mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ ∅ = ∅

pp: case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ Ioc (l ?m.47) (u ?m.47) = ∅

pp: ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ ι

Tactic:
]

Goal state after:
case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ Ioc (l ?m.47) (u ?m.47) = ∅

ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
i : ι
hi : ¬l i < u i
⊢ ι
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

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

Tactic:
]

Goal state after:
ι : Type u_2
G : Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : DecidableEq ι
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
b : Basis ι ℝ G
v : ι → G
this : FiniteDimensional ℝ G
⊢ parallelepiped v = parallelepiped (⇑((b.constr ℕ) v) ∘ ⇑b)
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

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

Tactic:
]

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 s ≤ span K (Set.range Subtype.val)

Tactic:
]

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
s : Set E
h_incl : s ⊆ ↑L
h_span : span K s = span K ↑L
h_lind : LinearIndependent K Subtype.val
⊢ span K s ≤ span K (Set.range Subtype.val)
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ∩ ↑J ⊆ ↑I

Tactic:
]

Goal state after:
ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ∩ ↑J ⊆ ↑I
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 ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar).IsAddLeftInvariant

Tactic:
]

Goal state after:
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 ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar).IsAddLeftInvariant
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ∩ ↑J ⊆ ↑J

Tactic:
]

Goal state after:
ι : Type u_1
I✝ J✝ : Box ι
x y : ι → ℝ
I J : WithBot (Box ι)
⊢ ↑I ∩ ↑J ⊆ ↑J
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
I J : Box ι
⊢ (↑I ∩ ↑J).Nonempty ↔ (↑I ∩ ↑J).Nonempty

Tactic:
]

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

Goal state before:
pp: case refine_1
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
⊢ (↑(Submodule.map (span ℤ (Set.range ⇑b)).mkQ L)).Finite

Tactic:
]

Goal state after:
case refine_1
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
⊢ (↑(Submodule.map (span ℤ (Set.range ⇑b)).mkQ L)).Finite
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 ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar)

Tactic:
]

Goal state after:
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 ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar)
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
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • closedBall 0 1)) / μ (closedBall x r) ≤ μ (s ∩ closedBall x r) / μ (closedBall x r)

Tactic:
]

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
t_bound : t ⊆ closedBall 0 1
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • closedBall 0 1)) / μ (closedBall x r) ≤ μ (s ∩ closedBall x r) / μ (closedBall x r)
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
r : ℝ
h : ∀ (i : ι), I.upper i - I.lower i = (J.upper i - J.lower i) / r
i : ι
hr : ¬0 < r
this : I.upper i - I.lower i ≤ 0
⊢ False

Tactic:
]

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
r : ℝ
h : ∀ (i : ι), I.upper i - I.lower i = (J.upper i - J.lower i) / r
i : ι
hr : ¬0 < r
this : I.upper i - I.lower i ≤ 0
⊢ False
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
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ closedBall x r) / μ (closedBall x r) ≤ μ (s ∩ closedBall x r) / μ (closedBall x r)

Tactic:
]

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

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
i : ι
⊢ nndist I.lower I.upper / nndist (I.lower i) (I.upper i) * nndist (I.lower i) (I.upper i) ≤
    nndist I.lower I.upper / nndist (I.lower i) (I.upper i) * nndist (I.lower i) (I.upper i)

Tactic:
]

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

Goal state before:
pp: case refine_1
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ fract b x ∈ fundamentalDomain b ∩ ↑L

Tactic:
]

Goal state after:
case refine_1
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ fract b x ∈ fundamentalDomain b ∩ ↑L
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I : Box ι
s t : Set ι
h : s ≠ t
⊢ ↑(I.splitCenterBox s) ⊓ ↑(I.splitCenterBox t) ⊆ ⊥

Tactic:
]

Goal state after:
ι : Type u_1
I : Box ι
s t : Set ι
h : s ≠ t
⊢ ↑(I.splitCenterBox s) ⊓ ↑(I.splitCenterBox t) ⊆ ⊥
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I : Box ι
s t : Set ι
h : s ≠ t
y : ι → ℝ
hs : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s
ht : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ t
i : ι
⊢ i ∈ s ↔ i ∈ t

Tactic:
]

Goal state after:
ι : Type u_1
I : Box ι
s t : Set ι
h : s ≠ t
y : ι → ℝ
hs : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s
ht : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ t
i : ι
⊢ i ∈ s ↔ i ∈ t
Module: Mathlib.Analysis.BoxIntegral.Box.SubboxInduction

Goal state before:
pp: ι : Type u_1
I : Box ι
s t : Set ι
h : s ≠ t
y : ι → ℝ
hs : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s
ht : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ t
i : ι
⊢ (I.lower i + I.upper i) / 2 < y i ↔ (I.lower i + I.upper i) / 2 < y i

Tactic:
]

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

Goal state before:
pp: case refine_1.refine_2
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ x + -↑(floor b x) ∈ L

Tactic:
]

Goal state after:
case refine_1.refine_2
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ x + -↑(floor b x) ∈ L
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_1.refine_2
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ span ℤ s ≤ L

Tactic:
]

Goal state after:
case refine_1.refine_2
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
b : Basis { x // x ∈ s } K E := Basis.mk h_lind ⋯
this✝ : Fintype ↑s
this : (fundamentalDomain b ∩ ↑L).Finite
x : E
h_mem : x ∈ ↑L
⊢ span ℤ s ≤ L
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_2
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 ℤ s).FG

Tactic:
]

Goal state after:
case refine_2
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 ℤ s).FG
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
f : ↥(span ℝ ↑L) →ₗ[ℝ] E := (span ℝ ↑L).subtype
L₀ : Submodule ℤ ↥(span ℝ ↑L) := comap (↑ℤ f) L
⊢ ↑L = ↑L

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
f : ↥(span ℝ ↑L) →ₗ[ℝ] E := (span ℝ ↑L).subtype
L₀ : Submodule ℤ ↥(span ℝ ↑L) := comap (↑ℤ f) L
⊢ L ≤ (↑ℤ f).range

Tactic:
]

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
f : ↥(span ℝ ↑L) →ₗ[ℝ] E := (span ℝ ↑L).subtype
L₀ : Submodule ℤ ↥(span ℝ ↑L) := comap (↑ℤ f) L
⊢ L ≤ (↑ℤ f).range
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
f : ↥(span ℝ ↑L) →ₗ[ℝ] E := (span ℝ ↑L).subtype
L₀ : Submodule ℤ ↥(span ℝ ↑L) := comap (↑ℤ f) L
h_img : ⇑f '' ↑L₀ = ↑L
this : DiscreteTopology ↥L₀
⊢ span ℝ ↑L = span ℝ ↑L

Tactic:
]

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)
B : Tendsto (fun r ↦ μ (closedBall x r) / μ ({x} + r • u)) (𝓝[>] 0) (𝓝 (μ (closedBall x 1) / μ ({x} + u)))
C : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t)) / μ (closedBall x r) * (μ (closedBall x r) / μ ({x} + r • u))) (𝓝[>] 0) (𝓝 0)
r : ℝ
rpos : 0 < r
⊢ μ (s ∩ ({x} + r • t)) / μ ({x} + r • u) = μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)

Tactic:
]

Goal state after:
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:
]

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

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
inst✝ : Fintype ι
x : ι → ℝ
⊢ #({J ∈ π.boxes | x ∈ Box.Icc J}) ≤ Fintype.card (Set ι)

Tactic:
]

Goal state after:
ι : Type u_1
I : Box ι
π : Prepartition I
inst✝ : Fintype ι
x : ι → ℝ
⊢ #({J ∈ π.boxes | x ∈ Box.Icc J}) ≤ Fintype.card (Set ι)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: case e'_2
ι : Type u_1
I : Box ι
π : Prepartition I
x : ι → ℝ
x✝ : Box ι
⊢ x✝ ∈ π ∧ x ∈ x✝ ↔ x✝ ∈ π ∧ x ∈ x✝

Tactic:
]

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

Goal state before:
pp: case mp
ι : Type u_1
I : Box ι
π₁ π₂ : Prepartition I
H : π₁ ≤ π₂
J : Box ι
hJ : J ∈ π₁
J' : Box ι
hJ' : J' ∈ π₂
J'' : Box ι
hJ'' : J'' ∈ π₂
Hle : J ≤ J''
x : ι → ℝ
hx : x ∈ ↑J
hx' : x ∈ ↑J'
⊢ J ≤ J''

Tactic:
]

Goal state after:
case mp
ι : Type u_1
I : Box ι
π₁ π₂ : Prepartition I
H : π₁ ≤ π₂
J : Box ι
hJ : J ∈ π₁
J' : Box ι
hJ' : J' ∈ π₂
J'' : Box ι
hJ'' : J'' ∈ π₂
Hle : J ≤ J''
x : ι → ℝ
hx : x ∈ ↑J
hx' : x ∈ ↑J'
⊢ J ≤ J''
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case e'_2
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
⊢ span ℤ (Subtype.val '' Set.range ⇑b₀) = span ℤ (⇑L.subtype '' Set.range ⇑b₀)

Tactic:
]

Goal state after:
case e'_2
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
⊢ span ℤ (Subtype.val '' Set.range ⇑b₀) = span ℤ (⇑L.subtype '' Set.range ⇑b₀)
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi : (J : Box ι) → Prepartition J
J₁' J₁ : Box ι
hJ₁ : J₁ ∈ π.boxes
hJ₁' : J₁' ∈ (πi J₁).boxes
J₂' J₂ : Box ι
hJ₂ : J₂ ∈ π.boxes
hJ₂' : J₂' ∈ (πi J₂).boxes
Hne : J₁' ≠ J₂'
⊢ ∀ ⦃a : ι → ℝ⦄, a ∈ ↑J₁' → a ∉ ↑J₂'

Tactic:
]

Goal state after:
ι : Type u_1
I J J₁✝ J₂✝ : Box ι
π π₁ π₂ : Prepartition I
x : ι → ℝ
πi : (J : Box ι) → Prepartition J
J₁' J₁ : Box ι
hJ₁ : J₁ ∈ π.boxes
hJ₁' : J₁' ∈ (πi J₁).boxes
J₂' J₂ : Box ι
hJ₂ : J₂ ∈ π.boxes
hJ₂' : J₂' ∈ (πi J₂).boxes
Hne : J₁' ≠ J₂'
⊢ ∀ ⦃a : ι → ℝ⦄, a ∈ ↑J₁' → a ∉ ↑J₂'
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'
ι : Type u_7
inst✝² : Fintype ι
f : ι → Type u_8
inst✝¹ : (i : ι) → NormedAddCommGroup (f i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (f i)
x y : PiLp 2 f
⊢ ∑ x_1, (starRingEnd 𝕜) ((inst✝ x_1).toInner.1 (y.ofLp x_1) (x.ofLp x_1)) =
    ∑ i, (inst✝ i).toInner.1 (x.ofLp i) (y.ofLp i)

Tactic:
]

Goal state after:
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'
ι : Type u_7
inst✝² : Fintype ι
f : ι → Type u_8
inst✝¹ : (i : ι) → NormedAddCommGroup (f i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (f i)
x y : PiLp 2 f
⊢ ∑ x_1, (starRingEnd 𝕜) ((inst✝ x_1).toInner.1 (y.ofLp x_1) (x.ofLp x_1)) =
    ∑ i, (inst✝ i).toInner.1 (x.ofLp i) (y.ofLp i)