mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : NormedField β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
u v : α → β
l : Filter α
inst✝ : ClosedIicTopology β
h : u ~[l] v
hv : ∀ᶠ (t : α) in l, v t < 0
⊢ ∀ᶠ (x : α) in l, u x < 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_mono : ∀ᵐ (x : α) ∂μ, MonotoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ MonotoneOn (fun b ↦ ∫ (x : α), f x b ∂μ) s

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
hp_pos : p ≠ 0
H : ∀ (f' : α →ₛ E) (δ : ℝ≥0∞), δ ≠ 0 → MemLp (⇑f') p μ → ∃ g, eLpNorm (⇑f' - g) p μ ≤ δ ∧ P g
⊢ ∃ g, eLpNorm (f - g) p μ ≤ ε ∧ P g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
u v t w : α → ℝ
l : Filter α
hu : 0 ≤ v
hw : 0 ≤ w
htu : u ~[l] v
hvw : t ~[l] w
⊢ u + t ~[l] v + w

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
x₀ : E
ε : ℝ≥0
ε_pos : 0 < ε
⊢ (fun x ↦ ‖(id - fun x ↦ x₀) x‖₊) x₀ < ε

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝ : SeminormedAddCommGroup E
x₀ : E
ε : ℝ
ε_pos : 0 < ε
⊢ (fun x ↦ ‖(id - fun x ↦ x₀) x‖) x₀ < ε

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : SeminormedAddCommGroup W
inst✝¹ : PseudoMetricSpace Q
inst✝ : NormedAddTorsor W Q
c : V
x y : P
⊢ dist (c +ᵥ x) (c +ᵥ y) = dist x y

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : SeminormedAddCommGroup W
inst✝¹ : PseudoMetricSpace Q
inst✝ : NormedAddTorsor W Q
x y : P × Q
⊢ dist x y = ‖x -ᵥ y‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : SeminormedAddCommGroup W
inst✝¹ : PseudoMetricSpace Q
inst✝ : NormedAddTorsor W Q
x y : P × Q
⊢ dist x y = ‖x -ᵥ y‖

Tactic:
by
    simp only [Prod.dist_eq, NormedAddTorsor.dist_eq_norm', Prod.norm_def, Prod.fst_vsub,
      Prod.snd_vsub]

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v₁ v₂ : V
x : P
⊢ dist (v₁ +ᵥ x) (v₂ +ᵥ x) = dist v₁ v₂

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v : V
x : P
⊢ dist (v +ᵥ x) x = ‖v‖

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v : V
x : P
⊢ dist x (v +ᵥ x) = ‖v‖

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y z : P
⊢ dist (x -ᵥ y) (x -ᵥ z) = dist y z

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : PartialOrder E
inst✝³ : IsOrderedAddMonoid E
inst✝² : IsOrderedModule ℝ E
inst✝¹ : ClosedIciTopology E
β : Type u_6
inst✝ : Preorder β
f : α → β → E
s : Set β
hf_anti : ∀ᵐ (x : α) ∂μ, AntitoneOn (f x) s
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ AntitoneOn (fun b ↦ ∫ (x : α), f x b ∂μ) s

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ dist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ dist p₁ p₃ + dist p₂ p₄

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ nndist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ nndist p₁ p₃ + nndist p₂ p₄

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
v v' : V
p p' : P
⊢ edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'

Tactic:
by

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

Goal state before:
pp: V : Type u_2
P : Type u_3
inst✝² : SeminormedAddCommGroup V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p₁ p₂ p₃ p₄ : P
⊢ edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x : P
⊢ ‖x -ᵥ x‖ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y : P
⊢ ‖x -ᵥ y‖ = ‖y -ᵥ x‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ ‖x -ᵥ z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x : P
⊢ ‖x -ᵥ x‖ = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x y : P
⊢ ‖x -ᵥ y‖ = ‖y -ᵥ x‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
Hs : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ μ ∅ < ∞

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x y z : P
⊢ ‖x -ᵥ z‖ ≤ ‖x -ᵥ y‖ + ‖y -ᵥ z‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x✝ y✝ : P
h : ‖x✝ -ᵥ y✝‖ = 0
⊢ x✝ = y✝

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q✝ : Type u_5
inst✝⁹ : SeminormedAddCommGroup V
inst✝⁸ : PseudoMetricSpace P
inst✝⁷ : NormedAddTorsor V P
inst✝⁶ : SeminormedAddCommGroup W
inst✝⁵ : PseudoMetricSpace Q✝
inst✝⁴ : NormedAddTorsor W Q✝
Q : Type u_6
inst✝³ : VAdd V Q
inst✝² : VSub V Q
inst✝¹ : Nonempty Q
inst✝ : PseudoMetricSpace Q
f : Q → P
hf : Injective f
vadd : ∀ (c : V) (x : Q), f (c +ᵥ x) = c +ᵥ f x
vsub : ∀ (x y : Q), x -ᵥ y = f x -ᵥ f y
norm : ∀ (x y : Q), dist x y = dist (f x) (f y)
x y : Q
⊢ dist x y = ‖x -ᵥ y‖

Tactic:
by

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

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q✝ : Type u_5
inst✝⁸ : SeminormedAddCommGroup V
inst✝⁷ : PseudoMetricSpace P
inst✝⁶ : NormedAddTorsor V P
inst✝⁵ : SeminormedAddCommGroup W
inst✝⁴ : PseudoMetricSpace Q✝
inst✝³ : NormedAddTorsor W Q✝
Q : Type u_6
inst✝² : VAdd V Q
inst✝¹ : VSub V Q
inst✝ : PseudoMetricSpace Q
f : P → Q
hf : Surjective f
vadd : ∀ (c : V) (x : P), f (c +ᵥ x) = c +ᵥ f x
vsub : ∀ (x y : P), x -ᵥ y = f x -ᵥ f y
norm : ∀ (x y : P), dist x y = dist (f x) (f y)
⊢ ∀ (x y : Q), dist x y = ‖x -ᵥ y‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ ∫ (x : α), f x (p • a + q • b) ∂μ ≤ ∫ (x : α), p • f x a + q • f x b ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ ∫ (x : α), p • f x a + q • f x b ∂μ = ∫ (x : α), p • f x a ∂μ + ∫ (x : α), q • f x b ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
a : β
ha : a ∈ s
b : β
hb : b ∈ s
p q : ℝ
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
⊢ ∫ (x : α), p • f x a ∂μ + ∫ (x : α), q • f x b ∂μ = p • ∫ (x : α), f x a ∂μ + q • ∫ (x : α), f x b ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ ConvexOn ℝ s fun b ↦ ∫ (x : α), f x b ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conc : ∀ᵐ (x : α) ∂μ, ConcaveOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ ConcaveOn ℝ s fun b ↦ ∫ (x : α), f x b ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hfi : Integrable (fun x ↦ ↑(f x)) μ
⊢ ∫⁻ (a : α), ↑(f a) ∂μ = ENNReal.ofReal (∫ (a : α), ↑(f a) ∂μ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hfi : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
⊢ ENNReal.ofReal (∫ (x : α), f x ∂μ) = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
hf : ∀ᵐ (x : α) ∂μ, f x < ∞
⊢ ∫ (a : α), (f a).toReal ∂μ = (∫⁻ (a : α), f a ∂μ).toReal

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
hfi : Integrable (fun x ↦ ↑(f x)) μ
b : ℝ≥0
⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ≥0
b : ℝ≥0
h : ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b
⊢ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ ∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ ∫ (x : α), f x ∂μ = 0 ↔ f =ᵐ[μ] 0

Tactic:
by
  simp_rw [integral_eq_lintegral_of_nonneg_ae hf hfi.1, ENNReal.toReal_eq_zero_iff,
    ← ENNReal.not_lt_top, ← hasFiniteIntegral_iff_ofReal hf, hfi.2, not_true_eq_false, or_false]
  rw [lintegral_eq_zero_iff']
  · rw [← hf.ge_iff_eq', Filter.EventuallyEq, Filter.EventuallyLE]
    simp only [Pi.zero_apply, ofReal_eq_zero]
  · exact (ENNReal.measurable_ofReal.comp_aemeasurable hfi.1.aemeasurable)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f g : α → ℝ
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
⊢ ∫ (a : α), f a ∂μ = ∫ (a : α), g a ∂μ ↔ f =ᵐ[μ] g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ 0 < ∫ (x : α), f x ∂μ ↔ 0 < μ (Function.support f)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hμ : NeZero μ
hf : Integrable (fun x ↦ Real.exp (f x)) μ
this : (Function.support fun x ↦ Real.exp (f x)) = univ
⊢ 0 < μ (Function.support fun x ↦ Real.exp (f x))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hμ : NeZero μ
hf : Integrable (fun x ↦ Real.exp (f x)) μ
⊢ 0 < ∫ (x : α), Real.exp (f x) ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
⊢ ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
this : Tendsto (fun n ↦ ∫ (x : α), f' n x ∂μ) atTop (𝓝 (∫ (x : α), (F - f 0) x ∂μ))
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
⊢ 0 ≤ᵐ[μ] fun x ↦ (F - f 0) x

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε : ℝ≥0∞
hε : ε ≠ 0
⊢ ∃ g, eLpNorm (f - g) p μ ≤ ε ∧ P g

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
f : ↥(α →₁ₛ[μ] E)
⊢ Integrable (⇑(Lp.simpleFunc.toSimpleFunc f)) μ

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f g : P →ᴬ[R] Q
h : ↑f = ↑g
⊢ f = g

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P →ᴬ[R] Q
h : Continuous (↑f).toFun
⊢ { toAffineMap := ↑f, cont := h } = f

Tactic:
by

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
P : (α → E) → Prop
h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → P (s.indicator fun x ↦ c)
h_add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → Integrable f μ → Integrable g μ → P f → P g → P (f + g)
h_closed : IsClosed {f | P ↑↑f}
h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → Integrable f μ → P f → P g
⊢ ∀ ⦃f : α → E⦄, Integrable f μ → P f

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
⊢ Nonempty Q

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace P
inst✝⁴ : AddTorsor V P
inst✝³ : AddCommGroup W
inst✝² : Module R W
inst✝¹ : TopologicalSpace Q
inst✝ : AddTorsor W Q
f : P →ᴬ[R] Q
p₀ p₁ : P
c : R
⊢ f ((AffineMap.lineMap p₀ p₁) c) = (AffineMap.lineMap (f p₀) (f p₁)) c

Tactic:
by

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

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹⁵ : Ring R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : Module R V
inst✝¹² : TopologicalSpace P
inst✝¹¹ : AddTorsor V P
inst✝¹⁰ : AddCommGroup W
inst✝⁹ : Module R W
inst✝⁸ : TopologicalSpace Q
inst✝⁷ : AddTorsor W Q
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace W
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : ContinuousVAdd V P
inst✝ : ContinuousVAdd W Q
f : P →ᴬ[R] Q
p₀ p₁ : P
c : R
⊢ f ((lineMap p₀ p₁) c) = (lineMap (f p₀) (f p₁)) c

Tactic:
by

Goal state after: