ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.affineCombination_vsub
theorem affineCombination_vsub (w₁ w₂ : ι → k) (p : ι → P) :
    s.affineCombination k p w₁ -ᵥ s.affineCombination k p w₂ = s.weightedVSub p (w₁ - w₂) 
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 : Finset ι) (w₁ w₂ : ι → k) (p : ι → P)
Goal: (Finset.affineCombination k s p) w₁ -ᵥ (Finset.affineCombination k s p) w₂ = (s.weightedVSub p) (w₁ - w₂)
-- Finset.attach_affineCombination_of_injective
theorem attach_affineCombination_of_injective [DecidableEq P] (s : Finset P) (w : P → k) (f : s → P)
    (hf : Function.Injective f) :
    s.attach.affineCombination k f (w ∘ f) = (image f univ).affineCombination k id w 
Arguments: {k : Type u_1} {V : Type u_2} {P : Type u_3} [Ring k] [AddCommGroup V] [Module k V] [S : AffineSpace V P] [DecidableEq P] (s : Finset P) (w : P → k) (f : { x // x ∈ s } → P) (hf : Function.Injective f)
Goal: (Finset.affineCombination k s.attach f) (w ∘ f) = (Finset.affineCombination k (Finset.image f Finset.univ) id) w
-- Finset.attach_affineCombination_coe
theorem attach_affineCombination_coe (s : Finset P) (w : P → k) :
    s.attach.affineCombination k ((↑) : s → P) (w ∘ (↑)) = s.affineCombination k id w 
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 : Finset P) (w : P → k)
Goal: (Finset.affineCombination k s.attach Subtype.val) (w ∘ Subtype.val) = (Finset.affineCombination k s id) w
-- Finset.weightedVSub_eq_linear_combination
@[simp]
theorem weightedVSub_eq_linear_combination {ι} (s : Finset ι) {w : ι → k} {p : ι → V}
    (hw : s.sum w = 0) : s.weightedVSub p w = ∑ i ∈ s, w i • p i 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {ι : Type u_6} (s : Finset ι) {w : ι → k} {p : ι → V} (hw : s.sum w = 0)
Goal: (s.weightedVSub p) w = ∑ i ∈ s, w i • p i
-- Finset.affineCombination_eq_linear_combination
@[simp]
theorem affineCombination_eq_linear_combination (s : Finset ι) (p : ι → V) (w : ι → k)
    (hw : ∑ i ∈ s, w i = 1) : s.affineCombination k p w = ∑ i ∈ s, w i • p i 
Arguments: {k : Type u_1} {V : Type u_2} [Ring k] [AddCommGroup V] [Module k V] {ι : Type u_4} (s : Finset ι) (p : ι → V) (w : ι → k) (hw : ∑ i ∈ s, w i = 1)
Goal: (Finset.affineCombination k s p) w = ∑ i ∈ s, w i • p i
-- Finset.affineCombination_of_eq_one_of_eq_zero
@[simp]
theorem affineCombination_of_eq_one_of_eq_zero (w : ι → k) (p : ι → P) {i : ι} (his : i ∈ s)
    (hwi : w i = 1) (hw0 : ∀ i2 ∈ s, i2 ≠ i → w i2 = 0) : s.affineCombination k p w = p 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 : Finset ι) (w : ι → k) (p : ι → P) {i : ι} (his : i ∈ s) (hwi : w i = 1) (hw0 : ∀ i2 ∈ s, i2 ≠ i → w i2 = 0)
Goal: (Finset.affineCombination k s p) w = p i
-- Finset.affineCombination_indicator_subset
theorem affineCombination_indicator_subset (w : ι → k) (p : ι → P) {s₁ s₂ : Finset ι}
    (h : s₁ ⊆ s₂) :
    s₁.affineCombination k p w = s₂.affineCombination k p (Set.indicator (↑s₁) w) 
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} (w : ι → k) (p : ι → P) {s₁ s₂ : Finset ι} (h : s₁ ⊆ s₂)
Goal: (Finset.affineCombination k s₁ p) w = (Finset.affineCombination k s₂ p) ((↑s₁).indicator w)
-- Finset.affineCombination_map
theorem affineCombination_map (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P) :
    (s₂.map e).affineCombination k p w = s₂.affineCombination k (p ∘ e) (w ∘ e) 
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} {ι₂ : Type u_5} (s₂ : Finset ι₂) (e : ι₂ ↪ ι) (w : ι → k) (p : ι → P)
Goal: (Finset.affineCombination k (Finset.map e s₂) p) w = (Finset.affineCombination k s₂ (p ∘ ⇑e)) (w ∘ ⇑e)
-- Finset.sum_smul_vsub_eq_affineCombination_vsub
theorem sum_smul_vsub_eq_affineCombination_vsub (w : ι → k) (p₁ p₂ : ι → P) :
    (∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) =
      s.affineCombination k p₁ w -ᵥ s.affineCombination k p₂ w 
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 : Finset ι) (w : ι → k) (p₁ p₂ : ι → P)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) = (Finset.affineCombination k s p₁) w -ᵥ (Finset.affineCombination k s p₂) w
-- Finset.sum_smul_vsub_const_eq_affineCombination_vsub
theorem sum_smul_vsub_const_eq_affineCombination_vsub (w : ι → k) (p₁ : ι → P) (p₂ : P)
    (h : ∑ i ∈ s, w i = 1) : (∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.affineCombination k p₁ w -ᵥ 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] {ι : Type u_4} (s : Finset ι) (w : ι → k) (p₁ : ι → P) (p₂ : P) (h : ∑ i ∈ s, w i = 1)
Goal: ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (Finset.affineCombination k s p₁) w -ᵥ p₂
-- Finset.sum_smul_const_vsub_eq_vsub_affineCombination
theorem sum_smul_const_vsub_eq_vsub_affineCombination (w : ι → k) (p₂ : ι → P) (p₁ : P)
    (h : ∑ i ∈ s, w i = 1) : (∑ i ∈ s, w i • (p₁ -ᵥ p₂ i)) = p₁ -ᵥ s.affineCombination k p₂ w 
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 : Finset ι) (w : ι → k) (p₂ : ι → P) (p₁ : P) (h : ∑ i ∈ s, w i = 1)
Goal: ∑ i ∈ s, w i • (p₁ -ᵥ p₂ i) = p₁ -ᵥ (Finset.affineCombination k s p₂) w
-- Finset.affineCombination_sdiff_sub
theorem affineCombination_sdiff_sub [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k)
    (p : ι → P) :
    (s \ s₂).affineCombination k p w -ᵥ s₂.affineCombination k p (-w) = s.weightedVSub p w 
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 : Finset ι) [DecidableEq ι] {s₂ : Finset ι} (h : s₂ ⊆ s) (w : ι → k) (p : ι → P)
Goal: (Finset.affineCombination k (s \ s₂) p) w -ᵥ (Finset.affineCombination k s₂ p) (-w) = (s.weightedVSub p) w
-- Finset.affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one
theorem affineCombination_eq_of_weightedVSub_eq_zero_of_eq_neg_one {w : ι → k} {p : ι → P}
    (hw : s.weightedVSub p w = (0 : V)) {i : ι} [DecidablePred (· ≠ i)] (his : i ∈ s)
    (hwi : w i = -1) : (s.filter (· ≠ i)).affineCombination k p w = p 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 : Finset ι) {w : ι → k} {p : ι → P} (hw : (s.weightedVSub p) w = 0) {i : ι} [DecidablePred fun x => x ≠ i] (his : i ∈ s) (hwi : w i = -1)
Goal: (Finset.affineCombination k (Finset.filter (fun x => x ≠ i) s) p) w = p i
-- Finset.affineCombination_subtype_eq_filter
theorem affineCombination_subtype_eq_filter (w : ι → k) (p : ι → P) (pred : ι → Prop)
    [DecidablePred pred] :
    ((s.subtype pred).affineCombination k (fun i => p i) fun i => w i) =
      (s.filter pred).affineCombination k p w 
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 : Finset ι) (w : ι → k) (p : ι → P) (pred : ι → Prop) [DecidablePred pred]
Goal: ((Finset.affineCombination k (Finset.subtype pred s) fun i => p ↑i) fun i => w ↑i) =
  (Finset.affineCombination k (Finset.filter pred s) p) w
-- Finset.affineCombination_filter_of_ne
theorem affineCombination_filter_of_ne (w : ι → k) (p : ι → P) {pred : ι → Prop}
    [DecidablePred pred] (h : ∀ i ∈ s, w i ≠ 0 → pred i) :
    (s.filter pred).affineCombination k p w = s.affineCombination k p w 
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 : Finset ι) (w : ι → k) (p : ι → P) {pred : ι → Prop} [DecidablePred pred] (h : ∀ i ∈ s, w i ≠ 0 → pred i)
Goal: (Finset.affineCombination k (Finset.filter pred s) p) w = (Finset.affineCombination k s p) w
-- Finset.eq_weightedVSubOfPoint_subset_iff_eq_weightedVSubOfPoint_subtype
theorem eq_weightedVSubOfPoint_subset_iff_eq_weightedVSubOfPoint_subtype {v : V} {x : k} {s : Set ι}
    {p : ι → P} {b : P} :
    (∃ fs : Finset ι, ↑fs ⊆ s ∧ ∃ w : ι → k, ∑ i ∈ fs, w i = x ∧
        v = fs.weightedVSubOfPoint p b w) ↔
      ∃ (fs : Finset s) (w : s → k), ∑ i ∈ fs, w i = x ∧
        v = fs.weightedVSubOfPoint (fun i : s => p i) b w 
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} {v : V} {x : k} {s : Set ι} {p : ι → P} {b : P}
Goal: (∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = x ∧ v = (fs.weightedVSubOfPoint p b) w) ↔
  ∃ fs w, ∑ i ∈ fs, w i = x ∧ v = (fs.weightedVSubOfPoint (fun i => p ↑i) b) w
-- Finset.eq_weightedVSub_subset_iff_eq_weightedVSub_subtype
theorem eq_weightedVSub_subset_iff_eq_weightedVSub_subtype {v : V} {s : Set ι} {p : ι → P} :
    (∃ fs : Finset ι, ↑fs ⊆ s ∧ ∃ w : ι → k, ∑ i ∈ fs, w i = 0 ∧
        v = fs.weightedVSub p w) ↔
      ∃ (fs : Finset s) (w : s → k), ∑ i ∈ fs, w i = 0 ∧
        v = fs.weightedVSub (fun i : s => p i) w 
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} {v : V} {s : Set ι} {p : ι → P}
Goal: (∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = 0 ∧ v = (fs.weightedVSub p) w) ↔
  ∃ fs w, ∑ i ∈ fs, w i = 0 ∧ v = (fs.weightedVSub fun i => p ↑i) w
-- Finset.eq_affineCombination_subset_iff_eq_affineCombination_subtype
theorem eq_affineCombination_subset_iff_eq_affineCombination_subtype {p0 : P} {s : Set ι}
    {p : ι → P} :
    (∃ fs : Finset ι, ↑fs ⊆ s ∧ ∃ w : ι → k, ∑ i ∈ fs, w i = 1 ∧
        p0 = fs.affineCombination k p w) ↔
      ∃ (fs : Finset s) (w : s → k), ∑ i ∈ fs, w i = 1 ∧
        p0 = fs.affineCombination k (fun i : s => p i) w 
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} {p0 : P} {s : Set ι} {p : ι → P}
Goal: (∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = 1 ∧ p0 = (Finset.affineCombination k fs p) w) ↔
  ∃ fs w, ∑ i ∈ fs, w i = 1 ∧ p0 = (Finset.affineCombination k fs fun i => p ↑i) w
-- Finset.map_affineCombination
theorem map_affineCombination {V₂ P₂ : Type*} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂]
    (p : ι → P) (w : ι → k) (hw : s.sum w = 1) (f : P →ᵃ[k] P₂) :
    f (s.affineCombination k p w) = s.affineCombination k (f ∘ p) w 
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 : Finset ι) {V₂ : Type u_6} {P₂ : Type u_7} [AddCommGroup V₂] [Module k V₂] [AffineSpace V₂ P₂] (p : ι → P) (w : ι → k) (hw : s.sum w = 1) (f : P →ᵃ[k] P₂)
Goal: f ((Finset.affineCombination k s p) w) = (Finset.affineCombination k s (⇑f ∘ p)) w
-- Finset.affineCombinationSingleWeights_apply_self
@[simp]
theorem affineCombinationSingleWeights_apply_self [DecidableEq ι] (i : ι) :
    affineCombinationSingleWeights k i i = 1 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] (i : ι)
Goal: Finset.affineCombinationSingleWeights k i i = 1
-- Finset.affineCombinationSingleWeights_apply_of_ne
@[simp]
theorem affineCombinationSingleWeights_apply_of_ne [DecidableEq ι] {i j : ι} (h : j ≠ i) :
    affineCombinationSingleWeights k i j = 0 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] {i j : ι} (h : j ≠ i)
Goal: Finset.affineCombinationSingleWeights k i j = 0
-- Finset.sum_affineCombinationSingleWeights
@[simp]
theorem sum_affineCombinationSingleWeights [DecidableEq ι] {i : ι} (h : i ∈ s) :
    ∑ j ∈ s, affineCombinationSingleWeights k i j = 1 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {i : ι} (h : i ∈ s)
Goal: ∑ j ∈ s, Finset.affineCombinationSingleWeights k i j = 1
-- Finset.weightedVSubVSubWeights_self
@[simp]
theorem weightedVSubVSubWeights_self [DecidableEq ι] (i : ι) :
    weightedVSubVSubWeights k i i = 0 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] (i : ι)
Goal: Finset.weightedVSubVSubWeights k i i = 0
-- Finset.weightedVSubVSubWeights_apply_left
@[simp]
theorem weightedVSubVSubWeights_apply_left [DecidableEq ι] {i j : ι} (h : i ≠ j) :
    weightedVSubVSubWeights k i j i = 1 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] {i j : ι} (h : i ≠ j)
Goal: Finset.weightedVSubVSubWeights k i j i = 1
-- Finset.weightedVSubVSubWeights_apply_right
@[simp]
theorem weightedVSubVSubWeights_apply_right [DecidableEq ι] {i j : ι} (h : i ≠ j) :
    weightedVSubVSubWeights k i j j = -1 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] {i j : ι} (h : i ≠ j)
Goal: Finset.weightedVSubVSubWeights k i j j = -1
-- Finset.weightedVSubVSubWeights_apply_of_ne
@[simp]
theorem weightedVSubVSubWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i) (hj : t ≠ j) :
    weightedVSubVSubWeights k i j t = 0 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} [DecidableEq ι] {i j t : ι} (hi : t ≠ i) (hj : t ≠ j)
Goal: Finset.weightedVSubVSubWeights k i j t = 0
-- Finset.sum_weightedVSubVSubWeights
@[simp]
theorem sum_weightedVSubVSubWeights [DecidableEq ι] {i j : ι} (hi : i ∈ s) (hj : j ∈ s) :
    ∑ t ∈ s, weightedVSubVSubWeights k i j t = 0 
Arguments: (k : Type u_1) [Ring k] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {i j : ι} (hi : i ∈ s) (hj : j ∈ s)
Goal: ∑ t ∈ s, Finset.weightedVSubVSubWeights k i j t = 0
-- Finset.affineCombinationLineMapWeights_self
@[simp]
theorem affineCombinationLineMapWeights_self [DecidableEq ι] (i : ι) (c : k) :
    affineCombinationLineMapWeights i i c = affineCombinationSingleWeights k i 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_4} [DecidableEq ι] (i : ι) (c : k)
Goal: Finset.affineCombinationLineMapWeights i i c = Finset.affineCombinationSingleWeights k i
-- Finset.affineCombinationLineMapWeights_apply_left
@[simp]
theorem affineCombinationLineMapWeights_apply_left [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k) :
    affineCombinationLineMapWeights i j c i = 1 - c 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_4} [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k)
Goal: Finset.affineCombinationLineMapWeights i j c i = 1 - c
-- Finset.affineCombinationLineMapWeights_apply_right
@[simp]
theorem affineCombinationLineMapWeights_apply_right [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k) :
    affineCombinationLineMapWeights i j c j = c 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_4} [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k)
Goal: Finset.affineCombinationLineMapWeights i j c j = c
-- Finset.affineCombinationLineMapWeights_apply_of_ne
@[simp]
theorem affineCombinationLineMapWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i)
    (hj : t ≠ j) (c : k) : affineCombinationLineMapWeights i j c t = 0 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_4} [DecidableEq ι] {i j t : ι} (hi : t ≠ i) (hj : t ≠ j) (c : k)
Goal: Finset.affineCombinationLineMapWeights i j c t = 0
-- Finset.sum_affineCombinationLineMapWeights
@[simp]
theorem sum_affineCombinationLineMapWeights [DecidableEq ι] {i j : ι} (hi : i ∈ s) (hj : j ∈ s)
    (c : k) : ∑ t ∈ s, affineCombinationLineMapWeights i j c t = 1 
Arguments: {k : Type u_1} [Ring k] {ι : Type u_4} (s : Finset ι) [DecidableEq ι] {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (c : k)
Goal: ∑ t ∈ s, Finset.affineCombinationLineMapWeights i j c t = 1
-- Finset.affineCombination_affineCombinationSingleWeights
@[simp]
theorem affineCombination_affineCombinationSingleWeights [DecidableEq ι] (p : ι → P) {i : ι}
    (hi : i ∈ s) : s.affineCombination k p (affineCombinationSingleWeights k i) = p 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 : Finset ι) [DecidableEq ι] (p : ι → P) {i : ι} (hi : i ∈ s)
Goal: (Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i) = p i
-- Finset.weightedVSub_weightedVSubVSubWeights
@[simp]
theorem weightedVSub_weightedVSubVSubWeights [DecidableEq ι] (p : ι → P) {i j : ι} (hi : i ∈ s)
    (hj : j ∈ s) : s.weightedVSub p (weightedVSubVSubWeights k i j) = p i -ᵥ p j 
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 : Finset ι) [DecidableEq ι] (p : ι → P) {i j : ι} (hi : i ∈ s) (hj : j ∈ s)
Goal: (s.weightedVSub p) (Finset.weightedVSubVSubWeights k i j) = p i -ᵥ p j
-- Finset.affineCombination_affineCombinationLineMapWeights
@[simp]
theorem affineCombination_affineCombinationLineMapWeights [DecidableEq ι] (p : ι → P) {i j : ι}
    (hi : i ∈ s) (hj : j ∈ s) (c : k) :
    s.affineCombination k p (affineCombinationLineMapWeights i j c) =
      AffineMap.lineMap (p i) (p j) c 
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 : Finset ι) [DecidableEq ι] (p : ι → P) {i j : ι} (hi : i ∈ s) (hj : j ∈ s) (c : k)
Goal: (Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i j c) = (AffineMap.lineMap (p i) (p j)) c
-- Finset.centroidWeights_apply
@[simp]
theorem centroidWeights_apply (i : ι) : s.centroidWeights k i = (card s : k)⁻¹ 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) (i : ι)
Goal: Finset.centroidWeights k s i = (↑s.card)⁻¹
-- Finset.centroidWeights_eq_const
theorem centroidWeights_eq_const : s.centroidWeights k = Function.const ι (card s : k)⁻¹ 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι)
Goal: Finset.centroidWeights k s = Function.const ι (↑s.card)⁻¹
-- Finset.sum_centroidWeights_eq_one_of_cast_card_ne_zero
theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
    ∑ i ∈ s, s.centroidWeights k i = 1 
Arguments: {k : Type u_1} [DivisionRing k] {ι : Type u_4} (s : Finset ι) (h : ↑s.card ≠ 0)
Goal: ∑ i ∈ s, Finset.centroidWeights k s i = 1
-- Finset.sum_centroidWeights_eq_one_of_card_ne_zero
theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s ≠ 0) :
    ∑ i ∈ s, s.centroidWeights k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] (h : s.card ≠ 0)
Goal: ∑ i ∈ s, Finset.centroidWeights k s i = 1
-- Finset.sum_centroidWeights_eq_one_of_nonempty
theorem sum_centroidWeights_eq_one_of_nonempty [CharZero k] (h : s.Nonempty) :
    ∑ i ∈ s, s.centroidWeights k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] (h : s.Nonempty)
Goal: ∑ i ∈ s, Finset.centroidWeights k s i = 1
-- Finset.sum_centroidWeights_eq_one_of_card_eq_add_one
theorem sum_centroidWeights_eq_one_of_card_eq_add_one [CharZero k] {n : ℕ} (h : card s = n + 1) :
    ∑ i ∈ s, s.centroidWeights k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] {n : ℕ} (h : s.card = n + 1)
Goal: ∑ i ∈ s, Finset.centroidWeights k s i = 1
-- Finset.centroid_def
theorem centroid_def (p : ι → P) : s.centroid k p = s.affineCombination k p (s.centroidWeights k) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (s : Finset ι) (p : ι → P)
Goal: Finset.centroid k s p = (Finset.affineCombination k s p) (Finset.centroidWeights k s)
-- Finset.centroid_univ
theorem centroid_univ (s : Finset P) : univ.centroid k ((↑) : s → P) = s.centroid k id 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] (s : Finset P)
Goal: Finset.centroid k Finset.univ Subtype.val = Finset.centroid k s id
-- Finset.centroid_singleton
@[simp]
theorem centroid_singleton (p : ι → P) (i : ι) : ({i} : Finset ι).centroid k p = p i 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (p : ι → P) (i : ι)
Goal: Finset.centroid k {i} p = p i
-- Finset.centroid_pair
theorem centroid_pair [DecidableEq ι] [Invertible (2 : k)] (p : ι → P) (i₁ i₂ : ι) :
    ({i₁, i₂} : Finset ι).centroid k p = (2⁻¹ : k) • (p i₂ -ᵥ p i₁) +ᵥ p i₁ 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} [DecidableEq ι] [Invertible 2] (p : ι → P) (i₁ i₂ : ι)
Goal: Finset.centroid k {i₁, i₂} p = 2⁻¹ • (p i₂ -ᵥ p i₁) +ᵥ p i₁
-- Finset.centroid_pair_fin
theorem centroid_pair_fin [Invertible (2 : k)] (p : Fin 2 → P) :
    univ.centroid k p = (2⁻¹ : k) • (p 1 -ᵥ p 0) +ᵥ p 0 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] [Invertible 2] (p : Fin 2 → P)
Goal: Finset.centroid k Finset.univ p = 2⁻¹ • (p 1 -ᵥ p 0) +ᵥ p 0
-- Finset.centroid_map
theorem centroid_map (e : ι₂ ↪ ι) (p : ι → P) :
    (s₂.map e).centroid k p = s₂.centroid k (p ∘ e) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} {ι₂ : Type u_5} (s₂ : Finset ι₂) (e : ι₂ ↪ ι) (p : ι → P)
Goal: Finset.centroid k (Finset.map e s₂) p = Finset.centroid k s₂ (p ∘ ⇑e)
-- Finset.centroidWeightsIndicator_def
theorem centroidWeightsIndicator_def :
    s.centroidWeightsIndicator k = Set.indicator (↑s) (s.centroidWeights k) 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι)
Goal: Finset.centroidWeightsIndicator k s = (↑s).indicator (Finset.centroidWeights k s)
-- Finset.sum_centroidWeightsIndicator
theorem sum_centroidWeightsIndicator [Fintype ι] :
    ∑ i, s.centroidWeightsIndicator k i = ∑ i ∈ s, s.centroidWeights k i 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [Fintype ι]
Goal: ∑ i : ι, Finset.centroidWeightsIndicator k s i = ∑ i ∈ s, Finset.centroidWeights k s i
-- Finset.sum_centroidWeightsIndicator_eq_one_of_card_ne_zero
theorem sum_centroidWeightsIndicator_eq_one_of_card_ne_zero [CharZero k] [Fintype ι]
    (h : card s ≠ 0) : ∑ i, s.centroidWeightsIndicator k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] [Fintype ι] (h : s.card ≠ 0)
Goal: ∑ i : ι, Finset.centroidWeightsIndicator k s i = 1
-- Finset.sum_centroidWeightsIndicator_eq_one_of_nonempty
theorem sum_centroidWeightsIndicator_eq_one_of_nonempty [CharZero k] [Fintype ι] (h : s.Nonempty) :
    ∑ i, s.centroidWeightsIndicator k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] [Fintype ι] (h : s.Nonempty)
Goal: ∑ i : ι, Finset.centroidWeightsIndicator k s i = 1
-- Finset.sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one
theorem sum_centroidWeightsIndicator_eq_one_of_card_eq_add_one [CharZero k] [Fintype ι] {n : ℕ}
    (h : card s = n + 1) : ∑ i, s.centroidWeightsIndicator k i = 1 
Arguments: (k : Type u_1) [DivisionRing k] {ι : Type u_4} (s : Finset ι) [CharZero k] [Fintype ι] {n : ℕ} (h : s.card = n + 1)
Goal: ∑ i : ι, Finset.centroidWeightsIndicator k s i = 1
-- Finset.centroid_eq_affineCombination_fintype
theorem centroid_eq_affineCombination_fintype [Fintype ι] (p : ι → P) :
    s.centroid k p = univ.affineCombination k p (s.centroidWeightsIndicator k) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (s : Finset ι) [Fintype ι] (p : ι → P)
Goal: Finset.centroid k s p = (Finset.affineCombination k Finset.univ p) (Finset.centroidWeightsIndicator k s)
-- Finset.centroid_eq_centroid_image_of_inj_on
theorem centroid_eq_centroid_image_of_inj_on {p : ι → P}
    (hi : ∀ i ∈ s, ∀ j ∈ s, p i = p j → i = j) {ps : Set P} [Fintype ps]
    (hps : ps = p '' ↑s) : s.centroid k p = (univ : Finset ps).centroid k fun x => (x : P) 
Arguments: (k : Type u_1) {V : Type u_2} {P : Type u_3} [DivisionRing k] [AddCommGroup V] [Module k V] [AffineSpace V P] {ι : Type u_4} (s : Finset ι) {p : ι → P} (hi : ∀ i ∈ s, ∀ j ∈ s, p i = p j → i = j) {ps : Set P} [Fintype ↑ps] (hps : ps = p '' ↑s)
Goal: Finset.centroid k s p = Finset.centroid k Finset.univ fun x => ↑x