ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.norm_conjTranspose
@[simp]
theorem norm_conjTranspose [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) : ‖Aᴴ‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α)
Goal: ‖Aᴴ‖ = ‖A‖
-- Matrix.nnnorm_col
@[simp]
theorem nnnorm_col (v : m → α) : ‖col ι v‖₊ = ‖v‖₊ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.col ι v‖₊ = ‖v‖₊
-- Matrix.norm_col
@[simp]
theorem norm_col (v : m → α) : ‖col ι v‖ = ‖v‖ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.col ι v‖ = ‖v‖
-- Matrix.nnnorm_row
@[simp]
theorem nnnorm_row (v : n → α) : ‖row ι v‖₊ = ‖v‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.row ι v‖₊ = ‖v‖₊
-- Matrix.norm_row
@[simp]
theorem norm_row (v : n → α) : ‖row ι v‖ = ‖v‖ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.row ι v‖ = ‖v‖
-- Matrix.nnnorm_diagonal
@[simp]
theorem nnnorm_diagonal [DecidableEq n] (v : n → α) : ‖diagonal v‖₊ = ‖v‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α)
Goal: ‖Matrix.diagonal v‖₊ = ‖v‖₊
-- Matrix.norm_diagonal
@[simp]
theorem norm_diagonal [DecidableEq n] (v : n → α) : ‖diagonal v‖ = ‖v‖ 
Arguments: {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α)
Goal: ‖Matrix.diagonal v‖ = ‖v‖
-- Matrix.boundedSMul
protected theorem boundedSMul [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α]
    [BoundedSMul R α] : BoundedSMul R (Matrix m n α) 
Arguments: {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α]
Goal: BoundedSMul R (Matrix m n α)
-- Matrix.linftyOpBoundedSMul
@[local instance]
protected theorem linftyOpBoundedSMul
    [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α] :
    BoundedSMul R (Matrix m n α) 
Arguments: {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α]
Goal: BoundedSMul R (Matrix m n α)
-- Matrix.linfty_opNorm_def
theorem linfty_opNorm_def (A : Matrix m n α) :
    ‖A‖ = ((Finset.univ : Finset m).sup fun i : m => ∑ j : n, ‖A i j‖₊ : ℝ≥0) 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖ = ↑(Finset.univ.sup fun i => ∑ j : n, ‖A i j‖₊)
-- Matrix.linfty_opNNNorm_def
theorem linfty_opNNNorm_def (A : Matrix m n α) :
    ‖A‖₊ = (Finset.univ : Finset m).sup fun i : m => ∑ j : n, ‖A i j‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖₊ = Finset.univ.sup fun i => ∑ j : n, ‖A i j‖₊
-- Matrix.linfty_opNNNorm_col
@[simp]
theorem linfty_opNNNorm_col (v : m → α) : ‖col ι v‖₊ = ‖v‖₊ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.col ι v‖₊ = ‖v‖₊
-- Matrix.linfty_opNorm_col
@[simp]
theorem linfty_opNorm_col (v : m → α) : ‖col ι v‖ = ‖v‖ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.col ι v‖ = ‖v‖
-- Matrix.linfty_opNNNorm_row
@[simp]
theorem linfty_opNNNorm_row (v : n → α) : ‖row ι v‖₊ = ∑ i, ‖v i‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.row ι v‖₊ = ∑ i : n, ‖v i‖₊
-- Matrix.linfty_opNorm_row
@[simp]
theorem linfty_opNorm_row (v : n → α) : ‖row ι v‖ = ∑ i, ‖v i‖ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.row ι v‖ = ∑ i : n, ‖v i‖
-- Matrix.linfty_opNNNorm_diagonal
@[simp]
theorem linfty_opNNNorm_diagonal [DecidableEq m] (v : m → α) : ‖diagonal v‖₊ = ‖v‖₊ 
Arguments: {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [DecidableEq m] (v : m → α)
Goal: ‖Matrix.diagonal v‖₊ = ‖v‖₊
-- Matrix.linfty_opNorm_diagonal
@[simp]
theorem linfty_opNorm_diagonal [DecidableEq m] (v : m → α) : ‖diagonal v‖ = ‖v‖ 
Arguments: {m : Type u_3} {α : Type u_5} [Fintype m] [SeminormedAddCommGroup α] [DecidableEq m] (v : m → α)
Goal: ‖Matrix.diagonal v‖ = ‖v‖
-- Matrix.linfty_opNNNorm_mul
theorem linfty_opNNNorm_mul (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊ 
Arguments: {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [NonUnitalSeminormedRing α] (A : Matrix l m α) (B : Matrix m n α)
Goal: ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊
-- Matrix.linfty_opNorm_mul
theorem linfty_opNorm_mul (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖ ≤ ‖A‖ * ‖B‖ 
Arguments: {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [NonUnitalSeminormedRing α] (A : Matrix l m α) (B : Matrix m n α)
Goal: ‖A * B‖ ≤ ‖A‖ * ‖B‖
-- Matrix.linfty_opNNNorm_mulVec
theorem linfty_opNNNorm_mulVec (A : Matrix l m α) (v : m → α) : ‖A *ᵥ v‖₊ ≤ ‖A‖₊ * ‖v‖₊ 
Arguments: {l : Type u_2} {m : Type u_3} {α : Type u_5} [Fintype l] [Fintype m] [NonUnitalSeminormedRing α] (A : Matrix l m α) (v : m → α)
Goal: ‖A *ᵥ v‖₊ ≤ ‖A‖₊ * ‖v‖₊
-- Matrix.linfty_opNorm_mulVec
theorem linfty_opNorm_mulVec (A : Matrix l m α) (v : m → α) : ‖A *ᵥ v‖ ≤ ‖A‖ * ‖v‖ 
Arguments: {l : Type u_2} {m : Type u_3} {α : Type u_5} [Fintype l] [Fintype m] [NonUnitalSeminormedRing α] (A : Matrix l m α) (v : m → α)
Goal: ‖A *ᵥ v‖ ≤ ‖A‖ * ‖v‖
-- _private.Mathlib.Analysis.Matrix.0.Matrix.norm_unitOf
private theorem norm_unitOf (a : α) : ‖unitOf a‖₊ = 1 
Arguments: {α : Type u_5} [NormedDivisionRing α] [NormedAlgebra ℝ α] (a : α)
Goal: ‖Matrix.unitOf a‖₊ = 1
-- _private.Mathlib.Analysis.Matrix.0.Matrix.mul_unitOf
private theorem mul_unitOf (a : α) : a * unitOf a = algebraMap _ _ (‖a‖₊ : ℝ)  
Arguments: {α : Type u_5} [NormedDivisionRing α] [NormedAlgebra ℝ α] (a : α)
Goal: a * Matrix.unitOf a = (algebraMap ℝ α) ↑‖a‖₊
-- Matrix.linfty_opNNNorm_eq_opNNNorm
lemma linfty_opNNNorm_eq_opNNNorm (A : Matrix m n α) :
    ‖A‖₊ = ‖ContinuousLinearMap.mk (Matrix.mulVecLin A)‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α)
Goal: ‖A‖₊ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖₊
-- Matrix.linfty_opNorm_eq_opNorm
lemma linfty_opNorm_eq_opNorm (A : Matrix m n α) :
    ‖A‖ = ‖ContinuousLinearMap.mk (Matrix.mulVecLin A)‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] (A : Matrix m n α)
Goal: ‖A‖ = ‖{ toLinearMap := A.mulVecLin, cont := ⋯ }‖
-- Matrix.linfty_opNNNorm_toMatrix
@[simp] lemma linfty_opNNNorm_toMatrix (f : (n → α) →L[α] (m → α)) :
    ‖LinearMap.toMatrix' (↑f : (n → α) →ₗ[α] (m → α))‖₊ = ‖f‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α)
Goal: ‖LinearMap.toMatrix' ↑f‖₊ = ‖f‖₊
-- Matrix.linfty_opNorm_toMatrix
@[simp] lemma linfty_opNorm_toMatrix (f : (n → α) →L[α] (m → α)) :
    ‖LinearMap.toMatrix' (↑f : (n → α) →ₗ[α] (m → α))‖ = ‖f‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [NontriviallyNormedField α] [NormedAlgebra ℝ α] [DecidableEq n] (f : (n → α) →L[α] m → α)
Goal: ‖LinearMap.toMatrix' ↑f‖ = ‖f‖
-- Matrix.frobeniusBoundedSMul
@[local instance]
theorem frobeniusBoundedSMul [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α]
    [BoundedSMul R α] :
    BoundedSMul R (Matrix m n α) 
Arguments: {R : Type u_1} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedRing R] [SeminormedAddCommGroup α] [Module R α] [BoundedSMul R α]
Goal: BoundedSMul R (Matrix m n α)
-- Matrix.frobenius_nnnorm_def
theorem frobenius_nnnorm_def (A : Matrix m n α) :
    ‖A‖₊ = (∑ i, ∑ j, ‖A i j‖₊ ^ (2 : ℝ)) ^ (1 / 2 : ℝ) 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖₊ = (∑ i : m, ∑ j : n, ‖A i j‖₊ ^ 2) ^ (1 / 2)
-- Matrix.frobenius_norm_def
theorem frobenius_norm_def (A : Matrix m n α) :
    ‖A‖ = (∑ i, ∑ j, ‖A i j‖ ^ (2 : ℝ)) ^ (1 / 2 : ℝ) 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖A‖ = (∑ i : m, ∑ j : n, ‖A i j‖ ^ 2) ^ (1 / 2)
-- Matrix.frobenius_nnnorm_map_eq
@[simp]
theorem frobenius_nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
    ‖A.map f‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊)
Goal: ‖A.map f‖₊ = ‖A‖₊
-- Matrix.frobenius_norm_map_eq
@[simp]
theorem frobenius_norm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖ = ‖a‖) :
    ‖A.map f‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} {β : Type u_6} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [SeminormedAddCommGroup β] (A : Matrix m n α) (f : α → β) (hf : ∀ (a : α), ‖f a‖ = ‖a‖)
Goal: ‖A.map f‖ = ‖A‖
-- Matrix.frobenius_nnnorm_transpose
@[simp]
theorem frobenius_nnnorm_transpose (A : Matrix m n α) : ‖Aᵀ‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖Aᵀ‖₊ = ‖A‖₊
-- Matrix.frobenius_norm_transpose
@[simp]
theorem frobenius_norm_transpose (A : Matrix m n α) : ‖Aᵀ‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] (A : Matrix m n α)
Goal: ‖Aᵀ‖ = ‖A‖
-- Matrix.frobenius_nnnorm_conjTranspose
@[simp]
theorem frobenius_nnnorm_conjTranspose [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) :
    ‖Aᴴ‖₊ = ‖A‖₊ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α)
Goal: ‖Aᴴ‖₊ = ‖A‖₊
-- Matrix.frobenius_norm_conjTranspose
@[simp]
theorem frobenius_norm_conjTranspose [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α) :
    ‖Aᴴ‖ = ‖A‖ 
Arguments: {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype m] [Fintype n] [SeminormedAddCommGroup α] [StarAddMonoid α] [NormedStarGroup α] (A : Matrix m n α)
Goal: ‖Aᴴ‖ = ‖A‖
-- Matrix.frobenius_norm_row
@[simp]
theorem frobenius_norm_row (v : m → α) : ‖row ι v‖ = ‖(WithLp.equiv 2 _).symm v‖ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.row ι v‖ = ‖(WithLp.equiv 2 (m → α)).symm v‖
-- Matrix.frobenius_nnnorm_row
@[simp]
theorem frobenius_nnnorm_row (v : m → α) : ‖row ι v‖₊ = ‖(WithLp.equiv 2 _).symm v‖₊ 
Arguments: {m : Type u_3} {α : Type u_5} {ι : Type u_7} [Fintype m] [Unique ι] [SeminormedAddCommGroup α] (v : m → α)
Goal: ‖Matrix.row ι v‖₊ = ‖(WithLp.equiv 2 (m → α)).symm v‖₊
-- Matrix.frobenius_norm_col
@[simp]
theorem frobenius_norm_col (v : n → α) : ‖col ι v‖ = ‖(WithLp.equiv 2 _).symm v‖ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.col ι v‖ = ‖(WithLp.equiv 2 (n → α)).symm v‖
-- Matrix.frobenius_nnnorm_col
@[simp]
theorem frobenius_nnnorm_col (v : n → α) : ‖col ι v‖₊ = ‖(WithLp.equiv 2 _).symm v‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} {ι : Type u_7} [Fintype n] [Unique ι] [SeminormedAddCommGroup α] (v : n → α)
Goal: ‖Matrix.col ι v‖₊ = ‖(WithLp.equiv 2 (n → α)).symm v‖₊
-- Matrix.frobenius_nnnorm_diagonal
@[simp]
theorem frobenius_nnnorm_diagonal [DecidableEq n] (v : n → α) :
    ‖diagonal v‖₊ = ‖(WithLp.equiv 2 _).symm v‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α)
Goal: ‖Matrix.diagonal v‖₊ = ‖(WithLp.equiv 2 (n → α)).symm v‖₊
-- Matrix.frobenius_norm_diagonal
@[simp]
theorem frobenius_norm_diagonal [DecidableEq n] (v : n → α) :
    ‖diagonal v‖ = ‖(WithLp.equiv 2 _).symm v‖ 
Arguments: {n : Type u_4} {α : Type u_5} [Fintype n] [SeminormedAddCommGroup α] [DecidableEq n] (v : n → α)
Goal: ‖Matrix.diagonal v‖ = ‖(WithLp.equiv 2 (n → α)).symm v‖
-- Matrix.frobenius_nnnorm_one
theorem frobenius_nnnorm_one [DecidableEq n] [SeminormedAddCommGroup α] [One α] :
    ‖(1 : Matrix n n α)‖₊ = NNReal.sqrt (Fintype.card n) * ‖(1 : α)‖₊ 
Arguments: {n : Type u_4} {α : Type u_5} [Fintype n] [DecidableEq n] [SeminormedAddCommGroup α] [One α]
Goal: ‖1‖₊ = NNReal.sqrt ↑(Fintype.card n) * ‖1‖₊
-- Matrix.frobenius_nnnorm_mul
theorem frobenius_nnnorm_mul (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊ 
Arguments: {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [RCLike α] (A : Matrix l m α) (B : Matrix m n α)
Goal: ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊
-- Matrix.frobenius_norm_mul
theorem frobenius_norm_mul (A : Matrix l m α) (B : Matrix m n α) : ‖A * B‖ ≤ ‖A‖ * ‖B‖ 
Arguments: {l : Type u_2} {m : Type u_3} {n : Type u_4} {α : Type u_5} [Fintype l] [Fintype m] [Fintype n] [RCLike α] (A : Matrix l m α) (B : Matrix m n α)
Goal: ‖A * B‖ ≤ ‖A‖ * ‖B‖
-- BoxIntegral.norm_volume_sub_integral_face_upper_sub_lower_smul_le
theorem norm_volume_sub_integral_face_upper_sub_lower_smul_le {f : (Fin (n + 1) → ℝ) → E}
    {f' : (Fin (n + 1) → ℝ) →L[ℝ] E} (hfc : ContinuousOn f (Box.Icc I)) {x : Fin (n + 1) → ℝ}
    (hxI : x ∈ (Box.Icc I)) {a : E} {ε : ℝ} (h0 : 0 < ε)
    (hε : ∀ y ∈ (Box.Icc I), ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0}
    (hc : I.distortion ≤ c) :
    ‖(∏ j, (I.upper j - I.lower j)) • f' (Pi.single i 1) -
      (integral (I.face i) ⊥ (f ∘ i.insertNth (α := fun _ ↦ ℝ) (I.upper i)) BoxAdditiveMap.volume -
        integral (I.face i) ⊥ (f ∘ i.insertNth (α := fun _ ↦ ℝ) (I.lower i))
          BoxAdditiveMap.volume)‖ ≤
      2 * ε * c * ∏ j, (I.upper j - I.lower j) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} [CompleteSpace E] (I : BoxIntegral.Box (Fin (n + 1))) {i : Fin (n + 1)} {f : (Fin (n + 1) → ℝ) → E} {f' : (Fin (n + 1) → ℝ) →L[ℝ] E} (hfc : ContinuousOn f (BoxIntegral.Box.Icc I)) {x : Fin (n + 1) → ℝ} (hxI : x ∈ BoxIntegral.Box.Icc I) {a : E} {ε : ℝ} (h0 : 0 < ε) (hε : ∀ y ∈ BoxIntegral.Box.Icc I, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0} (hc : I.distortion ≤ c)
Goal: ‖(∏ j : Fin (n + 1), (I.upper j - I.lower j)) • f' (Pi.single i 1) -
      (BoxIntegral.integral (I.face i) ⊥ (f ∘ i.insertNth (I.upper i)) BoxIntegral.BoxAdditiveMap.volume -
        BoxIntegral.integral (I.face i) ⊥ (f ∘ i.insertNth (I.lower i)) BoxIntegral.BoxAdditiveMap.volume)‖ ≤
  2 * ε * ↑c * ∏ j : Fin (n + 1), (I.upper j - I.lower j)
-- BoxIntegral.hasIntegral_GP_pderiv
theorem hasIntegral_GP_pderiv (f : (Fin (n + 1) → ℝ) → E)
    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ))
    (hs : s.Countable) (Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x)
    (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) (i : Fin (n + 1)) :
    HasIntegral.{0, u, u} I GP (fun x => f' x (Pi.single i 1)) BoxAdditiveMap.volume
      (integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x))
          BoxAdditiveMap.volume -
        integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.lower i) x))
          BoxAdditiveMap.volume) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} [CompleteSpace E] (I : BoxIntegral.Box (Fin (n + 1))) (f : (Fin (n + 1) → ℝ) → E) (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hs : ∀ x ∈ s, ContinuousWithinAt f (BoxIntegral.Box.Icc I) x) (Hd : ∀ x ∈ BoxIntegral.Box.Icc I \ s, HasFDerivWithinAt f (f' x) (BoxIntegral.Box.Icc I) x) (i : Fin (n + 1))
Goal: BoxIntegral.HasIntegral I BoxIntegral.IntegrationParams.GP (fun x => (f' x) (Pi.single i 1))
  BoxIntegral.BoxAdditiveMap.volume
  (BoxIntegral.integral (I.face i) BoxIntegral.IntegrationParams.GP (fun x => f (i.insertNth (I.upper i) x))
      BoxIntegral.BoxAdditiveMap.volume -
    BoxIntegral.integral (I.face i) BoxIntegral.IntegrationParams.GP (fun x => f (i.insertNth (I.lower i) x))
      BoxIntegral.BoxAdditiveMap.volume)
-- BoxIntegral.hasIntegral_GP_divergence_of_forall_hasDerivWithinAt
theorem hasIntegral_GP_divergence_of_forall_hasDerivWithinAt
    (f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E)
    (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → E))
    (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
    (Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x)
    (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) :
    HasIntegral.{0, u, u} I GP (fun x => ∑ i, f' x (Pi.single i 1) i) BoxAdditiveMap.volume
      (∑ i,
        (integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x) i)
            BoxAdditiveMap.volume -
          integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.lower i) x) i)
            BoxAdditiveMap.volume)) 
Arguments: {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E] {n : ℕ} [CompleteSpace E] (I : BoxIntegral.Box (Fin (n + 1))) (f : (Fin (n + 1) → ℝ) → Fin (n + 1) → E) (f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E) (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable) (Hs : ∀ x ∈ s, ContinuousWithinAt f (BoxIntegral.Box.Icc I) x) (Hd : ∀ x ∈ BoxIntegral.Box.Icc I \ s, HasFDerivWithinAt f (f' x) (BoxIntegral.Box.Icc I) x)
Goal: BoxIntegral.HasIntegral I BoxIntegral.IntegrationParams.GP (fun x => ∑ i : Fin (n + 1), (f' x) (Pi.single i 1) i)
  BoxIntegral.BoxAdditiveMap.volume
  (∑ i : Fin (n + 1),
    (BoxIntegral.integral (I.face i) BoxIntegral.IntegrationParams.GP (fun x => f (i.insertNth (I.upper i) x) i)
        BoxIntegral.BoxAdditiveMap.volume -
      BoxIntegral.integral (I.face i) BoxIntegral.IntegrationParams.GP (fun x => f (i.insertNth (I.lower i) x) i)
        BoxIntegral.BoxAdditiveMap.volume))
-- FormalMultilinearSeries.applyComposition_ones
theorem applyComposition_ones (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
    p.applyComposition (Composition.ones n) = fun v i =>
      p 1 fun _ => v (Fin.castLE (Composition.length_le _) i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ)
Goal: p.applyComposition (Composition.ones n) = fun v i => (p 1) fun x => v (Fin.castLE ⋯ i)
-- FormalMultilinearSeries.applyComposition_single
theorem applyComposition_single (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n)
    (v : Fin n → E) : p.applyComposition (Composition.single n hn) v = fun _j => p n v 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : 0 < n) (v : Fin n → E)
Goal: p.applyComposition (Composition.single n hn) v = fun _j => (p n) v
-- FormalMultilinearSeries.removeZero_applyComposition
@[simp]
theorem removeZero_applyComposition (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (c : Composition n) : p.removeZero.applyComposition c = p.applyComposition c 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
Goal: p.removeZero.applyComposition c = p.applyComposition c
-- FormalMultilinearSeries.applyComposition_update
theorem applyComposition_update (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n)
    (j : Fin n) (v : Fin n → E) (z : E) :
    p.applyComposition c (Function.update v j z) =
      Function.update (p.applyComposition c v) (c.index j)
        (p (c.blocksFun (c.index j))
          (Function.update (v ∘ c.embedding (c.index j)) (c.invEmbedding j) z)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [Module 𝕜 E] [Module 𝕜 F] [TopologicalSpace E] [TopologicalSpace F] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (c : Composition n) (j : Fin n) (v : Fin n → E) (z : E)
Goal: p.applyComposition c (Function.update v j z) =
  Function.update (p.applyComposition c v) (c.index j)
    ((p (c.blocksFun (c.index j))) (Function.update (v ∘ ⇑(c.embedding (c.index j))) (c.invEmbedding j) z))
-- FormalMultilinearSeries.compContinuousLinearMap_applyComposition
@[simp]
theorem compContinuousLinearMap_applyComposition {n : ℕ} (p : FormalMultilinearSeries 𝕜 F G)
    (f : E →L[𝕜] F) (c : Composition n) (v : Fin n → E) :
    (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (f ∘ v) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] {n : ℕ} (p : FormalMultilinearSeries 𝕜 F G) (f : E →L[𝕜] F) (c : Composition n) (v : Fin n → E)
Goal: (p.compContinuousLinearMap f).applyComposition c v = p.applyComposition c (⇑f ∘ v)
-- ContinuousMultilinearMap.compAlongComposition_apply
@[simp]
theorem compAlongComposition_apply {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n)
    (f : ContinuousMultilinearMap 𝕜 (fun _i : Fin c.length => F) G) (v : Fin n → E) :
    (f.compAlongComposition p c) v = f (p.applyComposition c v) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] {n : ℕ} (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (f : ContinuousMultilinearMap 𝕜 (fun _i => F) G) (v : Fin n → E)
Goal: (ContinuousMultilinearMap.compAlongComposition p c f) v = f (p.applyComposition c v)
-- FormalMultilinearSeries.compAlongComposition_apply
@[simp]
theorem compAlongComposition_apply {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G)
    (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (v : Fin n → E) :
    (q.compAlongComposition p c) v = q c.length (p.applyComposition c v) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [CommRing 𝕜] [AddCommGroup E] [AddCommGroup F] [AddCommGroup G] [Module 𝕜 E] [Module 𝕜 F] [Module 𝕜 G] [TopologicalSpace E] [TopologicalSpace F] [TopologicalSpace G] [TopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] [TopologicalAddGroup F] [ContinuousConstSMul 𝕜 F] [TopologicalAddGroup G] [ContinuousConstSMul 𝕜 G] {n : ℕ} (q : FormalMultilinearSeries 𝕜 F G) (p : FormalMultilinearSeries 𝕜 E F) (c : Composition n) (v : Fin n → E)
Goal: (q.compAlongComposition p c) v = (q c.length) (p.applyComposition c v)