mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.SymmetricAlgebra.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : CommSemiring A
inst✝ : Algebra R A
f : M →ₗ[R] A
h : IsSymmetricAlgebra f
a : M
⊢ h.equiv (h.equiv.symm (f a)) = h.equiv ((SymmetricAlgebra.ι R M) a)

Tactic:
by simp

Goal state after:
Module: Mathlib.Logic.Godel.GodelBetaFunction

Goal state before:
pp: m : ℕ
a : Fin m → ℕ
i : Fin m
⊢ i ∈ Finset.univ

Tactic:
by simp

Goal state after:
Module: Mathlib.Logic.Godel.GodelBetaFunction

Goal state before:
pp: l : List ℕ
i : Fin l.length
⊢ i ∈ Finset.univ

Tactic:
by simp

Goal state after:
Module: Mathlib.Logic.Hydra

Goal state before:
pp: α : Type u_1
r : α → α → Prop
inst✝ : Std.Irrefl r
p : α → Prop
h : ∀ {a' a : α}, r a' a → p a → p a'
s' s : Multiset α
⊢ CutExpand r s' s → (∀ a ∈ s, p a) → ∀ a ∈ s', p a

Tactic:
rw [cutExpand_iff]
  rintro ⟨t, a, hr, ha, rfl⟩ hsp a' h'
  obtain (h' | h') := mem_add.1 h'
  exacts [hsp a' (mem_of_mem_erase h'), h (hr a' h') (hsp a ha)]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e f✝ : V ≃ₗ[K] V
f : Dual K V
u : V
hf : (↑e).fixedSubmodule ⊔ K ∙ (e u - u) ≤ ker f
⊢ -1 • (e u - u) = u - e u

Tactic:
by simp

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
W : Submodule K V
v : V
hW : 1 < finrank K (V ⧸ W)
htop : W ⊔ K ∙ v = ⊤
h1 : finrank K (V ⧸ W) + finrank K ↥W = finrank K V
⊢ {v}.toFinset.card ≤ 1

Tactic:
by simp

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
n : ℕ
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
hind :
  ∀ {e : V ≃ₗ[K] V},
    e.fixedReduce = 1 → finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
f : Dual K V
v : V
hf : (↑e).fixedSubmodule ≤ ker f
hv : f v = 1
hv' : v ∈ (↑e).fixedSubmodule
⊢ False

Tactic:
apply one_ne_zero' K

Goal state after:
K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
n : ℕ
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
hind :
  ∀ {e : V ≃ₗ[K] V},
    e.fixedReduce = 1 → finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
f : Dual K V
v : V
hf : (↑e).fixedSubmodule ≤ ker f
hv : f v = 1
hv' : v ∈ (↑e).fixedSubmodule
⊢ 1 = 0
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : e.fixedReduce = 1
n : ℕ
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
hind :
  ∀ {e : V ≃ₗ[K] V},
    e.fixedReduce = 1 → finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
f : Dual K V
v : V
hf : (↑e).fixedSubmodule ≤ ker f
hv : f v = 1
hv_notMem : v ∉ (↑e).fixedSubmodule
hv' : v - e v ∈ (↑e).fixedSubmodule
t : V ≃ₗ[K] V := auxTransvection ⋯
ht : t = auxTransvection ⋯
⊢ t⁻¹ * (t * e) = e

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
⊢ ∀ (i : ι), univ ∈ {s | MeasurableSet s}

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
⊢ ∅.Countable ∧ ∅ = ∅.domRestrict ⁻¹' ∅

Tactic:
by simp

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

Goal state before:
pp: ι : Type u_2
α : ι → Type u_1
inst✝ : (i : ι) → MeasurableSpace (α i)
s : Set ((i : ι) → α i)
hs : MeasurableSet s
I : Set ι
t : Set ((a : ↑I) → α ↑a)
hI : I.Countable
⊢ (I.domRestrict ⁻¹' t)ᶜ = I.domRestrict ⁻¹' tᶜ

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
H : instWithTopOfOrderTopology = generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}
⊢ {⊤}.Countable

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
H : instWithTopOfOrderTopology = generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}
⊢ {↑x₁}.Countable

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
inhabited_h : Inhabited α
⊢ ∅ = Set.Ioc default default

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: case inr.a
ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := ⋯
d : Set (WithTop ι) := ⋯
⊢ instWithTopOfOrderTopology ≤ generateFrom {s | ∃ a ∈ d, s = Ioi a ∨ s = Iio a}

Tactic:
apply le_generateFrom_iff_subset_isOpen.2
    grind [isOpen_Iio', isOpen_Ioi']

Goal state after:
Module: Mathlib.MeasureTheory.SetSemiring

Goal state before:
pp: α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : Nonempty α
u v : α
huv : u ≤ v
u' v' : α
hu'v' : u' ≤ v'
hu : u < u'
hv : v' < v
⊢ Set.Ioc u u' ∪ Set.Ioc v' v = ⋃₀ ↑{Set.Ioc u u', Set.Ioc v' v}

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
⊢ 0 = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
⊢ ∀ (I : Finset (Set α)), ↑I ⊆ C → (↑I).PairwiseDisjoint id → ⋃₀ ↑I ∈ C → 0 = ∑ u ∈ I, 0

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: case hC
α : Type u_1
𝒜 : Set (Set α)
s t : Set α
ht : t ∈ generateSetAlgebra 𝒜
h : MeasurableSet t
⊢ MeasurableSet t

Tactic:
clear h
    induction ht with
    | base u u_mem => exact measurableSet_generateFrom u_mem
    | empty => exact @MeasurableSet.empty _ (generateFrom 𝒜)
    | compl u _ mu => exact mu.compl
    | union u v _ _ mu mv => exact MeasurableSet.union mu mv

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: α : Type u_1
𝒜 ℬ : Set (Set α)
h : 𝒜 ⊆ ℬ
hℬ : IsSetAlgebra ℬ
⊢ generateSetAlgebra 𝒜 ⊆ ℬ

Tactic:
intro s hs
  induction hs with
  | base t t_mem => exact h t_mem
  | empty => exact hℬ.empty_mem
  | compl t _ t_mem => exact hℬ.compl_mem t_mem
  | union t u _ _ t_mem u_mem => exact hℬ.union_mem t_mem u_mem

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: α : Type u_1
𝒜 : Set (Set α)
s u : Set α
u_mem : u ∈ 𝒜
⊢ u = ⋃ a ∈ {{u}}, ⋂ t ∈ a, t

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: α : Type u_1
𝒜 : Set (Set α)
s : Set α
⊢ ∅ = ⋃ a ∈ ∅, ⋂ t ∈ a, t

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb : ⊤ ∈ Ioi ↑a
ha : Ioi a = ∅
⊢ ⊤ ∈ Ioi ↑x₁

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝¹ : Preorder ι
ts : TopologicalSpace ι
ht : OrderTopology ι
inst✝ : SecondCountableTopology ι
x₀ : ι
c : Set ι
c_count : c.Countable
hc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}
c' : Set ι
c'_count : c'.Countable
hc' : Dense c'
x₁ : ι := if h : ∃ x, Ioi x = ∅ then h.choose else x₀
d : Set (WithTop ι) := WithTop.some '' c ∪ WithTop.some '' c' ∪ {⊤} ∪ {↑x₁}
basis : Set (Set ι) := {s | ∃ f g, f ⊆ c ∧ g ⊆ c ∧ f.Finite ∧ g.Finite ∧ s = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ g, Iio a}
h_basis : IsTopologicalBasis basis
a : ι
hb✝ : ⊤ ∈ Ioi ↑a
ha : (Ioi a).Nonempty
b : ι
bc' : b ∈ c'
hb : b ∈ Ioi a
⊢ ⊤ ∈ Ioi ↑b

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
s : Set α
hs : s ∈ J
this : s = ⋃ t ∈ J', s ∩ t
x✝ : Set α
⊢ s ∩ x✝ ⊆ id x✝

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.SetAlgebra

Goal state before:
pp: α : Type u_1
𝒜 : Set (Set α)
h : 𝒜.Countable
ℬ : Set (Set α) := {s | s ∈ 𝒜} ∪ {s | sᶜ ∈ 𝒜}
s : Set α
hs : sᶜ ∈ 𝒜
⊢ sᶜᶜ = s

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetSemiring C
J J' : Finset (Set α)
hJ : ↑J ⊆ C
hJdisj : (↑J).PairwiseDisjoint id
hJ' : ↑J' ⊆ C
hJ'disj : (↑J').PairwiseDisjoint id
h : ⋃₀ ↑J = ⋃₀ ↑J'
t : Set α
ht : t ∈ J'
this : t = ⋃ s ∈ J, s ∩ t
x✝ : Set α
⊢ x✝ ∩ t ⊆ id x✝

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
hs : s ∈ C
⊢ (↑{s}).PairwiseDisjoint id

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
hC : IsSetSemiring C
m : AddContent G C
s : Set α
hs : s ∈ C
⊢ s = ⋃₀ ↑{s}

Tactic:
by simp

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

Goal state before:
pp: case refine_2
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
s : Set α
hs : s ∈ f '' ↑I
⊢ ∀ ⦃y : Set α⦄, y ∈ f '' ↑I → s ≠ y → (Disjoint on id) s y

Tactic:
t

Goal state after:
case refine_2
α : Type u_1
C : Set (Set α)
m : AddContent ℝ≥0∞ C
hC : IsSetSemiring C
m_subadd :
  ∀ (f : ℕ → Set α), (∀ (i : ℕ), f i ∈ C) → ⋃ i, f i ∈ C → Pairwise (Disjoint on f) → m (⋃ i, f i) ≤ ∑' (i : ℕ), m (f i)
f : ℕ → Set α
hf : ∀ (i : ℕ), f i ∈ C
hf_Union : ⋃ i, f i ∈ C
hf_disj : Pairwise (Disjoint on f)
I : Finset ℕ
s : Set α
hs : s ∈ f '' ↑I
t : Set α
⊢ t ∈ f '' ↑I → s ≠ t → (Disjoint on id) s t
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
hn0 : n = 0
⊢ (auxTransvection hf)⁻¹ * (auxTransvection hf * e) = e

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Order.WithTop

Goal state before:
pp: ι : Type u_1
inst✝³ : LinearOrder ι
inst✝² : TopologicalSpace ι
inst✝¹ : OrderTopology ι
inst✝ : OrderTop ι
h_fr : frontier {⊤} = ∅
⊢ ContinuousOn (fun x ↦ Sum.inr ()) {x | x = ⊤}

Tactic:
by simp

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
hn_pos : n > 0
he' : auxTransvection hf * e ∈ transvections K V ^ n * dilatransvections K V
⊢ (auxTransvection hf)⁻¹ * (auxTransvection hf * e) = e

Tactic:
by simp

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

Goal state before:
pp: case succ
α : Type u_1
C : Set (Set α)
G : Type u_2
inst✝ : AddCommMonoid G
m : AddContent G C
hC : IsSetRing C
s : ℕ → Set α
hs_disj : Pairwise (Disjoint on s)
hsC : ∀ (i : ℕ), s i ∈ C
n : ℕ
hn : m (accumulate s n) = ∑ i ∈ Finset.range (n + 1), m (s i)
⊢ m (accumulate s (n + 1)) = ∑ i ∈ Finset.range (n + 1 + 1), m (s i)

Tactic:
rw [Finset.sum_range_succ, ← hn, Set.accumulate_succ, addContent_union hC _ (hsC _)]
    · exact Set.disjoint_accumulate hs_disj (Nat.lt_succ_self n)
    · exact hC.accumulate_mem hsC n

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Generation

Goal state before:
pp: K : Type u_1
inst✝³ : DivisionRing K
V : Type u_2
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : Module.Finite K V
n✝ : ℕ
e : V ≃ₗ[K] V
he : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x
n : ℕ
hind :
  ∀ {e : V ≃ₗ[K] V},
    (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →
      finrank K (V ⧸ (↑e).fixedSubmodule) = n + 1 → e ∈ transvections K V ^ n * dilatransvections K V
h : finrank K (V ⧸ (↑e).fixedSubmodule) = n + 2
f : Dual K V
v : V
hv : LinearIndependent K ![(↑e).fixedSubmodule.mkQ v, e.fixedReduce ((↑e).fixedSubmodule.mkQ v)]
hf : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker f
hfv : f v = 1
hv_notMem : v ∉ (↑e).fixedSubmodule
hrank : finrank K (V ⧸ (↑e).fixedSubmodule ⊔ K ∙ v) = n + 1
hn_pos : n > 0
he' : auxTransvection hf * e ∉ transvections K V ^ n * dilatransvections K V
hne_top : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v < ⊤
g : Dual K V
hg0 : g ≠ 0
hg2 : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ⊔ K ∙ v ≤ ker g
hg : (↑e).fixedSubmodule ⊔ K ∙ (e v - v) ≤ ker (f + g)
hfgv : (f + g) v = 1
⊢ (auxTransvection hg)⁻¹ * (auxTransvection hg * e) = e

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
C : Set (Set α)
hC : IsSetRing C
m : AddContent ℝ≥0∞ C
s : ℕ → Set α
hd : Pairwise (Disjoint on s)
hs : ∀ (i : ℕ), s i ∈ C
hm_iSup : ∀ ⦃s : ℕ → Set α⦄, (∀ (n : ℕ), s n ∈ C) → Monotone s → m (⋃ n, s n) = ⨆ n, m (s n)
⊢ m (⋃ i, s i) = m (⋃ i, accumulate s i)

Tactic:
by simp

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
F : Type u_3
G : Type u_4
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
inst✝¹ : CompleteSpace G
B : F →L[ℝ] E →L[ℝ] G
inst✝ : CompleteSpace E
hm : m ≤ mΩ
f : SimpleFunc Ω F
g : Ω → E
hg : Integrable g μ
this :
  ∀ (s : Set Ω) (c : F) (f : Ω → E),
    (fun ω ↦ (B (s.indicator (Function.const Ω c) ω)) (f ω)) = s.indicator fun ω ↦ (B c) (f ω)
g₁ g₂ : SimpleFunc Ω F
x✝ : Disjoint (Function.support ⇑g₁) (Function.support ⇑g₂)
h_eq₁ : μ[fun ω ↦ (B (g₁ ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (g₁ ω)) (μ[g | m] ω)
h_eq₂ : μ[fun ω ↦ (B (g₂ ω)) (g ω) | m] =ᵐ[μ] fun ω ↦ (B (g₂ ω)) (μ[g | m] ω)
h_add : ⇑(g₁ + g₂) = ⇑g₁ + ⇑g₂
⊢ (fun ω ↦ (B (g₁ ω)) (μ[g | m] ω) + (B (g₂ ω)) (μ[g | m] ω)) =ᵐ[μ] fun ω ↦ (B ((g₁ + g₂) ω)) (μ[g | m] ω)

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSpace α
p : ℝ≥0∞
f : ι → α → β
ε : ℝ
x✝² : 0 < ε
i : ι
s : Set α
x✝¹ : MeasurableSet s
x✝ : 0 s ≤ ENNReal.ofReal 1
⊢ eLpNorm (s.indicator (f i)) p 0 ≤ ENNReal.ofReal ε

Tactic:
by simp

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v w : VectorMeasure α M
h : ∀ (i : Set α), MeasurableSet i → v i = w i
i : Set α
hi : ¬MeasurableSet i
⊢ v i = w i

Tactic:
not_measurable _ hi

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

Goal state before:
pp: case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A \ B) (A ∩ B)

Tactic:
rw [disjoint_comm]

Goal state after:
case h
α : Type u_1
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
v : VectorMeasure α M
inst✝ : T2Space M
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h' : v (B \ A) = 0
⊢ Disjoint (A ∩ B) (A \ B)
Module: Mathlib.MeasureTheory.Function.UniformIntegrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
x : α
⊢ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator (fun a ↦ ‖f a‖ₑ) x ^ p.toReal =
    {x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun a ↦ ‖‖f a‖ ^ p.toReal‖ₑ) x

Tactic:
have hiff : M ^ (1 / p.toReal) ≤ ‖f x‖₊ ↔ M ≤ ‖‖f x‖ ^ p.toReal‖₊ := by
    rw [coe_nnnorm, coe_nnnorm, Real.norm_rpow_of_nonneg (norm_nonneg _), norm_norm,
      ← Real.rpow_le_rpow_iff hM' (by positivity)
        (one_div_pos.2 <| ENNReal.toReal_pos hp_ne_zero hp_ne_top), ← Real.rpow_mul (norm_nonneg _),
      mul_one_div_cancel (ENNReal.toReal_pos hp_ne_zero hp_ne_top).ne.symm, Real.rpow_one]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : MemLp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ∞
M : ℝ
hM' : 0 ≤ M
hM :
  ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦ ‖f x‖ ^ p.toReal) x‖ₑ ∂μ ≤
    ENNReal.ofReal (ε ^ p.toReal)
x : α
hiff : M ^ (1 / p.toReal) ≤ ↑‖f x‖₊ ↔ M ≤ ↑‖‖f x‖ ^ p.toReal‖₊
⊢ {x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator (fun a ↦ ‖f a‖ₑ) x ^ p.toReal =
    {x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun a ↦ ‖‖f a‖ ^ p.toReal‖ₑ) x
Module: Mathlib.MeasureTheory.VectorMeasure.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝⁴ : AddCommMonoid M
inst✝³ : TopologicalSpace M
R : Type u_4
inst✝² : Semiring R
inst✝¹ : DistribMulAction R M
inst✝ : ContinuousConstSMul R M
r : R
v : VectorMeasure α M
x✝ : Set α
hi : ¬MeasurableSet x✝
⊢ (r • ⇑v) x✝ = 0

Tactic:
by rw [Pi.smul_apply, v.not_measurable hi, smul_zero]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : ContinuousAdd M
v w : VectorMeasure α M
⊢ (⇑v + ⇑w) ∅ = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
v : VectorMeasure α M
⊢ (-⇑v) ∅ = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : TopologicalSpace M
inst✝ : IsTopologicalAddGroup M
v w : VectorMeasure α M
⊢ (⇑v - ⇑w) ∅ = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : MeasurableSpace β
x✝ : β
v✝ : M
s : Set β
x : β
v : M
⊢ (if MeasurableSet ∅ ∧ x ∈ ∅ then v else 0) = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
hμ : IsFiniteMeasure μ
⊢ (if MeasurableSet ∅ then μ.real ∅ else 0) = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
⊢ (if MeasurableSet ∅ then μ ∅ else 0) = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
v✝ v : VectorMeasure α M
f : α → β
hf : Measurable f
⊢ (if MeasurableSet ∅ then v (f ⁻¹' ∅) else 0) = 0

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
f : α → β
⊢ map 0 f = 0

Tactic:
by_cases hf : Measurable f

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
f : α → β
hf : Measurable f
⊢ map 0 f = 0

case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝² : MeasurableSpace β
M : Type u_3
inst✝¹ : AddCommMonoid M
inst✝ : TopologicalSpace M
f : α → β
hf : ¬Measurable f
⊢ map 0 f = 0
Module: Mathlib.MeasureTheory.Measure.Complex

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
c : ComplexMeasure α
μ : VectorMeasure α ℝ≥0∞
h : re c ≪ᵥ μ ∧ im c ≪ᵥ μ
i : Set α
hi : μ i = 0
⊢ ↑0 + ↑0 * Complex.I = 0

Tactic:
by simp

Goal state after: