mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p✝ : P
h : ∀ (i : Fin (n + 1)), p✝ ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p✝ = (p✝ -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p✝ -ᵥ s.centroid ∈ k ∙ u i
p : Fin (n + 1) → P := fun x ↦ if x = i₀ then s.centroid else s.points x
hindep : AffineIndependent k p
h1 :
  LinearIndependent k fun i ↦ (if ↑i = i₀ then s.centroid else s.points ↑i) -ᵥ if True then s.centroid else s.points i₀
f : ↥{i₀}ᶜ → { x // x ≠ i₀ } :=
  have h := ⋯;
  fun x ↦ ⟨↑x, ⋯⟩
f_inj : Function.Injective f
⊢ LinearIndependent k u

Tactic:
have h2 := h1.comp f f_inj

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p✝ : P
h : ∀ (i : Fin (n + 1)), p✝ ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p✝ = (p✝ -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p✝ -ᵥ s.centroid ∈ k ∙ u i
p : Fin (n + 1) → P := fun x ↦ if x = i₀ then s.centroid else s.points x
hindep : AffineIndependent k p
h1 :
  LinearIndependent k fun i ↦ (if ↑i = i₀ then s.centroid else s.points ↑i) -ᵥ if True then s.centroid else s.points i₀
f : ↥{i₀}ᶜ → { x // x ≠ i₀ } :=
  have h := ⋯;
  fun x ↦ ⟨↑x, ⋯⟩
f_inj : Function.Injective f
h2 :
  LinearIndependent k
    ((fun i ↦ (if ↑i = i₀ then s.centroid else s.points ↑i) -ᵥ if True then s.centroid else s.points i₀) ∘ f)
⊢ LinearIndependent k u
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
⊢ LinearIndependent k u

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ #s' = n

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s u : Set P
hu : IsOpen u
hsu : s ⊆ u
h : AffineIndependent ℝ Subtype.val
q : P
hq : q ∈ s
ε : ℝ
ε0 : 0 < ε
hεu : Metric.closedBall q ε ⊆ u
t : Set P
ht₁ : s ⊆ t
ht₂ : AffineIndependent ℝ fun p ↦ ↑p
ht₃ : affineSpan ℝ t = ⊤
f : P → P := fun y ↦ (lineMap q y) (ε / dist y q)
⊢ ∀ (y : P), f y ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
hcard : #s' = n
⊢ 1 < #s'

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
s : AffineSubspace k P
inst✝ : FiniteDimensional k ↥s.direction
p : P
⊢ FiniteDimensional k ↥(vectorSpan k (insert p ↑s))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
s : Set P
inst✝ : FiniteDimensional k ↥(vectorSpan k s)
p : P
⊢ FiniteDimensional k ↥(vectorSpan k (insert p s))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ ∃ i j, i ≠ j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
s : Set P
inst✝ : FiniteDimensional k ↥(affineSpan k s).direction
p : P
⊢ FiniteDimensional k ↥(affineSpan k (insert p s)).direction

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
i j : ↥s'
hij : i ≠ j
⊢ Disjoint {i} {j}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
s : Set P
inst✝ : FiniteDimensional k ↥(vectorSpan k s)
⊢ Collinear k s ↔ finrank k ↥(vectorSpan k s) ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
⊢ Collinear k ∅

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : P
⊢ Collinear k {p}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
i j : ↥s'
hij : i ≠ j
h_ij : Disjoint {i} {j}
⊢ Disjoint (k ∙ u i) (k ∙ u j)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
⊢ p = s.centroid

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
h : LinearIndependent k fun i ↦ s.points ↑i -ᵥ s.points 0
x✝ : { x // x ≠ 0 }
⊢ (-↑n)⁻¹ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
⊢ AffineIndependent k fun i ↦ s.faceOppositeCentroid i

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : DivisionRing k
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AffineSpace V P
m n : ℕ
inst✝² : NeZero m
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
e : Fin (n + 1) ≃ Fin (m + 1)
⊢ (s.reindex e).medial = s.medial.reindex e

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : DivisionRing k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : AffineSpace V P
V₂ : Type u_4
P₂ : Type u_5
inst✝⁴ : AddCommGroup V₂
inst✝³ : Module k V₂
inst✝² : AffineSpace V₂ P₂
inst✝¹ : CharZero k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
f : P →ᵃ[k] P₂
hf : Function.Injective ⇑f
⊢ (s.map f hf).medial = s.medial.map f hf

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
⊢ s.medial.points 0 ∈ Set.range s.medial.points

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
⊢ Set.range (s.faceOpposite 0).points ⊆ Set.range s.points

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
⊢ s.medial.points 0 ∈ affineSpan k (Set.range s.points)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
v : V
hp₀v : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀
w : V
hw : w ∈ vectorSpan k s
⊢ vectorSpan k s ≤ k ∙ v

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₀ : P
h : p₀ ∈ s
⊢ Collinear k s ↔ ∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
⊢ Collinear k s ↔ ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points)
this : Set.range s.medial.points -ᵥ Set.range s.medial.points = (-↑n)⁻¹ • (Set.range s.points -ᵥ Set.range s.points)
⊢ Submodule.span k (Set.range s.medial.points -ᵥ Set.range s.medial.points) =
    Submodule.span k (Set.range s.points -ᵥ Set.range s.points)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ : P
⊢ Collinear k {p₁, p₂}

Tactic:
rw [collinear_iff_exists_forall_eq_smul_vadd]
  use p₁, p₂ -ᵥ p₁
  intro p hp
  rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hp
  rcases hp with hp | hp
  · use 0
    simp [hp]
  · use 1
    simp [hp]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ : P
⊢ Collinear k {p₁, p₂}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
⊢ AffineIndependent k p ↔ ¬Collinear k (Set.range p)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
⊢ Collinear k (Set.range p) ↔ ¬AffineIndependent k p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s u : Set P
hu : IsOpen u
hsu : s ⊆ u
hne : s.Nonempty
h : AffineIndependent ℝ Subtype.val
⊢ ∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
⊢ AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k {p₁, p₂, p₃}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : Fin 3 → P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ Finset.univ = {i₁, i₂, i₃}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)
hmem2 : s.medial.points 0 ∈ affineSpan k (Set.range s.points)
v : V
this : (∃ a b, (↑n)⁻¹ • (s.points b -ᵥ s.points a) = v) ↔ ∃ a b, -((↑n)⁻¹ • (s.points a -ᵥ s.points b)) = v
⊢ v ∈ Set.range s.medial.points -ᵥ Set.range s.medial.points ↔ v ∈ (-↑n)⁻¹ • (Set.range s.points -ᵥ Set.range s.points)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
u : Set P
hu : IsOpen u
hne : u.Nonempty
⊢ ∃ s ⊆ u, AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ s = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
ι : Type u_3
inst✝ : Fintype ι
b : AffineBasis ι ℝ V
⊢ Finset.centroid ℝ Finset.univ ⇑b ∈ interior ((convexHull ℝ) (range ⇑b))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
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}

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₁ ≠ p₂

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
h : affineSpan ℝ s = ⊤
t : Set V
hts : t ⊆ s
b : AffineBasis ↑t ℝ V
hb : ⇑b = Subtype.val
this : (interior ((convexHull ℝ) t)).Nonempty
⊢ interior ((convexHull ℝ) t) ⊆ interior ((convexHull ℝ) s)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
⊢ affineSpan k (Set.range s.medial.points) = affineSpan k (Set.range s.points)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₁ ≠ p₃

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
h : affineSpan ℝ s = ⊤
t : Set V
hts : t ⊆ s
b : AffineBasis ↑t ℝ V
hb : ⇑b = Subtype.val
this : (interior ((convexHull ℝ) (range ⇑b))).Nonempty
⊢ (interior ((convexHull ℝ) s)).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : ¬Collinear k {p₁, p₂, p₃}
⊢ p₂ ≠ p₃

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
⊢ (interior ((convexHull ℝ) s)).Nonempty ↔ affineSpan ℝ s = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
hs : Convex ℝ s
⊢ (interior s).Nonempty ↔ affineSpan ℝ s = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
S : AffineSubspace k P
hS : affineSpan k (Set.range s.points) ≤ S
⊢ (s.restrict S hS).medial = s.medial.restrict S ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
c : E
f : ContDiffBump c
⊢ 1 < f.rOut / f.rIn

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
⊢ ↑f (c - x) = ↑f (c + x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
f : ContDiffBump 0
x : E
⊢ ↑f (-x) = ↑f x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
hx : x ∈ closedBall c f.rIn
⊢ ↑f x = 1

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
p₁ : P
hp₁ : p₁ ∈ s
v : V
h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁
r₂ : k
hp₂ : r₂ • v +ᵥ p₁ ∈ s
hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁
r₃ : k
hp₃ : r₃ • v +ᵥ p₁ ∈ s
⊢ r₂ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : 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
⊢ p₃ ∈ line[k, p₁, p₂]

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Basic

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
⊢ support ↑f = ball c f.rOut

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
E : Type u_5
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Group G
inst✝¹ : MeasurableInv G
f : G → E
μ : Measure G
inst✝ : μ.IsInvInvariant
⊢ ∫ (x : G), f x⁻¹ ∂μ = ∫ (x : G), f x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
F : Type u_6
inst✝⁴ : MeasurableSpace G
inst✝³ : NormedAddCommGroup F
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → F
s : Set G
hf : IntegrableOn f s μ
⊢ s⁻¹ = ⇑(MeasurableEquiv.inv G) ⁻¹' s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
F : Type u_6
inst✝⁴ : MeasurableSpace G
inst✝³ : NormedAddCommGroup F
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → F
s : Set G
hf : IntegrableOn f s μ
⊢ IntegrableOn (fun x ↦ f x⁻¹) s⁻¹ μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
F : Type u_6
inst✝⁶ : MeasurableSpace G
inst✝⁵ : NormedAddCommGroup F
μ : Measure G
inst✝⁴ : PartialOrder G
inst✝³ : CommGroup G
inst✝² : IsOrderedMonoid G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
c : G
f : G → F
hf : IntegrableOn f (Set.Ici c⁻¹) μ
⊢ IntegrableOn (fun x ↦ f x⁻¹) (Set.Iic c) μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
F : Type u_6
inst✝⁶ : MeasurableSpace G
inst✝⁵ : NormedAddCommGroup F
μ : Measure G
inst✝⁴ : PartialOrder G
inst✝³ : CommGroup G
inst✝² : IsOrderedMonoid G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
c : G
f : G → F
hf : IntegrableOn f (Set.Iic c⁻¹) μ
⊢ IntegrableOn (fun x ↦ f x⁻¹) (Set.Ici c) μ

Tactic:
by

Goal state after: