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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ normSq (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2)

Tactic:
← @ofReal_inj 𝕜,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ↑(normSq (⟪x, y⟫ • x - ⟪x, x⟫ • y)) = ↑(normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2))
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ ‖MemLp.toLp ↑↑f ⋯‖ ≤ ‖f‖

Tactic:
rw [Lp.norm_def, Lp.norm_def, eLpNorm_congr_ae (MemLp.coeFn_toLp _)]

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
s : Set X
f : ↥(Lp E p μ)
⊢ (eLpNorm (↑↑f) p (μ.restrict s)).toReal ≤ (eLpNorm (↑↑f) p μ).toReal
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ im ⟪x, x⟫ = 0

Tactic:
← @ofReal_inj 𝕜,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ↑(im ⟪x, x⟫) = ↑0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∫ (x : X) in support F, F x ∂ν = ∫ (x : X) in univ, F x ∂ν

Tactic:
rw [setIntegral_eq_of_subset_of_forall_sdiff_eq_zero MeasurableSet.univ (subset_univ (support F))]

Goal state after:
X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∀ x ∈ univ \ support F, F x = 0
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
⊢ g i * ⟪v i, v i⟫ = ⟪v i, ∑ j ∈ s, g j • v j⟫

Tactic:
rw [inner_sum]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hz : ∀ (i : ι), v i ≠ 0
ho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
s : Finset ι
g : ι → 𝕜
hg : ∑ i ∈ s, g i • v i = 0
i : ι
hi : i ∈ s
⊢ g i * ⟪v i, v i⟫ = ∑ i_1 ∈ s, ⟪v i, g i_1 • v i_1⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖ ^ 2 + ‖x - y‖ ^ 2 = 2 * (‖x‖ ^ 2 + ‖y‖ ^ 2)

Tactic:
sq,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖ * ‖x + y‖ + ‖x - y‖ * ‖x - y‖ = 2 * (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖)
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖₊ ^ 2 + ‖x - y‖₊ ^ 2 = 2 * (‖x‖₊ ^ 2 + ‖y‖₊ ^ 2)

Tactic:
sq,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖x + y‖₊ * ‖x + y‖₊ + ‖x - y‖₊ * ‖x - y‖₊ = 2 * (‖x‖₊ * ‖x‖₊ + ‖y‖₊ * ‖y‖₊)
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : Fintype ι
l : (i : ι) → G i
i : ι
v : G i
⊢ ⟪(V i) v, ∑ j, (V j) (l j)⟫ = ∑ j, ⟪(V i) v, (V j) (l j)⟫

Tactic:
rw [inner_sum]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
ι : Type u_4
v : ι → E
hv : Function.Injective v
f : ι ≃ ↑(Set.range v) := Equiv.ofInjective v hv
h : Orthonormal 𝕜 v
⊢ Orthonormal 𝕜 (v ∘ ⇑(Equiv.ofInjective v hv).symm)

Tactic:
exact h.comp f.symm f.symm.injective

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
E' : Type u_7
inst✝¹ : SeminormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
v : ι → E
hv : Orthonormal 𝕜 v
f : E →ₗᵢ[𝕜] E'
⊢ Orthonormal 𝕜 (⇑f ∘ v)

Tactic:
by rwa [f.orthonormal_comp_iff]

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

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
x : E
h : ‖T x‖ ^ 2 / ‖T x‖ ≤ C
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C

Tactic:
sq,

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
T : E →L[𝕜] F
C : ℝ
hC : 0 ≤ C
x : E
h : ‖T x‖ * ‖T x‖ / ‖T x‖ ≤ C
hx : ‖x‖ = 1
hTx : ¬‖T x‖ = 0
⊢ ‖T x‖ ≤ C
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
⊢ K ⊓ Kᗮ = ⊥

Tactic:
rw [eq_bot_iff]
  intro x
  rw [mem_inf]
  exact fun ⟨hx, ho⟩ => inner_self_eq_zero.1 (ho x hx)

Goal state after:
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
f : E →ₗ[𝕜] E
hf : f.IsSymmetric
hf' : f ≠ 0
α : 𝕜
h : (α • f).IsSymmetric
⊢ IsSelfAdjoint α

Tactic:
simp only [ne_eq, LinearMap.ext_iff, zero_apply, ext_iff_inner_left 𝕜 (E := E),
    inner_zero_right] at hf'

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
f : E →ₗ[𝕜] E
hf : f.IsSymmetric
α : 𝕜
h : (α • f).IsSymmetric
hf' : ¬∀ (x v : E), ⟪v, f x⟫ = 0
⊢ IsSelfAdjoint α
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

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
inst✝ : CompleteSpace E
K₁ K₂ : ClosedSubmodule 𝕜 E
⊢ K₁ᗮ ⊔ K₂ᗮ = (K₁ ⊓ K₂)ᗮ

Tactic:
simpa using congr($(inf_orthogonal K₁ᗮ K₂ᗮ)ᗮ).symm

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 : WithLp 0 (α × β)
⊢ edist f f = 0

Tactic:
classical
    simp

Goal state after:
Module: Mathlib.Data.Matrix.Reflection

Goal state before:
pp: l m : ℕ
α : Type u_1
inst✝ : NonUnitalNonAssocSemiring α
A : Matrix (Fin l) (Fin m) α
v : Fin m → α
⊢ A.mulVecᵣ v = A *ᵥ v

Tactic:
by
  simp [mulVecᵣ]
  rfl

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
x✝ : (↑f).range.topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
hd : (↑f).range.topologicalClosure = ⊤
c : ℝ≥0
hf : AntilipschitzWith c ⇑f
⊢ Surjective ⇑f

Tactic:
← f.coe_coe,

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝⁷ : RingHomInvPair σ σ'
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
inst✝⁴ : CompleteSpace E
inst✝³ : CompleteSpace F
inst✝² : RingHomInvPair σ' σ
inst✝¹ : RingHomIsometric σ
inst✝ : RingHomIsometric σ'
f : E →SL[σ] F
x✝ : (↑f).range.topologicalClosure = ⊤ ∧ ∃ c, AntilipschitzWith c ⇑f
hd : (↑f).range.topologicalClosure = ⊤
c : ℝ≥0
hf : AntilipschitzWith c ⇑f
⊢ Surjective ⇑↑f
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case mpr.refine_2.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
⊢ x i = x i / a i * a i

Tactic:
rw [hai, inf_idem, sup_idem, ← le_antisymm_iff] at h

Goal state after:
case mpr.refine_2.inl
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a x : ι → ℝ
i : ι
h : 0 = x i
hai : a i = 0
⊢ x i = x i / a i * a i
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
𝕜 : Type u_5
inst✝ : SeminormedCommRing 𝕜
b : Basis ι 𝕜 (PiLp p fun x ↦ 𝕜)
A : Matrix ι ι 𝕜
this :
  (b.toMatrix fun x ↦
        (WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm (((basisFun p 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜))) x)) *
      A =
    of fun i j ↦ (b.repr ((WithLp.addEquiv p (ι → 𝕜)).invFun (A.col j))) i
⊢ b.toMatrix ⇑(basisFun p 𝕜 ι) * A = of fun i j ↦ (b.repr (toLp p (Aᵀ j))) i

Tactic:
Basis.map_apply,

Goal state after:
p : ℝ≥0∞
ι : Type u_2
inst✝¹ : Fintype ι
𝕜 : Type u_5
inst✝ : SeminormedCommRing 𝕜
b : Basis ι 𝕜 (PiLp p fun x ↦ 𝕜)
A : Matrix ι ι 𝕜
this :
  (b.toMatrix fun x ↦ (WithLp.linearEquiv p 𝕜 (ι → 𝕜)).symm ((WithLp.linearEquiv p 𝕜 (ι → 𝕜)) ((basisFun p 𝕜 ι) x))) *
      A =
    of fun i j ↦ (b.repr ((WithLp.addEquiv p (ι → 𝕜)).invFun (A.col j))) i
⊢ b.toMatrix ⇑(basisFun p 𝕜 ι) * A = of fun i j ↦ (b.repr (toLp p (Aᵀ j))) i
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ico (f p.1) (g p.1)}

Tactic:
by
  dsimp only [regionBetween, Ico, mem_ofPred_eq, ofPred_and]
  refine
    MeasurableSet.inter ?_
      ((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter
        (measurableSet_lt measurable_snd (hg.comp measurable_fst)))
  exact measurable_fst hs

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Fintype ι
⊢ fundamentalDomain b ⊆ parallelepiped ⇑b

Tactic:
rw [fundamentalDomain, parallelepiped_basis_eq, Set.ofPred_subset_ofPred]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝ : Fintype ι
⊢ ∀ (a : E), (∀ (i : ι), (b.repr a) i ∈ Set.Ico 0 1) → ∀ (i : ι), (b.repr a) i ∈ Set.Icc 0 1
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
⊢ Option.isSome ⊥ = true ↔ (↑⊥).Nonempty

Tactic:
unfold Option.isSome
    simp

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

Goal state before:
pp: K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
⊢ ⊤ ≤ span K (Set.range Subtype.val)

Tactic:
by
      rw [← hs.span_top, ← h_span]
      exact span_mono (by simp only [Subtype.range_coe_subtype, Set.ofPred_mem_eq, subset_rfl])

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

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : FiniteDimensional ℝ E
μ : Measure E
inst✝⁷ : μ.IsAddHaarMeasure
s : Set E
ι : Type u_2
G : Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : FiniteDimensional ℝ G
n : ℕ
_i : Fact (finrank ℝ G = n)
ω : G [⋀^Fin n]→ₗ[ℝ] ℝ
⊢ ω.measure.IsAddLeftInvariant

Tactic:
by
  rw [AlternatingMap.measure]; infer_instance

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

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : FiniteDimensional ℝ E
μ : Measure E
inst✝⁷ : μ.IsAddHaarMeasure
s : Set E
ι : Type u_2
G : Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : FiniteDimensional ℝ G
n : ℕ
_i : Fact (finrank ℝ G = n)
ω : G [⋀^Fin n]→ₗ[ℝ] ℝ
⊢ IsLocallyFiniteMeasure ω.measure

Tactic:
by
  rw [AlternatingMap.measure]; infer_instance

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

Goal state before:
pp: K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
this✝² : Module.Finite ℤ ↥L
this✝¹ : Module ℚ E
this✝ : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
h_spanE : span K (Set.range b) = ⊤
⊢ Fintype.card (Free.ChooseBasisIndex ℤ ↥L) = (Set.range b).toFinset.card

Tactic:
Set.toFinset_range,

Goal state after:
K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
hs : IsZLattice K L
this✝² : Module.Finite ℤ ↥L
this✝¹ : Module ℚ E
this✝ : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
h_spanE : span K (Set.range b) = ⊤
⊢ Fintype.card (Free.ChooseBasisIndex ℤ ↥L) = (Finset.image b Finset.univ).card
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

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

Tactic:
rw [← π.restrict_biUnion πi hJ]
    exact restrict_mono H

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

Goal state before:
pp: case refine_1
ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
π' : Prepartition I
H : π' ≤ π.biUnion πi
J : Box ι
hJ : J ∈ π
⊢ π'.restrict J ≤ πi J

Tactic:
rw [← π.restrict_biUnion πi hJ]
    exact restrict_mono H

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
A : Tendsto (fun n ↦ μ (t \ closedBall 0 ↑n)) atTop (𝓝 (μ (⋂ n, t \ closedBall 0 ↑n)))
⊢ t ∩ ∅ = ∅

Tactic:
inter_empty

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
⊢ Orthonormal 𝕜 ⇑b

Tactic:
by
  classical
    rw [orthonormal_iff_ite]
    intro i j
    rw [← b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j,
      EuclideanSpace.inner_single_left, PiLp.single_apply, map_one, one_mul]

Goal state after:
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
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
b : OrthonormalBasis ι 𝕜 ↥U
x : E
⊢ U.orthogonalProjectionOnto x = ∑ i, ⟪↑(b i), x⟫ • b i

Tactic:
simpa only [b.repr_apply_apply, inner_orthogonalProjectionOnto_eq_of_mem_left] using
    (b.sum_repr (U.orthogonalProjectionOnto x)).symm

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
v : ι → E
inst✝ : DecidableEq E
v' : ι' → E
h : Orthonormal 𝕜 v'
s : Finset ι'
e₀' : Basis (↥s) 𝕜 ↥(span 𝕜 (range (v' ∘ Subtype.val))) := Basis.span ⋯
e₀ : OrthonormalBasis (↥s) 𝕜 ↥(span 𝕜 (range (v' ∘ Subtype.val))) := OrthonormalBasis.mk ⋯ ⋯
⊢ span 𝕜 ↑(Finset.image v' s) = span 𝕜 (range (v' ∘ Subtype.val))

Tactic:
rw [Finset.coe_image, image_eq_range]
        rfl

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

Goal state before:
pp: K : Type u_1
inst✝¹⁴ : NormedField K
inst✝¹³ : LinearOrder K
inst✝¹² : IsStrictOrderedRing K
inst✝¹¹ : HasSolidNorm K
inst✝¹⁰ : FloorRing K
E✝ : Type u_2
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : NormedSpace K E✝
inst✝⁷ : FiniteDimensional K E✝
inst✝⁶ : ProperSpace E✝
L✝ : Submodule ℤ E✝
inst✝⁵ : DiscreteTopology ↥L✝
ι : Type u_3
hs : IsZLattice K L✝
b : Basis ι ℤ ↥L✝
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
⊢ Countable ↥L

Tactic:
simp_rw [← (Module.Free.chooseBasis ℤ L).ofZLatticeBasis_span ℝ]
  infer_instance

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:
by
  classical
  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.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[μ.restrict ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε

Tactic:
have := Fact.mk (I.measure_coe_lt_top μ)

Goal state after:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hf : f =ᵐ[μ.restrict ↑I] 0
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (μ ↑I < ∞)
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case pos
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
C₀ : ℝ
h₀ : ∀ x ∈ Box.Icc B, ‖F x‖ ≤ C₀
x : ι → ℝ
hx : x ∈ Box.Icc B
hs : x ∈ s
⊢ ‖F x‖ ≤ max 0 C₀

pp: case neg
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
C₀ : ℝ
h₀ : ∀ x ∈ Box.Icc B, ‖F x‖ ≤ C₀
x : ι → ℝ
hx : x ∈ Box.Icc B
hs : x ∉ s
⊢ ‖0‖ ≤ max 0 C₀

Tactic:
· exact le_max_of_le_right (h₀ x hx)

Goal state after:
case neg
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
C₀ : ℝ
h₀ : ∀ x ∈ Box.Icc B, ‖F x‖ ≤ C₀
x : ι → ℝ
hx : x ∈ Box.Icc B
hs : x ∉ s
⊢ ‖0‖ ≤ max 0 C₀