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

Goal state before:
pp: case refine_3.refine_1
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 : p.ball 0 1 ∈ 𝓝 0
h' : IsVonNBounded 𝕜 (p.ball 0 1)
s : Set E
hs : s ∈ 𝓝 0
⊢ s ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).filter

Tactic:
obtain ⟨c, hc, c_ne⟩ : ∃ (c : 𝕜), p.ball 0 1 ⊆ c • s ∧ c ≠ 0 :=
      ((h' hs).and (eventually_ne_cobounded 0)).exists

Goal state after:
case refine_3.refine_1
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 : p.ball 0 1 ∈ 𝓝 0
h' : IsVonNBounded 𝕜 (p.ball 0 1)
s : Set E
hs : s ∈ 𝓝 0
c : 𝕜
hc : p.ball 0 1 ⊆ c • s
c_ne : c ≠ 0
⊢ s ∈ (SeminormFamily.moduleFilterBasis fun x ↦ p).filter
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✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
⊢ IsLeast {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖} ‖f‖

Tactic:
refine IsClosed.isLeast_csInf ?_ bounds_nonempty bounds_bddBelow

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
⊢ IsClosed {c | 0 ≤ c ∧ ∀ (x : E), ‖f x‖ ≤ c * ‖x‖}
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✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
x : E
a b : ℝ
hf : ‖f‖ ≤ a
hx : ‖x‖ ≤ b
⊢ ‖f‖ * ‖x‖ ≤ a * b

Tactic:
gcongr; exact (opNorm_nonneg f).trans hf

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

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝²¹ : NormedField 𝕜₁✝
inst✝²⁰ : NormedField 𝕜₂✝
σ✝ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝¹⁹ : AddCommGroup E✝
inst✝¹⁸ : Module 𝕜₁✝ E✝
inst✝¹⁷ : TopologicalSpace E✝
inst✝¹⁶ : AddCommGroup F✝
inst✝¹⁵ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
inst✝¹⁴ : NormedField 𝕜₁
inst✝¹³ : NormedField 𝕜₂
inst✝¹² : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹¹ : RingHomCompTriple σ τ ρ
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜₁ E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₂ F
inst✝⁶ : AddCommGroup G
inst✝⁵ : Module 𝕜₃ G
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace F
inst✝² : TopologicalSpace G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
L : E →SL[σ] F
hL : MapsTo (fun x ↦ ⇑L '' x) 𝔖 𝔗
⊢ Continuous[UniformConvergenceCLM.instTopologicalSpace τ G 𝔗, UniformConvergenceCLM.instTopologicalSpace ρ G 𝔖] fun f ↦
    f ∘SL L

Tactic:
let : UniformSpace G := IsTopologicalAddGroup.rightUniformSpace G

Goal state after:
𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝²¹ : NormedField 𝕜₁✝
inst✝²⁰ : NormedField 𝕜₂✝
σ✝ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝¹⁹ : AddCommGroup E✝
inst✝¹⁸ : Module 𝕜₁✝ E✝
inst✝¹⁷ : TopologicalSpace E✝
inst✝¹⁶ : AddCommGroup F✝
inst✝¹⁵ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
inst✝¹⁴ : NormedField 𝕜₁
inst✝¹³ : NormedField 𝕜₂
inst✝¹² : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹¹ : RingHomCompTriple σ τ ρ
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜₁ E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₂ F
inst✝⁶ : AddCommGroup G
inst✝⁵ : Module 𝕜₃ G
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace F
inst✝² : TopologicalSpace G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
L : E →SL[σ] F
hL : MapsTo (fun x ↦ ⇑L '' x) 𝔖 𝔗
this : UniformSpace G := IsTopologicalAddGroup.rightUniformSpace G
⊢ Continuous[UniformConvergenceCLM.instTopologicalSpace τ G 𝔗, UniformConvergenceCLM.instTopologicalSpace ρ G 𝔖] fun f ↦
    f ∘SL L
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝²¹ : NormedField 𝕜₁✝
inst✝²⁰ : NormedField 𝕜₂✝
σ✝ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝¹⁹ : AddCommGroup E✝
inst✝¹⁸ : Module 𝕜₁✝ E✝
inst✝¹⁷ : TopologicalSpace E✝
inst✝¹⁶ : AddCommGroup F✝
inst✝¹⁵ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
inst✝¹⁴ : NormedField 𝕜₁
inst✝¹³ : NormedField 𝕜₂
inst✝¹² : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹¹ : RingHomCompTriple σ τ ρ
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁰ : AddCommGroup E
inst✝⁹ : Module 𝕜₁ E
inst✝⁸ : AddCommGroup F
inst✝⁷ : Module 𝕜₂ F
inst✝⁶ : AddCommGroup G
inst✝⁵ : Module 𝕜₃ G
inst✝⁴ : TopologicalSpace E
inst✝³ : TopologicalSpace F
inst✝² : TopologicalSpace G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜₃ G
L : E →SL[σ] F
hL : MapsTo (fun x ↦ ⇑L '' x) 𝔖 𝔗
this✝ : UniformSpace G := IsTopologicalAddGroup.rightUniformSpace G
this : IsUniformAddGroup G
⊢ Continuous[UniformConvergenceCLM.instTopologicalSpace τ G 𝔗, UniformOnFun.topologicalSpace E G 𝔖]
    ((⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) ∘ fun f ↦ f ∘SL L)

Tactic:
exact (UniformOnFun.precomp_uniformContinuous hL).continuous.comp
        (UniformConvergenceCLM.isEmbedding_coeFn _ _ _).continuous

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_5
G : Type u_7
inst✝¹¹ : SeminormedAddCommGroup E
inst✝¹⁰ : SeminormedAddCommGroup F
inst✝⁹ : SeminormedAddCommGroup G
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : RingHomIsometric σ₂₃
h : F →SL[σ₂₃] G
f : E →SL[σ₁₂] F
x : E
⊢ ‖(h ∘SL f) x‖ ≤ ‖h‖ * (‖f‖ * ‖x‖)

Tactic:
exact h.le_opNorm_of_le (f.le_opNorm x)

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

Goal state before:
pp: 𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝²³ : NormedField 𝕜₁✝
inst✝²² : NormedField 𝕜₂✝
σ✝ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝²¹ : AddCommGroup E✝
inst✝²⁰ : Module 𝕜₁✝ E✝
inst✝¹⁹ : TopologicalSpace E✝
inst✝¹⁸ : AddCommGroup F✝
inst✝¹⁷ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
inst✝³ : IsTopologicalAddGroup F
inst✝² : IsTopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
⊢ Continuous[UniformConvergenceCLM.instTopologicalSpace σ F 𝔖, UniformConvergenceCLM.instTopologicalSpace ρ G 𝔖] fun f ↦
    L ∘SL f

Tactic:
let : UniformSpace G := IsTopologicalAddGroup.rightUniformSpace G

Goal state after:
𝕜₁✝ : Type u_1
𝕜₂✝ : Type u_2
inst✝²³ : NormedField 𝕜₁✝
inst✝²² : NormedField 𝕜₂✝
σ✝ : 𝕜₁✝ →+* 𝕜₂✝
E✝ : Type u_3
F✝ : Type u_4
G✝ : Type u_5
inst✝²¹ : AddCommGroup E✝
inst✝²⁰ : Module 𝕜₁✝ E✝
inst✝¹⁹ : TopologicalSpace E✝
inst✝¹⁸ : AddCommGroup F✝
inst✝¹⁷ : Module 𝕜₂✝ F✝
𝕜₁ : Type u_6
𝕜₂ : Type u_7
𝕜₃ : Type u_8
inst✝¹⁶ : NormedField 𝕜₁
inst✝¹⁵ : NormedField 𝕜₂
inst✝¹⁴ : NormedField 𝕜₃
σ : 𝕜₁ →+* 𝕜₂
τ : 𝕜₂ →+* 𝕜₃
ρ : 𝕜₁ →+* 𝕜₃
inst✝¹³ : RingHomCompTriple σ τ ρ
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹² : AddCommGroup E
inst✝¹¹ : Module 𝕜₁ E
inst✝¹⁰ : AddCommGroup F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜₃ G
inst✝⁶ : TopologicalSpace E
inst✝⁵ : TopologicalSpace F
inst✝⁴ : TopologicalSpace G
𝔖 : Set (Set E)
𝔗 : Set (Set F)
inst✝³ : IsTopologicalAddGroup F
inst✝² : IsTopologicalAddGroup G
inst✝¹ : ContinuousConstSMul 𝕜₃ G
inst✝ : ContinuousConstSMul 𝕜₂ F
L : F →SL[τ] G
this : UniformSpace G := IsTopologicalAddGroup.rightUniformSpace G
⊢ Continuous[UniformConvergenceCLM.instTopologicalSpace σ F 𝔖, UniformConvergenceCLM.instTopologicalSpace ρ G 𝔖] fun f ↦
    L ∘SL f
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
r : ℝ≥0
hr : r < ‖f‖₊
y : E
hy : r * ‖f y‖₊⁻¹ < ‖y‖₊⁻¹
hy' : ‖y‖₊ ≠ 0
hfy : ‖f y‖₊ ≠ 0
k : 𝕜
hk₁ : r * ‖f y‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < ‖y‖₊⁻¹
⊢ r < ‖σ₁₂ k • f y‖₊

Tactic:
nnnorm_smul,

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
r : ℝ≥0
hr : r < ‖f‖₊
y : E
hy : r * ‖f y‖₊⁻¹ < ‖y‖₊⁻¹
hy' : ‖y‖₊ ≠ 0
hfy : ‖f y‖₊ ≠ 0
k : 𝕜
hk₁ : r * ‖f y‖₊⁻¹ < ‖k‖₊
hk₂ : ‖k‖₊ < ‖y‖₊⁻¹
⊢ r < ‖σ₁₂ k‖₊ * ‖f y‖₊
Module: Mathlib.Analysis.Normed.Operator.NNNorm

Goal state before:
pp: case pos
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : DenselyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
r : ℝ
hr : r < ‖f‖
hr₀ : r < 0
⊢ ∃ x, ‖x‖ < 1 ∧ r < ‖f x‖

Tactic:
exact ⟨0, by simpa using hr₀⟩

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

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁸ : NormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : RingHomIsometric σ₁₂
inst✝¹ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
inst✝ : TopologicalSpace E
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
⊢ WithSeminorms (q.comp f)

Tactic:
rw [hf.eq_induced]

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁸ : NormedField 𝕜
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : NormedField 𝕜₂
inst✝⁴ : AddCommGroup F
inst✝³ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝² : RingHomIsometric σ₁₂
inst✝¹ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
inst✝ : TopologicalSpace E
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
⊢ WithSeminorms (q.comp f)
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:
simp [Measure.restrict_toMeasurable hnt]

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
x : ℝ
x✝ : 0 ≤ x
⊢ (∀ᶠ (a : ι) in f, ↑(u a) ≤ ↑(NNReal.mk x x✝)) → ↑(NNReal.mk ↑b.toNNReal ⋯) ≤ ↑(NNReal.mk x x✝)

Tactic:
by simpa [*] using hb _

Goal state after:
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:
· simp [ofReal_lt_coe_iff, *]

Goal state after:
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
i : ι
⊢ ∀ t ∈ (preimage fun b ↦ b i) '' C i, MeasurableSet t

Tactic:
rintro _ ⟨s, hs, rfl⟩

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
i : ι
s : Set (α i)
hs : s ∈ C i
⊢ MeasurableSet ((fun b ↦ b i) ⁻¹' s)
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

Tactic:
subst n
    exact ⟨_, (v x • (1 : s) : s).prop, mul_one _, mul_one _⟩

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

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Subsingleton α
s₁ s₂ : Subalgebra R C(α, R)
⊢ s₁ = s₂

Tactic:
cases isEmpty_or_nonempty α

Goal state after:
case inl
α : Type u_1
inst✝⁴ : TopologicalSpace α
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Subsingleton α
s₁ s₂ : Subalgebra R C(α, R)
h✝ : IsEmpty α
⊢ s₁ = s₂

case inr
α : Type u_1
inst✝⁴ : TopologicalSpace α
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Subsingleton α
s₁ s₂ : Subalgebra R C(α, R)
h✝ : Nonempty α
⊢ s₁ = s₂
Module: Mathlib.MeasureTheory.MeasurableSpace.Pi

Goal state before:
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
⊢ ∀ t ∈ univ.pi '' univ.pi C, MeasurableSet t

Tactic:
rintro _ ⟨s, hs, rfl⟩

Goal state after:
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
s : (i : ι) → Set (α i)
hs : s ∈ univ.pi C
⊢ MeasurableSet (univ.pi s)
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α →ₘ[μ] ℝ≥0∞
⊢ ∫⁻ (a : α), ↑f a ∂μ = f.lintegral

Tactic:
← lintegral_mk (hf := f.aestronglyMeasurable),

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α →ₘ[μ] ℝ≥0∞
⊢ (mk ↑f ⋯).lintegral = f.lintegral
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)
h₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i, (μ i) (s i)
h4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s
⊢ ∀ s ∈ univ.pi '' univ.pi C, (Measure.pi μ) s = μν s

Tactic:
rintro _ ⟨s, hs, rfl⟩

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝¹ : Fintype ι
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hC : ∀ (i : ι), generateFrom (C i) = inst✝ i
h2C : ∀ (i : ι), IsPiSystem (C i)
h3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
μν : Measure ((i : ι) → α i)
h₁ : ∀ (s : (i : ι) → Set (α i)), (∀ (i : ι), s i ∈ C i) → μν (univ.pi s) = ∏ i, (μ i) (s i)
h4C : ∀ (i : ι), ∀ s ∈ C i, MeasurableSet s
s : (i : ι) → Set (α i)
hs : s ∈ univ.pi C
⊢ (Measure.pi μ) (univ.pi s) = μν (univ.pi s)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
p : ι → Prop
inst✝ : DecidablePred p
e : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=
  (MeasurableEquiv.piEquivPiSubtypeProd α p).symm
s : (i : ι) → Set (α i)
x✝ : ∀ (i : ι), MeasurableSet (s i)
this : ⇑e ⁻¹' univ.pi s = (univ.pi fun i ↦ s ↑i) ×ˢ univ.pi fun i ↦ s ↑i
⊢ (Measure.map (⇑e) ((Measure.pi fun i ↦ μ ↑i).prod (Measure.pi fun i ↦ μ ↑i))) (univ.pi s) = ∏ i, (μ i) (s i)

Tactic:
rw [e.map_apply, this, prod_prod, pi_pi, pi_pi]

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
p : ι → Prop
inst✝ : DecidablePred p
e : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=
  (MeasurableEquiv.piEquivPiSubtypeProd α p).symm
s : (i : ι) → Set (α i)
x✝ : ∀ (i : ι), MeasurableSet (s i)
this : ⇑e ⁻¹' univ.pi s = (univ.pi fun i ↦ s ↑i) ×ˢ univ.pi fun i ↦ s ↑i
⊢ (∏ i, (μ ↑i) (s ↑i)) * ∏ i, (μ ↑i) (s ↑i) = ∏ i, (μ i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
inst✝² : Fintype ι
inst✝¹ : Fintype ι'
X : ι ⊕ ι' → Type u_4
m : (i : ι ⊕ ι') → MeasurableSpace (X i)
μ : (i : ι ⊕ ι') → Measure (X i)
inst✝ : ∀ (i : ι ⊕ ι'), SigmaFinite (μ i)
⊢ Measure.map (⇑(MeasurableEquiv.sumPiEquivProdPi X).symm)
      ((Measure.pi fun i ↦ μ (Sum.inl i)).prod (Measure.pi fun i ↦ μ (Sum.inr i))) =
    Measure.pi μ

Tactic:
by
    refine (pi_eq fun s _ => ?_).symm
    simp_rw [MeasurableEquiv.map_apply, MeasurableEquiv.coe_sumPiEquivProdPi_symm,
      Equiv.sumPiEquivProdPi_symm_preimage_univ_pi, Measure.prod_prod, Measure.pi_pi,
      Fintype.prod_sum_type]

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
𝕜 : Type u_5
inst✝ : RCLike 𝕜
f : α → 𝕜
hf : MemLp f p μ
⊢ MemLp (fun x ↦ RCLike.im (f x)) p μ

Tactic:
have : ∀ x, ‖RCLike.im (f x)‖ ≤ 1 * ‖f x‖ := by
    intro x
    rw [one_mul]
    exact RCLike.norm_im_le_norm (f x)
  refine hf.of_le_mul ?_ (Eventually.of_forall this)
  exact RCLike.continuous_im.comp_aestronglyMeasurable hf.1

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
⊢ (1 + s) ^ p < 1 + p * s

Tactic:
have hs2 : 0 < 1 + p * s := by
    rw [← neg_lt_iff_pos_add']
    rcases lt_or_gt_of_ne hs' with h | h
    · exact hs.trans (lt_mul_of_lt_one_left h hp2)
    · exact neg_one_lt_zero.trans (mul_pos hp1 h)

Goal state after:
case inr
s : ℝ
hs✝ : -1 ≤ s
hs' : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
⊢ (1 + s) ^ p < 1 + p * s
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : Ring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
f g : ι → α
⊢ Monovary f g ↔ ∀ (i j : ι), 0 ≤ (f j - f i) * (g j - g i)

Tactic:
by
  simp only [smul_eq_mul, monovary_iff_forall_smul_nonneg]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case refine_2
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
h : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p
⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p

Tactic:
simpa [Fin.sum_univ_succ] using h

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case refine_2
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ ((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ (∑ i ∈ s, w i * z i ^ p).toNNReal

Tactic:
toNNReal_sum h_top_rpow,

Goal state after:
case refine_2
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h_top : ∀ a ∈ s, w a * z a ≠ ∞
h_top_rpow : ∀ a ∈ s, w a * z a ^ p ≠ ∞
⊢ ((∑ i ∈ s, w i * z i) ^ p).toNNReal ≤ ∑ a ∈ s, (w a * z a ^ p).toNNReal
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case refine_2
w₁ w₂ z₁ z₂ : ℝ≥0∞
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
h : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p
⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p

Tactic:
simpa [Fin.sum_univ_succ] using h

Goal state after:
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p✝ : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ = 1
h2p : ∀ i ∈ s, p i = 0
⊢ (∫⁻ (a : α), f i₀ a ∂μ) ^ p i₀ * ∏ i ∈ s, 1 = (∫⁻ (a : α), f i₀ a ∂μ) ^ p i₀ * ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
by
            congr 1
            apply prod_congr rfl fun i hi ↦ by rw [h2p i hi, ENNReal.rpow_zero]

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

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
h_rw

Goal state after:
case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
g : α → ℝ≥0∞ := fun x ↦ 1
h_rw : ∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ = ∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ
⊢ (∫⁻ (a : α), (‖f a‖ₑ * g a) ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q) * μ Set.univ ^ (1 / p - 1 / q)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ ∫⁻ (a : α), (f a + g a) ^ p ∂μ ≤ ∫⁻ (a : α), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p ∂μ

Tactic:
refine lintegral_mono fun a => ?_

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
a : α
⊢ (f a + g a) ^ p ≤ 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p