.toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) A‖₊
-- Matrix.l2_opNorm_conjTranspose
lemma l2_opNorm_conjTranspose [DecidableEq m] (A : Matrix m n 𝕜) : ‖Aᴴ‖ = ‖A‖ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] [DecidableEq m] (A : Matrix m n 𝕜)
Goal: ‖Aᴴ‖ = ‖A‖
-- Matrix.l2_opNNNorm_conjTranspose
lemma l2_opNNNorm_conjTranspose [DecidableEq m] (A : Matrix m n 𝕜) : ‖Aᴴ‖₊ = ‖A‖₊ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] [DecidableEq m] (A : Matrix m n 𝕜)
Goal: ‖Aᴴ‖₊ = ‖A‖₊
-- Matrix.l2_opNorm_conjTranspose_mul_self
lemma l2_opNorm_conjTranspose_mul_self (A : Matrix m n 𝕜) : ‖Aᴴ * A‖ = ‖A‖ * ‖A‖ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜)
Goal: ‖Aᴴ * A‖ = ‖A‖ * ‖A‖
-- Matrix.l2_opNNNorm_conjTranspose_mul_self
lemma l2_opNNNorm_conjTranspose_mul_self (A : Matrix m n 𝕜) : ‖Aᴴ * A‖₊ = ‖A‖₊ * ‖A‖₊ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜)
Goal: ‖Aᴴ * A‖₊ = ‖A‖₊ * ‖A‖₊
-- Matrix.l2_opNorm_mulVec
lemma l2_opNorm_mulVec (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n) :
    ‖(EuclideanSpace.equiv m 𝕜).symm <| A *ᵥ x‖ ≤ ‖A‖ * ‖x‖ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n)
Goal: ‖(EuclideanSpace.equiv m 𝕜).symm (A *ᵥ x)‖ ≤ ‖A‖ * ‖x‖
-- Matrix.l2_opNNNorm_mulVec
lemma l2_opNNNorm_mulVec (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n) :
    ‖(EuclideanSpace.equiv m 𝕜).symm <| A *ᵥ x‖₊ ≤ ‖A‖₊ * ‖x‖₊ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m n 𝕜) (x : EuclideanSpace 𝕜 n)
Goal: ‖(EuclideanSpace.equiv m 𝕜).symm (A *ᵥ x)‖₊ ≤ ‖A‖₊ * ‖x‖₊
-- Matrix.l2_opNorm_mul
lemma l2_opNorm_mul (A : Matrix m n 𝕜) (B : Matrix n l 𝕜) :
    ‖A * B‖ ≤ ‖A‖ * ‖B‖ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} {l : Type u_4} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] [Fintype l] [DecidableEq l] (A : Matrix m n 𝕜) (B : Matrix n l 𝕜)
Goal: ‖A * B‖ ≤ ‖A‖ * ‖B‖
-- Matrix.l2_opNNNorm_mul
lemma l2_opNNNorm_mul (A : Matrix m n 𝕜) (B : Matrix n l 𝕜) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊ 
Arguments: {𝕜 : Type u_1} {m : Type u_2} {n : Type u_3} {l : Type u_4} [RCLike 𝕜] [Fintype m] [Fintype n] [DecidableEq n] [Fintype l] [DecidableEq l] (A : Matrix m n 𝕜) (B : Matrix n l 𝕜)
Goal: ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊
-- Matrix.cstar_norm_def
lemma cstar_norm_def (A : Matrix n n 𝕜) : ‖A‖ = ‖toEuclideanCLM (n := n) (𝕜 := 𝕜) A‖ 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] (A : Matrix n n 𝕜)
Goal: ‖A‖ = ‖Matrix.toEuclideanCLM A‖
-- Matrix.cstar_nnnorm_def
lemma cstar_nnnorm_def (A : Matrix n n 𝕜) : ‖A‖₊ = ‖toEuclideanCLM (n := n) (𝕜 := 𝕜) A‖₊ 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n] (A : Matrix n n 𝕜)
Goal: ‖A‖₊ = ‖Matrix.toEuclideanCLM A‖₊
-- Matrix.instCstarRing
lemma instCstarRing : CstarRing (Matrix n n 𝕜) 
Arguments: {𝕜 : Type u_1} {n : Type u_3} [RCLike 𝕜] [Fintype n] [DecidableEq n]
Goal: CstarRing (Matrix n n 𝕜)
-- gaugeRescale_def
theorem gaugeRescale_def (s t : Set E) (x : E) :
    gaugeRescale s t x = (gauge s x / gauge t x) • x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] (s t : Set E) (x : E)
Goal: gaugeRescale s t x = (gauge s x / gauge t x) • x
-- gaugeRescale_zero
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] (s t : Set E)
Goal: gaugeRescale s t 0 = 0
-- gaugeRescale_smul
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
    gaugeRescale s t (c • x) = c • gaugeRescale s t x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E)
Goal: gaugeRescale s t (c • x) = c • gaugeRescale s t x
-- gaugeRescale_self_apply
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
    (x : E) : gaugeRescale s s x = x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] {s : Set E} (hsa : Absorbent ℝ s) (hsb : Bornology.IsVonNBounded ℝ s) (x : E)
Goal: gaugeRescale s s x = x
-- gaugeRescale_self
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
    gaugeRescale s s = id 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] {s : Set E} (hsa : Absorbent ℝ s) (hsb : Bornology.IsVonNBounded ℝ s)
Goal: gaugeRescale s s = id
-- gauge_gaugeRescale'
theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
    gauge t (gaugeRescale s t x) = gauge s x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0)
Goal: gauge t (gaugeRescale s t x) = gauge s x
-- gauge_gaugeRescale
theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
    (x : E) : gauge t (gaugeRescale s t x) = gauge s x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : Bornology.IsVonNBounded ℝ t) (x : E)
Goal: gauge t (gaugeRescale s t x) = gauge s x
-- gauge_gaugeRescale_le
theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
    gauge t (gaugeRescale s t x) ≤ gauge s x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] (s t : Set E) (x : E)
Goal: gauge t (gaugeRescale s t x) ≤ gauge s x
-- gaugeRescale_gaugeRescale
theorem gaugeRescale_gaugeRescale {s t u : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
    (x : E) : gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] {s t u : Set E} (hta : Absorbent ℝ t) (htb : Bornology.IsVonNBounded ℝ t) (x : E)
Goal: gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x
-- mapsTo_gaugeRescale_interior
theorem mapsTo_gaugeRescale_interior (h₀ : t ∈ 𝓝 0) (hc : Convex ℝ t) :
    MapsTo (gaugeRescale s t) (interior s) (interior t) 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (h₀ : t ∈ 𝓝 0) (hc : Convex ℝ t)
Goal: Set.MapsTo (gaugeRescale s t) (interior s) (interior t)
-- mapsTo_gaugeRescale_closure
theorem mapsTo_gaugeRescale_closure {s t : Set E} (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
    (htc : Convex ℝ t) (ht₀ : 0 ∈ t) (hta : Absorbent ℝ t) :
    MapsTo (gaugeRescale s t) (closure s) (closure t) 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (htc : Convex ℝ t) (ht₀ : 0 ∈ t) (hta : Absorbent ℝ t)
Goal: Set.MapsTo (gaugeRescale s t) (closure s) (closure t)
-- continuous_gaugeRescale
theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
    (ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
    Continuous (gaugeRescale s t) 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : Bornology.IsVonNBounded ℝ t)
Goal: Continuous (gaugeRescale s t)
-- image_gaugeRescaleHomeomorph_interior
theorem image_gaugeRescaleHomeomorph_interior {s t : Set E}
    (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (hsb : IsVonNBounded ℝ s)
    (htc : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
    gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb '' interior s = interior t 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (hsb : Bornology.IsVonNBounded ℝ s) (htc : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : Bornology.IsVonNBounded ℝ t)
Goal: ⇑(gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb) '' interior s = interior t
-- image_gaugeRescaleHomeomorph_closure
theorem image_gaugeRescaleHomeomorph_closure {s t : Set E}
    (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (hsb : IsVonNBounded ℝ s)
    (htc : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
    gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb '' closure s = closure t 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (hsb : Bornology.IsVonNBounded ℝ s) (htc : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : Bornology.IsVonNBounded ℝ t)
Goal: ⇑(gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb) '' closure s = closure t
-- exists_homeomorph_image_eq
theorem exists_homeomorph_image_eq {s t : Set E}
    (hsc : Convex ℝ s) (hsne : (interior s).Nonempty) (hsb : IsVonNBounded ℝ s)
    (hst : Convex ℝ t) (htne : (interior t).Nonempty) (htb : IsVonNBounded ℝ t) :
    ∃ e : E ≃ₜ E, e '' interior s = interior t ∧ e '' closure s = closure t ∧
      e '' frontier s = frontier t 
Arguments: {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T1Space E] [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E} (hsc : Convex ℝ s) (hsne : (interior s).Nonempty) (hsb : Bornology.IsVonNBounded ℝ s) (hst : Convex ℝ t) (htne : (interior t).Nonempty) (htb : Bornology.IsVonNBounded ℝ t)
Goal: ∃ e, ⇑e '' interior s = interior t ∧ ⇑e '' closure s = closure t ∧ ⇑e '' frontier s = frontier t
-- exists_homeomorph_image_interior_closure_frontier_eq_unitBall
theorem exists_homeomorph_image_interior_closure_frontier_eq_unitBall {s : Set E}
    (hc : Convex ℝ s) (hne : (interior s).Nonempty) (hb : IsBounded s) :
    ∃ h : E ≃ₜ E, h '' interior s = ball 0 1 ∧ h '' closure s = closedBall 0 1 ∧
      h '' frontier s = sphere 0 1 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hc : Convex ℝ s) (hne : (interior s).Nonempty) (hb : Bornology.IsBounded s)
Goal: ∃ h, ⇑h '' interior s = Metric.ball 0 1 ∧ ⇑h '' closure s = Metric.closedBall 0 1 ∧ ⇑h '' frontier s = Metric.sphere 0 1
-- PiLp.inner_apply
@[simp]
theorem PiLp.inner_apply {ι : Type*} [Fintype ι] {f : ι → Type*} [∀ i, NormedAddCommGroup (f i)]
    [∀ i, InnerProductSpace 𝕜 (f i)] (x y : PiLp 2 f) : ⟪x, y⟫ = ∑ i, ⟪x i, y i⟫ 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {ι : Type u_8} [Fintype ι] {f : ι → Type u_9} [(i : ι) → NormedAddCommGroup (f i)] [(i : ι) → InnerProductSpace 𝕜 (f i)] (x y : PiLp 2 f)
Goal: ⟪x, y⟫_𝕜 = ∑ i : ι, ⟪x i, y i⟫_𝕜
-- EuclideanSpace.nnnorm_eq
theorem EuclideanSpace.nnnorm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x : EuclideanSpace 𝕜 n) : ‖x‖₊ = NNReal.sqrt (∑ i, ‖x i‖₊ ^ 2) 
Arguments: {𝕜 : Type u_8} [RCLike 𝕜] {n : Type u_9} [Fintype n] (x : EuclideanSpace 𝕜 n)
Goal: ‖x‖₊ = NNReal.sqrt (∑ i : n, ‖x i‖₊ ^ 2)
-- EuclideanSpace.norm_eq
theorem EuclideanSpace.norm_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x : EuclideanSpace 𝕜 n) : ‖x‖ = √(∑ i, ‖x i‖ ^ 2) 
Arguments: {𝕜 : Type u_8} [RCLike 𝕜] {n : Type u_9} [Fintype n] (x : EuclideanSpace 𝕜 n)
Goal: ‖x‖ = √(∑ i : n, ‖x i‖ ^ 2)
-- EuclideanSpace.dist_eq
theorem EuclideanSpace.dist_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x y : EuclideanSpace 𝕜 n) : dist x y = √(∑ i, dist (x i) (y i) ^ 2) 
Arguments: {𝕜 : Type u_8} [RCLike 𝕜] {n : Type u_9} [Fintype n] (x y : EuclideanSpace 𝕜 n)
Goal: dist x y = √(∑ i : n, dist (x i) (y i) ^ 2)
-- EuclideanSpace.nndist_eq
theorem EuclideanSpace.nndist_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x y : EuclideanSpace 𝕜 n) : nndist x y = NNReal.sqrt (∑ i, nndist (x i) (y i) ^ 2) 
Arguments: {𝕜 : Type u_8} [RCLike 𝕜] {n : Type u_9} [Fintype n] (x y : EuclideanSpace 𝕜 n)
Goal: nndist x y = NNReal.sqrt (∑ i : n, nndist (x i) (y i) ^ 2)
-- EuclideanSpace.edist_eq
theorem EuclideanSpace.edist_eq {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
    (x y : EuclideanSpace 𝕜 n) : edist x y = (∑ i, edist (x i) (y i) ^ 2) ^ (1 / 2 : ℝ) 
Arguments: {𝕜 : Type u_8} [RCLike 𝕜] {n : Type u_9} [Fintype n] (x y : EuclideanSpace 𝕜 n)
Goal: edist x y = (∑ i : n, edist (x i) (y i) ^ 2) ^ (1 / 2)
-- EuclideanSpace.ball_zero_eq
theorem EuclideanSpace.ball_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
    Metric.ball (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 < r ^ 2} 
Arguments: {n : Type u_8} [Fintype n] (r : ℝ) (hr : 0 ≤ r)
Goal: Metric.ball 0 r = {x | ∑ i : n, x i ^ 2 < r ^ 2}
-- EuclideanSpace.closedBall_zero_eq
theorem EuclideanSpace.closedBall_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
    Metric.closedBall (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 ≤ r ^ 2} 
Arguments: {n : Type u_8} [Fintype n] (r : ℝ) (hr : 0 ≤ r)
Goal: Metric.closedBall 0 r = {x | ∑ i : n, x i ^ 2 ≤ r ^ 2}
-- EuclideanSpace.sphere_zero_eq
theorem EuclideanSpace.sphere_zero_eq {n : Type*} [Fintype n] (r : ℝ) (hr : 0 ≤ r) :
    Metric.sphere (0 : EuclideanSpace ℝ n) r = {x | ∑ i, x i ^ 2 = r ^ 2} 
Arguments: {n : Type u_8} [Fintype n] (r : ℝ) (hr : 0 ≤ r)
Goal: Metric.sphere 0 r = {x | ∑ i : n, x i ^ 2 = r ^ 2}
-- finrank_euclideanSpace
@[simp]
theorem finrank_euclideanSpace :
    FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [Fintype ι]
Goal: FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 ι) = Fintype.card ι
-- finrank_euclideanSpace_fin
theorem finrank_euclideanSpace_fin {n : ℕ} :
    FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 (Fin n)) = n 
Arguments: {𝕜 : Type u_3} [RCLike 𝕜] {n : ℕ}
Goal: FiniteDimensional.finrank 𝕜 (EuclideanSpace 𝕜 (Fin n)) = n
-- EuclideanSpace.inner_eq_star_dotProduct
theorem EuclideanSpace.inner_eq_star_dotProduct (x y : EuclideanSpace 𝕜 ι) :
    ⟪x, y⟫ = Matrix.dotProduct (star <| WithLp.equiv _ _ x) (WithLp.equiv _ _ y) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [Fintype ι] (x y : EuclideanSpace 𝕜 ι)
Goal: ⟪x, y⟫_𝕜 = Matrix.dotProduct (star ((WithLp.equiv 2 (ι → 𝕜)) x)) ((WithLp.equiv 2 (ι → 𝕜)) y)
-- EuclideanSpace.inner_piLp_equiv_symm
theorem EuclideanSpace.inner_piLp_equiv_symm (x y : ι → 𝕜) :
    ⟪(WithLp.equiv 2 _).symm x, (WithLp.equiv 2 _).symm y⟫ = Matrix.dotProduct (star x) y 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [Fintype ι] (x y : ι → 𝕜)
Goal: ⟪(WithLp.equiv 2 (ι → 𝕜)).symm x, (WithLp.equiv 2 (ι → 𝕜)).symm y⟫_𝕜 = Matrix.dotProduct (star x) y
-- DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply
@[simp]
theorem DirectSum.IsInternal.isometryL2OfOrthogonalFamily_symm_apply [DecidableEq ι]
    {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V)
    (hV' : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) (w : PiLp 2 fun i => V i) :
    (hV.isometryL2OfOrthogonalFamily hV').symm w = ∑ i, (w i : E) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] {V : ι → Submodule 𝕜 E} (hV : DirectSum.IsInternal V) (hV' : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ) (w : PiLp 2 fun i => ↥(V i))
Goal: (hV.isometryL2OfOrthogonalFamily hV').symm w = ∑ i : ι, ↑(w i)
-- EuclideanSpace.projₗ_apply
@[simps!]
def EuclideanSpace.projₗ (i : ι) : EuclideanSpace 𝕜 ι →ₗ[𝕜] 𝕜 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] (i : ι)
Goal: ∀ (a : WithLp 2 (ι → 𝕜)), (EuclideanSpace.projₗ i) a = a i
-- EuclideanSpace.proj_coe
@[simps! apply coe]
def EuclideanSpace.proj (i : ι) : EuclideanSpace 𝕜 ι →L[𝕜] 𝕜 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] (i : ι)
Goal: ↑(EuclideanSpace.proj i) = EuclideanSpace.projₗ i
-- EuclideanSpace.proj_apply
@[simps! apply coe]
def EuclideanSpace.proj (i : ι) : EuclideanSpace 𝕜 ι →L[𝕜] 𝕜 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] (i : ι)
Goal: ∀ (a : WithLp 2 (ι → 𝕜)), (EuclideanSpace.proj i) a = a i
-- WithLp.equiv_single
@[simp]
theorem WithLp.equiv_single (i : ι) (a : 𝕜) :
    WithLp.equiv _ _ (EuclideanSpace.single i a) = Pi.single i a 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] (i : ι) (a : 𝕜)
Goal: (WithLp.equiv 2 ((i : ι) → (fun x => 𝕜) i)) (EuclideanSpace.single i a) = Pi.single i a
-- WithLp.equiv_symm_single
@[simp]
theorem WithLp.equiv_symm_single (i : ι) (a : 𝕜) :
    (WithLp.equiv _ _).symm (Pi.single i a) = EuclideanSpace.single i a 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] (i : ι) (a : 𝕜)
Goal: (WithLp.equiv 2 (ι → 𝕜)).symm (Pi.single i a) = EuclideanSpace.single i a
-- EuclideanSpace.single_apply
@[simp]
theorem EuclideanSpace.single_apply (i : ι) (a : 𝕜) (j : ι) :
    (EuclideanSpace.single i a) j = ite (j = i) a 0 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] (i : ι) (a : 𝕜) (j : ι)
Goal: EuclideanSpace.single i a j = if j = i then a else 0
-- EuclideanSpace.inner_single_left
theorem EuclideanSpace.inner_single_left (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) :
    ⟪EuclideanSpace.single i (a : 𝕜), v⟫ = conj a * v i 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι)
Goal: ⟪EuclideanSpace.single i a, v⟫_𝕜 = (starRingEnd 𝕜) a * v i
-- EuclideanSpace.inner_single_right
theorem EuclideanSpace.inner_single_right (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι) :
    ⟪v, EuclideanSpace.single i (a : 𝕜)⟫ = a * conj (v i) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a : 𝕜) (v : EuclideanSpace 𝕜 ι)
Goal: ⟪v, EuclideanSpace.single i a⟫_𝕜 = a * (starRingEnd ((fun x => 𝕜) i)) (v i)
-- EuclideanSpace.norm_single
@[simp]
theorem EuclideanSpace.norm_single (i : ι) (a : 𝕜) :
    ‖EuclideanSpace.single i (a : 𝕜)‖ = ‖a‖ 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a : 𝕜)
Goal: ‖EuclideanSpace.single i a‖ = ‖a‖
-- EuclideanSpace.nnnorm_single
@[simp]
theorem EuclideanSpace.nnnorm_single (i : ι) (a : 𝕜) :
    ‖EuclideanSpace.single i (a : 𝕜)‖₊ = ‖a‖₊ 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a : 𝕜)
Goal: ‖EuclideanSpace.single i a‖₊ = ‖a‖₊
-- EuclideanSpace.dist_single_same
@[simp]
theorem EuclideanSpace.dist_single_same (i : ι) (a b : 𝕜) :
    dist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = dist a b 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a b : 𝕜)
Goal: dist (EuclideanSpace.single i a) (EuclideanSpace.single i b) = dist a b
-- EuclideanSpace.nndist_single_same
@[simp]
theorem EuclideanSpace.nndist_single_same (i : ι) (a b : 𝕜) :
    nndist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = nndist a b 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a b : 𝕜)
Goal: nndist (EuclideanSpace.single i a) (EuclideanSpace.single i b) = nndist a b
-- EuclideanSpace.edist_single_same
@[simp]
theorem EuclideanSpace.edist_single_same (i : ι) (a b : 𝕜) :
    edist (EuclideanSpace.single i (a : 𝕜)) (EuclideanSpace.single i (b : 𝕜)) = edist a b 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] (i : ι) (a b : 𝕜)
Goal: edist (EuclideanSpace.single i a) (EuclideanSpace.single i b) = edist a b
-- EuclideanSpace.orthonormal_single
theorem EuclideanSpace.orthonormal_single :
    Orthonormal 𝕜 fun i : ι => EuclideanSpace.single i (1 : 𝕜) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι]
Goal: Orthonormal 𝕜 fun i => EuclideanSpace.single i 1
-- EuclideanSpace.piLpCongrLeft_single
theorem EuclideanSpace.piLpCongrLeft_single
    {ι' : Type*} [Fintype ι'] [DecidableEq ι'] (e : ι' ≃ ι) (i' : ι') (v : 𝕜) :
    LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e (EuclideanSpace.single i' v) =
      EuclideanSpace.single (e i') v 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] [DecidableEq ι] [Fintype ι] {ι' : Type u_8} [Fintype ι'] [DecidableEq ι'] (e : ι' ≃ ι) (i' : ι') (v : 𝕜)
Goal: (LinearIsometryEquiv.piLpCongrLeft 2 𝕜 𝕜 e) (EuclideanSpace.single i' v) = EuclideanSpace.single (e i') v
-- OrthonormalBasis.repr_injective
theorem repr_injective :
    Injective (repr : OrthonormalBasis ι 𝕜 E → E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι]
Goal: Function.Injective OrthonormalBasis.repr
-- OrthonormalBasis.coe_ofRepr
@[simp]
theorem coe_ofRepr [DecidableEq ι] (e : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι) :
    ⇑(OrthonormalBasis.ofRepr e) = fun i => e.symm (EuclideanSpace.single i (1 : 𝕜)) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] (e : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι)
Goal: ⇑{ repr := e } = fun i => e.symm (EuclideanSpace.single i 1)
-- OrthonormalBasis.repr_symm_single
@[simp]
protected theorem repr_symm_single [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι) :
    b.repr.symm (EuclideanSpace.single i (1 : 𝕜)) = b i 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι)
Goal: b.repr.symm (EuclideanSpace.single i 1) = b i
-- OrthonormalBasis.repr_self
@[simp]
protected theorem repr_self [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι) :
    b.repr (b i) = EuclideanSpace.single i (1 : 𝕜) 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] [DecidableEq ι] (b : OrthonormalBasis ι 𝕜 E) (i : ι)
Goal: b.repr (b i) = EuclideanSpace.single i 1
-- OrthonormalBasis.repr_apply_apply
protected theorem repr_apply_apply (b : OrthonormalBasis ι 𝕜 E) (v : E) (i : ι) :
    b.repr v i = ⟪b i, v⟫ 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (v : E) (i : ι)
Goal: b.repr v i = ⟪b i, v⟫_𝕜
-- OrthonormalBasis.orthonormal
@[simp]
protected theorem orthonormal (b : OrthonormalBasis ι 𝕜 E) : Orthonormal 𝕜 b 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E)
Goal: Orthonormal 𝕜 ⇑b
-- OrthonormalBasis.coe_toBasis
@[simp]
protected theorem coe_toBasis (b : OrthonormalBasis ι 𝕜 E) : (⇑b.toBasis : ι → E) = ⇑b 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E)
Goal: ⇑b.toBasis = ⇑b
-- OrthonormalBasis.coe_toBasis_repr
@[simp]
protected theorem coe_toBasis_repr (b : OrthonormalBasis ι 𝕜 E) :
    b.toBasis.equivFun = b.repr.toLinearEquiv 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E)
Goal: b.toBasis.equivFun = b.repr.toLinearEquiv
-- OrthonormalBasis.coe_toBasis_repr_apply
@[simp]
protected theorem coe_toBasis_repr_apply (b : OrthonormalBasis ι 𝕜 E) (x : E) (i : ι) :
    b.toBasis.repr x i = b.repr x i 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (x : E) (i : ι)
Goal: (b.toBasis.repr x) i = b.repr x i
-- OrthonormalBasis.sum_repr
protected theorem sum_repr (b : OrthonormalBasis ι 𝕜 E) (x : E) : ∑ i, b.repr x i • b i = x 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (x : E)
Goal: ∑ i : ι, b.repr x i • b i = x
-- OrthonormalBasis.sum_repr'
protected theorem sum_repr' (b : OrthonormalBasis ι 𝕜 E) (x : E) : ∑ i, ⟪b i, x⟫_𝕜 • b i = x 
Arguments: {ι : Type u_1} {𝕜 : Type u_3} [RCLike 𝕜] {E : Type u_4} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [Fintype ι] (b : OrthonormalBasis ι 𝕜 E) (x : E)
Goal: ∑ i : ι, ⟪b i, x⟫_𝕜 • b i = x