ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Matrix.toLin_transpose
@[simp]
theorem Matrix.toLin_transpose (M : Matrix ι₁ ι₂ K) : Matrix.toLin B₁.dualBasis B₂.dualBasis Mᵀ =
    Module.Dual.transpose (R := K) (Matrix.toLin B₂ B₁ M) 
Arguments: {K : Type u_1} {V₁ : Type u_2} {V₂ : Type u_3} {ι₁ : Type u_4} {ι₂ : Type u_5} [Field K] [AddCommGroup V₁] [Module K V₁] [AddCommGroup V₂] [Module K V₂] [Fintype ι₁] [Fintype ι₂] [DecidableEq ι₁] [DecidableEq ι₂] {B₁ : Basis ι₁ K V₁} {B₂ : Basis ι₂ K V₂} (M : Matrix ι₁ ι₂ K)
Goal: (Matrix.toLin B₁.dualBasis B₂.dualBasis) Mᵀ = Module.Dual.transpose ((Matrix.toLin B₂ B₁) M)
-- vectorSpan_def
theorem vectorSpan_def (s : Set P) : vectorSpan k s = Submodule.span k (s -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)
Goal: vectorSpan k s = Submodule.span k (s -ᵥ s)
-- vectorSpan_mono
theorem vectorSpan_mono {s₁ s₂ : Set P} (h : s₁ ⊆ s₂) : vectorSpan k s₁ ≤ vectorSpan k s₂ 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s₁ s₂ : Set P} (h : s₁ ⊆ s₂)
Goal: vectorSpan k s₁ ≤ vectorSpan k s₂
-- vectorSpan_empty
@[simp]
theorem vectorSpan_empty : vectorSpan k (∅ : Set P) = (⊥ : Submodule k V) 
Arguments: (k : Type u_1) {V : Type u_2} (P : Type u_3) [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P]
Goal: vectorSpan k ∅ = ⊥
-- vectorSpan_singleton
@[simp]
theorem vectorSpan_singleton (p : P) : vectorSpan k ({p} : 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] (p : P)
Goal: vectorSpan k {p} = ⊥
-- vsub_set_subset_vectorSpan
theorem vsub_set_subset_vectorSpan (s : Set P) : s -ᵥ s ⊆ ↑(vectorSpan k 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: s -ᵥ s ⊆ ↑(vectorSpan k s)
-- vsub_mem_vectorSpan
theorem vsub_mem_vectorSpan {s : Set P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
    p1 -ᵥ p2 ∈ vectorSpan k 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} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s)
Goal: p1 -ᵥ p2 ∈ vectorSpan k s
-- mem_spanPoints
theorem mem_spanPoints (p : P) (s : Set P) : p ∈ s → p ∈ spanPoints k s
  
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P) (s : Set P)
Goal: p ∈ s → p ∈ spanPoints k s
-- subset_spanPoints
theorem subset_spanPoints (s : Set P) : s ⊆ spanPoints k 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: s ⊆ spanPoints k s
-- spanPoints_nonempty
@[simp]
theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).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: (spanPoints k s).Nonempty ↔ s.Nonempty
-- vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan
theorem vadd_mem_spanPoints_of_mem_spanPoints_of_mem_vectorSpan {s : Set P} {p : P} {v : V}
    (hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s) : v +ᵥ p ∈ spanPoints k 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} {v : V} (hp : p ∈ spanPoints k s) (hv : v ∈ vectorSpan k s)
Goal: v +ᵥ p ∈ spanPoints k s
-- vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints
theorem vsub_mem_vectorSpan_of_mem_spanPoints_of_mem_spanPoints {s : Set P} {p1 p2 : P}
    (hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s) : p1 -ᵥ p2 ∈ vectorSpan k 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} {p1 p2 : P} (hp1 : p1 ∈ spanPoints k s) (hp2 : p2 ∈ spanPoints k s)
Goal: p1 -ᵥ p2 ∈ vectorSpan k s
-- AffineSubspace.smul_vsub_vadd_mem
structure AffineSubspace (k : Type*) {V : Type*} (P : Type*) [Ring k] [AddCommGroup V]
  [Module k V] [AffineSpace V P] where
    carrier : Set P
  smul_vsub_vadd_mem :
    ∀ (c : k) {p1 p2 p3 : P},
      p1 ∈ carrier → p2 ∈ carrier → p3 ∈ carrier → c • (p1 -ᵥ p2 : V) +ᵥ p3 ∈ carrier
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (self : AffineSubspace k P) (c : k) {p1 p2 p3 : P}
Goal: p1 ∈ self.carrier → p2 ∈ self.carrier → p3 ∈ self.carrier → c • (p1 -ᵥ p2) +ᵥ p3 ∈ self.carrier
-- AffineSubspace.mem_coe
-- Porting note: removed `simp`, proof is `simp only [SetLike.mem_coe]`
theorem mem_coe (p : P) (s : AffineSubspace k P) : p ∈ (s : Set P) ↔ 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] (p : P) (s : AffineSubspace k P)
Goal: p ∈ ↑s ↔ p ∈ s
-- AffineSubspace.direction_eq_vectorSpan
theorem direction_eq_vectorSpan (s : AffineSubspace k P) : s.direction = vectorSpan k (s : 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] (s : AffineSubspace k P)
Goal: s.direction = vectorSpan k ↑s
-- AffineSubspace.directionOfNonempty_eq_direction
theorem directionOfNonempty_eq_direction {s : AffineSubspace k P} (h : (s : Set P).Nonempty) :
    directionOfNonempty h = s.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P} (h : (↑s).Nonempty)
Goal: AffineSubspace.directionOfNonempty h = s.direction
-- AffineSubspace.coe_direction_eq_vsub_set
theorem coe_direction_eq_vsub_set {s : AffineSubspace k P} (h : (s : Set P).Nonempty) :
    (s.direction : Set V) = (s : Set 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 : AffineSubspace k P} (h : (↑s).Nonempty)
Goal: ↑s.direction = ↑s -ᵥ ↑s
-- AffineSubspace.mem_direction_iff_eq_vsub
theorem mem_direction_iff_eq_vsub {s : AffineSubspace k P} (h : (s : Set P).Nonempty) (v : V) :
    v ∈ s.direction ↔ ∃ p1 ∈ s, ∃ p2 ∈ s, v = 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] {s : AffineSubspace k P} (h : (↑s).Nonempty) (v : V)
Goal: v ∈ s.direction ↔ ∃ p1 ∈ s, ∃ p2 ∈ s, v = p1 -ᵥ p2
-- AffineSubspace.vadd_mem_of_mem_direction
theorem vadd_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P}
    (hp : p ∈ s) : 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 : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P} (hp : p ∈ s)
Goal: v +ᵥ p ∈ s
-- AffineSubspace.vsub_mem_direction
theorem vsub_mem_direction {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s) :
    p1 -ᵥ p2 ∈ s.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P} {p1 p2 : P} (hp1 : p1 ∈ s) (hp2 : p2 ∈ s)
Goal: p1 -ᵥ p2 ∈ s.direction
-- AffineSubspace.vadd_mem_iff_mem_direction
theorem vadd_mem_iff_mem_direction {s : AffineSubspace k P} (v : V) {p : P} (hp : p ∈ s) :
    v +ᵥ p ∈ s ↔ v ∈ s.direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {s : AffineSubspace k P} (v : V) {p : P} (hp : p ∈ s)
Goal: v +ᵥ p ∈ s ↔ v ∈ s.direction
-- AffineSubspace.vadd_mem_iff_mem_of_mem_direction
theorem vadd_mem_iff_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction)
    {p : P} : v +ᵥ p ∈ s ↔ 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 : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P}
Goal: v +ᵥ p ∈ s ↔ p ∈ s
-- AffineSubspace.coe_direction_eq_vsub_set_right
theorem coe_direction_eq_vsub_set_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) :
    (s.direction : Set 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 : AffineSubspace k P} {p : P} (hp : p ∈ s)
Goal: ↑s.direction = (fun x => x -ᵥ p) '' ↑s
-- AffineSubspace.coe_direction_eq_vsub_set_left
theorem coe_direction_eq_vsub_set_left {s : AffineSubspace k P} {p : P} (hp : p ∈ s) :
    (s.direction : Set V) = (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 : AffineSubspace k P} {p : P} (hp : p ∈ s)
Goal: ↑s.direction = (fun x => p -ᵥ x) '' ↑s
-- AffineSubspace.mem_direction_iff_eq_vsub_right
theorem mem_direction_iff_eq_vsub_right {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) :
    v ∈ s.direction ↔ ∃ p2 ∈ s, v = p2 -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] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V)
Goal: v ∈ s.direction ↔ ∃ p2 ∈ s, v = p2 -ᵥ p
-- AffineSubspace.mem_direction_iff_eq_vsub_left
theorem mem_direction_iff_eq_vsub_left {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V) :
    v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -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] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (v : V)
Goal: v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2
-- AffineSubspace.vsub_right_mem_direction_iff_mem
theorem vsub_right_mem_direction_iff_mem {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P) :
    p2 -ᵥ p ∈ s.direction ↔ 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] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P)
Goal: p2 -ᵥ p ∈ s.direction ↔ p2 ∈ s
-- AffineSubspace.vsub_left_mem_direction_iff_mem
theorem vsub_left_mem_direction_iff_mem {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P) :
    p -ᵥ p2 ∈ s.direction ↔ 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] {s : AffineSubspace k P} {p : P} (hp : p ∈ s) (p2 : P)
Goal: p -ᵥ p2 ∈ s.direction ↔ p2 ∈ s
-- AffineSubspace.coe_injective
theorem coe_injective : Function.Injective ((↑) : AffineSubspace k P → 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]
Goal: Function.Injective SetLike.coe
-- AffineSubspace.ext
@[ext]
theorem ext {p q : AffineSubspace k P} (h : ∀ x, x ∈ p ↔ x ∈ q) : p = q 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p q : AffineSubspace k P} (h : ∀ (x : P), x ∈ p ↔ x ∈ q)
Goal: p = q
-- AffineSubspace.ext_iff
protected theorem ext_iff (s₁ s₂ : AffineSubspace k P) : s₁ = s₂ ↔ (s₁ : Set 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] (s₁ s₂ : AffineSubspace k P)
Goal: s₁ = s₂ ↔ ↑s₁ = ↑s₂
-- AffineSubspace.ext_of_direction_eq
theorem ext_of_direction_eq {s1 s2 : AffineSubspace k P} (hd : s1.direction = s2.direction)
    (hn : ((s1 : Set P) ∩ s2).Nonempty) : 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] {s1 s2 : AffineSubspace k P} (hd : s1.direction = s2.direction) (hn : (↑s1 ∩ ↑s2).Nonempty)
Goal: s1 = s2
-- AffineSubspace.coe_vsub
@[simp, norm_cast]
theorem coe_vsub (s : AffineSubspace k P) [Nonempty s] (a b : s) : ↑(a -ᵥ b) = (a : P) -ᵥ (b : 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 : AffineSubspace k P) [Nonempty ↥s] (a b : ↥s)
Goal: ↑(a -ᵥ b) = ↑a -ᵥ ↑b
-- AffineSubspace.coe_vadd
@[simp, norm_cast]
theorem coe_vadd (s : AffineSubspace k P) [Nonempty s] (a : s.direction) (b : s) :
    ↑(a +ᵥ b) = (a : V) +ᵥ (b : 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 : AffineSubspace k P) [Nonempty ↥s] (a : ↥s.direction) (b : ↥s)
Goal: ↑(a +ᵥ b) = ↑a +ᵥ ↑b
-- AffineSubspace.subtype_linear
@[simp]
theorem subtype_linear (s : AffineSubspace k P) [Nonempty s] :
    s.subtype.linear = s.direction.subtype 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : AffineSubspace k P) [Nonempty ↥s]
Goal: s.subtype.linear = s.direction.subtype
-- AffineSubspace.subtype_apply
theorem subtype_apply (s : AffineSubspace k P) [Nonempty s] (p : s) : s.subtype 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 : AffineSubspace k P) [Nonempty ↥s] (p : ↥s)
Goal: s.subtype p = ↑p
-- AffineSubspace.coeSubtype
@[simp]
theorem coeSubtype (s : AffineSubspace k P) [Nonempty s] : (s.subtype : s → 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 : AffineSubspace k P) [Nonempty ↥s]
Goal: ⇑s.subtype = Subtype.val
-- AffineSubspace.injective_subtype
theorem injective_subtype (s : AffineSubspace k P) [Nonempty s] : Function.Injective s.subtype 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : AffineSubspace k P) [Nonempty ↥s]
Goal: Function.Injective ⇑s.subtype
-- AffineSubspace.eq_iff_direction_eq_of_mem
theorem eq_iff_direction_eq_of_mem {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁)
    (h₂ : p ∈ s₂) : 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] [AffineSpace V P] {s₁ s₂ : AffineSubspace k P} {p : P} (h₁ : p ∈ s₁) (h₂ : p ∈ s₂)
Goal: s₁ = s₂ ↔ s₁.direction = s₂.direction
-- AffineSubspace.self_mem_mk'
theorem self_mem_mk' (p : P) (direction : Submodule k V) : p ∈ mk' p direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P) (direction : Submodule k V)
Goal: p ∈ AffineSubspace.mk' p direction
-- AffineSubspace.vadd_mem_mk'
theorem vadd_mem_mk' {v : V} (p : P) {direction : Submodule k V} (hv : v ∈ direction) :
    v +ᵥ p ∈ mk' p direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {v : V} (p : P) {direction : Submodule k V} (hv : v ∈ direction)
Goal: v +ᵥ p ∈ AffineSubspace.mk' p direction
-- AffineSubspace.mk'_nonempty
theorem mk'_nonempty (p : P) (direction : Submodule k V) : (mk' p direction : Set P).Nonempty 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P) (direction : Submodule k V)
Goal: (↑(AffineSubspace.mk' p direction)).Nonempty
-- AffineSubspace.direction_mk'
@[simp]
theorem direction_mk' (p : P) (direction : Submodule k V) :
    (mk' p direction).direction = direction 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] (p : P) (direction : Submodule k V)
Goal: (AffineSubspace.mk' p direction).direction = direction
-- AffineSubspace.mem_mk'_iff_vsub_mem
theorem mem_mk'_iff_vsub_mem {p₁ p₂ : P} {direction : Submodule k V} :
    p₂ ∈ mk' p₁ direction ↔ p₂ -ᵥ p₁ ∈ direction 
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} {direction : Submodule k V}
Goal: p₂ ∈ AffineSubspace.mk' p₁ direction ↔ p₂ -ᵥ p₁ ∈ direction
-- AffineSubspace.mk'_eq
@[simp]
theorem mk'_eq {s : AffineSubspace k P} {p : P} (hp : p ∈ s) : mk' p s.direction = 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 : AffineSubspace k P} {p : P} (hp : p ∈ s)
Goal: AffineSubspace.mk' p s.direction = s
-- AffineSubspace.spanPoints_subset_coe_of_subset_coe
theorem spanPoints_subset_coe_of_subset_coe {s : Set P} {s1 : AffineSubspace k P} (h : s ⊆ s1) :
    spanPoints k s ⊆ s1 
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} {s1 : AffineSubspace k P} (h : s ⊆ ↑s1)
Goal: spanPoints k s ⊆ ↑s1
-- Submodule.mem_toAffineSubspace
@[simp]
theorem mem_toAffineSubspace {p : Submodule k V} {x : V} :
    x ∈ p.toAffineSubspace ↔ x ∈ p 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {p : Submodule k V} {x : V}
Goal: x ∈ p.toAffineSubspace ↔ x ∈ p
-- Submodule.toAffineSubspace_direction
@[simp]
theorem toAffineSubspace_direction (s : Submodule k V) : s.toAffineSubspace.direction = s 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] (s : Submodule k V)
Goal: s.toAffineSubspace.direction = s
-- AffineMap.lineMap_mem
theorem AffineMap.lineMap_mem {k V P : Type*} [Ring k] [AddCommGroup V] [Module k V]
    [AddTorsor V P] {Q : AffineSubspace k P} {p₀ p₁ : P} (c : k) (h₀ : p₀ ∈ Q) (h₁ : p₁ ∈ Q) :
    AffineMap.lineMap p₀ p₁ c ∈ Q 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {Q : AffineSubspace k P} {p₀ p₁ : P} (c : k) (h₀ : p₀ ∈ Q) (h₁ : p₁ ∈ Q)
Goal: (AffineMap.lineMap p₀ p₁) c ∈ Q
-- coe_affineSpan
@[simp]
theorem coe_affineSpan (s : Set P) : (affineSpan k s : Set P) = spanPoints k 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) = spanPoints k s
-- subset_affineSpan
theorem subset_affineSpan (s : Set P) : s ⊆ affineSpan k 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: s ⊆ ↑(affineSpan k s)
-- direction_affineSpan
theorem direction_affineSpan (s : Set P) : (affineSpan k s).direction = vectorSpan k 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).direction = vectorSpan k s
-- mem_affineSpan
theorem mem_affineSpan {p : P} {s : Set P} (hp : p ∈ s) : p ∈ affineSpan k s 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [AffineSpace V P] {p : P} {s : Set P} (hp : p ∈ s)
Goal: p ∈ affineSpan k s
-- AffineSubspace.le_def
theorem le_def (s1 s2 : AffineSubspace k P) : s1 ≤ s2 ↔ (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.le_def'
theorem le_def' (s1 s2 : AffineSubspace k 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] (s1 s2 : AffineSubspace k P)
Goal: s1 ≤ s2 ↔ ∀ p ∈ s1, p ∈ s2
-- AffineSubspace.lt_def
theorem lt_def (s1 s2 : AffineSubspace k P) : s1 < s2 ↔ (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.not_le_iff_exists
theorem not_le_iff_exists (s1 s2 : AffineSubspace k 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] (s1 s2 : AffineSubspace k P)
Goal: ¬s1 ≤ s2 ↔ ∃ p ∈ s1, p ∉ s2
-- AffineSubspace.exists_of_lt
theorem exists_of_lt {s1 s2 : AffineSubspace k P} (h : s1 < s2) : ∃ p ∈ s2, p ∉ s1 
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: ∃ p ∈ s2, p ∉ s1
-- AffineSubspace.lt_iff_le_and_exists
theorem lt_iff_le_and_exists (s1 s2 : AffineSubspace k P) :
    s1 < s2 ↔ s1 ≤ s2 ∧ ∃ p ∈ s2, p ∉ s1 
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 ∧ ∃ p ∈ s2, p ∉ s1
-- AffineSubspace.eq_of_direction_eq_of_nonempty_of_le
theorem eq_of_direction_eq_of_nonempty_of_le {s₁ s₂ : AffineSubspace k P}
    (hd : s₁.direction = s₂.direction) (hn : (s₁ : Set P).Nonempty) (hle : s₁ ≤ s₂) : s₁ = 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} (hd : s₁.direction = s₂.direction) (hn : (↑s₁).Nonempty) (hle : s₁ ≤ s₂)
Goal: s₁ = s₂
-- AffineSubspace.affineSpan_eq_sInf
theorem affineSpan_eq_sInf (s : Set P) :
    affineSpan k s = sInf { s' : AffineSubspace k P | s ⊆ 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)
Goal: affineSpan k s = sInf {s' | s ⊆ ↑s'}
-- AffineSubspace.span_empty
@[simp]
theorem span_empty : affineSpan k (∅ : 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]
Goal: affineSpan k ∅ = ⊥
-- AffineSubspace.span_univ
@[simp]
theorem span_univ : affineSpan k (Set.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]
Goal: affineSpan k Set.univ = ⊤
-- affineSpan_le
theorem _root_.affineSpan_le {s : Set P} {Q : AffineSubspace k P} :
    affineSpan k s ≤ Q ↔ s ⊆ (Q : 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} {Q : AffineSubspace k P}
Goal: affineSpan k s ≤ Q ↔ s ⊆ ↑Q
-- AffineSubspace.coe_affineSpan_singleton
@[simp 1001] -- Porting note: this needs to take priority over `coe_affineSpan`
theorem coe_affineSpan_singleton (p : P) : (affineSpan k ({p} : Set P) : Set P) = {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: ↑(affineSpan k {p}) = {p}
-- AffineSubspace.mem_affineSpan_singleton
@[simp]
theorem mem_affineSpan_singleton : p₁ ∈ affineSpan k ({p₂} : Set P) ↔ p₁ = p₂ 
Arguments: (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {p₁ p₂ : P}
Goal: p₁ ∈ affineSpan k {p₂} ↔ p₁ = p₂
-- AffineSubspace.preimage_coe_affineSpan_singleton
@[simp]
theorem preimage_coe_affineSpan_singleton (x : P) :
    ((↑) : affineSpan k ({x} : Set P) → P) ⁻¹' {x} = univ 
Arguments: (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] (x : P)
Goal: Subtype.val ⁻¹' {x} = Set.univ
-- AffineSubspace.span_union
theorem span_union (s t : Set P) : affineSpan k (s ∪ t) = affineSpan k s ⊔ affineSpan k t 
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 t : Set P)
Goal: affineSpan k (s ∪ t) = affineSpan k s ⊔ affineSpan k t
-- AffineSubspace.span_iUnion
theorem span_iUnion {ι : Type*} (s : ι → Set P) :
    affineSpan k (⋃ i, s i) = ⨆ i, affineSpan k (s i) 
Arguments: (k : Type u_1) (V : Type u_2) {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] {ι : Type u_4} (s : ι → Set P)
Goal: affineSpan k (⋃ i, s i) = ⨆ i, affineSpan k (s i)
-- AffineSubspace.top_coe
@[simp]
theorem top_coe : ((⊤ : AffineSubspace k P) : Set P) = Set.univ 
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: ↑⊤ = Set.univ
-- AffineSubspace.mem_top
@[simp]
theorem mem_top (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_top
@[simp]
theorem direction_top : (⊤ : 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.bot_coe
@[simp]
theorem bot_coe : ((⊥ : AffineSubspace k P) : 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]
Goal: ↑⊥ = ∅
-- AffineSubspace.bot_ne_top
theorem bot_ne_top : (⊥ : 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]
Goal: ⊥ ≠ ⊤