mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
hμs : μ' s ≠ ∞
this : ∀ {s : Set G}, μ' s ≠ ∞ → MeasurableSet s → ∀ᵐ (x : G) ∂μ', ν' ((fun x_1 ↦ x_1 * x) ⁻¹' s) < ∞
sm : ¬MeasurableSet s
x : G
⊢ ν' ((fun x_1 ↦ x_1 * x) ⁻¹' toMeasurable μ' s) < ∞ → ν' ((fun x_1 ↦ x_1 * x) ⁻¹' s) < ∞

Tactic:
hx

Goal state after:
G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
s : Set G
inst✝⁴ : MeasurableInv G
μ' ν' : Measure G
inst✝³ : SigmaFinite μ'
inst✝² : SigmaFinite ν'
inst✝¹ : μ'.IsMulLeftInvariant
inst✝ : ν'.IsMulLeftInvariant
hμs : μ' s ≠ ∞
this : ∀ {s : Set G}, μ' s ≠ ∞ → MeasurableSet s → ∀ᵐ (x : G) ∂μ', ν' ((fun x_1 ↦ x_1 * x) ⁻¹' s) < ∞
sm : ¬MeasurableSet s
x : G
hx : ν' ((fun x_1 ↦ x_1 * x) ⁻¹' toMeasurable μ' s) < ∞
⊢ ν' ((fun x_1 ↦ x_1 * x) ⁻¹' s) < ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
x : α
⊢ x ∈ s → x ∈ t ∪ {x | f x = 0}

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
s : Set α
hf : AEMeasurable f μ
hs : μ ({x | f x ≠ 0} ∩ s) = 0
t : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)
x : α
hx : x ∈ s
⊢ x ∈ t ∪ {x | f x = 0}
Module: Mathlib.MeasureTheory.Function.Egorov

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ENNReal.ofReal (ε / 2) * ENNReal.ofReal (∑' (n : ℕ), (1 / 2) ^ n) ≤ ENNReal.ofReal ε

pp: case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ∀ (n : ℕ), 0 ≤ 2⁻¹ ^ n

pp: case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow 2⁻¹)

Tactic:
tsum_geometric_two,

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ENNReal.ofReal (ε / 2) * ENNReal.ofReal 2 ≤ ENNReal.ofReal ε

case hf_nonneg
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ∀ (n : ℕ), 0 ≤ 2⁻¹ ^ n

case hf
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
inst✝³ : PseudoEMetricSpace β
μ : Measure α
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝² : SemilatticeSup ι
inst✝¹ : Nonempty ι
inst✝ : Countable ι
hε : 0 < ε
hf : ∀ (n : ι), Measurable fun a ↦ edist (f n a) (g a)
hsm : MeasurableSet s
hs : μ s ≠ ∞
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Summable (HPow.hPow 2⁻¹)
Module: Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
x : α
⊢ f x ≠ ∞ → ENNReal.ofReal (f x).toReal = f x

Tactic:
hx

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
x : α
hx : f x ≠ ∞
⊢ ENNReal.ofReal (f x).toReal = f x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
Z : ∀ᵐ (x : α) ∂μ.restrict {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x
x : α
⊢ g x = AEMeasurable.mk g hg x → (f' * g') x = (f' * g) x

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
Z : ∀ᵐ (x : α) ∂μ.restrict {x | AEMeasurable.mk f hf x ≠ 0}, g x = AEMeasurable.mk g hg x
x : α
hx : g x = AEMeasurable.mk g hg x
⊢ (f' * g') x = (f' * g) x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
x : α
⊢ x ∈ {x | f' x ≠ 0}ᶜ → (f' * g') x = (f' * g) x

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
M : MeasurableSet {x | f' x ≠ 0}ᶜ
x : α
hx : x ∈ {x | f' x ≠ 0}ᶜ
⊢ (f' * g') x = (f' * g) x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
x : α
⊢ f x = AEMeasurable.mk f hf x → (f' * g) x = (f * g) x

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
hg : AEMeasurable g (μ.withDensity f')
this : μ.withDensity f = μ.withDensity f'
g' : α → ℝ≥0∞ := AEMeasurable.mk g hg
x : α
hx : f x = AEMeasurable.mk f hf x
⊢ (f' * g) x = (f * g) x
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: ⊢ 1 < √2

Tactic:
← Real.sqrt_one

Goal state after:
⊢ √1 < √2
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
x : α
⊢ f x < ∞ → f x = AEMeasurable.mk f hf x → AEMeasurable.mk f hf x < ∞

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
x : α
hx : f x < ∞
⊢ f x = AEMeasurable.mk f hf x → AEMeasurable.mk f hf x < ∞
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
x : α
⊢ f x = AEMeasurable.mk f hf x → (f' * g) x = (f * g) x

Tactic:
hx

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hf : AEMeasurable f μ
h'f : ∀ᵐ (x : α) ∂μ, f x < ∞
g : α → ℝ≥0∞
f' : α → ℝ≥0∞ := AEMeasurable.mk f hf
x : α
hx : f x = AEMeasurable.mk f hf x
⊢ (f' * g) x = (f * g) x
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: case e_a.e_f
α : Type u_1
m0 : MeasurableSpace α
μ✝ : Measure α
f : α → ℝ≥0∞
hfm : Measurable f
μ : Measure α
inst✝ : SFinite μ
hμ : IsFiniteMeasure μ
s : Set α := {x | f x = ∞}
hs : MeasurableSet s
x : α
⊢ s.indicator f x = ∑' (i : ℕ), s.indicator 1 x

Tactic:
if hx : x ∈ s then simpa [hx, ENNReal.tsum_const_eq_top_of_ne_zero]
      else simp [hx]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
inst✝ : Countable α
⊢ SFinite μ

Tactic:
obtain ⟨s, h⟩ := exists_sum_smul_dirac μ
  rw [h]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Star.Unitary

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝³ : CommSemiring R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : StarMul A
a : A
U : ↥(unitary A)
⊢ spectrum R (star ↑U * a * ↑U) = spectrum R a

Tactic:
by
  simpa using spectrum_star_right_conjugate (U := star U)

Goal state after:
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝⁹ : Semiring R
inst✝⁸ : Semiring R₂
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R₂ M₂
inst✝³ : TopologicalSpace M
inst✝² : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝¹ : RingHomInvPair σ σ'
inst✝ : RingHomInvPair σ' σ
L : SummationFilter ι
f : ι → M
e : M ≃SL[σ] M₂
x : M
⊢ HasSum f (e.symm (e x)) L ↔ HasSum f x L

Tactic:
ContinuousLinearEquiv.symm_apply_apply

Goal state after:
ι : Type u_5
R : Type u_7
R₂ : Type u_8
M : Type u_9
M₂ : Type u_10
inst✝⁹ : Semiring R
inst✝⁸ : Semiring R₂
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : AddCommMonoid M₂
inst✝⁴ : Module R₂ M₂
inst✝³ : TopologicalSpace M
inst✝² : TopologicalSpace M₂
σ : R →+* R₂
σ' : R₂ →+* R
inst✝¹ : RingHomInvPair σ σ'
inst✝ : RingHomInvPair σ' σ
L : SummationFilter ι
f : ι → M
e : M ≃SL[σ] M₂
x : M
⊢ HasSum f x L ↔ HasSum f x L
Module: Mathlib.Topology.Algebra.InfiniteSum.Module

Goal state before:
pp: case h
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
g : Quotient (orbitRel α β) → R
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
⊢ Quotient.lift (fun b ↦ ∑' (a : α), g (Quotient.mk' (a • b)) • f (a • b)) ⋯ (π b) =
    g (π b) • Quotient.lift (fun b ↦ ∑' (a : α), f (a • b)) ⋯ (π b)

Tactic:
have H₁ : ∀ a : α, π (a • b) = π b := by
    intro a
    apply (@Quotient.eq _ (MulAction.orbitRel α β) (a • b) b).mpr
    use a

Goal state after:
case h
α : Type u_1
β : Type u_2
M : Type u_11
inst✝⁷ : TopologicalSpace M
inst✝⁶ : AddCommMonoid M
inst✝⁵ : T2Space M
R : Type u_12
inst✝⁴ : DivisionRing R
inst✝³ : Module R M
inst✝² : ContinuousConstSMul R M
inst✝¹ : Group α
inst✝ : MulAction α β
f : β → M
g : Quotient (orbitRel α β) → R
b : β
π : β → Quotient (orbitRel α β) := Quotient.mk (orbitRel α β)
H₁ : ∀ (a : α), π (a • b) = π b
⊢ Quotient.lift (fun b ↦ ∑' (a : α), g (Quotient.mk' (a • b)) • f (a • b)) ⋯ (π b) =
    g (π b) • Quotient.lift (fun b ↦ ∑' (a : α), f (a • b)) ⋯ (π b)
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
x : A
hx : IsSelfAdjoint x
⊢ skewAdjoint.negISMul 0 = 0

Tactic:
map_zero

Goal state after:
A : Type u_1
inst✝³ : AddCommGroup A
inst✝² : Module ℂ A
inst✝¹ : StarAddMonoid A
inst✝ : StarModule ℂ A
x : A
hx : IsSelfAdjoint x
⊢ 0 = 0
Module: Mathlib.Analysis.Complex.Basic

Goal state before:
pp: A : Type u_1
inst✝⁴ : SeminormedAddCommGroup A
inst✝³ : StarAddMonoid A
inst✝² : NormedSpace ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : NormedStarGroup A
x : A
⊢ ‖realPart x‖ ≤ 2⁻¹ * (2 * ‖x‖)

Tactic:
← AddSubgroup.norm_coe,

Goal state after:
A : Type u_1
inst✝⁴ : SeminormedAddCommGroup A
inst✝³ : StarAddMonoid A
inst✝² : NormedSpace ℂ A
inst✝¹ : StarModule ℂ A
inst✝ : NormedStarGroup A
x : A
⊢ ‖↑(realPart x)‖ ≤ 2⁻¹ * (2 * ‖x‖)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ normSq (-z) = normSq z

Tactic:
simp only [normSq_eq_def', norm_neg]

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
q : ℚ≥0
⊢ ‖↑q‖ = ↑q

Tactic:
rw [← ofReal_nnratCast]

Goal state after:
K : Type u_1
inst✝ : RCLike K
q : ℚ≥0
⊢ ‖↑↑q‖ = ↑q
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: A : Type u_1
inst✝⁵ : NonUnitalRing A
inst✝⁴ : StarRing A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
inst✝¹ : Module ℂ A
inst✝ : StarModule ℂ A
a : A
⊢ a ≤ 0 ↔ ℜ a ≤ 0 ∧ ℑ a = 0

Tactic:
by
  simpa using nonneg_iff_realPart_imaginaryPart (a := -a)

Goal state after:
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z w : K
hz : re 0 < re z ∧ im 0 = im z
hw : re 0 < re w ∧ im 0 = im w
⊢ re 0 < re (z * w) ∧ im 0 = im (z * w)

Tactic:
map_zero

Goal state after:
K : Type u_1
inst✝ : RCLike K
z w : K
hz : 0 < re z ∧ im 0 = im z
hw : 0 < re w ∧ im 0 = im w
⊢ 0 < re (z * w) ∧ im 0 = im (z * w)
Module: Mathlib.Analysis.Asymptotics.Theta

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : α → E
g : α → F
l : Filter α
l' : Filter β
⊢ f =Θ[l] g → (f ∘ Prod.snd) =Θ[l' ×ˢ l] (g ∘ Prod.snd)

Tactic:
simp only [IsTheta]
  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.comp_snd l', h₂.comp_snd l'⟩

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
x : α
⊢ ‖f x‖ ≤ c * ‖g x‖ → ‖f x‖ ≤ max c 1 * ‖g x‖

Tactic:
hx

Goal state after:
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
x : α
hx : ‖f x‖ ≤ c * ‖g x‖
⊢ ‖f x‖ ≤ max c 1 * ‖g x‖
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc_pos : c > 0
hc : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
x : α
⊢ ‖f x‖ ≤ c * ‖g x‖ → c⁻¹ * ‖f x‖ ≤ ‖g x‖

Tactic:
hx

Goal state after:
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc_pos : c > 0
hc : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
x : α
hx : ‖f x‖ ≤ c * ‖g x‖
⊢ c⁻¹ * ‖f x‖ ≤ ‖g x‖
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc_pos : c > 0
hc : ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
x : α
⊢ c * ‖f x‖ ≤ ‖g x‖ → ‖f x‖ ≤ c⁻¹ * ‖g x‖

Tactic:
hx

Goal state after:
α : Type u_1
E : Type u_3
E''' : Type u_12
inst✝¹ : Norm E
inst✝ : SeminormedAddGroup E'''
f : α → E
l : Filter α
g : α → E'''
c : ℝ
hc_pos : c > 0
hc : ∀ᶠ (x : α) in l, c * ‖f x‖ ≤ ‖g x‖
x : α
hx : c * ‖f x‖ ≤ ‖g x‖
⊢ ‖f x‖ ≤ c⁻¹ * ‖g x‖
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝² : Norm E
inst✝¹ : Norm F
inst✝ : Norm G
c c' : ℝ
f : α → E
g : α → F
k : α → G
l : Filter α
hc : 0 ≤ c
hfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
hgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖
x : α
⊢ ‖f x‖ ≤ c * ‖g x‖ → ‖g x‖ ≤ c' * ‖k x‖ → ‖f x‖ ≤ c * c' * ‖k x‖

Tactic:
hx

Goal state after:
α : Type u_1
E : Type u_3
F : Type u_4
G : Type u_5
inst✝² : Norm E
inst✝¹ : Norm F
inst✝ : Norm G
c c' : ℝ
f : α → E
g : α → F
k : α → G
l : Filter α
hc : 0 ≤ c
hfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖
hgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖
x : α
hx : ‖f x‖ ≤ c * ‖g x‖
⊢ ‖g x‖ ≤ c' * ‖k x‖ → ‖f x‖ ≤ c * c' * ‖k x‖
Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
⊢ ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n

Tactic:
simpa using sum_range_diag_flip K fun m n ↦ f m * g n

Goal state after:
Module: Mathlib.Analysis.Asymptotics.Defs

Goal state before:
pp: α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
C : ι → ℝ
s : Finset ι
B : ι → α → ℝ
hs : s.Nonempty
hAB : ∀ i ∈ s, ∀ᶠ (x : α) in l, ‖A i x‖ ≤ C i * ‖B i x‖
x : α
⊢ (∀ i ∈ s, ‖A i x‖ ≤ C i * ‖B i x‖) → ‖∑ i ∈ s, A i x‖ ≤ sSup (C '' ↑s) * ‖∑ i ∈ s, ‖B i x‖‖

Tactic:
hx

Goal state after:
α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
l : Filter α
ι : Type u_18
A : ι → α → E'
C : ι → ℝ
s : Finset ι
B : ι → α → ℝ
hs : s.Nonempty
hAB : ∀ i ∈ s, ∀ᶠ (x : α) in l, ‖A i x‖ ≤ C i * ‖B i x‖
x : α
hx : ∀ i ∈ s, ‖A i x‖ ≤ C i * ‖B i x‖
⊢ ‖∑ i ∈ s, A i x‖ ≤ sSup (C '' ↑s) * ‖∑ i ∈ s, ‖B i x‖‖
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: 𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
α : Type u_17
l : Filter α
f g : α → 𝕜
h : f =O[l] g
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖f x / g x‖

Tactic:
obtain ⟨c, h₀, hc⟩ := h.exists_nonneg

Goal state after:
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
α : Type u_17
l : Filter α
f g : α → 𝕜
h : f =O[l] g
c : ℝ
h₀ : c ≥ 0
hc : IsBigOWith c l f g
⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun x ↦ ‖f x / g x‖
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
C : ℝ
a : 𝕜
f g : α → 𝕜
l : Filter α
h : Tendsto (fun x ↦ g x / f x) l (𝓝 a)
hC : 0 < C
ha : C⁻¹ < ‖a‖
x : α
⊢ C⁻¹ ≤ ((fun a ↦ ‖a‖) ∘ fun x ↦ g x / f x) x → ‖f x‖ ≤ C * ‖g x‖

Tactic:
hx

Goal state after:
α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
C : ℝ
a : 𝕜
f g : α → 𝕜
l : Filter α
h : Tendsto (fun x ↦ g x / f x) l (𝓝 a)
hC : 0 < C
ha : C⁻¹ < ‖a‖
x : α
hx : C⁻¹ ≤ ((fun a ↦ ‖a‖) ∘ fun x ↦ g x / f x) x
⊢ ‖f x‖ ≤ C * ‖g x‖
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
⊢ ‖cexp ↑x‖ = Real.exp x

Tactic:
rw [← ofReal_exp]

Goal state after:
x : ℝ
⊢ ‖↑(Real.exp x)‖ = Real.exp x
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: x : ℝ
h : cos x = 1
n : ℤ
hn : ↑(n / 2 * 2) * π + π = x
hn1 : n % 2 = 1
⊢ ↑(n / 2) * (2 * π) = x

Tactic:
Int.cast_mul,

Goal state after:
x : ℝ
h : cos x = 1
n : ℤ
hn : ↑(n / 2) * ↑2 * π + π = x
hn1 : n % 2 = 1
⊢ ↑(n / 2) * (2 * π) = x
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
h : ↑S.topologicalClosure ≠ univ
⊢ 1 ≤ ‖S.normedMk‖

Tactic:
obtain ⟨x, hx⟩ : ∃ x : M, 0 < ‖(x : M ⧸ S)‖ := by
    refine (Set.nonempty_compl.2 h).imp fun x hx ↦ ?_
    exact (norm_nonneg _).lt_of_ne' <| mt norm_mk_eq_zero_iff_mem_closure.1 hx

Goal state after:
M : Type u_1
inst✝ : SeminormedAddCommGroup M
S : AddSubgroup M
h : ↑S.topologicalClosure ≠ univ
x : M
hx : 0 < ‖↑x‖
⊢ 1 ≤ ‖S.normedMk‖
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
R✝ : Type u_2
S : Type u_3
inst✝⁷ : Field R✝
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
v w : AbsoluteValue R✝ S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
_i : OrderTopology S
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
⊢ ‖x‖ < 1 → Summable fun n ↦ x ^ n

Tactic:
hx

Goal state after:
α : Type u_1
R✝ : Type u_2
S : Type u_3
inst✝⁷ : Field R✝
inst✝⁶ : Field S
inst✝⁵ : LinearOrder S
v w : AbsoluteValue R✝ S
inst✝⁴ : TopologicalSpace S
inst✝³ : IsStrictOrderedRing S
inst✝² : Archimedean S
_i : OrderTopology S
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : CompleteSpace R
x : R
hx : ‖x‖ < 1
⊢ Summable fun n ↦ x ^ n
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x y : E
δ ε : ℝ
hδ : 0 < δ
hε : 0 < ε
⊢ Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y

Tactic:
refine ⟨fun h => le_of_not_gt fun hxy => ?_, ball_disjoint_ball⟩
  rw [add_comm] at hxy
  obtain ⟨z, hxz, hzy⟩ := exists_dist_lt_lt hδ hε hxy
  rw [dist_comm] at hxz
  exact h.le_bot ⟨hxz, hzy⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

Goal state before:
pp: x : ℝ
⊢ x ∈ Ioo (-(π / 2)) (π / 2) → cos x ∈ Ioi 0

Tactic:
hx

Goal state after:
x : ℝ
hx : x ∈ Ioo (-(π / 2)) (π / 2)
⊢ cos x ∈ Ioi 0
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
hr₀ : 0 ≤ r
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
⊢ Summable f

Tactic:
← @summable_nat_add_iff α _ _ _ _ N

Goal state after:
case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
hr₀ : 0 ≤ r
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
⊢ Summable fun n ↦ f (n + N)
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: case right.inr
X : Type u_1
inst✝ : TopologicalSpace X
U V : Set X
hU : IsPathConnected U
hV : IsPathConnected V
x : X
xU : x ∈ U
xV : x ∈ V
y : X
yV : y ∈ V
⊢ JoinedIn (U ∪ V) x y

Tactic:
exact (hV.joinedIn x xV y yV).mono subset_union_right

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
N : ℕ
hNN₀ : N₀ ≤ N
hN : ‖f N‖ ≠ 0
⊢ ¬Summable f

Tactic:
← @summable_nat_add_iff α _ _ _ _ N

Goal state after:
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
N : ℕ
hNN₀ : N₀ ≤ N
hN : ‖f N‖ ≠ 0
⊢ ¬Summable fun n ↦ f (n + N)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case convert_2
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
N : ℕ
hNN₀ : N₀ ≤ N
hN : ‖f N‖ ≠ 0
h' : Tendsto (fun n ↦ f (n + N)) atTop (𝓝 0)
⊢ 0 < ((fun a ↦ ‖a‖) ∘ fun n ↦ f (n + N)) 0

Tactic:
refine lt_of_le_of_ne (norm_nonneg _) ?_
    intro h''
    specialize hN₀ N hNN₀
    simp only [comp_apply, zero_add] at h''
    exact hN h''.symm

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Antitone f
hf0 : Tendsto f atTop (𝓝 0)
hzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b
hfa' : Monotone fun n ↦ -f n
⊢ Tendsto (fun n ↦ -f n) atTop (𝓝 0)

Tactic:
convert! hf0.neg

Goal state after:
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : ℝ
f : ℕ → ℝ
z : ℕ → E
hfa : Antitone f
hf0 : Tendsto f atTop (𝓝 0)
hzb : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, z i‖ ≤ b
hfa' : Monotone fun n ↦ -f n
⊢ 0 = -0