mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
u v : E
hvm : v ∈ K
hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0
hvs : K.starProjection u - v ∈ K
⊢ ⟪u - v, K.starProjection u - v⟫ - ⟪u - K.starProjection u, K.starProjection u - v⟫ = 0

Tactic:
starProjection_inner_eq_zero u _ hvs,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
u v : E
hvm : v ∈ K
hvo : ∀ w ∈ K, ⟪u - v, w⟫ = 0
hvs : K.starProjection u - v ∈ K
⊢ ⟪u - v, K.starProjection u - v⟫ - 0 = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
hK : K ≠ ⊥
x : E
hxK : x ∈ K
hx_ne_zero : x ≠ 0
⊢ 1 ≤ ‖K.orthogonalProjectionOnto‖

Tactic:
simpa [K.norm_orthogonalProjectionOnto_apply hxK, norm_eq_zero, hx_ne_zero]
    using K.orthogonalProjectionOnto.ratio_le_opNorm x

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
w : E
⊢ K.starProjection w + Kᗮ.starProjection w = w

Tactic:
by
  simp

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

Goal state before:
pp: case inl
α : Type u_2
β : Type u_3
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
f g : WithLp 0 (α × β)
⊢ edist f g = edist g f

Tactic:
simp only [prod_edist_eq_card, edist_comm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case zero
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : Fintype ι
V : ι → Submodule 𝕜 E
inst✝ : ∀ (i : ι), CompleteSpace ↥(V i)
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
x : E
⊢ ∑ x, 0 = 0

Tactic:
Finset.sum_const_zero

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
v : Set E
inst✝ : FiniteDimensional 𝕜 E
hv : Orthonormal 𝕜 Subtype.val
hv_coe : Set.range Subtype.val = v
⊢ (∃ b, ⇑b = Subtype.val) → span 𝕜 v = ⊤

Tactic:
· rintro ⟨h, coe_h⟩
    rw [← h.span_eq, coe_h, hv_coe]

Goal state after:
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: case left
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A : Matrix n n α
hA : A ∈ unitaryGroup n α
⊢ star A.det * A.det = 1

Tactic:
simpa [star, det_transpose] using congr_arg det hA.1

Goal state after:
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: case left
n : Type u
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
R : Type u_1
m : Type u_2
inst✝³ : Semiring R
inst✝² : StarRing R
inst✝¹ : Fintype m
inst✝ : DecidableEq m
U₁ : Matrix n n R
U₂ : Matrix m m R
hU₁ : U₁ ∈ unitary (Matrix n n R)
hU₂ : U₂ ∈ unitary (Matrix m m R)
i✝ j✝ : n × m
⊢ ∑ j, star U₂ i✝.2 j.2 * star U₁ i✝.1 j.1 * U₁ j.1 j✝.1 * U₂ j.2 j✝.2 = 1 i✝ j✝

Tactic:
simp_rw [mul_assoc _ (star U₁ _ _), ← Finset.univ_product_univ, Finset.sum_product]
    rw [Finset.sum_comm]
    simp_rw [← Finset.sum_mul, ← Finset.mul_sum, ← Matrix.mul_apply, hU₁.1, Matrix.one_apply,
      mul_boole, ite_mul, zero_mul, Finset.sum_ite_irrel, ← Matrix.mul_apply, hU₂.1,
      Matrix.one_apply, Finset.sum_const_zero, ← ite_and, Prod.eq_iff_fst_eq_snd_eq]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: this : StieltjesFunction.id.measure.IsAddLeftInvariant
⊢ StieltjesFunction.id.measure ↑(stdOrthonormalBasis ℝ ℝ).toBasis.parallelepiped = 1

Tactic:
change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis ℝ ℝ)) = 1

Goal state after:
this : StieltjesFunction.id.measure.IsAddLeftInvariant
⊢ StieltjesFunction.id.measure (parallelepiped ⇑(stdOrthonormalBasis ℝ ℝ)) = 1
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: a r : ℝ
⊢ ofReal (r + r) = ofReal (2 * r)

Tactic:
two_mul

Goal state after:
a r : ℝ
⊢ ofReal (r + r) = ofReal (r + r)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case inr
a : ℝ
r : ℝ≥0
⊢ ↑r + ↑r = 2 * ↑r

Tactic:
two_mul

Goal state after:
case inr
a : ℝ
r : ℝ≥0
⊢ ↑r + ↑r = ↑r + ↑r
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case hmul
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ ofReal |A.det⁻¹ * B.det⁻¹| • volume = ofReal |(A.det * B.det)⁻¹| • volume

pp: case hmul.hg
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ Measurable ⇑(toLin' A)

pp: case hmul.hf
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ Measurable ⇑(toLin' B)

Tactic:
mul_inv

Goal state after:
case hmul
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ ofReal |A.det⁻¹ * B.det⁻¹| • volume = ofReal |A.det⁻¹ * B.det⁻¹| • volume

case hmul.hg
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ Measurable ⇑(toLin' A)

case hmul.hf
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
A B : Matrix ι ι ℝ
a✝¹ : A.det ≠ 0
a✝ : B.det ≠ 0
IHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume
IHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume
⊢ Measurable ⇑(toLin' B)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ

Tactic:
simp only [fg, LinearEquiv.coe_coe, LinearMap.coe_comp]

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
f : E →ₗ[ℝ] E
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
g : (ι → ℝ) →ₗ[ℝ] ι → ℝ
hf : LinearMap.det g ≠ 0
hg : g = ↑e ∘ₗ f ∘ₗ ↑e.symm
gdet : LinearMap.det g = LinearMap.det f
fg : f = ↑e.symm ∘ₗ g ∘ₗ ↑e
⊢ map (⇑e.symm ∘ ⇑g ∘ ⇑e) μ = ENNReal.ofReal |(LinearMap.det g)⁻¹| • μ
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case intro
E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ MeasurableSet (⇑↑b.equivFun ⁻¹' D)

pp: E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ fundamentalDomain b = ⇑↑b.equivFun ⁻¹' D

Tactic:
· refine measurableSet_preimage (LinearMap.continuous_of_finiteDimensional _).measurable ?_
    exact MeasurableSet.pi Set.countable_univ fun _ _ => measurableSet_Ico

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
⊢ fundamentalDomain b = ⇑↑b.equivFun ⁻¹' D
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ENNReal.ofReal (2 ^ finrank ℝ E) * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall x r)

Tactic:
addHaar_closedBall_center μ x,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
r : ℝ
hr : r ∈ Ioi 0
x : E
⊢ ENNReal.ofReal (2 ^ finrank ℝ E) * μ (closedBall 0 r) ≤ ↑(2 ^ finrank ℝ E) * μ (closedBall 0 r)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: ι : Type u_2
G : Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : DecidableEq ι
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
b : Basis ι ℝ G
v : ι → G
this : FiniteDimensional ℝ G
A : parallelepiped v = ⇑((b.constr ℕ) v) '' parallelepiped ⇑b
⊢ b.addHaar (parallelepiped v) = ENNReal.ofReal |b.det v|

Tactic:
rw [A, addHaar_image_linearMap, b.addHaar_self, mul_one, ← LinearMap.det_toMatrix b,
    ← Basis.toMatrix_eq_toMatrix_constr, Basis.det_apply]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | J₁.lower i = x i} = {i | J₂.lower i = x i}
⊢ J₁ = J₂

Tactic:
suffices ∀ i, (Ioc (J₁.lower i) (J₁.upper i) ∩ Ioc (J₂.lower i) (J₂.upper i)).Nonempty by
    choose y hy₁ hy₂ using this
    exact π.eq_of_mem_of_mem h₁ h₂ hy₁ hy₂

Goal state after:
ι : Type u_1
I : Box ι
π : Prepartition I
x : ι → ℝ
J₁ : Box ι
h₁ : J₁ ∈ π
hx₁ : x ∈ Box.Icc J₁
J₂ : Box ι
h₂ : J₂ ∈ π
hx₂ : x ∈ Box.Icc J₂
H : {i | J₁.lower i = x i} = {i | J₂.lower i = x i}
⊢ ∀ (i : ι), (Set.Ioc (J₁.lower i) (J₁.upper i) ∩ Set.Ioc (J₂.lower i) (J₂.upper i)).Nonempty
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case pos
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
J : Box ι
hJ : J ∈ π.biUnion πi
⊢ π.biUnionIndex πi J ≤ I

pp: case neg
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
J : Box ι
hJ : J ∉ π.biUnion πi
⊢ π.biUnionIndex πi J ≤ I

Tactic:
· exact π.le_of_mem (π.biUnionIndex_mem hJ)

Goal state after:
case neg
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
J : Box ι
hJ : J ∉ π.biUnion πi
⊢ π.biUnionIndex πi J ≤ I
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case h1
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
⊢ Tendsto (fun r ↦ R * r) (𝓝[>] 0) (𝓝 (R * 0))

pp: case h2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (x : ℝ) in 𝓝[>] 0, R * x ∈ Ioi (R * 0)

Tactic:
· exact (tendsto_const_nhds.mul tendsto_id).mono_left nhdsWithin_le_nhds

Goal state after:
case h2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := ⋯
ht' : t' = R⁻¹ • t
u' : Set E := ⋯
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
⊢ ∀ᶠ (x : ℝ) in 𝓝[>] 0, R * x ∈ Ioi (R * 0)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq ι
i : ι
a : 𝕜
⊢ (single i a).ofLp = Pi.single i a

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq ι
i : ι
a : 𝕜
⊢ toLp 2 (Pi.single i a) = single i a

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝¹ : RCLike 𝕜
inst✝ : DecidableEq ι
i : ι
a : 𝕜
j : ι
⊢ (single i a).ofLp j = if j = i then a else 0

Tactic:
by
  simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
⊢ ∃ n, 0 < n ∧ μ (t \ closedBall 0 ↑n) < ε / 2 * μ t

Tactic:
have A :
      Tendsto (fun n : ℕ => μ (t \ closedBall 0 n)) atTop
        (𝓝 (μ (⋂ n : ℕ, t \ closedBall 0 n))) := by
      have N : ∃ n : ℕ, μ (t \ closedBall 0 n) ≠ ∞ :=
        ⟨0, ((measure_mono sdiff_subset).trans_lt h''t.lt_top).ne⟩
      refine tendsto_measure_iInter_atTop
        (fun n ↦ (ht.diff measurableSet_closedBall).nullMeasurableSet) (fun m n hmn ↦ ?_) N
      exact sdiff_subset_sdiff Subset.rfl (by gcongr)

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
A : Tendsto (fun n ↦ μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))
⊢ ∃ n, 0 < n ∧ μ (t \ closedBall 0 ↑n) < ε / 2 * μ t
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
a b : 𝕜
⊢ dist (single i a) (single i b) = dist a b

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
a b : 𝕜
⊢ nndist (single i a) (single i b) = nndist a b

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
i : ι
a b : 𝕜
⊢ edist (single i a) (single i b) = edist a b

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
i : ι
⊢ ⟪b i, b i⟫ = 1

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: K : Type u_1
inst✝⁴ : NormedField K
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
L : Submodule ℤ E
hL : span K ↑L = ⊤
e : F →ₗ[K] E
he : ↑L ⊆ ↑e.range
⊢ comap e (span K ↑L) = ⊤

Tactic:
hL,

Goal state after:
K : Type u_1
inst✝⁴ : NormedField K
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
L : Submodule ℤ E
hL : span K ↑L = ⊤
e : F →ₗ[K] E
he : ↑L ⊆ ↑e.range
⊢ comap e ⊤ = ⊤
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
r : ℝ
x✝¹ : r > 0
x : E
x✝ : x ∈ S r
a : ℝ
ar : a > r
ha : ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε
⊢ eball x (ENNReal.ofReal ((a - r) / 2)) ⊆ S r

Tactic:
refine fun y hy ↦ ⟨a - (a - r) / 2, by linarith,
      le_trans (ediam_mono (image_mono fun z hz ↦ ?_)) ha⟩

Goal state after:
E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : PseudoEMetricSpace E
K : Set E
f : E → F
D : Set E
ε : ℝ≥0∞
comp : IsCompact K
hK : ∀ x ∈ K, oscillationWithin f D x < ε
S : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}
r : ℝ
x✝¹ : r > 0
x : E
x✝ : x ∈ S r
a : ℝ
ar : a > r
ha : ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε
y : E
hy : y ∈ eball x (ENNReal.ofReal ((a - r) / 2))
z : E
hz : z ∈ eball y (ENNReal.ofReal (a - (a - r) / 2)) ∩ D
⊢ z ∈ eball x (ENNReal.ofReal a) ∩ D
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_7
E : Type u_8
F : Type u_9
ι : Type u_10
ι' : Type u_11
inst✝⁷ : RCLike 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
x : E
y : F
b : Basis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 F
this : Fintype ι
⊢ (toMatrix b'.toBasis b) (LinearMap.toSpanSingleton 𝕜 E x ∘ₗ ↑((innerSL 𝕜) y)) =
    Matrix.vecMulVec (⇑(b.repr x)) (star (b'.repr y).ofLp)

Tactic:
toMatrix_comp _ (OrthonormalBasis.singleton Unit 𝕜).toBasis,

Goal state after:
𝕜 : Type u_7
E : Type u_8
F : Type u_9
ι : Type u_10
ι' : Type u_11
inst✝⁷ : RCLike 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
x : E
y : F
b : Basis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 F
this : Fintype ι
⊢ (toMatrix (OrthonormalBasis.singleton Unit 𝕜).toBasis b) (LinearMap.toSpanSingleton 𝕜 E x) *
      (toMatrix b'.toBasis (OrthonormalBasis.singleton Unit 𝕜).toBasis) ↑((innerSL 𝕜) y) =
    Matrix.vecMulVec (⇑(b.repr x)) (star (b'.repr y).ofLp)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Unique ι
h : finrank 𝕜 E = 1
v : E
hv : ‖v‖ = 1
⊢ range ⇑(orthonormalBasisSingleton ι 𝕜 h v hv) = {v}

Tactic:
by
  simp

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
⊢ f n = gramSchmidt 𝕜 f n + ∑ i < n, (𝕜 ∙ gramSchmidt 𝕜 f i).starProjection (f n)

Tactic:
rw [gramSchmidt_def, sub_add_cancel]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
⊢ gramSchmidt 𝕜 f n ≠ 0

Tactic:
by_contra h
  have h₁ : f n ∈ span 𝕜 (f '' Set.Iio n) := by
    rw [← span_gramSchmidt_Iio 𝕜 f n, gramSchmidt_def' 𝕜 f, h, zero_add]
    apply Submodule.sum_mem _ _
    intro a ha
    simp only [starProjection_singleton]
    apply Submodule.smul_mem _ _ _
    rw [Finset.mem_Iio] at ha
    exact subset_span ⟨a, ha, by rfl⟩
  have h₂ : (f ∘ ((↑) : Set.Iic n → ι)) ⟨n, le_refl n⟩ ∈
      span 𝕜 (f ∘ ((↑) : Set.Iic n → ι) '' Set.Iio ⟨n, le_refl n⟩) := by
    rw [image_comp]
    simpa using h₁
  apply LinearIndependent.notMem_span_image h₀ _ h₂
  simp only [Set.mem_Iio, lt_self_iff_false, not_false_iff]

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case intro
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
x : Orientation R M ι
f : M ≃ₗ[R] M
val✝ : Fintype ι
⊢ (Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x

Tactic:
let := Classical.decEq ι

Goal state after:
case intro
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : Finite ι
e : Basis ι R M
x : Orientation R M ι
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
⊢ (Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det (↑(∏ i, w i)⁻¹ • (∏ i, ↑(w i)) • (e.unitsSMul w).det)

Tactic:
smul_smul

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det ((↑(∏ i, w i)⁻¹ * ∏ i, ↑(w i)) • (e.unitsSMul w).det)
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
⊢ Orthonormal ℝ ⇑(e.toBasis.adjustToOrientation x)

Tactic:
apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg
  simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x

Goal state after: