mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h0 : NeZero μ
this : ∀ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * rexp (g x) = rexp ((f + g) x) / ∫ (x : α), rexp (f x) ∂μ
⊢ ∫ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * rexp (g x) ∂μ =
    (∫ (x : α), rexp ((f + g) x) ∂μ) / ∫ (x : α), rexp (f x) ∂μ

Tactic:
simp_rw [this, div_eq_mul_inv]

Goal state after:
case inr
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f g : α → ℝ
h0 : NeZero μ
this : ∀ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * rexp (g x) = rexp ((f + g) x) / ∫ (x : α), rexp (f x) ∂μ
⊢ ∫ (x : α), rexp ((f + g) x) * (∫ (x : α), rexp (f x) ∂μ)⁻¹ ∂μ =
    (∫ (x : α), rexp ((f + g) x) ∂μ) * (∫ (x : α), rexp (f x) ∂μ)⁻¹
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ
hf : Integrable (fun x ↦ rexp (f x)) μ
g : α → E
hμ : ¬μ = 0
hf_meas : AEMeasurable f μ
⊢ Integrable (fun x ↦ (ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)).toReal • g x) μ ↔
    Integrable (fun x ↦ rexp (f x) • g x) μ

Tactic:
calc Integrable (fun x ↦ (ENNReal.ofReal (exp (f x) / ∫ a, exp (f a) ∂μ)).toReal • g x) μ
  _ ↔ Integrable (fun x ↦ (exp (f x) / ∫ a, exp (f a) ∂μ) • g x) μ := by
    congr! with a
    rw [ENNReal.toReal_ofReal]
    positivity
  _ ↔ Integrable (fun x ↦ (∫ a, exp (f a) ∂μ)⁻¹ • exp (f x) • g x) μ := by
    congr! 2 with a
    rw [smul_smul, div_eq_inv_mul]
  _ ↔ Integrable (fun x ↦ exp (f x) • g x) μ := by
    rw [integrable_fun_smul_iff]
    simp only [ne_eq, inv_eq_zero]
    have : NeZero μ := ⟨hμ⟩
    exact (integral_exp_pos hf).ne'

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 ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal) ν ↔ Integrable (llr μ ν) μ

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 ↦ 1 - (μ.rnDeriv ν x).toReal) ν

Tactic:
integrable_rnDeriv_mul_log_iff hμν

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 (llr μ ν) μ ↔ Integrable (llr μ ν) μ

α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
⊢ Integrable (fun x ↦ 1 - (μ.rnDeriv ν x).toReal) ν
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: case hf
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
⊢ Integrable (fun a ↦ (μ.rnDeriv ν a).toReal * log (μ.rnDeriv ν a).toReal) ν

Tactic:
integrable_rnDeriv_mul_log_iff hμν

Goal state after:
case hf
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
⊢ Integrable (llr μ ν) μ
Module: Mathlib.InformationTheory.KullbackLeibler.KLFun

Goal state before:
pp: case hf
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
⊢ Integrable (fun x ↦ (μ.rnDeriv ν x).toReal * log (μ.rnDeriv ν x).toReal) ν

Tactic:
integrable_rnDeriv_mul_log_iff hμν

Goal state after:
case hf
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
⊢ Integrable (llr μ ν) μ
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
⊢ limsup f (ae (μ.trim hm)) = limsup f (ae μ)

Tactic:
by
  simp_rw [limsup_eq]
  suffices h_set_eq : { a : ℝ≥0∞ | ∀ᵐ n ∂μ.trim hm, f n ≤ a } = { a : ℝ≥0∞ | ∀ᵐ n ∂μ, f n ≤ a } by
    rw [h_set_eq]
  ext1 a
  suffices h_meas_eq : μ { x | ¬f x ≤ a } = μ.trim hm { x | ¬f x ≤ a } by
    simp_rw [Set.mem_ofPred_eq, ae_iff, h_meas_eq]
  refine (trim_measurableSet_eq hm ?_).symm
  exact (measurableSet_le hf measurable_const).compl

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_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:
rw [← not_iff_not, ne_eq, Decidable.not_not] at h_int_iff

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: case pos
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
⊢ (klDiv μ ν).toReal = ∫ (a : α), llr μ ν a ∂μ

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
⊢ (klDiv μ ν).toReal = ∫ (a : α), llr μ ν a ∂μ

Tactic:
· simp [toReal_klDiv h h_int, h_eq, measureReal_def]

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_eq : μ univ = ν univ
h_int : ¬Integrable (llr μ ν) μ
⊢ (klDiv μ ν).toReal = ∫ (a : α), llr μ ν a ∂μ
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✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(lpMeas F 𝕜 m p μ)

Tactic:
rw [(lpMeasSubgroupToLpMeasIso F 𝕜 p μ).symm.completeSpace_iff]

Goal state after:
α : Type u_1
F : Type u_2
𝕜 : Type u_3
p : ℝ≥0∞
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
hm : Fact (m ≤ m0)
inst✝ : CompleteSpace F
hp : Fact (1 ≤ p)
⊢ CompleteSpace ↥(lpMeasSubgroup F m p μ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: α : Type u_1
𝕜 : Type u_7
inst✝ : RCLike 𝕜
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp 𝕜 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
h_eq_inner : ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) f) x ∂μ = ⟪indicatorConstLp 2 ⋯ hμs 1, ↑((condExpL2 𝕜 𝕜 hm) f)⟫
⊢ ⟪indicatorConstLp 2 ⋯ hμs 1, ↑((condExpL2 𝕜 𝕜 hm) f)⟫ = ⟪indicatorConstLp 2 ⋯ hμs 1, f⟫

Tactic:
← inner_condExpL2_left_eq_right,

Goal state after:
α : Type u_1
𝕜 : Type u_7
inst✝ : RCLike 𝕜
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
f : ↥(Lp 𝕜 2 μ)
hs : MeasurableSet s
hμs : μ s ≠ ∞
h_eq_inner : ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) f) x ∂μ = ⟪indicatorConstLp 2 ⋯ hμs 1, ↑((condExpL2 𝕜 𝕜 hm) f)⟫
⊢ ⟪↑((condExpL2 𝕜 𝕜 hm) (indicatorConstLp 2 ⋯ hμs 1)), f⟫ = ⟪indicatorConstLp 2 ⋯ hμs 1, f⟫
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case pos
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)
x y : G
hs : MeasurableSet s
hμs : μ s = ∞
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

pp: case neg
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)
x y : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

Tactic:
· simp_rw [condExpIndL1_of_measure_eq_top hμs]; rw [zero_add]

Goal state after:
case neg
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)
x y : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
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)
x y : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1 hm μ s (x + y) = condExpIndL1 hm μ s x + condExpIndL1 hm μ s y

Tactic:
simp_rw [condExpIndL1_of_measurableSet_of_measure_ne_top hs hμs]

Goal state after:
case neg
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)
x y : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1Fin hm hs hμs (x + y) = condExpIndL1Fin hm hs hμs x + condExpIndL1Fin hm hs hμs y
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
⊢ ENNReal.ofReal (μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ) ≤ klDiv μ ν

Tactic:
by
  by_cases hμν : μ ≪ ν
  swap; · simp [hμν]
  by_cases h_int : Integrable (llr μ ν) μ
  swap; · simp [h_int]
  rw [← ENNReal.ofReal_toReal (a := klDiv μ ν)]
  · exact ENNReal.ofReal_le_ofReal (mul_log_le_toReal_klDiv hμν h_int)
  · rw [klDiv_ne_top_iff]
    exact ⟨hμν, h_int⟩

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

Goal state before:
pp: case neg
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)
c : ℝ
x : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1 hm μ s (c • x) = c • condExpIndL1 hm μ s x

Tactic:
simp_rw [condExpIndL1_of_measurableSet_of_measure_ne_top hs hμs]

Goal state after:
case neg
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)
c : ℝ
x : G
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1Fin hm hs hμs (c • x) = c • condExpIndL1Fin hm hs hμs x
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
F : Type u_2
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
inst✝² : SigmaFinite (μ.trim hm)
inst✝¹ : NormedSpace ℝ F
inst✝ : SMulCommClass ℝ 𝕜 F
c : 𝕜
x : F
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1 hm μ s (c • x) = c • condExpIndL1 hm μ s x

Tactic:
simp_rw [condExpIndL1_of_measurableSet_of_measure_ne_top hs hμs]

Goal state after:
case neg
α : Type u_1
F : Type u_2
𝕜 : Type u_6
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
hm : m ≤ m0
inst✝² : SigmaFinite (μ.trim hm)
inst✝¹ : NormedSpace ℝ F
inst✝ : SMulCommClass ℝ 𝕜 F
c : 𝕜
x : F
hs : MeasurableSet s
hμs : ¬μ s = ∞
⊢ condExpIndL1Fin hm hs hμs (c • x) = c • condExpIndL1Fin hm hs hμs x
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
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) x ∂μ = ⟪indicatorConstLp 2 ⋯ ⋯ c, ↑((condExpL2 E 𝕜 hm) f)⟫

Tactic:
← inner_condExpL2_left_eq_right,

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
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯) x ∂μ = ⟪↑((condExpL2 E 𝕜 hm) (indicatorConstLp 2 ⋯ ⋯ c)), f⟫
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Goal state before:
pp: α : Type u_1
E : Type u_3
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
f : α → E
m : MeasurableSpace α
⊢ μ[c • f | m] =ᵐ[μ] c • μ[f | m]

Tactic:
by
  by_cases hm : m ≤ m₀
  swap; · simp_rw [condExp_of_not_le hm]; simp
  by_cases hμm : SigmaFinite (μ.trim hm)
  swap; · simp_rw [condExp_of_not_sigmaFinite hm hμm]; simp
  refine (condExp_ae_eq_condExpL1 hm _).trans ?_
  rw [condExpL1_smul c f]
  refine (condExp_ae_eq_condExpL1 hm f).mp ?_
  refine (coeFn_smul c (condExpL1 hm μ f)).mono fun x hx1 hx2 => ?_
  simp only [hx1, hx2, Pi.smul_apply]

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

Goal state before:
pp: case pos.refine_1
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
⊢ Integrable (⇑T ∘ f) μ

pp: case pos.refine_2
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
s : Set α
ms : MeasurableSet s
hs : μ s < ∞
⊢ IntegrableOn (⇑T ∘ μ[f | m]) s μ

pp: case pos.refine_3
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
s : Set α
ms : MeasurableSet s
hs : μ s < ∞
⊢ ∫ (x : α) in s, (⇑T ∘ μ[f | m]) x ∂μ = ∫ (x : α) in s, (⇑T ∘ f) x ∂μ

pp: case pos.refine_4
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
⊢ AEStronglyMeasurable (⇑T ∘ μ[f | m]) μ

Tactic:
· exact T.integrable_comp hf_int

Goal state after:
case pos.refine_2
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
s : Set α
ms : MeasurableSet s
hs : μ s < ∞
⊢ IntegrableOn (⇑T ∘ μ[f | m]) s μ

case pos.refine_3
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
s : Set α
ms : MeasurableSet s
hs : μ s < ∞
⊢ ∫ (x : α) in s, (⇑T ∘ μ[f | m]) x ∂μ = ∫ (x : α) in s, (⇑T ∘ f) x ∂μ

case pos.refine_4
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
inst✝ : NormedSpace ℝ F
hf_int : Integrable f μ
T : E →L[ℝ] F
hm : m ≤ m₀
hμ : SigmaFinite (μ.trim hm)
⊢ AEStronglyMeasurable (⇑T ∘ μ[f | m]) μ
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M
⊢ b.flag 0 = ⊥

Tactic:
by simp [flag]

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✝⁷ : AddCommGroup V
inst✝⁶ : AffineSpace V P
inst✝⁵ : Ring k
inst✝⁴ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝³ : Finite ι
inst✝² : Fintype ι'
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial k
p : ι' → P
A : Matrix ι ι' k
hA : A * b.toMatrix p = 1
val✝ : Fintype ι
i : ι
hAi : ∑ j, A i j = 1
j : ι
⊢ (if j = i then 1 else 0) = 1 i j

Tactic:
Matrix.one_apply,

Goal state after:
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁷ : AddCommGroup V
inst✝⁶ : AffineSpace V P
inst✝⁵ : Ring k
inst✝⁴ : Module k V
b : AffineBasis ι k P
ι' : Type u_1
inst✝³ : Finite ι
inst✝² : Fintype ι'
inst✝¹ : DecidableEq ι
inst✝ : Nontrivial k
p : ι' → P
A : Matrix ι ι' k
hA : A * b.toMatrix p = 1
val✝ : Fintype ι
i : ι
hAi : ∑ j, A i j = 1
j : ι
⊢ (if j = i then 1 else 0) = if i = j then 1 else 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : MultilinearMap R' (fun x ↦ Mᵢ) N₁
b : MultilinearMap R' (fun x ↦ Mᵢ) N₂
⊢ (↑(alternatization a)).domCoprod ↑(alternatization b) =
    ∑ σa, ∑ σb, Perm.sign σa • Perm.sign σb • (domDomCongr σa a).domCoprod (domDomCongr σb b)

Tactic:
← MultilinearMap.domCoprod'_apply,

Goal state after:
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : MultilinearMap R' (fun x ↦ Mᵢ) N₁
b : MultilinearMap R' (fun x ↦ Mᵢ) N₂
⊢ domCoprod' (↑(alternatization a) ⊗ₜ[R'] ↑(alternatization b)) =
    ∑ σa, ∑ σb, Perm.sign σa • Perm.sign σb • domCoprod' (domDomCongr σa a ⊗ₜ[R'] domDomCongr σb b)
Module: Mathlib.LinearAlgebra.FixedSubmodule

Goal state before:
pp: R : Type u_4
V : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
e : V ≃ₗ[R] V
a : R
⊢ (∀ (x : V ⧸ (↑e).fixedSubmodule), e.fixedReduce x = a • x) ↔ ∀ (v : V), e v - a • v ∈ (↑e).fixedSubmodule

Tactic:
simp only [← e.fixedSubmodule.ker_mkQ, mem_ker, map_sub, ← fixedReduce_mkQ, sub_eq_zero]
  constructor
  · intro H x; simp [H]
  · intro H x
    have ⟨y, hy⟩ := e.fixedSubmodule.mkQ_surjective x
    rw [← hy]
    apply H

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case h.left
V : Type u_2
inst✝³ : AddCommGroup V
K : Type u_3
inst✝² : DivisionRing K
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
x✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
he' : e.fixedReduce = 1
he_one : ¬e = 1
hefixed_ne_top : (↑e).fixedSubmodule ≠ ⊤
w : V
hw : w ∉ (↑e).fixedSubmodule
f : Dual K V
hfw : f w ≠ 0
hf : map f (↑e).fixedSubmodule = ⊥
hf' : (↑e).fixedSubmodule = ker f
eq_top : (↑e).fixedSubmodule ⊔ K ∙ w = ⊤
v : V := (f w)⁻¹ • (e w - w)
hfv : f v = 0
x : V
hx : x ∈ (↑e).fixedSubmodule
⊢ x ∈ (↑e - LinearMap.transvection f v).ker

Tactic:
suffices f x = 0 by
          simpa [this, LinearMap.transvection.apply, sub_eq_zero] using hx

Goal state after:
case h.left
V : Type u_2
inst✝³ : AddCommGroup V
K : Type u_3
inst✝² : DivisionRing K
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
x✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
he' : e.fixedReduce = 1
he_one : ¬e = 1
hefixed_ne_top : (↑e).fixedSubmodule ≠ ⊤
w : V
hw : w ∉ (↑e).fixedSubmodule
f : Dual K V
hfw : f w ≠ 0
hf : map f (↑e).fixedSubmodule = ⊥
hf' : (↑e).fixedSubmodule = ker f
eq_top : (↑e).fixedSubmodule ⊔ K ∙ w = ⊤
v : V := (f w)⁻¹ • (e w - w)
hfv : f v = 0
x : V
hx : x ∈ (↑e).fixedSubmodule
⊢ f x = 0
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ (ι Q) a * (ι Q) b * ⅟((ι Q) a) = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
invOf_ι,

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ (ι Q) a * (ι Q) b * (ι Q) (⅟(Q a) • a) = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ ⅟((ι Q) a) * (ι Q) b * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
invOf_ι,

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ (ι Q) (⅟(Q a) • a) * (ι Q) b * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Inversion

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
a b : M
inst✝¹ : Invertible ((ι Q) a)
inst✝ : Invertible (Q a)
⊢ ⅟((ι Q) a) * (ι Q) b * (ι Q) a = (ι Q) ((⅟(Q a) * QuadraticMap.polar (⇑Q) a b) • a - b)

Tactic:
rw [invOf_ι, map_smul, smul_mul_assoc, smul_mul_assoc, ι_mul_ι_mul_ι, ← map_smul, smul_sub,
    smul_smul, smul_smul, invOf_mul_self, one_smul]

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: case inv_mem
R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
x✝ : (CliffordAlgebra Q)ˣ
inst✝ : Invertible 2
x : (CliffordAlgebra Q)ˣ
hx : x ∈ Units.val ⁻¹' Set.range ⇑(ι Q)
m : M
⊢ ↑x⁻¹ * (ι Q) m * ↑x⁻¹⁻¹ ∈ (ι Q).range

Tactic:
| inv_mem x hx =>

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: case inv_mem
R : Type u_1
inst✝³ : CommRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
Q : QuadraticForm R M
inst✝ : Invertible 2
x✝ x : (CliffordAlgebra Q)ˣ
hx : x ∈ Units.val ⁻¹' Set.range ⇑(ι Q)
b : M
⊢ involute ↑x⁻¹ * (ι Q) b * ↑x⁻¹⁻¹ ∈ (ι Q).range

Tactic:
| inv_mem x hx =>

Goal state after: