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

Goal state before:
pp: case succ
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
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 < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ (μ.restrict ↑I) (N ⁻¹' {n + 1}) < ↑(δ (n + 1)) / ↑(n + 1)

Tactic:
refine (measure_mono_null ?_ hf).le.trans_lt ?_

Goal state after:
case succ.refine_1
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
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 < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ N ⁻¹' {n + 1} ⊆ {x | f x ≠ 0}

case succ.refine_2
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
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 < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ 0 < ↑(δ (n + 1)) / ↑(n + 1)
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
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 < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), (μ.restrict ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
⊢ ∃ 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:
refine ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ _ => ?_⟩

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
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 < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), (μ.restrict ↑I) (U n) < ↑(δ n) / ↑n
r : (ι → ℝ) → ↑(Set.Ioi 0)
hrU : ∀ (x : ι → ℝ), closedBall x ↑(r x) ⊆ U (N x)
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c ((fun x ↦ r) c) π
x✝ : π.IsPartition
⊢ dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f

Tactic:
cases isEmpty_or_nonempty ι

Goal state after:
case inl
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f

case inr
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional R M
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ (map ι f) x = x ↔ 0 < LinearMap.det ↑f
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
⊢ (map ι f) x = -x ↔ LinearMap.det ↑f < 0

Tactic:
cases isEmpty_or_nonempty ι

Goal state after:
case inl
R : Type u_1
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : IsEmpty ι
⊢ (map ι f) x = -x ↔ LinearMap.det ↑f < 0

case inr
R : Type u_1
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : Fintype ι
x : Orientation R M ι
f : M ≃ₗ[R] M
h : Fintype.card ι = finrank R M
h✝ : Nonempty ι
⊢ (map ι f) x = -x ↔ LinearMap.det ↑f < 0
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ.h0
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))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
⊢ i ∈ Finset.univ → 0 ≤ |⟪b i, v i⟫|

Tactic:
_

Goal state after:
case succ.h0
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))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
a✝ : i ∈ Finset.univ
⊢ 0 ≤ |⟪b i, v i⟫|
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ.h1
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))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
⊢ i ∈ Finset.univ → |⟪b i, v i⟫| ≤ ‖v i‖

Tactic:
_

Goal state after:
case succ.h1
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))
v : Fin (n + 1) → E
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
a✝ : i ∈ Finset.univ
⊢ |⟪b i, v i⟫| ≤ ‖v i‖
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_2
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
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑ε

Tactic:
grw [← hcε]

Goal state after:
case refine_2
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
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
      (∑ y ∈ Finset.image (fun J ↦ Nx (π.tag J)) π.boxes,
        ∑ J ∈ (π.filter fun J ↦ Nx (π.tag J) = y).boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) ≤
    ↑c✝
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
⊢ ‖∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
        μ.toBoxAdditive J •
          (f ((π₁.infPrepartition π₂.toPrepartition).tag J) - f ((π₂.infPrepartition π₁.toPrepartition).tag J))‖ ≤
    ε

Tactic:
have μI : μ I < ⊤ := lt_of_le_of_lt (μ.mono I.coe_subset_Icc) I.isCompact_Icc.measure_lt_top

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
⊢ ‖∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
        μ.toBoxAdditive J •
          (f ((π₁.infPrepartition π₂.toPrepartition).tag J) - f ((π₂.infPrepartition π₁.toPrepartition).tag J))‖ ≤
    ε
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
⊢ ENNReal.ofReal |LinearMap.det ↑(Basis.ofZLatticeBasis ℝ L b).equivFun| * volume s =
    (ENNReal.ofReal |↑(LinearEquiv.det (Basis.ofZLatticeBasis ℝ L b).equivFun)|)⁻¹⁻¹ * volume s

Tactic:
inv_inv,

Goal state after:
ι : Type u_2
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
⊢ ENNReal.ofReal |LinearMap.det ↑(Basis.ofZLatticeBasis ℝ L b).equivFun| * volume s =
    ENNReal.ofReal |↑(LinearEquiv.det (Basis.ofZLatticeBasis ℝ L b).equivFun)| * volume s
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
u : ℕ → ℕ
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n k : ℕ
⊢ k ∈ range n → (u (k + 2) - u (k + 1)) • f (u (k + 1)) ≤ C • (u (k + 1) - u k) • f (u (k + 1))

Tactic:
_

Goal state after:
M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
u : ℕ → ℕ
C : ℕ
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
h_nonneg : ∀ (n : ℕ), 0 ≤ f n
hu : Monotone u
h_succ_diff : SuccDiffBounded C u
n k : ℕ
a✝ : k ∈ range n
⊢ (u (k + 2) - u (k + 1)) • f (u (k + 1)) ≤ C • (u (k + 1) - u k) • f (u (k + 1))
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ : ℕ
⊢ ∀ ⦃n_1 : ℕ⦄, 0 < m✝ → m✝ ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m✝ (n + m))

Tactic:
_

Goal state after:
case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ n✝ : ℕ
⊢ 0 < m✝ → m✝ ≤ n✝ → f (max n✝ (n + m)) ≤ f (max m✝ (n + m))
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ n✝ : ℕ
⊢ 0 < m✝ → m✝ ≤ n✝ → f (max n✝ (n + m)) ≤ f (max m✝ (n + m))

Tactic:
_

Goal state after:
case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ n✝ : ℕ
a✝ : 0 < m✝
⊢ m✝ ≤ n✝ → f (max n✝ (n + m)) ≤ f (max m✝ (n + m))
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ n✝ : ℕ
a✝ : 0 < m✝
⊢ m✝ ≤ n✝ → f (max n✝ (n + m)) ≤ f (max m✝ (n + m))

Tactic:
_

Goal state after:
case convert_2
f : ℕ → ℝ
n : ℕ
hn : ∀ (b : ℕ), n ≤ b → 0 b ≤ f b
m : ℕ
hm : ∀ (b : ℕ), m ≤ b → f (b + 1) ≤ f b
m✝ n✝ : ℕ
a✝¹ : 0 < m✝
a✝ : m✝ ≤ n✝
⊢ f (max n✝ (n + m)) ≤ f (max m✝ (n + m))
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
b : Basis ι ℤ ↥L
x : ↥L
n : ℕ
i : ι
hi : ↑n ≤ |(b.repr x) i|
⊢ normBound b * ↑n ≤ ‖x‖

Tactic:
by
  contrapose! hi
  exact abs_repr_lt_of_norm_lt b x n hi i

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

Goal state before:
pp: ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ (volume s).toReal / covolume L volume = (volume s).toReal / (ENNReal.ofReal (covolume L volume)).toReal

Tactic:
ENNReal.toReal_ofReal (covolume_pos L volume).le

Goal state after:
ι : Type u_1
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
s : Set (ι → ℝ)
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
⊢ (volume s).toReal / covolume L volume = (volume s).toReal / covolume L volume
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume =
    (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / (ENNReal.ofReal (covolume L volume)).toReal

Tactic:
ENNReal.toReal_ofReal (covolume_pos L volume).le

Goal state after:
ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume = (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ (volume s).toReal / covolume L volume = (volume s).toReal / (ENNReal.ofReal (covolume L volume)).toReal

Tactic:
ENNReal.toReal_ofReal (covolume_pos L volume).le

Goal state after:
case e'_5
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace ℝ E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ (volume s).toReal / covolume L volume = (volume s).toReal / covolume L volume
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume =
    (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / (ENNReal.ofReal (covolume L volume)).toReal

Tactic:
ENNReal.toReal_ofReal (covolume_pos L volume).le

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
b : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
⊢ (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume = (volume {x | x ∈ X ∧ F x ≤ 1}).toReal / covolume L volume
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case pos
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
I : Type u_1 := Free.ChooseBasisIndex ℤ ↥L
this : Fintype I
b : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L
d : ℕ := Fintype.card I
hd : d ≠ 0
ε : ℝ := normBound b
A : ℝ
hA : A > 0
B : ℝ
hB : B > 0
H : ∀ r < -↑d, ∀ (s : Finset ↥L), ∑ z ∈ s, ‖z‖ ^ r ≤ A * B ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)
hA' : A ≤ 1
r : ℝ
hr : r < -↑d
s : Finset ↥L
⊢ A * (B ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)) ≤ B ^ r * ∑' (k : ℕ), ↑k ^ (↑d - 1 + r)

Tactic:
exact mul_le_of_le_one_left (mul_nonneg (by positivity) (by positivity)) hA'

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
b✝ : σ → ℕ
⊢ a.support.sum b✝ = n →
    (∀ (i : σ), ¬b✝ i = 0 → ¬a i = 0) →
      ¬b✝ = ⇑s →
        (s = indicator a.support fun i x ↦ b✝ i) → ↑(Nat.multinomial a.support b✝) * ∏ x ∈ a.support, a x ^ b✝ x = 0

Tactic:
_

Goal state after:
case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
b✝ : σ → ℕ
a✝ : a.support.sum b✝ = n
⊢ (∀ (i : σ), ¬b✝ i = 0 → ¬a i = 0) →
    ¬b✝ = ⇑s →
      (s = indicator a.support fun i x ↦ b✝ i) → ↑(Nat.multinomial a.support b✝) * ∏ x ∈ a.support, a x ^ b✝ x = 0
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
b✝ : σ → ℕ
a✝ : a.support.sum b✝ = n
⊢ (∀ (i : σ), ¬b✝ i = 0 → ¬a i = 0) →
    ¬b✝ = ⇑s →
      (s = indicator a.support fun i x ↦ b✝ i) → ↑(Nat.multinomial a.support b✝) * ∏ x ∈ a.support, a x ^ b✝ x = 0

Tactic:
_

Goal state after:
case h₀
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
s : σ →₀ ℕ
a : σ →₀ R
n : ℕ
hs : (s.sum fun x m ↦ m) = n
b✝ : σ → ℕ
a✝¹ : a.support.sum b✝ = n
a✝ : ∀ (i : σ), ¬b✝ i = 0 → ¬a i = 0
⊢ ¬b✝ = ⇑s →
    (s = indicator a.support fun i x ↦ b✝ i) → ↑(Nat.multinomial a.support b✝) * ∏ x ∈ a.support, a x ^ b✝ x = 0