mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Constructions.Polish.Basic

Goal state before:
pp: case right
X : Type u_6
E : Type u_7
ι : Type u_8
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : CommMonoid E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : IsCompletelyPseudoMetrizableSpace E
inst✝⁶ : SecondCountableTopology E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : MeasurableMul₂ E
inst✝² : Countable ι
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f g : ι → X → E
hg_meas : ∀ (i : ι), Measurable (g i)
hg_eq_f : ∀ (i : ι), f i =ᵐ[μ] g i
⊢ (fun x ↦ ∏'[L] (i : ι), f i x) =ᵐ[μ] fun x ↦ ∏'[L] (i : ι), g i x

Tactic:
filter_upwards [ae_all_iff.mpr hg_eq_f] with x h_eq using tprod_congr h_eq

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

Goal state before:
pp: G : Type u_1
H : Type u_2
inst✝³ : MeasurableSpace G
inst✝² : MeasurableSpace H
inst✝¹ : Group G
μ : Measure G
inst✝ : MeasurableInv G
⊢ count.inv = count

Tactic:
by ext s hs; rw [count_apply hs, inv_apply, count_apply hs.inv, encard_inv]

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

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : MeasurableSpace α
inst✝³ : MeasurableSpace β
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : SFinite ν
ρ : Measure (α × β)
inst✝ : IsZeroOrProbabilityMeasure ρ
h : ρ = 0
⊢ IsZeroOrProbabilityMeasure ρ.snd

Tactic:
simp only [h, snd_zero]
    infer_instance

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ✝ : Measure α
inst✝¹ : TopologicalSpace α
ι : Type u_3
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).InnerRegular
a : Finset ι
⊢ (∑ i ∈ a, μ i).InnerRegular

Tactic:
induction a using Finset.induction with

Goal state after:
Module: Mathlib.MeasureTheory.Group.Prod

Goal state before:
pp: G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
g : G
this : map (fun x ↦ g * x) (map Inv.inv μ) = map (fun x ↦ g * x⁻¹) μ
⊢ μ ≪ map (fun h ↦ g * h⁻¹) μ

Tactic:
← this

Goal state after:
G : Type u_1
inst✝⁵ : MeasurableSpace G
inst✝⁴ : Group G
inst✝³ : MeasurableMul₂ G
μ : Measure G
inst✝² : SFinite μ
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulLeftInvariant
g : G
this : map (fun x ↦ g * x) (map Inv.inv μ) = map (fun x ↦ g * x⁻¹) μ
⊢ μ ≪ map (fun x ↦ g * x) (map Inv.inv μ)
Module: Mathlib.MeasureTheory.Measure.Regular

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ✝ : Measure α
inst✝¹ : TopologicalSpace α
ι : Type u_3
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).InnerRegular
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
hr : r < (sum μ) s
this : Tendsto (fun a ↦ ∑ i ∈ a, (μ i) s) atTop (𝓝 ((sum μ) s))
a : Finset ι
ha : r < (∑ i ∈ a, μ i) s
K : Set α
Ks : K ⊆ s
hK : IsCompact K
h'K : r < (∑ i ∈ a, μ i) K
⊢ ∃ K ⊆ s, IsCompact K ∧ r < (sum μ) K

Tactic:
refine ⟨K, Ks, hK, h'K.trans_le ?_⟩

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ✝ : Measure α
inst✝¹ : TopologicalSpace α
ι : Type u_3
μ : ι → Measure α
inst✝ : ∀ (i : ι), (μ i).InnerRegular
s : Set α
hs : MeasurableSet s
r : ℝ≥0∞
hr : r < (sum μ) s
this : Tendsto (fun a ↦ ∑ i ∈ a, (μ i) s) atTop (𝓝 ((sum μ) s))
a : Finset ι
ha : r < (∑ i ∈ a, μ i) s
K : Set α
Ks : K ⊆ s
hK : IsCompact K
h'K : r < (∑ i ∈ a, μ i) K
⊢ (∑ i ∈ a, μ i) K ≤ (sum μ) K
Module: Mathlib.MeasureTheory.Measure.WithDensity

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∑' (i : ι), ∫⁻ (x : α) in s, f i x ∂μ

Tactic:
rw [← lintegral_tsum fun i => (h i).aemeasurable]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ι : Type u_2
inst✝ : Countable ι
f : ι → α → ℝ≥0∞
h : ∀ (i : ι), Measurable (f i)
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, (∑' (n : ι), f n) x ∂μ = ∫⁻ (a : α) in s, ∑' (i : ι), f i a ∂μ
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
⊢ (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) s → ∀ r < μ s, ∃ K ⊆ s, (fun s ↦ IsCompact s ∧ IsClosed[inst✝³] s) K ∧ r < μ K

Tactic:
⟨s_meas, μs⟩

Goal state after:
G : Type u_1
inst✝⁴ : MeasurableSpace G
inst✝³ : TopologicalSpace G
inst✝² : BorelSpace G
μ : Measure G
inst✝¹ : Group G
inst✝ : IsTopologicalGroup G
h : μ.InnerRegularCompactLTTop
s : Set G
s_meas : MeasurableSet s
μs : μ s ≠ ∞
⊢ ∀ r < μ s, ∃ K ⊆ s, (fun s ↦ IsCompact s ∧ IsClosed[inst✝³] s) K ∧ r < μ K
Module: Mathlib.MeasureTheory.Group.Measure

Goal state before:
pp: case inl
G : Type u_1
H : Type u_2
inst✝¹⁰ : MeasurableSpace G
inst✝⁹ : MeasurableSpace H
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
μ✝ : Measure G
inst✝⁶ : μ✝.IsHaarMeasure
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : T1Space G
inst✝² : WeaklyLocallyCompactSpace G
inst✝¹ : (𝓝[≠] 1).NeBot
μ : Measure G
inst✝ : μ.IsHaarMeasure
h : μ 1 = 0
⊢ ∀ (x : G), μ {x} = 0

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x : ℝ
h : 0 ≤ x
⊢ ↑(NNReal.sqrt x.toNNReal * NNReal.sqrt x.toNNReal) = x

Tactic:
NNReal.mul_self_sqrt,

Goal state after:
x : ℝ
h : 0 ≤ x
⊢ ↑x.toNNReal = x
Module: Mathlib.Analysis.Real.Sqrt

Goal state before:
pp: x y : ℝ
⊢ √x = y ↔ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0

Tactic:
constructor
  · rintro rfl
    rcases le_or_gt 0 x with hle | hlt
    · exact Or.inl ⟨mul_self_sqrt hle, sqrt_nonneg x⟩
    · exact Or.inr ⟨hlt, sqrt_eq_zero_of_nonpos hlt.le⟩
  · rintro (⟨rfl, hy⟩ | ⟨hx, rfl⟩)
    exacts [sqrt_mul_self hy, sqrt_eq_zero_of_nonpos hx.le]

Goal state after:
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
hf_ne_zero : ∀ᵐ (x : α) ∂μ, f x ≠ 0
hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ∞
⊢ ((μ.withDensity f).withDensity fun x ↦ (f x)⁻¹) = μ

Tactic:
by
  rw [← withDensity_mul₀ hf hf.fun_inv]
  suffices (f * fun x ↦ (f x)⁻¹) =ᵐ[μ] 1 by
    rw [withDensity_congr_ae this, withDensity_one]
  filter_upwards [hf_ne_zero, hf_ne_top] with x hf_ne_zero hf_ne_top
  simp only [Pi.mul_apply]
  rw [ENNReal.mul_inv_cancel hf_ne_zero hf_ne_top, Pi.one_apply]

Goal state after:
Module: Mathlib.Analysis.Complex.Norm

Goal state before:
pp: x : ℝ
hx : ‖x‖ ≤ 1
⊢ ‖-x‖ ≤ 1

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
s : Finset ι
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs_nonempty : s.Nonempty
hs : ∀ i ∈ s, p (f i)
⊢ m (𝔼 i ∈ s, f i) ≤ 𝔼 i ∈ s, m (f i)

Tactic:
simp only [expect, h_div _ _ (sum_induction_nonempty _ _ hp_add hs_nonempty hs)]

Goal state after:
ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
p : M → Prop
f : ι → M
s : Finset ι
h_add : ∀ (a b : M), p a → p b → m (a + b) ≤ m a + m b
hp_add : ∀ (a b : M), p a → p b → p (a + b)
h_div : ∀ (n : ℕ) (a : M), p a → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs_nonempty : s.Nonempty
hs : ∀ i ∈ s, p (f i)
⊢ (↑(#s))⁻¹ • m (∑ x ∈ s, f x) ≤ (↑(#s))⁻¹ • ∑ x ∈ s, m (f x)
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
f : ι → M
s : Finset ι
m : M → N
h_mul : ∀ (a b : M), m (a + b) ≤ m a + m b
h_div : ∀ (n : ℕ) (a : M), m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : s.Nonempty
⊢ ∀ (a b : M), True → True → m (a + b) ≤ m a + m b

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
f : ι → M
s : Finset ι
m : M → N
h_mul : ∀ (a b : M), m (a + b) ≤ m a + m b
h_div : ∀ (n : ℕ) (a : M), m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
hs : s.Nonempty
⊢ ∀ (n : ℕ) (a : M), True → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
f : ι → M
s : Finset ι
h_zero : m 0 = 0
h_add : ∀ (a b : M), m (a + b) ≤ m a + m b
h_div : ∀ (n : ℕ) (a : M), m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
⊢ ∀ (a b : M), True → True → m (a + b) ≤ m a + m b

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
M : Type u_4
N : Type u_5
inst✝⁶ : AddCommMonoid M
inst✝⁵ : Module ℚ≥0 M
inst✝⁴ : AddCommMonoid N
inst✝³ : PartialOrder N
inst✝² : IsOrderedAddMonoid N
inst✝¹ : Module ℚ≥0 N
inst✝ : PosSMulMono ℚ≥0 N
m : M → N
f : ι → M
s : Finset ι
h_zero : m 0 = 0
h_add : ∀ (a b : M), m (a + b) ≤ m a + m b
h_div : ∀ (n : ℕ) (a : M), m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a
⊢ ∀ (n : ℕ) (a : M), True → m ((↑n)⁻¹ • a) = (↑n)⁻¹ • m a

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.BigOperators.Expect

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝⁴ : AddCommMonoid α
inst✝³ : PartialOrder α
inst✝² : IsOrderedCancelAddMonoid α
inst✝¹ : Module ℚ≥0 α
inst✝ : PosSMulStrictMono ℚ≥0 α
s : Finset ι
f g : ι → α
hfg : ∀ i ∈ s, f i ≤ g i
i : ι
hi : i ∈ s
this : s.Nonempty
⊢ 0 < (↑(#s))⁻¹

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Hom

Goal state before:
pp: V : Type u_1
inst✝¹ : SeminormedAddCommGroup V
inst✝ : NontrivialTopology V
x : V
hx : ‖x‖ ≠ 0
this : ‖(id V) x‖ / ‖x‖ ≤ ‖id V‖
⊢ 1 ≤ ‖id V‖

Tactic:
rwa [id_apply, div_self hx] at this

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

Goal state before:
pp: E : Type u_2
inst✝¹ : NonUnitalNormedRing E
inst✝ : StarRing E
h : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖
this : NormedStarGroup E
⊢ ∀ (x : E), ‖x⋆‖ * ‖x⋆‖ ≤ ‖x⋆⋆ * x⋆‖

Tactic:
simpa

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Algebra R ℝ
r : R
x✝ : ℂ
xr xi : ℝ
⊢ ((ofRealHom.comp (algebraMap R ℝ)) r * { re := xr, im := xi }).re =
    ({ re := xr, im := xi } * (ofRealHom.comp (algebraMap R ℝ)) r).re

Tactic:
simp [Algebra.commutes]

Goal state after:
Module: Mathlib.LinearAlgebra.Complex.Module

Goal state before:
pp: case a
R : Type u_1
S : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Algebra R ℝ
r : R
x✝ : ℂ
xr xi : ℝ
⊢ ((ofRealHom.comp (algebraMap R ℝ)) r * { re := xr, im := xi }).im =
    ({ re := xr, im := xi } * (ofRealHom.comp (algebraMap R ℝ)) r).im

Tactic:
simp [Algebra.commutes]

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

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
⊢ 1 + z ∈ Metric.ball 1 1

Tactic:
simpa

Goal state after:
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‖ ≤ ‖x‖

Tactic:
← inv_mul_cancel_left₀ two_ne_zero ‖x‖,

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: case inr
K : Type u_1
inst✝ : RCLike K
z : K
h₀ : z ≠ 0
⊢ ↑‖z‖ ^ 2 ≠ 0

Tactic:
simpa

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

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
z : K
⊢ z * I⁻¹ = -(z * I)

Tactic:
inv_I,

Goal state after:
K : Type u_1
inst✝ : RCLike K
z : K
⊢ z * -I = -(z * I)
Module: Mathlib.Topology.PartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : PartialEquiv X Y
h : IsHomeomorph ⇑f.toEquiv
⊢ ContinuousOn (↑f) f.source

Tactic:
continuousOn_iff_continuous_domRestrict,

Goal state after:
X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : PartialEquiv X Y
h : IsHomeomorph ⇑f.toEquiv
⊢ Continuous[instTopologicalSpaceSubtype, inst✝³] (f.source.domRestrict ↑f)
Module: Mathlib.Topology.PartialHomeomorph.Basic

Goal state before:
pp: X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : PartialEquiv X Y
h : IsHomeomorph ⇑f.toEquiv
⊢ ContinuousOn f.invFun f.target

Tactic:
continuousOn_iff_continuous_domRestrict,

Goal state after:
X : Type u_1
X' : Type u_2
Y : Type u_3
Y' : Type u_4
Z : Type u_5
Z' : Type u_6
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace X'
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Y'
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace Z'
e : PartialHomeomorph X Y
f : PartialEquiv X Y
h : IsHomeomorph ⇑f.toEquiv
⊢ Continuous[instTopologicalSpaceSubtype, inst✝⁵] (f.target.domRestrict f.invFun)
Module: Mathlib.Analysis.RCLike.Basic

Goal state before:
pp: K : Type u_1
inst✝ : RCLike K
x y : K
⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ

Tactic:
by rw [map_sub re x y]

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

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ (x y : ℝ), (algebraMap ℝ 𝕜) (x + y) = (algebraMap ℝ 𝕜) x + (algebraMap ℝ 𝕜) y

Tactic:
exact h.algebraMap.map_add'

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

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝ : RCLike K
𝕜 : Type u_3
h : RCLike 𝕜
__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField
⊢ ∀ {f : Filter 𝕜}, Cauchy f → ∃ x, f ≤ nhds x

Tactic:
exact h.complete

Goal state after:
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
l : Filter α
f : α × β → E
g : α × β → F
l' : Filter β
⊢ f =Θ[l ×ˢ l'] g → ∀ᶠ (x : α) in l, (fun x_1 ↦ f (x, x_1)) =Θ[l'] fun x_1 ↦ g (x, x_1)

Tactic:
simp only [IsTheta, eventually_and]
  exact fun ⟨h₁, h₂⟩ ↦ ⟨h₁.fiberwise_right, h₂.fiberwise_right⟩

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

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝¹ : Norm E
inst✝ : Norm F
f : ℤ → E
g : ℤ → F
hf : f =O[cofinite] g
⊢ (fun n ↦ f (-n)) =O[cofinite] fun n ↦ g (-n)

Tactic:
by
  rw [← Equiv.neg_apply]
  exact hf.comp_tendsto (Equiv.neg ℤ).injective.tendsto_cofinite

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

Goal state before:
pp: case hbc
α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
c₁ c₂ : ℝ
l : Filter α
f₁ f₂ : α → E'
g₁ g₂ : α → ℝ
h₁ : ∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖
h₂ : ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖
x : α
hx₁ : ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖
hx₂ : ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖
⊢ c₁ ≤ max c₁ c₂

Tactic:
simp [le_max_left _ _, le_max_right _ _]

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

Goal state before:
pp: case hbc
α : Type u_1
E' : Type u_6
inst✝ : SeminormedAddCommGroup E'
c₁ c₂ : ℝ
l : Filter α
f₁ f₂ : α → E'
g₁ g₂ : α → ℝ
h₁ : ∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖
h₂ : ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖
x : α
hx₁ : ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖
hx₂ : ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖
⊢ c₂ ≤ max c₁ c₂

Tactic:
simp [le_max_left _ _, le_max_right _ _]

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

Goal state before:
pp: α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
c : E
c' : F''
hc' : c' ≠ 0
l : Filter α
⊢ IsBigOWith (‖c‖ / ‖c'‖) l (fun _x ↦ c) fun _x ↦ c'

Tactic:
by
  simp only [IsBigOWith_def]
  apply univ_mem'
  intro x
  rw [mem_ofPred, div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hc')]

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

Goal state before:
pp: α : Type u_1
R : Type u_13
inst✝³ : SeminormedRing R
S : Type u_17
inst✝² : NormedRing S
inst✝¹ : NormMulClass S
c : ℝ
l : Filter α
inst✝ : NormOneClass S
f : α → R
g : α → S
h : IsBigOWith c l f g
⊢ IsBigOWith (Nat.casesOn 1 ‖1‖ fun n ↦ c ^ (n + 1)) l (fun x ↦ f x ^ 1) fun x ↦ g x ^ 1

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ cosh (-x) = cosh x

Tactic:
by simp [add_comm, cosh, exp_neg]

Goal state after:
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: case hba
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n j : ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ range (j - n)
⊢ n.factorial * n.succ ^ x ≤ (x + n).factorial

Tactic:
add_comm

Goal state after:
case hba
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
n j : ℕ
hn : 0 < n
x : ℕ
a✝ : x ∈ range (j - n)
⊢ n.factorial * n.succ ^ x ≤ (n + x).factorial
Module: Mathlib.Analysis.Asymptotics.Lemmas

Goal state before:
pp: α : Type u_1
𝕜 : Type u_15
inst✝ : NormedDivisionRing 𝕜
l : Filter α
f g : α → 𝕜
a : 𝕜
h : Tendsto (fun x ↦ g x / f x) l (𝓝 a)
ha : a ≠ 0
⊢ 1 < 1 + ‖a‖

Tactic:
simpa

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

Goal state before:
pp: α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
f : α → E
g'' : α → F''
h : f =O[cofinite] g''
C : ℝ
C₀ : C > 0
hC : {x | ¬‖f x‖ ≤ C * ‖g'' x‖}.Finite
C' : ℝ
hC' : ∀ i ∈ Finset.image (fun x ↦ ‖f x‖ / ‖g'' x‖) hC.toFinset, i ≤ C'
this : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'
x : α
h₀ : g'' x ≠ 0
⊢ ‖f x‖ ≤ C * ‖g'' x‖ ∨ ‖f x‖ ≤ C' * ‖g'' x‖

Tactic:
or_iff_not_imp_left,

Goal state after:
α : Type u_1
E : Type u_3
F'' : Type u_10
inst✝¹ : Norm E
inst✝ : NormedAddCommGroup F''
f : α → E
g'' : α → F''
h : f =O[cofinite] g''
C : ℝ
C₀ : C > 0
hC : {x | ¬‖f x‖ ≤ C * ‖g'' x‖}.Finite
C' : ℝ
hC' : ∀ i ∈ Finset.image (fun x ↦ ‖f x‖ / ‖g'' x‖) hC.toFinset, i ≤ C'
this : ∀ (x : α), C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C'
x : α
h₀ : g'' x ≠ 0
⊢ ¬‖f x‖ ≤ C * ‖g'' x‖ → ‖f x‖ ≤ C' * ‖g'' x‖
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ sin x ^ 2 + cos x ^ 2 = cos x ^ 2 + sin x ^ 2

Tactic:
add_comm

Goal state after:
x : ℂ
⊢ cos x ^ 2 + sin x ^ 2 = cos x ^ 2 + sin x ^ 2
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
⊢ cos x ^ 2 = 1 / 2 + cos (2 * x) / 2

Tactic:
by
  simp [cos_two_mul, ← add_div, mul_div_cancel_left₀, -one_div]

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
hx : ‖x‖ ≤ 1
⊢ ‖-x * I‖ ≤ 1

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℂ
hx : ‖x‖ ≤ 1
⊢ ‖x * I‖ ≤ 1

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Complex.Exponential

Goal state before:
pp: x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
⊢ |x| = x

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
⊢ ↑(cos (3 * x)) = ↑(4 * cos x ^ 3 - 3 * cos x)

Tactic:
simp [cos_three_mul]

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
hx : |x| ≤ 1
⊢ ‖↑x‖ ≤ 1

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Complex.Trigonometric

Goal state before:
pp: x : ℝ
hx0 : 0 < x
hx : x ≤ 2
⊢ x ≤ 1 * 2

Tactic:
simpa

Goal state after:
Module: Mathlib.Algebra.Order.Field.Power

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
n : ℤ
hn : Odd n
⊢ a ^ n < 0 ∨ a ^ n = 0 ↔ a < 0 ∨ a = 0

Tactic:
hn.zpow_neg_iff,

Goal state after:
α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a : α
n : ℤ
hn : Odd n
⊢ a < 0 ∨ a ^ n = 0 ↔ a < 0 ∨ a = 0
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
hz : ‖z‖ ≤ 1
this : ‖cexp (x + z) - cexp x - z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖cexp (x + z) - cexp x‖ - ‖z‖ * ‖cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2

Tactic:
← norm_smul z

Goal state after:
r : ℝ
hr_nonneg : 0 ≤ r
hr_le : r ≤ 1
x y : ℂ
hyx : ‖y - x‖ < r
hy_eq : y = x + (y - x)
hyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖
z : ℂ
hz : ‖z‖ ≤ 1
this : ‖cexp (x + z) - cexp x - z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
⊢ ‖cexp (x + z) - cexp x‖ - ‖z • cexp x‖ ≤ ‖cexp x‖ * ‖z‖ ^ 2
Module: Mathlib.Analysis.SpecialFunctions.Exp

Goal state before:
pp: a : ℝ
⊢ rexp '' Set.Iio a = Set.Ioo 0 (rexp a)

Tactic:
rw [← coe_comp_expOrderIso, image_comp, expOrderIso.image_Iio, image_subtype_val_Ioi_Iio,
    Function.comp_apply]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Quotient

Goal state before:
pp: M : Type u_1
inst✝ : SeminormedCommGroup M
S : Subgroup M
m : M
ε : ℝ
hε : 0 < ε
n : M
hn : ↑n = (mk' S) m
hn' : ‖n‖ < ‖(mk' S) m‖ + ε
⊢ ‖m * (m⁻¹ * n)‖ < ‖(mk' S) m‖ + ε

Tactic:
simpa

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Pointwise

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedCommGroup E
δ : ℝ
s : Set E
⊢ ball 1 δ * s = thickening δ s

Tactic:
rw [mul_comm, mul_ball_one]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RCLike.Real

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
x : E
r : ℝ
hr : r ≠ 0
⊢ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball x r) = closedBall x r

Tactic:
by
  refine Subset.antisymm closure_ball_subset_closedBall fun y hy => ?_
  have : ContinuousWithinAt (fun c : ℝ => c • (y - x) + x) (Ico 0 1) 1 := by fun_prop
  convert! this.mem_closure _ _
  · rw [one_smul, sub_add_cancel]
  · simp [closure_Ico zero_ne_one, zero_le_one]
  · rintro c ⟨hc0, hc1⟩
    rw [mem_ball, dist_eq_norm, add_sub_cancel_right, norm_smul, Real.norm_eq_abs,
      abs_of_nonneg hc0, mul_comm, ← mul_one r]
    rw [mem_closedBall, dist_eq_norm] at hy
    replace hr : 0 < r := ((norm_nonneg _).trans hy).lt_of_ne hr.symm
    apply mul_lt_mul' <;> assumption

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

Goal state before:
pp: case neg
R : Type u_2
inst✝ : NormedRing R
k : ℕ
r₁ : R
r₂ : ℝ
h : ‖r₁‖ < r₂
h0 : ¬r₁ = 0
⊢ (fun n ↦ ↑n ^ k * r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n

Tactic:
← Ne,

Goal state after:
case neg
R : Type u_2
inst✝ : NormedRing R
k : ℕ
r₁ : R
r₂ : ℝ
h : ‖r₁‖ < r₂
h0 : r₁ ≠ 0
⊢ (fun n ↦ ↑n ^ k * r₁ ^ n) =o[atTop] fun n ↦ r₂ ^ n
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ ∑' (i : ℕ), x ^ (i + 1) + 1 = 1 + ∑' (b : ℕ), x ^ (b + 1)

Tactic:
add_comm

Goal state after:
R : Type u_4
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : R
h : ‖x‖ < 1
⊢ 1 + ∑' (i : ℕ), x ^ (i + 1) = 1 + ∑' (b : ℕ), x ^ (b + 1)