mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
B : Tendsto (fun r ↦ R * r) (𝓝[>] 0) (𝓝[>] 0)
r : ℝ
⊢ (r * R) • R⁻¹ • t = r • t

Tactic:
smul_smul,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
B : Tendsto (fun r ↦ R * r) (𝓝[>] 0) (𝓝[>] 0)
r : ℝ
⊢ (r * R * R⁻¹) • t = r • t
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
B : Tendsto (fun r ↦ R * r) (𝓝[>] 0) (𝓝[>] 0)
r : ℝ
T : (R * r) • t' = r • t
⊢ (r * R) • R⁻¹ • u = r • u

Tactic:
smul_smul,

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)
t u : Set E
h'u : μ u ≠ 0
R : ℝ
Rpos : 0 < R
t_bound : t ⊆ closedBall 0 R
t' : Set E := R⁻¹ • t
ht' : t' = R⁻¹ • t
u' : Set E := R⁻¹ • u
hu' : u' = R⁻¹ • u
A : Tendsto (fun r ↦ μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0)
B : Tendsto (fun r ↦ R * r) (𝓝[>] 0) (𝓝[>] 0)
r : ℝ
T : (R * r) • t' = r • t
⊢ (r * R * R⁻¹) • u = r • u
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x : E
⊢ ∑ i, ‖⟪x, b i⟫‖ ^ 2 = ‖x‖ ^ 2

Tactic:
by
  convert! sum_sq_norm_inner_right b x using 2 with i -
  rw [← inner_conj_symm, RCLike.norm_conj]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ ∀ (u v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u

Tactic:
intro u

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
hs : MeasurableSet s
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
u : Set E
⊢ ∀ (v : Set E), μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v → μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case inr
η : Type u_7
inst✝⁵ : Fintype η
inst✝⁴ : DecidableEq η
ι : η → Type u_8
inst✝³ : (i : η) → Fintype (ι i)
𝕜 : Type u_9
inst✝² : RCLike 𝕜
E : η → Type u_10
inst✝¹ : (i : η) → NormedAddCommGroup (E i)
inst✝ : (i : η) → InnerProductSpace 𝕜 (E i)
B : (i : η) → OrthonormalBasis (ι i) 𝕜 (E i)
k i : η
j : ι i
hi : i ≠ k
⊢ (B k).repr.symm (toLp 2 (Pi.single i (Pi.single j 1) k)) = Pi.single i ((B i) j) k

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
s : Finset (ι × ℝ)
⊢ π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s

Tactic:
induction s using Finset.induction_on with

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: case e_a
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
this : Tendsto (fun r ↦ μ (toMeasurable μ s ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1)
r : ℝ
⊢ MeasurableSet ((fun x_1 ↦ -x + x_1) ⁻¹' (r • t))

Tactic:
apply (continuous_const_add (-x)).measurable (ht.const_smul₀ r)

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Additive

Goal state before:
pp: ι : Type u_1
M : Type u_2
n : ℕ
N : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : AddCommMonoid N
I₀✝ : WithTop (Box ι)
I✝ : Box ι
i : ι
inst✝ : Finite ι
f : Box ι → M
I₀ : WithTop (Box ι)
hf :
  ∀ (I : Box ι),
    ↑I ≤ I₀ →
      ∀ {i : ι} {x : ℝ},
        x ∈ Set.Ioo (I.lower i) (I.upper i) →
          Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) = f I
I : Box ι
hI : ↑I ≤ I₀
s : Finset (ι × ℝ)
⊢ ∑ J ∈ (splitMany I s).boxes, f J = f I

Tactic:
induction s using Finset.induction_on with

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝³ : LinearOrder ι
inst✝² : LocallyFiniteOrder ι
inst✝¹ : OrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
⊢ gramSchmidt 𝕜 f ⊥ = f ⊥

Tactic:
by
  rw [gramSchmidt_def, Iio_eq_Ico, Finset.Ico_self, Finset.sum_empty, sub_zero]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: case refine_1
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₁.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₁.tag J))

pp: case refine_2
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₂.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₂.tag J))

Tactic:
· rw [disjUnion_tag_of_mem_left _ hJ]

Goal state after:
case refine_2
ι : Type u
E : Type v
F : Type w
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
I : Box ι
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
π₁ π₂ : TaggedPrepartition I
h : Disjoint π₁.iUnion π₂.iUnion
J : Box ι
hJ : J ∈ π₂.boxes
⊢ (vol J) (f ((π₁.disjUnion π₂ h).tag J)) = (vol J) (f (π₂.tag J))
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
I : Box ι
inst✝ : Fintype ι
l : IntegrationParams
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
α : Type u_1
s : Finset α
f : α → (ι → ℝ) → E
g : α → F
h : ∀ i ∈ s, HasIntegral I l (f i) vol (g i)
⊢ HasIntegral I l (fun x ↦ ∑ i ∈ s, f i x) vol (∑ i ∈ s, g i)

Tactic:
induction s using Finset.induction_on with

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤ (μ.real ↑I + 1 + 1) * ↑ε

Tactic:
refine (dist_triangle4 _ (∑ J ∈ π.boxes, μ.real J • f (Nx <| π.tag J) (π.tag J))
    (∑ J ∈ π.boxes, ∫ x in J, f (Nx <| π.tag J) x ∂μ) _).trans ?_

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)) +
        dist (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J))
          (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) +
      dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
    (μ.real ↑I + 1 + 1) * ↑ε
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : Fintype ι₁
inst✝ : Fintype ι₂
v : OrthonormalBasis ι₁ 𝕜 E
w : OrthonormalBasis ι₂ 𝕜 F
⊢ ∀ (i : ι₁ ⊕ ι₂),
    (v.prod w) i =
      Sum.elim (WithLp.toLp 2 ∘ ⇑(LinearMap.inl 𝕜 E F) ∘ ⇑v) (WithLp.toLp 2 ∘ ⇑(LinearMap.inr 𝕜 E F) ∘ ⇑w) i

Tactic:
rw [Sum.forall]
  unfold OrthonormalBasis.prod
  aesop

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
hdim : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin (n + 1)
hi : v i = 0
⊢ ‖v i‖ = 0

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case pos
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
v : Fin (n + 1) → E
hv : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0
this : FiniteDimensional ℝ E
hdim : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis hdim v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin (n + 1)
hi : v i = 0
⊢ |⟪b i, v i⟫| = 0

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x : E
y : E ⧸ K
⊢ ‖(K.quotientEquivOfIsCompl Kᗮ ⋯) y‖ = ‖y‖

Tactic:
set f := K.quotientEquivOfIsCompl Kᗮ K.isCompl_orthogonal
    rw [coe_norm, ← norm_orthogonalProjectionOnto_apply Kᗮ (f y).2,
      orthogonalProjectionOnto_orthogonal, coe_norm, starProjection_minimal, eq_comm]
    have h : ‖Quotient.mk (f y).val‖ = sInf ((fun (x : E) ↦ ‖(f y).val + x‖) '' K.toAddSubgroup) :=
      quotient_norm_mk_eq K.toAddSubgroup (f y).1
    convert! h using 2
    · simp [f]
    · rw [sInf_image', ← Equiv.iInf_comp (Equiv.neg K)]
      simp

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := ⋯
b₂ : Basis ι ℤ ↥L₂ := ⋯
⊢ covolume L₁ volume / covolume L₂ volume = |(Basis.ofZLatticeBasis ℝ L₂ b₂).det ⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)|

Tactic:
rw [← Basis.det_mul_det _ (Pi.basisFun ℝ ι) _, abs_mul, Pi.basisFun_det_apply,
      ← Basis.det_inv, Units.val_inv_eq_inv_val, IsUnit.unit_spec, Pi.basisFun_det_apply,
      covolume_eq_det _ b₁, covolume_eq_det _ b₂, mul_comm, abs_inv]

Goal state after:
ι : Type u_2
inst✝⁴ : Fintype ι
L₁ L₂ : Submodule ℤ (ι → ℝ)
inst✝³ : DiscreteTopology ↥L₁
inst✝² : IsZLattice ℝ L₁
inst✝¹ : DiscreteTopology ↥L₂
inst✝ : IsZLattice ℝ L₂
h : L₁ ≤ L₂
b₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁
b₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂
⊢ |(Matrix.of (Subtype.val ∘ ⇑b₁)).det| / |(Matrix.of (Subtype.val ∘ ⇑b₂)).det| =
    |(Matrix.of ⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)).det| * |(Matrix.of ⇑(Basis.ofZLatticeBasis ℝ L₂ b₂)).det|⁻¹
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ k ∈ range (2 ^ n), f k ≤ f 0 + ∑ k ∈ Ico 1 (2 ^ n), f k

Tactic:
← sum_range_add_sum_Ico _ n.one_le_two_pow,

Goal state after:
M : Type u_1
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : IsOrderedAddMonoid M
f : ℕ → M
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
n : ℕ
⊢ ∑ k ∈ range 1, f k + ∑ k ∈ Ico 1 (2 ^ n), f k ≤ f 0 + ∑ k ∈ Ico 1 (2 ^ n), f k
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
n : ℕ
npos : n ∈ Set.Ici 1
nposℝ : 0 < ↑n
⊢ (↑n)⁻¹ • ↑n • l = l

Tactic:
smul_smul,

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
u : ℕ → E
l : E
h : Tendsto u atTop (𝓝 l)
this : (fun n ↦ ∑ i ∈ Finset.range n, (u i - l)) =o[atTop] fun n ↦ ↑n
n : ℕ
npos : n ∈ Set.Ici 1
nposℝ : 0 < ↑n
⊢ ((↑n)⁻¹ * ↑n) • l = l
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun k ↦ 2 ^ k * ((2 ^ p) ^ k)⁻¹) ↔ 1 < p

Tactic:
← inv_pow,

Goal state after:
case inl
p : ℝ
hp : 0 ≤ p
⊢ (Summable fun k ↦ 2 ^ k * (2 ^ p)⁻¹ ^ k) ↔ 1 < p
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
⊢ x ∈ Ideal.span (⇑(of k G) '' s) ↔ ∀ m ∈ x.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'

Tactic:
let RHS : Ideal (MonoidAlgebra k G) :=
    { carrier := { p | ∀ m : G, m ∈ p.coeff.support → ∃ m' ∈ s, ∃ d, m = d * m' }
      add_mem' {x y} hx hy m hm := (Finset.mem_union.1 <| Finsupp.support_add hm).elim (hx m) (hy m)
      zero_mem' := by simp
      smul_mem' x y hy m hm := by
        simp only [smul_eq_mul, mul_def, coeff_finsuppSum] at hm
        obtain ⟨xm, -, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
        obtain ⟨ym, hym, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
        obtain rfl := Finset.mem_singleton.mp (Finsupp.support_single_subset hm)
        refine (hy _ hym).imp fun sm p => And.imp_right ?_ p
        rintro ⟨d, rfl⟩
        exact ⟨xm * d, (mul_assoc _ _ _).symm⟩ }

Goal state after:
k : Type u_1
G : Type u_3
inst✝¹ : Monoid G
inst✝ : Semiring k
s : Set G
x : k[G]
RHS : Ideal k[G] :=
  { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ x ∈ Ideal.span (⇑(of k G) '' s) ↔ ∀ m ∈ x.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'
Module: Mathlib.Algebra.MonoidAlgebra.ToDirectSum

Goal state before:
pp: ι : Type u_1
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : Semiring M
f g : M[ι]
to_hom : M[ι] →+ ⨁ (x : ι), M := { toFun := toDirectSum, map_zero' := ⋯, map_add' := ⋯ }
⊢ (f * g).toDirectSum = f.toDirectSum * g.toDirectSum

Tactic:
change to_hom (f * g) = to_hom f * to_hom g

Goal state after:
ι : Type u_1
M : Type u_3
inst✝² : DecidableEq ι
inst✝¹ : AddMonoid ι
inst✝ : Semiring M
f g : M[ι]
to_hom : M[ι] →+ ⨁ (x : ι), M := { toFun := toDirectSum, map_zero' := ⋯, map_add' := ⋯ }
⊢ to_hom (f * g) = to_hom f * to_hom g
Module: Mathlib.Data.List.GetD

Goal state before:
pp: α : Type u
l : List α
inst✝ : Inhabited α
n : ℕ
⊢ l.getI n = l[n]?.getD default

Tactic:
rw [← getD_default_eq_getI, getD_eq_getElem?_getD]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case e'_3.a.e'_3
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
h✝ : Nontrivial ↥L
I : Type u_1 := Free.ChooseBasisIndex ℤ ↥L
this : Fintype I
b : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L
d : ℕ := Fintype.card I
hd : d ≠ 0
ε : ℝ := normBound b
r : ℝ
hr : r < -↑d
e : (I → ℤ) ≃ₗ[ℤ] ↥L := (b.repr ≪≫ₗ Finsupp.linearEquivFunOnFinite ℤ ℤ I).symm
u : Finset (I → ℤ)
n : ℕ
hn : u ⊆ Fintype.piFinset fun x ↦ Finset.Icc (-↑n) ↑n
x : I → ℤ
a✝ : x ∈ Fintype.piFinset fun x ↦ Finset.Icc (-↑n) ↑n
⊢ e x = ∑ i, x i • b i

Tactic:
simp [e, Finsupp.linearCombination]

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
r : ℝ
hr : r < -↑(finrank ℤ ↥L)
x : E
h✝ : Nontrivial ↥L
H : IsClosed ↑L
⊢ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x_1}ᶜ ⊆
    ⇑(Function.Embedding.subtype fun x ↦ x ∈ L) ⁻¹' (Metric.closedBall 0 (2 * ‖x‖) ∩ ↑L)

Tactic:
intro t ht

Goal state after:
case inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
r : ℝ
hr : r < -↑(finrank ℤ ↥L)
x : E
h✝ : Nontrivial ↥L
H : IsClosed ↑L
t : ↥L
ht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ‖i‖ ^ r) x_1}ᶜ
⊢ t ∈ ⇑(Function.Embedding.subtype fun x ↦ x ∈ L) ⁻¹' (Metric.closedBall 0 (2 * ‖x‖) ∩ ↑L)
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
f : α → ℕ
a b : α
inst✝ : DecidableEq α
h : a ≠ b
⊢ multinomial {a, b} f = (f a + f b)! / ((f a)! * (f b)!)

Tactic:
by
  simp [multinomial, Finset.sum_pair h, Finset.prod_pair h]

Goal state after:
Module: Mathlib.Algebra.MvPolynomial.Coeff

Goal state before:
pp: R : Type u_1
σ : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Fintype σ
a : σ → R
s : σ →₀ ℕ
n : ℕ
⊢ coeff s ((∑ i, a i • X i) ^ n) = if (s.sum fun x m ↦ m) = n then ↑s.multinomial * s.prod fun r m ↦ a r ^ m else 0

Tactic:
by
  rw [← ofSupportFinite_coe (f := a) (hf := Set.toFinite _),
    prod_congr (fun r _ ↦ rfl), ← coeff_linearCombination_X_pow]
  simp [linearCombination_apply, sum_of_support_subset (s := Finset.univ)]

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f)) (n : ℕ),
    (∑ i ∈ s, f i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f)
n : ℕ
this :
  ∀ p ∈ antidiagonal n,
    ∑ g ∈ s.piAntidiag p.2,
        ↑((g a + p.1 + s.sum g).choose (g a + p.1)) * ↑(multinomial s (g + fun i ↦ if i = a then p.1 else 0)) *
          (f a ^ (g a + p.1) * s.noncommProd (fun i ↦ f i ^ (g i + if i = a then p.1 else 0)) ⋯) =
      ∑ g ∈ s.piAntidiag p.2, ↑(n.choose p.1) * ↑(multinomial s g) * (f a ^ p.1 * s.noncommProd (fun i ↦ f i ^ g i) ⋯)
i : α
hi : i ∈ s
⊢ i ∈ ↑(cons a s has)

Tactic:
simp [hi]

Goal state after:
Module: Mathlib.Data.Nat.Choose.Multinomial

Goal state before:
pp: case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
⊢ ∀ a_1 ∈ s, (f + fun i ↦ if i = a then p.1 else 0) a_1 = f a_1

Tactic:
intro t ht

Goal state after:
case cons.e_a.e_a
α : Type u_1
R : Type u_2
inst✝¹ : DecidableEq α
inst✝ : Semiring R
f✝ : α → R
a : α
s : Finset α
has : a ∉ s
ih :
  ∀ (hc : (↑s).Pairwise (Commute on f✝)) (n : ℕ),
    (∑ i ∈ s, f✝ i) ^ n = ∑ k ∈ s.piAntidiag n, ↑(multinomial s k) * s.noncommProd (fun i ↦ f✝ i ^ k i) ⋯
hc : (↑(cons a s has)).Pairwise (Commute on f✝)
n : ℕ
p : ℕ × ℕ
hp : p ∈ antidiagonal n
f : α → ℕ
hf : s.sum f = p.2 ∧ ∀ (i : α), f i ≠ 0 → i ∈ s
t : α
ht : t ∈ s
⊢ (f + fun i ↦ if i = a then p.1 else 0) t = f t