mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
Eₗ : Type u_5
F : Type u_6
Fₗ : Type u_7
G : Type u_8
Gₗ : Type u_9
𝓕 : Type u_10
inst✝³² : SeminormedAddCommGroup E
inst✝³¹ : SeminormedAddCommGroup Eₗ
inst✝³⁰ : SeminormedAddCommGroup F
inst✝²⁹ : SeminormedAddCommGroup Fₗ
inst✝²⁸ : SeminormedAddCommGroup G
inst✝²⁷ : SeminormedAddCommGroup Gₗ
inst✝²⁶ : NontriviallyNormedField 𝕜
inst✝²⁵ : NontriviallyNormedField 𝕜₂
inst✝²⁴ : NontriviallyNormedField 𝕜₃
inst✝²³ : NormedSpace 𝕜 E
inst✝²² : NormedSpace 𝕜 Eₗ
inst✝²¹ : NormedSpace 𝕜₂ F
inst✝²⁰ : NormedSpace 𝕜 Fₗ
inst✝¹⁹ : NormedSpace 𝕜₃ G
inst✝¹⁸ : NormedSpace 𝕜 Gₗ
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁶ : FunLike 𝓕 E F
σ₂₁ : 𝕜₂ →+* 𝕜
inst✝¹⁵ : RingHomInvPair σ₁₂ σ₂₁
inst✝¹⁴ : RingHomInvPair σ₂₁ σ₁₂
E' : Type u_11
F' : Type u_12
inst✝¹³ : SeminormedAddCommGroup E'
inst✝¹² : SeminormedAddCommGroup F'
𝕜₁' : Type u_13
𝕜₂' : Type u_14
inst✝¹¹ : NontriviallyNormedField 𝕜₁'
inst✝¹⁰ : NontriviallyNormedField 𝕜₂'
inst✝⁹ : NormedSpace 𝕜₁' E'
inst✝⁸ : NormedSpace 𝕜₂' F'
σ₁' : 𝕜₁' →+* 𝕜
σ₁₃' : 𝕜₁' →+* 𝕜₃
σ₂' : 𝕜₂' →+* 𝕜₂
σ₂₃' : 𝕜₂' →+* 𝕜₃
inst✝⁷ : RingHomCompTriple σ₁' σ₁₃ σ₁₃'
inst✝⁶ : RingHomCompTriple σ₂' σ₂₃ σ₂₃'
inst✝⁵ : RingHomIsometric σ₂₃
inst✝⁴ : RingHomIsometric σ₁₃'
inst✝³ : RingHomIsometric σ₂₃'
inst✝² : RingHomIsometric σ₁₃
inst✝¹ : RingHomIsometric σ₁'
inst✝ : RingHomIsometric σ₂'
m : 𝕜
c : StrongDual 𝕜 E
x : Fₗ
x✝ : E
⊢ ((smulRightₗ (m • c)) x) x✝ = (((RingHom.id 𝕜) m • smulRightₗ c) x) x✝

Tactic:
simp [smul_smul]

Goal state after:
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inl
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s = 0
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1

pp: case inr
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s ≠ 0
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s = 0
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inr.inl
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s ≠ 0
h' : μ s = ∞
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1

pp: case inr.inr
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s ≠ 0
h' : μ s ≠ ∞
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1

Tactic:
·

Goal state after:
case inr.inl
Ω : Type u_1
Ω' : Type u_2
α : Type u_3
m : MeasurableSpace Ω
m' : MeasurableSpace Ω'
μ : Measure Ω
s t : Set Ω
h : μ s ≠ 0
h' : μ s = ∞
⊢ μ s / μ s = 0 ∨ μ s / μ s = 1
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hnt : μ s = ∞
⊢ (μ (toMeasurable μ s))⁻¹ • μ.restrict (toMeasurable μ s) = (μ s)⁻¹ • μ.restrict s

pp: case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hnt : ¬μ s = ∞
⊢ (μ (toMeasurable μ s))⁻¹ • μ.restrict (toMeasurable μ s) = (μ s)⁻¹ • μ.restrict s

Tactic:
·

Goal state after:
case pos
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hnt : μ s = ∞
⊢ (μ (toMeasurable μ s))⁻¹ • μ.restrict (toMeasurable μ s) = (μ s)⁻¹ • μ.restrict s
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hnt : ¬μ s = ∞
⊢ (μ (toMeasurable μ s))⁻¹ • μ.restrict (toMeasurable μ s) = (μ s)⁻¹ • μ.restrict s

Tactic:
·

Goal state after:
case neg
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hnt : ¬μ s = ∞
⊢ (μ (toMeasurable μ s))⁻¹ • μ.restrict (toMeasurable μ s) = (μ s)⁻¹ • μ.restrict s
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hmt : MeasurableSet t
hcs : μ s ≠ ∞
u : Set Ω
a✝ : MeasurableSet u
hst : μ (s ∩ t) = 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]

pp: case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hmt : MeasurableSet t
hcs : μ s ≠ ∞
u : Set Ω
a✝ : MeasurableSet u
hst : μ (s ∩ t) ≠ 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hmt : MeasurableSet t
hcs : μ s ≠ ∞
u : Set Ω
a✝ : MeasurableSet u
hst : μ (s ∩ t) = 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≤ b) → ∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, ↑(u a) ≤ x

pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, ↑(u a) ≤ x) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≤ b

Tactic:
·

Goal state after:
case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≤ b) → ∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, ↑(u a) ≤ x
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hmt : MeasurableSet t
hcs : μ s ≠ ∞
u : Set Ω
a✝ : MeasurableSet u
hst : μ (s ∩ t) ≠ 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]

Tactic:
·

Goal state after:
case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s t : Set Ω
hms : MeasurableSet s
hmt : MeasurableSet t
hcs : μ s ≠ ∞
u : Set Ω
a✝ : MeasurableSet u
hst : μ (s ∩ t) ≠ 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hms : MeasurableSet s
hcs' : μ s ≠ ∞
t : Set Ω
hcs : μ s = 0
⊢ μ[t | s] * μ s = μ (s ∩ t)

pp: case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hms : MeasurableSet s
hcs' : μ s ≠ ∞
t : Set Ω
hcs : μ s ≠ 0
⊢ μ[t | s] * μ s = μ (s ∩ t)

Tactic:
·

Goal state after:
case inl
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hms : MeasurableSet s
hcs' : μ s ≠ ∞
t : Set Ω
hcs : μ s = 0
⊢ μ[t | s] * μ s = μ (s ∩ t)
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hms : MeasurableSet s
hcs' : μ s ≠ ∞
t : Set Ω
hcs : μ s ≠ 0
⊢ μ[t | s] * μ s = μ (s ∩ t)

Tactic:
·

Goal state after:
case inr
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
s : Set Ω
hms : MeasurableSet s
hcs' : μ s ≠ ∞
t : Set Ω
hcs : μ s ≠ 0
⊢ μ[t | s] * μ s = μ (s ∩ t)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, ↑(u a) ≤ x) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≤ b

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, ↑(u a) ≤ x) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≤ b
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≥ b) → ∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, x ≤ ↑(u a)

pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, x ≤ ↑(u a)) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≥ b

Tactic:
·

Goal state after:
case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≥ b) → ∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, x ≤ ↑(u a)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, x ≤ ↑(u a)) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≥ b

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x, ∃ (_ : 0 ≤ x), ∀ᶠ (a : ι) in f, x ≤ ↑(u a)) → ∃ b, ∀ᶠ (a : ι) in f, ↑(u a) ≥ b
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
⊢ (∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a) →
    ∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x_1 hx_1)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk x_1 hx_1)

pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x_1 hx_1)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk x_1 hx_1)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a

Tactic:
·

Goal state after:
case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
⊢ (∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a) →
    ∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x_1 hx_1)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk x_1 hx_1)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x_1 hx_1)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk x_1 hx_1)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x_1 hx_1)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk x_1 hx_1)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a) →
    ∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(NNReal.mk x_1 hx_1) ≤ ↑(u a)) → ↑(NNReal.mk x_1 hx_1) ≤ ↑(NNReal.mk x hx)

pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(NNReal.mk x_1 hx_1) ≤ ↑(u a)) → ↑(NNReal.mk x_1 hx_1) ≤ ↑(NNReal.mk x hx)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a

Tactic:
·

Goal state after:
case mp
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a) →
    ∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(NNReal.mk x_1 hx_1) ≤ ↑(u a)) → ↑(NNReal.mk x_1 hx_1) ≤ ↑(NNReal.mk x hx)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(NNReal.mk x_1 hx_1) ≤ ↑(u a)) → ↑(NNReal.mk x_1 hx_1) ≤ ↑(NNReal.mk x hx)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a

Tactic:
·

Goal state after:
case mpr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
⊢ (∃ x,
      ∃ (hx : 0 ≤ x),
        ∀ (x_1 : ℝ) (hx_1 : 0 ≤ x_1),
          (∀ᶠ (a : ι) in f, ↑(NNReal.mk x_1 hx_1) ≤ ↑(u a)) → ↑(NNReal.mk x_1 hx_1) ≤ ↑(NNReal.mk x hx)) →
    ∃ b, ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr.inl
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(NNReal.mk x hx) ≤ ↑(u a)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk b hb₀)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≥ x
hx₀ : x ≤ 0
⊢ b ≥ x

pp: case mpr.inr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(NNReal.mk x hx) ≤ ↑(u a)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk b hb₀)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≥ x
hx₀ : 0 ≤ x
⊢ b ≥ x

Tactic:
·

Goal state after:
case mpr.inl
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(NNReal.mk x hx) ≤ ↑(u a)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk b hb₀)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≥ x
hx₀ : x ≤ 0
⊢ b ≥ x
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr.inr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(NNReal.mk x hx) ≤ ↑(u a)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk b hb₀)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≥ x
hx₀ : 0 ≤ x
⊢ b ≥ x

Tactic:
·

Goal state after:
case mpr.inr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(NNReal.mk x hx) ≤ ↑(u a)) → ↑(NNReal.mk x hx) ≤ ↑(NNReal.mk b hb₀)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≥ x
hx₀ : 0 ≤ x
⊢ b ≥ x
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case neg
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : ¬IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ liminf (fun i ↦ ↑(u i)) f = ↑(liminf u f)

pp: case pos
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ liminf (fun i ↦ ↑(u i)) f = ↑(liminf u f)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : ¬IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
⊢ liminf (fun i ↦ ↑(u i)) f = ↑(liminf u f)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case inl
ι : Type u_1
u : ι → ℝ≥0
⊢ limsup (fun i ↦ ↑(u i)) ⊥ = ↑(limsup u ⊥)

pp: case inr
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : f.NeBot
⊢ limsup (fun i ↦ ↑(u i)) f = ↑(limsup u f)

Tactic:
·

Goal state after:
case inl
ι : Type u_1
u : ι → ℝ≥0
⊢ limsup (fun i ↦ ↑(u i)) ⊥ = ↑(limsup u ⊥)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case neg
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf✝ : f.NeBot
hf : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ limsup (fun i ↦ ↑(u i)) f = ↑(limsup u f)

pp: case pos
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf✝ : f.NeBot
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ limsup (fun i ↦ ↑(u i)) f = ↑(limsup u f)

Tactic:
·

Goal state after:
case neg
ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf✝ : f.NeBot
hf : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
⊢ limsup (fun i ↦ ↑(u i)) f = ↑(limsup u f)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : a = 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

pp: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : ¬a = 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

Tactic:
·

Goal state after:
case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : a = 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case pos
α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : a = 0
⊢ liminf (fun x ↦ a * u x) f = a * liminf u f

pp: case neg
α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : ¬a = 0
⊢ liminf (fun x ↦ a * u x) f = a * liminf u f

Tactic:
·

Goal state after:
case pos
α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
ha₀ : a = 0
⊢ liminf (fun x ↦ a * u x) f = a * liminf u f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

pp: case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

Tactic:
·

Goal state after:
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ∞
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

pp: case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : ¬u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

Tactic:
·

Goal state after:
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : ¬u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f

Tactic:
·

Goal state after:
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : ¬u =ᶠ[f] 0
⊢ limsup (fun x ↦ a * u x) f = a * limsup u f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y

pp: case mpr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y

Tactic:
·

Goal state after:
case mp
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mp.none
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
hx : none > ↑r
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < none

pp: case mp.some
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < some x

Tactic:
·

Goal state after:
case mp.none
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
hx : none > ↑r
⊢ ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < none
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ∃ b, f = mk (const α b) ⋯
⊢ StronglyMeasurable (const α (Classical.choose h))

pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ¬∃ b, f = mk (const α b) ⋯
⊢ StronglyMeasurable (AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ∃ b, f = mk (const α b) ⋯
⊢ StronglyMeasurable (const α (Classical.choose h))
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ¬∃ b, f = mk (const α b) ⋯
⊢ StronglyMeasurable (AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ¬∃ b, f = mk (const α b) ⋯
⊢ StronglyMeasurable (AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯)
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case inl
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
a : α
ha : u a < ↑x
ha₀ : u a ≤ 0
⊢ ENNReal.ofReal (u a) < some x

pp: case inr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
a : α
ha : u a < ↑x
ha₀ : 0 ≤ u a
⊢ ENNReal.ofReal (u a) < some x

Tactic:
·

Goal state after:
case inl
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
a : α
ha : u a < ↑x
ha₀ : u a ≤ 0
⊢ ENNReal.ofReal (u a) < some x
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ∃ b, f = mk (const α b) ⋯
⊢ mk (const α (Classical.choose h)) ⋯ = f

pp: case neg
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ¬∃ b, f = mk (const α b) ⋯
⊢ mk (AEStronglyMeasurable.mk ↑(Quotient.out f) ⋯) ⋯ = f

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α →ₘ[μ] β
h : ∃ b, f = mk (const α b) ⋯
⊢ mk (const α (Classical.choose h)) ⋯ = f
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case inr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
a : α
ha : u a < ↑x
ha₀ : 0 ≤ u a
⊢ ENNReal.ofReal (u a) < some x

Tactic:
·

Goal state after:
case inr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
h : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
x : ℝ≥0
hx : some x > ↑r
a : α
ha : u a < ↑x
ha₀ : 0 ≤ u a
⊢ ENNReal.ofReal (u a) < some x
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case mpr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y

Tactic:
·

Goal state after:
case mpr
α : Type u_1
f : Filter α
u : α → ℝ
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
r : ℝ≥0
⊢ (∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y) → ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y
Module: Mathlib.MeasureTheory.MeasurableSpace.Pi

Goal state before:
pp: case intro.a
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
⊢ MeasurableSpace.pi ≤ generateFrom (univ.pi '' univ.pi C)

pp: case a
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
⊢ generateFrom (univ.pi '' univ.pi C) ≤ MeasurableSpace.pi

Tactic:
·

Goal state after:
case intro.a
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), IsCountablySpanning (C i)
val✝ : Encodable ι
⊢ MeasurableSpace.pi ≤ generateFrom (univ.pi '' univ.pi C)
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : TopologicalSpace α
𝕜 : Type u_5
inst✝² : TopologicalSpace 𝕜
inst✝¹ : Field 𝕜
inst✝ : IsTopologicalRing 𝕜
s : Subalgebra 𝕜 C(α, 𝕜)
h : s.SeparatesPoints
v : α → 𝕜
x y : α
n : x = y
⊢ ∃ f ∈ ↑s, f x = v x ∧ f y = v y

pp: case neg
α : Type u_1
inst✝³ : TopologicalSpace α
𝕜 : Type u_5
inst✝² : TopologicalSpace 𝕜
inst✝¹ : Field 𝕜
inst✝ : IsTopologicalRing 𝕜
s : Subalgebra 𝕜 C(α, 𝕜)
h : s.SeparatesPoints
v : α → 𝕜
x y : α
n : ¬x = y
⊢ ∃ f ∈ ↑s, f x = v x ∧ f y = v y

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝³ : TopologicalSpace α
𝕜 : Type u_5
inst✝² : TopologicalSpace 𝕜
inst✝¹ : Field 𝕜
inst✝ : IsTopologicalRing 𝕜
s : Subalgebra 𝕜 C(α, 𝕜)
h : s.SeparatesPoints
v : α → 𝕜
x y : α
n : x = y
⊢ ∃ f ∈ ↑s, f x = v x ∧ f y = v y