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

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I * I = -1
⊢ ⟪T x, y⟫ =
    (⟪T x, x⟫ + ⟪T x, y⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) - (⟪T x, x⟫ - ⟪T x, y⟫ - (⟪T y, x⟫ - ⟪T y, y⟫)) -
          I * (⟪T x, x⟫ + I * ⟪T x, y⟫ + (-I * ⟪T y, x⟫ + -I * (I * ⟪T y, y⟫))) +
        I * (⟪T x, x⟫ - I * ⟪T x, y⟫ - (-I * ⟪T y, x⟫ - -I * (I * ⟪T y, y⟫)))) /
      4

Tactic:
mul_add,

Goal state after:
case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
x y : E
h : I * I = -1
⊢ ⟪T x, y⟫ =
    (⟪T x, x⟫ + ⟪T x, y⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) - (⟪T x, x⟫ - ⟪T x, y⟫ - (⟪T y, x⟫ - ⟪T y, y⟫)) -
          (I * ⟪T x, x⟫ + I * (I * ⟪T x, y⟫) + (I * (-I * ⟪T y, x⟫) + I * (-I * (I * ⟪T y, y⟫)))) +
        I * (⟪T x, x⟫ - I * ⟪T x, y⟫ - (-I * ⟪T y, x⟫ - -I * (I * ⟪T y, y⟫)))) /
      4
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

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
x : E
⊢ K.reflection x = x ↔ K.starProjection x = x

Tactic:
reflection_apply,

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
x : E
⊢ 2 • K.starProjection x - x = x ↔ K.starProjection x = x
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
p q : E →ₗ[𝕜] E
hp : p.IsSymmetricProjection
hq : q.IsSymmetricProjection
hqp : q ∘ₗ p = p
x y : E
⊢ ⟪y, (p * q) x⟫ = ⟪y, p x⟫

Tactic:
simp_rw [Module.End.mul_apply, ← hp.isSymmetric _, ← hq.isSymmetric _, ← comp_apply, hqp]

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

Goal state before:
pp: case hvm
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ⟪v, w⟫ • v ∈ 𝕜 ∙ v

Tactic:
rw [Submodule.mem_span_singleton]
    use ⟪v, w⟫

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
U V : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
h : U.orthogonalProjectionOnto ∘SL V.subtypeL = 0
u : E
hu : u ∈ U
v : E
hv : v ∈ V
this : U.orthogonalProjectionOnto v = 0
⊢ v = v - U.starProjection v

Tactic:
rw [starProjection_apply, this, Submodule.coe_zero, sub_zero]

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

Goal state before:
pp: case mem
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
i : ι
x : E
hx : x ∈ V i
⊢ ∑ i, (V i).starProjection x = x

Tactic:
refine
      (Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => ?_).trans
        (starProjection_eq_self_iff.mpr hx)

Goal state after:
case mem
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✝¹ : E
i : ι
x : E
hx : x ∈ V i
j : ι
x✝ : j ∈ Finset.univ
hij : j ≠ i
⊢ (V j).starProjection x = 0
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
C : ℝ
Cpos : C > 0
hC : ∀ (y : F), ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
y : F
x : E
xs : x ∈ s
fxy : f x = y
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ s
z : F
hz : z ∈ ball y (ε / C)
w : E
wim : f w = z - y
wnorm : ‖w‖ ≤ C * ‖z - y‖
⊢ z ∈ ⇑f '' s

Tactic:
have : f (x + w) = z := by rw [f.map_add, wim, fxy, add_sub_cancel]

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
C : ℝ
Cpos : C > 0
hC : ∀ (y : F), ∃ x, f x = y ∧ ‖x‖ ≤ C * ‖y‖
y : F
x : E
xs : x ∈ s
fxy : f x = y
ε : ℝ
εpos : ε > 0
hε : ball x ε ⊆ s
z : F
hz : z ∈ ball y (ε / C)
w : E
wim : f w = z - y
wnorm : ‖w‖ ≤ C * ‖z - y‖
this : f (x + w) = z
⊢ z ∈ ⇑f '' s
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(c • f).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal ≤ ‖c‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal)

Tactic:
mul_add,

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(c • f).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal ≤
    ‖c‖₊ ^ p.toReal * ‖f.fst‖₊ ^ p.toReal + ‖c‖₊ ^ p.toReal * ‖f.snd‖₊ ^ p.toReal
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → AddCommGroup (β i)
p : ℝ≥0∞
i : ι
a b : β i
⊢ single p i (a - b) = single p i a - single p i b

Tactic:
simp_rw [← toLp_single, Pi.single_sub, toLp_sub]

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

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(c • f).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal = ‖c‖₊ ^ p.toReal * (‖f.fst‖₊ ^ p.toReal + ‖f.snd‖₊ ^ p.toReal)

Tactic:
mul_add,

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖(c • f).fst‖₊ ^ p.toReal + ‖(c • f).snd‖₊ ^ p.toReal =
    ‖c‖₊ ^ p.toReal * ‖f.fst‖₊ ^ p.toReal + ‖c‖₊ ^ p.toReal * ‖f.snd‖₊ ^ p.toReal
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : CompleteSpace E
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace F
f : E →L[𝕜] F
G : Submodule 𝕜 F
h : IsCompl (↑f).range G
inst✝ : CompleteSpace ↥G
hker : (↑f).ker = ⊥
⊢ (↑f).range = Submodule.map (↑↑(f.coprodSubtypeLEquivOfIsCompl h hker)) (⊤.prod ⊥)

Tactic:
by
  rw [coprodSubtypeLEquivOfIsCompl, ← ContinuousLinearEquiv.toLinearMap_toContinuousLinearMap,
    ContinuousLinearEquiv.coe_ofBijective, coe_coprod, LinearMap.coprod_map_prod, Submodule.map_bot,
    sup_bot_eq, Submodule.map_top]

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

Goal state before:
pp: ι : Type u_2
β : ι → Type u_4
inst✝¹ : Fintype ι
inst✝ : (i : ι) → PseudoEMetricSpace (β i)
x y : WithLp ∞ ((i : ι) → β i)
⊢ edist x.ofLp y.ofLp ≤ edist x y

Tactic:
simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed[instTopologicalSpaceProd] ↑g.graph
this✝ : CompleteSpace ↥g.graph := completeSpace_coe_iff_isComplete.mpr (IsClosed.isComplete hg)
φ₀ : E →ₗ[𝕜] E × F := id.prod g
this : LeftInverse Prod.fst ⇑φ₀
φ : E ≃ₗ[𝕜] ↥g.graph := LinearEquiv.ofLeftInverse this ≪≫ₗ LinearEquiv.ofEq φ₀.range g.graph ⋯
ψ : ↥g.graph ≃L[𝕜] E := φ.symm.toContinuousLinearEquivOfContinuous ⋯
⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace]
    ⇑g

Tactic:
exact (continuous_subtype_val.comp ψ.symm.continuous).snd

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

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ∑ i, ‖(c • f).ofLp i‖₊ ^ p.toReal ≤ ∑ x, (‖c‖₊ * ‖f.ofLp x‖₊) ^ p.toReal

Tactic:
smul_apply

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ∑ i, ‖c • f.ofLp i‖₊ ^ p.toReal ≤ ∑ x, (‖c‖₊ * ‖f.ofLp x‖₊) ^ p.toReal
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case mpr.refine_1.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)
hai : a i ≤ 0
⊢ 0 ≤ (fun i ↦ x i / a i) i ∧ (fun i ↦ x i / a i) i ≤ 1

Tactic:
rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
        exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁴ : NormedField K
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
b : Basis ι K E
F : Type u_4
inst✝¹ : AddCommGroup F
inst✝ : Module K F
f : E ≃ₗ[K] F
⊢ Submodule.map (↑(LinearEquiv.restrictScalars ℤ f)) (span ℤ (Set.range ⇑b)) = span ℤ (Set.range ⇑(b.map f))

Tactic:
by
  simp_rw [Submodule.map_span, LinearEquiv.coe_coe, LinearEquiv.restrictScalars_apply,
    Basis.coe_map, Set.range_comp]

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ x ∈ fundamentalDomain (b.map f)

Tactic:
mem_fundamentalDomain,

Goal state after:
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁵ : NormedField K
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
b : Basis ι K E
inst✝² : LinearOrder K
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
f : E ≃ₗ[K] F
x : F
⊢ x ∈ ⇑f '' fundamentalDomain b ↔ ∀ (i : ι), ((b.map f).repr x) i ∈ Set.Ico 0 1
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 (fundamentalDomain b)

Tactic:
(_ : fundamentalDomain b = b.equivFun.toLinearMap ⁻¹' D)

Goal state after:
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)

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.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ x ∉ fundamentalDomain b
⊢ x ∈ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))

Tactic:
mem_fundamentalDomain,

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this : FiniteDimensional ℝ E
x : E
hx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ¬∀ (i : ι), (b.repr x) i ∈ Set.Ico 0 1
⊢ x ∈ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'

Tactic:
simp [biUnion]

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

Goal state before:
pp: case inl
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
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (s ∩ ({x} + b • t)) / μ ({x} + b • t) < ε

Tactic:
filter_upwards with r

Goal state after:
case inl
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
r : ℝ
⊢ μ (s ∩ ({x} + r • t)) / μ ({x} + r • t) < ε
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
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
u v : Set E
uzero : μ u ≠ 0
utop : μ u ≠ ∞
vmeas : MeasurableSet v
⊢ μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u

Tactic:
simp_rw [div_eq_mul_inv]

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
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
u v : Set E
uzero : μ u ≠ 0
utop : μ u ≠ ∞
vmeas : MeasurableSet v
⊢ μ u * (μ u)⁻¹ - μ (vᶜ ∩ u) * (μ u)⁻¹ = μ (v ∩ u) * (μ u)⁻¹
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝ : AddCommMonoid M
I : Box ι
i : ι
x : ℝ
f : Box ι → M
⊢ ∑ J ∈ {I.splitLower i x, I.splitUpper i x}, Option.elim' 0 f J =
    Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)

Tactic:
Finset.sum_pair (I.splitLower_ne_splitUpper i x)

Goal state after:
ι : Type u_1
M : Type u_3
inst✝ : AddCommMonoid M
I : Box ι
i : ι
x : ℝ
f : Box ι → M
⊢ Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) =
    Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)
Module: Mathlib.Analysis.BoxIntegral.Partition.Measure

Goal state before:
pp: ι : Type u_1
inst✝¹ : Finite ι
I : Box ι
π : Prepartition I
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∀ b ∈ π.boxes, MeasurableSet ↑b

Tactic:
exact fun J _ => J.measurableSet_coe

Goal state after:
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
x : E
⊢ oscillationWithin f univ x = 0 ↔ ContinuousWithinAt f univ x

Tactic:
exact OscillationWithin.eq_zero_iff_continuousWithinAt f (mem_univ x)

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

Goal state before:
pp: ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I : Box ι
i : ι
inst✝ : Finite ι
f : Box ι → M
I₀ : WithTop (Box ι)
hf :
  ∀ (I : Box ι),
    ↑I ≤ I₀ →
      ∀ {i : ι} {x : ℝ},
        x ∈ Set.Ioo (I.lower i) (I.upper i) →
          Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I
⊢ ι →ᵇᵃ[I₀] M

Tactic:
refine ⟨f, ?_⟩
  replace hf (I : Box ι) (hI : ↑I ≤ I₀) (s) : ∑ J ∈ (splitMany I s).boxes, f J = f I := by
    induction s using Finset.induction_on with
    | empty => simp
    | insert a s _ ihs =>
      rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
      refine Finset.sum_congr rfl fun J' hJ' => ?_
      by_cases h : a.2 ∈ Ioo (J'.lower a.1) (J'.upper a.1)
      · rw [sum_split_boxes]
        exact hf _ ((WithTop.coe_le_coe.2 <| le_of_mem _ hJ').trans hI) h
      · rw [split_of_notMem_Ioo h, top_boxes, Finset.sum_singleton]
  intro I hI π hπ
  have Hle : ∀ J ∈ π, ↑J ≤ I₀ := fun J hJ => (WithTop.coe_le_coe.2 <| π.le_of_mem hJ).trans hI
  rcases hπ.exists_splitMany_le with ⟨s, hs⟩
  rw [← hf _ hI, ← inf_of_le_right hs, inf_splitMany, biUnion_boxes, sum_biUnion_boxes]
  exact Finset.sum_congr rfl fun J hJ => (hf _ (Hle _ hJ) _).symm

Goal state after:
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
y : E
hy : y ∈ eball x (ENNReal.ofReal ((a - r) / 2))
z : E
hz : z ∈ eball y (ENNReal.ofReal (a - (a - r) / 2)) ∩ D
⊢ ENNReal.ofReal (a - (a - r) / 2) + ENNReal.ofReal ((a - r) / 2) = ENNReal.ofReal a

Tactic:
← ofReal_add (by linarith) (by linarith),

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
⊢ ENNReal.ofReal (a - (a - r) / 2 + (a - r) / 2) = ENNReal.ofReal a
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
ι : Type u_7
inst✝ : Fintype ι
card_ι : finrank 𝕜 E = Fintype.card ι
v : ι → E
s : Set ι
hv : Orthonormal 𝕜 (s.domRestrict v)
⊢ ∃ b, ∀ i ∈ s, b i = v i

Tactic:
by
  have hsv : Injective (s.domRestrict v) := hv.linearIndependent.injective
  have hX : Orthonormal 𝕜 ((↑) : Set.range (s.domRestrict v) → E) := by
    rwa [orthonormal_subtype_range hsv]
  obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormalBasis_extension
  have hιY : Fintype.card ι = Y.card := by
    refine card_ι.symm.trans ?_
    exact Module.finrank_eq_card_finset_basis b₀.toBasis
  have hvsY : s.MapsTo v Y := (s.mapsTo_image v).mono_right (by rwa [← range_domRestrict])
  have hsv' : Set.InjOn v s := by
    rw [Set.injOn_iff_injective]
    exact hsv
  obtain ⟨g, hg⟩ := hvsY.exists_equiv_extend_of_card_eq hιY hsv'
  use b₀.reindex g.symm
  intro i hi
  simp [hb₀, hg i hi]

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : Fintype ι
inst✝¹ : FiniteDimensional 𝕜 E
n : ℕ
hn : finrank 𝕜 E = n
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
⊢ {a | subordinateOrthonormalBasisIndex hn hV a hV' = i}.card = finrank 𝕜 ↥(V i)

Tactic:
apply Finset.card_eq_of_equiv_fin
  simpa using hV.subordinateOrthonormalBasisIndexFiberEquiv hn hV' i

Goal state after: