mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr (c⁻¹ • μ) ν) μ

Tactic:
integrable_congr (llr_smul_nnreal_left hμν c⁻¹ (by simpa))

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ llr μ ν x + log ↑c⁻¹) μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ ‖condExpIndL1Fin hm hs hμs x‖ ≤ μ.real s * ‖x‖

Tactic:
by
  rw [L1.norm_eq_integral_norm, ← ENNReal.toReal_ofReal (norm_nonneg x), measureReal_def,
    ← ENNReal.toReal_mul,
    ← ENNReal.ofReal_le_iff_le_toReal (ENNReal.mul_ne_top hμs ENNReal.ofReal_ne_top),
    ofReal_integral_norm_eq_lintegral_enorm]
  swap; · rw [← memLp_one_iff_integrable]; exact Lp.memLp _
  have h_eq :
    ∫⁻ a, ‖condExpIndL1Fin hm hs hμs x a‖ₑ ∂μ = ∫⁻ a, ‖condExpIndSMul hm hs hμs x a‖ₑ ∂μ := by
    refine lintegral_congr_ae ?_
    filter_upwards [condExpIndL1Fin_ae_eq_condExpIndSMul hm hs hμs x] with z hz
    rw [hz]
  rw [h_eq, ofReal_norm]
  exact lintegral_nnnorm_condExpIndSMul_le hm hs hμs x

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
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

pp: case pos
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
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

Tactic:
· simp [h_int]

Goal state after:
case pos
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
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : G
⊢ ↑↑((condExpInd G hm μ s) x) =ᵐ[μ] ↑↑(condExpIndL1Fin hm hs hμs x)

Tactic:
simp [condExpInd, condExpIndL1, hs, hμs]

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

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
⊢ MeasurableSet s → μ s < ∞ → IntegrableOn (↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp)) s μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_2
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 < ∞ →
        ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) (MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯)) x ∂μ =
          ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) x ∂μ

Tactic:
intro s

Goal state after:
case refine_2
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 < ∞ →
      ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) (MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯)) x ∂μ =
        ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : InnerProductSpace 𝕜 E'
inst✝⁶ : CompleteSpace E'
inst✝⁵ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : RCLike 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2 μ)
⊢ ∀ (s : Set α),
    MeasurableSet s →
      μ s < ∞ →
        ∫ (x : α) in s, ↑↑↑((condExpL2 E'' 𝕜' hm) (T.compLp f)) x ∂μ =
          ∫ (x : α) in s, ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) x ∂μ

Tactic:
intro s

Goal state after:
case refine_1
α : Type u_1
E' : Type u_3
𝕜 : Type u_7
inst✝⁹ : RCLike 𝕜
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : InnerProductSpace 𝕜 E'
inst✝⁶ : CompleteSpace E'
inst✝⁵ : NormedSpace ℝ E'
m m0 : MeasurableSpace α
μ : Measure α
E'' : Type u_8
𝕜' : Type u_9
inst✝⁴ : RCLike 𝕜'
inst✝³ : NormedAddCommGroup E''
inst✝² : InnerProductSpace 𝕜' E''
inst✝¹ : CompleteSpace E''
inst✝ : NormedSpace ℝ E''
hm : m ≤ m0
T : E' →L[ℝ] E''
f : ↥(Lp E' 2 μ)
s : Set α
⊢ MeasurableSet s →
    μ s < ∞ →
      ∫ (x : α) in s, ↑↑↑((condExpL2 E'' 𝕜' hm) (T.compLp f)) x ∂μ =
        ∫ (x : α) in s, ↑↑(T.compLp ↑((condExpL2 E' 𝕜 hm) f)) x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : Integrable f μ
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)

pp: case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : ¬Integrable f μ
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)

Tactic:
· rw [if_pos hfi]
    by_cases hfm : StronglyMeasurable[m] f
    · rw [if_pos hfm]
      exact (condExpL1_of_aestronglyMeasurable' hfm.aestronglyMeasurable hfi).symm
    · rw [if_neg hfm]
      exact aestronglyMeasurable_condExpL1.ae_eq_mk.symm

Goal state after:
case neg
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
f : α → E
hfi : ¬Integrable f μ
⊢ (if Integrable f μ then if StronglyMeasurable f then f else AEStronglyMeasurable.mk ↑↑(condExpL1 hm μ f) ⋯
    else 0) =ᵐ[μ]
    ↑↑(condExpL1 hm μ f)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
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
hμ : NeZero μ
f : α → E
hμ_finite : IsFiniteMeasure μ
h_meas : StronglyMeasurable μ[f | ⊥]
c : E
h_eq : μ[f | ⊥] = fun x ↦ c
h_integral : ∫ (x : α), c ∂μ = ∫ (x : α), f x ∂μ
⊢ (fun x ↦ c) = fun x ↦ (μ.real Set.univ)⁻¹ • ∫ (x : α), f x ∂μ

Tactic:
integral_const

Goal state after:
case pos
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
hμ : NeZero μ
f : α → E
hμ_finite : IsFiniteMeasure μ
h_meas : StronglyMeasurable μ[f | ⊥]
c : E
h_eq : μ[f | ⊥] = fun x ↦ c
h_integral : μ.real Set.univ • c = ∫ (x : α), f x ∂μ
⊢ (fun x ↦ c) = fun x ↦ (μ.real Set.univ)⁻¹ • ∫ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: case pos
α : Type u_1
E : Type u_3
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m₀ : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m₀
inst✝ : SigmaFinite (μ.trim hm₂)
hμm₁ : SigmaFinite (μ.trim ⋯)
hf : Integrable f μ
⊢ ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, μ[μ[f | m₂] | m₁] x ∂μ = ∫ (x : α) in s, μ[f | m₁] x ∂μ

Tactic:
intro s

Goal state after:
case pos
α : Type u_1
E : Type u_3
f : α → E
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
m₁ m₂ m₀ : MeasurableSpace α
μ : Measure α
hm₁₂ : m₁ ≤ m₂
hm₂ : m₂ ≤ m₀
inst✝ : SigmaFinite (μ.trim hm₂)
hμm₁ : SigmaFinite (μ.trim ⋯)
hf : Integrable f μ
s : Set α
⊢ MeasurableSet s → μ s < ∞ → ∫ (x : α) in s, μ[μ[f | m₂] | m₁] x ∂μ = ∫ (x : α) in s, μ[f | m₁] x ∂μ
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)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ↑↑(condExpL1 hm μ f) =ᵐ[μ] f

Tactic:
condExpL1_eq hfi

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)
f : α → F'
inst✝ : CompleteSpace F'
hfm : AEStronglyMeasurable f μ
hfi : Integrable f μ
⊢ ↑↑((condExpL1CLM F' hm μ) (Integrable.toL1 f hfi)) =ᵐ[μ] f
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
hY : AEMeasurable Y (P.trim hm)
hXY : ∀ (s : Set Ω), MeasurableSet s → ∫⁻ (ω : Ω) in s, Y ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P
⊢ ∀ (s : Set Ω),
    MeasurableSet s → (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, Y x ∂P.trim hm = ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x ∂P.trim hm

Tactic:
intro s

Goal state after:
case h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
hY : AEMeasurable Y (P.trim hm)
hXY : ∀ (s : Set Ω), MeasurableSet s → ∫⁻ (ω : Ω) in s, Y ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P
s : Set Ω
⊢ MeasurableSet s → (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, Y x ∂P.trim hm = ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x ∂P.trim hm
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: case pos.refine_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
⊢ ∀ (s : Set 𝓧), MeasurableSet s → μ s < ∞ → ∫ (x : 𝓧) in s, (μ⁻[f | m] x).toReal ∂μ = ∫ (x : 𝓧) in s, (f x).toReal ∂μ

Tactic:
intro s

Goal state after:
case pos.refine_3
𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
μ : Measure 𝓧
f : 𝓧 → ℝ≥0∞
hf_meas : AEMeasurable f μ
hf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞
hm : m ≤ m𝓧
hμ : SigmaFinite (μ.trim hm)
s : Set 𝓧
⊢ MeasurableSet s → μ s < ∞ → ∫ (x : 𝓧) in s, (μ⁻[f | m] x).toReal ∂μ = ∫ (x : 𝓧) in s, (f x).toReal ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X ≤ᵐ[P] Y
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω),
    MeasurableSet s →
      (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[Y | mΩ] x ∂P.trim hm

Tactic:
intro s

Goal state after:
case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hXY : X ≤ᵐ[P] Y
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s →
    (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[Y | mΩ] x ∂P.trim hm
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω),
    MeasurableSet s →
      (P.trim hm) s < ∞ →
        ∫⁻ (x : Ω) in s, (P⁻[X | mΩ] + P⁻[Y | mΩ]) x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[X + Y | mΩ] x ∂P.trim hm

Tactic:
intro s

Goal state after:
case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s →
    (P.trim hm) s < ∞ →
      ∫⁻ (x : Ω) in s, (P⁻[X | mΩ] + P⁻[Y | mΩ]) x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[X + Y | mΩ] x ∂P.trim hm
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω), MeasurableSet s → ∫⁻ (ω : Ω) in s, (P⁻[X | mΩ] + P⁻[Y | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (X + Y) ω ∂P

Tactic:
intro s

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hX : AEMeasurable X P
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s → ∫⁻ (ω : Ω) in s, (P⁻[X | mΩ] + P⁻[Y | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (X + Y) ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hX : AEMeasurable X P
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω), MeasurableSet s → ∫⁻ (ω : Ω) in s, (c • P⁻[X | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (c • X) ω ∂P

Tactic:
intro s

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
hX : AEMeasurable X P
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s → ∫⁻ (ω : Ω) in s, (c • P⁻[X | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (c • X) ω ∂P
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
x : ↥Y
⊢ x ∈ ⇑Y.subtypeL ⁻¹' s → Y.subtypeL x ∈ s

Tactic:
by
          #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
          (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this
          goal. It is not yet clear whether this is due to defeq abuse in Mathlib or a problem
          in the new canonicalizer; a minimization would help. The original proof was: `grind` -/
          simp

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

Goal state before:
pp: case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω),
    MeasurableSet s →
      (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, (c • P⁻[X | mΩ]) x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[c • X | mΩ] x ∂P.trim hm

Tactic:
intro s

Goal state after:
case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s →
    (P.trim hm) s < ∞ → ∫⁻ (x : Ω) in s, (c • P⁻[X | mΩ]) x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[c • X | mΩ] x ∂P.trim hm
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω), MeasurableSet s → ∫⁻ (ω : Ω) in s, (c • P⁻[X | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (c • X) ω ∂P

Tactic:
intro s

Goal state after:
case pos
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X : Ω → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s → ∫⁻ (ω : Ω) in s, (c • P⁻[X | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (c • X) ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
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
hσ : SigmaFinite (P.trim hm)
⊢ ∀ (s : Set Ω),
    MeasurableSet s → ∫⁻ (ω : Ω) in s, (∑' (i : ι), P⁻[X i | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (∑' (i : ι), X i) ω ∂P

Tactic:
intro s

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
hσ : SigmaFinite (P.trim hm)
s : Set Ω
⊢ MeasurableSet s → ∫⁻ (ω : Ω) in s, (∑' (i : ι), P⁻[X i | mΩ]) ω ∂P = ∫⁻ (ω : Ω) in s, (∑' (i : ι), X i) ω ∂P
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hcomm : ∀ (i j : ι) (r : R), i ≠ j → Commute f ((b.coord i).transvection (r • b j))
⊢ ∃ x, ∃ (h : ∀ (g : R), g * x = x * g), f = ⟨x, ⋯⟩ • 1

Tactic:
rcases subsingleton_or_nontrivial V with hV | hV

Goal state after:
case inl
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hcomm : ∀ (i j : ι) (r : R), i ≠ j → Commute f ((b.coord i).transvection (r • b j))
hV : Subsingleton V
⊢ ∃ x, ∃ (h : ∀ (g : R), g * x = x * g), f = ⟨x, ⋯⟩ • 1

case inr
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hcomm : ∀ (i j : ι) (r : R), i ≠ j → Commute f ((b.coord i).transvection (r • b j))
hV : Nontrivial V
⊢ ∃ x, ∃ (h : ∀ (g : R), g * x = x * g), f = ⟨x, ⋯⟩ • 1
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid V
inst✝² : Module R V
W : Type u_3
inst✝¹ : AddCommMonoid W
inst✝ : Module R W
f : Dual R V
v : V
e : V ≃ₗ[R] W
⊢ ↑e ∘ₗ transvection f v ∘ₗ ↑e.symm = (f ∘ₗ ↑e.symm).transvection (e v)

Tactic:
ext; simp [transvection.apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
feq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i
i j : ι
hij : i ≠ j
r : R
x : V := b.repr.symm ((Finsupp.single i 1).update j r)
s t : R
hst : s = 0 → ¬t = 0
h : ∀ (i : ι), s * (b.repr x) i + t * (b.repr (f x)) i = 0
hx : x = b i + r • b j
this : s + t * ((b.repr (f (b i))) i + r * (b.repr (f (b j))) i) = 0
⊢ s + t * (b.coord i) (f (b i)) = 0

Tactic:
rw [mul_add, ← add_assoc, ← mul_assoc, add_eq_zero_iff_eq_neg] at this

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
feq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i
i j : ι
hij : i ≠ j
r : R
x : V := b.repr.symm ((Finsupp.single i 1).update j r)
s t : R
hst : s = 0 → ¬t = 0
h : ∀ (i : ι), s * (b.repr x) i + t * (b.repr (f x)) i = 0
hx : x = b i + r • b j
this : s + t * (b.repr (f (b i))) i = -(t * r * (b.repr (f (b j))) i)
⊢ s + t * (b.coord i) (f (b i)) = 0
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
⊢ transvections R V ⊆ dilatransvections R V

Tactic:
by
  rintro e ⟨f, v, hfv, rfl⟩
  exact ⟨f, v, by simp⟩

Goal state after:
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
h' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.coord j) (f (b j))
feq : ∀ (i j : ι), f (b j) = (b.coord i) (f (b i)) • b j
i : ι := Classical.ofNonempty
r : R
j : ι
hij : j ≠ i
⊢ (b.coord i) (f (b i)) * r = r * (b.coord i) (f (b i))

Tactic:
h' i j (Ne.symm hij),

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
h' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.coord j) (f (b j))
feq : ∀ (i j : ι), f (b j) = (b.coord i) (f (b i)) • b j
i : ι := Classical.ofNonempty
r : R
j : ι
hij : j ≠ i
⊢ r * (b.coord j) (f (b j)) = r * (b.coord i) (f (b i))
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : IsDomain R
inst✝³ : StrongRankCondition R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
hV1 : finrank R V ≠ 1
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
⊢ ∃ a, f = a • 1

Tactic:
rcases subsingleton_or_nontrivial V with hV | hV

Goal state after:
case inl
R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : IsDomain R
inst✝³ : StrongRankCondition R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
hV1 : finrank R V ≠ 1
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV : Subsingleton V
⊢ ∃ a, f = a • 1

case inr
R : Type u_1
V : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : IsDomain R
inst✝³ : StrongRankCondition R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
hV1 : finrank R V ≠ 1
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV : Nontrivial V
⊢ ∃ a, f = a • 1