ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.adjugate_fin_three
theorem adjugate_fin_three (A : Matrix (Fin 3) (Fin 3) α) :
    adjugate A =
    !![A 1 1 * A 2 2 - A 1 2 * A 2 1,
      -(A 0 1 * A 2 2) + A 0 2 * A 2 1,
      A 0 1 * A 1 2 - A 0 2 * A 1 1;
      -(A 1 0 * A 2 2) + A 1 2 * A 2 0,
      A 0 0 * A 2 2 - A 0 2 * A 2 0,
      -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
      A 1 0 * A 2 1 - A 1 1 * A 2 0,
      -(A 0 0 * A 2 1) + A 0 1 * A 2 0,
      A 0 0 * A 1 1 - A 0 1 * A 1 0] 
Arguments: {α : Type w} [CommRing α] (A : Matrix (Fin 3) (Fin 3) α)
Goal: A.adjugate =
  !![A 1 1 * A 2 2 - A 1 2 * A 2 1, -(A 0 1 * A 2 2) + A 0 2 * A 2 1, A 0 1 * A 1 2 - A 0 2 * A 1 1;
    -(A 1 0 * A 2 2) + A 1 2 * A 2 0, A 0 0 * A 2 2 - A 0 2 * A 2 0, -(A 0 0 * A 1 2) + A 0 2 * A 1 0;
    A 1 0 * A 2 1 - A 1 1 * A 2 0, -(A 0 0 * A 2 1) + A 0 1 * A 2 0, A 0 0 * A 1 1 - A 0 1 * A 1 0]
-- Matrix.adjugate_fin_three_of
@[simp]
theorem adjugate_fin_three_of (a b c d e f g h i : α) :
    adjugate !![a, b, c; d, e, f; g, h, i] =
      !![  e * i  - f * h, -(b * i) + c * h,   b * f  - c * e;
         -(d * i) + f * g,   a * i  - c * g, -(a * f) + c * d;
           d * h  - e * g, -(a * h) + b * g,   a * e  - b * d] 
Arguments: {α : Type w} [CommRing α] (a b c d e f g h i : α)
Goal: !![a, b, c; d, e, f; g, h, i].adjugate =
  !![e * i - f * h, -(b * i) + c * h, b * f - c * e;
    -(d * i) + f * g, a * i - c * g, -(a * f) + c * d;
    d * h - e * g, -(a * h) + b * g, a * e - b * d]
-- Matrix.det_eq_sum_mul_adjugate_row
theorem det_eq_sum_mul_adjugate_row (A : Matrix n n α) (i : n) :
    det A = ∑ j : n, A i j * adjugate A j i 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α) (i : n)
Goal: A.det = ∑ j : n, A i j * A.adjugate j i
-- Matrix.det_eq_sum_mul_adjugate_col
theorem det_eq_sum_mul_adjugate_col (A : Matrix n n α) (j : n) :
    det A = ∑ i : n, A i j * adjugate A j i 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α) (j : n)
Goal: A.det = ∑ i : n, A i j * A.adjugate j i
-- Matrix.adjugate_conjTranspose
theorem adjugate_conjTranspose [StarRing α] (A : Matrix n n α) : A.adjugateᴴ = adjugate Aᴴ 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] [StarRing α] (A : Matrix n n α)
Goal: A.adjugate.conjTranspose = A.conjTranspose.adjugate
-- Matrix.isRegular_of_isLeftRegular_det
theorem isRegular_of_isLeftRegular_det {A : Matrix n n α} (hA : IsLeftRegular A.det) :
    IsRegular A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] {A : Matrix n n α} (hA : IsLeftRegular A.det)
Goal: IsRegular A
-- Matrix.adjugate_mul_distrib_aux
theorem adjugate_mul_distrib_aux (A B : Matrix n n α) (hA : IsLeftRegular A.det)
    (hB : IsLeftRegular B.det) : adjugate (A * B) = adjugate B * adjugate A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A B : Matrix n n α) (hA : IsLeftRegular A.det) (hB : IsLeftRegular B.det)
Goal: (A * B).adjugate = B.adjugate * A.adjugate
-- Matrix.adjugate_mul_distrib
theorem adjugate_mul_distrib (A B : Matrix n n α) : adjugate (A * B) = adjugate B * adjugate A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A B : Matrix n n α)
Goal: (A * B).adjugate = B.adjugate * A.adjugate
-- Matrix.adjugate_pow
@[simp]
theorem adjugate_pow (A : Matrix n n α) (k : ℕ) : adjugate (A ^ k) = adjugate A ^ k 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α) (k : ℕ)
Goal: (A ^ k).adjugate = A.adjugate ^ k
-- Matrix.det_smul_adjugate_adjugate
theorem det_smul_adjugate_adjugate (A : Matrix n n α) :
    det A • adjugate (adjugate A) = det A ^ (Fintype.card n - 1) • A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α)
Goal: A.det • A.adjugate.adjugate = A.det ^ (Fintype.card n - 1) • A
-- Matrix.adjugate_adjugate
theorem adjugate_adjugate (A : Matrix n n α) (h : Fintype.card n ≠ 1) :
    adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α) (h : Fintype.card n ≠ 1)
Goal: A.adjugate.adjugate = A.det ^ (Fintype.card n - 2) • A
-- Matrix.adjugate_adjugate'
theorem adjugate_adjugate' (A : Matrix n n α) [Nontrivial n] :
    adjugate (adjugate A) = det A ^ (Fintype.card n - 2) • A 
Arguments: {n : Type v} {α : Type w} [DecidableEq n] [Fintype n] [CommRing α] (A : Matrix n n α) [Nontrivial n]
Goal: A.adjugate.adjugate = A.det ^ (Fintype.card n - 2) • A
-- Matrix.inv_pow'
@[simp]
theorem inv_pow' (A : M) (n : ℕ) : A⁻¹ ^ n = (A ^ n)⁻¹ 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℕ)
Goal: A⁻¹ ^ n = (A ^ n)⁻¹
-- Matrix.pow_sub'
theorem pow_sub' (A : M) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m) :
    A ^ (m - n) = A ^ m * (A ^ n)⁻¹ 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m)
Goal: A ^ (m - n) = A ^ m * (A ^ n)⁻¹
-- Matrix.pow_inv_comm'
theorem pow_inv_comm' (A : M) (m n : ℕ) : A⁻¹ ^ m * A ^ n = A ^ n * A⁻¹ ^ m 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (m n : ℕ)
Goal: A⁻¹ ^ m * A ^ n = A ^ n * A⁻¹ ^ m
-- Matrix.one_zpow
@[simp]
theorem one_zpow : ∀ n : ℤ, (1 : M) ^ n = 1
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (n : ℤ)
Goal: 1 ^ n = 1
-- Matrix.zero_zpow
theorem zero_zpow : ∀ z : ℤ, z ≠ 0 → (0 : M) ^ z = 0
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (z : ℤ)
Goal: z ≠ 0 → 0 ^ z = 0
-- Matrix.zero_zpow_eq
theorem zero_zpow_eq (n : ℤ) : (0 : M) ^ n = if n = 0 then 1 else 0 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (n : ℤ)
Goal: 0 ^ n = if n = 0 then 1 else 0
-- Matrix.inv_zpow
theorem inv_zpow (A : M) : ∀ n : ℤ, A⁻¹ ^ n = (A ^ n)⁻¹
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℤ)
Goal: A⁻¹ ^ n = (A ^ n)⁻¹
-- Matrix.zpow_neg_one
@[simp]
theorem zpow_neg_one (A : M) : A ^ (-1 : ℤ) = A⁻¹ 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R)
Goal: A ^ (-1) = A⁻¹
-- Matrix.zpow_neg_natCast
@[simp]
theorem zpow_neg_natCast (A : M) (n : ℕ) : A ^ (-n : ℤ) = (A ^ n)⁻¹ 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℕ)
Goal: A ^ (-↑n) = (A ^ n)⁻¹
-- IsUnit.det_zpow
theorem _root_.IsUnit.det_zpow {A : M} (h : IsUnit A.det) (n : ℤ) : IsUnit (A ^ n).det 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ℤ)
Goal: IsUnit (A ^ n).det
-- Matrix.isUnit_det_zpow_iff
theorem isUnit_det_zpow_iff {A : M} {z : ℤ} : IsUnit (A ^ z).det ↔ IsUnit A.det ∨ z = 0 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} {z : ℤ}
Goal: IsUnit (A ^ z).det ↔ IsUnit A.det ∨ z = 0
-- Matrix.zpow_neg
theorem zpow_neg {A : M} (h : IsUnit A.det) : ∀ n : ℤ, A ^ (-n) = (A ^ n)⁻¹
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ℤ)
Goal: A ^ (-n) = (A ^ n)⁻¹
-- Matrix.inv_zpow'
theorem inv_zpow' {A : M} (h : IsUnit A.det) (n : ℤ) : A⁻¹ ^ n = A ^ (-n) 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ℤ)
Goal: A⁻¹ ^ n = A ^ (-n)
-- Matrix.zpow_add_one
theorem zpow_add_one {A : M} (h : IsUnit A.det) : ∀ n : ℤ, A ^ (n + 1) = A ^ n * A
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ℤ)
Goal: A ^ (n + 1) = A ^ n * A
-- Matrix.zpow_sub_one
theorem zpow_sub_one {A : M} (h : IsUnit A.det) (n : ℤ) : A ^ (n - 1) = A ^ n * A⁻¹ 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ℤ)
Goal: A ^ (n - 1) = A ^ n * A⁻¹
-- Matrix.zpow_add
theorem zpow_add {A : M} (ha : IsUnit A.det) (m n : ℤ) : A ^ (m + n) = A ^ m * A ^ n 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (ha : IsUnit A.det) (m n : ℤ)
Goal: A ^ (m + n) = A ^ m * A ^ n
-- Matrix.zpow_add_of_nonpos
theorem zpow_add_of_nonpos {A : M} {m n : ℤ} (hm : m ≤ 0) (hn : n ≤ 0) :
    A ^ (m + n) = A ^ m * A ^ n 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} {m n : ℤ} (hm : m ≤ 0) (hn : n ≤ 0)
Goal: A ^ (m + n) = A ^ m * A ^ n
-- Matrix.zpow_add_of_nonneg
theorem zpow_add_of_nonneg {A : M} {m n : ℤ} (hm : 0 ≤ m) (hn : 0 ≤ n) :
    A ^ (m + n) = A ^ m * A ^ n 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} {m n : ℤ} (hm : 0 ≤ m) (hn : 0 ≤ n)
Goal: A ^ (m + n) = A ^ m * A ^ n
-- Matrix.zpow_one_add
theorem zpow_one_add {A : M} (h : IsUnit A.det) (i : ℤ) : A ^ (1 + i) = A * A ^ i 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : IsUnit A.det) (i : ℤ)
Goal: A ^ (1 + i) = A * A ^ i
-- Matrix.SemiconjBy.zpow_right
theorem SemiconjBy.zpow_right {A X Y : M} (hx : IsUnit X.det) (hy : IsUnit Y.det)
    (h : SemiconjBy A X Y) : ∀ m : ℤ, SemiconjBy A (X ^ m) (Y ^ m)
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A X Y : Matrix n' n' R} (hx : IsUnit X.det) (hy : IsUnit Y.det) (h : SemiconjBy A X Y) (m : ℤ)
Goal: SemiconjBy A (X ^ m) (Y ^ m)
-- Matrix.Commute.zpow_right
theorem Commute.zpow_right {A B : M} (h : Commute A B) (m : ℤ) : Commute A (B ^ m) 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A B : Matrix n' n' R} (h : Commute A B) (m : ℤ)
Goal: Commute A (B ^ m)
-- Matrix.Commute.zpow_left
theorem Commute.zpow_left {A B : M} (h : Commute A B) (m : ℤ) : Commute (A ^ m) B 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A B : Matrix n' n' R} (h : Commute A B) (m : ℤ)
Goal: Commute (A ^ m) B
-- Matrix.Commute.zpow_zpow
theorem Commute.zpow_zpow {A B : M} (h : Commute A B) (m n : ℤ) : Commute (A ^ m) (B ^ n) 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A B : Matrix n' n' R} (h : Commute A B) (m n : ℤ)
Goal: Commute (A ^ m) (B ^ n)
-- Matrix.Commute.zpow_self
theorem Commute.zpow_self (A : M) (n : ℤ) : Commute (A ^ n) A 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℤ)
Goal: Commute (A ^ n) A
-- Matrix.Commute.self_zpow
theorem Commute.self_zpow (A : M) (n : ℤ) : Commute A (A ^ n) 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℤ)
Goal: Commute A (A ^ n)
-- Matrix.Commute.zpow_zpow_self
theorem Commute.zpow_zpow_self (A : M) (m n : ℤ) : Commute (A ^ m) (A ^ n) 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (m n : ℤ)
Goal: Commute (A ^ m) (A ^ n)
-- Matrix.zpow_add_one_of_ne_neg_one
theorem zpow_add_one_of_ne_neg_one {A : M} : ∀ n : ℤ, n ≠ -1 → A ^ (n + 1) = A ^ n * A
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (n : ℤ)
Goal: n ≠ -1 → A ^ (n + 1) = A ^ n * A
-- Matrix.zpow_mul
theorem zpow_mul (A : M) (h : IsUnit A.det) : ∀ m n : ℤ, A ^ (m * n) = (A ^ m) ^ n
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (h : IsUnit A.det) (m n : ℤ)
Goal: A ^ (m * n) = (A ^ m) ^ n
-- Matrix.zpow_mul'
theorem zpow_mul' (A : M) (h : IsUnit A.det) (m n : ℤ) : A ^ (m * n) = (A ^ n) ^ m 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (h : IsUnit A.det) (m n : ℤ)
Goal: A ^ (m * n) = (A ^ n) ^ m
-- Matrix.coe_units_zpow
@[simp, norm_cast]
theorem coe_units_zpow (u : Mˣ) : ∀ n : ℤ, ((u ^ n : Mˣ) : M) = (u : M) ^ n
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (u : (Matrix n' n' R)ˣ) (n : ℤ)
Goal: ↑(u ^ n) = ↑u ^ n
-- Matrix.zpow_ne_zero_of_isUnit_det
theorem zpow_ne_zero_of_isUnit_det [Nonempty n'] [Nontrivial R] {A : M} (ha : IsUnit A.det)
    (z : ℤ) : A ^ z ≠ 0 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] [Nonempty n'] [Nontrivial R] {A : Matrix n' n' R} (ha : IsUnit A.det) (z : ℤ)
Goal: A ^ z ≠ 0
-- Matrix.zpow_sub
theorem zpow_sub {A : M} (ha : IsUnit A.det) (z1 z2 : ℤ) : A ^ (z1 - z2) = A ^ z1 / A ^ z2 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (ha : IsUnit A.det) (z1 z2 : ℤ)
Goal: A ^ (z1 - z2) = A ^ z1 / A ^ z2
-- Matrix.Commute.mul_zpow
theorem Commute.mul_zpow {A B : M} (h : Commute A B) : ∀ i : ℤ, (A * B) ^ i = A ^ i * B ^ i
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A B : Matrix n' n' R} (h : Commute A B) (i : ℤ)
Goal: (A * B) ^ i = A ^ i * B ^ i
-- Matrix.zpow_neg_mul_zpow_self
theorem zpow_neg_mul_zpow_self (n : ℤ) {A : M} (h : IsUnit A.det) : A ^ (-n) * A ^ n = 1 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (n : ℤ) {A : Matrix n' n' R} (h : IsUnit A.det)
Goal: A ^ (-n) * A ^ n = 1
-- Matrix.one_div_pow
theorem one_div_pow {A : M} (n : ℕ) : (1 / A) ^ n = 1 / A ^ n 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (n : ℕ)
Goal: (1 / A) ^ n = 1 / A ^ n
-- Matrix.one_div_zpow
theorem one_div_zpow {A : M} (n : ℤ) : (1 / A) ^ n = 1 / A ^ n 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (n : ℤ)
Goal: (1 / A) ^ n = 1 / A ^ n
-- Matrix.transpose_zpow
@[simp]
theorem transpose_zpow (A : M) : ∀ n : ℤ, (A ^ n)ᵀ = Aᵀ ^ n
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] (A : Matrix n' n' R) (n : ℤ)
Goal: (A ^ n)ᵀ = Aᵀ ^ n
-- Matrix.conjTranspose_zpow
@[simp]
theorem conjTranspose_zpow [StarRing R] (A : M) : ∀ n : ℤ, (A ^ n)ᴴ = Aᴴ ^ n
  
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] [StarRing R] (A : Matrix n' n' R) (n : ℤ)
Goal: (A ^ n)ᴴ = Aᴴ ^ n
-- Matrix.IsSymm.zpow
theorem IsSymm.zpow {A : M} (h : A.IsSymm) (k : ℤ) :
    (A ^ k).IsSymm 
Arguments: {n' : Type u_1} [DecidableEq n'] [Fintype n'] {R : Type u_2} [CommRing R] {A : Matrix n' n' R} (h : A.IsSymm) (k : ℤ)
Goal: (A ^ k).IsSymm
-- Matrix.dotProduct_stdBasis_eq_mul
@[simp]
theorem dotProduct_stdBasis_eq_mul [DecidableEq n] (v : n → R) (c : R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i c) = v i * c 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] [DecidableEq n] (v : n → R) (c : R) (i : n)
Goal: Matrix.dotProduct v ((LinearMap.stdBasis R (fun x => R) i) c) = v i * c
-- Matrix.dotProduct_stdBasis_one
theorem dotProduct_stdBasis_one [DecidableEq n] (v : n → R) (i : n) :
    dotProduct v (LinearMap.stdBasis R (fun _ => R) i 1) = v i 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] [DecidableEq n] (v : n → R) (i : n)
Goal: Matrix.dotProduct v ((LinearMap.stdBasis R (fun x => R) i) 1) = v i
-- Matrix.dotProduct_eq
theorem dotProduct_eq (v w : n → R) (h : ∀ u, dotProduct v u = dotProduct w u) : v = w 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] (v w : n → R) (h : ∀ (u : n → R), Matrix.dotProduct v u = Matrix.dotProduct w u)
Goal: v = w
-- Matrix.dotProduct_eq_iff
theorem dotProduct_eq_iff {v w : n → R} : (∀ u, dotProduct v u = dotProduct w u) ↔ v = w 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] {v w : n → R}
Goal: (∀ (u : n → R), Matrix.dotProduct v u = Matrix.dotProduct w u) ↔ v = w
-- Matrix.dotProduct_eq_zero
theorem dotProduct_eq_zero (v : n → R) (h : ∀ w, dotProduct v w = 0) : v = 0 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] (v : n → R) (h : ∀ (w : n → R), Matrix.dotProduct v w = 0)
Goal: v = 0
-- Matrix.dotProduct_eq_zero_iff
theorem dotProduct_eq_zero_iff {v : n → R} : (∀ w, dotProduct v w = 0) ↔ v = 0 
Arguments: {n : Type u_2} {R : Type u_4} [Semiring R] [Fintype n] {v : n → R}
Goal: (∀ (w : n → R), Matrix.dotProduct v w = 0) ↔ v = 0
-- Matrix.dotProduct_nonneg_of_nonneg
lemma dotProduct_nonneg_of_nonneg {v w : n → R} (hv : 0 ≤ v) (hw : 0 ≤ w) : 0 ≤ dotProduct v w 
Arguments: {n : Type u_2} {R : Type u_4} [OrderedSemiring R] [Fintype n] {v w : n → R} (hv : 0 ≤ v) (hw : 0 ≤ w)
Goal: 0 ≤ Matrix.dotProduct v w
-- Matrix.dotProduct_le_dotProduct_of_nonneg_right
lemma dotProduct_le_dotProduct_of_nonneg_right {u v w : n → R} (huv : u ≤ v) (hw : 0 ≤ w) :
    dotProduct u w ≤ dotProduct v w 
Arguments: {n : Type u_2} {R : Type u_4} [OrderedSemiring R] [Fintype n] {u v w : n → R} (huv : u ≤ v) (hw : 0 ≤ w)
Goal: Matrix.dotProduct u w ≤ Matrix.dotProduct v w
-- Matrix.dotProduct_le_dotProduct_of_nonneg_left
lemma dotProduct_le_dotProduct_of_nonneg_left {u v w : n → R} (huv : u ≤ v) (hw : 0 ≤ w) :
    dotProduct w u ≤ dotProduct w v 
Arguments: {n : Type u_2} {R : Type u_4} [OrderedSemiring R] [Fintype n] {u v w : n → R} (huv : u ≤ v) (hw : 0 ≤ w)
Goal: Matrix.dotProduct w u ≤ Matrix.dotProduct w v
-- Matrix.dotProduct_self_eq_zero
@[simp]
theorem dotProduct_self_eq_zero [LinearOrderedRing R] {v : n → R} : dotProduct v v = 0 ↔ v = 0 
Arguments: {n : Type u_2} {R : Type u_4} [Fintype n] [LinearOrderedRing R] {v : n → R}
Goal: Matrix.dotProduct v v = 0 ↔ v = 0
-- Matrix.dotProduct_star_self_eq_zero
@[simp]
theorem dotProduct_star_self_eq_zero {v : n → R} : dotProduct (star v) v = 0 ↔ v = 0 
Arguments: {n : Type u_2} {R : Type u_4} [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {v : n → R}
Goal: Matrix.dotProduct (star v) v = 0 ↔ v = 0
-- Matrix.dotProduct_self_star_eq_zero
@[simp]
theorem dotProduct_self_star_eq_zero {v : n → R} : dotProduct v (star v) = 0 ↔ v = 0 
Arguments: {n : Type u_2} {R : Type u_4} [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {v : n → R}
Goal: Matrix.dotProduct v (star v) = 0 ↔ v = 0
-- Matrix.conjTranspose_mul_self_eq_zero
@[simp]
lemma conjTranspose_mul_self_eq_zero {A : Matrix m n R} : Aᴴ * A = 0 ↔ A = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype m] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {A : Matrix m n R}
Goal: A.conjTranspose * A = 0 ↔ A = 0
-- Matrix.self_mul_conjTranspose_eq_zero
@[simp]
lemma self_mul_conjTranspose_eq_zero {A : Matrix m n R} : A * Aᴴ = 0 ↔ A = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {A : Matrix m n R}
Goal: A * A.conjTranspose = 0 ↔ A = 0
-- Matrix.conjTranspose_mul_self_mul_eq_zero
lemma conjTranspose_mul_self_mul_eq_zero (A : Matrix m n R) (B : Matrix n p R) :
    (Aᴴ * A) * B = 0 ↔ A * B = 0 
Arguments: {m : Type u_1} {n : Type u_2} {p : Type u_3} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (B : Matrix n p R)
Goal: A.conjTranspose * A * B = 0 ↔ A * B = 0
-- Matrix.self_mul_conjTranspose_mul_eq_zero
lemma self_mul_conjTranspose_mul_eq_zero (A : Matrix m n R) (B : Matrix m p R) :
    (A * Aᴴ) * B = 0 ↔ Aᴴ * B = 0 
Arguments: {m : Type u_1} {n : Type u_2} {p : Type u_3} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (B : Matrix m p R)
Goal: A * A.conjTranspose * B = 0 ↔ A.conjTranspose * B = 0
-- Matrix.mul_self_mul_conjTranspose_eq_zero
lemma mul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (B : Matrix p m R) :
    B * (A * Aᴴ) = 0 ↔ B * A = 0 
Arguments: {m : Type u_1} {n : Type u_2} {p : Type u_3} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (B : Matrix p m R)
Goal: B * (A * A.conjTranspose) = 0 ↔ B * A = 0
-- Matrix.mul_conjTranspose_mul_self_eq_zero
lemma mul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (B : Matrix p n R) :
    B * (Aᴴ * A) = 0 ↔ B * Aᴴ = 0 
Arguments: {m : Type u_1} {n : Type u_2} {p : Type u_3} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (B : Matrix p n R)
Goal: B * (A.conjTranspose * A) = 0 ↔ B * A.conjTranspose = 0
-- Matrix.conjTranspose_mul_self_mulVec_eq_zero
lemma conjTranspose_mul_self_mulVec_eq_zero (A : Matrix m n R) (v : n → R) :
    (Aᴴ * A) *ᵥ v = 0 ↔ A *ᵥ v = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (v : n → R)
Goal: (A.conjTranspose * A).mulVec v = 0 ↔ A.mulVec v = 0
-- Matrix.self_mul_conjTranspose_mulVec_eq_zero
lemma self_mul_conjTranspose_mulVec_eq_zero (A : Matrix m n R) (v : m → R) :
    (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (v : m → R)
Goal: (A * A.conjTranspose).mulVec v = 0 ↔ A.conjTranspose.mulVec v = 0
-- Matrix.vecMul_conjTranspose_mul_self_eq_zero
lemma vecMul_conjTranspose_mul_self_eq_zero (A : Matrix m n R) (v : n → R) :
    v ᵥ* (Aᴴ * A) = 0 ↔ v ᵥ* Aᴴ = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (v : n → R)
Goal: Matrix.vecMul v (A.conjTranspose * A) = 0 ↔ Matrix.vecMul v A.conjTranspose = 0
-- Matrix.vecMul_self_mul_conjTranspose_eq_zero
lemma vecMul_self_mul_conjTranspose_eq_zero (A : Matrix m n R) (v : m → R) :
    v ᵥ* (A * Aᴴ) = 0 ↔ v ᵥ* A = 0 
Arguments: {m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype m] [Fintype n] [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (v : m → R)
Goal: Matrix.vecMul v (A * A.conjTranspose) = 0 ↔ Matrix.vecMul v A = 0
-- Basis.toMatrix_apply
theorem toMatrix_apply : e.toMatrix v i j = e.repr (v j) i 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_5} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (e : Basis ι R M) (v : ι' → M) (i : ι) (j : ι')
Goal: e.toMatrix v i j = (e.repr (v j)) i
-- Basis.toMatrix_transpose_apply
theorem toMatrix_transpose_apply : (e.toMatrix v)ᵀ j = e.repr (v j) 
Arguments: {ι : Type u_1} {ι' : Type u_2} {R : Type u_5} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (e : Basis ι R M) (v : ι' → M) (j : ι')
Goal: (e.toMatrix v)ᵀ j = ⇑(e.repr (v j))
-- Basis.toMatrix_eq_toMatrix_constr
theorem toMatrix_eq_toMatrix_constr [Fintype ι] [DecidableEq ι] (v : ι → M) :
    e.toMatrix v = LinearMap.toMatrix e e (e.constr ℕ v) 
Arguments: {ι : Type u_1} {R : Type u_5} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (e : Basis ι R M) [Fintype ι] [DecidableEq ι] (v : ι → M)
Goal: e.toMatrix v = (LinearMap.toMatrix e e) ((e.constr ℕ) v)
-- Basis.coePiBasisFun.toMatrix_eq_transpose
theorem coePiBasisFun.toMatrix_eq_transpose [Finite ι] :
    ((Pi.basisFun R ι).toMatrix : Matrix ι ι R → Matrix ι ι R) = Matrix.transpose 
Arguments: {ι : Type u_1} {R : Type u_5} [CommSemiring R] [Finite ι]
Goal: (Pi.basisFun R ι).toMatrix = Matrix.transpose
-- Basis.toMatrix_self
@[simp]
theorem toMatrix_self [DecidableEq ι] : e.toMatrix e = 1 
Arguments: {ι : Type u_1} {R : Type u_5} {M : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] (e : Basis ι R M) [DecidableEq ι]
Goal: e.toMatrix ⇑e = 1