ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineSubspace.nonempty_of_affineSpan_eq_top
theorem nonempty_of_affineSpan_eq_top {s : Set P} (h : affineSpan k s = ⊤) : 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] {s : Set P} (h : affineSpan k s = ⊤)
Goal: s.Nonempty
-- AffineSubspace.vectorSpan_eq_top_of_affineSpan_eq_top
theorem vectorSpan_eq_top_of_affineSpan_eq_top {s : Set P} (h : affineSpan k s = ⊤) :
    vectorSpan k s = ⊤ 
Arguments: (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s : Set P} (h : affineSpan k s = ⊤)
Goal: vectorSpan k s = ⊤
-- AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty
theorem affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty {s : Set P} (hs : s.Nonempty) :
    affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤ 
Arguments: (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s : Set P} (hs : s.Nonempty)
Goal: affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤
-- AffineSubspace.affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial
theorem affineSpan_eq_top_iff_vectorSpan_eq_top_of_nontrivial {s : Set P} [Nontrivial P] :
    affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤ 
Arguments: (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s : Set P} [Nontrivial P]
Goal: affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤
-- AffineSubspace.card_pos_of_affineSpan_eq_top
theorem card_pos_of_affineSpan_eq_top {ι : Type*} [Fintype ι] {p : ι → P}
    (h : affineSpan k (range p) = ⊤) : 0 < Fintype.card ι 
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} [Fintype ι] {p : ι → P} (h : affineSpan k (Set.range p) = ⊤)
Goal: 0 < Fintype.card ι
-- AffineSubspace.linear_topEquiv
@[simps! linear apply symm_apply_coe]
def topEquiv : (⊤ : AffineSubspace k P) ≃ᵃ[k] 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]
Goal: (AffineSubspace.topEquiv k V P).linear = (LinearEquiv.ofEq ⊤.direction ⊤ ⋯).trans Submodule.topEquiv
-- AffineSubspace.topEquiv_symm_apply_coe
@[simps! linear apply symm_apply_coe]
def topEquiv : (⊤ : AffineSubspace k P) ≃ᵃ[k] 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] (a : P)
Goal: ↑((AffineSubspace.topEquiv k V P).symm a) = a
-- AffineSubspace.topEquiv_apply
@[simps! linear apply symm_apply_coe]
def topEquiv : (⊤ : AffineSubspace k P) ≃ᵃ[k] 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] (self : { x // x ∈ Set.univ })
Goal: (AffineSubspace.topEquiv k V P) self = ↑self
-- AffineSubspace.not_mem_bot
theorem not_mem_bot (p : P) : p ∉ (⊥ : AffineSubspace k 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] (p : P)
Goal: p ∉ ⊥
-- AffineSubspace.direction_bot
@[simp]
theorem direction_bot : (⊥ : AffineSubspace k P).direction = ⊥ 
Arguments: (k : Type u_1) (V : Type u_2) (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P]
Goal: ⊥.direction = ⊥
-- AffineSubspace.coe_eq_bot_iff
@[simp]
theorem coe_eq_bot_iff (Q : AffineSubspace k P) : (Q : Set P) = ∅ ↔ Q = ⊥ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (Q : AffineSubspace k P)
Goal: ↑Q = ∅ ↔ Q = ⊥
-- AffineSubspace.coe_eq_univ_iff
@[simp]
theorem coe_eq_univ_iff (Q : AffineSubspace k P) : (Q : Set P) = univ ↔ Q = ⊤ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (Q : AffineSubspace k P)
Goal: ↑Q = Set.univ ↔ Q = ⊤
-- AffineSubspace.nonempty_iff_ne_bot
theorem nonempty_iff_ne_bot (Q : AffineSubspace k P) : (Q : Set P).Nonempty ↔ Q ≠ ⊥ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (Q : AffineSubspace k P)
Goal: (↑Q).Nonempty ↔ Q ≠ ⊥
-- AffineSubspace.eq_bot_or_nonempty
theorem eq_bot_or_nonempty (Q : AffineSubspace k P) : Q = ⊥ ∨ (Q : Set P).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] (Q : AffineSubspace k P)
Goal: Q = ⊥ ∨ (↑Q).Nonempty
-- AffineSubspace.subsingleton_of_subsingleton_span_eq_top
theorem subsingleton_of_subsingleton_span_eq_top {s : Set P} (h₁ : s.Subsingleton)
    (h₂ : affineSpan k s = ⊤) : Subsingleton 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] {s : Set P} (h₁ : s.Subsingleton) (h₂ : affineSpan k s = ⊤)
Goal: Subsingleton P
-- AffineSubspace.eq_univ_of_subsingleton_span_eq_top
theorem eq_univ_of_subsingleton_span_eq_top {s : Set P} (h₁ : s.Subsingleton)
    (h₂ : affineSpan k s = ⊤) : s = (univ : Set 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] {s : Set P} (h₁ : s.Subsingleton) (h₂ : affineSpan k s = ⊤)
Goal: s = Set.univ
-- AffineSubspace.direction_eq_top_iff_of_nonempty
@[simp]
theorem direction_eq_top_iff_of_nonempty {s : AffineSubspace k P} (h : (s : Set P).Nonempty) :
    s.direction = ⊤ ↔ s = ⊤ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s : AffineSubspace k P} (h : (↑s).Nonempty)
Goal: s.direction = ⊤ ↔ s = ⊤
-- AffineSubspace.inf_coe
@[simp]
theorem inf_coe (s1 s2 : AffineSubspace k P) : (s1 ⊓ s2 : Set P) = (s1 : Set P) ∩ s2 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (s1 s2 : AffineSubspace k P)
Goal: ↑s1 ⊓ ↑s2 = ↑s1 ∩ ↑s2
-- AffineSubspace.mem_inf_iff
theorem mem_inf_iff (p : P) (s1 s2 : AffineSubspace k P) : p ∈ s1 ⊓ s2 ↔ p ∈ s1 ∧ p ∈ s2 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (p : P) (s1 s2 : AffineSubspace k P)
Goal: p ∈ s1 ⊓ s2 ↔ p ∈ s1 ∧ p ∈ s2
-- AffineSubspace.direction_inf
theorem direction_inf (s1 s2 : AffineSubspace k P) :
    (s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (s1 s2 : AffineSubspace k P)
Goal: (s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction
-- AffineSubspace.direction_inf_of_mem
theorem direction_inf_of_mem {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂) :
    (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂)
Goal: (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction
-- AffineSubspace.direction_inf_of_mem_inf
theorem direction_inf_of_mem_inf {s₁ s₂ : AffineSubspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂) :
    (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h : p ∈ s₁ ⊓ s₂)
Goal: (s₁ ⊓ s₂).direction = s₁.direction ⊓ s₂.direction
-- AffineSubspace.direction_le
theorem direction_le {s1 s2 : AffineSubspace k P} (h : s1 ≤ s2) : s1.direction ≤ s2.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s1 s2 : AffineSubspace k P} (h : s1 ≤ s2)
Goal: s1.direction ≤ s2.direction
-- AffineSubspace.direction_lt_of_nonempty
theorem direction_lt_of_nonempty {s1 s2 : AffineSubspace k P} (h : s1 < s2)
    (hn : (s1 : Set P).Nonempty) : s1.direction < s2.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s1 s2 : AffineSubspace k P} (h : s1 < s2) (hn : (↑s1).Nonempty)
Goal: s1.direction < s2.direction
-- AffineSubspace.sup_direction_le
theorem sup_direction_le (s1 s2 : AffineSubspace k P) :
    s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (s1 s2 : AffineSubspace k P)
Goal: s1.direction ⊔ s2.direction ≤ (s1 ⊔ s2).direction
-- AffineSubspace.sup_direction_lt_of_nonempty_of_inter_empty
theorem sup_direction_lt_of_nonempty_of_inter_empty {s1 s2 : AffineSubspace k P}
    (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty) (he : (s1 ∩ s2 : Set P) = ∅) :
    s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {s1 s2 : AffineSubspace k P} (h1 : (↑s1).Nonempty) (h2 : (↑s2).Nonempty) (he : ↑s1 ∩ ↑s2 = ∅)
Goal: s1.direction ⊔ s2.direction < (s1 ⊔ s2).direction
-- AffineSubspace.inter_nonempty_of_nonempty_of_sup_direction_eq_top
theorem inter_nonempty_of_nonempty_of_sup_direction_eq_top {s1 s2 : AffineSubspace k P}
    (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty)
    (hd : s1.direction ⊔ s2.direction = ⊤) : ((s1 : Set P) ∩ s2).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] {s1 s2 : AffineSubspace k P} (h1 : (↑s1).Nonempty) (h2 : (↑s2).Nonempty) (hd : s1.direction ⊔ s2.direction = ⊤)
Goal: (↑s1 ∩ ↑s2).Nonempty
-- AffineSubspace.inter_eq_singleton_of_nonempty_of_isCompl
theorem inter_eq_singleton_of_nonempty_of_isCompl {s1 s2 : AffineSubspace k P}
    (h1 : (s1 : Set P).Nonempty) (h2 : (s2 : Set P).Nonempty)
    (hd : IsCompl s1.direction s2.direction) : ∃ p, (s1 : Set P) ∩ s2 = {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] {s1 s2 : AffineSubspace k P} (h1 : (↑s1).Nonempty) (h2 : (↑s2).Nonempty) (hd : IsCompl s1.direction s2.direction)
Goal: ∃ p, ↑s1 ∩ ↑s2 = {p}
-- AffineSubspace.affineSpan_coe
@[simp]
theorem affineSpan_coe (s : AffineSubspace k P) : affineSpan k (s : Set P) = s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (s : AffineSubspace k P)
Goal: affineSpan k ↑s = s
-- vectorSpan_eq_span_vsub_set_left
theorem vectorSpan_eq_span_vsub_set_left {s : Set P} {p : P} (hp : p ∈ s) :
    vectorSpan k s = Submodule.span k ((p -ᵥ ·) '' s) 
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} {p : P} (hp : p ∈ s)
Goal: vectorSpan k s = Submodule.span k ((fun x => p -ᵥ x) '' s)
-- vectorSpan_eq_span_vsub_set_right
theorem vectorSpan_eq_span_vsub_set_right {s : Set P} {p : P} (hp : p ∈ s) :
    vectorSpan k s = Submodule.span k ((· -ᵥ p) '' s) 
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} {p : P} (hp : p ∈ s)
Goal: vectorSpan k s = Submodule.span k ((fun x => x -ᵥ p) '' s)
-- vectorSpan_eq_span_vsub_set_left_ne
theorem vectorSpan_eq_span_vsub_set_left_ne {s : Set P} {p : P} (hp : p ∈ s) :
    vectorSpan k s = Submodule.span k ((p -ᵥ ·) '' (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 : Set P} {p : P} (hp : p ∈ s)
Goal: vectorSpan k s = Submodule.span k ((fun x => p -ᵥ x) '' (s \ {p}))
-- vectorSpan_eq_span_vsub_set_right_ne
theorem vectorSpan_eq_span_vsub_set_right_ne {s : Set P} {p : P} (hp : p ∈ s) :
    vectorSpan k s = Submodule.span k ((· -ᵥ p) '' (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 : Set P} {p : P} (hp : p ∈ s)
Goal: vectorSpan k s = Submodule.span k ((fun x => x -ᵥ p) '' (s \ {p}))
-- vectorSpan_eq_span_vsub_finset_right_ne
theorem vectorSpan_eq_span_vsub_finset_right_ne [DecidableEq P] [DecidableEq V] {s : Finset P}
    {p : P} (hp : p ∈ s) :
    vectorSpan k (s : Set P) = Submodule.span k ((s.erase p).image (· -ᵥ p)) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] [DecidableEq P] [DecidableEq V] {s : Finset P} {p : P} (hp : p ∈ s)
Goal: vectorSpan k ↑s = Submodule.span k ↑(Finset.image (fun x => x -ᵥ p) (s.erase p))
-- vectorSpan_image_eq_span_vsub_set_left_ne
theorem vectorSpan_image_eq_span_vsub_set_left_ne (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s) :
    vectorSpan k (p '' s) = Submodule.span k ((p i -ᵥ ·) '' (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} (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s)
Goal: vectorSpan k (p '' s) = Submodule.span k ((fun x => p i -ᵥ x) '' (p '' (s \ {i})))
-- vectorSpan_image_eq_span_vsub_set_right_ne
theorem vectorSpan_image_eq_span_vsub_set_right_ne (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s) :
    vectorSpan k (p '' s) = Submodule.span k ((· -ᵥ p i) '' (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} (p : ι → P) {s : Set ι} {i : ι} (hi : i ∈ s)
Goal: vectorSpan k (p '' s) = Submodule.span k ((fun x => x -ᵥ p i) '' (p '' (s \ {i})))
-- vectorSpan_range_eq_span_range_vsub_left
theorem vectorSpan_range_eq_span_range_vsub_left (p : ι → P) (i0 : ι) :
    vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : ι => p i0 -ᵥ 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) (i0 : ι)
Goal: vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i => p i0 -ᵥ p i)
-- vectorSpan_range_eq_span_range_vsub_right
theorem vectorSpan_range_eq_span_range_vsub_right (p : ι → P) (i0 : ι) :
    vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i : ι => p i -ᵥ p i0) 
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) (i0 : ι)
Goal: vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i => p i -ᵥ p i0)
-- vectorSpan_range_eq_span_range_vsub_left_ne
theorem vectorSpan_range_eq_span_range_vsub_left_ne (p : ι → P) (i₀ : ι) :
    vectorSpan k (Set.range p) =
      Submodule.span k (Set.range fun i : { x // x ≠ i₀ } => p i₀ -ᵥ 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) (i₀ : ι)
Goal: vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i => p i₀ -ᵥ p ↑i)
-- vectorSpan_range_eq_span_range_vsub_right_ne
theorem vectorSpan_range_eq_span_range_vsub_right_ne (p : ι → P) (i₀ : ι) :
    vectorSpan k (Set.range p) =
      Submodule.span k (Set.range fun i : { x // x ≠ i₀ } => p i -ᵥ p i₀) 
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) (i₀ : ι)
Goal: vectorSpan k (Set.range p) = Submodule.span k (Set.range fun i => p ↑i -ᵥ p i₀)
-- affineSpan_nonempty
theorem affineSpan_nonempty : (affineSpan k s : Set P).Nonempty ↔ s.Nonempty 
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}
Goal: (↑(affineSpan k s)).Nonempty ↔ s.Nonempty
-- affineSpan_eq_bot
@[simp]
theorem affineSpan_eq_bot : affineSpan k s = ⊥ ↔ s = ∅ 
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}
Goal: affineSpan k s = ⊥ ↔ s = ∅
-- bot_lt_affineSpan
@[simp]
theorem bot_lt_affineSpan : ⊥ < affineSpan k s ↔ s.Nonempty 
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}
Goal: ⊥ < affineSpan k s ↔ s.Nonempty
-- affineSpan_induction
theorem affineSpan_induction {x : P} {s : Set P} {p : P → Prop} (h : x ∈ affineSpan k s)
    (mem : ∀ x : P, x ∈ s → p x)
    (smul_vsub_vadd : ∀ (c : k) (u v w : P), p u → p v → p w → p (c • (u -ᵥ v) +ᵥ w)) : p x 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {x : P} {s : Set P} {p : P → Prop} (h : x ∈ affineSpan k s) (mem : ∀ x ∈ s, p x) (smul_vsub_vadd : ∀ (c : k) (u v w : P), p u → p v → p w → p (c • (u -ᵥ v) +ᵥ w))
Goal: p x
-- affineSpan_induction'
@[elab_as_elim]
theorem affineSpan_induction' {s : Set P} {p : ∀ x, x ∈ affineSpan k s → Prop}
    (mem : ∀ (y) (hys : y ∈ s), p y (subset_affineSpan k _ hys))
    (smul_vsub_vadd :
      ∀ (c : k) (u hu v hv w hw),
        p u hu →
          p v hv → p w hw → p (c • (u -ᵥ v) +ᵥ w) (AffineSubspace.smul_vsub_vadd_mem _ _ hu hv hw))
    {x : P} (h : x ∈ affineSpan k s) : p x h 
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} {p : (x : P) → x ∈ affineSpan k s → Prop} (mem : ∀ (y : P) (hys : y ∈ s), p y ⋯) (smul_vsub_vadd :
  ∀ (c : k) (u : P) (hu : u ∈ affineSpan k s) (v : P) (hv : v ∈ affineSpan k s) (w : P) (hw : w ∈ affineSpan k s),
    p u hu → p v hv → p w hw → p (c • (u -ᵥ v) +ᵥ w) ⋯) {x : P} (h : x ∈ affineSpan k s)
Goal: p x h
-- affineSpan_coe_preimage_eq_top
@[simp]
theorem affineSpan_coe_preimage_eq_top (A : Set P) [Nonempty A] :
    affineSpan k (((↑) : affineSpan k A → P) ⁻¹' A) = ⊤ 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (A : Set P) [Nonempty ↑A]
Goal: affineSpan k (Subtype.val ⁻¹' A) = ⊤
-- affineSpan_singleton_union_vadd_eq_top_of_span_eq_top
theorem affineSpan_singleton_union_vadd_eq_top_of_span_eq_top {s : Set V} (p : P)
    (h : Submodule.span k (Set.range ((↑) : s → V)) = ⊤) :
    affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s) = ⊤ 
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} (p : P) (h : Submodule.span k (Set.range Subtype.val) = ⊤)
Goal: affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s) = ⊤
-- vectorSpan_pair
theorem vectorSpan_pair (p₁ p₂ : P) : vectorSpan k ({p₁, p₂} : Set P) = k ∙ p₁ -ᵥ p₂ 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: vectorSpan k {p₁, p₂} = Submodule.span k {p₁ -ᵥ p₂}
-- vectorSpan_pair_rev
theorem vectorSpan_pair_rev (p₁ p₂ : P) : vectorSpan k ({p₁, p₂} : Set P) = k ∙ p₂ -ᵥ p₁ 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: vectorSpan k {p₁, p₂} = Submodule.span k {p₂ -ᵥ p₁}
-- vsub_mem_vectorSpan_pair
theorem vsub_mem_vectorSpan_pair (p₁ p₂ : P) : p₁ -ᵥ p₂ ∈ vectorSpan k ({p₁, p₂} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: p₁ -ᵥ p₂ ∈ vectorSpan k {p₁, p₂}
-- vsub_rev_mem_vectorSpan_pair
theorem vsub_rev_mem_vectorSpan_pair (p₁ p₂ : P) : p₂ -ᵥ p₁ ∈ vectorSpan k ({p₁, p₂} : Set P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: p₂ -ᵥ p₁ ∈ vectorSpan k {p₁, p₂}
-- smul_vsub_mem_vectorSpan_pair
theorem smul_vsub_mem_vectorSpan_pair (r : k) (p₁ p₂ : P) :
    r • (p₁ -ᵥ p₂) ∈ vectorSpan k ({p₁, p₂} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: r • (p₁ -ᵥ p₂) ∈ vectorSpan k {p₁, p₂}
-- smul_vsub_rev_mem_vectorSpan_pair
theorem smul_vsub_rev_mem_vectorSpan_pair (r : k) (p₁ p₂ : P) :
    r • (p₂ -ᵥ p₁) ∈ vectorSpan k ({p₁, p₂} : Set P) 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: r • (p₂ -ᵥ p₁) ∈ vectorSpan k {p₁, p₂}
-- mem_vectorSpan_pair
theorem mem_vectorSpan_pair {p₁ p₂ : P} {v : V} :
    v ∈ vectorSpan k ({p₁, p₂} : Set P) ↔ ∃ r : k, r • (p₁ -ᵥ p₂) = v 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ : P} {v : V}
Goal: v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v
-- mem_vectorSpan_pair_rev
theorem mem_vectorSpan_pair_rev {p₁ p₂ : P} {v : V} :
    v ∈ vectorSpan k ({p₁, p₂} : Set P) ↔ ∃ r : k, r • (p₂ -ᵥ 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] {p₁ p₂ : P} {v : V}
Goal: v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₂ -ᵥ p₁) = v
-- left_mem_affineSpan_pair
theorem left_mem_affineSpan_pair (p₁ p₂ : P) : p₁ ∈ line[k, p₁, p₂] 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: p₁ ∈ affineSpan k {p₁, p₂}
-- right_mem_affineSpan_pair
theorem right_mem_affineSpan_pair (p₁ p₂ : P) : p₂ ∈ line[k, p₁, p₂] 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p₁ p₂ : P)
Goal: p₂ ∈ affineSpan k {p₁, p₂}
-- AffineMap.lineMap_mem_affineSpan_pair
theorem AffineMap.lineMap_mem_affineSpan_pair (r : k) (p₁ p₂ : P) :
    AffineMap.lineMap p₁ p₂ r ∈ line[k, p₁, p₂] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: (AffineMap.lineMap p₁ p₂) r ∈ affineSpan k {p₁, p₂}
-- AffineMap.lineMap_rev_mem_affineSpan_pair
theorem AffineMap.lineMap_rev_mem_affineSpan_pair (r : k) (p₁ p₂ : P) :
    AffineMap.lineMap p₂ p₁ r ∈ line[k, p₁, p₂] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: (AffineMap.lineMap p₂ p₁) r ∈ affineSpan k {p₁, p₂}
-- smul_vsub_vadd_mem_affineSpan_pair
theorem smul_vsub_vadd_mem_affineSpan_pair (r : k) (p₁ p₂ : P) :
    r • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ line[k, p₁, p₂] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: r • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ affineSpan k {p₁, p₂}
-- smul_vsub_rev_vadd_mem_affineSpan_pair
theorem smul_vsub_rev_vadd_mem_affineSpan_pair (r : k) (p₁ p₂ : P) :
    r • (p₁ -ᵥ p₂) +ᵥ p₂ ∈ line[k, p₁, p₂] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (r : k) (p₁ p₂ : P)
Goal: r • (p₁ -ᵥ p₂) +ᵥ p₂ ∈ affineSpan k {p₁, p₂}
-- vadd_left_mem_affineSpan_pair
theorem vadd_left_mem_affineSpan_pair {p₁ p₂ : P} {v : V} :
    v +ᵥ p₁ ∈ line[k, p₁, p₂] ↔ ∃ r : k, r • (p₂ -ᵥ 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] {p₁ p₂ : P} {v : V}
Goal: v +ᵥ p₁ ∈ affineSpan k {p₁, p₂} ↔ ∃ r, r • (p₂ -ᵥ p₁) = v
-- vadd_right_mem_affineSpan_pair
theorem vadd_right_mem_affineSpan_pair {p₁ p₂ : P} {v : V} :
    v +ᵥ p₂ ∈ line[k, p₁, p₂] ↔ ∃ r : k, r • (p₁ -ᵥ p₂) = v 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ : P} {v : V}
Goal: v +ᵥ p₂ ∈ affineSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v
-- affineSpan_pair_le_of_mem_of_mem
theorem affineSpan_pair_le_of_mem_of_mem {p₁ p₂ : P} {s : AffineSubspace k P} (hp₁ : p₁ ∈ s)
    (hp₂ : p₂ ∈ s) : line[k, p₁, p₂] ≤ s 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ : P} {s : AffineSubspace k P} (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s)
Goal: affineSpan k {p₁, p₂} ≤ s
-- affineSpan_pair_le_of_left_mem
theorem affineSpan_pair_le_of_left_mem {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) :
    line[k, p₁, p₃] ≤ line[k, p₂, p₃] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : p₁ ∈ affineSpan k {p₂, p₃})
Goal: affineSpan k {p₁, p₃} ≤ affineSpan k {p₂, p₃}
-- affineSpan_pair_le_of_right_mem
theorem affineSpan_pair_le_of_right_mem {p₁ p₂ p₃ : P} (h : p₁ ∈ line[k, p₂, p₃]) :
    line[k, p₂, p₁] ≤ line[k, p₂, p₃] 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p₁ p₂ p₃ : P} (h : p₁ ∈ affineSpan k {p₂, p₃})
Goal: affineSpan k {p₂, p₁} ≤ affineSpan k {p₂, p₃}