mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))

Tactic:
convert! indicator 0 MeasurableSet.empty (by simp) using 1

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = ∅.indicator fun x ↦ 0
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
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
⊢ ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) = ∅.indicator fun x ↦ 0

Tactic:
ext

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
x✝ : α
⊢ (SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) x✝ = ∅.indicator (fun x ↦ 0) x✝
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h : c ≠ 0
⊢ Tendsto (u - fun x ↦ c) l (𝓝 0) ↔ Tendsto u l (𝓝 c)

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h : c ≠ 0
⊢ Tendsto (u - fun x ↦ c) l (𝓝 0) → Tendsto u l (𝓝 c)

case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
h : c ≠ 0
⊢ Tendsto u l (𝓝 c) → Tendsto (u - fun x ↦ c) l (𝓝 0)
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v) ↔ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v) → ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ (∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v) → ∃ φ, Tendsto φ l (𝓝 0) ∧ u - v =ᶠ[l] φ * v
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
⊢ u ~[l] v ↔ Tendsto (u / v) l (𝓝 1)

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
⊢ u ~[l] v → Tendsto (u / v) l (𝓝 1)

case mpr
α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
hz : ∀ᶠ (x : α) in l, v x ≠ 0
⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case e'_5
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : a ~[l] b
huv : u ~[l] v
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
⊢ (fun x ↦ b x • (φ x • u x - v x)) = fun i ↦ (φ * b) i • u i - b i • v i

Tactic:
ext

Goal state after:
case e'_5
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : a ~[l] b
huv : u ~[l] v
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
x✝ : α
⊢ b x✝ • (φ x✝ • u x✝ - v x✝) = (φ * b) x✝ • u x✝ - b x✝ • v x✝
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✝ : OrderTopology β
huv : u ~[l] v
hu : Tendsto u l atBot
⊢ v = Neg.neg ∘ fun x ↦ -v x

Tactic:
ext

Goal state after:
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✝ : OrderTopology β
huv : u ~[l] v
hu : Tendsto u l atBot
x✝ : α
⊢ v x✝ = (Neg.neg ∘ fun x ↦ -v x) x✝
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:
simp [integral_smul]

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:
ext

Goal state after:
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
x✝ : P
⊢ { toAffineMap := ↑f, cont := h } x✝ = f x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : f.contLinear = 0
⊢ (↑f).linear = 0

Tactic:
ext

Goal state after:
case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : f.contLinear = 0
x✝ : V
⊢ (↑f).linear x✝ = 0 x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : (↑f).linear = 0
⊢ f.contLinear = 0

Tactic:
ext

Goal state after:
case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : (↑f).linear = 0
x✝ : V
⊢ f.contLinear x✝ = 0 x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h₁ : f.contLinear = 0 ↔ (↑f).linear = 0
q : Q
h : f = const R P q
⊢ ↑f = AffineMap.const R P q

Tactic:
ext

Goal state after:
case refine_1
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h₁ : f.contLinear = 0 ↔ (↑f).linear = 0
q : Q
h : f = const R P q
p✝ : P
⊢ ↑f p✝ = (AffineMap.const R P q) p✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h₁ : f.contLinear = 0 ↔ (↑f).linear = 0
q : Q
h : ↑f = AffineMap.const R P q
⊢ f = const R P q

Tactic:
ext

Goal state after:
case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h₁ : f.contLinear = 0 ↔ (↑f).linear = 0
q : Q
h : ↑f = AffineMap.const R P q
x✝ : P
⊢ f x✝ = (const R P q) x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹⁴ : Ring S
inst✝¹³ : Ring R
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module R W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : Module S W
inst✝⁴ : SMulCommClass R S W
inst✝³ : ContinuousConstSMul S W
inst✝² : AddTorsor W Q
inst✝¹ : TopologicalSpace Q
inst✝ : IsTopologicalAddTorsor Q
this : IsTopologicalAddGroup W
v : Q
f : V →L[R] W
⊢ (fun f ↦ (f 0, f.contLinear)) ((fun p ↦ p.2.toContinuousAffineMap +ᵥ const R V p.1) (v, f)) = (v, f)

Tactic:
ext

Goal state after:
case fst
R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹⁴ : Ring S
inst✝¹³ : Ring R
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module R W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : Module S W
inst✝⁴ : SMulCommClass R S W
inst✝³ : ContinuousConstSMul S W
inst✝² : AddTorsor W Q
inst✝¹ : TopologicalSpace Q
inst✝ : IsTopologicalAddTorsor Q
this : IsTopologicalAddGroup W
v : Q
f : V →L[R] W
⊢ ((fun f ↦ (f 0, f.contLinear)) ((fun p ↦ p.2.toContinuousAffineMap +ᵥ const R V p.1) (v, f))).1 = (v, f).1

case snd
R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹⁴ : Ring S
inst✝¹³ : Ring R
inst✝¹² : AddCommGroup V
inst✝¹¹ : Module R V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module R W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : Module S W
inst✝⁴ : SMulCommClass R S W
inst✝³ : ContinuousConstSMul S W
inst✝² : AddTorsor W Q
inst✝¹ : TopologicalSpace Q
inst✝ : IsTopologicalAddTorsor Q
this : IsTopologicalAddGroup W
v : Q
f : V →L[R] W
x✝ : V
⊢ ((fun f ↦ (f 0, f.contLinear)) ((fun p ↦ p.2.toContinuousAffineMap +ᵥ const R V p.1) (v, f))).2 x✝ = (v, f).2 x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
V : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹⁰ : Ring R
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module R V
inst✝⁷ : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module R W
inst✝³ : TopologicalSpace W
inst✝² : AddTorsor W Q
inst✝¹ : TopologicalSpace Q
inst✝ : IsTopologicalAddTorsor Q
p : Q × (V →L[R] W)
this : IsTopologicalAddGroup W
⊢ ((decompEquiv R V Q).symm p).contLinear = p.2

Tactic:
ext

Goal state after:
R : Type u_1
V : Type u_3
W : Type u_4
Q : Type u_5
inst✝¹⁰ : Ring R
inst✝⁹ : AddCommGroup V
inst✝⁸ : Module R V
inst✝⁷ : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module R W
inst✝³ : TopologicalSpace W
inst✝² : AddTorsor W Q
inst✝¹ : TopologicalSpace Q
inst✝ : IsTopologicalAddTorsor Q
p : Q × (V →L[R] W)
this : IsTopologicalAddGroup W
x✝ : V
⊢ ((decompEquiv R V Q).symm p).contLinear x✝ = p.2 x✝
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
inst✝¹² : Ring S
inst✝¹¹ : Ring R
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module R V
inst✝⁸ : TopologicalSpace V
inst✝⁷ : IsTopologicalAddGroup V
inst✝⁶ : AddCommGroup W
inst✝⁵ : Module R W
inst✝⁴ : TopologicalSpace W
inst✝³ : Module S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : IsTopologicalAddGroup W
p : W × (V →L[R] W)
⊢ ((decompLinearEquiv R S V W).symm p).contLinear = p.2

Tactic:
ext

Goal state after:
R : Type u_1
S : Type u_2
V : Type u_3
W : Type u_4
inst✝¹² : Ring S
inst✝¹¹ : Ring R
inst✝¹⁰ : AddCommGroup V
inst✝⁹ : Module R V
inst✝⁸ : TopologicalSpace V
inst✝⁷ : IsTopologicalAddGroup V
inst✝⁶ : AddCommGroup W
inst✝⁵ : Module R W
inst✝⁴ : TopologicalSpace W
inst✝³ : Module S W
inst✝² : SMulCommClass R S W
inst✝¹ : ContinuousConstSMul S W
inst✝ : IsTopologicalAddGroup W
p : W × (V →L[R] W)
x✝ : V
⊢ ((decompLinearEquiv R S V W).symm p).contLinear x✝ = p.2 x✝
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[𝕜] P₂
⊢ f.linear = f.linearIsometry.toLinearMap

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[𝕜] P₂
x✝ : V
⊢ f.linear x✝ = f.linearIsometry.toLinearMap x✝
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
f : V →ₗᵢ[𝕜] V₂
⊢ f.toAffineIsometry.linearIsometry = f

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
f : V →ₗᵢ[𝕜] V₂
x✝ : V
⊢ f.toAffineIsometry.linearIsometry x✝ = f x✝
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_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f' i a ≤ F' a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
have h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a := by
    unfold f'
    ext i
    rw [ENNReal.toReal_ofReal]
    · abel
    · simp [ha_mono zero_le]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
h_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f' i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f' i a ≤ F' a
h : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
a : α
ha : Tendsto (fun i ↦ f' i a) atTop (𝓝 (F' a))
ha_mono : Monotone fun i ↦ f i a
ha_bound : ∀ (i : ℕ), f i a ≤ F a
h1 : (fun i ↦ f i a) = fun i ↦ (f' i a).toReal + f 0 a
⊢ Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P ≃ᵃⁱ[𝕜] P₂
⊢ e.linear = e.linearIsometryEquiv.toLinearEquiv

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P ≃ᵃⁱ[𝕜] P₂
x✝ : V
⊢ e.linear x✝ = e.linearIsometryEquiv.toLinearEquiv x✝
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V₁ : Type u_3
V₂ : Type u_5
P₁ : Type u_8
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V₁
inst✝⁶ : NormedSpace 𝕜 V₁
inst✝⁵ : PseudoMetricSpace P₁
inst✝⁴ : NormedAddTorsor V₁ P₁
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P₁ → P₂
e' : V₁ ≃ₗᵢ[𝕜] V₂
p : P₁
h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p
⊢ (mk' e e' p h).linearIsometryEquiv = e'

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V₁ : Type u_3
V₂ : Type u_5
P₁ : Type u_8
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V₁
inst✝⁶ : NormedSpace 𝕜 V₁
inst✝⁵ : PseudoMetricSpace P₁
inst✝⁴ : NormedAddTorsor V₁ P₁
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
e : P₁ → P₂
e' : V₁ ≃ₗᵢ[𝕜] V₂
p : P₁
h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p
x✝ : V₁
⊢ (mk' e e' p h).linearIsometryEquiv x✝ = e' x✝
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
e : V ≃ₗᵢ[𝕜] V₂
⊢ e.toAffineIsometryEquiv.linearIsometryEquiv = e

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : NormedSpace 𝕜 V₂
e : V ≃ₗᵢ[𝕜] V₂
x✝ : V
⊢ e.toAffineIsometryEquiv.linearIsometryEquiv x✝ = e x✝
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p : P
⊢ (constVSub 𝕜 p).symm = (LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p)

Tactic:
ext

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p : P
x✝ : V
⊢ (constVSub 𝕜 p).symm x✝ = ((LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p)) x✝
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: case e_f
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : IntegrableOn f s μ
hi : Countable { x // x ∈ s }
hf' : Integrable (fun x ↦ f ↑x) (Measure.comap Subtype.val μ)
a : ↑s
⊢ (μ (Subtype.val '' {a})).toReal • f ↑a = μ.real {↑a} • f ↑a

Tactic:
simp [measureReal_def]

Goal state after: