mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Manifold.BumpFunction

Goal state before:
pp: E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
H : Type uH
inst✝³ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝² : TopologicalSpace M
inst✝¹ : ChartedSpace H M
c : M
f : SmoothBumpFunction I c
x : M
inst✝ : FiniteDimensional ℝ E
⊢ ↑f x ∈ Icc 0 1

Tactic:
by
  have : f x = 0 ∨ f x = _ := indicator_eq_zero_or_self _ _ _
  rcases this with h | h <;> rw [h]
  exacts [left_mem_Icc.2 zero_le_one, ⟨f.nonneg, f.le_one⟩]

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: case e'_21
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
⊢ p / q = p * q⁻¹

Tactic:
simp [div_eq_mul_inv]

Goal state after:
Module: Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions

Goal state before:
pp: case e'_24
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
e_21✝ : p / q = p * q⁻¹
x✝ : TangentSpace I z
⊢ ((1 / q z) • p' - (p z / q z ^ 2) • q') x✝ = (p z • -(q z ^ 2)⁻¹ • q' + q⁻¹ z • p') x✝

Tactic:
simp [div_eq_mul_inv]

Goal state after:
case e'_24
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
H : Type u_3
inst✝⁴ : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝³ : TopologicalSpace M
inst✝² : ChartedSpace H M
s : Set M
z : M
F' : Type u_21
inst✝¹ : NormedField F'
inst✝ : NormedAlgebra 𝕜 F'
p q : M → F'
p' q' : TangentSpace I z →L[𝕜] F'
hp : HasMFDerivAt[s] p z p'
hq : HasMFDerivAt[s] q z q'
hq_ne : q z ≠ 0
e_21✝ : p / q = p * q⁻¹
x✝ : TangentSpace I z
⊢ (q z)⁻¹ * p' x✝ - p z * (q z ^ 2)⁻¹ * q' x✝ = -(p z * ((q z ^ 2)⁻¹ * q' x✝)) + (q z)⁻¹ * p' x✝
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: ι : Type u
X : Type v
inst✝¹ : TopologicalSpace X
inst✝ : ParacompactSpace X
s : Set X
hs : IsClosed[inst✝¹] s
u : ι → Set X
uo : ∀ (i : ι), IsOpen[inst✝¹] (u i)
us : s ⊆ ⋃ i, u i
uc : ⋃ i, Option.elim' sᶜ u i = univ
v : Option ι → Set X
vo : ∀ (a : Option ι), IsOpen[inst✝¹] (v a)
vf : LocallyFinite v
vu : ∀ (a : Option ι), v a ⊆ Option.elim' sᶜ u a
vc : (v none)ᶜ ⊆ ⋃ i, v (some i)
⊢ s ⊆ ⋃ i, (v ∘ some) i

Tactic:
exact Subset.trans (subset_compl_comm.1 <| vu Option.none) vc

Goal state after:
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: X : Type v
inst✝ : TopologicalSpace X
ι : X → Sort u_1
p : (x : X) → ι x → Prop
s : (x : X) → ι x → Set X
hb : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h :
  ∀ (f : (x : X) → ι x),
    (∀ (x : X), p x (f x)) →
      ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)
α : Type v
S : α → Set X
ho : ∀ (a : α), IsOpen[inst✝] (S a)
hu : ⋃ a, S a = univ
a : X → α
f : (x : X) → ι x
hp : ∀ (x : X), p x (f x)
hsub : ∀ (x : X), s x (f x) ⊆ S (a x)
β : Type u
t : β → Set X
hto : ∀ (b : β), IsOpen[inst✝] (t b)
ht : ⋃ b, t b = univ
htf : LocallyFinite t
hts : ∀ (b : β), ∃ x, t b ⊆ s x (f x)
⊢ ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ S a

Tactic:
refine ⟨range t, Subtype.val, forall_subtype_range_iff.2 hto, ?_, htf.on_range,
      forall_subtype_range_iff.2 fun b ↦ ?_⟩

Goal state after:
case refine_1
X : Type v
inst✝ : TopologicalSpace X
ι : X → Sort u_1
p : (x : X) → ι x → Prop
s : (x : X) → ι x → Set X
hb : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h :
  ∀ (f : (x : X) → ι x),
    (∀ (x : X), p x (f x)) →
      ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)
α : Type v
S : α → Set X
ho : ∀ (a : α), IsOpen[inst✝] (S a)
hu : ⋃ a, S a = univ
a : X → α
f : (x : X) → ι x
hp : ∀ (x : X), p x (f x)
hsub : ∀ (x : X), s x (f x) ⊆ S (a x)
β : Type u
t : β → Set X
hto : ∀ (b : β), IsOpen[inst✝] (t b)
ht : ⋃ b, t b = univ
htf : LocallyFinite t
hts : ∀ (b : β), ∃ x, t b ⊆ s x (f x)
⊢ ⋃ b, ↑b = univ

case refine_2
X : Type v
inst✝ : TopologicalSpace X
ι : X → Sort u_1
p : (x : X) → ι x → Prop
s : (x : X) → ι x → Set X
hb : ∀ (x : X), (𝓝 x).HasBasis (p x) (s x)
h :
  ∀ (f : (x : X) → ι x),
    (∀ (x : X), p x (f x)) →
      ∃ β t, (∀ (b : β), IsOpen[inst✝] (t b)) ∧ ⋃ b, t b = univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)
α : Type v
S : α → Set X
ho : ∀ (a : α), IsOpen[inst✝] (S a)
hu : ⋃ a, S a = univ
a : X → α
f : (x : X) → ι x
hp : ∀ (x : X), p x (f x)
hsub : ∀ (x : X), s x (f x) ⊆ S (a x)
β : Type u
t : β → Set X
hto : ∀ (b : β), IsOpen[inst✝] (t b)
ht : ⋃ b, t b = univ
htf : LocallyFinite t
hts : ∀ (b : β), ∃ x, t b ⊆ s x (f x)
b : β
⊢ ∃ a, ↑⟨t b, ⋯⟩ ⊆ S a
Module: Mathlib.Topology.Compactness.Paracompact

Goal state before:
pp: case refine_1
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : CompactSpace X
inst✝ : ParacompactSpace Y
α : Type (max v w)
s : α → Set (X × Y)
ho : ∀ (a : α), IsOpen[instTopologicalSpaceProd] (s a)
hu : ⋃ a, s a = univ
a : X → Y → α
U : X → Y → Set X
V : X → Y → Set Y
hUo : ∀ (x : X) (y : Y), IsOpen[inst✝³] (U x y)
hVo : ∀ (x : X) (y : Y), IsOpen[inst✝²] (V x y)
hxU : ∀ (x : X) (y : Y), x ∈ U x y
hyV : ∀ (x : X) (y : Y), y ∈ V x y
hUV : ∀ (x : X) (y : Y), U x y ×ˢ V x y ⊆ s (a x y)
T : Y → Finset X
hT : ∀ (y : Y), ⋃ x ∈ T y, U x y = ⊤
W : Y → Set Y := fun y ↦ ⋂ x ∈ T y, V x y
hWo : ∀ (y : Y), IsOpen[inst✝²] (W y)
hW : ∀ (y : Y), y ∈ W y
E : Y → Set Y
hEo : ∀ (a : Y), IsOpen[inst✝²] (E a)
hE : ⋃ i, E i = univ
hEf : LocallyFinite E
hEA : ∀ (a : Y), E a ⊆ W a
x✝ : X × Y
x : X
y : Y
⊢ ∃ i, (x, y) ∈ U (↑i.snd) i.fst ×ˢ E i.fst

Tactic:
rcases iUnion_eq_univ_iff.1 hE y with ⟨b, hb⟩
      rcases iUnion₂_eq_univ_iff.1 (hT b) x with ⟨a, ha, hx⟩
      exact ⟨⟨b, a, ha⟩, hx, hb⟩

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
x : X
⊢ ∑ᶠ (i : ι), f.toPOUFun i x = 1 - ∏ᶠ (i : ι), (1 - (f i) x)

Tactic:
set s := (f.point_finite x).toFinset
  have hs : (s : Set ι) = { i | f i x ≠ 0 } := Finite.coe_toFinset _
  have A : (support fun i => toPOUFun f i x) ⊆ s := by
    rw [hs]
    exact fun i hi => f.support_toPOUFun_subset i hi
  have B : (mulSupport fun i => 1 - f i x) ⊆ s := by
    rw [hs, mulSupport_one_sub]
    exact fun i => id
  classical
  let : LinearOrder ι := linearOrderOfSTO WellOrderingRel
  rw [finsum_eq_sum_of_support_subset _ A, finprod_eq_prod_of_mulSupport_subset _ B,
    Finset.prod_one_sub_ordered, sub_sub_cancel]
  refine Finset.sum_congr rfl fun i _ => ?_
  convert! f.toPOUFun_eq_mul_prod _ _ _ fun j _ hj => _
  rwa [Finite.mem_toFinset]

Goal state after:
Module: Mathlib.Topology.PartitionOfUnity

Goal state before:
pp: ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι
x : X
U : Set X
hU : U ∈ 𝓝 x
hf : {i | ((fun i ↦ support ⇑(f i)) i ∩ U).Nonempty}.Finite
y : X
hyU : y ∈ U
⊢ f.toPOUFun i y = (⇑(f i) * ⇑(∏ j ∈ hf.toFinset with WellOrderingRel j i, (1 - f j))) y

Tactic:
simp only [ContinuousMap.coe_prod, Pi.mul_apply, Finset.prod_apply]

Goal state after:
ι : Type u
X : Type v
inst✝ : TopologicalSpace X
s : Set X
f : BumpCovering ι X s
i : ι
x : X
U : Set X
hU : U ∈ 𝓝 x
hf : {i | ((fun i ↦ support ⇑(f i)) i ∩ U).Nonempty}.Finite
y : X
hyU : y ∈ U
⊢ f.toPOUFun i y = (f i) y * ∏ c ∈ hf.toFinset with WellOrderingRel c i, (1 - f c) y
Module: Mathlib.Geometry.Manifold.PartitionOfUnity

Goal state before:
pp: case refine_3
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
H : Type uH
inst✝⁶ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H M
inst✝³ : FiniteDimensional ℝ E
inst✝² : IsManifold I ∞ M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
n : ℕ∞
s t : Set M
hs : IsOpen[inst✝⁵] s
ht : IsClosed[inst✝⁵] t
h : t ⊆ s
f : M → ℝ
f_supp : support f = s
f_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑n) f
f_pos : ∀ (x : M), 0 ≤ f x
g : M → ℝ
g_supp : support g = tᶜ
g_diff : ContMDiff I 𝓘(ℝ, ℝ) (↑n) g
g_pos : ∀ (x : M), 0 ≤ g x
A : ∀ (x : M), 0 < f x + g x
B : (support fun x ↦ f x + g x) = univ
⊢ (support fun x ↦ f x / (f x + g x)) = s

Tactic:
simp only [support_div, f_supp, B, inter_univ]

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
⊢ ∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∫ (x : E), g x • ∑ i ∈ s, (a i • fun x ↦ lineDeriv ℝ f x (v i)) x ∂μ

Tactic:
Finset.smul_sum

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
C : ℝ≥0
f : E → ℝ
μ : Measure E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
ι : Type u_3
s : Finset ι
a : ι → ℝ
v : ι → E
g : E → ℝ
g_smooth : ContDiff ℝ (↑⊤) g
g_comp : HasCompactSupport g
⊢ ∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ =
    ∫ (x : E), ∑ x_1 ∈ s, g x • (a x_1 • fun x ↦ lineDeriv ℝ f x (v x_1)) x ∂μ
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc a b)
g : ℝ → ℝ := fun x ↦ f (max a (min x b))
hg : Monotone g
hfg : EqOn (deriv f) (deriv g) (Ioo a b)
G : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)
G_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))
n : ℕ
⊢ ∫ (t : ℝ) in Ioc a b, G (↑n)⁻¹ t ≤ g b - g a

Tactic:
← intervalIntegral.integral_of_le hab

Goal state after:
f : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc a b)
g : ℝ → ℝ := fun x ↦ f (max a (min x b))
hg : Monotone g
hfg : EqOn (deriv f) (deriv g) (Ioo a b)
G : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)
G_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))
n : ℕ
⊢ ∫ (x : ℝ) in a..b, G (↑n)⁻¹ x ≤ g b - g a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc a b)
g : ℝ → ℝ := fun x ↦ f (max a (min x b))
hg : Monotone g
hfg : EqOn (deriv f) (deriv g) (Ioo a b)
G : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)
G_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))
n : ℕ
⊢ ∫ (x : ℝ) in Icc a b, G (↑n)⁻¹ x ≤ g b - g a

Tactic:
rw [integral_Icc_eq_integral_Ioc, ← intervalIntegral.integral_of_le hab]

Goal state after:
f : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hf : MonotoneOn f (Icc a b)
g : ℝ → ℝ := fun x ↦ f (max a (min x b))
hg : Monotone g
hfg : EqOn (deriv f) (deriv g) (Ioo a b)
G : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)
G_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))
n : ℕ
⊢ ∫ (x : ℝ) in a..b, G (↑n)⁻¹ x ≤ g b - g a
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable

Goal state before:
pp: case right
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
ebound : ∫ (t : ℝ) in Ioc a b, ‖deriv f t‖ ∂volume ≤ f b - f a
⊢ ∫ (x : ℝ) in a..b, deriv f x ≤ f b - f a

Tactic:
← intervalIntegral.integral_of_le hab

Goal state after:
case right
f : ℝ → ℝ
a b : ℝ
hf : MonotoneOn f (Icc a b)
hab : a ≤ b
G : ℕ → ℝ → ℝ
hGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))
hG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))
hG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.ofReal (f b - f a)
hG'₀ : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ≠ ⊤
integrable_f_deriv : Integrable (deriv f) (volume.restrict (Icc a b))
this : f a ≤ f b
f_deriv_nonneg : ∀ {x : ℝ}, x ∈ Ioo a b → 0 ≤ deriv f x
ebound : ∫ (x : ℝ) in a..b, ‖deriv f x‖ ≤ f b - f a
⊢ ∫ (x : ℝ) in a..b, deriv f x ≤ f b - f a
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: case h₁
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ dist (f (I i).1 • g (I i).1) (f (I i).1 • g (I i).2) ≤ C * dist (g (I i).1) (g (I i).2)

pp: case h₂
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ dist (f (I i).1 • g (I i).2) (f (I i).2 • g (I i).2) ≤ D * dist (f (I i).1) (f (I i).2)

Tactic:
· rw [dist_smul₀]
      gcongr
      exact hC _ (hnI.left i hi |>.left)

Goal state after:
case h₂
F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
hg :
  Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (g (E.2 i).1) (g (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))
    (𝓝 0)
C : ℝ
hC : ∀ x ∈ uIcc a b, ‖f x‖ ≤ C
D : ℝ
hD : ∀ x ∈ uIcc a b, ‖g x‖ ≤ D
n : ℕ
I : ℕ → ℝ × ℝ
hnI :
  (∀ i ∈ Finset.range n, (I i).1 ∈ uIcc a b ∧ (I i).2 ∈ uIcc a b) ∧
    (↑(Finset.range n)).PairwiseDisjoint fun i ↦ uIoc (I i).1 (I i).2
i : ℕ
hi : i ∈ Finset.range n
⊢ dist (f (I i).1 • g (I i).2) (f (I i).2 • g (I i).2) ≤ D * dist (f (I i).1) (f (I i).2)
Module: Mathlib.MeasureTheory.Function.AbsolutelyContinuous

Goal state before:
pp: F : Type u_2
inst✝³ : SeminormedAddCommGroup F
a b : ℝ
M : Type u_3
inst✝² : SeminormedRing M
inst✝¹ : Module M F
inst✝ : NormSMulClass M F
f : ℝ → M
g : ℝ → F
hf : AbsolutelyContinuousOnInterval f a b
hg : AbsolutelyContinuousOnInterval g a b
⊢ AbsolutelyContinuousOnInterval (f • g) a b

Tactic:
by
  obtain ⟨C, hC⟩ := hf.exists_bound
  obtain ⟨D, hD⟩ := hg.exists_bound
  unfold AbsolutelyContinuousOnInterval at hf hg
  apply squeeze_zero' ?_ ?_
    (by simpa using (hg.const_mul C).add (hf.const_mul D))
  · exact Filter.Eventually.of_forall <| fun _ ↦ Finset.sum_nonneg (fun i hi ↦ dist_nonneg)
  rw [eventually_inf_principal]
  filter_upwards with (n, I) hnI
  simp only [Finset.mul_sum, ← Finset.sum_add_distrib]
  gcongr with i hi
  trans dist (f (I i).1 • g (I i).1) (f (I i).1 • g (I i).2) +
    dist (f (I i).1 • g (I i).2) (f (I i).2 • g (I i).2)
  · exact dist_triangle _ _ _
  · simp only [disjWithin, mem_ofPred_eq] at hnI
    gcongr
    · rw [dist_smul₀]
      gcongr
      exact hC _ (hnI.left i hi |>.left)
    · rw [mul_comm]
      grw [dist_pair_smul]
      gcongr
      rw [dist_zero_right]
      exact hD _ (hnI.left i hi |>.right)

Goal state after:
Module: Mathlib.Analysis.Calculus.Rademacher

Goal state before:
pp: E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
C : ℝ≥0
s : Set E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : FiniteDimensional ℝ F
inst✝ : μ.IsAddHaarMeasure
f : E → F
hf : LipschitzOnWith C f s
A : F ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ F) → ℝ
⊢ ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ (⇑A ∘ f) s x

Tactic:
apply ae_differentiableWithinAt_of_mem_pi

Goal state after:
case hf
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
C : ℝ≥0
s : Set E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : FiniteDimensional ℝ F
inst✝ : μ.IsAddHaarMeasure
f : E → F
hf : LipschitzOnWith C f s
A : F ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ F) → ℝ
⊢ LipschitzOnWith ?C (⇑A ∘ f) s

case C
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
F : Type u_2
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
C : ℝ≥0
s : Set E
μ : Measure E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : FiniteDimensional ℝ F
inst✝ : μ.IsAddHaarMeasure
f : E → F
hf : LipschitzOnWith C f s
A : F ≃L[ℝ] ↑(Basis.ofVectorSpaceIndex ℝ F) → ℝ
⊢ ℝ≥0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun

Goal state before:
pp: X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
u_coe_sum : ∀ (s : Finset ↑u) (g : ℝ → ℝ → ℝ), ∑ b ∈ s, g (↑b).1 (↑b).2 = ∑ z ∈ u_coe s, g z.1 z.2
hu₃ : Tendsto T atTop (totalLengthFilter ⊓ 𝓟 (disjWithin d b))
hu₄ : Tendsto (fun s ↦ ∑ z ∈ u_coe s, dist (f z.1) (f z.2)) (SummationFilter.unconditional ↑u).filter (𝓝 y)
⊢ dist (f d) (f b) ≤ y

Tactic:
have sum_tendsto := hf.comp hu₃ |>.add hu₄

Goal state after:
X : Type u_1
inst✝ : PseudoMetricSpace X
f : ℝ → X
d b y : ℝ
hdb : d ≤ b
hf : AbsolutelyContinuousOnInterval f d b
u : Set (ℝ × ℝ)
u_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtype.val s
hu₁ : ∀ (s : Finset ↑u) ⦃z : ℝ × ℝ⦄, z ∈ u_coe s → d ≤ z.1 ∧ z.1 ≤ z.2 ∧ z.2 ≤ b
hu₂ : ∀ (s : Finset ↑u), (↑(u_coe s)).PairwiseDisjoint fun z ↦ Icc z.1 z.2
T : Finset ↑u → ℕ × (ℕ → ℝ × ℝ) := fun s ↦ ((u_coe s).card + 1, fun i ↦ (u_coe s).intervalGapsWithin ⋯ d b ↑i)
hT₁ : ∀ (s : Finset ↑u) (i : ℕ), d ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1
hT₂ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).1 ≤ ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2
hT₃ : ∀ (s : Finset ↑u) (i : ℕ), ((u_coe s).intervalGapsWithin ⋯ d b ↑i).2 ≤ b
hT₄ :
  ∀ (s : Finset ↑u),
    (Iio ((u_coe s).card + 1)).PairwiseDisjoint fun j ↦
      Ioc ((u_coe s).intervalGapsWithin ⋯ d b ↑j).1 ((u_coe s).intervalGapsWithin ⋯ d b ↑j).2
hT : MapsTo T univ (disjWithin d b)
u_coe_sum : ∀ (s : Finset ↑u) (g : ℝ → ℝ → ℝ), ∑ b ∈ s, g (↑b).1 (↑b).2 = ∑ z ∈ u_coe s, g z.1 z.2
hu₃ : Tendsto T atTop (totalLengthFilter ⊓ 𝓟 (disjWithin d b))
hu₄ : Tendsto (fun s ↦ ∑ z ∈ u_coe s, dist (f z.1) (f z.2)) (SummationFilter.unconditional ↑u).filter (𝓝 y)
sum_tendsto :
  Tendsto
    (fun x ↦
      ((fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) ∘ T) x + ∑ z ∈ u_coe x, dist (f z.1) (f z.2))
    atTop (𝓝 (0 + y))
⊢ dist (f d) (f b) ≤ y
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hne : x ≠ y
⊢ ‖(1 / 2) • x + (1 / 2) • y‖ ≠ 1

Tactic:
rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
    div_eq_one_iff_eq (two_ne_zero' ℝ)]

Goal state after:
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2
x y : E
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hne : x ≠ y
⊢ ¬‖x + y‖ = 2
Module: Mathlib.Analysis.Convex.StrictConvexSpace

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y
⊢ StrictConvexSpace ℝ E

Tactic:
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h₂ => ?_
  rw [mem_sphere_zero_iff_norm] at hx hy
  exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
f : α → β
inst✝ : DecidablePred fun x ↦ f x ≤ 0
s : Finset α
h0 : Even #({x ∈ s | f x ≤ 0})
⊢ (∏ x ∈ s with f x ≤ 0, -1) * 1 * ∏ x ∈ s, f x = ∏ x ∈ s, f x

Tactic:
mul_one,

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : CommRing β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
f : α → β
inst✝ : DecidablePred fun x ↦ f x ≤ 0
s : Finset α
h0 : Even #({x ∈ s | f x ≤ 0})
⊢ (∏ x ∈ s with f x ≤ 0, -1) * ∏ x ∈ s, f x = ∏ x ∈ s, f x
Module: Mathlib.Analysis.Convex.SpecificFunctions.Deriv

Goal state before:
pp: case succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ ∏ k ∈ Finset.range (2 * n + 2 * 1), (m - ↑k)

Tactic:
mul_one,

Goal state after:
case succ
m : ℤ
n : ℕ
ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)
⊢ 0 ≤ ∏ k ∈ Finset.range (2 * n + 2), (m - ↑k)
Module: Mathlib.Analysis.Complex.AbsMax

Goal state before:
pp: F : Type v
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℂ F
inst✝ : CompleteSpace F
f : ℂ → F
z w : ℂ
r : ℝ := dist w z
hd : DiffContOnCl ℂ f (ball z r)
hz : IsMaxOn (norm ∘ f) (closedBall z r) z
hw : w ∈ closedBall z r
hw_lt : ‖f w‖ < ‖f z‖
hr : 0 < r
this : ‖∮ (ζ : ℂ) in C(z, r), (ζ - z)⁻¹ • f ζ‖ < 2 * π * ‖f z‖
⊢ False

Tactic:
refine this.ne ?_
    have A : (∮ ζ in C(z, r), (ζ - z)⁻¹ • f ζ) = (2 * π * I : ℂ) • f z :=
      hd.circleIntegral_sub_inv_smul (mem_ball_self hr)
    simp [A, norm_smul, Real.pi_pos.le]

Goal state after: