mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: 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
A : ↥(unitaryGroup n α)
x : n → α
⊢ (toLin' (A * A⁻¹)) x = x

Tactic:
by rw [mul_inv_cancel, toLin'_one, id_apply]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
⊢ IsUnit f → Bijective ⇑f

Tactic:
rintro ⟨f, rfl⟩

Goal state after:
case mp
𝕜 : Type u_1
inst✝³ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace E
f : (E →L[𝕜] E)ˣ
⊢ Bijective ⇑↑f
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 (↑↑(ContinuousLinearEquiv.ofBijective (f.coprod G.subtypeL) ⋯ ⋯)) (⊤.prod ⊥)

Tactic:
ContinuousLinearEquiv.coe_ofBijective,

Goal state after:
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.coprod G.subtypeL)) (⊤.prod ⊥)
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✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
x' : PiLp p fun x ↦ E
h : 1 ≤ p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 (ι → E) ≪≫ₗ
          (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv p 𝕜 (ι' → E)).symm))
        x'‖ =
    ‖x'‖

Tactic:
simp only [norm_eq_sum (zero_lt_one.trans_le h)]

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✝⁸ : Fintype ι
inst✝⁷ : Semiring 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (α i)
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝⁴ : (i : ι) → Module 𝕜 (α i)
inst✝³ : (i : ι) → Module 𝕜 (β i)
c : 𝕜
ι' : Type u_5
inst✝² : Fintype ι'
E : Type u_6
inst✝¹ : SeminormedAddCommGroup E
inst✝ : Module 𝕜 E
e : ι ≃ ι'
x' : PiLp p fun x ↦ E
h : 1 ≤ p.toReal
⊢ (∑ i,
        ‖((WithLp.linearEquiv p 𝕜 (ι → E) ≪≫ₗ
                    (LinearEquiv.piCongrLeft' 𝕜 (fun x ↦ E) e ≪≫ₗ (WithLp.linearEquiv p 𝕜 (ι' → E)).symm))
                  x').ofLp
              i‖ ^
          p.toReal) ^
      (1 / p.toReal) =
    (∑ i, ‖x'.ofLp i‖ ^ p.toReal) ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)
⊢ (∫⋯∫⁻_{t.i}, fun a ↦ (univ.pi s).indicator 1 ((toLin' t.toMatrix) a) ∂fun x ↦ volume) =
    ∫⋯∫⁻_{t.i}, (univ.pi s).indicator 1 ∂fun x ↦ volume

Tactic:
lmarginal_singleton

Goal state after:
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
t : TransvectionStruct ι ℝ
ht : Measurable ⇑(toLin' t.toMatrix)
s : ι → Set ℝ
hs : ∀ (i : ι), MeasurableSet (s i)
h2s : MeasurableSet (univ.pi s)
⊢ (fun x ↦ ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 ((toLin' t.toMatrix) (Function.update x t.i xᵢ))) = fun x ↦
    ∫⁻ (xᵢ : ℝ), (univ.pi s).indicator 1 (Function.update x t.i xᵢ)
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁷ : NormedField K
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace K E
b : Basis ι K E
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorRing K
inst✝¹ : Fintype ι
inst✝ : HasSolidNorm K
m : E
⊢ ‖fract b m‖ = ‖∑ i, (b.repr (fract b m)) i • b i‖

Tactic:
rw [b.sum_repr]

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

Goal state before:
pp: ι : Type u_2
inst✝ : Finite ι
⊢ DiscreteTopology ↥(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))

Tactic:
by
  cases nonempty_fintype ι
  refine discreteTopology_iff_isOpen_singleton_zero.mpr ⟨Metric.ball 0 1, Metric.isOpen_ball, ?_⟩
  ext x
  rw [Set.mem_preimage, mem_ball_zero_iff, pi_norm_lt_iff zero_lt_one, Set.mem_singleton_iff]
  simp_rw [← coe_eq_zero, funext_iff, Pi.zero_apply, Real.norm_eq_abs]
  refine forall_congr' (fun i => ?_)
  rsuffices ⟨y, hy⟩ : ∃ (y : ℤ), (y : ℝ) = (x : ι → ℝ) i
  · rw [← hy, ← Int.cast_abs, ← Int.cast_one, Int.cast_lt, Int.abs_lt_one_iff, Int.cast_eq_zero]
  exact ((Pi.basisFun ℝ ι).mem_span_iff_repr_mem ℤ x).mp (SetLike.coe_mem x) i

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 u : Set E
h'u : μ u ≠ 0
t_bound : t ⊆ closedBall 0 1
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t)) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
this : closedBall x r = {x} + r • closedBall 0 1
⊢ μ (closedBall x 1) / μ ({x} + u) = μ (closedBall 0 1) / μ u

Tactic:
simp only [addHaar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]

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

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

Tactic:
have : Module ℚ E := Module.compHom E (algebraMap ℚ ℝ)

Goal state after:
K : Type u_1
inst✝¹³ : NormedField K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsStrictOrderedRing K
inst✝¹⁰ : HasSolidNorm K
inst✝⁹ : FloorRing K
E✝ : Type u_2
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace K E✝
inst✝⁶ : FiniteDimensional K E✝
inst✝⁵ : ProperSpace E✝
L✝ : Submodule ℤ E✝
inst✝⁴ : DiscreteTopology ↥L✝
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
this : Module ℚ E
⊢ Free ℤ ↥L
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 u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
⊢ Tendsto (fun r ↦ μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0)

Tactic:
set t' := R⁻¹ • t with ht'

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 u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
⊢ Tendsto (fun r ↦ μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 0)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case inl
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
i : ι
H : ∀ (x_1 : ι), J₁.lower x_1 = x x_1 ↔ J₂.lower x_1 = x x_1
hi₁ : J₁.lower i = x i
hi₂ : J₂.lower i = x i
H₁ : x i < J₁.upper i
H₂ : x i < J₂.upper i
⊢ x i < min (J₁.upper i) (J₂.upper i)

Tactic:
exact lt_min H₁ H₂

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
π : Prepartition I
πi : (J : Box ι) → TaggedPrepartition J
hJ : J ∈ π
J' : Box ι
hJ' : J' ∈ πi J
⊢ (π.biUnionTagged πi).tag J' = (πi J).tag J'

Tactic:
by
  rw [← π.biUnionIndex_of_mem (πi := fun J => (πi J).toPrepartition) hJ hJ']
  rfl

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

Goal state before:
pp: case ofRepr
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
g : OrthonormalBasis ι 𝕜 E
repr✝ : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι
h : { repr := repr✝ }.repr = g.repr
⊢ { repr := repr✝ } = g

Tactic:
cases g

Goal state after:
case ofRepr.ofRepr
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
repr✝¹ repr✝ : E ≃ₗᵢ[𝕜] EuclideanSpace 𝕜 ι
h : { repr := repr✝¹ }.repr = { repr := repr✝ }.repr
⊢ { repr := repr✝¹ } = { repr := repr✝ }
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a b : OrthonormalBasis ι ℝ F
⊢ a.toBasis.det ⇑b = 1 ∨ a.toBasis.det ⇑b = -1

Tactic:
← sq_eq_one_iff

Goal state after:
ι : Type u_1
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
a b : OrthonormalBasis ι ℝ F
⊢ a.toBasis.det ⇑b ^ 2 = 1
Module: Mathlib.Analysis.BoxIntegral.Partition.Additive

Goal state before:
pp: case mk
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid N
I₀ : WithTop (Box ι)
I : Box ι
i : ι
g : ι →ᵇᵃ[I₀] M
toFun✝ : Box ι → M
sum_partition_boxes'✝ :
  ∀ (J : Box ι), ↑J ≤ I₀ → ∀ (π : Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, toFun✝ Ji = toFun✝ J
h : { toFun := toFun✝, sum_partition_boxes' := sum_partition_boxes'✝ }.toFun = g.toFun
⊢ { toFun := toFun✝, sum_partition_boxes' := sum_partition_boxes'✝ } = g

Tactic:
cases g

Goal state after:
case mk.mk
ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : AddCommMonoid N
I₀ : WithTop (Box ι)
I : Box ι
i : ι
toFun✝¹ : Box ι → M
sum_partition_boxes'✝¹ :
  ∀ (J : Box ι), ↑J ≤ I₀ → ∀ (π : Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, toFun✝¹ Ji = toFun✝¹ J
toFun✝ : Box ι → M
sum_partition_boxes'✝ :
  ∀ (J : Box ι), ↑J ≤ I₀ → ∀ (π : Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, toFun✝ Ji = toFun✝ J
h :
  { toFun := toFun✝¹, sum_partition_boxes' := sum_partition_boxes'✝¹ }.toFun =
    { toFun := toFun✝, sum_partition_boxes' := sum_partition_boxes'✝ }.toFun
⊢ { toFun := toFun✝¹, sum_partition_boxes' := sum_partition_boxes'✝¹ } =
    { toFun := toFun✝, sum_partition_boxes' := sum_partition_boxes'✝ }
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝ : NeZero n
ν : ι → ℤ
x : ι → ℝ
⊢ x ∈ box n ν ↔ ∀ (i : ι), ↑(ν i) / ↑n < x i ∧ x i ≤ (↑(ν i) + 1) / ↑n

Tactic:
simp_rw [Box.mem_def, box, Set.mem_Ioc]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
i : ι
⊢ Metric.ediam (Set.Icc ((box n ν).lower i) ((box n ν).upper i)) ≤ ENNReal.ofReal (1 / ↑n)

Tactic:
simp_rw [Real.ediam_Icc, box.upper_sub_lower, le_rfl]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
⊢ (if hI : ∃ ν_1 ∈ admissibleIndex n B, box n ν = box n ν_1 then tag n hI.choose else ⋯.choose) = tag n ν

Tactic:
have h : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν' := ⟨ν, hν, rfl⟩

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
h : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν'
⊢ (if hI : ∃ ν_1 ∈ admissibleIndex n B, box n ν = box n ν_1 then tag n hI.choose else ⋯.choose) = tag n ν
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
x : ι → ℝ
hx : x ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))
⊢ tag n (index n x) = x

Tactic:
mem_smul_span_iff

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Finite ι
x : ι → ℝ
hx : ∀ (i : ι), ↑n * x i ∈ Set.range ⇑(algebraMap ℤ ℝ)
⊢ tag n (index n x) = x
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_1
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ (prepartition n B).tag (box n (index n x✝)) ∈ s

Tactic:
prepartition_tag n (mem_admissibleIndex_of_mem_box n hB (hs₀ hx.1)),

Goal state after:
case refine_1
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ tag n (index n x✝) ∈ s
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
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
⊢ ⟪f i, gramSchmidt 𝕜 f j⟫_𝕜 = 0

Tactic:
← mem_orthogonal_singleton_iff_inner_right

Goal state after:
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
hf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0
i j : ι
hj : j ∈ Finset.Iio i
this : span 𝕜 (f '' Set.Iic j) ⟂ 𝕜 ∙ f i
⊢ gramSchmidt 𝕜 f j ∈ (𝕜 ∙ f i)ᗮ
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = (1 / ↑n ^ card ι).toReal * F ((prepartition n B).tag (box n (index n x✝)))

Tactic:
prepartition_tag n (mem_admissibleIndex_of_mem_box n hB (hs₀ hx.1)),

Goal state after:
case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = (1 / ↑n ^ card ι).toReal * F (tag n (index n x✝))
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : WellFoundedLT ι
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional 𝕜 E
h : finrank 𝕜 E = Fintype.card ι
f : ι → E
i : ι
hi : gramSchmidtNormed 𝕜 f i = 0
j : ι
⊢ ⟪(gramSchmidtOrthonormalBasis h f) i, f j⟫_𝕜 = 0

Tactic:
← mem_orthogonal_singleton_iff_inner_right

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : WellFoundedLT ι
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional 𝕜 E
h : finrank 𝕜 E = Fintype.card ι
f : ι → E
i : ι
hi : gramSchmidtNormed 𝕜 f i = 0
j : ι
⊢ f j ∈ (𝕜 ∙ (gramSchmidtOrthonormalBasis h f) i)ᗮ
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case intro.h
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
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
g : M [⋀^ι]→ₗ[R] R
hg : g ≠ 0
⊢ SameRay R (g.compLinearMap ↑f.symm) (↑(LinearEquiv.det f)⁻¹ • g)

Tactic:
(g.compLinearMap f.symm).eq_smul_basis_det e,

Goal state after:
case intro.h
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
f : M ≃ₗ[R] M
val✝ : Fintype ι
this : DecidableEq ι := Classical.decEq ι
g : M [⋀^ι]→ₗ[R] R
hg : g ≠ 0
⊢ SameRay R ((g.compLinearMap ↑f.symm) ⇑e • e.det) (↑(LinearEquiv.det f)⁻¹ • g)
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
⊢ ¬-rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯ =
      rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯

Tactic:
simp only [neg_rayOfNeZero]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
⊢ ¬rayOfNeZero ℝ (-AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯ =
      rayOfNeZero ℝ (AlternatingMap.constLinearEquivOfIsEmpty 1) ⋯
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n✝ : ℕ
_i : Fact (finrank ℝ E = n✝ + 1)
o : Orientation ℝ E (Fin (n✝ + 1))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

Tactic:
simp_rw [volumeForm]

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n✝ : ℕ
_i : Fact (finrank ℝ E = n✝ + 1)
o : Orientation ℝ E (Fin (n✝ + 1))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ Nat.casesAuxOn (motive := fun a ↦ n✝ + 1 = a → E [⋀^Fin (n✝ + 1)]→ₗ[ℝ] ℝ) (n✝ + 1)
      (fun h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = 0)] o ↦
            ⋯.by_cases (fun x ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1) fun x ↦
              -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          ⋯ o)
      (fun n h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n + 1)] o ↦ (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det) ⋯ o)
      ⋯ =
    b.toBasis.det
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ o.volumeForm = -b.toBasis.det

Tactic:
simp_rw [volumeForm]

Goal state after:
case succ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
b : OrthonormalBasis (Fin (n + 1)) ℝ E
hb : b.toBasis.orientation ≠ o
e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o
⊢ Nat.casesAuxOn (motive := fun a ↦ n + 1 = a → E [⋀^Fin (n + 1)]→ₗ[ℝ] ℝ) (n + 1)
      (fun h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = 0)] o ↦
            ⋯.by_cases (fun x ↦ AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1) fun x ↦
              -AlternatingMap.constOfIsEmpty ℝ E (Fin 0) 1)
          ⋯ o)
      (fun n_1 h ↦
        Eq.rec (motive := fun x x_1 ↦ [_i : Fact (finrank ℝ E = x)] → Orientation ℝ E (Fin x) → E [⋀^Fin x]→ₗ[ℝ] ℝ)
          (fun [Fact (finrank ℝ E = n_1 + 1)] o ↦ (Orientation.finOrthonormalBasis ⋯ ⋯ o).toBasis.det) ⋯ o)
      ⋯ =
    -b.toBasis.det
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)) +
        dist (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
          (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) +
      dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
    (μ.real ↑I + 1 + 1) * ↑ε

Tactic:
add_mul,

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)) +
        dist (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
          (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) +
      dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
    (μ.real ↑I + 1) * ↑ε + 1 * ↑ε