ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineIndependent.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one
theorem AffineIndependent.vectorSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one
    [DecidableEq P] {p : ι → P}
    (hi : AffineIndependent k p) {s : Finset ι} {sm : Submodule k V} [FiniteDimensional k sm]
    (hle : vectorSpan k (s.image p : Set P) ≤ sm) (hc : Finset.card s = finrank k sm + 1) :
    vectorSpan k (s.image p : Set P) = sm 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [DecidableEq P] {p : ι → P} (hi : AffineIndependent k p) {s : Finset ι} {sm : Submodule k V} [FiniteDimensional k ↥sm] (hle : vectorSpan k ↑(Finset.image p s) ≤ sm) (hc : s.card = FiniteDimensional.finrank k ↥sm + 1)
Goal: vectorSpan k ↑(Finset.image p s) = sm
-- AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one
theorem AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one [Fintype ι] {p : ι → P}
    (hi : AffineIndependent k p) {sm : Submodule k V} [FiniteDimensional k sm]
    (hle : vectorSpan k (Set.range p) ≤ sm) (hc : Fintype.card ι = finrank k sm + 1) :
    vectorSpan k (Set.range p) = sm 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [Fintype ι] {p : ι → P} (hi : AffineIndependent k p) {sm : Submodule k V} [FiniteDimensional k ↥sm] (hle : vectorSpan k (Set.range p) ≤ sm) (hc : Fintype.card ι = FiniteDimensional.finrank k ↥sm + 1)
Goal: vectorSpan k (Set.range p) = sm
-- AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one
theorem AffineIndependent.affineSpan_image_finset_eq_of_le_of_card_eq_finrank_add_one
    [DecidableEq P] {p : ι → P}
    (hi : AffineIndependent k p) {s : Finset ι} {sp : AffineSubspace k P}
    [FiniteDimensional k sp.direction] (hle : affineSpan k (s.image p : Set P) ≤ sp)
    (hc : Finset.card s = finrank k sp.direction + 1) : affineSpan k (s.image p : Set P) = sp 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [DecidableEq P] {p : ι → P} (hi : AffineIndependent k p) {s : Finset ι} {sp : AffineSubspace k P} [FiniteDimensional k ↥sp.direction] (hle : affineSpan k ↑(Finset.image p s) ≤ sp) (hc : s.card = FiniteDimensional.finrank k ↥sp.direction + 1)
Goal: affineSpan k ↑(Finset.image p s) = sp
-- AffineIndependent.affineSpan_eq_of_le_of_card_eq_finrank_add_one
theorem AffineIndependent.affineSpan_eq_of_le_of_card_eq_finrank_add_one [Fintype ι] {p : ι → P}
    (hi : AffineIndependent k p) {sp : AffineSubspace k P} [FiniteDimensional k sp.direction]
    (hle : affineSpan k (Set.range p) ≤ sp) (hc : Fintype.card ι = finrank k sp.direction + 1) :
    affineSpan k (Set.range p) = sp 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [Fintype ι] {p : ι → P} (hi : AffineIndependent k p) {sp : AffineSubspace k P} [FiniteDimensional k ↥sp.direction] (hle : affineSpan k (Set.range p) ≤ sp) (hc : Fintype.card ι = FiniteDimensional.finrank k ↥sp.direction + 1)
Goal: affineSpan k (Set.range p) = sp
-- AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one
theorem AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one [FiniteDimensional k V]
    [Fintype ι] {p : ι → P} (hi : AffineIndependent k p) :
    affineSpan k (Set.range p) = ⊤ ↔ Fintype.card ι = finrank k V + 1 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} {ι : Type u_4} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [FiniteDimensional k V] [Fintype ι] {p : ι → P} (hi : AffineIndependent k p)
Goal: affineSpan k (Set.range p) = ⊤ ↔ Fintype.card ι = FiniteDimensional.finrank k V + 1
-- Affine.Simplex.span_eq_top
theorem Affine.Simplex.span_eq_top [FiniteDimensional k V] {n : ℕ} (T : Affine.Simplex k V n)
    (hrank : finrank k V = n) : affineSpan k (Set.range T.points) = ⊤ 
Arguments: {k : Type u_1} {V : Type u_2} [DivisionRing k] [AddCommGroup V] [Module k V] [FiniteDimensional k V] {n : ℕ} (T : Affine.Simplex k V n) (hrank : FiniteDimensional.finrank k V = n)
Goal: affineSpan k (Set.range T.points) = ⊤
-- collinear_iff_rank_le_one
theorem collinear_iff_rank_le_one (s : Set P) :
    Collinear k s ↔ Module.rank k (vectorSpan k s) ≤ 1 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Set P)
Goal: Collinear k s ↔ Module.rank k ↥(vectorSpan k s) ≤ 1
-- collinear_iff_finrank_le_one
theorem collinear_iff_finrank_le_one {s : Set P} [FiniteDimensional k (vectorSpan k s)] :
    Collinear k s ↔ finrank k (vectorSpan k s) ≤ 1 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} [FiniteDimensional k ↥(vectorSpan k s)]
Goal: Collinear k s ↔ FiniteDimensional.finrank k ↥(vectorSpan k s) ≤ 1
-- Collinear.subset
theorem Collinear.subset {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Collinear k s₂) : Collinear k s₁ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Collinear k s₂)
Goal: Collinear k s₁
-- Collinear.finiteDimensional_vectorSpan
theorem Collinear.finiteDimensional_vectorSpan {s : Set P} (h : Collinear k s) :
    FiniteDimensional k (vectorSpan k s) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s)
Goal: FiniteDimensional k ↥(vectorSpan k s)
-- Collinear.finiteDimensional_direction_affineSpan
theorem Collinear.finiteDimensional_direction_affineSpan {s : Set P} (h : Collinear k s) :
    FiniteDimensional k (affineSpan k s).direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s)
Goal: FiniteDimensional k ↥(affineSpan k s).direction
-- collinear_empty
theorem collinear_empty : Collinear k (∅ : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} (P : Type u_3) [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
Goal: Collinear k ∅
-- collinear_singleton
theorem collinear_singleton (p : P) : Collinear k ({p} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P)
Goal: Collinear k {p}
-- collinear_iff_of_mem
theorem collinear_iff_of_mem {s : Set P} {p₀ : P} (h : p₀ ∈ s) :
    Collinear k s ↔ ∃ v : V, ∀ p ∈ s, ∃ r : k, p = r • v +ᵥ p₀ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} {p₀ : P} (h : p₀ ∈ s)
Goal: Collinear k s ↔ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
-- collinear_iff_exists_forall_eq_smul_vadd
theorem collinear_iff_exists_forall_eq_smul_vadd (s : Set P) :
    Collinear k s ↔ ∃ (p₀ : P) (v : V), ∀ p ∈ s, ∃ r : k, p = r • v +ᵥ p₀ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Set P)
Goal: Collinear k s ↔ ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
-- collinear_pair
theorem collinear_pair (p₁ p₂ : P) : Collinear k ({p₁, p₂} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: Collinear k {p₁, p₂}
-- affineIndependent_iff_not_collinear
theorem affineIndependent_iff_not_collinear {p : Fin 3 → P} :
    AffineIndependent k p ↔ ¬Collinear k (Set.range p) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : Fin 3 → P}
Goal: AffineIndependent k p ↔ ¬Collinear k (Set.range p)
-- collinear_iff_not_affineIndependent
theorem collinear_iff_not_affineIndependent {p : Fin 3 → P} :
    Collinear k (Set.range p) ↔ ¬AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : Fin 3 → P}
Goal: Collinear k (Set.range p) ↔ ¬AffineIndependent k p
-- affineIndependent_iff_not_collinear_set
theorem affineIndependent_iff_not_collinear_set {p₁ p₂ p₃ : P} :
    AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k ({p₁, p₂, p₃} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P}
Goal: AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k {p₁, p₂, p₃}
-- collinear_iff_not_affineIndependent_set
theorem collinear_iff_not_affineIndependent_set {p₁ p₂ p₃ : P} :
    Collinear k ({p₁, p₂, p₃} : Set P) ↔ ¬AffineIndependent k ![p₁, p₂, p₃] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P}
Goal: Collinear k {p₁, p₂, p₃} ↔ ¬AffineIndependent k ![p₁, p₂, p₃]
-- affineIndependent_iff_not_collinear_of_ne
theorem affineIndependent_iff_not_collinear_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    AffineIndependent k p ↔ ¬Collinear k ({p i₁, p i₂, p i₃} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
Goal: AffineIndependent k p ↔ ¬Collinear k {p i₁, p i₂, p i₃}
-- collinear_iff_not_affineIndependent_of_ne
theorem collinear_iff_not_affineIndependent_of_ne {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂)
    (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃) :
    Collinear k ({p i₁, p i₂, p i₃} : Set P) ↔ ¬AffineIndependent k p 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : Fin 3 → P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) (h₁₃ : i₁ ≠ i₃) (h₂₃ : i₂ ≠ i₃)
Goal: Collinear k {p i₁, p i₂, p i₃} ↔ ¬AffineIndependent k p
-- ne₁₂_of_not_collinear
theorem ne₁₂_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₂ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : ¬Collinear k {p₁, p₂, p₃})
Goal: p₁ ≠ p₂
-- ne₁₃_of_not_collinear
theorem ne₁₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₁ ≠ p₃ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : ¬Collinear k {p₁, p₂, p₃})
Goal: p₁ ≠ p₃
-- ne₂₃_of_not_collinear
theorem ne₂₃_of_not_collinear {p₁ p₂ p₃ : P} (h : ¬Collinear k ({p₁, p₂, p₃} : Set P)) :
    p₂ ≠ p₃ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : ¬Collinear k {p₁, p₂, p₃})
Goal: p₂ ≠ p₃
-- Collinear.mem_affineSpan_of_mem_of_ne
theorem Collinear.mem_affineSpan_of_mem_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
    (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : p₃ ∈ line[k, p₁, p₂] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₁p₂ : p₁ ≠ p₂)
Goal: p₃ ∈ affineSpan k {p₁, p₂}
-- Collinear.affineSpan_eq_of_ne
theorem Collinear.affineSpan_eq_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) (hp₁p₂ : p₁ ≠ p₂) : line[k, p₁, p₂] = affineSpan k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s) {p₁ p₂ : P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) (hp₁p₂ : p₁ ≠ p₂)
Goal: affineSpan k {p₁, p₂} = affineSpan k s
-- Collinear.collinear_insert_iff_of_ne
theorem Collinear.collinear_insert_iff_of_ne {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P}
    (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₃ : p₂ ≠ p₃) :
    Collinear k (insert p₁ s) ↔ Collinear k ({p₁, p₂, p₃} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s) {p₁ p₂ p₃ : P} (hp₂ : p₂ ∈ s) (hp₃ : p₃ ∈ s) (hp₂p₃ : p₂ ≠ p₃)
Goal: Collinear k (insert p₁ s) ↔ Collinear k {p₁, p₂, p₃}
-- collinear_insert_iff_of_mem_affineSpan
theorem collinear_insert_iff_of_mem_affineSpan {s : Set P} {p : P} (h : p ∈ affineSpan k s) :
    Collinear k (insert p s) ↔ Collinear k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} {p : P} (h : p ∈ affineSpan k s)
Goal: Collinear k (insert p s) ↔ Collinear k s
-- collinear_insert_of_mem_affineSpan_pair
theorem collinear_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) :
    Collinear k ({p₁, p₂, p₃} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : p₁ ∈ affineSpan k {p₂, p₃})
Goal: Collinear k {p₁, p₂, p₃}
-- collinear_insert_insert_of_mem_affineSpan_pair
theorem collinear_insert_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ p₄ : P} (h₁ : p₁ ∈ line[k, p₃, p₄])
    (h₂ : p₂ ∈ line[k, p₃, p₄]) : Collinear k ({p₁, p₂, p₃, p₄} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ p₄ : P} (h₁ : p₁ ∈ affineSpan k {p₃, p₄}) (h₂ : p₂ ∈ affineSpan k {p₃, p₄})
Goal: Collinear k {p₁, p₂, p₃, p₄}
-- collinear_insert_insert_insert_of_mem_affineSpan_pair
theorem collinear_insert_insert_insert_of_mem_affineSpan_pair {p₁ p₂ p₃ p₄ p₅ : P}
    (h₁ : p₁ ∈ line[k, p₄, p₅]) (h₂ : p₂ ∈ line[k, p₄, p₅]) (h₃ : p₃ ∈ line[k, p₄, p₅]) :
    Collinear k ({p₁, p₂, p₃, p₄, p₅} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ p₄ p₅ : P} (h₁ : p₁ ∈ affineSpan k {p₄, p₅}) (h₂ : p₂ ∈ affineSpan k {p₄, p₅}) (h₃ : p₃ ∈ affineSpan k {p₄, p₅})
Goal: Collinear k {p₁, p₂, p₃, p₄, p₅}
-- collinear_insert_insert_insert_left_of_mem_affineSpan_pair
theorem collinear_insert_insert_insert_left_of_mem_affineSpan_pair {p₁ p₂ p₃ p₄ p₅ : P}
    (h₁ : p₁ ∈ line[k, p₄, p₅]) (h₂ : p₂ ∈ line[k, p₄, p₅]) (h₃ : p₃ ∈ line[k, p₄, p₅]) :
    Collinear k ({p₁, p₂, p₃, p₄} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ p₄ p₅ : P} (h₁ : p₁ ∈ affineSpan k {p₄, p₅}) (h₂ : p₂ ∈ affineSpan k {p₄, p₅}) (h₃ : p₃ ∈ affineSpan k {p₄, p₅})
Goal: Collinear k {p₁, p₂, p₃, p₄}
-- collinear_triple_of_mem_affineSpan_pair
theorem collinear_triple_of_mem_affineSpan_pair {p₁ p₂ p₃ p₄ p₅ : P} (h₁ : p₁ ∈ line[k, p₄, p₅])
    (h₂ : p₂ ∈ line[k, p₄, p₅]) (h₃ : p₃ ∈ line[k, p₄, p₅]) :
    Collinear k ({p₁, p₂, p₃} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ p₄ p₅ : P} (h₁ : p₁ ∈ affineSpan k {p₄, p₅}) (h₂ : p₂ ∈ affineSpan k {p₄, p₅}) (h₃ : p₃ ∈ affineSpan k {p₄, p₅})
Goal: Collinear k {p₁, p₂, p₃}
-- Coplanar.finiteDimensional_vectorSpan
theorem Coplanar.finiteDimensional_vectorSpan {s : Set P} (h : Coplanar k s) :
    FiniteDimensional k (vectorSpan k s) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Coplanar k s)
Goal: FiniteDimensional k ↥(vectorSpan k s)
-- Coplanar.finiteDimensional_direction_affineSpan
theorem Coplanar.finiteDimensional_direction_affineSpan {s : Set P} (h : Coplanar k s) :
    FiniteDimensional k (affineSpan k s).direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Coplanar k s)
Goal: FiniteDimensional k ↥(affineSpan k s).direction
-- coplanar_iff_finrank_le_two
theorem coplanar_iff_finrank_le_two {s : Set P} [FiniteDimensional k (vectorSpan k s)] :
    Coplanar k s ↔ finrank k (vectorSpan k s) ≤ 2 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} [FiniteDimensional k ↥(vectorSpan k s)]
Goal: Coplanar k s ↔ FiniteDimensional.finrank k ↥(vectorSpan k s) ≤ 2
-- Coplanar.subset
theorem Coplanar.subset {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Coplanar k s₂) : Coplanar k s₁ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P} (hs : s₁ ⊆ s₂) (h : Coplanar k s₂)
Goal: Coplanar k s₁
-- Collinear.coplanar
theorem Collinear.coplanar {s : Set P} (h : Collinear k s) : Coplanar k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s)
Goal: Coplanar k s
-- coplanar_empty
theorem coplanar_empty : Coplanar k (∅ : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} (P : Type u_3) [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P]
Goal: Coplanar k ∅
-- coplanar_singleton
theorem coplanar_singleton (p : P) : Coplanar k ({p} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P)
Goal: Coplanar k {p}
-- coplanar_pair
theorem coplanar_pair (p₁ p₂ : P) : Coplanar k ({p₁, p₂} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: Coplanar k {p₁, p₂}
-- coplanar_insert_iff_of_mem_affineSpan
theorem coplanar_insert_iff_of_mem_affineSpan {s : Set P} {p : P} (h : p ∈ affineSpan k s) :
    Coplanar k (insert p s) ↔ Coplanar k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} {p : P} (h : p ∈ affineSpan k s)
Goal: Coplanar k (insert p s) ↔ Coplanar k s
-- finrank_vectorSpan_insert_le
theorem finrank_vectorSpan_insert_le (s : AffineSubspace k P) (p : P) :
    finrank k (vectorSpan k (insert p (s : Set P))) ≤ finrank k s.direction + 1 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : AffineSubspace k P) (p : P)
Goal: FiniteDimensional.finrank k ↥(vectorSpan k (insert p ↑s)) ≤ FiniteDimensional.finrank k ↥s.direction + 1
-- finrank_vectorSpan_insert_le_set
theorem finrank_vectorSpan_insert_le_set (s : Set P) (p : P) :
    finrank k (vectorSpan k (insert p s)) ≤ finrank k (vectorSpan k s) + 1 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Set P) (p : P)
Goal: FiniteDimensional.finrank k ↥(vectorSpan k (insert p s)) ≤ FiniteDimensional.finrank k ↥(vectorSpan k s) + 1
-- Collinear.coplanar_insert
theorem Collinear.coplanar_insert {s : Set P} (h : Collinear k s) (p : P) :
    Coplanar k (insert p s) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : Set P} (h : Collinear k s) (p : P)
Goal: Coplanar k (insert p s)
-- coplanar_of_finrank_eq_two
theorem coplanar_of_finrank_eq_two (s : Set P) (h : finrank k V = 2) : Coplanar k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Set P) (h : FiniteDimensional.finrank k V = 2)
Goal: Coplanar k s
-- coplanar_of_fact_finrank_eq_two
theorem coplanar_of_fact_finrank_eq_two (s : Set P) [h : Fact (finrank k V = 2)] : Coplanar k s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Set P) [h : Fact (FiniteDimensional.finrank k V = 2)]
Goal: Coplanar k s
-- coplanar_triple
theorem coplanar_triple (p₁ p₂ p₃ : P) : Coplanar k ({p₁, p₂, p₃} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ p₃ : P)
Goal: Coplanar k {p₁, p₂, p₃}
-- AffineBasis.finiteDimensional
protected theorem finiteDimensional [Finite ι] (b : AffineBasis ι k P) : FiniteDimensional k V 
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [Finite ι] (b : AffineBasis ι k P)
Goal: FiniteDimensional k V
-- AffineBasis.finite
protected theorem finite [FiniteDimensional k V] (b : AffineBasis ι k P) : Finite ι 
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [FiniteDimensional k V] (b : AffineBasis ι k P)
Goal: Finite ι
-- AffineBasis.finite_set
protected theorem finite_set [FiniteDimensional k V] {s : Set ι} (b : AffineBasis s k P) :
    s.Finite 
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [FiniteDimensional k V] {s : Set ι} (b : AffineBasis (↑s) k P)
Goal: s.Finite
-- AffineBasis.card_eq_finrank_add_one
theorem card_eq_finrank_add_one [Fintype ι] (b : AffineBasis ι k P) :
    Fintype.card ι = FiniteDimensional.finrank k V + 1 
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [Fintype ι] (b : AffineBasis ι k P)
Goal: Fintype.card ι = FiniteDimensional.finrank k V + 1
-- AffineBasis.exists_affineBasis_of_finiteDimensional
theorem exists_affineBasis_of_finiteDimensional [Fintype ι] [FiniteDimensional k V]
    (h : Fintype.card ι = FiniteDimensional.finrank k V + 1) : Nonempty (AffineBasis ι k P) 
Arguments: {ι : Type u₁} {k : Type u₂} {V : Type u₃} {P : Type u₄} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [Fintype ι] [FiniteDimensional k V] (h : Fintype.card ι = FiniteDimensional.finrank k V + 1)
Goal: Nonempty (AffineBasis ι k P)
-- eigenvalue_mem_ball
theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
    ∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) 
Arguments: {K : Type u_1} {n : Type u_2} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K} {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ)
Goal: ∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖)
-- det_ne_zero_of_sum_row_lt_diag
theorem det_ne_zero_of_sum_row_lt_diag (h : ∀ k, ∑ j ∈ Finset.univ.erase k, ‖A k j‖ < ‖A k k‖) :
    A.det ≠ 0 
Arguments: {K : Type u_1} {n : Type u_2} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K} (h : ∀ (k : n), ∑ j ∈ Finset.univ.erase k, ‖A k j‖ < ‖A k k‖)
Goal: A.det ≠ 0
-- det_ne_zero_of_sum_col_lt_diag
theorem det_ne_zero_of_sum_col_lt_diag (h : ∀ k, ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖) :
    A.det ≠ 0 
Arguments: {K : Type u_1} {n : Type u_2} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K} (h : ∀ (k : n), ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖)
Goal: A.det ≠ 0
-- Matrix.circulant_apply
@[simp]
theorem circulant_apply [Sub n] (v : n → α) (i j) : circulant v i j = v (i - j) 
Arguments: {α : Type u_1} {n : Type u_4} [Sub n] (v : n → α) (i j : n)
Goal: Matrix.circulant v i j = v (i - j)
-- Matrix.circulant_col_zero_eq
theorem circulant_col_zero_eq [AddGroup n] (v : n → α) (i : n) : circulant v i 0 = v i 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n] (v : n → α) (i : n)
Goal: Matrix.circulant v i 0 = v i
-- Matrix.circulant_injective
theorem circulant_injective [AddGroup n] : Injective (circulant : (n → α) → Matrix n n α) 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n]
Goal: Function.Injective Matrix.circulant
-- Matrix.Fin.circulant_injective
theorem Fin.circulant_injective : ∀ n, Injective fun v : Fin n → α => circulant v
  
Arguments: {α : Type u_1} (n : ℕ)
Goal: Function.Injective fun v => Matrix.circulant v
-- Matrix.circulant_inj
@[simp]
theorem circulant_inj [AddGroup n] {v w : n → α} : circulant v = circulant w ↔ v = w 
Arguments: {α : Type u_1} {n : Type u_4} [AddGroup n] {v w : n → α}
Goal: Matrix.circulant v = Matrix.circulant w ↔ v = w