mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
h_Iio : s ∩ Iio a = ∅
h_Ioi : (s ∩ Ioi a).Nonempty
⊢ 𝓝[s \ {a}] a = 𝓝[>] a

Tactic:
sdiff_eq,

Goal state after:
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
h_Iio : s ∩ Iio a = ∅
h_Ioi : (s ∩ Ioi a).Nonempty
⊢ 𝓝[s ∩ {a}ᶜ] a = 𝓝[>] a
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: case inr
𝕜 : Type u_4
inst✝² : LinearOrder 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
a : 𝕜
h : ∀ x ∈ closure[inst✝¹] s, Ioo x a ⊆ s
b : 𝕜
hbs : b ∈ s
hba : b ∈ Iio a
⊢ ∀ᶠ (x : 𝕜) in 𝓝[s ∩ Iio a] a, s \ {a} ∈ 𝓝 x

Tactic:
have : Ioo b a ⊆ s := h b (subset_closure hbs)

Goal state after:
case inr
𝕜 : Type u_4
inst✝² : LinearOrder 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
a : 𝕜
h : ∀ x ∈ closure[inst✝¹] s, Ioo x a ⊆ s
b : 𝕜
hbs : b ∈ s
hba : b ∈ Iio a
this : Ioo b a ⊆ s
⊢ ∀ᶠ (x : 𝕜) in 𝓝[s ∩ Iio a] a, s \ {a} ∈ 𝓝 x
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
| 𝓝[s \ {a}] a

Tactic:
sdiff_eq,

Goal state after:
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
| 𝓝[s ∩ {a}ᶜ] a
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ ∀ x ∈ s ∪ s', x ∉ s' → (if x ∈ s' then b * w' x else 0) • v x = 0

pp: case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ ∀ x ∈ s ∪ s', x ∉ s → (if x ∈ s then a * w x else 0) • v x = 0

Tactic:
· intro i _ hi'
        simp [hi']

Goal state after:
case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ ∀ x ∈ s ∪ s', x ∉ s → (if x ∈ s then a * w x else 0) • v x = 0
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_2.refine_2
R : Type u_1
E : Type u_3
ι : Type u_5
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
v : ι → E
s : Finset ι
w : ι → R
hw₀ : ∀ i ∈ s, 0 ≤ w i
hw₁ : s.sum w = 1
s' : Finset ι
w' : ι → R
hw₀' : ∀ i ∈ s', 0 ≤ w' i
hw₁' : s'.sum w' = 1
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
W : ι → R := fun i ↦ (if i ∈ s then a * w i else 0) + if i ∈ s' then b * w' i else 0
hW₁ : (s ∪ s').sum W = 1
⊢ ∀ x ∈ s ∪ s', x ∉ s → (if x ∈ s then a * w x else 0) • v x = 0

Tactic:
· intro i _ hi'
        simp [hi']

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Convex

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
hs : Convex ℝ s
δ : ℝ
⊢ Convex ℝ (Metric.thickening δ s)

Tactic:
rw [← add_ball_zero]

Goal state after:
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
hs : Convex ℝ s
δ : ℝ
⊢ Convex ℝ (s + ball 0 δ)
Module: Mathlib.Analysis.Convex.Jensen

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
β : Type u_4
ι : Type u_5
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : AddCommGroup β
inst✝⁴ : PartialOrder β
inst✝³ : IsOrderedAddMonoid β
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜 β
inst✝ : IsStrictOrderedModule 𝕜 β
s : Set E
f : E → β
w : ι → 𝕜
p : ι → E
hf : StrictConvexOn 𝕜 s f
j k : ι
hjk : p j ≠ p k
u : Finset ι
hj✝ : j ∉ u
hk✝ : k ∉ u
h₀ : ∀ i ∈ cons j (cons k u hk✝) ⋯, 0 < w i
h₁ : ∑ i ∈ cons j (cons k u hk✝) ⋯, w i = 1
hmem : ∀ i ∈ cons j (cons k u hk✝) ⋯, p i ∈ s
hj : j ∈ cons j (cons k u hk✝) ⋯
hk : k ∈ cons j (cons k u hk✝) ⋯
this✝¹ : k ∈ (cons j (cons k u hk✝) ⋯).erase j
this✝ : 0 < w j
this : 0 < w k
c : 𝕜 := w j + w k
hc : w j / c + w k / c = 1
⊢ c + ∑ i ∈ u, w i = 1

Tactic:
by simpa [-cons_eq_insert, ← add_assoc] using h₁

Goal state after:
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s t : Set E
⊢ (convexHull R) (s + -t) = (convexHull R) s + -(convexHull R) t

Tactic:
convexHull_add,

Goal state after:
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s t : Set E
⊢ (convexHull R) s + (convexHull R) (-t) = (convexHull R) s + -(convexHull R) t
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_1
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
s : Set ι
t✝ : (i : ι) → Set (E i)
x : (i : ι) → E i
val✝ : Fintype ι
t : (i : ι) → Set (E i)
κ : ι → Type
x✝ : (i : ι) → Fintype (κ i)
w : (i : ι) → κ i → 𝕜
f : (i : ι) → κ i → E i
hw₀ : ∀ (i : ι) (i_1 : κ i), 0 ≤ w i i_1
hw₁ : ∀ (i : ι), ∑ i_1, w i i_1 = 1
hft : ∀ (i : ι) (i_1 : κ i), f i i_1 ∈ t i
hf : ∀ (i : ι), ∑ i_1, w i i_1 • f i i_1 = x i
⊢ ∑ i, ∏ i_1, w i_1 (i i_1) = 1

Tactic:
← Fintype.prod_sum

Goal state after:
case refine_1
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
inst✝⁵ : Finite ι
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : (i : ι) → AddCommGroup (E i)
inst✝ : (i : ι) → Module 𝕜 (E i)
s : Set ι
t✝ : (i : ι) → Set (E i)
x : (i : ι) → E i
val✝ : Fintype ι
t : (i : ι) → Set (E i)
κ : ι → Type
x✝ : (i : ι) → Fintype (κ i)
w : (i : ι) → κ i → 𝕜
f : (i : ι) → κ i → E i
hw₀ : ∀ (i : ι) (i_1 : κ i), 0 ≤ w i i_1
hw₁ : ∀ (i : ι), ∑ i_1, w i i_1 = 1
hft : ∀ (i : ι) (i_1 : κ i), f i i_1 ∈ t i
hf : ∀ (i : ι), ∑ i_1, w i i_1 • f i i_1 = x i
⊢ ∏ i, ∑ j, w i j = 1
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_11
E : Type u_12
inst✝² : NormedField 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
⊢ WithSeminorms fun x ↦ normSeminorm 𝕜 E

Tactic:
rw [SeminormFamily.withSeminorms_iff_nhds_eq_iInf, iInf_const, coe_normSeminorm,
    comap_norm_nhds_zero]

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

Goal state before:
pp: 𝕝 : Type u_4
𝕝₂ : Type u_5
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕝
inst✝⁷ : Module 𝕝 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : NormedField 𝕝₂
inst✝⁴ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝³ : RingHomIsometric τ₁₂
q : SeminormFamily 𝕝₂ F ι'
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : ∀ (i : ι'), Continuous[inst✝², _] ⇑((q i).comp f)
this : IsTopologicalAddGroup F
⊢ ContinuousAt (⇑f) 0

Tactic:
simp_rw [ContinuousAt, f.map_zero, q.withSeminorms_iff_nhds_eq_iInf.mp hq, Filter.tendsto_iInf,
    Filter.tendsto_comap_iff]

Goal state after:
𝕝 : Type u_4
𝕝₂ : Type u_5
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕝
inst✝⁷ : Module 𝕝 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : NormedField 𝕝₂
inst✝⁴ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝³ : RingHomIsometric τ₁₂
q : SeminormFamily 𝕝₂ F ι'
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : ∀ (i : ι'), Continuous[inst✝², _] ⇑((q i).comp f)
this : IsTopologicalAddGroup F
⊢ ∀ (i : ι'), Tendsto (⇑(q i) ∘ ⇑f) (𝓝 0) (𝓝 0)
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 : AddCommGroup 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
x x' : E
y : F
⊢ (f x - f x') y = (f x) y - (f x') y

Tactic:
sub_apply

Goal state after:
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 : AddCommGroup 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
x x' : E
y : F
⊢ (f x) y - (f x') y = (f x) y - (f x') y
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
s : Finset ι
⊢ ((fun s ↦ s.sup p) s).comp LinearMap.id ≤ 1 • s.sup p

Tactic:
rw [one_smul]

Goal state after:
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
s : Finset ι
⊢ ((fun s ↦ s.sup p) s).comp LinearMap.id ≤ s.sup p
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✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ ((fun i ↦ (Finset.Iic i).sup p) i).comp LinearMap.id ≤ 1 • (Finset.Iic i).sup p

Tactic:
rw [one_smul]

Goal state after:
case refine_1
𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
i : ι
⊢ ((fun i ↦ (Finset.Iic i).sup p) i).comp LinearMap.id ≤ (Finset.Iic i).sup p
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: case neg
𝕜 : Type u_2
F : Type u_7
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
q : Seminorm 𝕜 F
hq : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑q
hq' : Tendsto (⇑q) (𝓝 0) (𝓝 0)
ε : ℝ
ε_pos : 0 < ε
hε : {y | ‖y‖ < ε} ⊆ ⇑q ⁻¹' Iio 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < ‖c‖ / ε
x✝ : F
hx : ¬‖x✝‖ = 0
x : F
hle : ε / ‖c‖ ≤ (normSeminorm 𝕜 F) x
hlt : (normSeminorm 𝕜 F) x < ε
⊢ 1 ≤ ‖c‖ / ε * (normSeminorm 𝕜 F) x

Tactic:
rwa [← div_le_iff₀' this, one_div_div]

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

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
f : C(α, β)
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
inst✝ : WeaklyLocallyCompactSpace α
⊢ Tendsto F p (𝓝 f) ↔ TendstoLocallyUniformly (fun i a ↦ (F i) a) (⇑f) p

Tactic:
refine ⟨fun h V hV x ↦ ?_, tendsto_of_tendstoLocallyUniformly⟩
  rw [tendsto_iff_forall_isCompact_tendstoUniformlyOn] at h
  obtain ⟨n, hn₁, hn₂⟩ := exists_compact_mem_nhds x
  exact ⟨n, hn₂, h n hn₁ V hV⟩

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]

Goal state after:
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 ↑{f | Continuous[inst✝³, inst✝².toTopologicalSpace] f}
Module: Mathlib.Probability.ConditionalProbability

Goal state before:
pp: Ω : Type u_1
m : MeasurableSpace Ω
s : Set Ω
hms : MeasurableSet s
μ : Measure Ω
t : Set Ω
⊢ (μ s)⁻¹ • (μ.restrict s) t = (μ s)⁻¹ * μ (s ∩ t)

Tactic:
Measure.restrict_apply' hms,

Goal state after:
Ω : Type u_1
m : MeasurableSpace Ω
s : Set Ω
hms : MeasurableSet s
μ : Measure Ω
t : Set Ω
⊢ (μ s)⁻¹ • μ (t ∩ s) = (μ s)⁻¹ * μ (s ∩ t)
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

Tactic:
rintro h (_ | x) hx

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

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
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: case inr.inr.mpr
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
r : ℝ
hr : 0 ≤ r
⊢ (∀ y > r, ∀ᶠ (a : α) in f, (u a).toReal < y) → ∀ y > ENNReal.ofReal r, ∀ᶠ (a : α) in f, u a < y

Tactic:
rintro h (_ | x) hx

Goal state after:
case inr.inr.mpr.none
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
r : ℝ
hr : 0 ≤ r
h : ∀ y > r, ∀ᶠ (a : α) in f, (u a).toReal < y
hx : none > ENNReal.ofReal r
⊢ ∀ᶠ (a : α) in f, u a < none

case inr.inr.mpr.some
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
r : ℝ
hr : 0 ≤ r
h : ∀ y > r, ∀ᶠ (a : α) in f, (u a).toReal < y
x : ℝ≥0
hx : some x > ENNReal.ofReal r
⊢ ∀ᶠ (a : α) in f, u a < some x
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s : Set Ω
hs : s.Finite
hs' : s.Nonempty
⊢ (uniformOn s) s = 1

Tactic:
rw [uniformOn, cond_apply hs.measurableSet, Set.inter_self, ENNReal.inv_mul_cancel]
  · rwa [Measure.count_ne_zero_iff]
  · exact (Measure.count_apply_lt_top.2 hs).ne

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

Goal state before:
pp: ι : Type u_1
inst✝¹ : Fintype ι
X : ι → Type u_4
Y : ι → Type u_5
mX : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝ : (i : ι) → MeasurableSpace (Y i)
f : (i : ι) → X i → Y i
hμ : ∀ (i : ι), SigmaFinite (map (f i) (μ i))
hf : ∀ (i : ι), AEMeasurable (f i) (μ i)
this : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (Y i)
hs : ∀ (i : ι), MeasurableSet (s i)
⊢ AEMeasurable (fun x i ↦ f i (x i)) (Measure.pi μ)

Tactic:
exact aemeasurable_pi_lambda _
      fun i ↦ (hf i).comp_quasiMeasurePreserving (quasiMeasurePreserving_eval _ i)

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

Goal state before:
pp: α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ
hp : 0 < p
⊢ ((∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p ≤ (c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p

Tactic:
ENNReal.smul_def,

Goal state after:
α : Type u_1
F : Type u_3
G : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
g : α → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊
p : ℝ
hp : 0 < p
⊢ ((∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p ≤ (↑c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
c : E
hμsc : c = 0 ∨ μ s ≠ ∞
⊢ MemLp (s.indicator fun x ↦ c) p μ

Tactic:
memLp_indicator_iff_restrict hs

Goal state after:
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
c : E
hμsc : c = 0 ∨ μ s ≠ ∞
⊢ MemLp (fun x ↦ c) p (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
⊢ ((∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p ≤ (c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p

Tactic:
ENNReal.smul_def,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_5
ε' : Type u_6
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
f : α → ε
g : α → ε'
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ↑c * ‖g x‖ₑ
p : ℝ
hp : 0 < p
⊢ ((∫⁻ (a : α), ‖f a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p ≤ (↑c • (∫⁻ (a : α), ‖g a‖ₑ ^ p ∂μ) ^ (1 / p)) ^ p
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case hq0_lt
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hfp : eLpNorm f p μ < ∞
⊢ 0 < p.toReal

pp: case hfq
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hfp : eLpNorm f p μ < ∞
⊢ eLpNorm' f p.toReal μ < ∞

Tactic:
· exact ENNReal.toReal_pos hp_ne_zero hp_ne_top

Goal state after:
case hfq
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hfp : eLpNorm f p μ < ∞
⊢ eLpNorm' f p.toReal μ < ∞
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), IsFiniteMeasureOnCompacts (μ i)
K : Set ((i : ι) → α i)
hK : IsCompact K
⊢ (Measure.pi μ) (univ.pi fun j ↦ eval j '' K) < ∞

Tactic:
Measure.pi_pi

Goal state after:
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝⁴ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → TopologicalSpace (α i)
inst✝ : ∀ (i : ι), IsFiniteMeasureOnCompacts (μ i)
K : Set ((i : ι) → α i)
hK : IsCompact K
⊢ ∏ i, (μ i) (eval i '' K) < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: α : Type u_1
F : Type u_3
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
hp : p ≠ 0
hp' : p ≠ ∞
f : α → F
C : ℝ≥0
s : Set α
hs : MeasurableSet s
hf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊
⊢ C • μ s ^ (1 / p.toReal) ≤ eLpNorm f p μ

Tactic:
ENNReal.smul_def,

Goal state after:
α : Type u_1
F : Type u_3
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
hp : p ≠ 0
hp' : p ≠ ∞
f : α → F
C : ℝ≥0
s : Set α
hs : MeasurableSet s
hf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊
⊢ ↑C • μ s ^ (1 / p.toReal) ≤ eLpNorm f p μ
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case inl.inl
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) < (f y - f a) / (y - a)

pp: case inl.inr
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : a < y
⊢ (f x - f a) / (x - a) < (f y - f a) / (y - a)

Tactic:
· convert! hf.secant_strict_mono_aux3 hx ha hxy hya using 1 <;> rw [← neg_div_neg_eq] <;>
        simp

Goal state after:
case inl.inr
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : a < y
⊢ (f x - f a) / (x - a) < (f y - f a) / (y - a)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
⊢ p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)

Tactic:
· have q : 0 < z - y := by rwa [sub_pos]
    rw [lt_div_iff₀ q, ← div_lt_div_iff_of_pos_right hy', _root_.sub_div, div_self hy'.ne',
      ← div_rpow hz hy.le, lt_sub_iff_add_lt', ← add_sub_cancel_right (z / y) 1, add_comm _ 1,
      add_sub_assoc, ← div_mul_eq_mul_div, mul_div_assoc, ← rpow_sub hy, sub_sub_cancel_left,
      rpow_neg_one, mul_assoc, ← div_eq_inv_mul, _root_.sub_div, div_self hy.ne']
    apply one_add_mul_self_lt_rpow_one_add _ _ hp
    · rw [le_sub_iff_add_le, neg_add_cancel, div_nonneg_iff]
      exact Or.inl ⟨hz, hy.le⟩
    · rw [sub_ne_zero]
      exact ((one_lt_div hy).mpr hyz).ne'

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp : ∞ ≠ 0
hbdd : ∀ (n : ℕ), eLpNormEssSup (f n) μ ≤ ↑R
this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖f n x‖ₑ < ↑R + 1
⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞

Tactic:
rw [← ae_all_iff] at this

Goal state after:
case pos
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp : ∞ ≠ 0
hbdd : ∀ (n : ℕ), eLpNormEssSup (f n) μ ≤ ↑R
this : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), ‖f i a‖ₑ < ↑R + 1
⊢ ∀ᵐ (x : α) ∂μ, liminf (fun n ↦ ‖f n x‖ₑ) atTop < ∞