ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.IsHermitian.spectral_theorem
theorem spectral_theorem :
    A = (eigenvectorUnitary hA : Matrix n n 𝕜) * diagonal (RCLike.ofReal ∘ hA.eigenvalues)
      * (star (eigenvectorUnitary hA : Matrix n n 𝕜)) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} [DecidableEq n] (hA : A.IsHermitian)
Goal: A = ↑hA.eigenvectorUnitary * Matrix.diagonal (RCLike.ofReal ∘ hA.eigenvalues) * star ↑hA.eigenvectorUnitary
-- Matrix.IsHermitian.eigenvalues_eq
theorem eigenvalues_eq (i : n) :
    (hA.eigenvalues i) = RCLike.re (Matrix.dotProduct (star ⇑(hA.eigenvectorBasis i))
    (A *ᵥ ⇑(hA.eigenvectorBasis i))) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} [DecidableEq n] (hA : A.IsHermitian) (i : n)
Goal: hA.eigenvalues i =
  RCLike.re
    (Matrix.dotProduct (star ((WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis i)))
      (A.mulVec ((WithLp.equiv 2 ((i : n) → (fun x => 𝕜) i)) (hA.eigenvectorBasis i))))
-- Matrix.IsHermitian.det_eq_prod_eigenvalues
theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} [DecidableEq n] (hA : A.IsHermitian)
Goal: A.det = ∏ i : n, ↑(hA.eigenvalues i)
-- Matrix.IsHermitian.rank_eq_rank_diagonal
lemma rank_eq_rank_diagonal : A.rank = (Matrix.diagonal hA.eigenvalues).rank 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} [DecidableEq n] (hA : A.IsHermitian)
Goal: A.rank = (Matrix.diagonal hA.eigenvalues).rank
-- Matrix.IsHermitian.rank_eq_card_non_zero_eigs
lemma rank_eq_card_non_zero_eigs : A.rank = Fintype.card {i // hA.eigenvalues i ≠ 0} 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} [DecidableEq n] (hA : A.IsHermitian)
Goal: A.rank = Fintype.card { i // hA.eigenvalues i ≠ 0 }
-- Matrix.IsHermitian.exists_eigenvector_of_ne_zero
lemma exists_eigenvector_of_ne_zero (hA : IsHermitian A) (h_ne : A ≠ 0) :
    ∃ (v : n → 𝕜) (t : ℝ), t ≠ 0 ∧ v ≠ 0 ∧ A *ᵥ v = t • v 
Arguments: {𝕜 : Type u_1} [RCLike 𝕜] {n : Type u_2} [Fintype n] {A : Matrix n n 𝕜} (hA : A.IsHermitian) (h_ne : A ≠ 0)
Goal: ∃ v t, t ≠ 0 ∧ v ≠ 0 ∧ A.mulVec v = t • v
-- Matrix.Nondegenerate.eq_zero_of_ortho
theorem Nondegenerate.eq_zero_of_ortho {M : Matrix m m R} (hM : Nondegenerate M) {v : m → R}
    (hv : ∀ w, Matrix.dotProduct v (M *ᵥ w) = 0) : v = 0 
Arguments: {m : Type u_1} {R : Type u_2} [Fintype m] [CommRing R] {M : Matrix m m R} (hM : M.Nondegenerate) {v : m → R} (hv : ∀ (w : m → R), Matrix.dotProduct v (M.mulVec w) = 0)
Goal: v = 0
-- Matrix.Nondegenerate.exists_not_ortho_of_ne_zero
theorem Nondegenerate.exists_not_ortho_of_ne_zero {M : Matrix m m R} (hM : Nondegenerate M)
    {v : m → R} (hv : v ≠ 0) : ∃ w, Matrix.dotProduct v (M *ᵥ w) ≠ 0 
Arguments: {m : Type u_1} {R : Type u_2} [Fintype m] [CommRing R] {M : Matrix m m R} (hM : M.Nondegenerate) {v : m → R} (hv : v ≠ 0)
Goal: ∃ w, Matrix.dotProduct v (M.mulVec w) ≠ 0
-- Matrix.nondegenerate_of_det_ne_zero
theorem nondegenerate_of_det_ne_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) :
    Nondegenerate M 
Arguments: {m : Type u_1} {A : Type u_3} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0)
Goal: M.Nondegenerate
-- Matrix.eq_zero_of_vecMul_eq_zero
theorem eq_zero_of_vecMul_eq_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A}
    (hv : v ᵥ* M = 0) : v = 0 
Arguments: {m : Type u_1} {A : Type u_3} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : Matrix.vecMul v M = 0)
Goal: v = 0
-- Matrix.eq_zero_of_mulVec_eq_zero
theorem eq_zero_of_mulVec_eq_zero [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A}
    (hv : M *ᵥ v = 0) : v = 0 
Arguments: {m : Type u_1} {A : Type u_3} [Fintype m] [CommRing A] [IsDomain A] [DecidableEq m] {M : Matrix m m A} (hM : M.det ≠ 0) {v : m → A} (hv : M.mulVec v = 0)
Goal: v = 0
-- Finset.univ_fin2
theorem univ_fin2 : (univ : Finset (Fin 2)) = {0, 1} 
Goal: Finset.univ = {0, 1}
-- Finset.weightedVSubOfPoint_apply
@[simp]
theorem weightedVSubOfPoint_apply (w : ι → k) (p : ι → P) (b : P) :
    s.weightedVSubOfPoint p b w = ∑ i ∈ s, w i • (p i -ᵥ b) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (b : P)
Goal: (s.weightedVSubOfPoint p b) w = ∑ i ∈ s, w i • (p i -ᵥ b)
-- Finset.weightedVSubOfPoint_apply_const
@[simp (high)]
theorem weightedVSubOfPoint_apply_const (w : ι → k) (p : P) (b : P) :
    s.weightedVSubOfPoint (fun _ => p) b w = (∑ i ∈ s, w i) • (p -ᵥ b) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p b : P)
Goal: (s.weightedVSubOfPoint (fun x => p) b) w = (∑ i ∈ s, w i) • (p -ᵥ b)
-- Finset.weightedVSubOfPoint_vadd
lemma weightedVSubOfPoint_vadd (s : Finset ι) (w : ι → k) (p : ι → P) (b : P) (v : V) :
    s.weightedVSubOfPoint (v +ᵥ p) b w = s.weightedVSubOfPoint p (-v +ᵥ b) w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (b : P) (v : V)
Goal: (s.weightedVSubOfPoint (v +ᵥ p) b) w = (s.weightedVSubOfPoint p (-v +ᵥ b)) w
-- Finset.weightedVSubOfPoint_congr
theorem weightedVSubOfPoint_congr {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P}
    (hp : ∀ i ∈ s, p₁ i = p₂ i) (b : P) :
    s.weightedVSubOfPoint p₁ b w₁ = s.weightedVSubOfPoint p₂ b w₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P} (hp : ∀ i ∈ s, p₁ i = p₂ i) (b : P)
Goal: (s.weightedVSubOfPoint p₁ b) w₁ = (s.weightedVSubOfPoint p₂ b) w₂
-- Finset.weightedVSubOfPoint_eq_of_weights_eq
theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
    (hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
    s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (p : ι → P) (j : ι) (w₁ w₂ : ι → k) (hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i)
Goal: (s.weightedVSubOfPoint p (p j)) w₁ = (s.weightedVSubOfPoint p (p j)) w₂
-- Finset.weightedVSubOfPoint_eq_of_sum_eq_zero
theorem weightedVSubOfPoint_eq_of_sum_eq_zero (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0)
    (b₁ b₂ : P) : s.weightedVSubOfPoint p b₁ w = s.weightedVSubOfPoint p b₂ w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0) (b₁ b₂ : P)
Goal: (s.weightedVSubOfPoint p b₁) w = (s.weightedVSubOfPoint p b₂) w
-- Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one
theorem weightedVSubOfPoint_vadd_eq_of_sum_eq_one (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 1)
    (b₁ b₂ : P) : s.weightedVSubOfPoint p b₁ w +ᵥ b₁ = s.weightedVSubOfPoint p b₂ w +ᵥ b₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 1) (b₁ b₂ : P)
Goal: (s.weightedVSubOfPoint p b₁) w +ᵥ b₁ = (s.weightedVSubOfPoint p b₂) w +ᵥ b₂
-- Finset.weightedVSubOfPoint_erase
@[simp (high)]
theorem weightedVSubOfPoint_erase [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι) :
    (s.erase i).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι)
Goal: ((s.erase i).weightedVSubOfPoint p (p i)) w = (s.weightedVSubOfPoint p (p i)) w
-- Finset.weightedVSubOfPoint_insert
@[simp (high)]
theorem weightedVSubOfPoint_insert [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι) :
    (insert i s).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι)
Goal: ((insert i s).weightedVSubOfPoint p (p i)) w = (s.weightedVSubOfPoint p (p i)) w
-- Finset.weightedVSubOfPoint_indicator_subset
theorem weightedVSubOfPoint_indicator_subset (w : ι → k) (p : ι → P) (b : P) {s₁ s₂ : Finset ι}
    (h : s₁ ⊆ s₂) :
    s₁.weightedVSubOfPoint p b w = s₂.weightedVSubOfPoint p b (Set.indicator (↑s₁) w) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (w : ι → k) (p : ι → P) (b : P) {s₁ s₂ : Finset ι} (h : s₁ ⊆ s₂)
Goal: (s₁.weightedVSubOfPoint p b) w = (s₂.weightedVSubOfPoint p b) ((↑s₁).indicator w)
-- Finset.weightedVSubOfPoint_map
theorem weightedVSubOfPoint_map (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P) (b : P) :
    (s₂.map e).weightedVSubOfPoint p b w = s₂.weightedVSubOfPoint (p ∘ e) b (w ∘ e) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} {ι₂ : Type u_5} (s₂ : Finset ι₂) (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P) (b : P)
Goal: ((Finset.map e s₂).weightedVSubOfPoint p b) w = (s₂.weightedVSubOfPoint (p ∘ ⇑e) b) (w ∘ ⇑e)
-- Finset.sum_smul_vsub_eq_weightedVSubOfPoint_sub
theorem sum_smul_vsub_eq_weightedVSubOfPoint_sub (w : ι → k) (p₁ p₂ : ι → P) (b : P) :
    (∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) =
      s.weightedVSubOfPoint p₁ b w - s.weightedVSubOfPoint p₂ b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₁ p₂ : ι → P) (b : P)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) = (s.weightedVSubOfPoint p₁ b) w - (s.weightedVSubOfPoint p₂ b) w
-- Finset.sum_smul_vsub_const_eq_weightedVSubOfPoint_sub
theorem sum_smul_vsub_const_eq_weightedVSubOfPoint_sub (w : ι → k) (p₁ : ι → P) (p₂ b : P) :
    (∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.weightedVSubOfPoint p₁ b w - (∑ i ∈ s, w i) • (p₂ -ᵥ b) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₁ : ι → P) (p₂ b : P)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSubOfPoint p₁ b) w - (∑ i ∈ s, w i) • (p₂ -ᵥ b)
-- Finset.sum_smul_const_vsub_eq_sub_weightedVSubOfPoint
theorem sum_smul_const_vsub_eq_sub_weightedVSubOfPoint (w : ι → k) (p₂ : ι → P) (p₁ b : P) :
    (∑ i ∈ s, w i • (p₁ -ᵥ p₂ i)) = (∑ i ∈ s, w i) • (p₁ -ᵥ b) - s.weightedVSubOfPoint p₂ b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₂ : ι → P) (p₁ b : P)
Goal: ∑ i ∈ s, w i • (p₁ -ᵥ p₂ i) = (∑ i ∈ s, w i) • (p₁ -ᵥ b) - (s.weightedVSubOfPoint p₂ b) w
-- Finset.weightedVSubOfPoint_sdiff
theorem weightedVSubOfPoint_sdiff [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k)
    (p : ι → P) (b : P) :
    (s \ s₂).weightedVSubOfPoint p b w + s₂.weightedVSubOfPoint p b w =
      s.weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P) (b : P)
Goal: ((s \ s₂).weightedVSubOfPoint p b) w + (s₂.weightedVSubOfPoint p b) w = (s.weightedVSubOfPoint p b) w
-- Finset.weightedVSubOfPoint_sdiff_sub
theorem weightedVSubOfPoint_sdiff_sub [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k)
    (p : ι → P) (b : P) :
    (s \ s₂).weightedVSubOfPoint p b w - s₂.weightedVSubOfPoint p b (-w) =
      s.weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P) (b : P)
Goal: ((s \ s₂).weightedVSubOfPoint p b) w - (s₂.weightedVSubOfPoint p b) (-w) = (s.weightedVSubOfPoint p b) w
-- Finset.weightedVSubOfPoint_subtype_eq_filter
theorem weightedVSubOfPoint_subtype_eq_filter (w : ι → k) (p : ι → P) (b : P) (pred : ι → Prop)
    [DecidablePred pred] :
    ((s.subtype pred).weightedVSubOfPoint (fun i => p i) b fun i => w i) =
      (s.filter pred).weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (b : P) (pred : ι → Prop) [DecidablePred pred]
Goal: (((Finset.subtype pred s).weightedVSubOfPoint (fun i => p ↑i) b) fun i => w ↑i) =
  ((Finset.filter pred s).weightedVSubOfPoint p b) w
-- Finset.weightedVSubOfPoint_filter_of_ne
theorem weightedVSubOfPoint_filter_of_ne (w : ι → k) (p : ι → P) (b : P) {pred : ι → Prop}
    [DecidablePred pred] (h : ∀ i ∈ s, w i ≠ 0 → pred i) :
    (s.filter pred).weightedVSubOfPoint p b w = s.weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (b : P) {pred : ι → Prop} [DecidablePred pred] (h : ∀ i ∈ s, w i ≠ 0 → pred i)
Goal: ((Finset.filter pred s).weightedVSubOfPoint p b) w = (s.weightedVSubOfPoint p b) w
-- Finset.weightedVSubOfPoint_const_smul
theorem weightedVSubOfPoint_const_smul (w : ι → k) (p : ι → P) (b : P) (c : k) :
    s.weightedVSubOfPoint p b (c • w) = c • s.weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (b : P) (c : k)
Goal: (s.weightedVSubOfPoint p b) (c • w) = c • (s.weightedVSubOfPoint p b) w
-- Finset.weightedVSub_apply
theorem weightedVSub_apply (w : ι → k) (p : ι → P) :
    s.weightedVSub p w = ∑ i ∈ s, w i • (p i -ᵥ Classical.choice S.nonempty) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P)
Goal: (s.weightedVSub p) w = ∑ i ∈ s, w i • (p i -ᵥ Classical.choice ⋯)
-- Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero
theorem weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero (w : ι → k) (p : ι → P)
    (h : ∑ i ∈ s, w i = 0) (b : P) : s.weightedVSub p w = s.weightedVSubOfPoint p b w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 0) (b : P)
Goal: (s.weightedVSub p) w = (s.weightedVSubOfPoint p b) w
-- Finset.weightedVSub_apply_const
@[simp]
theorem weightedVSub_apply_const (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 0) :
    s.weightedVSub (fun _ => p) w = 0 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 0)
Goal: (s.weightedVSub fun x => p) w = 0
-- Finset.weightedVSub_empty
@[simp]
theorem weightedVSub_empty (w : ι → k) (p : ι → P) : (∅ : Finset ι).weightedVSub p w = (0 : V) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (w : ι → k) (p : ι → P)
Goal: (∅.weightedVSub p) w = 0
-- Finset.weightedVSub_vadd
lemma weightedVSub_vadd {s : Finset ι} {w : ι → k} (h : ∑ i ∈ s, w i = 0) (p : ι → P) (v : V) :
    s.weightedVSub (v +ᵥ p) w = s.weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} {s : Finset ι} {w : ι → k} (h : ∑ i ∈ s, w i = 0) (p : ι → P) (v : V)
Goal: (s.weightedVSub (v +ᵥ p)) w = (s.weightedVSub p) w
-- Finset.weightedVSub_congr
theorem weightedVSub_congr {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P}
    (hp : ∀ i ∈ s, p₁ i = p₂ i) : s.weightedVSub p₁ w₁ = s.weightedVSub p₂ w₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P} (hp : ∀ i ∈ s, p₁ i = p₂ i)
Goal: (s.weightedVSub p₁) w₁ = (s.weightedVSub p₂) w₂
-- Finset.weightedVSub_indicator_subset
theorem weightedVSub_indicator_subset (w : ι → k) (p : ι → P) {s₁ s₂ : Finset ι} (h : s₁ ⊆ s₂) :
    s₁.weightedVSub p w = s₂.weightedVSub p (Set.indicator (↑s₁) w) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (w : ι → k) (p : ι → P) {s₁ s₂ : Finset ι} (h : s₁ ⊆ s₂)
Goal: (s₁.weightedVSub p) w = (s₂.weightedVSub p) ((↑s₁).indicator w)
-- Finset.weightedVSub_map
theorem weightedVSub_map (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P) :
    (s₂.map e).weightedVSub p w = s₂.weightedVSub (p ∘ e) (w ∘ e) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} {ι₂ : Type u_5} (s₂ : Finset ι₂) (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P)
Goal: ((Finset.map e s₂).weightedVSub p) w = (s₂.weightedVSub (p ∘ ⇑e)) (w ∘ ⇑e)
-- Finset.sum_smul_vsub_eq_weightedVSub_sub
theorem sum_smul_vsub_eq_weightedVSub_sub (w : ι → k) (p₁ p₂ : ι → P) :
    (∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) = s.weightedVSub p₁ w - s.weightedVSub p₂ w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₁ p₂ : ι → P)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) = (s.weightedVSub p₁) w - (s.weightedVSub p₂) w
-- Finset.sum_smul_vsub_const_eq_weightedVSub
theorem sum_smul_vsub_const_eq_weightedVSub (w : ι → k) (p₁ : ι → P) (p₂ : P)
    (h : ∑ i ∈ s, w i = 0) : (∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.weightedVSub p₁ w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₁ : ι → P) (p₂ : P) (h : ∑ i ∈ s, w i = 0)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSub p₁) w
-- Finset.sum_smul_const_vsub_eq_neg_weightedVSub
theorem sum_smul_const_vsub_eq_neg_weightedVSub (w : ι → k) (p₂ : ι → P) (p₁ : P)
    (h : ∑ i ∈ s, w i = 0) : (∑ i ∈ s, w i • (p₁ -ᵥ p₂ i)) = -s.weightedVSub p₂ w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₂ : ι → P) (p₁ : P) (h : ∑ i ∈ s, w i = 0)
Goal: ∑ i ∈ s, w i • (p₁ -ᵥ p₂ i) = -(s.weightedVSub p₂) w
-- Finset.weightedVSub_sdiff
theorem weightedVSub_sdiff [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P) :
    (s \ s₂).weightedVSub p w + s₂.weightedVSub p w = s.weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P)
Goal: ((s \ s₂).weightedVSub p) w + (s₂.weightedVSub p) w = (s.weightedVSub p) w
-- Finset.weightedVSub_sdiff_sub
theorem weightedVSub_sdiff_sub [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k)
    (p : ι → P) : (s \ s₂).weightedVSub p w - s₂.weightedVSub p (-w) = s.weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P)
Goal: ((s \ s₂).weightedVSub p) w - (s₂.weightedVSub p) (-w) = (s.weightedVSub p) w
-- Finset.weightedVSub_subtype_eq_filter
theorem weightedVSub_subtype_eq_filter (w : ι → k) (p : ι → P) (pred : ι → Prop)
    [DecidablePred pred] :
    ((s.subtype pred).weightedVSub (fun i => p i) fun i => w i) =
      (s.filter pred).weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (pred : ι → Prop) [DecidablePred pred]
Goal: (((Finset.subtype pred s).weightedVSub fun i => p ↑i) fun i => w ↑i) = ((Finset.filter pred s).weightedVSub p) w
-- Finset.weightedVSub_filter_of_ne
theorem weightedVSub_filter_of_ne (w : ι → k) (p : ι → P) {pred : ι → Prop} [DecidablePred pred]
    (h : ∀ i ∈ s, w i ≠ 0 → pred i) : (s.filter pred).weightedVSub p w = s.weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) {pred : ι → Prop} [DecidablePred pred] (h : ∀ i ∈ s, w i ≠ 0 → pred i)
Goal: ((Finset.filter pred s).weightedVSub p) w = (s.weightedVSub p) w
-- Finset.weightedVSub_const_smul
theorem weightedVSub_const_smul (w : ι → k) (p : ι → P) (c : k) :
    s.weightedVSub p (c • w) = c • s.weightedVSub p w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (c : k)
Goal: (s.weightedVSub p) (c • w) = c • (s.weightedVSub p) w
-- Finset.affineCombination_linear
@[simp]
theorem affineCombination_linear (p : ι → P) :
    (s.affineCombination k p).linear = s.weightedVSub p 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (p : ι → P)
Goal: (Finset.affineCombination k s p).linear = s.weightedVSub p
-- Finset.affineCombination_apply
theorem affineCombination_apply (w : ι → k) (p : ι → P) :
    (s.affineCombination k p) w =
      s.weightedVSubOfPoint p (Classical.choice S.nonempty) w +ᵥ Classical.choice S.nonempty 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P)
Goal: (Finset.affineCombination k s p) w = (s.weightedVSubOfPoint p (Classical.choice ⋯)) w +ᵥ Classical.choice ⋯
-- Finset.affineCombination_apply_const
@[simp]
theorem affineCombination_apply_const (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 1) :
    s.affineCombination k (fun _ => p) w = p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 1)
Goal: (Finset.affineCombination k s fun x => p) w = p
-- Finset.affineCombination_congr
theorem affineCombination_congr {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P}
    (hp : ∀ i ∈ s, p₁ i = p₂ i) : s.affineCombination k p₁ w₁ = s.affineCombination k p₂ w₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) {w₁ w₂ : ι → k} (hw : ∀ i ∈ s, w₁ i = w₂ i) {p₁ p₂ : ι → P} (hp : ∀ i ∈ s, p₁ i = p₂ i)
Goal: (Finset.affineCombination k s p₁) w₁ = (Finset.affineCombination k s p₂) w₂
-- Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
theorem affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one (w : ι → k) (p : ι → P)
    (h : ∑ i ∈ s, w i = 1) (b : P) :
    s.affineCombination k p w = s.weightedVSubOfPoint p b w +ᵥ b 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p : ι → P) (h : ∑ i ∈ s, w i = 1) (b : P)
Goal: (Finset.affineCombination k s p) w = (s.weightedVSubOfPoint p b) w +ᵥ b
-- Finset.weightedVSub_vadd_affineCombination
theorem weightedVSub_vadd_affineCombination (w₁ w₂ : ι → k) (p : ι → P) :
    s.weightedVSub p w₁ +ᵥ s.affineCombination k p w₂ = s.affineCombination k p (w₁ + w₂) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : Finset ι) (w₁ w₂ : ι → k) (p : ι → P)
Goal: (s.weightedVSub p) w₁ +ᵥ (Finset.affineCombination k s p) w₂ = (Finset.affineCombination k s p) (w₁ + w₂)