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

Goal state before:
pp: 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✝² : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
f : β × γ → E
h1f : StronglyMeasurable f
this : ∀ (x : β), ∀ᵐ (y : γ) ∂η (a, x), 0 ≤ ‖f (x, y)‖
⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ‖f (b, c)‖ₑ ∂η (a, b) ∂κ a < ∞ ↔
    (∀ᵐ (x : β) ∂κ a, ∫⁻ (a : γ), ‖f (x, a)‖ₑ ∂η (a, x) < ∞) ∧
      ∫⁻ (a_1 : β), ENNReal.ofReal (∫⁻ (a : γ), ‖f (a_1, a)‖ₑ ∂η (a, a_1)).toReal ∂κ a < ∞

Tactic:
have : ∀ {p q r : Prop} (_ : r → p), (r ↔ p ∧ q) ↔ p → (r ↔ q) := fun {p q r} h1 => by
    rw [← and_congr_right_iff, and_iff_right_of_imp h1]

Goal state after:
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✝² : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
f : β × γ → E
h1f : StronglyMeasurable f
this✝ : ∀ (x : β), ∀ᵐ (y : γ) ∂η (a, x), 0 ≤ ‖f (x, y)‖
this : ∀ {p q r : Prop}, (r → p) → ((r ↔ p ∧ q) ↔ p → (r ↔ q))
⊢ ∫⁻ (b : β), ∫⁻ (c : γ), ‖f (b, c)‖ₑ ∂η (a, b) ∂κ a < ∞ ↔
    (∀ᵐ (x : β) ∂κ a, ∫⁻ (a : γ), ‖f (x, a)‖ₑ ∂η (a, x) < ∞) ∧
      ∫⁻ (a_1 : β), ENNReal.ofReal (∫⁻ (a : γ), ‖f (a_1, a)‖ₑ ∂η (a, a_1)).toReal ∂κ a < ∞
Module: Mathlib.Probability.Kernel.MeasurableIntegral

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
a : α
E : Type u_4
inst✝² : NormedAddCommGroup E
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
hf : StronglyMeasurable f
⊢ StronglyMeasurable fun x ↦ ∫ (y : γ), f (x, y) ∂η (a, x)

Tactic:
change
    StronglyMeasurable
      ((fun x => ∫ y, (fun u : (α × β) × γ => f (u.1.2, u.2)) (x, y) ∂η x) ∘ fun x => (a, x))

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
a : α
E : Type u_4
inst✝² : NormedAddCommGroup E
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
hf : StronglyMeasurable f
⊢ StronglyMeasurable ((fun x ↦ ∫ (y : γ), (fun u ↦ f (u.1.2, u.2)) (x, y) ∂η x) ∘ fun x ↦ (a, x))
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: case h₁
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✝² : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
f : β × γ → E
h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)
⊢ ∀ᵐ (x : β) ∂κ a,
    HasFiniteIntegral (fun y ↦ AEStronglyMeasurable.mk f h1f (x, y)) (η (a, x)) ↔
      HasFiniteIntegral (fun y ↦ f (x, y)) (η (a, x))

Tactic:
filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using
      hasFiniteIntegral_congr hx

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : ¬AEMeasurable f μ
⊢ ¬AEMeasurable (fun x ↦ rexp (f x)) μ

Tactic:
exact fun h ↦ hf (aemeasurable_of_aemeasurable_exp h)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
⊢ ∫⁻ (x : α) in s, g x ∂μ.tilted f = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

Tactic:
by
  by_cases hf : AEMeasurable f μ
  · rw [Measure.tilted, setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable₀]
    · simp only [Pi.mul_apply]
    · refine AEMeasurable.restrict ?_
      exact ((measurable_exp.comp_aemeasurable hf).div_const _).ennreal_ofReal
    · exact hs
    · filter_upwards
      simp only [ENNReal.ofReal_lt_top, implies_true]
  · have hf' : ¬ Integrable (fun x ↦ exp (f x)) μ := by
      exact fun h ↦ hf (aemeasurable_of_aemeasurable_exp h.1.aemeasurable)
    simp only [hf, not_false_eq_true, tilted_of_not_aemeasurable, Measure.restrict_zero,
      lintegral_zero_measure]
    rw [integral_undef hf']
    simp

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
⊢ (fun a ↦ f (μ.rnDeriv ν a).toReal) ≤ᵐ[ν] fun a ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)).toReal ∂η a

Tactic:
have h_integrable : Integrable (fun x ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) x).toReal) (ν ⊗ₘ η) :=
    Measure.integrable_toReal_rnDeriv

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int : Integrable (fun p ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) p).toReal) (ν ⊗ₘ η)
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable : Integrable (fun x ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) x).toReal) (ν ⊗ₘ η)
⊢ (fun a ↦ f (μ.rnDeriv ν a).toReal) ≤ᵐ[ν] fun a ↦ ∫ (b : 𝓨), f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)).toReal ∂η a
Module: Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝ : SigmaFinite μ
⊢ (fun x ↦ rexp (llr μ ν x)) =ᵐ[ν] fun x ↦ if μ.rnDeriv ν x = 0 then 1 else (μ.rnDeriv ν x).toReal

Tactic:
by
  filter_upwards [Measure.rnDeriv_lt_top μ ν] with x hx
  by_cases h_zero : μ.rnDeriv ν x = 0
  · simp only [llr, h_zero, ENNReal.toReal_zero, log_zero, exp_zero, ite_true]
  · rw [llr, exp_log, if_neg h_zero]
    exact ENNReal.toReal_pos h_zero hx.ne

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
𝓨 : Type u_2
m𝓨 : MeasurableSpace 𝓨
κ η : Kernel 𝓧 𝓨
f : ℝ → ℝ
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
hf : StronglyMeasurable f
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont_at : ContinuousWithinAt f (Ici 0) 0
h_int✝ :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
hκη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
hf_cont : ContinuousOn f (Ici 0)
h_lt_top✝ : ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_integrable :
  (∀ᵐ (x : 𝓧) ∂ν, Integrable (fun y ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal) (η x)) ∧
    Integrable (fun x ↦ ∫ (y : 𝓨), ‖((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (x, y)).toReal‖ ∂η x) ν
h_ae1 : ∀ᵐ (a : 𝓧) ∂ν, μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = μ.rnDeriv ν a
h_ae2 :
  ∀ᵐ (a : 𝓧) ∂ν, ∀ᵐ (b : 𝓨) ∂η a, μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
a : 𝓧
h_eq_one : μ.rnDeriv ν a * ∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a = μ.rnDeriv ν a
h_mul_eq : ∀ᵐ (b : 𝓨) ∂η a, μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) = (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b)
h_lt_top : ∀ᵐ (b : 𝓨) ∂η a, (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) < ∞
h_int' : Integrable (fun y ↦ ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) (η a)
h_int : Integrable (fun y ↦ f ((μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, y)).toReal) (η a)
⊢ f (∫⁻ (b : 𝓨), μ.rnDeriv ν a * (μ ⊗ₘ κ).rnDeriv (μ ⊗ₘ η) (a, b) ∂η a).toReal =
    f (∫⁻ (b : 𝓨), (μ ⊗ₘ κ).rnDeriv (ν ⊗ₘ η) (a, b) ∂η a).toReal

Tactic:
congr 2
    refine lintegral_congr_ae ?_
    filter_upwards [h_mul_eq] with b hb using hb

Goal state after:
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 : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
h0 : NeZero μ
x : α
hx : ((μ.tilted f).rnDeriv ν x).toReal = (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * (μ.rnDeriv ν x).toReal
hx_pos : 0 < μ.rnDeriv ν x
⊢ μ.rnDeriv ν x < ∞ → llr (μ.tilted f) ν x = f x - log (∫ (z : α), rexp (f z) ∂μ) + llr μ ν x

Tactic:
hx_lt_top

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) μ
hfν : AEMeasurable f ν
h0 : NeZero μ
x : α
hx : ((μ.tilted f).rnDeriv ν x).toReal = (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * (μ.rnDeriv ν x).toReal
hx_pos : 0 < μ.rnDeriv ν x
hx_lt_top : μ.rnDeriv ν x < ∞
⊢ llr (μ.tilted f) ν x = f x - log (∫ (z : α), rexp (f z) ∂μ) + llr μ ν x
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 : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h0 : NeZero ν
x : α
hx : (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
hx_pos : 0 < μ.rnDeriv ν x
⊢ μ.rnDeriv ν x < ∞ → llr μ (ν.tilted f) x = -f x + log (∫ (z : α), rexp (f z) ∂ν) + llr μ ν x

Tactic:
hx_lt_top

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hf : Integrable (fun x ↦ rexp (f x)) ν
h0 : NeZero ν
x : α
hx : (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal
hx_pos : 0 < μ.rnDeriv ν x
hx_lt_top : μ.rnDeriv ν x < ∞
⊢ llr μ (ν.tilted f) x = -f x + log (∫ (z : α), rexp (f z) ∂ν) + llr μ ν x
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique

Goal state before:
pp: α : Type u_1
E' : Type u_2
𝕜 : Type u_4
p : ℝ≥0∞
m m0 : MeasurableSpace α
μ : Measure α
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E'
inst✝² : InnerProductSpace 𝕜 E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
hm : m ≤ m0
f g : ↥(Lp E' p μ)
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑f) s μ
hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑g) s μ
hfg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ
hf_meas : AEStronglyMeasurable (↑↑f) μ
hg_meas : AEStronglyMeasurable (↑↑g) μ
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (a : α) in s, ↑↑f a ∂μ - ∫ (a : α) in s, ↑↑g a ∂μ = 0

Tactic:
exact sub_eq_zero.mpr (hfg s hs hμs)

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

Goal state before:
pp: α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
f : ↥(Lp F p μ)
⊢ f ∈ lpMeas F 𝕜 m p μ ↔ AEStronglyMeasurable (↑↑f) μ

Tactic:
rw [← SetLike.mem_coe, ← Submodule.mem_carrier, lpMeas, Set.mem_ofPred_eq]

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

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s t : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
ht : MeasurableSet t
hμt : μ t ≠ ∞
x : α
hx : ↑↑(indicatorConstLp 2 hs hμs 1) x = s.indicator (fun x ↦ 1) x
⊢ ↑‖s.indicator (fun x ↦ 1) x‖₊ = s.indicator (fun x ↦ 1) x

Tactic:
classical
    simp_rw [Set.indicator_apply]
    split_ifs <;> simp

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

Goal state before:
pp: case refine_1
α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp E 2 μ)
c : E
h_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) 2 μ
h_eq : ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) =ᵐ[μ] fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫
⊢ ∀ (s : Set α),
    MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp)) s μ

Tactic:
intro s _ hμs

Goal state after:
case refine_1
α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp E 2 μ)
c : E
h_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) 2 μ
h_eq : ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) =ᵐ[μ] fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫
s : Set α
a✝ : MeasurableSet s
hμs : μ s < ∞
⊢ IntegrableOn (↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp)) s μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
⊢ μ[f | m] =
    if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯ else 0

Tactic:
rw [condExp, dif_pos hm]
  grind

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

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ (if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯) = f

Tactic:
if_pos hf

Goal state after:
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hf : StronglyMeasurable f
hfi : Integrable f μ
⊢ f = f
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
t : Set α
⊢ IntegrableOn (AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h) t (μ.trim hm)

Tactic:
refine @Integrable.integrableOn _ _ m _ _ _ _ _ ?_

Goal state after:
case refine_1
α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
hs : MeasurableSet s
hμs : μ s ≠ ∞
inst✝ : SigmaFinite (μ.trim hm)
h : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ
t : Set α
⊢ Integrable (AEStronglyMeasurable.mk (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) h) (μ.trim hm)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos.refine_3
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
⊢ ∀ (s_1 : Set α),
    MeasurableSet s_1 →
      (μ.restrict s) s_1 < ∞ → ∫ (x : α) in s_1, μ[f | m] x ∂μ.restrict s = ∫ (x : α) in s_1, 0 x ∂μ.restrict s

pp: case pos.refine_4
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 μ[f | m] (μ.restrict s)

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:
· intro t ht _
    rw [Measure.restrict_restrict (hm _ ht), setIntegral_condExp hm hf_int (ht.inter hs), ←
      Measure.restrict_restrict (hm _ ht)]
    refine setIntegral_congr_ae (hm _ ht) ?_
    filter_upwards [hf] with x hx _ using hx

Goal state after:
case pos.refine_4
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 μ[f | m] (μ.restrict s)

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)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
this : Nontrivial ↥(Lp F' 1 μ)
⊢ CompleteSpace F'

Tactic:
exact completeSpace_of_completeSpace_Lp F' 1 μ

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∞
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] + P⁻[Y | mΩ] ≤ᵐ[P] P⁻[X + Y | mΩ]

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] + P⁻[Y | mΩ] ≤ᵐ[P] P⁻[X + Y | mΩ]

Tactic:
· simp_rw [condLExp_of_not_le hm]; filter_upwards; simp

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ P⁻[X | mΩ] + P⁻[Y | mΩ] ≤ᵐ[P] P⁻[X + Y | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

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

pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ c • P⁻[X | mΩ] ≤ᵐ[P] P⁻[c • X | mΩ]

Tactic:
· simp_rw [condLExp_of_not_le hm]; filter_upwards; simp

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
⊢ c • P⁻[X | mΩ] ≤ᵐ[P] P⁻[c • X | mΩ]
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : ¬mΩ ≤ mΩ₀
⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]

pp: case pos
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : mΩ ≤ mΩ₀
⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]

Tactic:
· simp_rw [condLExp_of_not_le hm]; filter_upwards; simp

Goal state after:
case pos
Ω : Type u_1
mΩ₀ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
mΩ : MeasurableSpace Ω
inst✝ : Countable ι
X : ι → Ω → ℝ≥0∞
hX : ∀ (i : ι), AEMeasurable (X i) P
hm : mΩ ≤ mΩ₀
⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ s φ
hφ_cont : UpperSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
a : α
h : ((-φ) ∘ μ[f | m]) a ≤ μ[(-φ) ∘ f | m] a
ha : μ[-φ ∘ f | m] a = (-μ[φ ∘ f | m]) a
⊢ μ[φ ∘ f | m] a ≤ (φ ∘ μ[f | m]) a

Tactic:
simp_all [Pi.neg_comp]

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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hφ_cont : UpperSemicontinuous φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
a : α
h : ((-φ) ∘ μ[f | m]) a ≤ μ[(-φ) ∘ f | m] a
ha : μ[-φ ∘ f | m] a = (-μ[φ ∘ f | m]) a
⊢ μ[φ ∘ f | m] a ≤ (φ ∘ μ[f | m]) a

Tactic:
simp_all [Pi.neg_comp]

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

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
inst✝¹ : FiniteDimensional ℝ E
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
hφ_cvx : ConcaveOn ℝ univ φ
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
a : α
h : ((-φ) ∘ μ[f | m]) a ≤ μ[(-φ) ∘ f | m] a
ha : μ[-φ ∘ f | m] a = (-μ[φ ∘ f | m]) a
⊢ μ[φ ∘ f | m] a ≤ (φ ∘ μ[f | m]) a

Tactic:
simp_all [Pi.neg_comp]

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

Goal state before:
pp: case mp
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial k
p : ι → P
⊢ IsUnit (b.toMatrix p) → AffineIndependent k p ∧ affineSpan k (range p) = ⊤

Tactic:
rintro ⟨⟨B, A, hA, hA'⟩, rfl : B = b.toMatrix p⟩
    exact ⟨b.affineIndependent_of_toMatrix_right_inv p hA,
      b.affineSpan_eq_top_of_toMatrix_left_inv p hA'⟩

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

Goal state before:
pp: case mpr
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁶ : AddCommGroup V
inst✝⁵ : AffineSpace V P
inst✝⁴ : Ring k
inst✝³ : Module k V
b : AffineBasis ι k P
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial k
p : ι → P
⊢ AffineIndependent k p ∧ affineSpan k (range p) = ⊤ → IsUnit (b.toMatrix p)

Tactic:
rintro ⟨h_tot, h_ind⟩
    let b' : AffineBasis ι k P := ⟨p, h_tot, h_ind⟩
    change IsUnit (b.toMatrix b')
    exact b.isUnit_toMatrix b'

Goal state after: