ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- BilinForm.toMatrix_mul
theorem BilinForm.toMatrix_mul (B : BilinForm R₂ M₂) (M : Matrix n n R₂) :
    BilinForm.toMatrix b B * M = BilinForm.toMatrix b (B.compRight (Matrix.toLin b b M)) 
Arguments: {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} [Fintype n] [DecidableEq n] (b : Basis n R₂ M₂) (B : LinearMap.BilinForm R₂ M₂) (M : Matrix n n R₂)
Goal: (BilinForm.toMatrix b) B * M = (BilinForm.toMatrix b) (B.compRight ((Matrix.toLin b b) M))
-- Matrix.toBilin_comp
theorem Matrix.toBilin_comp (M : Matrix n n R₂) (P Q : Matrix n o R₂) :
    (Matrix.toBilin b M).comp (toLin c b P) (toLin c b Q) = Matrix.toBilin c (Pᵀ * M * Q) 
Arguments: {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {n : Type u_11} {o : Type u_12} [Fintype n] [Fintype o] [DecidableEq n] (b : Basis n R₂ M₂) {M₂' : Type u_13} [AddCommMonoid M₂'] [Module R₂ M₂'] (c : Basis o R₂ M₂') [DecidableEq o] (M : Matrix n n R₂) (P Q : Matrix n o R₂)
Goal: ((Matrix.toBilin b) M).comp ((Matrix.toLin c b) P) ((Matrix.toLin c b) Q) = (Matrix.toBilin c) (P.transpose * M * Q)
-- isAdjointPair_toBilin'
@[simp]
theorem isAdjointPair_toBilin' [DecidableEq n] :
    BilinForm.IsAdjointPair (Matrix.toBilin' J) (Matrix.toBilin' J₃) (Matrix.toLin' A)
        (Matrix.toLin' A') ↔
      Matrix.IsAdjointPair J J₃ A A' 
Arguments: {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J J₃ A A' : Matrix n n R₃) [DecidableEq n]
Goal: (Matrix.toBilin' J).IsAdjointPair (Matrix.toBilin' J₃) (Matrix.toLin' A) (Matrix.toLin' A') ↔ J.IsAdjointPair J₃ A A'
-- isAdjointPair_toBilin
@[simp]
theorem isAdjointPair_toBilin [DecidableEq n] :
    BilinForm.IsAdjointPair (Matrix.toBilin b J) (Matrix.toBilin b J₃) (Matrix.toLin b b A)
        (Matrix.toLin b b A') ↔
      Matrix.IsAdjointPair J J₃ A A' 
Arguments: {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {n : Type u_11} [Fintype n] (b : Basis n R₃ M₃) (J J₃ A A' : Matrix n n R₃) [DecidableEq n]
Goal: ((Matrix.toBilin b) J).IsAdjointPair ((Matrix.toBilin b) J₃) ((Matrix.toLin b b) A) ((Matrix.toLin b b) A') ↔
  J.IsAdjointPair J₃ A A'
-- Matrix.isAdjointPair_equiv'
theorem Matrix.isAdjointPair_equiv' [DecidableEq n] (P : Matrix n n R₃) (h : IsUnit P) :
    (Pᵀ * J * P).IsAdjointPair (Pᵀ * J * P) A A' ↔
      J.IsAdjointPair J (P * A * P⁻¹) (P * A' * P⁻¹) 
Arguments: {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J A A' : Matrix n n R₃) [DecidableEq n] (P : Matrix n n R₃) (h : IsUnit P)
Goal: (P.transpose * J * P).IsAdjointPair (P.transpose * J * P) A A' ↔ J.IsAdjointPair J (P * A * P⁻¹) (P * A' * P⁻¹)
-- mem_pairSelfAdjointMatricesSubmodule'
theorem mem_pairSelfAdjointMatricesSubmodule' :
    A ∈ pairSelfAdjointMatricesSubmodule J J₃ ↔ Matrix.IsAdjointPair J J₃ A A 
Arguments: {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J J₃ A : Matrix n n R₃) [DecidableEq n]
Goal: A ∈ pairSelfAdjointMatricesSubmodule J J₃ ↔ J.IsAdjointPair J₃ A A
-- mem_selfAdjointMatricesSubmodule'
theorem mem_selfAdjointMatricesSubmodule' :
    A ∈ selfAdjointMatricesSubmodule J ↔ J.IsSelfAdjoint A 
Arguments: {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J A : Matrix n n R₃) [DecidableEq n]
Goal: A ∈ selfAdjointMatricesSubmodule J ↔ J.IsSelfAdjoint A
-- mem_skewAdjointMatricesSubmodule'
theorem mem_skewAdjointMatricesSubmodule' :
    A ∈ skewAdjointMatricesSubmodule J ↔ J.IsSkewAdjoint A 
Arguments: {R₃ : Type u_7} [CommRing R₃] {n : Type u_11} [Fintype n] (J A : Matrix n n R₃) [DecidableEq n]
Goal: A ∈ skewAdjointMatricesSubmodule J ↔ J.IsSkewAdjoint A
-- Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin
theorem _root_.Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin {M : Matrix ι ι R₂}
    (b : Basis ι R₂ M₂) : M.toBilin'.Nondegenerate ↔ (Matrix.toBilin b M).Nondegenerate 
Arguments: {R₂ : Type u_5} {M₂ : Type u_6} [CommSemiring R₂] [AddCommMonoid M₂] [Module R₂ M₂] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₂} (b : Basis ι R₂ M₂)
Goal: (Matrix.toBilin' M).Nondegenerate ↔ ((Matrix.toBilin b) M).Nondegenerate
-- Matrix.Nondegenerate.toBilin'
theorem _root_.Matrix.Nondegenerate.toBilin' {M : Matrix ι ι R₃} (h : M.Nondegenerate) :
    M.toBilin'.Nondegenerate 
Arguments: {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (h : M.Nondegenerate)
Goal: (Matrix.toBilin' M).Nondegenerate
-- Matrix.nondegenerate_toBilin'_iff
@[simp]
theorem _root_.Matrix.nondegenerate_toBilin'_iff {M : Matrix ι ι R₃} :
    M.toBilin'.Nondegenerate ↔ M.Nondegenerate 
Arguments: {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃}
Goal: (Matrix.toBilin' M).Nondegenerate ↔ M.Nondegenerate
-- Matrix.Nondegenerate.toBilin
theorem _root_.Matrix.Nondegenerate.toBilin {M : Matrix ι ι R₃} (h : M.Nondegenerate)
    (b : Basis ι R₃ M₃) : (Matrix.toBilin b M).Nondegenerate 
Arguments: {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (h : M.Nondegenerate) (b : Basis ι R₃ M₃)
Goal: ((Matrix.toBilin b) M).Nondegenerate
-- Matrix.nondegenerate_toBilin_iff
@[simp]
theorem _root_.Matrix.nondegenerate_toBilin_iff {M : Matrix ι ι R₃} (b : Basis ι R₃ M₃) :
    (Matrix.toBilin b M).Nondegenerate ↔ M.Nondegenerate 
Arguments: {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι R₃} (b : Basis ι R₃ M₃)
Goal: ((Matrix.toBilin b) M).Nondegenerate ↔ M.Nondegenerate
-- LinearMap.BilinForm.nondegenerate_toMatrix'_iff
@[simp]
theorem nondegenerate_toMatrix'_iff {B : BilinForm R₃ (ι → R₃)} :
    B.toMatrix'.Nondegenerate (m := ι) ↔ B.Nondegenerate 
Arguments: {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : LinearMap.BilinForm R₃ (ι → R₃)}
Goal: (LinearMap.BilinForm.toMatrix' B).Nondegenerate ↔ B.Nondegenerate
-- LinearMap.BilinForm.Nondegenerate.toMatrix'
theorem Nondegenerate.toMatrix' {B : BilinForm R₃ (ι → R₃)} (h : B.Nondegenerate) :
    B.toMatrix'.Nondegenerate 
Arguments: {R₃ : Type u_7} [CommRing R₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : LinearMap.BilinForm R₃ (ι → R₃)} (h : B.Nondegenerate)
Goal: (LinearMap.BilinForm.toMatrix' B).Nondegenerate
-- LinearMap.BilinForm.nondegenerate_toMatrix_iff
@[simp]
theorem nondegenerate_toMatrix_iff {B : BilinForm R₃ M₃} (b : Basis ι R₃ M₃) :
    (BilinForm.toMatrix b B).Nondegenerate ↔ B.Nondegenerate 
Arguments: {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : LinearMap.BilinForm R₃ M₃} (b : Basis ι R₃ M₃)
Goal: ((BilinForm.toMatrix b) B).Nondegenerate ↔ B.Nondegenerate
-- LinearMap.BilinForm.Nondegenerate.toMatrix
theorem Nondegenerate.toMatrix {B : BilinForm R₃ M₃} (h : B.Nondegenerate) (b : Basis ι R₃ M₃) :
    (BilinForm.toMatrix b B).Nondegenerate 
Arguments: {R₃ : Type u_7} {M₃ : Type u_8} [CommRing R₃] [AddCommGroup M₃] [Module R₃ M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : LinearMap.BilinForm R₃ M₃} (h : B.Nondegenerate) (b : Basis ι R₃ M₃)
Goal: ((BilinForm.toMatrix b) B).Nondegenerate
-- LinearMap.BilinForm.nondegenerate_toBilin'_iff_det_ne_zero
theorem nondegenerate_toBilin'_iff_det_ne_zero {M : Matrix ι ι A} :
    M.toBilin'.Nondegenerate ↔ M.det ≠ 0 
Arguments: {A : Type u_11} [CommRing A] [IsDomain A] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {M : Matrix ι ι A}
Goal: (Matrix.toBilin' M).Nondegenerate ↔ M.det ≠ 0
-- LinearMap.BilinForm.nondegenerate_toBilin'_of_det_ne_zero'
theorem nondegenerate_toBilin'_of_det_ne_zero' (M : Matrix ι ι A) (h : M.det ≠ 0) :
    M.toBilin'.Nondegenerate 
Arguments: {A : Type u_11} [CommRing A] [IsDomain A] {ι : Type u_12} [DecidableEq ι] [Fintype ι] (M : Matrix ι ι A) (h : M.det ≠ 0)
Goal: (Matrix.toBilin' M).Nondegenerate
-- LinearMap.BilinForm.nondegenerate_iff_det_ne_zero
theorem nondegenerate_iff_det_ne_zero {B : BilinForm A M₃} (b : Basis ι A M₃) :
    B.Nondegenerate ↔ (BilinForm.toMatrix b B).det ≠ 0 
Arguments: {M₃ : Type u_8} [AddCommGroup M₃] {A : Type u_11} [CommRing A] [IsDomain A] [Module A M₃] {ι : Type u_12} [DecidableEq ι] [Fintype ι] {B : LinearMap.BilinForm A M₃} (b : Basis ι A M₃)
Goal: B.Nondegenerate ↔ ((BilinForm.toMatrix b) B).det ≠ 0
-- LinearMap.BilinForm.nondegenerate_of_det_ne_zero
theorem nondegenerate_of_det_ne_zero (b : Basis ι A M₃) (h : (BilinForm.toMatrix b B₃).det ≠ 0) :
    B₃.Nondegenerate 
Arguments: {M₃ : Type u_8} [AddCommGroup M₃] {A : Type u_11} [CommRing A] [IsDomain A] [Module A M₃] (B₃ : LinearMap.BilinForm A M₃) {ι : Type u_12} [DecidableEq ι] [Fintype ι] (b : Basis ι A M₃) (h : ((BilinForm.toMatrix b) B₃).det ≠ 0)
Goal: B₃.Nondegenerate
-- affineIndependent_def
theorem affineIndependent_def (p : ι → P) :
    AffineIndependent k p ↔
      ∀ (s : Finset ι) (w : ι → k),
        ∑ i ∈ s, w i = 0 → s.weightedVSub p w = (0 : V) → ∀ i ∈ s, w i = 0 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (p : ι → P)
Goal: AffineIndependent k p ↔ ∀ (s : Finset ι) (w : ι → k), ∑ i ∈ s, w i = 0 → (s.weightedVSub p) w = 0 → ∀ i ∈ s, w i = 0
-- affineIndependent_of_subsingleton
theorem affineIndependent_of_subsingleton [Subsingleton ι] (p : ι → P) : AffineIndependent k p 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Subsingleton ι] (p : ι → P)
Goal: AffineIndependent k p
-- affineIndependent_iff_of_fintype
theorem affineIndependent_iff_of_fintype [Fintype ι] (p : ι → P) :
    AffineIndependent k p ↔
      ∀ w : ι → k, ∑ i, w i = 0 → Finset.univ.weightedVSub p w = (0 : V) → ∀ i, w i = 0 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Fintype ι] (p : ι → P)
Goal: AffineIndependent k p ↔ ∀ (w : ι → k), ∑ i : ι, w i = 0 → (Finset.univ.weightedVSub p) w = 0 → ∀ (i : ι), w i = 0
-- affineIndependent_vadd
@[simp] lemma affineIndependent_vadd {p : ι → P} {v : V} :
    AffineIndependent k (v +ᵥ p) ↔ AffineIndependent k p 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} {v : V}
Goal: AffineIndependent k (v +ᵥ p) ↔ AffineIndependent k p
-- affineIndependent_iff_linearIndependent_vsub
theorem affineIndependent_iff_linearIndependent_vsub (p : ι → P) (i1 : ι) :
    AffineIndependent k p ↔ LinearIndependent k fun i : { x // x ≠ i1 } => (p i -ᵥ p i1 : V) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (p : ι → P) (i1 : ι)
Goal: AffineIndependent k p ↔ LinearIndependent k fun i => p ↑i -ᵥ p i1
-- affineIndependent_set_iff_linearIndependent_vsub
theorem affineIndependent_set_iff_linearIndependent_vsub {s : Set P} {p₁ : P} (hp₁ : p₁ ∈ s) :
    AffineIndependent k (fun p => p : s → P) ↔
      LinearIndependent k (fun v => v : (fun p => (p -ᵥ p₁ : V)) '' (s \ {p₁}) → V) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} {p₁ : P} (hp₁ : p₁ ∈ s)
Goal: (AffineIndependent k fun p => ↑p) ↔ LinearIndependent k fun (v : ↑((fun p => p -ᵥ p₁) '' (s \ {p₁}))) => ↑v
-- linearIndependent_set_iff_affineIndependent_vadd_union_singleton
theorem linearIndependent_set_iff_affineIndependent_vadd_union_singleton {s : Set V}
    (hs : ∀ v ∈ s, v ≠ (0 : V)) (p₁ : P) : LinearIndependent k (fun v => v : s → V) ↔
    AffineIndependent k (fun p => p : ({p₁} ∪ (fun v => v +ᵥ p₁) '' s : Set P) → P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set V} (hs : ∀ v ∈ s, v ≠ 0) (p₁ : P)
Goal: (LinearIndependent k fun (v : ↑s) => ↑v) ↔ AffineIndependent k fun p => ↑p
-- affineIndependent_iff_indicator_eq_of_affineCombination_eq
theorem affineIndependent_iff_indicator_eq_of_affineCombination_eq (p : ι → P) :
    AffineIndependent k p ↔
      ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
        ∑ i ∈ s1, w1 i = 1 →
          ∑ i ∈ s2, w2 i = 1 →
            s1.affineCombination k p w1 = s2.affineCombination k p w2 →
              Set.indicator (↑s1) w1 = Set.indicator (↑s2) w2 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (p : ι → P)
Goal: AffineIndependent k p ↔
  ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k),
    ∑ i ∈ s1, w1 i = 1 →
      ∑ i ∈ s2, w2 i = 1 →
        (Finset.affineCombination k s1 p) w1 = (Finset.affineCombination k s2 p) w2 →
          (↑s1).indicator w1 = (↑s2).indicator w2
-- affineIndependent_iff_eq_of_fintype_affineCombination_eq
theorem affineIndependent_iff_eq_of_fintype_affineCombination_eq [Fintype ι] (p : ι → P) :
    AffineIndependent k p ↔ ∀ w1 w2 : ι → k, ∑ i, w1 i = 1 → ∑ i, w2 i = 1 →
    Finset.univ.affineCombination k p w1 = Finset.univ.affineCombination k p w2 → w1 = w2 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Fintype ι] (p : ι → P)
Goal: AffineIndependent k p ↔
  ∀ (w1 w2 : ι → k),
    ∑ i : ι, w1 i = 1 →
      ∑ i : ι, w2 i = 1 →
        (Finset.affineCombination k Finset.univ p) w1 = (Finset.affineCombination k Finset.univ p) w2 → w1 = w2
-- AffineIndependent.units_lineMap
theorem AffineIndependent.units_lineMap {p : ι → P} (hp : AffineIndependent k p) (j : ι)
    (w : ι → Units k) : AffineIndependent k fun i => AffineMap.lineMap (p j) (p i) (w i : k) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} (hp : AffineIndependent k p) (j : ι) (w : ι → kˣ)
Goal: AffineIndependent k fun i => (AffineMap.lineMap (p j) (p i)) ↑(w i)
-- AffineIndependent.indicator_eq_of_affineCombination_eq
theorem AffineIndependent.indicator_eq_of_affineCombination_eq {p : ι → P}
    (ha : AffineIndependent k p) (s₁ s₂ : Finset ι) (w₁ w₂ : ι → k) (hw₁ : ∑ i ∈ s₁, w₁ i = 1)
    (hw₂ : ∑ i ∈ s₂, w₂ i = 1) (h : s₁.affineCombination k p w₁ = s₂.affineCombination k p w₂) :
    Set.indicator (↑s₁) w₁ = Set.indicator (↑s₂) w₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} (ha : AffineIndependent k p) (s₁ s₂ : Finset ι) (w₁ w₂ : ι → k) (hw₁ : ∑ i ∈ s₁, w₁ i = 1) (hw₂ : ∑ i ∈ s₂, w₂ i = 1) (h : (Finset.affineCombination k s₁ p) w₁ = (Finset.affineCombination k s₂ p) w₂)
Goal: (↑s₁).indicator w₁ = (↑s₂).indicator w₂
-- AffineIndependent.injective
protected theorem AffineIndependent.injective [Nontrivial k] {p : ι → P}
    (ha : AffineIndependent k p) : Function.Injective p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Nontrivial k] {p : ι → P} (ha : AffineIndependent k p)
Goal: Function.Injective p
-- AffineIndependent.comp_embedding
theorem AffineIndependent.comp_embedding {ι2 : Type*} (f : ι2 ↪ ι) {p : ι → P}
    (ha : AffineIndependent k p) : AffineIndependent k (p ∘ f) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {ι2 : Type u_5} (f : ι2 ↪ ι) {p : ι → P} (ha : AffineIndependent k p)
Goal: AffineIndependent k (p ∘ ⇑f)
-- AffineIndependent.subtype
protected theorem AffineIndependent.subtype {p : ι → P} (ha : AffineIndependent k p) (s : Set ι) :
    AffineIndependent k fun i : s => p i 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} (ha : AffineIndependent k p) (s : Set ι)
Goal: AffineIndependent k fun i => p ↑i
-- AffineIndependent.range
protected theorem AffineIndependent.range {p : ι → P} (ha : AffineIndependent k p) :
    AffineIndependent k (fun x => x : Set.range p → P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} (ha : AffineIndependent k p)
Goal: AffineIndependent k fun x => ↑x
-- affineIndependent_equiv
theorem affineIndependent_equiv {ι' : Type*} (e : ι ≃ ι') {p : ι' → P} :
    AffineIndependent k (p ∘ e) ↔ AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {ι' : Type u_5} (e : ι ≃ ι') {p : ι' → P}
Goal: AffineIndependent k (p ∘ ⇑e) ↔ AffineIndependent k p
-- AffineIndependent.mono
protected theorem AffineIndependent.mono {s t : Set P}
    (ha : AffineIndependent k (fun x => x : t → P)) (hs : s ⊆ t) :
    AffineIndependent k (fun x => x : s → P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s t : Set P} (ha : AffineIndependent k fun x => ↑x) (hs : s ⊆ t)
Goal: AffineIndependent k fun x => ↑x
-- AffineIndependent.of_set_of_injective
theorem AffineIndependent.of_set_of_injective {p : ι → P}
    (ha : AffineIndependent k (fun x => x : Set.range p → P)) (hi : Function.Injective p) :
    AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {p : ι → P} (ha : AffineIndependent k fun x => ↑x) (hi : Function.Injective p)
Goal: AffineIndependent k p
-- AffineIndependent.of_comp
theorem AffineIndependent.of_comp {p : ι → P} (f : P →ᵃ[k] P₂) (hai : AffineIndependent k (f ∘ p)) :
    AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {p : ι → P} (f : P →ᵃ[k] P₂) (hai : AffineIndependent k (⇑f ∘ p))
Goal: AffineIndependent k p
-- AffineIndependent.map'
theorem AffineIndependent.map' {p : ι → P} (hai : AffineIndependent k p) (f : P →ᵃ[k] P₂)
    (hf : Function.Injective f) : AffineIndependent k (f ∘ p) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {p : ι → P} (hai : AffineIndependent k p) (f : P →ᵃ[k] P₂) (hf : Function.Injective ⇑f)
Goal: AffineIndependent k (⇑f ∘ p)
-- AffineMap.affineIndependent_iff
theorem AffineMap.affineIndependent_iff {p : ι → P} (f : P →ᵃ[k] P₂) (hf : Function.Injective f) :
    AffineIndependent k (f ∘ p) ↔ AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {p : ι → P} (f : P →ᵃ[k] P₂) (hf : Function.Injective ⇑f)
Goal: AffineIndependent k (⇑f ∘ p) ↔ AffineIndependent k p
-- AffineEquiv.affineIndependent_iff
theorem AffineEquiv.affineIndependent_iff {p : ι → P} (e : P ≃ᵃ[k] P₂) :
    AffineIndependent k (e ∘ p) ↔ AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {p : ι → P} (e : P ≃ᵃ[k] P₂)
Goal: AffineIndependent k (⇑e ∘ p) ↔ AffineIndependent k p
-- AffineEquiv.affineIndependent_set_of_eq_iff
theorem AffineEquiv.affineIndependent_set_of_eq_iff {s : Set P} (e : P ≃ᵃ[k] P₂) :
    AffineIndependent k ((↑) : e '' s → P₂) ↔ AffineIndependent k ((↑) : s → P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {V₂ : Type u_5} {P₂ : Type u_6} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] {s : Set P} (e : P ≃ᵃ[k] P₂)
Goal: AffineIndependent k Subtype.val ↔ AffineIndependent k Subtype.val
-- AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan
theorem AffineIndependent.exists_mem_inter_of_exists_mem_inter_affineSpan [Nontrivial k] {p : ι → P}
    (ha : AffineIndependent k p) {s1 s2 : Set ι} {p0 : P} (hp0s1 : p0 ∈ affineSpan k (p '' s1))
    (hp0s2 : p0 ∈ affineSpan k (p '' s2)) : ∃ i : ι, i ∈ s1 ∩ s2 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Nontrivial k] {p : ι → P} (ha : AffineIndependent k p) {s1 s2 : Set ι} {p0 : P} (hp0s1 : p0 ∈ affineSpan k (p '' s1)) (hp0s2 : p0 ∈ affineSpan k (p '' s2))
Goal: ∃ i, i ∈ s1 ∩ s2
-- AffineIndependent.affineSpan_disjoint_of_disjoint
theorem AffineIndependent.affineSpan_disjoint_of_disjoint [Nontrivial k] {p : ι → P}
    (ha : AffineIndependent k p) {s1 s2 : Set ι} (hd : Disjoint s1 s2) :
    Disjoint (affineSpan k (p '' s1) : Set P) (affineSpan k (p '' s2)) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Nontrivial k] {p : ι → P} (ha : AffineIndependent k p) {s1 s2 : Set ι} (hd : Disjoint s1 s2)
Goal: Disjoint ↑(affineSpan k (p '' s1)) ↑(affineSpan k (p '' s2))
-- AffineIndependent.mem_affineSpan_iff
@[simp]
protected theorem AffineIndependent.mem_affineSpan_iff [Nontrivial k] {p : ι → P}
    (ha : AffineIndependent k p) (i : ι) (s : Set ι) : p i ∈ affineSpan k (p '' s) ↔ i ∈ s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Nontrivial k] {p : ι → P} (ha : AffineIndependent k p) (i : ι) (s : Set ι)
Goal: p i ∈ affineSpan k (p '' s) ↔ i ∈ s
-- AffineIndependent.not_mem_affineSpan_diff
theorem AffineIndependent.not_mem_affineSpan_diff [Nontrivial k] {p : ι → P}
    (ha : AffineIndependent k p) (i : ι) (s : Set ι) : p i ∉ affineSpan k (p '' (s \ {i})) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [Nontrivial k] {p : ι → P} (ha : AffineIndependent k p) (i : ι) (s : Set ι)
Goal: p i ∉ affineSpan k (p '' (s \ {i}))
-- exists_nontrivial_relation_sum_zero_of_not_affine_ind
theorem exists_nontrivial_relation_sum_zero_of_not_affine_ind {t : Finset V}
    (h : ¬AffineIndependent k ((↑) : t → V)) :
    ∃ f : V → k, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {t : Finset V} (h : ¬AffineIndependent k Subtype.val)
Goal: ∃ f, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, f x ≠ 0
-- affineIndependent_iff
theorem affineIndependent_iff {ι} {p : ι → V} :
    AffineIndependent k p ↔
      ∀ (s : Finset ι) (w : ι → k), s.sum w = 0 → ∑ e ∈ s, w e • p e = 0 → ∀ e ∈ s, w e = 0 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {ι : Type u_5} {p : ι → V}
Goal: AffineIndependent k p ↔ ∀ (s : Finset ι) (w : ι → k), s.sum w = 0 → ∑ e ∈ s, w e • p e = 0 → ∀ e ∈ s, w e = 0
-- AffineIndependent.eq_zero_of_sum_eq_zero
lemma AffineIndependent.eq_zero_of_sum_eq_zero (hp : AffineIndependent k p)
    (hw₀ : ∑ i ∈ s, w i = 0) (hw₁ : ∑ i ∈ s, w i • p i = 0) : ∀ i ∈ s, w i = 0 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {ι : Type u_4} {s : Finset ι} {w : ι → k} {p : ι → V} (hp : AffineIndependent k p) (hw₀ : ∑ i ∈ s, w i = 0) (hw₁ : ∑ i ∈ s, w i • p i = 0) (i : ι)
Goal: i ∈ s → w i = 0
-- AffineIndependent.eq_of_sum_eq_sum
lemma AffineIndependent.eq_of_sum_eq_sum (hp : AffineIndependent k p)
    (hw : ∑ i ∈ s, w₁ i = ∑ i ∈ s, w₂ i) (hwp : ∑ i ∈ s, w₁ i • p i = ∑ i ∈ s, w₂ i • p i) :
    ∀ i ∈ s, w₁ i = w₂ i 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {ι : Type u_4} {s : Finset ι} {w₁ w₂ : ι → k} {p : ι → V} (hp : AffineIndependent k p) (hw : ∑ i ∈ s, w₁ i = ∑ i ∈ s, w₂ i) (hwp : ∑ i ∈ s, w₁ i • p i = ∑ i ∈ s, w₂ i • p i) (i : ι)
Goal: i ∈ s → w₁ i = w₂ i
-- AffineIndependent.eq_zero_of_sum_eq_zero_subtype
lemma AffineIndependent.eq_zero_of_sum_eq_zero_subtype {s : Finset V}
    (hp : AffineIndependent k ((↑) : s → V)) {w : V → k} (hw₀ : ∑ x ∈ s, w x = 0)
    (hw₁ : ∑ x ∈ s, w x • x = 0) : ∀ x ∈ s, w x = 0 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {s : Finset V} (hp : AffineIndependent k Subtype.val) {w : V → k} (hw₀ : ∑ x ∈ s, w x = 0) (hw₁ : ∑ x ∈ s, w x • x = 0) (x : V)
Goal: x ∈ s → w x = 0