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
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp E p μ)
hf : ↑↑f =ᵐ[μ] 0
⊢ 0 =ᵐ[μ] 0

Tactic:
]

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)) 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 < ∞
⊢ -f x + log (∫ (x : α), rexp (f x) ∂ν) + log (μ.rnDeriv ν x).toReal =
    -f x + log (∫ (z : α), rexp (f z) ∂ν) + log (μ.rnDeriv ν x).toReal

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
hx_lt_top : μ.rnDeriv ν x < ∞
⊢ ∫ (x : α), rexp (f x) ∂ν ≠ 0

pp: case hx
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 < ∞
⊢ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ≠ 0

pp: case hy
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 < ∞
⊢ (μ.rnDeriv ν x).toReal ≠ 0

Tactic:
]

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 < ∞
⊢ ∫ (x : α), rexp (f x) ∂ν ≠ 0

case hx
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 < ∞
⊢ rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν ≠ 0

case hy
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 < ∞
⊢ (μ.rnDeriv ν x).toReal ≠ 0
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: case inl
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
μ : Measure α
inst✝ : Nontrivial ↥(Lp E ∞ μ)
f : ↥(Lp E ∞ μ)
hf : f ≠ 0
hfne : MemLp.toLp (fun x ↦ 1) ⋯ = 0
this : (fun x ↦ 1) =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0

Tactic:
]

Goal state after:
case inl
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
μ : Measure α
inst✝ : Nontrivial ↥(Lp E ∞ μ)
f : ↥(Lp E ∞ μ)
hf : f ≠ 0
hfne : MemLp.toLp (fun x ↦ 1) ⋯ = 0
this : (fun x ↦ 1) =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Trim

Goal state before:
pp: α : Type u_1
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : α → ℝ≥0∞
hf : Measurable f
h_set_eq : {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} = {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}
⊢ sInf {a | ∀ᵐ (n : α) ∂μ, f n ≤ a} = sInf {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: case inr.inl
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
μ : Measure α
inst✝ : Nontrivial ↥(Lp E 0 μ)
f : ↥(Lp E 0 μ)
hf : f ≠ 0
h'p : 0 < ∞
this✝ : MemLp (fun x ↦ 1) 0 μ
hfne : MemLp.toLp (fun x ↦ 1) this✝ = 0
this : (fun x ↦ 1) =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0

Tactic:
]

Goal state after:
case inr.inl
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
μ : Measure α
inst✝ : Nontrivial ↥(Lp E 0 μ)
f : ↥(Lp E 0 μ)
hf : f ≠ 0
h'p : 0 < ∞
this✝ : MemLp (fun x ↦ 1) 0 μ
hfne : MemLp.toLp (fun x ↦ 1) this✝ = 0
this : (fun x ↦ 1) =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0
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
⊢ llr μ (ν.tilted f) =ᵐ[μ] fun x ↦ -f x + log (∫ (z : α), rexp (f z) ∂ν) + llr μ ν x

Tactic:
cases eq_zero_or_neZero ν with
  | inl h =>
    have hμ : μ = 0 := by ext s _; exact hμν (by simp [h])
    simp only [hμ, ae_zero, Filter.EventuallyEq, Filter.eventually_bot]
  | inr h0 =>
    filter_upwards [hμν.ae_le (toReal_rnDeriv_tilted_right μ ν hf), Measure.rnDeriv_pos hμν,
      hμν.ae_le (Measure.rnDeriv_lt_top μ ν)] with x hx hx_pos hx_lt_top
    rw [llr, hx, log_mul, log_mul (exp_pos _).ne', log_exp, llr]
    · exact (integral_exp_pos hf).ne'
    · refine (mul_pos (exp_pos _) (integral_exp_pos hf)).ne'
    · simp [ENNReal.toReal_eq_zero_iff, hx_lt_top.ne, hx_pos.ne']

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 : 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:
]

Goal state after:
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
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 : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a + log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ

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

Tactic:
]

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

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ -f x + log (∫ (x : α), rexp (f x) ∂ν)) μ
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
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_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν =
    ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν

pp: case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν

pp: case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ᵐ[ν] fun x ↦ klFun (μ.rnDeriv ν x).toReal

Tactic:
]

Goal state after:
case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν

case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ᵐ[ν] fun x ↦ klFun (μ.rnDeriv ν x).toReal
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp E p μ)
hf : f ≠ 0
hfne : ¬↑↑f =ᵐ[μ] 0
h'p : p < ∞
hp : p ≠ 0
h'f : AEFinStronglyMeasurable (↑↑f) μ
s : Set α
s_meas : MeasurableSet s
s_pos : 0 < μ s
s_top : μ s < ∞
hzero : indicatorConstLp p s_meas ⋯ 1 = 0
⊢ ‖0‖ = ‖0‖

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp E p μ)
hf : f ≠ 0
hfne : ¬↑↑f =ᵐ[μ] 0
h'p : p < ∞
hp : p ≠ 0
h'f : AEFinStronglyMeasurable (↑↑f) μ
s : Set α
s_meas : MeasurableSet s
s_pos : 0 < μ s
s_top : μ s < ∞
hzero : indicatorConstLp p s_meas ⋯ 1 = 0
⊢ ‖indicatorConstLp p s_meas ⋯ 1‖ = ‖0‖

Tactic:
rw [hzero]

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

Goal state before:
pp: case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr μ ν) μ

Tactic:
]

Goal state after:
case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν ≠ ∞ ↔ Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : Integrable (llr μ ν) μ
⊢ 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 : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (a : α), -f a ∂μ + ∫ (a : α), log (∫ (x : α), rexp (f x) ∂ν) ∂μ + ∫ (a : α), llr μ ν a ∂μ =
    ∫ (a : α), -f a ∂μ + log (∫ (x : α), rexp (f x) ∂ν) + ∫ (x : α), llr μ ν x ∂μ

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a ↦ -f a) μ

Tactic:
]

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

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsProbabilityMeasure μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
hfμ : Integrable f μ
hfν : Integrable (fun x ↦ rexp (f x)) ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (fun a ↦ -f a) μ
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: case inr.inr
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp E p μ)
hf : f ≠ 0
hfne : ¬↑↑f =ᵐ[μ] 0
h'p : p < ∞
hp : p ≠ 0
h'f : AEFinStronglyMeasurable (↑↑f) μ
s : Set α
s_meas : MeasurableSet s
s_pos : 0 < μ s
s_top : μ s < ∞
hzero : indicatorConstLp p s_meas ⋯ 1 = 0
this : μ s = 0
⊢ False

Tactic:
]

Goal state after:
case inr.inr
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp E p μ)
hf : f ≠ 0
hfne : ¬↑↑f =ᵐ[μ] 0
h'p : p < ∞
hp : p ≠ 0
h'f : AEFinStronglyMeasurable (↑↑f) μ
s : Set α
s_meas : MeasurableSet s
s_pos : 0 < μ s
s_top : μ s < ∞
hzero : indicatorConstLp p s_meas ⋯ 1 = 0
this : μ s = 0
⊢ False
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Trim

Goal state before:
pp: α : Type u_1
ε : Type u_3
m m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
hm : m ≤ m0
f : α → ε
hf : AEStronglyMeasurable f (μ.trim hm)
⊢ eLpNorm (AEStronglyMeasurable.mk f hf) p (μ.trim hm) = eLpNorm (AEStronglyMeasurable.mk f hf) p μ

Tactic:
]

Goal state after:
α : Type u_1
ε : Type u_3
m m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
hm : m ≤ m0
f : α → ε
hf : AEStronglyMeasurable f (μ.trim hm)
⊢ eLpNorm (AEStronglyMeasurable.mk f hf) p (μ.trim hm) = eLpNorm (AEStronglyMeasurable.mk f hf) p μ
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_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔
    ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

Tactic:
]

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_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔
    ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞
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
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν

Tactic:
]

Goal state after:
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
⊢ 0 ≤ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν
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
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν

Tactic:
exact integral_nonneg fun x ↦ klFun_nonneg ENNReal.toReal_nonneg

Goal state after:
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
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (x : α), llr μ ν x ∂μ + ν.real univ - μ.real univ = ∫ (a : α), llr μ ν a ∂μ + ν.real univ - μ.real univ

pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), llr μ ν x ∂μ + ν.real univ - μ.real univ

Tactic:
]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), llr μ ν x ∂μ + ν.real univ - μ.real univ
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
h : μ ≪ ν
h_eq : μ univ = ν univ
h_int : ¬Integrable (llr μ ν) μ
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν = ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν

pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν

Tactic:
]

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ 0 ≤ ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν
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
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (klDiv μ ν).toReal = 0

pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (klDiv μ ν).toReal = 0

case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
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
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ 0 = 0

Tactic:
]

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
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ¬Integrable (llr μ ν) μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
h : μ ≪ ν
h_int : ¬Integrable (llr μ ν) μ
⊢ ¬Integrable (llr μ ν) μ
Module: Mathlib.MeasureTheory.Function.LpSpace.CompleteOfCompleteLp

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedSpace ℝ E
hp : Fact (1 ≤ p)
inst✝¹ : CompleteSpace ↥(Lp E p μ)
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp ℝ p μ)
m : E →L[ℝ] ↥(Lp E p μ) := (compLpL₂ p μ (lsmul ℝ ℝ).flip).flip f
u : ℕ → E
hu : CauchySeq u
g : ↥(Lp E p μ)
hg : Tendsto (⇑m ∘ u) atTop (𝓝 g)
f' : ℕ → α → E := fun n ↦ ↑↑((⇑m ∘ u) n)
ns : ℕ → ℕ
hns : StrictMono ns
nslim : ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f' (ns i) x) atTop (𝓝 (↑↑g x))
hf : ↑↑f =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_2
inst✝³ : NormedAddCommGroup E
mα : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedSpace ℝ E
hp : Fact (1 ≤ p)
inst✝¹ : CompleteSpace ↥(Lp E p μ)
inst✝ : Nontrivial ↥(Lp E p μ)
f : ↥(Lp ℝ p μ)
m : E →L[ℝ] ↥(Lp E p μ) := (compLpL₂ p μ (lsmul ℝ ℝ).flip).flip f
u : ℕ → E
hu : CauchySeq u
g : ↥(Lp E p μ)
hg : Tendsto (⇑m ∘ u) atTop (𝓝 g)
f' : ℕ → α → E := fun n ↦ ↑↑((⇑m ∘ u) n)
ns : ℕ → ℕ
hns : StrictMono ns
nslim : ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f' (ns i) x) atTop (𝓝 (↑↑g x))
hf : ↑↑f =ᵐ[μ] 0
⊢ ↑↑f =ᵐ[μ] 0
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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ ∫ (a : α), llr (c • μ) ν a ∂c • μ + ν.real univ - (c • μ).real univ =
    ↑c * (∫ (a : α), llr μ ν a ∂μ + ν.real univ - μ.real univ) + (1 - ↑c) * ν.real univ + ↑c * log ↑c * μ.real univ

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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ Integrable (llr (c • μ) ν) (c • μ)

Tactic:
]

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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ ∫ (a : α), llr (c • μ) ν a ∂c • μ + ν.real univ - (c • μ).real univ =
    ↑c * (∫ (a : α), llr μ ν a ∂μ + ν.real univ - μ.real univ) + (1 - ↑c) * ν.real univ + ↑c * log ↑c * μ.real univ

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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ Integrable (llr (c • μ) ν) (c • μ)
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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ Integrable (fun x ↦ llr μ ν x + log ↑c) μ

Tactic:
]

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 : llr (c • μ) ν =ᵐ[μ] fun x ↦ llr μ ν x + log ↑c
⊢ Integrable (fun x ↦ llr μ ν x + log ↑c) μ