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

Goal state before:
pp: A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
e : A
x✝ :
  (0 ≤ e ∧ ‖e‖ ≤ 1) ∧ ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
h1 : 0 ≤ e
h2 : ‖e‖ ≤ 1
h3 : ∀ (x₁ : A), 0 ≤ x₁ → ‖x₁‖ ≤ 1 → ∀ (x₂ : A), 0 ≤ x₂ → ‖x₂‖ ≤ 1 → e ∈ openSegment ℝ x₁ x₂ → x₁ = e
this : ↑e ≤ 1
⊢ 0 ≤ 2 - ↑e

Tactic:
grw [sub_nonneg, this, one_le_two]

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

Goal state before:
pp: case right
ι : Type u_1
B : Type u_2
F : Type u_3
inst✝¹ : TopologicalSpace B
inst✝ : TopologicalSpace F
Z : FiberBundleCore ι B F
i j : ι
⊢ EqOn (↑((Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j))) (↑(Z.trivChange i j).toPartialEquiv)
    ((Z.localTrivAsPartialEquiv i).symm.trans (Z.localTrivAsPartialEquiv j)).source

Tactic:
· rintro ⟨x, v⟩ hx
    simp only [trivChange, localTrivAsPartialEquiv, PartialEquiv.symm,
      Prod.mk_inj, prodMk_mem_set_prod_eq, PartialEquiv.trans_source, mem_inter_iff,
      mem_preimage, proj, mem_univ, (· ∘ ·),
      PartialEquiv.coe_trans] at hx ⊢
    simp only [Z.coordChange_comp, hx, mem_inter_iff, and_self_iff, mem_baseSet_at]

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

Goal state before:
pp: case inl
B : Type u_1
F : Type u_2
E : B → Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace B
inst✝² : TopologicalSpace F
proj : Z → B
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace (TotalSpace F E)
e✝ : Trivialization F proj
x : Z
e'✝ : Trivialization F TotalSpace.proj
b : B
y : E b
e e' : Trivialization F proj
H : Disjoint e.baseSet e'.baseSet
p : Z
hp : p ∈ e.source
⊢ (↑(e.disjointUnion e'.toOpenPartialHomeomorph ⋯ ⋯) p).1 = proj p

Tactic:
change (e.source.piecewise e e' p).1 = proj p
      rw [piecewise_eq_of_mem, e.coe_fst] <;> exact hp

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
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
g : G
⊢ (hf.fiberEquivGroup e) e' = g ↔ ↑e' = g • ↑e

Tactic:
fiberEquivGroup,

Goal state after:
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
hf : IsQuotientCoveringMap f G
x : X
e e' : ↑(f ⁻¹' {x})
g : G
⊢ (Equiv.ofBijective (fun g ↦ ⟨g • ↑e, ⋯⟩) ⋯).symm e' = g ↔ ↑e' = g • ↑e
Module: Mathlib.Topology.Covering.Basic

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
f : E → X
s : Set X
Y : Type u_3
inst✝ : TopologicalSpace Y
g : X ≃ₜ Y
h : IsCoveringMapOn (⇑g ∘ f) (⇑g.symm ⁻¹' s)
⊢ IsCoveringMapOn f s

Tactic:
convert! h.homeomorph_comp g.symm <;> (ext; simp)

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
this : ContinuousConstSMul G E
⊢ f '' {e | MulAction.stabilizer G e = ⊥} = Set.univ

Tactic:
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⟩

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.Circle

Goal state before:
pp: x y : Circle
⊢ range ⇑(x.path y) ⊂ univ

Tactic:
by
  rw [ssubset_univ_iff_nonempty_compl]
  obtain rfl | hne := eq_or_ne x y
  · use -x, by simp [neg_ne_self]
  rw [compl_range_path hne]
  use y.path x ⟨2⁻¹, by simp only [mem_Icc, inv_nonneg, Nat.ofNat_nonneg, true_and]; linarith⟩
  refine mem_image_of_mem _ ⟨by simp [← unitInterval.coe_pos], unitInterval.coe_lt_one.mp ?_⟩
  linarith

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f
y : 𝕜
h : y • 1 = ↑(↑g).symm ∘SL ↑↑f
hy : y ≠ 0
hfg : ↑↑f = y • ↑↑g
hgf : ↑↑g = star y • ↑↑f
⊢ ∃ α, ∀ (x : H), f x = (α • g) x

Tactic:
have : (g : H →L[𝕜] K) = (starRingEnd 𝕜 y * y) • g := by
    simp [← smul_smul, ← hfg, ← star_def, ← hgf]

Goal state after:
case neg
𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
H : Type u_5
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
K : Type u_6
inst✝² : NormedAddCommGroup K
inst✝¹ : InnerProductSpace 𝕜 K
inst✝ : CompleteSpace K
f g : H ≃ₗᵢ[𝕜] K
x✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f
y : 𝕜
h : y • 1 = ↑(↑g).symm ∘SL ↑↑f
hy : y ≠ 0
hfg : ↑↑f = y • ↑↑g
hgf : ↑↑g = star y • ↑↑f
this : ↑↑g = ((starRingEnd 𝕜) y * y) • ↑↑g
⊢ ∃ α, ∀ (x : H), f x = (α • g) x
Module: Mathlib.Analysis.Normed.Affine.AddTorsorBases

Goal state before:
pp: case inr
ι : Type u_1
E : Type u_2
inst✝² : Finite ι
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
b : AffineBasis ι ℝ E
h✝ : Nontrivial ι
this✝ : FiniteDimensional ℝ E
this : (convexHull ℝ) (range ⇑b) = ⋂ i, ⇑(b.coord i) ⁻¹' Ici 0
x✝ : E
⊢ x✝ ∈ interior ((convexHull ℝ) (range ⇑b)) ↔ x✝ ∈ {x | ∀ (i : ι), 0 < (b.coord i) x}

Tactic:
simp only [this, interior_iInter_of_finite, ←
      IsOpenMap.preimage_interior_eq_interior_preimage (isOpenMap_barycentric_coord b _)
        (continuous_barycentric_coord b _),
      interior_Ici, mem_iInter, mem_ofPred_eq, mem_Ioi, mem_preimage]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
⊢ x ∈ (fun x ↦ f.rIn⁻¹ • (x - c)) ⁻¹' ball 0 (f.rOut / f.rIn) ↔ x ∈ ball c f.rOut

Tactic:
simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv,
    abs_of_pos f.rIn_pos, ← div_eq_inv_mul, div_lt_div_iff_of_pos_right f.rIn_pos]

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

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf : FiniteDimensional k ↥s.direction
hs : s = ⊥
⊢ finrank k ↥(vectorSpan k (insert p ∅)) = finrank k ↥⊥

Tactic:
convert! rfl <;> simp

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

Goal state before:
pp: case inr.inr.inl
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p : ℝ≥0∞
f : (i : α) → E i
hp : 0 < p.toReal
hfq : Memℓp f 0
hpq : 0 ≤ p
⊢ Memℓp f p

Tactic:
apply memℓp_gen
    have : ∀ i ∉ hfq.finite_dsupport.toFinset, ‖f i‖ ^ p.toReal = 0 := by
      intro i hi
      have : f i = 0 := by simpa using hi
      simp [this, Real.zero_rpow hp.ne']
    exact summable_of_ne_finset_zero this

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

Goal state before:
pp: case inr.inr.inr.inr.inr
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ Summable fun i ↦ ‖f i‖ ^ p.toReal

Tactic:
refine .of_norm_bounded_eventually hf' (@Set.Finite.subset _ { i | 1 ≤ ‖f i‖ } ?_ _ ?_)

Goal state after:
case inr.inr.inr.inr.inr.refine_1
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ {i | 1 ≤ ‖f i‖}.Finite

case inr.inr.inr.inr.inr.refine_2
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
f : (i : α) → E i
hfq : Memℓp f q
hpq : q ≤ p
hq : 0 < q.toReal
left✝ : 0 < p.toReal
hpq' : q.toReal ≤ p.toReal
hf' : Summable fun i ↦ ‖f i‖ ^ q.toReal
⊢ {x | (fun i ↦ ‖‖f i‖ ^ p.toReal‖ ≤ ‖f i‖ ^ q.toReal) x}ᶜ ⊆ {i | 1 ≤ ‖f i‖}
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)
f : ↥(lp E p)
⊢ ‖-f‖ = ‖f‖

Tactic:
rcases p.trichotomy with (rfl | rfl | hp)
  · simp only [norm_eq_card_dsupport, coeFn_neg, Pi.neg_apply, ne_eq, neg_eq_zero]
  · cases isEmpty_or_nonempty α
    · simp only [lp.eq_zero' f, neg_zero, norm_zero]
    apply (lp.isLUB_norm (-f)).unique
    simpa only [coeFn_neg, Pi.neg_apply, norm_neg] using lp.isLUB_norm f
  · suffices ‖-f‖ ^ p.toReal = ‖f‖ ^ p.toReal by
      exact Real.rpow_left_injOn hp.ne' (norm_nonneg' _) (norm_nonneg' _) this
    apply (lp.hasSum_norm hp (-f)).unique
    simpa only [coeFn_neg, Pi.neg_apply, _root_.norm_neg] using lp.hasSum_norm hp f

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

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : p.toReal.HolderConjugate q.toReal
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) (‖f‖ ^ p.toReal)
hg₂ : HasSum (fun i ↦ ‖↑g i‖ ^ q.toReal) (‖g‖ ^ q.toReal)
⊢ (Summable fun i ↦ ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖

Tactic:
obtain ⟨C, -, hC', hC⟩ :=
    Real.inner_le_Lp_mul_Lq_hasSum_of_nonneg hpq (norm_nonneg' _) (norm_nonneg' _) hf₁ hg₁ hf₂ hg₂

Goal state after:
α : Type u_3
E : α → Type u_4
inst✝ : (i : α) → NormedAddCommGroup (E i)
p q : ℝ≥0∞
hpq : p.toReal.HolderConjugate q.toReal
f : ↥(lp E p)
g : ↥(lp E q)
hf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖
hg₁ : ∀ (i : α), 0 ≤ ‖↑g i‖
hf₂ : HasSum (fun i ↦ ‖↑f i‖ ^ p.toReal) (‖f‖ ^ p.toReal)
hg₂ : HasSum (fun i ↦ ‖↑g i‖ ^ q.toReal) (‖g‖ ^ q.toReal)
C : ℝ
hC' : C ≤ ‖f‖ * ‖g‖
hC : HasSum (fun i ↦ ‖↑f i‖ * ‖↑g i‖) C
⊢ (Summable fun i ↦ ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖
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
inst✝ : NormedSpace 𝕜 F
f : α → E → F
v : ℕ → α → ℝ
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 (↑N) (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ), ↑k ≤ N → ∀ᶠ (i : α) in cofinite, ∀ (x : E), ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
⊢ ContDiff 𝕜 ↑N fun x ↦ ∑' (i : α), f i x

Tactic:
by
  refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
  let t : Set α :=
    { i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
  have ht : Set.Finite t :=
    haveI A :
      ∀ᶠ i in (Filter.cofinite : Filter α),
        ∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
      rw [eventually_all_finset]
      intro i hi
      apply h'f
      simp only [Finset.mem_range_succ_iff] at hi
      exact (WithTop.coe_le_coe.2 hi).trans hm
    eventually_cofinite.2 A
  let T : Finset α := ht.toFinset
  have : (fun x => ∑' i, f i x) = (fun x => ∑ i ∈ T, f i x) +
      fun x => ∑' i : { i // i ∉ T }, f i x := by
    ext1 x
    refine (Summable.sum_add_tsum_subtype_compl ?_ T).symm
    refine .of_norm_bounded_eventually (hv 0 zero_le) ?_
    filter_upwards [h'f 0 zero_le] with i hi
    simpa only [norm_iteratedFDeriv_zero] using hi x
  rw [this]
  apply (ContDiff.sum fun i _ => (hf i).of_le (mod_cast hm)).add
  have h'u : ∀ k : ℕ, (k : ℕ∞) ≤ m → Summable (v k ∘ ((↑) : { i // i ∉ T } → α)) := fun k hk =>
    (hv k (hk.trans hm)).subtype _
  refine contDiff_tsum (fun i => (hf i).of_le (mod_cast hm)) h'u ?_
  rintro k ⟨i, hi⟩ x hk
  simp only [t, T, Finite.mem_toFinset, mem_ofPred_eq, Finset.mem_range, not_forall, not_le,
    exists_prop, not_exists, not_and, not_lt] at hi
  exact hi k (Nat.lt_succ_iff.2 (WithTop.coe_le_coe.1 hk)) x

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
⊢ ContDiff 𝕜 n f ↔ ∀ (i : ι), ContDiff 𝕜 n fun x ↦ (f x).ofLp i

Tactic:
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_contDiff_iff, contDiff_pi]
  rfl

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

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
C : ℝ
F : ι → ↥(lp E ∞)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
a : α
this : Tendsto (fun k ↦ ‖↑(F k) a‖) l (𝓝 ‖f a‖)
⊢ ∀ (x : ι), ‖F x‖ ≤ C → ‖↑(F x) a‖ ≤ C

Tactic:
intro k hCFk

Goal state after:
α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
C : ℝ
F : ι → ↥(lp E ∞)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
a : α
this : Tendsto (fun k ↦ ‖↑(F k) a‖) l (𝓝 ‖f a‖)
k : ι
hCFk : ‖F k‖ ≤ C
⊢ ‖↑(F k) a‖ ≤ C
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)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ∞
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
s : Finset α
hp' : p ≠ 0
hp'' : 0 < p.toReal
G : ((a : α) → E a) → ℝ := fun f ↦ ∑ a ∈ s, ‖f a‖ ^ p.toReal
hG : Continuous[Pi.topologicalSpace, _] G
⊢ ∀ (x : ι), ‖F x‖ ≤ C → (G ∘ id fun i ↦ ↑(F i)) x ≤ C ^ p.toReal

Tactic:
intro k hCFk

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ∞
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
s : Finset α
hp' : p ≠ 0
hp'' : 0 < p.toReal
G : ((a : α) → E a) → ℝ := fun f ↦ ∑ a ∈ s, ‖f a‖ ^ p.toReal
hG : Continuous[Pi.topologicalSpace, _] G
k : ι
hCFk : ‖F k‖ ≤ C
⊢ (G ∘ id fun i ↦ ↑(F i)) k ≤ C ^ p.toReal
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: G : Type uG
x : G
inst✝¹ : MeasurableSpace G
μ : Measure G
inst✝ : AddGroup G
f g g' : G → ℝ
hfg : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ
hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ
hf : ∀ (x : G), 0 ≤ f x
hg : ∀ (x : G), g x ≤ g' x
⊢ (fun t ↦ f t * g (x - t)) ≤ fun t ↦ f t * g' (x - t)

Tactic:
intro t

Goal state after:
G : Type uG
x : G
inst✝¹ : MeasurableSpace G
μ : Measure G
inst✝ : AddGroup G
f g g' : G → ℝ
hfg : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ
hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ
hf : ∀ (x : G), 0 ≤ f x
hg : ∀ (x : G), g x ≤ g' x
t : G
⊢ (fun t ↦ f t * g (x - t)) t ≤ (fun t ↦ f t * g' (x - t)) 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✝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
hfg : ConvolutionExistsAt f g x₀ L μ
⊢ ∀ (t : G), dist ((L (f t)) (g (x₀ - t))) ((L (f t)) z₀) ≤ ‖L (f t)‖ * ε

Tactic:
intro t

Goal state after:
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
hfg : ConvolutionExistsAt f g x₀ L μ
t : G
⊢ dist ((L (f t)) (g (x₀ - t))) ((L (f t)) z₀) ≤ ‖L (f t)‖ * ε
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 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
hfg : ConvolutionExistsAt f g x₀ L μ
h2 : ∀ (t : G), ‖(L (f t)) (g (x₀ - t)) - (L (f t)) z₀‖ ≤ ‖L (f t)‖ * ε
⊢ ∀ (t : G), ‖L (f t)‖ ≤ ‖L‖ * ‖f t‖

Tactic:
intro t

Goal state after:
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
hfg : ConvolutionExistsAt f g x₀ L μ
h2 : ∀ (t : G), ‖(L (f t)) (g (x₀ - t)) - (L (f t)) z₀‖ ≤ ‖L (f t)‖ * ε
t : G
⊢ ‖L (f t)‖ ≤ ‖L‖ * ‖f t‖
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
k : Set α
⊢ k ⊆ s \ μ.everywherePosSubset s → IsCompact k → μ k = 0

Tactic:
hk

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegular
hs : MeasurableSet s
k : Set α
hk : k ⊆ s \ μ.everywherePosSubset s
⊢ IsCompact k → μ k = 0
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ∞
A : μ (s \ μ.everywherePosSubset s) ≠ ∞
k : Set α
⊢ k ⊆ s \ μ.everywherePosSubset s → IsCompact k → μ k = 0

Tactic:
hk

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : OpensMeasurableSpace α
inst✝ : μ.InnerRegularCompactLTTop
hs : MeasurableSet s
h's : μ s ≠ ∞
A : μ (s \ μ.everywherePosSubset s) ≠ ∞
k : Set α
hk : k ⊆ s \ μ.everywherePosSubset s
⊢ IsCompact k → μ k = 0
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ
hi : Integrable (uncurry fun x y ↦ (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)
t : G
ht : ConvolutionExistsAt g k (x₀ - t) L₄ μ
⊢ (fun s ↦ ∫ (u : G), (L₃ (f s)) ((L₄ (g u)) (k (x₀ - s - u))) ∂μ) t =
    (fun s ↦ (L₃ (f s)) (∫ (u : G), (L₄ (g u)) (k (x₀ - s - u)) ∂μ)) t

Tactic:
exact (L₃ (f t)).integral_comp_comm ht

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
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
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
A : ∀ (n : ℕ), μ ((x * z⁻¹) • k \ k) ≤ u n
B : μ ((x * z⁻¹) • k \ k) = 0
C : μ (k \ (z * x⁻¹) • k) = 0
H : ¬x ∈ k
this : k ∩ ((z * x⁻¹) • k)ᶜ ∈ 𝓝[k] z
⊢ False

Tactic:
have : 0 < μ (k \ ((z * x⁻¹) • k)) := h z zk _ this

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
z : G
zk : z ∈ k
hz : MapClusterPt z atTop y
A : ∀ (n : ℕ), μ ((x * z⁻¹) • k \ k) ≤ u n
B : μ ((x * z⁻¹) • k \ k) = 0
C : μ (k \ (z * x⁻¹) • k) = 0
H : ¬x ∈ k
this✝ : k ∩ ((z * x⁻¹) • k)ᶜ ∈ 𝓝[k] z
this : 0 < μ (k \ (z * x⁻¹) • k)
⊢ False