mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case h_add
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
a✝ : Disjoint (support f) (support g)
f_int : Integrable f μ.variation
g_int : Integrable g 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]
hg : C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]
⊢ C ∫ᵛ (y : X), (f + g) y ∂[B; μ] = ∫ᵛ (y : X), (f + g) y ∂[(compL ℝ F G H) C ∘SL B; μ]

Tactic:
simp only [Pi.add_apply]

Goal state after:
case h_add
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
a✝ : Disjoint (support f) (support g)
f_int : Integrable f μ.variation
g_int : Integrable g 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]
hg : C ∫ᵛ (y : X), g y ∂[B; μ] = ∫ᵛ (y : X), g y ∂[(compL ℝ F G H) C ∘SL B; μ]
⊢ C ∫ᵛ (y : X), f y + g y ∂[B; μ] = ∫ᵛ (y : X), f y + g y ∂[(compL ℝ F G H) C ∘SL B; μ]
Module: Mathlib.MeasureTheory.VectorMeasure.Prod

Goal state before:
pp: case refine_1
X : Type u_2
Y : Type u_3
E : Type u_4
F : Type u_5
mX : MeasurableSpace X
mY : MeasurableSpace Y
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
μ : VectorMeasure X E
ν : VectorMeasure Y F
inst✝ : IsFiniteMeasure μ.variation
s : Set (X × Y)
hs : MeasurableSet s
⊢ AEStronglyMeasurable (fun x ↦ ν (Prod.mk x ⁻¹' s)) μ.variation

Tactic:
exact (stronglyMeasurable_vectorMeasure_prodMk_left hs).aestronglyMeasurable

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

Goal state before:
pp: case pos
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
C : ℝ
h : IsFiniteMeasure (μ.restrict s).variation
hC : ∀ᵐ (x : X) ∂(μ.restrict s).variation, ‖f x‖ ≤ C
hs : MeasurableSet s
⊢ ‖∫ᵛ (x : X) in s, f x ∂[B; μ]‖ ≤ C * ‖B‖ * μ.variation.real s

Tactic:
apply (norm_integral_le_of_norm_le_const hC).trans_eq

Goal state after:
case pos
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
C : ℝ
h : IsFiniteMeasure (μ.restrict s).variation
hC : ∀ᵐ (x : X) ∂(μ.restrict s).variation, ‖f x‖ ≤ C
hs : MeasurableSet s
⊢ C * ‖B‖ * (μ.restrict s).variation.real univ = C * ‖B‖ * μ.variation.real s
Module: Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
i j : Set X
him : MeasurableSet i
hi : μ ≤[i] 0
hjm : MeasurableSet j
⊢ (μ.toMeasureOfLEZero i him hi) j = ‖μ (i ∩ j)‖ₑ

Tactic:
have : μ (i ∩ j) ≤ 0 :=
    μ.nonpos_of_restrict_le_zero (μ.restrict_le_zero_subset ‹_› Set.inter_subset_left ‹_›)

Goal state after:
X : Type u_1
mX : MeasurableSpace X
μ : SignedMeasure X
i j : Set X
him : MeasurableSet i
hi : μ ≤[i] 0
hjm : MeasurableSet j
this : μ (i ∩ j) ≤ 0
⊢ (μ.toMeasureOfLEZero i him hi) j = ‖μ (i ∩ j)‖ₑ
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos
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
C : ℝ
h : IsFiniteMeasure (μ.variation.restrict s)
hC : ∀ x ∈ s, ‖f x‖ ≤ C
x : X
hx : x ∈ s
hs : MeasurableSet s
⊢ ∀ᵐ (x : X) ∂μ.variation.restrict s, ‖f x‖ ≤ C

Tactic:
filter_upwards [ae_restrict_mem hs] with x hx using hC x hx

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

Goal state before:
pp: case refine_1
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
this : ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' = ∫ᵛ (x : α), s.indicator 1 x ∂•m'
⊢ ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' = m s

Tactic:
this,

Goal state after:
case refine_1
α : Type u_1
hα : MeasurableSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : CompleteSpace E
μ : Measure α
inst✝¹ : NormedSpace ℝ E
inst✝ : IsFiniteMeasure μ
C : Set (Set α)
m : AddContent E C
hC : IsSetSemiring C
hm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s
h'C : ∀ s ∈ C, MeasurableSet s
M : MeasurableSpace α := generateFrom C
Mle : M ≤ hα
μ' : Measure α := μ.trim Mle
hμ' : μ' = μ.trim Mle
m' : VectorMeasure α E
m'C : ∀ s ∈ C, m' s = m s
hm' : ∀ (s : Set α), ‖m' s‖ₑ ≤ μ' s
m'_le : m'.variation ≤ μ'
this✝ : MeasurableSpace α := hα
m'' : VectorMeasure α E :=
  { measureOf' := fun s ↦ if MeasurableSet s then ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' else 0, empty' := ⋯,
    not_measurable' := ⋯, m_iUnion' := ⋯ }
s : Set α
hs : s ∈ C
this : ∫ᵛ (x : α), μ[s.indicator 1 | M] x ∂•m' = ∫ᵛ (x : α), s.indicator 1 x ∂•m'
⊢ ∫ᵛ (x : α), s.indicator 1 x ∂•m' = m s
Module: Mathlib.MeasureTheory.VectorMeasure.SetIntegral

Goal state before:
pp: case pos
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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
⊢ Tendsto (fun x ↦ ‖∑ i ∈ Finset.range x, ∫ᵛ (x : X) in s i, f x ∂[B; μ] - ∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ]‖ₑ) atTop
    (𝓝 0)

Tactic:
enorm_sub_rev

Goal state after:
case pos
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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
⊢ Tendsto (fun x ↦ ‖∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ] - ∑ i ∈ Finset.range x, ∫ᵛ (x : X) in s i, f x ∂[B; μ]‖ₑ) atTop
    (𝓝 0)
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: 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
⊢ (μ.transpose B).variation ≤ ‖B‖₊ • μ.variation

Tactic:
apply variation_le_of_forall_enorm_le (fun s hs ↦ ?_)
  apply opENorm_le_bound _ (fun x ↦ ?_)
  simp only [transpose, mapRange_apply, LinearMap.toAddMonoidHom_coe, coe_coe, flip_apply,
    Measure.smul_apply, Measure.nnreal_smul_coe_apply]
  grw [le_opENorm, le_opENorm, enorm_measure_le_variation, ← enorm_eq_nnnorm]
  exact le_of_eq (by ring)

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this✝ : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
N : ℕ
this : ⋃ n, s n = (⋃ n ∈ Finset.range N, s n) ∪ ⋃ n, s (n + N)
⊢ ‖∫ᵛ (x : X) in ⋃ n, s n, f x ∂[B; μ] - ∑ i ∈ Finset.range N, ∫ᵛ (x : X) in s i, f x ∂[B; μ]‖ₑ ≤
    ∑' (k : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s (k + N), ‖f x‖ₑ ∂μ.variation

Tactic:
this,

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this✝ : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
N : ℕ
this : ⋃ n, s n = (⋃ n ∈ Finset.range N, s n) ∪ ⋃ n, s (n + N)
⊢ ‖∫ᵛ (x : X) in (⋃ n ∈ Finset.range N, s n) ∪ ⋃ n, s (n + N), f x ∂[B; μ] -
        ∑ i ∈ Finset.range N, ∫ᵛ (x : X) in s i, f x ∂[B; μ]‖ₑ ≤
    ∑' (k : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s (k + N), ‖f x‖ₑ ∂μ.variation
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
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
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
this : μ.variation ≤ C⁻¹ • (μ.transpose B).variation
⊢ C • μ.variation ≤ (μ.transpose B).variation

Tactic:
this,

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
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
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
this : μ.variation ≤ C⁻¹ • (μ.transpose B).variation
⊢ C • C⁻¹ • (μ.transpose B).variation ≤ (μ.transpose B).variation
Module: Mathlib.MeasureTheory.VectorMeasure.Integral

Goal state before:
pp: X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
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
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
s : Set X
hs : MeasurableSet s
this : ‖μ s‖ₑ ≤ C⁻¹ • ‖(μ.transpose B) s‖ₑ
⊢ ‖μ s‖ₑ ≤ (C⁻¹ • (μ.transpose B).variation) s

Tactic:
this,

Goal state after:
X : Type u_2
E : Type u_4
F : Type u_5
G : Type u_6
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
B : E →L[ℝ] F →L[ℝ] G
inst✝ : Nontrivial E
C : ℝ≥0
hB : ∀ (x : E) (y : F), ‖(B x) y‖₊ = C * ‖x‖₊ * ‖y‖₊
hC : C ≠ 0
s : Set X
hs : MeasurableSet s
this : ‖μ s‖ₑ ≤ C⁻¹ • ‖(μ.transpose B) s‖ₑ
⊢ C⁻¹ • ‖(μ.transpose B) s‖ₑ ≤ (C⁻¹ • (μ.transpose B).variation) s
Module: Mathlib.MeasureTheory.VectorMeasure.WithDensityVec

Goal state before:
pp: case pos
X : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
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
f g : X → E
B : E →L[ℝ] F →L[ℝ] G
h : f =ᵐ[μ.variation] g
hf : μ.Integrable f
s : Set X
⊢ ∀ᵐ (x : X) ∂μ.variation, x ∈ s → f x = g x

Tactic:
filter_upwards [h] with x hx xs using hx

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this✝ : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
N : ℕ
this : ⋃ n, s n = (⋃ n ∈ Finset.range N, s n) ∪ ⋃ n, s (n + N)
⊢ ‖∑ i ∈ Finset.range N, ∫ᵛ (x : X) in s i, f x ∂[B; μ] + ∫ᵛ (x : X) in ⋃ n, s (n + N), f x ∂[B; μ] -
        ∑ i ∈ Finset.range N, ∫ᵛ (x : X) in s i, f x ∂[B; μ]‖ₑ ≤
    ∑' (k : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s (k + N), ‖f x‖ₑ ∂μ.variation

Tactic:
simp only [add_sub_cancel_left]

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
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedSpace ℝ F
inst✝ : NormedSpace ℝ G
B : E →L[ℝ] F →L[ℝ] G
s : ℕ → Set X
hm : ∀ (i : ℕ), MeasurableSet (s i)
hd : Pairwise (Disjoint on s)
hfi : μ.IntegrableOn f (⋃ i, s i)
hG : CompleteSpace G
I : ∑' (i : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s i, ‖f x‖ₑ ∂μ.variation < ∞
this✝ : Summable fun n ↦ ∫ᵛ (x : X) in s n, f x ∂[B; μ]
N : ℕ
this : ⋃ n, s n = (⋃ n ∈ Finset.range N, s n) ∪ ⋃ n, s (n + N)
⊢ ‖∫ᵛ (x : X) in ⋃ n, s (n + N), f x ∂[B; μ]‖ₑ ≤ ∑' (k : ℕ), ‖B‖ₑ * ∫⁻ (x : X) in s (k + N), ‖f x‖ₑ ∂μ.variation
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Definability

Goal state before:
pp: α : Type u_1
A : Set ℕ
inst✝ : Finite α
v : α → ℕ
t : Finset (α → ℕ)
x : α → ℕ
a : ↥t → ℕ
⊢ v + ∑ i ∈ t.attach, ↑(a i) * ↑i = x ↔ ∀ (b : α), x b = v b + ∑ i ∈ t.attach, ↑i b * a i

Tactic:
Eq.comm (b := x),

Goal state after:
α : Type u_1
A : Set ℕ
inst✝ : Finite α
v : α → ℕ
t : Finset (α → ℕ)
x : α → ℕ
a : ↥t → ℕ
⊢ x = v + ∑ i ∈ t.attach, ↑(a i) * ↑i ↔ ∀ (b : α), x b = v b + ∑ i ∈ t.attach, ↑i b * a i
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Definability

Goal state before:
pp: α : Type u_1
A : Set ℕ
inst✝ : Finite α
φ : presburger[[↑A]].Formula α
⊢ IsSemilinearSet (ofPred φ.Realize)

Tactic:
let e := Equiv.sumEmpty α (Fin 0)

Goal state after:
α : Type u_1
A : Set ℕ
inst✝ : Finite α
φ : presburger[[↑A]].Formula α
e : α ⊕ Fin 0 ≃ α := Equiv.sumEmpty α (Fin 0)
⊢ IsSemilinearSet (ofPred φ.Realize)
Module: Mathlib.ModelTheory.Arithmetic.Presburger.Definability

Goal state before:
pp: α : Type u_1
A : Set ℕ
inst✝ : Finite α
φ : presburger[[↑A]].Formula α
e : α ⊕ Fin 0 ≃ α := Equiv.sumEmpty α (Fin 0)
x : α → ℕ
⊢ x ∈ ofPred φ.Realize ↔
    x ∈ ⇑(LinearMap.funLeft ℕ ℕ ⇑e.symm) '' {v | BoundedFormula.Realize φ (v ∘ Sum.inl) (v ∘ Sum.inr)}

Tactic:
simp only [mem_ofPred_eq, mem_image]

Goal state after:
α : Type u_1
A : Set ℕ
inst✝ : Finite α
φ : presburger[[↑A]].Formula α
e : α ⊕ Fin 0 ≃ α := Equiv.sumEmpty α (Fin 0)
x : α → ℕ
⊢ φ.Realize x ↔
    ∃ x_1, BoundedFormula.Realize φ (x_1 ∘ Sum.inl) (x_1 ∘ Sum.inr) ∧ (LinearMap.funLeft ℕ ℕ ⇑e.symm) x_1 = x
Module: Mathlib.ModelTheory.Complexity

Goal state before:
pp: case of_isQF
L : Language
α : Type u'
n : ℕ
φ : L.BoundedFormula α n
n✝ : ℕ
φ✝ : L.BoundedFormula α n✝
hφ : φ✝.IsQF
ψ : L.BoundedFormula α n✝
hψ : ψ.IsPrenex
⊢ (φ✝.toPrenexImpRight ψ).IsPrenex

Tactic:
exact isPrenex_toPrenexImpRight hφ hψ

Goal state after:
Module: Mathlib.ModelTheory.Equivalence

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
n : ℕ
φ : L.BoundedFormula α n
M : T.ModelType
v : α → ↑M
xs : Fin n → ↑M
⊢ (φ ⊓ ∼φ ⇔ ⊥).Realize v xs

Tactic:
by
  simp only [realize_iff, realize_inf, realize_not, and_not_self, realize_bot]

Goal state after:
Module: Mathlib.ModelTheory.Complexity

Goal state before:
pp: case ex
L : Language
α : Type u'
n : ℕ
M : Type u_1
inst✝⁴ : L.Structure M
N : Type u_2
inst✝³ : L.Structure N
F : Type u_3
inst✝² : FunLike F M N
inst✝¹ : EmbeddingLike F M N
inst✝ : L.StrongHomClass F M N
φ : L.BoundedFormula α n
f : F
v : α → M
n✝ : ℕ
φ✝ : L.BoundedFormula α (n✝ + 1)
h✝ : φ✝.IsExistential
ih : ∀ {xs : Fin (n✝ + 1) → M}, φ✝.Realize v xs → φ✝.Realize (⇑f ∘ v) (⇑f ∘ xs)
xs : Fin n✝ → M
⊢ (∃' φ✝).Realize v xs → (∃' φ✝).Realize (⇑f ∘ v) (⇑f ∘ xs)

Tactic:
simp only [realize_ex, Nat.succ_eq_add_one]
    refine fun ⟨a, ha⟩ => ⟨f a, ?_⟩
    rw [← Fin.comp_snoc]
    exact ih ha

Goal state after:
Module: Mathlib.ModelTheory.DirectLimit

Goal state before:
pp: L : Language
ι : Type v
inst✝² : Preorder ι
G : ι → Type w
inst✝¹ : (i : ι) → L.Structure (G i)
f : (i j : ι) → i ≤ j → G i ↪[L] G j
G' : ℕ → Type w
inst✝ : (i : ℕ) → L.Structure (G' i)
f' : (n : ℕ) → G' n ↪[L] G' (n + 1)
x✝² x✝¹ x✝ : ℕ
hij : x✝ ≤ x✝¹
hjk : x✝¹ ≤ x✝²
⊢ ∀ (x : G' x✝), (natLERec f' x✝¹ x✝² hjk) ((natLERec f' x✝ x✝¹ hij) x) = (natLERec f' x✝ x✝² ⋯) x

Tactic:
by simp [Nat.leRecOn_trans hij hjk]

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

Goal state before:
pp: case a.hf
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
δ : NNReal := ε / 3
δpos : 0 < δ
g : X →ₛ E
hg : eLpNorm (f - ⇑g) 1 (μ.transpose B).variation ≤ ↑δ
gmem : MemLp (⇑g) 1 μ.variation
I1 : ∫⁻ (a : X) in s, ‖f a‖ₑ ∂(μ.transpose B).variation ≤ ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation + ↑δ
I2 :
  ∫⁻ (a : X) in s, ‖g a‖ₑ ∂(μ.transpose B).variation =
    ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i})
ρ : ℝ≥0∞
ρpos : ρ > 0
hρ : ∑ i ∈ g.range, ‖i‖ₑ * ρ ≤ ↑δ
P : E → Finset (Set X)
Pg : ∀ (i : E), ∀ t ∈ P i, t ⊆ ⇑g ⁻¹' {i}
Pdisj : ∀ (i : E), (↑(P i)).PairwiseDisjoint id
Pmeas : ∀ (i : E), ∀ t ∈ P i, MeasurableSet t
hP :
  ∀ (i : E),
    ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
      ‖i‖ₑ * (∑ p ∈ P i, ‖((μ.transpose B).restrict s) p‖ₑ + ρ)
I3 :
  ∑ i ∈ g.range, ‖i‖ₑ * ((μ.transpose B).restrict s).variation (⇑g ⁻¹' {i}) ≤
    ∑ i ∈ g.range.sigma P, ‖i.fst‖ₑ * ‖((μ.transpose B).restrict s) i.snd‖ₑ + ↑δ
i : (_ : E) × Set X
hi : i ∈ g.range.sigma P
⊢ ((μ.restrict s).restrict i.snd).Integrable fun x ↦ g x - f x

Tactic:
apply Integrable.restrict
        apply Integrable.restrict
        apply Integrable.sub (memLp_one_iff_integrable.1 gmem) hf

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

Goal state before:
pp: ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
this : Fintype ι
x : ι → ℕ
hx : x ∈ hs.fundamentalDomain
i : ↑hs.basisSet
hi : i ∈ Finset.univ
⊢ (hs.basis.repr (toRatVec x - toRatVec hs.base)) i • hs.basis i ≤ hs.basis i

Tactic:
apply smul_le_of_le_one_left

Goal state after:
case hb
ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
this : Fintype ι
x : ι → ℕ
hx : x ∈ hs.fundamentalDomain
i : ↑hs.basisSet
hi : i ∈ Finset.univ
⊢ 0 ≤ hs.basis i

case h
ι : Type u_3
s : Set (ι → ℕ)
hs : IsProperLinearSet s
inst✝ : Finite ι
this : Fintype ι
x : ι → ℕ
hx : x ∈ hs.fundamentalDomain
i : ↑hs.basisSet
hi : i ∈ Finset.univ
⊢ (hs.basis.repr (toRatVec x - toRatVec hs.base)) i ≤ 1
Module: Mathlib.ModelTheory.PartialEquiv

Goal state before:
pp: case refine_1
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
⊢ ∃ g, f = g.comp (Substructure.inclusion ⋯)

Tactic:
obtain ⟨⟨f', hf'⟩, mf', ff'1, ff'2⟩ := h ⟨⟨S, _, f.equivRange⟩, S_FG⟩ m

Goal state after:
case refine_1
L : Language
M : Type w
N : Type w'
inst✝¹ : L.Structure M
inst✝ : L.Structure N
h : L.IsExtensionPair M N
S : L.Substructure M
S_FG : S.FG
f : ↥S ↪[L] N
m : M
f' : M ≃ₚ[L] N
hf' : f'.dom.FG
mf' : m ∈ (↑⟨f', hf'⟩).dom
ff'1 : (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).dom ≤ (↑⟨f', hf'⟩).dom
ff'2 :
  (↑⟨f', hf'⟩).cod.subtype.comp ((↑⟨f', hf'⟩).toEquiv.toEmbedding.comp (Substructure.inclusion ff'1)) =
    (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).cod.subtype.comp
      (↑⟨{ dom := S, cod := f.toHom.range, toEquiv := f.equivRange }, S_FG⟩).toEquiv.toEmbedding
⊢ ∃ g, f = g.comp (Substructure.inclusion ⋯)
Module: Mathlib.Order.CountableDenseLinearOrder

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : DenselyOrdered β
inst✝² : NoMinOrder β
inst✝¹ : NoMaxOrder β
inst✝ : Nonempty β
f : PartialIso α β
a : α
b : β
hb : (a, b) ∈ ↑f
⊢ ∃ b, ∀ p ∈ ↑f, cmp p.1 a = cmp p.2 b

Tactic:
exact ⟨b, fun p hp ↦ f.prop _ hp _ hb⟩

Goal state after:
Module: Mathlib.Order.CountableDenseLinearOrder

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : LinearOrder α
inst✝⁴ : LinearOrder β
inst✝³ : DenselyOrdered β
inst✝² : NoMinOrder β
inst✝¹ : NoMaxOrder β
inst✝ : Nonempty β
f : PartialIso α β
a : α
⊢ ∃ b, ∀ p ∈ ↑f, cmp p.1 a = cmp p.2 b

Tactic:
by_cases h : ∃ b, (a, b) ∈ f.val
  · obtain ⟨b, hb⟩ := h
    exact ⟨b, fun p hp ↦ f.prop _ hp _ hb⟩
  have :
    ∀ x ∈ {p ∈ f.val | p.fst < a}.image Prod.snd,
      ∀ y ∈ {p ∈ f.val | a < p.fst}.image Prod.snd, x < y := by
    intro x hx y hy
    rw [Finset.mem_image] at hx hy
    rcases hx with ⟨p, hp1, rfl⟩
    rcases hy with ⟨q, hq1, rfl⟩
    rw [Finset.mem_filter] at hp1 hq1
    rw [← lt_iff_lt_of_cmp_eq_cmp (f.prop _ hp1.1 _ hq1.1)]
    exact lt_trans hp1.right hq1.right
  obtain ⟨b, hb⟩ := exists_between_finsets _ _ this
  use b
  rintro ⟨p1, p2⟩ hp
  have : p1 ≠ a := fun he ↦ h ⟨p2, he ▸ hp⟩
  rcases lt_or_gt_of_ne this with hl | hr
  · have : p1 < a ∧ p2 < b :=
      ⟨hl, hb.1 _ (Finset.mem_image.mpr ⟨(p1, p2), Finset.mem_filter.mpr ⟨hp, hl⟩, rfl⟩)⟩
    rw [← cmp_eq_lt_iff, ← cmp_eq_lt_iff] at this
    exact this.1.trans this.2.symm
  · have : a < p1 ∧ b < p2 :=
      ⟨hr, hb.2 _ (Finset.mem_image.mpr ⟨(p1, p2), Finset.mem_filter.mpr ⟨hp, hr⟩, rfl⟩)⟩
    rw [← cmp_eq_gt_iff, ← cmp_eq_gt_iff] at this
    exact this.1.trans this.2.symm

Goal state after: