mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ x ∈ range ⇑φ

Tactic:
assumption

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ u ∈ tangentConeAt ℝ (range ⇑φ) x

Tactic:
assumption

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare

Goal state before:
pp: case refine_4.a
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
a b c : E
s : Set E
ω : E → E →L[𝕜] F
dω : E → E →L[ℝ] E →L[𝕜] F
hs : Convex ℝ s
hω : ∀ x ∈ s, HasFDerivWithinAt ω (dω x) s x
hdω : ∀ a ∈ s, ∀ x ∈ tangentConeAt ℝ s a, ∀ y ∈ tangentConeAt ℝ s a, ((dω a) x) y = ((dω a) y) x
ha : a ∈ s
hb : b ∈ s
hc : c ∈ s
φ : (↑(Path.segment a b)).Homotopy ↑(Path.segment a c) :=
  ContinuousMap.Homotopy.affine ↑(Path.segment a b) ↑(Path.segment a c)
hφs : range ⇑φ ⊆ s
x : E
hx : x ∈ range ⇑φ
u : E
hu : u ∈ tangentConeAt ℝ (range ⇑φ) x
v : E
hv : v ∈ tangentConeAt ℝ (range ⇑φ) x
⊢ v ∈ tangentConeAt ℝ (range ⇑φ) x

Tactic:
assumption

Goal state after:
Module: Mathlib.MeasureTheory.Measure.TightNormed

Goal state before:
pp: E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖(starRingEnd 𝕜) (↑‖y‖)⁻¹ * ⟪y, x⟫_𝕜‖

Tactic:
simp only [map_inv₀, RCLike.conj_ofReal, norm_mul, norm_inv, norm_algebraMap', norm_norm]

Goal state after:
E : Type u_1
mE : MeasurableSpace E
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)
h : ∀ (y : E), ‖y‖ = 1 → Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | r < ‖⟪y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
y : E
hy : ¬y = 0
h' : Tendsto (fun r ↦ limsup (fun n ↦ (μ n) {x | ‖y‖⁻¹ * r < ‖⟪(↑‖y‖)⁻¹ • y, x⟫_𝕜‖}) atTop) atTop (𝓝 0)
r : ℝ
n : ℕ
x : E
⊢ r < ‖⟪y, x⟫_𝕜‖ ↔ ‖y‖⁻¹ * r < ‖y‖⁻¹ * ‖⟪y, x⟫_𝕜‖
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_3
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
r : ℝ
hr : 0 < r
hr' : -(2 * r⁻¹) ≤ 2 * r⁻¹
⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Set.Ioc (-(2 * r⁻¹)) (2 * r⁻¹)),
    Tendsto (fun n ↦ 1 - charFun (μ n) (a • z)) atTop (𝓝 (1 - f (a • z)))

Tactic:
exact ae_of_all _ fun _ ↦ tendsto_const_nhds.sub (h _)

Goal state after:
Module: Mathlib.MeasureTheory.Measure.LevyConvergence

Goal state before:
pp: case refine_3
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : ℕ → Measure E
inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)
f : E → ℂ
hf : ContinuousAt f 0
h : ∀ (t : E), Tendsto (fun n ↦ charFun (μ n) t) atTop (𝓝 (f t))
z : E
hz : ‖z‖ = 1
h_le_4 : ∀ (n : ℕ) (r : ℝ), 0 < r → 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖ ≤ 4
h_le :
  ∀ (n : ℕ) (r : ℝ),
    0 < r → (μ n).real {x | r < |⟪z, x⟫|} ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - charFun (μ n) (t • z)‖
h_limsup_le :
  ∀ (r : ℝ),
    0 < r →
      limsup (fun n ↦ (μ n).real {x | r < |⟪z, x⟫|}) atTop ≤ 2⁻¹ * r * ‖∫ (t : ℝ) in -2 * r⁻¹..2 * r⁻¹, 1 - f (t • z)‖
⊢ ∀ᶠ (b : ℝ) in atTop,
    limsup (fun n ↦ (μ n).real {x | b < ‖⟪z, x⟫‖}) atTop ≤ 2⁻¹ * b * ‖∫ (t : ℝ) in -2 * b⁻¹..2 * b⁻¹, 1 - f (t • z)‖

Tactic:
filter_upwards [eventually_gt_atTop 0] with r hr using h_limsup_le r hr

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
⊢ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ ≤ ↑(u n)

Tactic:
have : ∑ i ∈ Finset.range (m + 1), (ν i : Measure E) (K n)ᶜ
        = ∑ i ∈ Finset.Ioc n m, (ν i : Measure E) (K n)ᶜ := by
      apply (Finset.sum_subset (by grind) _).symm
      simp +contextual only [Finset.mem_range_succ_iff, Finset.mem_Ioc, not_and,
        not_true_eq_false, imp_false, not_lt, ← null_iff_toMeasure_null]
      intro i hi h'i
      apply (ν i).mono_null _ (νK i)
      rw [Monotone.partialSups_eq h]
      exact compl_subset_compl.2 (h h'i)

Goal state after:
E : Type u_1
inst✝³ : MeasurableSpace E
inst✝² : TopologicalSpace E
inst✝¹ : T2Space E
inst✝ : BorelSpace E
u : ℕ → ℝ≥0
K : ℕ → Set E
C : ℝ≥0
hu : Tendsto u atTop (𝓝 0)
hK : ∀ (n : ℕ), IsCompact (K n)
I :
  ∀ (μ : FiniteMeasure E) (n : ℕ),
    ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = μ.restrict ((partialSups K) n)
A : ∀ (n : ℕ), IsCompact ((partialSups K) n)
f : Ultrafilter (FiniteMeasure E)
hf : {μ | μ.mass ≤ C ∧ ∀ (n : ℕ), μ (K n)ᶜ ≤ u n} ∈ f
ν : ℕ → FiniteMeasure E
ν_reg : ∀ (n : ℕ), (↑(ν n)).InnerRegular
νK : ∀ (n : ℕ), (ν n) ((partialSups K) n)ᶜ = 0
hν : ∀ (n : ℕ), Tendsto (fun ρ ↦ ρ.restrict (disjointed K n)) (↑f) (𝓝 (ν n))
B : (Measure.sum fun n ↦ ↑(ν n)) univ ≤ ↑C
μ : FiniteMeasure E := ⟨Measure.sum fun n ↦ ↑(ν n), ⋯⟩
L : Tendsto id (↑f) (𝓝 μ)
n : ℕ
h : Monotone K
this✝ : Tendsto (fun m ↦ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ) atTop (𝓝 (∑' (i : ℕ), ↑(ν i) (K n)ᶜ))
m : ℕ
hm : n ≤ m
this : ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ = ∑ i ∈ Finset.Ioc n m, ↑(ν i) (K n)ᶜ
⊢ ∑ i ∈ Finset.range (m + 1), ↑(ν i) (K n)ᶜ ≤ ↑(u n)
Module: Mathlib.MeasureTheory.Measure.Support

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : OpensMeasurableSpace X
s : Set X
⊢ (μ.restrict s) (closure[inst✝²] s)ᶜ = 0

Tactic:
μ.restrict_apply isClosed_closure.isOpen_compl.measurableSet

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : MeasurableSpace X
μ : Measure X
inst✝ : OpensMeasurableSpace X
s : Set X
⊢ μ ((closure[inst✝²] s)ᶜ ∩ s) = 0
Module: Mathlib.MeasureTheory.Measure.WithDensityFinite

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
h✝ : IsFiniteMeasure μ
⊢ IsFiniteMeasure μ

Tactic:
assumption

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Prokhorov

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ 1 - ε
⊢ Tendsto (fun n ↦ μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) atTop (𝓝 (μlim (⋃ i, U i)))

Tactic:
← Set.accumulate_def,

Goal state after:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
inst✝² : PseudoMetricSpace 𝓧
inst✝¹ : OpensMeasurableSpace 𝓧
inst✝ : SecondCountableTopology 𝓧
S : Set (ProbabilityMeasure 𝓧)
U : ℕ → Set 𝓧
O : ∀ (i : ℕ), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U i)
Cov : ⋃ i, U i = univ
hcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)
ε : ℝ
hε' : 0 ≤ ε
hε : 0 < ε
hεbound : ε ≤ 1
μ : ℕ → ProbabilityMeasure 𝓧
hμInS : ∀ (k : ℕ), μ k ∈ S
hcontradiction : ∀ (k : ℕ), ↑((μ k) (⋃ i, ⋃ (_ : i ≤ k), U i)) ≤ 1 - ↑(NNReal.mk ε hε')
μlim : ProbabilityMeasure 𝓧
left✝ : μlim ∈ closure[ProbabilityMeasure.instTopologicalSpace] S
sub : ℕ → ℕ
hsubmono : StrictMono sub
hμconverges : Tendsto (μ ∘ sub) atTop (𝓝 μlim)
Measurebound : ∀ (n : ℕ), ↑(μlim (⋃ i, ⋃ (_ : i ≤ n), U i)) ≤ 1 - ε
⊢ Tendsto (fun n ↦ μlim (accumulate U n)) atTop (𝓝 (μlim (⋃ i, U i)))
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Basic

Goal state before:
pp: X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := Finpartition.ofPairwiseDisjoint P hP₂
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := Q.ofSubset ⋯ ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
⊢ ∑ p ∈ P, ‖μ p‖ₑ ≤ μ.variation s

Tactic:
have hQ' : ∀ t ∈ Q'.parts, t ⊆ s := by simp [Q', Q]; grind

Goal state after:
X : Type u_1
V : Type u_2
mX : MeasurableSpace X
inst✝² : TopologicalSpace V
inst✝¹ : ENormedAddCommMonoid V
inst✝ : T2Space V
μ : VectorMeasure X V
s : Set X
hs : MeasurableSet s
P : Finset (Set X)
hP₁ : ∀ t ∈ P, t ⊆ s
hP₂ : (↑P).PairwiseDisjoint id
Q : Finpartition (P.sup id) := Finpartition.ofPairwiseDisjoint P hP₂
defQ : Q = Finpartition.ofPairwiseDisjoint P hP₂
Q' : Finpartition ((filter MeasurableSet Q.parts).sup id) := Q.ofSubset ⋯ ⋯
defQ' : Q' = Q.ofSubset ⋯ ⋯
hQ' : ∀ t ∈ Q'.parts, t ⊆ s
⊢ ∑ p ∈ P, ‖μ p‖ₑ ≤ μ.variation s
Module: Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
s : Set X
μ ν : Measure X
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hs : IsHahnDecomposition μ ν s
hmeas : MeasurableSet s
⊢ (ν.restrict s).toSignedMeasure = VectorMeasure.restrict ν.toSignedMeasure s

Tactic:
exact (toSignedMeasure_restrict_eq_restrict_toSignedMeasure _ _ hmeas).symm

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
mX : MeasurableSpace X
μ : VectorMeasure X E
h : μ.semivariation univ = ∞
t : Set X → Set X
t_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)
t_subs : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → t s ⊆ s
t_var : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → μ.semivariation (t s) = ∞
ht : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → 1 ≤ ‖μ (s \ t s)‖ₑ
s : ℕ → Set X := fun n ↦ t^[n] univ
hs : ∀ (n : ℕ), MeasurableSet (s n) ∧ μ.semivariation (s n) = ∞
u : ℕ → Set X := fun n ↦ s n \ s (n + 1)
hu : ∀ (n : ℕ), 1 ≤ ‖μ (u n)‖ₑ
s_anti : Antitone s
m n : ℕ
hmn : m < n
this : Disjoint (u m) (s (m + 1))
⊢ s n ⊆ s (n + 1) ∪ s (m + 1)

Tactic:
exact Subset.trans (s_anti (by grind)) subset_union_right

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf : Integrable f μ.variation
⊢ ∀ ⦃f g : X → E⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ] →
          C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
intro f g hfg _ hf

Goal state after:
case h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedAddCommGroup H
μ : VectorMeasure X F
f✝ : X → E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedSpace ℝ G
inst✝² : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : CompleteSpace H
C : G →L[ℝ] H
hf✝ : Integrable f✝ μ.variation
f g : X → E
hfg : f =ᵐ[μ.variation] g
a✝ : Integrable f μ.variation
hf : C ∫ᵛ (y : X), f y ∂[B; μ] = ∫ᵛ (y : X), f y ∂[(compL ℝ F G H) C ∘SL B; μ]
⊢ C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
h : ν.HasProd μ B.flip
⊢ μ.HasProd ν B

Tactic:
by simpa using HasProd.flip (μ := ν) (ν := μ) (B := B.flip)

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f : X → H
C : H →L[ℝ] E
hf : Integrable f μ.variation
hG : CompleteSpace G
⊢ ∀ ⦃f g : X → H⦄,
    f =ᵐ[μ.variation] g →
      Integrable f μ.variation →
        ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ] →
          ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]

Tactic:
intro f g hfg _ hf

Goal state after:
case pos.h_ae
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
H : Type u_6
mX : MeasurableSpace X
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedAddCommGroup H
μ : VectorMeasure X F
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
inst✝ : NormedSpace ℝ H
B : E →L[ℝ] F →L[ℝ] G
f✝ : X → H
C : H →L[ℝ] E
hf✝ : Integrable f✝ μ.variation
hG : CompleteSpace G
f g : X → H
hfg : f =ᵐ[μ.variation] g
a✝ : Integrable f μ.variation
hf : ∫ᵛ (y : X), C (f y) ∂[B; μ] = ∫ᵛ (y : X), f y ∂[B ∘SL C; μ]
⊢ ∫ᵛ (y : X), C (g y) ∂[B; μ] = ∫ᵛ (y : X), g y ∂[B ∘SL C; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace ℝ G
μ : VectorMeasure X E
ν : VectorMeasure Y F
B : E →L[ℝ] F →L[ℝ] G
inst✝¹ : CompleteSpace G
inst✝ : IsFiniteMeasure μ.variation
s : Set (X × Y)
hs : MeasurableSet s
⊢ μ.prod ν B = μ.prodOfIsFiniteMeasureLeft ν B

Tactic:
classical
    apply prod_eq_of_forall_apply_prod (fun s t hs ht ↦ ?_)
    simp [prodOfIsFiniteMeasureLeft, hs.prod ht, ↓reduceIte, mk_preimage_prod_right_eq_if,
      of_if, integral_indicator hs, ContinuousLinearMap.flip_apply, restrict_apply, hs]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
⊢ ‖B‖ₑ * ∫⁻ (a : X), ‖f a‖ₑ ∂(μ.restrict s).variation ≤ ‖B‖ₑ * ∫⁻ (x : X) in s, ‖f x‖ₑ ∂μ.variation

Tactic:
variation_restrict_le

Goal state after:
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
⊢ ‖B‖ₑ * ∫⁻ (a : X) in s, ‖f a‖ₑ ∂μ.variation ≤ ‖B‖ₑ * ∫⁻ (x : X) in s, ‖f x‖ₑ ∂μ.variation
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
⊢ ∫⁻ (a : X), ‖f a‖ₑ ∂((μ.transpose B).restrict s).variation ≤ ∫⁻ (x : X) in s, ‖f x‖ₑ ∂(μ.transpose B).variation

Tactic:
variation_restrict_le

Goal state after:
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
s : Set X
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
⊢ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (x : X) in s, ‖f x‖ₑ ∂(μ.transpose B).variation
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: case pos
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
μ : VectorMeasure X F
B : E →L[ℝ] F →L[ℝ] G
s : Set X
hs : MeasurableSet s
⊢ (μ.restrict s).transpose B = (μ.transpose B).restrict s

Tactic:
ext t ht : 1
    simp [VectorMeasure.restrict_apply, hs, ht, transpose]

Goal state after:
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Basic

Goal state before:
pp: α : Type u_1
M : Type u_2
v : α → M
inst✝ : AddCommMonoidWithOne M
β : Type u_3
s : Finset β
f : β → presburger.Term α
| ∑ i ∈ s, Term.realize v (f i)

Tactic:
← s.toList_toFinset,

Goal state after:
α : Type u_1
M : Type u_2
v : α → M
inst✝ : AddCommMonoidWithOne M
β : Type u_3
s : Finset β
f : β → presburger.Term α
| ∑ i ∈ s.toList.toFinset, Term.realize v (f i)
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: M : Type u_1
N : Type u_2
F : Type u_5
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : FunLike F M N
inst✝ : AddMonoidHomClass F M N
s : Set M
hs : IsLinearSet s
f : F
⊢ IsLinearSet (⇑f '' s)

Tactic:
by
  rcases hs with ⟨a, t, ht, rfl⟩
  refine ⟨f a, f '' t, ht.image f, ?_⟩
  simp [image_vadd_distrib, ← AddMonoidHom.map_mclosure]

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
⊢ ∀ (ε : NNReal),
    0 < ε →
      (μ.withDensity f B).variation s < ∞ →
        ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ (μ.withDensity f B).variation s + ↑ε

Tactic:
rintro ε εpos -

Goal state after:
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
μ : VectorMeasure X F
f : X → E
B : E →L[ℝ] F →L[ℝ] G
inst✝ : CompleteSpace G
hf : μ.Integrable f
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = ‖B.flip y‖₊ * ‖x‖₊
s : Set X
hs : MeasurableSet s
ε : NNReal
εpos : 0 < ε
⊢ ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ (μ.withDensity f B).variation s + ↑ε
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs

Goal state before:
pp: S : Finset (Set ℕ)
hS : ∀ t ∈ S, IsProperLinearSet t
a b : ℕ
ht : LinearIndepOn ℕ id ↑{b}
⊢ b ≠ 0

Tactic:
simpa [ne_comm] using ht.zero_notMem_image

Goal state after:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case inr
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
inst✝ : Countable ι
s : ι → Set X
hm : ∀ (i : ι), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hι : Infinite ι
e : ι ≃ ℕ
⊢ HasSum (fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∫ᵛ (x : X) in ⋃ x, s (e.symm x), f x ∂[B; μ]

Tactic:
← e.symm.hasSum_iff

Goal state after:
case inr
X : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
mX : MeasurableSpace X
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedAddCommGroup G
μ : VectorMeasure X F
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
ι : Type u_7
inst✝ : Countable ι
s : ι → Set X
hm : ∀ (i : ι), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hι : Infinite ι
e : ι ≃ ℕ
⊢ HasSum ((fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]) ∘ ⇑e.symm) ∫ᵛ (x : X) in ⋃ x, s (e.symm x), f x ∂[B; μ]