mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
⊢ (fun x ↦ (↑x)⁻¹ • ∑ i ∈ Finset.range x, (u i - l)) =ᶠ[atTop] fun x ↦ (↑x)⁻¹ • ∑ i ∈ Finset.range x, u i - l

pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
⊢ (fun x ↦ (↑x)⁻¹ • ↑x) =ᶠ[atTop] fun _x ↦ 1

Tactic:
·

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
⊢ (fun x ↦ (↑x)⁻¹ • ∑ i ∈ Finset.range x, (u i - l)) =ᶠ[atTop] fun x ↦ (↑x)⁻¹ • ∑ i ∈ Finset.range x, u i - l
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
⊢ (fun x ↦ (↑x)⁻¹ • ↑x) =ᶠ[atTop] fun _x ↦ 1

Tactic:
·

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
⊢ (fun x ↦ (↑x)⁻¹ • ↑x) =ᶠ[atTop] fun _x ↦ 1
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hb
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
⊢ 0 ≤ f (u (k + 1))

pp: case ha
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) ≤ C * (u (k + 1) - u k)

Tactic:
·

Goal state after:
case hb
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
⊢ 0 ≤ f (u (k + 1))
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
⊢ Bornology.IsBounded (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

pp: case refine_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

Tactic:
·

Goal state after:
case refine_2
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
⊢ Bornology.IsBounded (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)

Tactic:
·

Goal state after:
case refine_3
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁵ : DiscreteTopology ↥L
inst✝⁴ : IsZLattice ℝ L
ι : Type u_2
inst✝³ : Fintype ι
b : Basis ι ℤ ↥L
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
s : Set E
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)) = 0
⊢ MeasurableSet (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' s)
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case mp
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h : Bornology.IsBounded (range f)
⊢ f =O[cocompact D] 1

pp: case mpr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h : f =O[cocompact D] 1
⊢ Bornology.IsBounded (range f)

Tactic:
·

Goal state after:
case mp
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h : Bornology.IsBounded (range f)
⊢ f =O[cocompact D] 1
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
⊢ ∑' (b : ℕ), ↑(f b) = ∞

pp: case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞

Tactic:
·

Goal state after:
case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
⊢ ∑' (b : ℕ), ↑(f b) = ∞
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case mpr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h : f =O[cocompact D] 1
⊢ Bornology.IsBounded (range f)

Tactic:
·

Goal state after:
case mpr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
h : f =O[cocompact D] 1
⊢ Bornology.IsBounded (range f)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞

Tactic:
·

Goal state after:
case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(f b) = ∞
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case mpr.hs
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' t)

pp: case mpr.ht
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' tᶜ)

Tactic:
·

Goal state after:
case mpr.hs
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' t)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case e'_1
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
⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable fun k ↦ 2 ^ k * f (max (2 ^ k) (n + m))

pp: case e'_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
⊢ Summable f ↔ Summable fun k ↦ f (max k (n + m))

pp: case convert_1
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
⊢ ∀ (n_1 : ℕ), 0 ≤ f (max n_1 (n + m))

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_1 n_1 : ℕ⦄, 0 < m_1 → m_1 ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m_1 (n + m))

Tactic:
·

Goal state after:
case e'_1
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
⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable fun k ↦ 2 ^ k * f (max (2 ^ k) (n + m))
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case mpr.ht
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' tᶜ)

Tactic:
·

Goal state after:
case mpr.ht
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
c : ℝ
t : Set D
hcompact : IsCompact t
h : tᶜ ⊆ {x | ‖f x‖ ≤ c}
⊢ Bornology.IsBounded (f '' tᶜ)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case e'_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
⊢ Summable f ↔ Summable fun k ↦ f (max k (n + m))

pp: case convert_1
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
⊢ ∀ (n_1 : ℕ), 0 ≤ f (max n_1 (n + m))

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_1 n_1 : ℕ⦄, 0 < m_1 → m_1 ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m_1 (n + m))

Tactic:
·

Goal state after:
case e'_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
⊢ Summable f ↔ Summable fun k ↦ f (max k (n + m))
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case convert_1
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
⊢ ∀ (n_1 : ℕ), 0 ≤ f (max n_1 (n + m))

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_1 n_1 : ℕ⦄, 0 < m_1 → m_1 ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m_1 (n + m))

Tactic:
·

Goal state after:
case convert_1
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
⊢ ∀ (n_1 : ℕ), 0 ≤ f (max n_1 (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_1 n_1 : ℕ⦄, 0 < m_1 → m_1 ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m_1 (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_1 n_1 : ℕ⦄, 0 < m_1 → m_1 ≤ n_1 → f (max n_1 (n + m)) ≤ f (max m_1 (n + m))
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p

pp: case inr
p : ℝ
hp : p < 0
⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p

Tactic:
·

Goal state after:
case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun k ↦ 2 ^ k * (↑(2 ^ k) ^ p)⁻¹) ↔ 1 < p

pp: case inl.h_nonneg
p : ℝ
hp : 0 ≤ p
⊢ ∀ (n : ℕ), 0 ≤ (↑n ^ p)⁻¹

pp: case inl.h_mono
p : ℝ
hp : 0 ≤ p
⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹

Tactic:
·

Goal state after:
case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun k ↦ 2 ^ k * (↑(2 ^ k) ^ p)⁻¹) ↔ 1 < p
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl.h_nonneg
p : ℝ
hp : 0 ≤ p
⊢ ∀ (n : ℕ), 0 ≤ (↑n ^ p)⁻¹

pp: case inl.h_mono
p : ℝ
hp : 0 ≤ p
⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹

Tactic:
·

Goal state after:
case inl.h_nonneg
p : ℝ
hp : 0 ≤ p
⊢ ∀ (n : ℕ), 0 ≤ (↑n ^ p)⁻¹
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl.h_mono
p : ℝ
hp : 0 ≤ p
⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹

Tactic:
·

Goal state after:
case inl.h_mono
p : ℝ
hp : 0 ≤ p
⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inr
p : ℝ
hp : p < 0
⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p

Tactic:
·

Goal state after:
case inr
p : ℝ
hp : p < 0
⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: ⊢ ¬Summable fun i ↦ 1 / (↑i + 1)

pp: ⊢ ∀ (n : ℕ), 0 ≤ 1 / (↑n + 1)

Tactic:
·

Goal state after:
⊢ ¬Summable fun i ↦ 1 / (↑i + 1)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: ⊢ ∀ (n : ℕ), 0 ≤ 1 / (↑n + 1)

Tactic:
·

Goal state after:
⊢ ∀ (n : ℕ), 0 ≤ 1 / (↑n + 1)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: ⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, 1 / (↑i + 1)) atTop atTop

Tactic:
by
  rw [← not_summable_iff_tendsto_nat_atTop_of_nonneg]
  · exact_mod_cast mt (_root_.summable_nat_add_iff 1).1 not_summable_one_div_natCast
  · exact fun i => by positivity

Goal state after:
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case refine_1
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
hk : k ≠ 0
h : k ≤ n
⊢ ∑ i ∈ Ioc k k, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑k)⁻¹

pp: case refine_2
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
hk : k ≠ 0
h : k ≤ n
⊢ ∀ (n : ℕ),
    k ≤ n → ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹ → ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
hk : k ≠ 0
h : k ≤ n
⊢ ∑ i ∈ Ioc k k, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑k)⁻¹
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case inr
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_2
b : Basis ι ℤ ↥L
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : Submodule ℤ E}
    [inst_3 : DiscreteTopology ↥L] {ι : Type u_2} (b : Basis ι ℤ ↥L),
    IsZLattice ℝ L → ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
H : ¬IsZLattice ℝ L
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖

pp: E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
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_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖

Tactic:
·

Goal state after:
case inr
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_2
b : Basis ι ℤ ↥L
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : Submodule ℤ E}
    [inst_3 : DiscreteTopology ↥L] {ι : Type u_2} (b : Basis ι ℤ ↥L),
    IsZLattice ℝ L → ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
H : ¬IsZLattice ℝ L
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case h
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ Ioo k n ⊆ Ioc k (max (k + 1) n)

pp: case hf
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ ∀ i ∈ Ioc k (max (k + 1) n), i ∉ Ioo k n → 0 ≤ (↑i ^ 2)⁻¹

Tactic:
·

Goal state after:
case h
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ Ioo k n ⊆ Ioc k (max (k + 1) n)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hf
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ ∀ i ∈ Ioc k (max (k + 1) n), i ∉ Ioo k n → 0 ≤ (↑i ^ 2)⁻¹

Tactic:
·

Goal state after:
case hf
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ ∀ i ∈ Ioc k (max (k + 1) n), i ∉ Ioo k n → 0 ≤ (↑i ^ 2)⁻¹
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case hab
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ k + 1 < max (k + 1) n + 1

pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ (↑(k + 1) ^ 2)⁻¹ + ∑ k ∈ Ico (k + 1 + 1) (max (k + 1) n + 1), (↑k ^ 2)⁻¹ ≤
    ((↑k + 1) ^ 2)⁻¹ + ∑ i ∈ Ioc k.succ (max (k + 1) n), (↑i ^ 2)⁻¹

Tactic:
·

Goal state after:
case hab
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ k + 1 < max (k + 1) n + 1
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case refine_1
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ k.succ ≠ 0

pp: case refine_2
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ (↑k.succ)⁻¹ - (↑(max k.succ n))⁻¹ ≤ (↑k + 1)⁻¹

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ k.succ ≠ 0
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case refine_2
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ (↑k.succ)⁻¹ - (↑(max k.succ n))⁻¹ ≤ (↑k + 1)⁻¹

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ (↑k.succ)⁻¹ - (↑(max k.succ n))⁻¹ ≤ (↑k + 1)⁻¹
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case pos
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
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_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
this✝ : Finite ι
b' : Basis ι ℝ E := Basis.ofZLatticeBasis ℝ L b
e : E ≃L[ℝ] ι → ℝ := (b'.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℝ ℝ ι).toContinuousLinearEquiv
this : IsOpen (⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1)
ε : ℝ
hε : ε > 0
hε' : Metric.ball 0 ε ⊆ ⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1
x : ↥L
i : ι
hx : x = 0
⊢ ε / 2 * ↑|(b.repr x) i| ≤ ‖x‖

pp: case neg
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
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_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
this✝ : Finite ι
b' : Basis ι ℝ E := Basis.ofZLatticeBasis ℝ L b
e : E ≃L[ℝ] ι → ℝ := (b'.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℝ ℝ ι).toContinuousLinearEquiv
this : IsOpen (⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1)
ε : ℝ
hε : ε > 0
hε' : Metric.ball 0 ε ⊆ ⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1
x : ↥L
i : ι
hx : ¬x = 0
⊢ ε / 2 * ↑|(b.repr x) i| ≤ ‖x‖

Tactic:
·

Goal state after:
case pos
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
L✝ : Submodule ℤ E✝
ι✝ : Type u_2
b✝ : Basis ι✝ ℤ ↥L✝
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_2
b : Basis ι ℤ ↥L
H : IsZLattice ℝ L
this✝ : Finite ι
b' : Basis ι ℝ E := Basis.ofZLatticeBasis ℝ L b
e : E ≃L[ℝ] ι → ℝ := (b'.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℝ ℝ ι).toContinuousLinearEquiv
this : IsOpen (⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1)
ε : ℝ
hε : ε > 0
hε' : Metric.ball 0 ε ⊆ ⇑e ⁻¹' Set.univ.pi fun x ↦ Set.Ioo (-1) 1
x : ↥L
i : ι
hx : x = 0
⊢ ε / 2 * ↑|(b.repr x) i| ≤ ‖x‖
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: a s : ℝ
hnorm : Tendsto (fun n ↦ ‖↑n‖) atTop atTop
⊢ (fun x ↦ ↑x + ?m.78) =ᶠ[atTop] fun n ↦ |↑n + a|

pp: a s : ℝ
hnorm : Tendsto (fun n ↦ ‖↑n‖) atTop atTop
⊢ ℝ

Tactic:
·

Goal state after:
a s : ℝ
hnorm : Tendsto (fun n ↦ ‖↑n‖) atTop atTop
⊢ (fun x ↦ ↑x + ?m.78) =ᶠ[atTop] fun n ↦ |↑n + a|
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: a s : ℝ
hnorm : Tendsto (fun n ↦ ‖↑n‖) atTop atTop
⊢ (fun x ↦ ↑x + ?m.78) =ᶠ[atTop] fun n ↦ |↑n + a|

Tactic:
filter_upwards [eventually_gt_atTop (Nat.ceil |a|)] with _ hn

Goal state after:
a s : ℝ
hnorm : Tendsto (fun n ↦ ‖↑n‖) atTop atTop
a✝ : ℕ
hn : ⌈|a|⌉₊ < a✝
⊢ ↑a✝ + ?m.78 = |↑a✝ + a|
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: case mp
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] := ⋯
⊢ x ∈ Ideal.span (⇑(of k G) '' s) → x ∈ RHS

pp: case mpr
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (⇑(of k G) '' s)

Tactic:
·

Goal state after:
case mp
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] := ⋯
⊢ x ∈ Ideal.span (⇑(of k G) '' s) → x ∈ RHS