mathlib-initiative/mathlib-tactics huggingface dataset:
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.Ioi c⁻¹) μ
⊢ IntegrableOn (fun x ↦ f x⁻¹) (Set.Iio c) μ

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
⊢ tsupport ↑f = closedBall c f.rOut

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 ∈ ball c f.rOut
⊢ ↑f x ≠ 0

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.Iio c⁻¹) μ
⊢ IntegrableOn (fun x ↦ f x⁻¹) (Set.Ioi c) μ

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 : f.rOut ≤ dist x c
⊢ ↑f x = 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
c : E
f : ContDiffBump c
inst✝ : FiniteDimensional ℝ E
⊢ HasCompactSupport ↑f

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
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₂p₃ : p₂ ≠ p₃
⊢ vectorSpan k (insert p₁ s) = vectorSpan k {p₁, p₂, p₃}

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
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulLeftInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (g * x) ∂μ = ∫ (x : G), f x ∂μ

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
hp₂ : p₂ ∈ s
hp₃ : p₃ ∈ s
hp₂p₃ : p₂ ≠ p₃
⊢ Collinear k (insert p₁ s) ↔ Collinear k {p₁, p₂, p₃}

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
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (x * g) ∂μ = ∫ (x : G), f x ∂μ

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
p : P
h : p ∈ affineSpan k s
⊢ Collinear k (insert p s) ↔ Collinear k 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
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p₁ p₂ p₃ : P
h : p₁ ∈ line[k, p₂, p₃]
⊢ Collinear k {p₁, p₂, p₃}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace ℝ X
inst✝ : HasContDiffBump E
n : ℕ∞
c g : X → E
s : Set X
f : (x : X) → ContDiffBump (c x)
x : X
hc : ContDiffWithinAt ℝ (↑n) c s x
hr : ContDiffWithinAt ℝ (↑n) (fun x ↦ (f x).rIn) s x
hR : ContDiffWithinAt ℝ (↑n) (fun x ↦ (f x).rOut) s x
hg : ContDiffWithinAt ℝ (↑n) g s x
⊢ ContDiffWithinAt ℝ (↑n) (fun x ↦ ↑(f x) (g x)) s x

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
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
f : G → E
g : G
⊢ ∫ (x : G), f (x / g) ∂μ = ∫ (x : G), f x ∂μ

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 p₄ : P
h₁ : p₁ ∈ line[k, p₃, p₄]
h₂ : p₂ ∈ line[k, p₃, p₄]
⊢ Collinear k {p₁, p₂, p₃, p₄}

Tactic:
by

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

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace ℝ X
inst✝ : HasContDiffBump E
n : ℕ∞
c g : X → E
f : (x : X) → ContDiffBump (c x)
hc : ContDiff ℝ (↑n) c
hr : ContDiff ℝ ↑n fun x ↦ (f x).rIn
hR : ContDiff ℝ ↑n fun x ↦ (f x).rOut
hg : ContDiff ℝ (↑n) g
⊢ ContDiff ℝ ↑n fun x ↦ ↑(f x) (g x)

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
μ : Measure G
f : G → E
g : G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulLeftInvariant
hf' : ∀ (x : G), f (g * x) = -f x
⊢ ∫ (x : G), f x ∂μ = 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
p₁ p₂ p₃ p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₁, p₂, p₃, p₄, p₅}

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
μ : Measure G
f : G → E
g : G
inst✝² : Group G
inst✝¹ : MeasurableMul G
inst✝ : μ.IsMulRightInvariant
hf' : ∀ (x : G), f (x * g) = -f x
⊢ ∫ (x : G), f x ∂μ = 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
p₁ p₂ p₃ p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₁, p₂, p₃, p₄}

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 : MeasurableMul G
f : G → F
inst✝ : μ.IsMulRightInvariant
hf : Integrable f μ
g : G
⊢ Integrable (fun t ↦ f (t / g)) 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 : Group G
inst✝³ : MeasurableMul G
inst✝² : MeasurableInv G
f : G → F
inst✝¹ : μ.IsInvInvariant
inst✝ : μ.IsMulLeftInvariant
g : G
⊢ Integrable (fun t ↦ f (g / t)) μ ↔ Integrable f μ

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 p₄ p₅ : P
h₁ : p₁ ∈ line[k, p₄, p₅]
h₂ : p₂ ∈ line[k, p₄, p₅]
h₃ : p₃ ∈ line[k, p₄, p₅]
⊢ Collinear k {p₁, p₂, p₃}

Tactic:
by

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

Goal state before:
pp: G : Type u_4
E : Type u_5
inst✝⁷ : MeasurableSpace G
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : Group G
inst✝³ : MeasurableMul G
inst✝² : MeasurableInv G
f : G → E
μ : Measure G
inst✝¹ : μ.IsInvInvariant
inst✝ : μ.IsMulLeftInvariant
x' : G
⊢ ∫ (x : G), f (x' / x) ∂μ = ∫ (x : G), f x ∂μ

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 : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₂ ∈ {p₁, p₂, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p ∈ {p₁, p₂, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₁ ∈ {p₁, p₂, p}

Tactic:
by

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

Goal state before:
pp: α : Type u_3
E : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
G : Type u_7
inst✝⁴ : Group G
inst✝³ : MeasurableSpace α
inst✝² : MulAction G α
inst✝¹ : MeasurableConstSMul G α
μ : Measure α
inst✝ : SMulInvariantMeasure G α μ
f : α → E
g : G
⊢ ∫ (x : α), f (g • x) ∂μ = ∫ (x : α), f x ∂μ

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 : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₂ ∈ {p₂, p₃, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p ∈ {p₂, p₃, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ p₃ ∈ {p₂, p₃, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
⊢ Collinear k {p₁, p₃, p₂, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
h : Collinear k {p₁, p₂, p}
h1 : Collinear k {p₁, p₃, p₂, p}
⊢ {p₁, p₂, p₃} ⊆ {p₁, p₃, p₂, 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
p₁ p₂ p₃ p : P
ha : AffineIndependent k ![p₁, p₂, p₃]
hcol : Collinear k {p₂, p₃, p}
hne : p₂ ≠ p
⊢ AffineIndependent k ![p₁, p₂, 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
p₁ p₂ p₃ p : P
hcol : Collinear k {p₂, p, p₃}
hne1 : p₂ ≠ p
hne2 : p₂ ≠ p₃
⊢ AffineIndependent k ![p₁, p₂, p] ↔ AffineIndependent 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
s : Set P
h : Coplanar k s
⊢ FiniteDimensional k ↥(vectorSpan k 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
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)
⊢ Coplanar k s ↔ finrank k ↥(vectorSpan k s) ≤ 2

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
p : P
h : p ∈ affineSpan k s
⊢ Coplanar k (insert p s) ↔ Coplanar k 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
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : ¬FiniteDimensional k ↥s.direction
h : FiniteDimensional k ↥(vectorSpan k (insert p ↑s))
⊢ s.direction ≤ 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
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : ¬FiniteDimensional k ↥s.direction
⊢ ¬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
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
⊢ finrank k ↥(vectorSpan k (insert p ↑s)) ≤ finrank k ↥s.direction + 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
p : P
⊢ finrank k ↥(vectorSpan k (insert p s)) ≤ finrank k ↥(vectorSpan k s) + 1

Tactic:
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan, direction_affineSpan,
    ← direction_affineSpan _ s]
  exact finrank_vectorSpan_insert_le ..

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
p : P
⊢ finrank k ↥(vectorSpan k (insert p 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
s : Set P
h : Collinear k s
p : P
⊢ Coplanar 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
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : Set P
h : finrank k V = 2
⊢ Coplanar k s

Tactic:
have : FiniteDimensional k V := .of_finrank_eq_succ h
  rw [coplanar_iff_finrank_le_two, ← h]
  exact Submodule.finrank_le _

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 : finrank k V = 2
⊢ Coplanar k 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
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : CharZero k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
⊢ s.points i = -↑n • (s.faceOppositeCentroid i -ᵥ s.centroid) +ᵥ s.centroid

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✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : CharZero k
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
⊢ Collinear k {s.points i, s.centroid, s.faceOppositeCentroid i}

Tactic:
by

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

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : DivisionRing k
inst✝² : Module k V
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional k V
h : Fintype.card ι = Module.finrank k V + 1
⊢ Nonempty (AffineBasis ι k P)

Tactic:
by

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

Goal state before:
pp: R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
P : Type u_5
inst✝¹⁴ : Ring R
inst✝¹³ : Ring S
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : Module.Finite R V
inst✝⁹ : Module.Free R V
inst✝⁸ : AffineSpace V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : Module S W
inst✝⁴ : Module.Finite S W
inst✝³ : SMulCommClass R S W
inst✝² : Module.Free S W
inst✝¹ : StrongRankCondition R
inst✝ : StrongRankCondition S
⊢ Module.finrank S (W × (V →ₗ[R] W)) = (Module.finrank R V + 1) * Module.finrank S W

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
⊢ Memℓp f 0 ↔ {i | f i ≠ 0}.Finite

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
⊢ Memℓp f ∞ ↔ BddAbove (Set.range fun i ↦ ‖f i‖)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : (i : α) → E i
⊢ Memℓp f p ↔ Summable fun i ↦ ‖f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ENNReal.toReal 0
⊢ Summable fun x ↦ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ENNReal.toReal 0
H : Summable fun x ↦ 1
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ∞.toReal
⊢ Summable fun x ↦ 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ ∞.toReal
H : Summable fun x ↦ 1
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Summable fun i ↦ ‖f i‖ ^ p.toReal
⊢ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
C : ℝ
f : (i : α) → E i
hf : ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C
⊢ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hp : 0 < p.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hf : Summable fun i ↦ ‖f i‖ ^ p.toReal
⊢ ∀ (i : α), 0 ≤ ‖f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hp : 0 < p.toReal
⊢ Memℓp f p ↔ ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ ∑' (i : α), ‖f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hp : 0 < p.toReal
hf : Memℓp f p
i : α
⊢ 0 ≤ ‖f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hp : 0 < p.toReal
⊢ Memℓp f p ↔ ∃ C, 0 ≤ C ∧ ∀ (s : Finset α), ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : Finite α
f : (i : α) → E i
⊢ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
⊢ Memℓp 0 p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
⊢ Memℓp (fun x ↦ ‖f x‖) p ↔ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
g : α → ℝ
hg : Summable fun i ↦ ‖g i‖ ^ p.toReal
hfg : ∀ (i : α), ‖f i‖ ≤ ‖g i‖
hp : 0 < p.toReal
i : α
⊢ 0 ≤ ‖f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
g : α → ℝ
hg : Memℓp g p
hfg : ∀ (i : α), ‖f i‖ ≤ g i
⊢ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
x : α → E
hx : Memℓp x 1
⊢ Summable fun a ↦ ‖x a‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hf : Memℓp f p
⊢ Memℓp (-f) p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
f : (i : α) → E i
hp : 0 < p.toReal
hfq : Memℓp f 0
hpq : 0 ≤ p
i : α
hi : i ∉ ⋯.toFinset
⊢ f i = 0

Tactic:
by

Goal state after: