mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I J : Box ι
i : ι
x : ℝ
y : ι → ℝ
h₁ : J ∈ split I i x
h₂ : y ∈ J
h₃ : x < y i
⊢ ↑J = ↑I ∩ {y | x < y i}

Tactic:
by

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
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), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u

Tactic:
by

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
⊢ finrank ℤ ↥L = finrank K E

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
h : I ≤ J
i : ι
x : ℝ
⊢ (split J i x).restrict I = split I i x

Tactic:
by

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✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
by

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
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
⊢ (v.toOrthonormalBasis hv).toBasis = v

Tactic:
by

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
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
⊢ ⇑(v.toOrthonormalBasis hv) = ⇑(v.toOrthonormalBasis hv).toBasis

Tactic:
by

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
v : Basis ι 𝕜 E
hv : Orthonormal 𝕜 ⇑v
⊢ ⇑(v.toOrthonormalBasis hv).toBasis = ⇑v

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
p : ι × ℝ
⊢ splitMany I (insert p s) = splitMany I s ⊓ split I p.1 p.2

Tactic:
by

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✝⁹ : RCLike 𝕜✝
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜✝ E
F : Type u_5
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : InnerProductSpace ℝ F
F' : Type u_6
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : InnerProductSpace ℝ F'
inst✝² : Fintype ι✝
ι : Type u_7
𝕜 : Type u_8
inst✝¹ : Unique ι
inst✝ : RCLike 𝕜
⊢ Orthonormal 𝕜 ⇑(Basis.singleton ι 𝕜)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
⊢ (splitMany I ∅).IsPartition

Tactic:
by

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

Goal state before:
pp: ι : Type u_7
𝕜 : Type u_8
inst✝¹ : Unique ι
inst✝ : RCLike 𝕜
⊢ ⇑(OrthonormalBasis.singleton ι 𝕜) = 1

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
s✝ : Finset (ι × ℝ)
a : ι × ℝ
s : Finset (ι × ℝ)
x✝ : a ∉ s
hs : (splitMany I s).IsPartition
⊢ (splitMany I (insert a s)).IsPartition

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
⊢ (splitMany I s).IsPartition

Tactic:
by

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

Goal state before:
pp: η : Type u_7
inst✝⁴ : Fintype η
ι : η → Type u_8
inst✝³ : (i : η) → Fintype (ι i)
𝕜 : Type u_9
inst✝² : RCLike 𝕜
E : η → Type u_10
inst✝¹ : (i : η) → NormedAddCommGroup (E i)
inst✝ : (i : η) → InnerProductSpace 𝕜 (E i)
B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)
⊢ (Pi.orthonormalBasis B).toBasis = (Pi.basis fun i ↦ (B i).toBasis).map (WithLp.linearEquiv 2 𝕜 ((j : η) → E j)).symm

Tactic:
by

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

Goal state before:
pp: η : Type u_7
inst✝⁵ : Fintype η
inst✝⁴ : DecidableEq η
ι : η → Type u_8
inst✝³ : (i : η) → Fintype (ι i)
𝕜 : Type u_9
inst✝² : RCLike 𝕜
E : η → Type u_10
inst✝¹ : (i : η) → NormedAddCommGroup (E i)
inst✝ : (i : η) → InnerProductSpace 𝕜 (E i)
B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)
j : (i : η) × ι i
⊢ (Pi.orthonormalBasis B) j = PiLp.single 2 j.fst ((B j.fst) j.snd)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
s : Finset (ι × ℝ)
⊢ π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s

Tactic:
by

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✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
v : ι → E
hon : Orthonormal 𝕜 v
hsp : ⊤ ≤ span 𝕜 (range v)
⊢ Orthonormal 𝕜 ⇑(Basis.mk ⋯ hsp)

Tactic:
by

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
v : ι → E
hon : Orthonormal 𝕜 v
hsp : ⊤ ≤ span 𝕜 (range v)
⊢ ⇑(OrthonormalBasis.mk hon hsp) = v

Tactic:
by

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 ⋯
⊢ Orthonormal 𝕜 ⇑e₀'

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
⊢ Basis ι K E

Tactic:
by

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
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 ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
⊢ Tendsto (fun r ↦ μ (closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
i : ι
⊢ (ofZLatticeBasis K L b) i = ↑(b i)

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
x : ↥L
i : ι
this : ↑ℤ ↑(ofZLatticeBasis K L b).repr ∘ₗ L.subtype = Finsupp.mapRange.linearMap (Algebra.linearMap ℤ K) ∘ₗ ↑b.repr
⊢ ((ofZLatticeBasis K L b).repr ↑x) i = ↑((b.repr x) i)

Tactic:
by

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:
by

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

Goal state before:
pp: ι' : Type u_2
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : DecidableEq E
v' : ι' → E
h : Orthonormal 𝕜 v'
s : Finset ι'
i : ↥s
⊢ ↑((OrthonormalBasis.span h s) i) = v' ↑i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I J Js : Box ι
s : Finset (ι × ℝ)
H : ∀ (i : ι), {(i, J.lower i), (i, J.upper i)} ⊆ s
HJs : Js ∈ splitMany I s
Hn : ¬Disjoint ↑J ↑Js
⊢ Js ≤ J

Tactic:
by

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✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
v : ι → E
hon : Orthonormal 𝕜 v
hsp : (span 𝕜 (range v))ᗮ = ⊥
⊢ ⊤ ≤ span 𝕜 (range v)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
s : Finset (Box ι)
⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop, ∀ (I J : Box ι), J ∈ s → ∀ J' ∈ splitMany I t, ¬Disjoint ↑J ↑J' → J' ≤ J

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
x : ↥L
i : ι
⊢ ((ofZLatticeBasis K L b).repr ↑x) i = ↑((b.repr x) i)

Tactic:
by

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✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
i' : ι'
⊢ (b.reindex e) i' = b (e.symm i')

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
⊢ span ℤ (Set.range ⇑(ofZLatticeBasis K L b)) = span ℤ (⇑L.subtype '' Set.range ⇑b)

Tactic:
by

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
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 ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
⊢ Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
⊢ span ℤ (⇑L.subtype '' Set.range ⇑b) = Submodule.map L.subtype (span ℤ (Set.range ⇑b))

Tactic:
by

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✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
x : E
i' : ι'
⊢ ((b.reindex e).repr x).ofLp i' = (b.repr x).ofLp (e.symm i')

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
⊢ Submodule.map L.subtype (span ℤ (Set.range ⇑b)) = L

Tactic:
by

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✝ : Fintype ι
x : EuclideanSpace 𝕜 ι
i : ι
⊢ ⟪(basisFun ι 𝕜) i, x⟫ = x.ofLp i

Tactic:
by

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
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
⊢ span ℤ (Set.range ⇑(ofZLatticeBasis K L b)) = L

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x : EuclideanSpace ℝ ι
i : ι
⊢ inner ℝ x ((basisFun ι ℝ) i) = x.ofLp i

Tactic:
by

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✝⁶ : RCLike 𝕜
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : Fintype ι
E' : Type u_7
inst✝² : Fintype ι'
inst✝¹ : NormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
b : OrthonormalBasis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 E'
e : ι ≃ ι'
⊢ (b.equiv b' e).symm = b'.equiv b e.symm

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
⊢ ∀ᶠ (t : Finset (ι × ℝ)) in atTop, π ⊓ splitMany I t = (splitMany I t).filter fun J ↦ ↑J ⊆ π.iUnion

Tactic:
by

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

Goal state before:
pp: ι : Type u_3
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
inst✝² : Finite ι
b : Basis ι ℤ ↥L
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
⊢ IsAddFundamentalDomain (↥L) (fundamentalDomain (Basis.ofZLatticeBasis ℝ L b)) μ

Tactic:
by

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✝⁶ : RCLike 𝕜
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : Fintype ι
E' : Type u_7
inst✝² : Fintype ι'
inst✝¹ : NormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
b : OrthonormalBasis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 E'
e : ι ≃ ι'
i : ι
⊢ (b.equiv b' e) (b i) = b' (e i)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
I : Box ι
π : Prepartition I
h : π.IsPartition
⊢ ∃ s, splitMany I s ≤ π

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Finite ι
I : Box ι
π : Prepartition I
h : π.IsPartition
⊢ ∃ s, splitMany I s ≤ π

Tactic:
by
  refine (eventually_splitMany_inf_eq_filter π).exists.imp fun s hs => ?_
  rwa [h.iUnion_eq, filter_of_true, inf_eq_right] at hs
  exact fun J hJ => le_of_mem _ hJ

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
⊢ b.equiv b (Equiv.refl ι) = LinearIsometryEquiv.refl 𝕜 E

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
⊢ ∃ π', π'.iUnion = ↑I \ π.iUnion

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π₁ π₂ : Prepartition I
h : π₁.iUnion = π₂.iUnion
⊢ π₁.compl = π₂.compl

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
h : π.IsPartition
⊢ π.compl = ⊥

Tactic:
by

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:
by

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

Goal state before:
pp: ⊢ Henstock ≤ Riemann

Tactic:
by

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

Goal state before:
pp: ⊢ Henstock ≤ McShane

Tactic:
by

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

Goal state before:
pp: ι : Type u_2
l : IntegrationParams
hl : l.bRiemann = false
r : (ι → ℝ) → ↑(Set.Ioi 0)
⊢ l.RCond r

Tactic:
by

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

Goal state before:
pp: E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : DiscreteTopology ↥(span ℤ s)
F : Submodule ℝ E := span ℝ s
L : Submodule ℤ ↥(span ℝ s) := comap (Submodule.restrictScalars ℤ F).subtype (span ℤ s)
f : ↥(comap (Submodule.restrictScalars ℤ (span ℝ s)).subtype (span ℤ s)) ≃ₗ[ℤ] ↥(span ℤ s) := comapSubtypeEquivOfLe ⋯
⊢ DiscreteTopology ↥L

Tactic:
by

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
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 ≠ ∞
I : ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
L : Tendsto (fun r ↦ μ (sᶜ ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
L' : Tendsto (fun r ↦ μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0)
⊢ Tendsto (fun r ↦ μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)

Tactic:
by

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✝⁶ : RCLike 𝕜
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : Fintype ι
E' : Type u_7
inst✝² : Fintype ι'
inst✝¹ : NormedAddCommGroup E'
inst✝ : InnerProductSpace 𝕜 E'
b : OrthonormalBasis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 E'
e : ι ≃ ι'
x : E
⊢ (b.equiv b' e) x = ∑ i, (b.repr x).ofLp i • b' (e i)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c₁ c₂ : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ r₁ π₁
h₂ : l.MemBaseSet I c₂ r₂ π₂
hU : π₁.iUnion = π₂.iUnion
⊢ ∃ π,
    π.iUnion = ↑I \ π₁.iUnion ∧ (l.bDistortion = true → π.distortion ≤ c₁) ∧ (l.bDistortion = true → π.distortion ≤ c₂)

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)
π₁ : TaggedPrepartition I
hπ₁ : l.MemBaseSet I c r₁ π₁
hle : ∀ x ∈ Box.Icc I, r₂ x ≤ r₁ x
π₂ : Prepartition I
hU : π₂.iUnion = ↑I \ π₁.iUnion
hc : l.bDistortion = true → π₂.distortion ≤ c
x✝ : l.bDistortion = true
⊢ ⊥.iUnion = ↑I \ (π₁.unionComplToSubordinate π₂ hU r₂).iUnion ∧ ⊥.distortion ≤ c

Tactic:
by

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
x : EuclideanSpace 𝕜 ι
⊢ ((EuclideanSpace.basisFun ι 𝕜).equiv b (Equiv.refl ι)) x = ∑ i, x.ofLp i • b i

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
c : ℝ≥0
l : IntegrationParams
π : TaggedPrepartition I
r : (ι → ℝ) → ↑(Set.Ioi 0)
hπ : l.MemBaseSet I c r π
p : Box ι → Prop
hD : l.bDistortion = true
π₁ : Prepartition I
hπ₁U : π₁.iUnion = ↑I \ π.iUnion
hc : π₁.distortion ≤ c
π₂ : TaggedPrepartition I := π.filter fun J ↦ ¬p J
⊢ Disjoint π₁.iUnion π₂.iUnion

Tactic:
by

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
⊢ ⇑((EuclideanSpace.basisFun ι 𝕜).equiv b (Equiv.refl ι)) = fun x ↦ ∑ i, x.ofLp i • b i

Tactic:
by

Goal state after: