mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.CStarMatrix

Goal state before:
pp: case hbc
A : Type u_1
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : PartialOrder A
inst✝² : StarOrderedRing A
m : Type u_2
n : Type u_3
inst✝¹ : Fintype m
inst✝ : Fintype n
M : CStarMatrix n n A
v : C⋆ᵐᵒᵈ(A, n → A)
⊢ ‖(toCLM Mᴴ ∘SL toCLM M) v‖ ≤ ‖toCLM Mᴴ ∘SL toCLM M‖ * ‖v‖

Tactic:
exact le_opNorm ((toCLM Mᴴ).comp (toCLM M)) v

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.RealImaginaryPart

Goal state before:
pp: A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (quasispectrum ℝ ↑(ℑ a))
hf0 : f 0 = 0
ha : IsStarNormal a
⊢ cfcₙ (fun x ↦ ↑(f x.im)) a = cfcₙ f ↑(ℑ a)

Tactic:
have : ContinuousOn (fun x ↦ (f x.re) : ℂ → ℂ) ((im · : ℂ → ℂ) '' quasispectrum ℂ a) := by
    rw [quasispectrum_imaginaryPart' a] at hf
    refine continuous_ofReal.comp_continuousOn <| hf.comp (by fun_prop) ?_
    simpa [Set.mapsTo_image_iff, Function.comp_def] using Set.mapsTo_image ..

Goal state after:
A : Type u_1
inst✝⁹ : TopologicalSpace A
inst✝⁸ : NonUnitalRing A
inst✝⁷ : StarRing A
inst✝⁶ : Module ℂ A
inst✝⁵ : IsScalarTower ℂ A A
inst✝⁴ : SMulCommClass ℂ A A
inst✝³ : StarModule ℂ A
inst✝² : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal
inst✝¹ : ContinuousMapZero.UniqueHom ℂ A
inst✝ : T2Space A
f : ℝ → ℝ
a : A
hf : ContinuousOn f (quasispectrum ℝ ↑(ℑ a))
hf0 : f 0 = 0
ha : IsStarNormal a
this : ContinuousOn (fun x ↦ ↑(f x.re)) ((fun x ↦ ↑x.im) '' quasispectrum ℂ a)
⊢ cfcₙ (fun x ↦ ↑(f x.im)) a = cfcₙ f ↑(ℑ a)
Module: Mathlib.Analysis.CStarAlgebra.Hom

Goal state before:
pp: F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
hφ : Function.Injective ⇑φ
a✝ : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
hψ : Function.Injective ⇑ψ
a : Unitization ℂ A
⊢ ‖ψ (star a * a)‖ = ‖star a * a‖

Tactic:
have ha : IsSelfAdjoint (star a * a) := .star_mul_self a

Goal state after:
F : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : NonUnitalCStarAlgebra A
inst✝³ : NonUnitalCStarAlgebra B
inst✝² : FunLike F A B
inst✝¹ : NonUnitalAlgHomClass F ℂ A B
inst✝ : StarHomClass F A B
φ : F
hφ : Function.Injective ⇑φ
a✝ : A
ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B
hψ : Function.Injective ⇑ψ
a : Unitization ℂ A
ha : IsSelfAdjoint (star a * a)
⊢ ‖ψ (star a * a)‖ = ‖star a * a‖
Module: Mathlib.Analysis.CStarAlgebra.Projection

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p q : A
hp : IsStarProjection p
hq : IsStarProjection q
⊢ [p ≤ q, q * p = p, p * q = p, IsStarProjection (q - p), IsIdempotentElem (q - p)].TFAE

Tactic:
tfae_have 1 → 2 := fun h ↦ (hq.mul_right_and_mul_left_of_nonneg_of_le hp.nonneg h).2
  tfae_have 2 → 3 := fun h ↦ by
    simpa [hp.isSelfAdjoint.star_eq, hq.isSelfAdjoint.star_eq] using congr(star $h)
  tfae_have 3 → 4 := hp.sub_of_mul_eq_left hq
  tfae_have 4 → 1 := fun h ↦ by simpa using h.nonneg
  tfae_have 4 ↔ 5 := by simp [isStarProjection_iff, hq.isSelfAdjoint.sub hp.isSelfAdjoint]
  tfae_finish

Goal state after:
Module: Mathlib.Topology.OpenPartialHomeomorph.Constructions

Goal state before:
pp: X : Type u_1
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
e : OpenPartialHomeomorph X Y
U V : Opens X
hU : Nonempty ↥U
hV : Nonempty ↥V
hUV : U ≤ V
i : ↑↑U → ↑↑V := inclusion hUV
y : Y
hy : y ∈ e.target ∩ ↑e.symm ⁻¹' ↑U
⊢ ↑e.symm y ∈ ↑V

Tactic:
exact hUV hy.2

Goal state after:
Module: Mathlib.Topology.FiberBundle.Basic

Goal state before:
pp: B : Type u_2
F : Type u_3
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace F
E : B → Type u_5
inst✝² : TopologicalSpace (TotalSpace F E)
inst✝¹ : (b : B) → TopologicalSpace (E b)
inst✝ : FiberBundle F E
s : (x : B) → E x
a : Set B
x₀ : B
a✝ : ContinuousWithinAt (fun x ↦ (↑(trivializationAt F E x₀) (TotalSpace.mk' F x (s x))).2) a x₀
⊢ ContinuousWithinAt (fun x ↦ x) a x₀

Tactic:
exact continuousWithinAt_id

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

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd ^ k).1 y
x y : A
⊢ (a.toProd.2 ^ (k + 1)) x * y = x * (a.toProd ^ (k + 1)).1 y

Tactic:
Prod.pow_fst

Goal state after:
case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd.2 ^ k) x * y = x * (a.toProd.1 ^ k) y
x y : A
⊢ (a.toProd.2 ^ (k + 1)) x * y = x * (a.toProd.1 ^ (k + 1)) y
Module: Mathlib.Analysis.CStarAlgebra.Multiplier

Goal state before:
pp: case succ
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
k : ℕ
hk : ∀ (x y : A), (a.toProd ^ k).2 x * y = x * (a.toProd ^ k).1 y
x y : A
⊢ (a.toProd ^ (k + 1)).2 x * y = x * (a.toProd ^ (k + 1)).1 y

Tactic:
| succ k hk =>

Goal state after:
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
s : Set X
I : Type u_3
inst✝¹ : TopologicalSpace I
x : X
inst✝ : Nonempty I
h✝ : DiscreteTopology I
U : Set X
hxU : x ∈ U
hU : IsOpen[inst✝²] U
hfU : IsOpen[inst✝³] (f ⁻¹' U)
H : ↑(f ⁻¹' U) ≃ₜ ↑U × I
hH : ∀ (x : ↑(f ⁻¹' U)), ↑(H x).1 = f ↑x
e : E
he : f e ∈ U
⊢ (↑{ toFun := fun e ↦ if he : f e ∈ U then (↑(H ⟨e, he⟩).1, (H ⟨e, he⟩).2) else (x, Classical.arbitrary I),
            invFun := fun xi ↦ ↑(H.symm (if hx : xi.1 ∈ U then ⟨xi.1, hx⟩ else ⟨x, hxU⟩, xi.2)), source := f ⁻¹' U,
            target := U ×ˢ Set.univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯,
            continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, open_source := hfU, open_target := ⋯ }
        e).1 =
    f e

Tactic:
by simp [he, hH]

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

Goal state before:
pp: case refine_2.refine_1
𝕜 : Type u_1
A : Type u_2
inst✝⁸ : DenselyNormedField 𝕜
inst✝⁷ : StarRing 𝕜
inst✝⁶ : NonUnitalNormedRing A
inst✝⁵ : StarRing A
inst✝⁴ : CStarRing A
inst✝³ : NormedSpace 𝕜 A
inst✝² : SMulCommClass 𝕜 A A
inst✝¹ : IsScalarTower 𝕜 A A
inst✝ : StarModule 𝕜 A
a : 𝓜(𝕜, A)
hball : (Metric.closedBall 0 1).Nonempty
key : ∀ (x y : A), ‖x‖₊ ≤ 1 → ‖y‖₊ ≤ 1 → ‖a.toProd.2 (star (a.toProd.1 (star x))) * y‖₊ ≤ ‖a‖₊ * ‖a‖₊
r : ℝ≥0
hr : r < ‖a‖₊ * ‖a‖₊
hr' : sqrt r < sSup ((fun x ↦ ‖a.toProd.1 x‖₊) '' Metric.closedBall 0 1)
x : A
hx : x ∈ Metric.closedBall 0 1
hxr : sqrt r < (fun x ↦ ‖a.toProd.1 x‖₊) x
hx' : ‖x‖₊ ≤ 1
⊢ BddAbove ((fun x_1 ↦ ‖a.toProd.2 (star (a.toProd.1 (star (star x)))) * x_1‖₊) '' Metric.closedBall 0 1)

Tactic:
refine ⟨‖a‖₊ * ‖a‖₊, ?_⟩
          rintro - ⟨y, hy, rfl⟩
          exact key (star x) y ((nnnorm_star x).trans_le hx') (mem_closedBall_zero_iff.1 hy)

Goal state after:
Module: Mathlib.Topology.Covering.Quotient

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
f : E → X
G : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G E
h : IsQuotientCoveringMap f G
⊢ IsCoveringMapOn f Set.univ

Tactic:
have := h.toContinuousConstSMul
    convert! ← h.isCoveringMapOn_of_smul_disjoint h.apply_eq_iff_mem_orbit fun e ↦ ?_
    · refine Set.eq_univ_of_forall fun x ↦ ?_
      obtain ⟨e, rfl⟩ := h.surjective x
      have ⟨U, hU, hGU⟩ := h.disjoint e
      replace hU := mem_of_mem_nhds hU
      exact ⟨e, (Subgroup.eq_bot_iff_forall _).mpr fun g hg ↦ hGU g (⟨e, ⟨e, hU, hg⟩, hU⟩), rfl⟩
    · have ⟨U, hU, hGU⟩ := h.disjoint e
      exact ⟨U, hU, fun g hg ↦ by rw [hGU g hg, one_smul]⟩

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Span

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : A
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2

Tactic:
let _ := CStarAlgebra.spectralOrder

Goal state after:
A : Type u_1
inst✝ : CStarAlgebra A
x : A
x✝ : (A : Type ?u.39) → [NonUnitalCStarAlgebra A] → PartialOrder A := spectralOrder
⊢ ∃ u c, x = ∑ i, c i • ↑(u i) ∧ ∀ (i : Fin 4), ‖c i‖ ≤ ‖x‖ / 2
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
hx : x ∈ ball 0 π
⊢ ‖x‖ < π

Tactic:
by simpa using hx

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Adjoint

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : CompleteSpace E
inst✝ : CompleteSpace F
T : E →L[𝕜] F
⊢ Function.Injective (⇑(adjoint T) ∘ ⇑T) ↔ Function.Injective ⇑T

Tactic:
rw [← coe_comp, ← coe_coe, ← LinearMap.ker_eq_bot, ← coe_coe, ← LinearMap.ker_eq_bot,
    ker_adjoint_comp_self]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional

Goal state before:
pp: case mpr
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
hc : Fintype.card ι = finrank k ↥⊤ + 1
⊢ affineSpan k (Set.range p) = ⊤

Tactic:
← direction_top k V P

Goal state after:
case mpr
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
inst✝¹ : FiniteDimensional k V
inst✝ : Fintype ι
p : ι → P
hi : AffineIndependent k p
hc : Fintype.card ι = finrank k ↥⊤.direction + 1
⊢ affineSpan k (Set.range p) = ⊤
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: case refine_2
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s u : Set P
hu : IsOpen u
hsu : s ⊆ u
h : AffineIndependent ℝ Subtype.val
q : P
hq : q ∈ s
ε : ℝ
ε0 : 0 < ε
hεu : Metric.closedBall q ε ⊆ u
t : Set P
ht₁ : s ⊆ t
ht₂ : AffineIndependent ℝ fun p ↦ ↑p
ht₃ : affineSpan ℝ t = ⊤
f : P → P := fun y ↦ (lineMap q y) (ε / dist y q)
hf : ∀ (y : P), f y ∈ u
hεyq : ∀ y ∉ s, ε / dist y q ≠ 0
w : ↑t → ℝˣ := fun p ↦ if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) ⋯
p : P
hp : p ∈ t
⊢ (fun p ↦ (lineMap q ↑p) ↑(w p)) ⟨p, hp⟩ ∈ u

Tactic:
by_cases hps : p ∈ s

Goal state after:
case pos
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s u : Set P
hu : IsOpen u
hsu : s ⊆ u
h : AffineIndependent ℝ Subtype.val
q : P
hq : q ∈ s
ε : ℝ
ε0 : 0 < ε
hεu : Metric.closedBall q ε ⊆ u
t : Set P
ht₁ : s ⊆ t
ht₂ : AffineIndependent ℝ fun p ↦ ↑p
ht₃ : affineSpan ℝ t = ⊤
f : P → P := fun y ↦ (lineMap q y) (ε / dist y q)
hf : ∀ (y : P), f y ∈ u
hεyq : ∀ y ∉ s, ε / dist y q ≠ 0
w : ↑t → ℝˣ := fun p ↦ if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) ⋯
p : P
hp : p ∈ t
hps : p ∈ s
⊢ (fun p ↦ (lineMap q ↑p) ↑(w p)) ⟨p, hp⟩ ∈ u

case neg
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s u : Set P
hu : IsOpen u
hsu : s ⊆ u
h : AffineIndependent ℝ Subtype.val
q : P
hq : q ∈ s
ε : ℝ
ε0 : 0 < ε
hεu : Metric.closedBall q ε ⊆ u
t : Set P
ht₁ : s ⊆ t
ht₂ : AffineIndependent ℝ fun p ↦ ↑p
ht₃ : affineSpan ℝ t = ⊤
f : P → P := fun y ↦ (lineMap q y) (ε / dist y q)
hf : ∀ (y : P), f y ∈ u
hεyq : ∀ y ∉ s, ε / dist y q ≠ 0
w : ↑t → ℝˣ := fun p ↦ if hp : ↑p ∈ s then 1 else Units.mk0 (ε / dist (↑p) q) ⋯
p : P
hp : p ∈ t
hps : p ∉ s
⊢ (fun p ↦ (lineMap q ↑p) ↑(w p)) ⟨p, hp⟩ ∈ u
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
h : affineSpan ℝ s = ⊤
t : Set V
hts : t ⊆ s
b : AffineBasis ↑t ℝ V
hb : ⇑b = Subtype.val
⊢ (interior ((convexHull ℝ) (range ⇑b))).Nonempty

Tactic:
lift t to Finset V using b.finite_set

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : FiniteDimensional ℝ V
s : Set V
h : affineSpan ℝ s = ⊤
t : Finset V
hts : ↑t ⊆ s
b : AffineBasis ↑↑t ℝ V
hb : ⇑b = Subtype.val
⊢ (interior ((convexHull ℝ) (range ⇑b))).Nonempty
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
g : α → ℝ
hg : Memℓp g p
hfg : ∀ (i : α), ‖f i‖ ≤ ‖g i‖
hp : 0 < p.toReal
⊢ Memℓp f p

Tactic:
rw [memℓp_gen_iff hp] at hg ⊢

Goal state after:
case inr.inr
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
g : α → ℝ
hg : Summable fun i ↦ ‖g i‖ ^ p.toReal
hfg : ∀ (i : α), ‖f i‖ ≤ ‖g i‖
hp : 0 < p.toReal
⊢ Summable fun i ↦ ‖f i‖ ^ p.toReal
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
this : Summable fun i ↦ ↑(‖c‖₊ ^ p.toReal * ‖f i‖₊ ^ p.toReal)
⊢ Summable fun i ↦ ↑(‖c • f i‖₊ ^ p.toReal)

Tactic:
NNReal.summable_coe,

Goal state after:
case inr.inr
𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
f : (i : α) → E i
hf : Memℓp f p
c : 𝕜
hp : 0 < p.toReal
this : Summable fun a ↦ ‖c‖₊ ^ p.toReal * ‖f a‖₊ ^ p.toReal
⊢ Summable fun a ↦ ‖c • f a‖₊ ^ p.toReal
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
s : Set E
x₀ x : E
hu : Summable u
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf : ∀ (n : α), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α), ∀ x ∈ s, ‖f' n x‖ ≤ u n
hx₀ : x₀ ∈ s
hf0 : Summable fun n ↦ f n x₀
hx : x ∈ s
y : E
hy : y ∈ s
⊢ Summable (fun b ↦ f b y) { filter := sInf (range fun a ↦ 𝓟 (Ici a)) }

Tactic:
exact summable_of_summable_hasFDerivAt_of_isPreconnected hu hs h's hf hf' hx₀ hf0 hy

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
⊢ ∑ i ∈ s, ‖↑f i‖ ^ p.toReal ≤ ‖f‖ ^ p.toReal

Tactic:
rw [lp.norm_rpow_eq_tsum hp f]
  have : ∀ i, 0 ≤ ‖f i‖ ^ p.toReal := fun i ↦ by positivity
  refine Summable.sum_le_tsum _ (fun i _ => this i) ?_
  exact (lp.memℓp f).summable hp

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
inst✝ : DecidableEq α
hp : 0 < p.toReal
f : ↥(lp E p)
s : Finset α
F : α → ℝ := fun i ↦ ‖↑f i‖ ^ p.toReal - ‖↑(f - ∑ i ∈ s, lp.single p i (↑f i)) i‖ ^ p.toReal
i : α
hi : i ∉ s
this : ‖↑f i‖ ^ p.toReal - ‖↑f i - if i ∈ s then ↑f i else 0‖ ^ p.toReal = 0
⊢ F i = 0

Tactic:
simpa only [coeFn_sub, coeFn_sum, lp.coeFn_single, Pi.sub_apply, Finset.sum_apply,
        Finset.sum_pi_single, F] using this

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:

Tactic:
dist (f x i) (f y i) ≤ dist (f x) (f y) := by
        simp only [dist_eq_norm]
        exact lp.norm_apply_le_norm top_ne_zero (f x - f y) i
      _ ≤ K * dist x y := hfl x hx y hy

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case inr.inl
p : ℝ[X]
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval 0⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0

pp: case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Tactic:
· rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]

Goal state after:
case inr.inr
p : ℝ[X]
x : ℝ
hx : 0 < x
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
x₀ : G
h : HasCompactSupport fun t ↦ (L (f t)) (g (x₀ - t))
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
u : G ≃ₜ G := (Homeomorph.neg G).trans (Homeomorph.addRight x₀)
v : G ≃ₜ G := (Homeomorph.neg G).trans (Homeomorph.addLeft x₀)
⊢ ConvolutionExistsAt f g x₀ L μ

Tactic:
apply ((u.isCompact_preimage.mpr h).bddAbove_image hg.norm.continuousOn).convolutionExistsAt' L
    isClosed_closure.measurableSet subset_closure (hf.integrableOn_isCompact h)

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
x₀ : G
h : HasCompactSupport fun t ↦ (L (f t)) (g (x₀ - t))
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
u : G ≃ₜ G := ⋯
v : G ≃ₜ G := ⋯
⊢ AEStronglyMeasurable g
    (Measure.map (fun t ↦ ↑(toAddUnits x₀) - t) (μ.restrict (tsupport fun t ↦ (L (f t)) (g (x₀ - t)))))
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedAddCommGroup E'
inst✝⁹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedSpace 𝕜 E
inst✝⁶ : NormedSpace 𝕜 E'
inst✝⁵ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : MeasurableSpace G
μ : Measure G
inst✝³ : AddGroup G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : BorelSpace G
hcf : HasCompactSupport f
hf : LocallyIntegrable f μ
hg : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ t : G
ht : f t = 0
⊢ (fun t ↦ (L (f t)) (g (x₀ - t))) t = 0

Tactic:
simp_rw [ht, L.map_zero₂]

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case neg
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : SeminormedAddCommGroup G
x₀ : G
R : ℝ
hf : support f ⊆ ball 0 R
hg : ∀ x ∈ ball x₀ R, g x = g x₀
t : G
ht : f t = 0
⊢ (L (f t)) (g (x₀ - t)) = (L (f t)) (g x₀)

Tactic:
simp_rw [ht, L.map_zero₂]

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case refine_2
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : SFinite μ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hif : Integrable f μ
hf : support f ⊆ ball 0 R
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
t : G
ht : f t = 0
⊢ (fun t ↦ (L (f t)) (g (x₀ - t))) t = 0

Tactic:
simp_rw [ht, L.map_zero₂]

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

Goal state before:
pp: G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
⊢ x ∈ k

Tactic:
choose v hv y hy hvy using mem_iInter.1 hx

Goal state after:
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
k : Set G
h : μ.IsEverywherePos k
hk : IsCompact k
h'k : IsClosed[inst✝⁷] k
u : ℕ → ℝ≥0∞
u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1
u_lim : Tendsto u atTop (𝓝 0)
W : ℕ → Set G
W_open : ∀ (n : ℕ), IsOpen[inst✝⁷] (W n)
mem_W : ∀ (n : ℕ), 1 ∈ W n
hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \ k) < u n
V : ℕ → Set G := fun n ↦ ⋂ i ∈ Finset.range n, W i
x : G
hx : x ∈ ⋂ n, V n * k
v : ℕ → G
hv : ∀ (i : ℕ), v i ∈ V i
y : ℕ → G
hy : ∀ (i : ℕ), y i ∈ k
hvy : ∀ (i : ℕ), (fun x1 x2 ↦ x1 * x2) (v i) (y i) = x
⊢ x ∈ k
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : HasContDiffBump E
inst✝ : MeasurableSpace E
μ : Measure E
f : ContDiffBump 0
x : E
⊢ f.normed μ (-x) = f.normed μ x

Tactic:
by
  simp_rw [f.normed_def, f.neg]

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
K_pos : 0 < K
⊢ f.rOut ^ finrank ℝ E * μ.real (closedBall 0 1) ≤ (f.rIn * K) ^ finrank ℝ E * μ.real (closedBall 0 1)

Tactic:
mul_comm _ K

Goal state after:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : HasContDiffBump E
inst✝⁴ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : μ.IsAddHaarMeasure
K : ℝ
h : f.rOut ≤ K * f.rIn
K_pos : 0 < K
⊢ f.rOut ^ finrank ℝ E * μ.real (closedBall 0 1) ≤ (K * f.rIn) ^ finrank ℝ E * μ.real (closedBall 0 1)