mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →L[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hxU : x ∈ U
hfU : ContDiffOn 𝕜 (↑i) f (insert x s ∩ U)
⊢ iteratedFDerivWithin 𝕜 i (⇑g ∘ f) (s ∩ U) x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f (s ∩ U) x)

Tactic:
rw [insert_eq_of_mem hx] at hfU

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
x : E
n : ℕ∞ω
f : E → F
g : F →L[𝕜] G
hf : ContDiffWithinAt 𝕜 n f s x
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
i : ℕ
hi : ↑i ≤ n
U : Set E
hU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hxU : x ∈ U
hfU : ContDiffOn 𝕜 (↑i) f (s ∩ U)
⊢ iteratedFDerivWithin 𝕜 i (⇑g ∘ f) (s ∩ U) x = g.compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f (s ∩ U) x)
Module: Mathlib.Analysis.Calculus.ContDiff.Comp

Goal state before:
pp: case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
s : Set E
f : E → F
x₀ : E
n : ℕ∞ω
hf : ContDiffWithinAt 𝕜 n f s x₀
hs : UniqueDiffOn 𝕜 s
hx₀s : x₀ ∈ s
m : ℕ∞ω
hmn : m ≤ n
⊢ ContDiffWithinAt 𝕜 m (iteratedFDerivWithin 𝕜 0 f s) s x₀

Tactic:
exact (hf.of_le hmn).continuousLinearMap_comp
      ((continuousMultilinearCurryFin0 𝕜 E F).symm : _ →L[𝕜] E [×0]→L[𝕜] F)

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
s : Set 𝕜
⊢ iteratedDerivWithin n f s = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDerivWithin 𝕜 n f s

Tactic:
by
  ext x; rfl

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Defs

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
f : 𝕜 → F
⊢ iteratedDeriv n f = ⇑(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f

Tactic:
by
  ext x; rfl

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

Goal state before:
pp: e : ℂ → ℂ
z : ℝ
n : WithTop ℕ∞
h : ContDiffAt ℂ n e ↑z
A : ContDiffAt ℝ n ofReal z
⊢ ContDiffAt ℝ n (fun x ↦ (e ↑x).re) z

Tactic:
have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ

Goal state after:
e : ℂ → ℂ
z : ℝ
n : WithTop ℕ∞
h : ContDiffAt ℂ n e ↑z
A : ContDiffAt ℝ n ofReal z
B : ContDiffAt ℝ n e ↑z
⊢ ContDiffAt ℝ n (fun x ↦ (e ↑x).re) z
Module: Mathlib.Analysis.Normed.Algebra.Exponential

Goal state before:
pp: 𝔸 : Type u_2
inst✝³ : Ring 𝔸
inst✝² : TopologicalSpace 𝔸
inst✝¹ : IsTopologicalRing 𝔸
inst✝ : T2Space 𝔸
x : 𝔸ᵐᵒᵖ
⊢ exp (MulOpposite.unop x) = MulOpposite.unop (exp x)

Tactic:
by
  induction x; simp

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

Goal state before:
pp: case pos.partSize.succ
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n✝ : ℕ
c✝ : OrderedFinpartition n✝
n : ℕ
c : OrderedFinpartition (n + 1)
h : range (c.emb 0) = {0}
A : c.length - 1 + 1 = c.length
i : Fin (c.length - 1)
a✝ :
  { length := c.length - 1 + 1, partSize := cons 1 fun i ↦ c.partSize (Fin.cast ⋯ i.succ), partSize_pos := ⋯,
          emb := fun i ↦
            cases (motive := fun x ↦ Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) x) → Fin (n + 1)) (fun x ↦ 0)
              (fun m ↦ succ ∘ fun j ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯) i,
          emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.partSize
      i.castSucc =
    c.partSize ⟨↑i.castSucc, ⋯⟩
⊢ { length := c.length - 1 + 1, partSize := cons 1 fun i ↦ c.partSize (Fin.cast ⋯ i.succ), partSize_pos := ⋯,
          emb := fun i ↦
            cases (motive := fun x ↦ Fin (cons 1 (fun i ↦ c.partSize (Fin.cast ⋯ i.succ)) x) → Fin (n + 1)) (fun x ↦ 0)
              (fun m ↦ succ ∘ fun j ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯) i,
          emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }.partSize
      i.succ =
    c.partSize ⟨↑i.succ, ⋯⟩

Tactic:
simp only [cons_succ, val_succ]; rfl

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p✝ : E → FormalMultilinearSeries 𝕜 E F
n : ℕ
c : OrderedFinpartition n
f : F [×c.length]→L[𝕜] G
p : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F
⊢ Continuous (MultilinearMap.mk' (fun v ↦ f (c.applyOrderedFinpartition p v)) ⋯ ⋯).toFun

Tactic:
by
    apply f.cont.comp
    change Continuous (fun v m ↦ p m (v ∘ c.emb m))
    fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
n : ℕ
x : 𝕜
s : Set 𝕜
hx : x ∈ s
h : UniqueDiffOn 𝕜 s
f : 𝕜 → F
R : Type u_3
inst✝² : DistribSMul R F
inst✝¹ : SMulCommClass 𝕜 R F
inst✝ : ContinuousConstSMul R F
c : R
hf : ContDiffWithinAt 𝕜 (↑n) f s x
⊢ iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x

Tactic:
by
  simp [iteratedDerivWithin, iteratedFDerivWithin_const_smul_apply hf h hx]

Goal state after:
Module: Mathlib.Topology.MetricSpace.CauSeqFilter

Goal state before:
pp: β : Type v
inst✝¹ : NormedRing β
hn : IsAbsoluteValue norm
f : CauSeq β norm
inst✝ : IsComplete β norm
s : Set β
os : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
lfs : f.lim ∈ s
⊢ ∃ a, ∀ b ≥ a, ↑f b ∈ s

Tactic:
rcases Metric.isOpen_iff.1 os _ lfs with ⟨ε, ⟨hε, hεs⟩⟩

Goal state after:
β : Type v
inst✝¹ : NormedRing β
hn : IsAbsoluteValue norm
f : CauSeq β norm
inst✝ : IsComplete β norm
s : Set β
os : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
lfs : f.lim ∈ s
ε : ℝ
hε : ε > 0
hεs : Metric.ball f.lim ε ⊆ s
⊢ ∃ a, ∀ b ≥ a, ↑f b ∈ s
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: case pos.refine_1
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
c : ℝ≥0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
ε0 : ε ≠ 0
f : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)
h : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞
hc : c = 0
⊢ ∀ (x : α), (piecewise s hs (const α c) (const α 0)) x ≤ (fun x ↦ 0) x

Tactic:
classical
        simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
          SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
          SimpleFunc.coe_piecewise, le_zero_iff]

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C : ConvexCone R M
h : C.IsReproducing
x : M
⊢ x ∈ Submodule.span R ↑C

Tactic:
rw [IsReproducing, Set.eq_univ_iff_forall] at h

Goal state after:
R : Type u_7
M : Type u_8
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C : ConvexCone R M
h : ∀ (x : M), x ∈ ↑C - ↑C
x : M
⊢ x ∈ Submodule.span R ↑C
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: case inr
n : ℕ∞ω
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : 0 < x
⊢ ContDiffAt ℝ n log x

Tactic:
exact A x hx

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: f : ℝ → ℝ
x f' : ℝ
s : Set ℝ
hf : HasDerivWithinAt f f' s x
hx : f x ≠ 0
⊢ HasDerivWithinAt (fun y ↦ Real.log (f y)) (f' / f x) s x

Tactic:
by
  rw [div_eq_inv_mul]
  exact (hasDerivAt_log hx).comp_hasDerivWithinAt x hf

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
h : |x| < 1
i : ℕ
⊢ |x| ^ (i + 1) / (↑i + 1) ≤ |x| ^ (i + 1) / (0 + 1)

Tactic:
by
      gcongr
      positivity

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
⊢ HasFDerivAt f f' x

Tactic:
simp only [hasFDerivAt_iff_isLittleO_nhds_zero, isLittleO_iff]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝 0, ‖f (x + x_1) - f x - f' x_1‖ ≤ c * ‖x_1‖
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
fs : α →ₛ ℝ≥0
fs_le_f : ∀ (x : α), fs x ≤ f x
int_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(fs x) ∂μ + ε / 2
int_fs_lt_top : ∫⁻ (x : α), ↑(fs x) ∂μ ≠ ∞
g : α → ℝ≥0
g_le_fs : ∀ (x : α), g x ≤ fs x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(fs x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε / 2
⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε

Tactic:
refine ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, ?_⟩

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : MeasurableSpace α
inst✝¹ : BorelSpace α
μ : Measure α
inst✝ : μ.WeaklyRegular
f : α → ℝ≥0
int_f : ∫⁻ (x : α), ↑(f x) ∂μ ≠ ∞
ε : ℝ≥0∞
ε0 : ε ≠ 0
fs : α →ₛ ℝ≥0
fs_le_f : ∀ (x : α), fs x ≤ f x
int_fs : ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(fs x) ∂μ + ε / 2
int_fs_lt_top : ∫⁻ (x : α), ↑(fs x) ∂μ ≠ ∞
g : α → ℝ≥0
g_le_fs : ∀ (x : α), g x ≤ fs x
gcont : UpperSemicontinuous g
gint : ∫⁻ (x : α), ↑(fs x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ ≤ ∫⁻ (x : α), ↑(g x) ∂μ + ε
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} = a⁻¹ • sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s}

Tactic:
← Real.sInf_smul_of_nonneg (inv_nonneg.2 ha)

Goal state after:
case inr
E : Type u_2
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module ℝ E
α : Type u_3
inst✝⁸ : Field α
inst✝⁷ : LinearOrder α
inst✝⁶ : IsStrictOrderedRing α
inst✝⁵ : MulActionWithZero α ℝ
inst✝⁴ : IsStrictOrderedModule α ℝ
inst✝³ : MulActionWithZero α E
inst✝² : SMulCommClass α ℝ ℝ
inst✝¹ : IsScalarTower α ℝ ℝ
inst✝ : IsScalarTower α ℝ E
s : Set E
a : α
ha : 0 ≤ a
ha' : 0 < a
x : E
⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ a • s} = sInf (a⁻¹ • {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ s})
Module: Mathlib.Analysis.Convex.Gauge

Goal state before:
pp: E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : 0 ∈ s
ha : Absorbent ℝ s
h : gauge s x ≤ 1
r : ℝ
hr₀ : 0 ≤ r
hr₁ : r < 1
⊢ r • x ∈ {x | gauge s x < 1}

Tactic:
mem_ofPred_eq,

Goal state after:
E : Type u_2
inst✝³ : AddCommGroup E
inst✝² : Module ℝ E
s : Set E
x : E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
hc : Convex ℝ s
hs₀ : 0 ∈ s
ha : Absorbent ℝ s
h : gauge s x ≤ 1
r : ℝ
hr₀ : 0 ≤ r
hr₁ : r < 1
⊢ gauge s (r • x) < 1
Module: Mathlib.Analysis.Calculus.FDeriv.Measurable

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e ↦ L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
k : ℕ
k_gt : n e < k
m : ℕ := k - 1
h'k : ‖y‖ ≤ (1 / 2) ^ (m + 1)
hk : (1 / 2) ^ (m + 1 + 1) < ‖y‖
m_ge : n e ≤ m
km : k = m + 1
J1 : ‖f (x + y) - f x - (L e (n e) m) (x + y - x)‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m
⊢ ‖f (x + y) - f x - (L e (n e) m) y‖ ≤ (1 / 2) ^ e * (1 / 2) ^ m

Tactic:
by
          simpa only [add_sub_cancel_left] using J1

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case refine_2
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
μ : Measure ℝ
inst✝ : PseudoMetrizableSpace ε
a : ℕ → ℝ
m n : ℕ
hmn : m ≤ n
⊢ ∀ (n : ℕ),
    m ≤ n →
      ((∀ k ∈ Ico m n, IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a n)) →
        (∀ k ∈ Ico m (n + 1), IntervalIntegrable f μ (a k) (a (k + 1))) → IntervalIntegrable f μ (a m) (a (n + 1))

Tactic:
· intro p hp IH h
    exact (IH fun k hk => h k (Ico_subset_Ico_right p.le_succ hk)).trans (h p (by simp [hp]))

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a✝ b✝ : ℝ
inst✝ : PseudoMetrizableSpace ε
c a b : ℝ
hf : IntegrableOn f [[a, b]] (Measure.map (fun x ↦ x + c) volume)
h : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞
hab : a ≤ b
A : MeasurableEmbedding fun x ↦ x + c
⊢ IntegrableOn (fun x ↦ f (x + c)) [[a - c, b - c]] volume

Tactic:
convert! (MeasurableEmbedding.integrableOn_map_iff A).mp hf using 1

Goal state after:
case e'_7
ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a✝ b✝ : ℝ
inst✝ : PseudoMetrizableSpace ε
c a b : ℝ
hf : IntegrableOn f [[a, b]] (Measure.map (fun x ↦ x + c) volume)
h : ‖f (min a b)‖ₑ ≠ ∞
h' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞
hab : a ≤ b
A : MeasurableEmbedding fun x ↦ x + c
⊢ [[a - c, b - c]] = (fun x ↦ x + c) ⁻¹' [[a, b]]
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: ε : Type u_3
inst✝² : TopologicalSpace ε
inst✝¹ : ENormedAddMonoid ε
f : ℝ → ε
a b : ℝ
inst✝ : PseudoMetrizableSpace ε
c : ℝ
h : ‖f (min a b + c)‖ₑ ≠ ∞
hf : IntervalIntegrable (fun x ↦ f (x + c)) volume a b
⊢ IntervalIntegrable f volume (a + c) (b + c)

Tactic:
simpa using hf.comp_add_right (-c)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ ∫ (x : ℝ) in Iic b, f x ∂μ = ∫ (x : ℝ) in Iic a ∪ Ioc a b, f x ∂μ

pp: case ht
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ MeasurableSet (Ioc a b)

pp: case hfs
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ IntegrableOn f (Iic a) μ

pp: case hft
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b) μ

Tactic:
Iic_union_Ioc_eq_Iic hab

Goal state after:
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ ∫ (x : ℝ) in Iic b, f x ∂μ = ∫ (x : ℝ) in Iic b, f x ∂μ

case ht
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ MeasurableSet (Ioc a b)

case hfs
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ IntegrableOn f (Iic a) μ

case hft
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a✝ b✝ : ℝ
f : ℝ → E
μ : Measure ℝ
a b : ℝ
ha : IntegrableOn f (Iic a) μ
hb : IntegrableOn f (Iic b) μ
hab : a ≤ b
⊢ IntegrableOn f (Ioc a b) μ
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := ⋯
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν ((fun x ↦ g * x) ⁻¹' (π_preA ∩ (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕))

Tactic:
rw [preimage_inter]

Goal state after:
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : MeasurableSpace G
ν : Measure G
Γ : Subgroup G
μ : Measure (G ⧸ Γ)
inst✝⁷ : QuotientMeasureEqMeasurePreimage ν μ
inst✝⁶ : TopologicalSpace G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : BorelSpace G
inst✝³ : PolishSpace G
inst✝² : T2Space (G ⧸ Γ)
inst✝¹ : SecondCountableTopology (G ⧸ Γ)
inst✝ : ν.IsMulLeftInvariant
hasFun : HasFundamentalDomain (↥Γ.op) G ν
g : G
A : Set (G ⧸ Γ)
hA : MeasurableSet A
meas_π : Measurable QuotientGroup.mk
𝓕 : Set G
h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν
h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν
π_preA : Set G := QuotientGroup.mk ⁻¹' A
this : QuotientGroup.mk ⁻¹' (fun x ↦ g • x) ⁻¹' A = (fun x ↦ g * x) ⁻¹' π_preA
⊢ ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ 𝓕) = ν ((fun x ↦ g * x) ⁻¹' π_preA ∩ (fun x ↦ g * x) ⁻¹' (fun x ↦ g⁻¹ * x) ⁻¹' 𝓕)
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: K : Type u_1
inst✝⁵ : Field K
inst✝⁴ : LinearOrder K
inst✝³ : IsStrictOrderedRing K
inst✝² : FloorSemiring K
inst✝¹ : TopologicalSpace K
inst✝ : OrderTopology K
a c : K
d : ℕ
⊢ Tendsto (fun n ↦ a * c ^ n / ↑(n - d)!) atTop (𝓝 0)

Tactic:
by
  rw [tendsto_order]
  constructor
  all_goals
    intro ε hε
    filter_upwards [eventually_mul_pow_lt_factorial_sub (a * ε⁻¹) c d] with n h
    rw [mul_right_comm, ← div_eq_mul_inv] at h
  · rw [div_lt_iff_of_neg hε] at h
    rwa [lt_div_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]
  · rw [div_lt_iff₀ hε] at h
    rwa [div_lt_iff₀' (Nat.cast_pos.mpr (Nat.factorial_pos _))]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case ctop
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := PositiveCompacts.map QuotientGroup.mk ⋯ ⋯ K
this : μ.IsMulLeftInvariant
s : Set G
finiteCovol : ν s ≠ ∞
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
⊢ ν (QuotientGroup.mk ⁻¹' QuotientGroup.mk '' ↑K ∩ s) ≤ ν s

Tactic:
apply measure_mono

Goal state after:
case ctop
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := ⋯
this : μ.IsMulLeftInvariant
s : Set G
finiteCovol : ν s ≠ ∞
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
⊢ QuotientGroup.mk ⁻¹' QuotientGroup.mk '' ↑K ∩ s ⊆ s
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: case inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
n : ℕ
⊢ (2 * π)⁻¹ * (|0|⁻¹ ^ n * (|0| * (|0|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c 0 x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c 0 θ)‖) * |0|⁻¹ ^ n

pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n

Tactic:
· cases n <;> simp [-mul_inv_rev]

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R : ℝ
n : ℕ
hR : R ≠ 0
⊢ (2 * π)⁻¹ * (|R|⁻¹ ^ n * (|R| * (|R|⁻¹ * ∫ (x : ℝ) in 0..2 * π, ‖f (circleMap c R x)‖))) ≤
    ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c R θ)‖) * |R|⁻¹ ^ n