mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
  (fun x ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) s) = fun x ↦
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * ∫⁻ (a : α), (Ici s).indicator (fun x ↦ 1) (f a) ∂μ =
    (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
  (fun x ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) s) = fun x ↦
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * ∫⁻ (a : α), (Ici s).indicator (fun x ↦ 1) (f a) ∂μ =
    (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case neg.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
h0 : ¬w₁ i = 0
h0' : ¬w₂ i = 0
h : SignType.sign (w₁ i) = -SignType.sign (w₂ i)
h' : 0 < w₁ i
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WOppSide ((Finset.affineCombination R Finset.univ s.points) w₁)
    ((Finset.affineCombination R Finset.univ s.points) w₂)

Tactic:
rw [sign_pos h', eq_comm, neg_eq_iff_eq_neg, sign_eq_neg_one_iff] at h
        exact (s.sOppSide_affineSpan_faceOpposite_of_pos_of_neg hw₁ hw₂ h' h).wOppSide

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
  (fun x ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) s) = fun x ↦
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun _x ↦ 1 * μ {a | s ≤ f a}) s =
    (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a}) s * ENNReal.ofReal (g s)

Tactic:
[

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
s : ℝ
aux₁ :
  (fun x ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) s) = fun x ↦
    ENNReal.ofReal (g s) * (Ioi 0).indicator (fun x ↦ 1) s * (Ici s).indicator (fun x ↦ 1) (f x)
⊢ ENNReal.ofReal (g s) * (Ioi 0).indicator (fun _x ↦ 1 * μ {a | s ≤ f a}) s =
    (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a}) s * ENNReal.ofReal (g s)
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: r : ℝ
n : ℕ
hnr : ↑n < r
hr : 0 < r
h_int : ∀ x ∈ Ioc 0 1, ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n) ≤ ENNReal.ofReal (x ^ (-(r⁻¹ * ↑n)))
⊢ IntegrableOn (fun x ↦ x ^ (-(r⁻¹ * ↑n))) (Ioc 0 1) volume

Tactic:
← intervalIntegrable_iff_integrableOn_Ioc_of_le zero_le_one

Goal state after:
r : ℝ
n : ℕ
hnr : ↑n < r
hr : 0 < r
h_int : ∀ x ∈ Ioc 0 1, ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n) ≤ ENNReal.ofReal (x ^ (-(r⁻¹ * ↑n)))
⊢ IntervalIntegrable (fun x ↦ x ^ (-(r⁻¹ * ↑n))) volume 0 1
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : g =ᵐ[volume.restrict (Ioi 0)] 0
ω : α
⊢ ∫ (t : ℝ) in 0..f ω, g t = ∫ (t : ℝ) in 0..f ω, 0

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : g =ᵐ[volume.restrict (Ioi 0)] 0
ω : α
⊢ ∫ (t : ℝ) in 0..f ω, g t = ∫ (t : ℝ) in 0..f ω, 0
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-r)‖ₑ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-r)‖ₑ ∂μ < ∞

Tactic:
this

Goal state after:
case right
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
this : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ (-r)‖ₑ ∂μ = ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ
⊢ ∫⁻ (a : E), ENNReal.ofReal ((1 + ‖a‖) ^ (-r)) ∂μ < ∞
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
this✝ : ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0
Hg : g =ᵐ[volume.restrict (⋃ n, Ioc 0 ↑n)] 0
this : ⋃ n, Ioc 0 ↑n = Ioi 0
⊢ g =ᵐ[volume.restrict (Ioi 0)] 0

Tactic:
this

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
this✝ : ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0
Hg : g =ᵐ[volume.restrict (Ioi 0)] 0
this : ⋃ n, Ioc 0 ↑n = Ioi 0
⊢ g =ᵐ[volume.restrict (Ioi 0)] 0
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
H3 : ∫ (x : ℝ) in Ioc 0 (u n), g x ∂volume ≤ 0
g_nn_ae : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioc 0 (u n)), 0 ≤ g t
Ig : ∫ (t : ℝ) in Ioc 0 (u n), g t = 0
J : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioc 0 (u n)), g t = 0
⊢ False

Tactic:
have : u n ≤ M := le_csSup M_bdd J

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
H3 : ∫ (x : ℝ) in Ioc 0 (u n), g x ∂volume ≤ 0
g_nn_ae : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioc 0 (u n)), 0 ≤ g t
Ig : ∫ (t : ℝ) in Ioc 0 (u n), g t = 0
J : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioc 0 (u n)), g t = 0
this : u n ≤ M
⊢ False
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
⊢ HasTemperateGrowth f ↔ ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
[

Goal state after:
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
⊢ HasTemperateGrowth f ↔ ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F✝ : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F✝
inst✝ : NormedSpace ℝ F✝
f : E → F✝
hf_temperate : HasTemperateGrowth f
n k : ℕ
hk : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
F : E → (N : Fin (n + 1)) → E [×↑N]→L[ℝ] F✝ := fun x N ↦ iteratedFDeriv ℝ (↑N) f x
⊢ F =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
[

Goal state after:
E : Type u_5
F✝ : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F✝
inst✝ : NormedSpace ℝ F✝
f : E → F✝
hf_temperate : HasTemperateGrowth f
n k : ℕ
hk : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
F : E → (N : Fin (n + 1)) → E [×↑N]→L[ℝ] F✝ := fun x N ↦ iteratedFDeriv ℝ (↑N) f x
⊢ F =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
meas : MeasurableSet {a | M < f a}
s : α
hs : s ∈ {a | M < f a}ᶜ
⊢ ∫ (t : ℝ) in 0..f s, g t = ∫ (t : ℝ) in 0..f s, 0

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
meas : MeasurableSet {a | M < f a}
s : α
hs : s ∈ {a | M < f a}ᶜ
⊢ ∫ (t : ℝ) in 0..f s, g t = ∫ (t : ℝ) in 0..f s, 0
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
i✝ : ℕ
hi✝ : i✝ ≤ n
i : ℕ
hi : i ∈ Finset.range (k₂ + 1)
⊢ C₂ * ((C₁ * (1 + ‖x‖) ^ k₁) ^ i * ↑(k₂.choose i)) ≤ C₂ * ↑(k₂.choose i) * C₁ ^ i * (1 + ‖x‖) ^ (k₁ * k₂)

Tactic:
[

Goal state after:
D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
i✝ : ℕ
hi✝ : i✝ ≤ n
i : ℕ
hi : i ∈ Finset.range (k₂ + 1)
⊢ C₂ * ((C₁ * (1 + ‖x‖) ^ k₁) ^ i * ↑(k₂.choose i)) ≤ C₂ * ↑(k₂.choose i) * C₁ ^ i * (1 + ‖x‖) ^ (k₁ * k₂)
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
⊢ Continuous[_, t] ⇑f ↔ ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑f ∘ ofSupportedIn K_sub_Ω)

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
⊢ Continuous[_, t] ⇑f ↔ ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑f ∘ ofSupportedIn K_sub_Ω)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
eq₂ : ∀ᵐ (ω : α) ∂μ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
eq₂ : ∀ᵐ (ω : α) ∂μ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
this✝¹ : IsTopologicalAddGroup 𝓓^{n}(Ω, F)
this✝ : ContinuousSMul ℝ 𝓓^{n}(Ω, F)
this : LocallyConvexSpace ℝ 𝓓^{n}(Ω, F)
⊢ topologicalSpace Ω F n ≤ induced (⇑(↑ℝ f)) t ↔
    ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑(↑ℝ f) ∘ ofSupportedIn K_sub_Ω)

Tactic:
[

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
this✝¹ : IsTopologicalAddGroup 𝓓^{n}(Ω, F)
this✝ : ContinuousSMul ℝ 𝓓^{n}(Ω, F)
this : LocallyConvexSpace ℝ 𝓓^{n}(Ω, F)
⊢ topologicalSpace Ω F n ≤ induced (⇑(↑ℝ f)) t ↔
    ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑(↑ℝ f) ∘ ofSupportedIn K_sub_Ω)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key :
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, cst t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (cst t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}

Tactic:
[

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key :
  ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, cst t) ∂μ =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (cst t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
T : (K : Compacts E) → ↑K ⊆ ↑Ω → 𝓓^{n}_{K}(E, F) →L[𝕜] V
toFun_eq_T :
  ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω) (f : 𝓓^{n}_{K}(E, F)), toFun (ofSupportedIn K_sub_Ω f) = (T K K_sub_Ω) f
f g : 𝓓^{n}(Ω, F)
K : Compacts E := { carrier := tsupport ⇑f ∪ tsupport ⇑g, isCompact' := ⋯ }
⊢ toFun (f + g) = toFun f + toFun g

Tactic:
have K_sub_Ω : (K : Set E) ⊆ Ω := union_subset f.tsupport_subset g.tsupport_subset

Goal state after:
𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
T : (K : Compacts E) → ↑K ⊆ ↑Ω → 𝓓^{n}_{K}(E, F) →L[𝕜] V
toFun_eq_T :
  ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω) (f : 𝓓^{n}_{K}(E, F)), toFun (ofSupportedIn K_sub_Ω f) = (T K K_sub_Ω) f
f g : 𝓓^{n}(Ω, F)
K : Compacts E := { carrier := tsupport ⇑f ∪ tsupport ⇑g, isCompact' := ⋯ }
K_sub_Ω : ↑K ⊆ ↑Ω
⊢ toFun (f + g) = toFun f + toFun g
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
𝕜 : Type u_2
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
n k1 : ℕ
h1 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k1
k2 : ℕ
h2 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k2
estimate :
  ∀ (x : D),
    ‖iteratedFDeriv ℝ n (fun x ↦ (B (f x)) (g x)) x‖ ≤
      ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i f x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ (fun a ↦ ↑(n.choose i) * ‖iteratedFDeriv ℝ i f a‖ * ‖iteratedFDeriv ℝ (n - i) g a‖) =O[⊤] fun x ↦
    (1 + ‖x‖) ^ (k1 + k2)

Tactic:
[

Goal state after:
case h
𝕜 : Type u_2
D : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NormedAlgebra ℝ 𝕜
inst✝⁶ : NormedAddCommGroup D
inst✝⁵ : NormedSpace ℝ D
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
hf : HasTemperateGrowth f
hg : HasTemperateGrowth g
n k1 : ℕ
h1 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k1
k2 : ℕ
h2 : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k2
estimate :
  ∀ (x : D),
    ‖iteratedFDeriv ℝ n (fun x ↦ (B (f x)) (g x)) x‖ ≤
      ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv ℝ i f x‖ * ‖iteratedFDeriv ℝ (n - i) g x‖
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ (fun a ↦ ↑(n.choose i) * ‖iteratedFDeriv ℝ i f a‖ * ‖iteratedFDeriv ℝ (n - i) g a‖) =O[⊤] fun x ↦
    (1 + ‖x‖) ^ (k1 + k2)