mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Composition.ParallelComp

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel γ δ
x : α × γ
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
a : α × γ
⊢ (κ a.1) Set.univ * (η a.2) Set.univ ≤ κ.bound * η.bound

Tactic:
gcongr
    · exact measure_le_bound κ a.1 Set.univ
    · exact measure_le_bound η a.2 Set.univ

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel (α × β) γ
a : β × α
g : γ → ℝ≥0∞
⊢ ∫⁻ (c : γ), g c ∂κ.swapLeft a = ∫⁻ (c : γ), g c ∂κ a.swap

Tactic:
rw [swapLeft_apply]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
inst✝¹ : MeasurableSingletonClass α
a : α
inst✝ : IsSFiniteKernel κ
s : Set (α × β)
hs : MeasurableSet s
⊢ (dirac a ⊗ₘ κ) s = (κ a) (Prod.mk a ⁻¹' s)

Tactic:
by
  rw [compProd_apply hs, lintegral_dirac]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
hμν : μ ⟂ₘ ν
κ η : Kernel α β
hμ : SFinite μ
hν : ¬SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
hμν : μ ⟂ₘ ν
κ η : Kernel α β
hμ : SFinite μ
hν : SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
· rw [compProd_of_not_sfinite _ _ hν]; simp

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
hμν : μ ⟂ₘ ν
κ η : Kernel α β
hμ : SFinite μ
hν : SFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.CompProd

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b ↦ ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
c : ℝ≥0∞
s : Set (β × γ)
hs : MeasurableSet s
⊢ c * ((κ ⊗ₖ η) a) s = ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), s.indicator (fun x ↦ c) (a_1, c_1) ∂η (a, a_1) ∂κ a

Tactic:
rw [compProd_apply hs, ← lintegral_const_mul c _]

Goal state after:
case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b ↦ ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
c : ℝ≥0∞
s : Set (β × γ)
hs : MeasurableSet s
⊢ ∫⁻ (a_1 : β), c * (η (a, a_1)) (Prod.mk a_1 ⁻¹' s) ∂κ a =
    ∫⁻ (a_1 : β), ∫⁻ (c_1 : γ), s.indicator (fun x ↦ c) (a_1, c_1) ∂η (a, a_1) ∂κ a

α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b ↦ ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
c : ℝ≥0∞
s : Set (β × γ)
hs : MeasurableSet s
⊢ Measurable fun b ↦ (η (a, b)) (Prod.mk b ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.RadonNikodym

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel η
h :
  (μ.singularPart ν ⊗ₘ κ + ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) =ᵐ[ν ⊗ₘ η]
    (μ.singularPart ν ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) + (ν.withDensity (μ.rnDeriv ν) ⊗ₘ κ).rnDeriv (ν ⊗ₘ η)
⊢ μ.singularPart ν ⊗ₘ κ ⟂ₘ ν ⊗ₘ η

Tactic:
exact Measure.MutuallySingular.compProd_of_left (μ.mutuallySingular_singularPart _) _ _

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝¹ : NormedAddCommGroup E
a : α
κ : Kernel α β
η : Kernel β γ
inst✝ : NormedSpace ℝ E
g : ↥(Lp E 1 ((η ∘ₖ κ) a))
this : ∀ (i : ↥(Lp E 1 ((η ∘ₖ κ) a))), Measurable fun z ↦ ‖↑↑i z - ↑↑g z‖ₑ
⊢ Tendsto (fun i ↦ ENNReal.ofReal ‖i - g‖) (𝓝 g) (𝓝 (ENNReal.ofReal 0))

Tactic:
exact (continuous_ofReal.tendsto 0).comp (tendsto_iff_norm_sub_tendsto_zero.1 tendsto_id)

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: ⊢ ¬DifferentiableWithinAt ℝ klFun (Ioi 0) 0

Tactic:
refine not_differentiableWithinAt_of_deriv_tendsto_atBot_Ioi _ ?_
  rw [deriv_klFun]
  exact tendsto_log_nhdsGT_zero

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν ↔ Integrable (llr μ ν) μ

Tactic:
suffices Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal
      + (1 - (μ.rnDeriv ν x).toReal)) ν ↔ Integrable (llr μ ν) μ by
    convert! this using 3 with x
    rw [klFun, add_sub_assoc]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal + (1 - (μ.rnDeriv ν x).toReal)) ν ↔
    Integrable (llr μ ν) μ
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
h_int : Integrable (llr μ ν) μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
⊢ Integrable (fun x ↦ -f x + log (∫ (z : α), rexp (f z) ∂ν) + llr μ ν x) μ

Tactic:
exact Integrable.add (hfμ.neg.add (integrable_const _)) h_int

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
c : ℝ≥0
hc : ¬c = 0
h_llr_left : llr (c⁻¹ • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c⁻¹
h_llr_right : llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c
⊢ (klDiv μ (c • ν)).toReal = ↑c * (klDiv (c⁻¹ • μ) ν).toReal

Tactic:
toReal_klDiv,

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
c : ℝ≥0
hc : ¬c = 0
h_llr_left : llr (c⁻¹ • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c⁻¹
h_llr_right : llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c
⊢ ∫ (a : α), llr μ (c • ν) a ∂μ + (c • ν).real univ - μ.real univ = ↑c * (klDiv (c⁻¹ • μ) ν).toReal

case neg.h
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
c : ℝ≥0
hc : ¬c = 0
h_llr_left : llr (c⁻¹ • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c⁻¹
h_llr_right : llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c
⊢ μ ≪ c • ν

case neg.h_int
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
c : ℝ≥0
hc : ¬c = 0
h_llr_left : llr (c⁻¹ • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c⁻¹
h_llr_right : llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c
⊢ Integrable (llr μ (c • ν)) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: case refine_2
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(f + g) =ᵐ[μ] ↑↑↑f + ↑↑↑g

Tactic:
← Lp.coeFn_add

Goal state after:
case refine_2
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ↑↑↑(f + g) =ᵐ[μ] ↑↑(↑f + ↑g)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp ℝ 2 μ)
hf : ↑↑f =ᵐ[μ.restrict s] 0
h_nnnorm_eq_zero : (fun x ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊) =ᵐ[μ.restrict s] 0
x : α
hx : ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊ = 0
⊢ ↑↑↑((condExpL2 ℝ ℝ hm) f) x = 0

Tactic:
ENNReal.coe_eq_zero,

Goal state after:
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
f : ↥(Lp ℝ 2 μ)
hf : ↑↑f =ᵐ[μ.restrict s] 0
h_nnnorm_eq_zero : (fun x ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊) =ᵐ[μ.restrict s] 0
x : α
hx : ‖↑↑↑((condExpL2 ℝ ℝ hm) f) x‖₊ = 0
⊢ ↑↑↑((condExpL2 ℝ ℝ hm) f) x = 0
Module: Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum

Goal state before:
pp: case pos
X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
ι : Type u_3
inst✝ : Countable ι
p : ℝ≥0∞
hp : 1 ≤ p
f : ι → X → E
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
h'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞
h'p : p < ∞
A : ∀ (s : Set X), MeasurableSet s → μ s ≠ ∞ → ∀ᵐ (x : X) ∂μ, x ∈ s → ∑' (n : ι), ‖f n x‖ₑ < ∞
s : ι → Set X
s_meas : ∀ (n : ι), MeasurableSet (s n)
hs : ∀ (n : ι), f n =ᵐ[μ.restrict (s n)ᶜ] 0
h's : ∀ (n : ι), SigmaFinite (μ.restrict (s n))
C : ∀ᵐ (x : X) ∂μ, ∀ (n : ι), x ∈ s n → ∑' (n : ι), ‖f n x‖ₑ < ∞
D : ∀ᵐ (x : X) ∂μ, ∀ (n : ι), x ∉ s n → f n x = 0
x : X
hx : ∀ (n : ι), x ∈ s n → ∑' (n : ι), ‖f n x‖ₑ < ∞
h'x : ∀ (n : ι), x ∉ s n → f n x = 0
h : ∃ n, x ∈ s n
⊢ ∑' (n : ι), ‖f n x‖ₑ < ∞

Tactic:
rcases h with ⟨n, hn⟩

Goal state after:
case pos
X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
ι : Type u_3
inst✝ : Countable ι
p : ℝ≥0∞
hp : 1 ≤ p
f : ι → X → E
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
h'f : ∑' (n : ι), eLpNorm (f n) p μ ≠ ∞
h'p : p < ∞
A : ∀ (s : Set X), MeasurableSet s → μ s ≠ ∞ → ∀ᵐ (x : X) ∂μ, x ∈ s → ∑' (n : ι), ‖f n x‖ₑ < ∞
s : ι → Set X
s_meas : ∀ (n : ι), MeasurableSet (s n)
hs : ∀ (n : ι), f n =ᵐ[μ.restrict (s n)ᶜ] 0
h's : ∀ (n : ι), SigmaFinite (μ.restrict (s n))
C : ∀ᵐ (x : X) ∂μ, ∀ (n : ι), x ∈ s n → ∑' (n : ι), ‖f n x‖ₑ < ∞
D : ∀ᵐ (x : X) ∂μ, ∀ (n : ι), x ∉ s n → f n x = 0
x : X
hx : ∀ (n : ι), x ∈ s n → ∑' (n : ι), ‖f n x‖ₑ < ∞
h'x : ∀ (n : ι), x ∉ s n → f n x = 0
n : ι
hn : x ∈ s n
⊢ ∑' (n : ι), ‖f n x‖ₑ < ∞
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
hf : MemLp f p μ
hg : MemLp g p μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (MemLp.toLp f hf)
hPg : P (MemLp.toLp g hg)
s_f : Set α := Function.support (AEStronglyMeasurable.mk f hfm)
hs_f : MeasurableSet s_f
hs_f_eq : s_f =ᵐ[μ] Function.support f
s_g : Set α := Function.support (AEStronglyMeasurable.mk g hgm)
hs_g : MeasurableSet s_g
hs_g_eq : s_g =ᵐ[μ] Function.support g
h_inter_empty : s_f ∩ s_g =ᵐ[μ] ∅
f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
⊢ P (MemLp.toLp f hf + MemLp.toLp g hg)

Tactic:
let g' := (s_g \ s_f).indicator (hgm.mk g)

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : ↥(Lp F p μ) → Prop
h_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p ⋯ ⋯ c)
h_add :
  ∀ ⦃f g : α → F⦄ (hf : MemLp f p μ) (hg : MemLp g p μ),
    StronglyMeasurable f →
      StronglyMeasurable g →
        Disjoint (Function.support f) (Function.support g) →
          P (MemLp.toLp f hf) → P (MemLp.toLp g hg) → P (MemLp.toLp f hf + MemLp.toLp g hg)
h_closed : IsClosed {f | P ↑f}
f✝ : ↥(Lp F p μ)
hf✝ : AEStronglyMeasurable (↑↑f✝) μ
f g : α → F
hf : MemLp f p μ
hg : MemLp g p μ
hfm : AEStronglyMeasurable f μ
hgm : AEStronglyMeasurable g μ
h_disj : Disjoint (Function.support f) (Function.support g)
hPf : P (MemLp.toLp f hf)
hPg : P (MemLp.toLp g hg)
s_f : Set α := Function.support (AEStronglyMeasurable.mk f hfm)
hs_f : MeasurableSet s_f
hs_f_eq : s_f =ᵐ[μ] Function.support f
s_g : Set α := Function.support (AEStronglyMeasurable.mk g hgm)
hs_g : MeasurableSet s_g
hs_g_eq : s_g =ᵐ[μ] Function.support g
h_inter_empty : s_f ∩ s_g =ᵐ[μ] ∅
f' : α → F := (s_f \ s_g).indicator (AEStronglyMeasurable.mk f hfm)
hff' : f =ᵐ[μ] f'
hf'_meas : StronglyMeasurable f'
hf'_Lp : MemLp f' p μ
g' : α → F := (s_g \ s_f).indicator (AEStronglyMeasurable.mk g hgm)
⊢ P (MemLp.toLp f hf + MemLp.toLp g hg)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
⊢ ∀ ⦃f : α → F⦄, MemLp f p μ → AEStronglyMeasurable f μ → P f

Tactic:
intro f hf hfm

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝² : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
inst✝¹ : Fact (1 ≤ p)
inst✝ : NormedSpace ℝ F
hm : m ≤ m0
hp_ne_top : p ≠ ∞
P : (α → F) → Prop
h_ind : ∀ (c : F) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add :
  ∀ ⦃f g : α → F⦄,
    Disjoint (Function.support f) (Function.support g) →
      MemLp f p μ → MemLp g p μ → StronglyMeasurable f → StronglyMeasurable g → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑↑f}
h_ae : ∀ ⦃f g : α → F⦄, f =ᵐ[μ] g → MemLp f p μ → P f → P g
f : α → F
hf : MemLp f p μ
hfm : AEStronglyMeasurable f μ
⊢ P f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E'
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
a : α
ha :
  ↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a =
    (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖↑↑↑((condExpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a‖₊ =
    ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
rw [ha]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case refine_1
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
c : F'
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ AEStronglyMeasurable (↑↑((condExpInd F' hm μ s) c)) μ

Tactic:
exact aestronglyMeasurable_condExpInd hs hμs.ne c

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : NormedSpace ℝ G
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
t : Set α
ht : MeasurableSet t
hμt : μ t ≠ ∞
a : α
ha : ↑↑(condExpIndSMul hm hs hμs x) a = (fun a ↦ ↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x) a
x✝ : a ∈ t
⊢ ↑‖↑↑(condExpIndSMul hm hs hμs x) a‖₊ = ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a • x‖₊

Tactic:
rw [ha]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : CompleteSpace F'
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
⊢ (condExpL1CLM F' hm μ) ↑f = ↑f

Tactic:
have hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g := by
    simp only [g, LinearIsometryEquiv.symm_apply_apply]

Goal state after:
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
inst✝ : CompleteSpace F'
f : ↥(lpMeas F' ℝ m 1 μ)
g : ↥(Lp F' 1 (μ.trim hm)) := (lpMeasToLpTrimLie F' ℝ 1 μ hm) f
hfg : f = (lpMeasToLpTrimLie F' ℝ 1 μ hm).symm g
⊢ (condExpL1CLM F' hm μ) ↑f = ↑f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case insert
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
ι : Type u_5
f : ι → α → E
m : MeasurableSpace α
i : ι
s : Finset ι
his : i ∉ s
heq : (∀ i ∈ s, Integrable (f i) μ) → μ[∑ i ∈ s, f i | m] =ᵐ[μ] ∑ i ∈ s, μ[f i | m]
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1) μ
⊢ μ[∑ i ∈ insert i s, f i | m] =ᵐ[μ] ∑ i ∈ insert i s, μ[f i | m]

Tactic:
Finset.sum_insert his,

Goal state after:
case insert
α : Type u_1
E : Type u_3
m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
ι : Type u_5
f : ι → α → E
m : MeasurableSpace α
i : ι
s : Finset ι
his : i ∉ s
heq : (∀ i ∈ s, Integrable (f i) μ) → μ[∑ i ∈ s, f i | m] =ᵐ[μ] ∑ i ∈ s, μ[f i | m]
hf : ∀ i_1 ∈ insert i s, Integrable (f i_1) μ
⊢ μ[f i + ∑ x ∈ s, f x | m] =ᵐ[μ] ∑ i ∈ insert i s, μ[f i | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos.refine_5
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : SigmaFinite ((μ.restrict s).trim hm)
hf_int : Integrable f μ
⊢ AEStronglyMeasurable 0 (μ.restrict s)

Tactic:
exact stronglyMeasurable_zero.aestronglyMeasurable

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[X + Y | mΩ] =ᵐ[P] P⁻[X | mΩ] + P⁻[Y | mΩ]

Tactic:
simp_rw [condLExp_of_not_le hm]; simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
⊢ φ ∘ μ[f | m] =ᵐ[μ] φY ∘ μ[fY | m]

Tactic:
filter_upwards [lem2] with a ha

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
lem2 : μ[f | m] =ᵐ[μ] ⇑Y.subtypeL ∘ μ[fY | m]
lem3 : φ ∘ f =ᵐ[μ] φY ∘ fY
a : α
ha : μ[f | m] a = (⇑Y.subtypeL ∘ μ[fY | m]) a
⊢ (φ ∘ μ[f | m]) a = (φY ∘ μ[fY | m]) a
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
f : ℝ → ℝ
hm : m ≤ m𝓧
hμν : μ ≪ ν
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun x ↦ f ((∂μ/∂ν) x).toReal) ν
h : Integrable (fun x ↦ f ((∂μ.trim hm/∂ν.trim hm) x).toReal) (ν.trim hm)
a : 𝓧
ha : ((∂μ.trim hm/∂ν.trim hm) a).toReal = ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] a
⊢ f ((∂μ.trim hm/∂ν.trim hm) a).toReal = f (ν[fun x ↦ ((∂μ/∂ν) x).toReal | m] a)

Tactic:
rw [ha]

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
a : 𝓧
ha : ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal = ν[fun a ↦ ((∂μ/∂ν) a).toReal | MeasurableSpace.comap g m𝓨] a
⊢ klFun ((∂Measure.map g μ/∂Measure.map g ν) (g a)).toReal =
    klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] a)

Tactic:
rw [ha]

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

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Ring k
inst✝¹ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝ : Fintype ι
q : ι' → P
i : ι'
⊢ ∑ j, b.toMatrix q i j = 1

Tactic:
by
  simp

Goal state after: