mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
h : ∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0
⊢ T1Space E

Tactic:
have := hp.topologicalAddGroup

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
h : ∀ (x : E), x ≠ 0 → ∃ i, (p i) x ≠ 0
this : IsTopologicalAddGroup E
⊢ T1Space E
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
i : ι
⊢ Continuous[t, _] ⇑(p i)

Tactic:
have := hp.topologicalAddGroup

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
i : ι
this : IsTopologicalAddGroup E
⊢ Continuous[t, _] ⇑(p i)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
⊢ WithSeminorms fun p ↦ ↑p

Tactic:
have := hp.topologicalAddGroup

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ WithSeminorms fun p ↦ ↑p
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
h : WithSeminorms fun x ↦ p
⊢ (Finset.univ.sup fun x ↦ p).ball 0 1 ⊆ p.ball 0 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
h : WithSeminorms fun x ↦ p
i : Fin 1
⊢ ∀ x ∈ p.ball 0 1, p x < 1

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
x✝ : E
⊢ ‖0 x✝‖ ≤ 0 * ‖x✝‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝² : SeminormedAddCommGroup E
inst✝¹ : NontriviallyNormedField 𝕜
inst✝ : NormedSpace 𝕜 E
x : E
⊢ ‖(ContinuousLinearMap.id 𝕜 E) x‖ ≤ 1 * ‖x‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₂ : Type u_2
inst✝¹⁶ : NormedField 𝕜₂
E : Type u_3
F : Type u_4
G : Type u_5
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : TopologicalSpace E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜₂ F
𝕜₁ : Type u_6
inst✝¹¹ : NontriviallyNormedField 𝕜₁
σ : 𝕜₁ →+* 𝕜₂
inst✝¹⁰ : Module 𝕜₁ E
inst✝⁹ : AddCommGroup G
𝕜₃ : Type u_7
inst✝⁸ : NormedField 𝕜₃
inst✝⁷ : Module 𝕜₃ G
τ : 𝕜₃ →+* 𝕜₂
inst✝⁶ : RingHomSurjective τ
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
inst✝³ : ContinuousConstSMul 𝕜₂ F
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
𝔖 : Set (Set E)
h𝔖 : ∀ s ∈ 𝔖, IsVonNBounded 𝕜₁ s
B : G →ₛₗ[τ] E →SLᵤ[σ, 𝔖] F
hB : Continuous[instTopologicalSpaceProd, inst✝⁵] fun p ↦ (B p.1) p.2
S : Set E
hS : S ∈ 𝔖
U : Set F
hU : U ∈ 𝓝 0
⊢ (B 0.1) 0.2 = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Goal state before:
pp: R : Type u_1
𝕜₂ : Type u_3
𝕜₃ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
inst✝¹³ : Semiring R
inst✝¹² : NormedField 𝕜₂
inst✝¹¹ : NormedField 𝕜₃
inst✝¹⁰ : AddCommMonoid E
inst✝⁹ : Module R E
inst✝⁸ : TopologicalSpace E
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜₃ G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
σ₁₃ : R →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
f : E →SL[σ₁₃] F →SL[σ₂₃] G
c : R
x : E
y : F
⊢ (f (c • x)) y = σ₁₃ c • (f x) y

Tactic:
by rw [f.map_smulₛₗ, smul_apply]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap

Goal state before:
pp: case e'_4
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : Module 𝕜 E
inst✝¹⁰ : ContinuousSMul 𝕜 E
F : Type u_3
inst✝⁹ : AddCommGroup F
inst✝⁸ : UniformSpace F
inst✝⁷ : IsUniformAddGroup F
inst✝⁶ : Module 𝕜 F
𝕜' : Type u_4
inst✝⁵ : NontriviallyNormedField 𝕜'
inst✝⁴ : NormedAlgebra 𝕜' 𝕜
inst✝³ : Module 𝕜' E
inst✝² : IsScalarTower 𝕜' 𝕜 E
inst✝¹ : Module 𝕜' F
inst✝ : IsScalarTower 𝕜' 𝕜 F
s : Set E
⊢ IsVonNBounded 𝕜' s ↔ IsVonNBounded 𝕜 s

Tactic:
exact ⟨fun h ↦ h.extend_scalars _, fun h ↦ h.restrict_scalars _⟩

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case refine_1
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ Seminorm.IsBounded p (fun s ↦ s.sup p) LinearMap.id

pp: case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ Seminorm.IsBounded (fun s ↦ s.sup p) p LinearMap.id

Tactic:
· intro s
    refine ⟨s, 1, ?_⟩
    rw [one_smul]
    rfl

Goal state after:
case refine_2
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ Seminorm.IsBounded (fun s ↦ s.sup p) p LinearMap.id
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : SeminormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →ₛₗᵢ[σ₁₂] F
x : E
⊢ ‖f.toContinuousLinearMap x‖ ≤ 1 * ‖x‖

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
⊢ LocallyConvexSpace ℝ E

Tactic:
have := hp.topologicalAddGroup

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁶ : NormedField 𝕜
inst✝⁵ : NormedSpace ℝ 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Module ℝ E
inst✝¹ : IsScalarTower ℝ 𝕜 E
inst✝ : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ LocallyConvexSpace ℝ E
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f g h : α → β
M : Type u_5
G : Type u_6
inst✝ : AddGroupWithOne G
x✝ : ℕ
⊢ ↑↑x✝ = ↑x✝

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Countable ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ FirstCountableTopology E

Tactic:
have := hp.topologicalAddGroup

Goal state after:
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Countable ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
this : IsTopologicalAddGroup E
⊢ FirstCountableTopology E
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝ g✝ h✝ : α → β
inst✝² : Mul β
inst✝¹ : LE β
inst✝ : ExistsMulOfLE β
x y : l.Germ β
f g : α → β
h : f ≤ᶠ[l] g
⊢ ∃ c, ↑g = ↑f * c

Tactic:
choose c hc using fun x (hx : f x ≤ g x) ↦ exists_mul_of_le hx
    refine ⟨ofFun fun x ↦ if hx : f x ≤ g x then c x hx else f x, coe_eq.2 ?_⟩
    filter_upwards [h] with x hx
    rw [dif_pos hx, hc]

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h :
  ∀ (s : Set X),
    (∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X),
        Continuous[_, inst✝] f → IsOpen (f ⁻¹' s)) →
      IsOpen[inst✝] s
⊢ CompactlyCoherentSpace X

Tactic:
by
  refine of_isOpen fun A hA ↦ h A fun K _ _ f hf ↦ ?_
  specialize hA (range f) (isCompact_range hf)
  have := hA.preimage (hf.codRestrict mem_range_self)
  rwa [← preimage_comp] at this

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Bilinear

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
Eₗ : Type u_5
Fₗ : Type u_7
Gₗ : Type u_9
inst✝⁸ : SeminormedAddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : SeminormedAddCommGroup Fₗ
inst✝⁵ : SeminormedAddCommGroup Gₗ
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜 Eₗ
inst✝¹ : NormedSpace 𝕜 Fₗ
inst✝ : NormedSpace 𝕜 Gₗ
L : E →L[𝕜] Fₗ →L[𝕜] Gₗ
⊢ 1 * ‖L‖ = ‖L‖

Tactic:
rw [one_mul]

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝³ : TopologicalSpace α
inst✝² : UniformSpace β
K : Set α
V : Set (β × β)
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝¹ : CompleteSpace β
inst✝ : CompactlyCoherentSpace α
⊢ CompleteSpace C(α, β)

Tactic:
rw [completeSpace_iff_isComplete_range
    isUniformEmbedding_toUniformOnFunIsCompact.isUniformInducing,
    range_toUniformOnFunIsCompact, ← completeSpace_coe_iff_isComplete]
  exact (UniformOnFun.isClosed_setOfPred_continuous
    CompactlyCoherentSpace.isCoherentWith).completeSpace_coe

Goal state after:
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
this : μ (s ∩ t ∩ u) = 0
⊢ μ[|s][u | t] = μ[u | s ∩ t]

Tactic:
simp [cond_apply, *, ← Set.inter_assoc]

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb : ∀ᶠ (a : ι) in f, ↑(u a) ≤ b
⊢ 0 ≤ ↑b.toNNReal

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb : ∀ᶠ (a : ι) in f, ↑(u a) ≥ b
⊢ 0 ≤ ↑b.toNNReal

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
b : ℝ
hb : ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≤ a) → b ≤ a
⊢ 0 ≤ ↑b.toNNReal

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
inst✝ : f.NeBot
b : ℝ
hb₀ : 0 ≤ b
hb : ∀ (x : ℝ) (hx : 0 ≤ x), (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x hx)) → ↑(NNReal.mk b hb₀) ≤ ↑(NNReal.mk x hx)
x : ℝ
hx : ∀ᶠ (a : ι) in f, ↑(u a) ≤ x
⊢ 0 ≤ ↑(u ⋯.choose)

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
b : ℝ
hb : ∀ (a : ℝ), (∀ᶠ (a_1 : ι) in f, ↑(u a_1) ≥ a) → b ≥ a
⊢ ∀ᶠ (a : ι) in f, ↑(u a) ≥ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
c : ℝ
⊢ ∀ᶠ (x : ℝ) in map (fun i ↦ ↑(u i)) f, (fun x1 x2 ↦ x1 ≥ x2) x 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf : IsCoboundedUnder (fun x1 x2 ↦ x1 ≥ x2) f u
c r : ℝ
hr✝ : r < c
hr : r < 0
i : ι
⊢ 0 ≤ ↑(u i)

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: ι : Type u_1
f : Filter ι
u : ι → ℝ≥0
hf✝ : f.NeBot
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
this : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u
c r : ℝ
hr✝ : r < c
hr : r < 0
i : ι
⊢ 0 ≤ ↑(u i)

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : CommMonoid β
inst✝ : ContinuousMul β
ι : Type u_3
s : Finset ι
f : ι → C(α, β)
a : α
⊢ (∏ i ∈ s, f i) a = ∏ i ∈ s, (f i) a

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: 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, ENNReal.ofReal (u a) < y
x : ℝ
hx : x > ↑r
⊢ 0 ≤ ↑r

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: α : Type u_1
f : Filter α
inst✝ : f.NeBot
u : α → ℝ≥0∞
C : ℝ≥0
hf : ∀ᶠ (a : α) in f, u a ≤ ↑C
h₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
x : α
hx : u x ≤ ↑C
⊢ ↑C ≠ ∞

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case inr
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
h₁ : ∀ᶠ (a : α) in f, u a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
hf : f.NeBot
⊢ (limsup u f).toReal = limsup (fun a ↦ (u a).toReal) f

Tactic:
have : IsCoboundedUnder (· ≤ ·) f fun a ↦ (u a).toReal := .of_frequently_ge (a := 0) (by simpa)

Goal state after:
case inr
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
h₁ : ∀ᶠ (a : α) in f, u a ≠ ∞
h₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
hf : f.NeBot
this : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
⊢ (limsup u f).toReal = limsup (fun a ↦ (u a).toReal) f
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))
val✝ : Encodable ι
i : ι
s : Set (α i)
hs : s ∈ C i
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
this : univ.pi (update (fun i' ↦ iUnion (t i')) i (⋃ x, s)) = univ.pi fun k ↦ ⋃ j, update (fun i' ↦ t i' j) i s k
n : ι → ℕ
⊢ MeasurableSet (univ.pi fun a ↦ update (fun i' ↦ t i' (n a)) i s a)

Tactic:
apply measurableSet_generateFrom

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))
val✝ : Encodable ι
i : ι
s : Set (α i)
hs : s ∈ C i
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
this : univ.pi (update (fun i' ↦ iUnion (t i')) i (⋃ x, s)) = univ.pi fun k ↦ ⋃ j, update (fun i' ↦ t i' j) i s k
n : ι → ℕ
⊢ (univ.pi fun a ↦ update (fun i' ↦ t i' (n a)) i s a) ∈ univ.pi '' univ.pi C
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝ : MeasurableSpace Ω
s : Set Ω
⊢ (uniformOn s) ∅ = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: α : Type u_1
f : Filter α
u : α → ℝ≥0∞
h₁ : ∀ᶠ (a : α) in f, u a ≠ ∞
hf : f.NeBot
this : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal
r : ℝ
hr : 0 ≤ r
x : ℝ
hx : ∀ᶠ (a : α) in f, (fun x1 x2 ↦ x1 ≤ x2) (u a).toReal x
hx₀ : 0 ≤ x
⊢ ¬ENNReal.ofReal x = ∞

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
g : β → γ
hg : Continuous[inst✝¹, inst✝] g
f✝ : α →ₘ[μ] β
f : α → β
x✝ : AEStronglyMeasurable f μ
⊢ (comp g hg (mk f x✝)).toGerm = Germ.map g (mk f x✝).toGerm

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁹ : MeasurableSpace α
μ : Measure α
inst✝⁸ : TopologicalSpace β
inst✝⁷ : TopologicalSpace γ
inst✝⁶ : MeasurableSpace β
inst✝⁵ : BorelSpace β
inst✝⁴ : PseudoMetrizableSpace β
inst✝³ : PseudoMetrizableSpace γ
inst✝² : SecondCountableTopology γ
inst✝¹ : MeasurableSpace γ
inst✝ : OpensMeasurableSpace γ
g : β → γ
hg : Measurable g
f✝ : α →ₘ[μ] β
f : α → β
x✝ : AEStronglyMeasurable f μ
⊢ (compMeasurable g hg (mk f x✝)).toGerm = Germ.map g (mk f x✝).toGerm

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
g : β → γ → δ
hg : Continuous[instTopologicalSpaceProd, inst✝²] (uncurry g)
f₁✝ : α →ₘ[μ] β
f₂✝ : α →ₘ[μ] γ
f₁ : α → β
x✝¹ : AEStronglyMeasurable f₁ μ
f₂ : α → γ
x✝ : AEStronglyMeasurable f₂ μ
⊢ (comp₂ g hg (mk f₁ x✝¹) (mk f₂ x✝)).toGerm = Germ.map₂ g (mk f₁ x✝¹).toGerm (mk f₂ x✝).toGerm

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝² : ConditionallyCompleteLattice β
f : α → β
inst✝¹ : Nonempty α
inst✝ : MeasurableSingletonClass α
hf : BddAbove (range f)
⊢ ∀ (a : α), Measure.count {a} ≠ 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝² : ConditionallyCompleteLattice β
f : α → β
inst✝¹ : Nonempty α
inst✝ : MeasurableSingletonClass α
hf : BddBelow (range f)
⊢ ∀ (a : α), Measure.count {a} ≠ 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝¹⁴ : MeasurableSpace α
μ : Measure α
inst✝¹³ : TopologicalSpace δ
inst✝¹² : TopologicalSpace β
inst✝¹¹ : TopologicalSpace γ
inst✝¹⁰ : PseudoMetrizableSpace β
inst✝⁹ : MeasurableSpace β
inst✝⁸ : BorelSpace β
inst✝⁷ : PseudoMetrizableSpace γ
inst✝⁶ : SecondCountableTopologyEither β γ
inst✝⁵ : MeasurableSpace γ
inst✝⁴ : BorelSpace γ
inst✝³ : PseudoMetrizableSpace δ
inst✝² : SecondCountableTopology δ
inst✝¹ : MeasurableSpace δ
inst✝ : OpensMeasurableSpace δ
g : β → γ → δ
hg : Measurable (uncurry g)
f₁✝ : α →ₘ[μ] β
f₂✝ : α →ₘ[μ] γ
f₁ : α → β
x✝¹ : AEStronglyMeasurable f₁ μ
f₂ : α → γ
x✝ : AEStronglyMeasurable f₂ μ
⊢ (comp₂Measurable g hg (mk f₁ x✝¹) (mk f₂ x✝)).toGerm = Germ.map₂ g (mk f₁ x✝¹).toGerm (mk f₂ x✝).toGerm

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : CompleteLattice β
m : MeasurableSpace α
f : α → β
⊢ ⊥ ∈ {a | ∀ᶠ (n : β) in map f (ae 0), n ≤ a}

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : CompleteLattice β
inst✝ : MeasurableSingletonClass α
f : α → β
⊢ ∀ (a : α), Measure.count {a} ≠ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u t : Set Ω
hs : s.Finite
⊢ s = s ∩ u ∪ s ∩ uᶜ

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case nil
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ✝ : (i : ι) → Measure (α i)
δ : Type u_4
X : δ → Type u_5
inst✝ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
⊢ Measure (TProd X [])

Tactic:
exact dirac PUnit.unit

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
inst✝ : ENorm ε
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
⊢ eLpNorm f p μ = eLpNorm' f p.toReal μ

Tactic:
by simp [eLpNorm, hp_ne_zero, hp_ne_top]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
inst✝ : ENorm ε
μ : Measure α
f : α → ε
⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ 1 ∂μ) ^ (1 / 1) = ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
one_div_one,

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
inst✝ : ENorm ε
μ : Measure α
f : α → ε
⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ 1 ∂μ) ^ 1 = ∫⁻ (x : α), ‖f x‖ₑ ∂μ
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
s : Finset ι
f : ι → α →ₘ[μ] γ
x : α
⊢ (∏ i ∈ s, ↑(f i)) x = ∏ i ∈ s, ↑(f i) x

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
s : Set (α i)
hs : MeasurableSet s
⊢ i ∈ Finset.univ

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α → E
hℒp : MemLp f p μ
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
ε : ℝ≥0∞
hε : ε ≠ 0
h : ¬ε = ∞
⊢ ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
f : α → ε'
hℒp : MemLp f p μ
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
ε : ℝ≥0
hε : ε ≠ 0
⊢ ↑ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : SeminormedAddGroup E
s t : Set α
h : s ⊆ t
f : α → E
a : α
⊢ ‖s.indicator f a‖ ≤ ‖t.indicator f a‖

Tactic:
simp only [norm_indicator_eq_indicator_norm]

Goal state after:
α : Type u_1
E : Type u_2
inst✝ : SeminormedAddGroup E
s t : Set α
h : s ⊆ t
f : α → E
a : α
⊢ s.indicator (fun a ↦ ‖f a‖) a ≤ t.indicator (fun a ↦ ‖f a‖) a
Module: Mathlib.Analysis.Normed.Group.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : SeminormedAddGroup E
s : Set α
f : α → E
a : α
⊢ s.indicator (fun a ↦ ‖f a‖) a ≤ ‖f a‖

Tactic:
apply indicator_norm_le_norm_self

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

Goal state before:
pp: α : Type u_1
q : ℝ
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
inst✝ : MeasurableSpace α
f : α → ε
hq_neg : q < 0
⊢ eLpNorm' f q 0 = ∞

Tactic:
simp [eLpNorm', hq_neg]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.SMul

Goal state before:
pp: α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
c : 𝕜
f : α → F
⊢ essSup (fun x ↦ ‖c • f x‖ₑ) μ = ‖c‖ₑ * essSup (fun x ↦ ‖f x‖ₑ) μ

Tactic:
enorm_smul,

Goal state after:
α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
𝕜 : Type u_3
inst✝² : NormedDivisionRing 𝕜
inst✝¹ : Module 𝕜 F
inst✝ : NormSMulClass 𝕜 F
c : 𝕜
f : α → F
⊢ essSup (fun x ↦ ‖c‖ₑ * ‖f x‖ₑ) μ = ‖c‖ₑ * essSup (fun x ↦ ‖f x‖ₑ) μ