mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Unique ι
h : finrank 𝕜 E = 1
v : E
hv : ‖v‖ = 1
w : E
⊢ (orthonormalBasisSingleton ι 𝕜 h v hv).repr w = EuclideanSpace.single default ⟪v, w⟫

Tactic:
ext

Goal state after:
ι : Type u_1
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Unique ι
h : finrank 𝕜 E = 1
v : E
hv : ‖v‖ = 1
w : E
i✝ : ι
⊢ ((orthonormalBasisSingleton ι 𝕜 h v hv).repr w).ofLp i✝ = (EuclideanSpace.single default ⟪v, w⟫).ofLp i✝
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
⊢ (∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1) ∧
    Pairwise fun i j ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0

Tactic:
constructor

Goal state after:
case left
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
⊢ ∀ (i : ι), ‖gramSchmidtNormed 𝕜 f i‖ = 1

case right
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
h₀ : LinearIndependent 𝕜 f
⊢ Pairwise fun i j ↦ ⟪gramSchmidtNormed 𝕜 f i, gramSchmidtNormed 𝕜 f j⟫_𝕜 = 0
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
⊢ e.toBasis.det = f.toBasis.det ↔ e.toBasis.orientation = f.toBasis.orientation

Tactic:
constructor

Goal state after:
case mp
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
⊢ e.toBasis.det = f.toBasis.det → e.toBasis.orientation = f.toBasis.orientation

case mpr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e f : OrthonormalBasis ι ℝ E
⊢ e.toBasis.orientation = f.toBasis.orientation → e.toBasis.det = f.toBasis.det
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case zero
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
⊢ E [⋀^Fin 0]→ₗ[ℝ] ℝ

Tactic:
let opos : E [⋀^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
      exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬e.orientation = x
⊢ -1 • e.det = -e.det

Tactic:
ext

Goal state after:
case neg
R : Type u_1
inst✝⁷ : CommRing R
inst✝⁶ : LinearOrder R
inst✝⁵ : IsStrictOrderedRing R
M : Type u_2
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
ι : Type u_3
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : Nonempty ι
e : Basis ι R M
x : Orientation R M ι
h✝ : ¬e.orientation = x
x✝ : ι → M
⊢ (-1 • e.det) x✝ = (-e.det) x✝
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
⊢ Orthonormal 𝕜 ⇑((v.toBasis.prod w.toBasis).map (WithLp.linearEquiv 2 𝕜 (E × F)).symm)

Tactic:
constructor

Goal state after:
case left
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.toBasis.prod w.toBasis).map (WithLp.linearEquiv 2 𝕜 (E × F)).symm) i‖ = 1

case right
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
⊢ Pairwise fun i j ↦
    ⟪((v.toBasis.prod w.toBasis).map (WithLp.linearEquiv 2 𝕜 (E × F)).symm) i,
        ((v.toBasis.prod w.toBasis).map (WithLp.linearEquiv 2 𝕜 (E × F)).symm) j⟫_𝕜 =
      0
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ ↑↑K.orthogonalDecomposition =
    ↑(WithLp.prodContinuousLinearEquiv 2 𝕜 ↥K ↥Kᗮ).symm ∘SL K.orthogonalProjectionOnto.prod Kᗮ.orthogonalProjectionOnto

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x✝ : E
⊢ ↑↑K.orthogonalDecomposition x✝ =
    (↑(WithLp.prodContinuousLinearEquiv 2 𝕜 ↥K ↥Kᗮ).symm ∘SL
        K.orthogonalProjectionOnto.prod Kᗮ.orthogonalProjectionOnto)
      x✝
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ WithLp.fstL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition = K.orthogonalProjectionOnto

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x✝ : E
⊢ ↑((WithLp.fstL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition) x✝) = ↑(K.orthogonalProjectionOnto x✝)
Module: Mathlib.Analysis.InnerProductSpace.ProdL2

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ WithLp.sndL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition = Kᗮ.orthogonalProjectionOnto

Tactic:
ext

Goal state after:
𝕜 : Type u_1
E : Type u_4
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
x✝ : E
⊢ ↑((WithLp.sndL 2 𝕜 ↥K ↥Kᗮ ∘SL ↑↑K.orthogonalDecomposition) x✝) = ↑(Kᗮ.orthogonalProjectionOnto x✝)
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
x : Fin (n + 1) → E
this : FiniteDimensional ℝ E
⊢ x = ⇑φ.symm ∘ ⇑φ ∘ x

Tactic:
ext

Goal state after:
case e'_3
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ : E ≃ₗᵢ[ℝ] E
hφ : 0 < LinearMap.det ↑φ.toLinearEquiv
n : ℕ
_i : Fact (finrank ℝ E = n + 1)
o : Orientation ℝ E (Fin (n + 1))
x : Fin (n + 1) → E
this : FiniteDimensional ℝ E
x✝ : Fin (n + 1)
⊢ x x✝ = (⇑φ.symm ∘ ⇑φ ∘ x) x✝
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : FiniteDimensional ℝ E
h : finrank ℝ E = 1
v : E
hv : v ≠ 0
⊢ volume = map (fun x ↦ x • v) (map (fun x ↦ ‖v‖⁻¹ • x) volume)

Tactic:
have hv' : Submodule.span ℝ {‖v‖⁻¹ • v} = ⊤ := by
      rw [Submodule.span_singleton_eq_top_iff]
      apply exists_smul_eq_of_finrank_eq_one h
      simpa
    let f : ℝ ≃ₗᵢ[ℝ] E := (LinearIsometryEquiv.toSpanUnitSingleton (‖v‖⁻¹ • v)
      (by simp [norm_smul, hv])).trans (LinearIsometryEquiv.ofTop E _ hv')
    rw [map_map (by fun_prop) (by fun_prop)]
    convert! f.measurePreserving.map_eq.symm
    ext x
    simp [f, mul_comm, smul_smul]

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

Goal state before:
pp: E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
L : Submodule ℤ E
inst✝⁸ : DiscreteTopology ↥L
inst✝⁷ : IsZLattice ℝ L
μ : Measure E
inst✝⁶ : μ.IsAddHaarMeasure
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
ν : Measure F
inst✝ : ν.IsAddHaarMeasure
e : F ≃L[ℝ] E
he : MeasurePreserving (⇑e) ν μ
⊢ Basis.ofZLatticeBasis ℝ (ZLattice.comap ℝ L ↑↑e) (Basis.ofZLatticeComap ℝ L (↑e) (Free.chooseBasis ℤ ↥L)) =
    (Basis.ofZLatticeBasis ℝ L (Free.chooseBasis ℤ ↥L)).map ↑e.symm

Tactic:
ext

Goal state after:
E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
L : Submodule ℤ E
inst✝⁸ : DiscreteTopology ↥L
inst✝⁷ : IsZLattice ℝ L
μ : Measure E
inst✝⁶ : μ.IsAddHaarMeasure
F : Type u_2
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
inst✝³ : FiniteDimensional ℝ F
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
ν : Measure F
inst✝ : ν.IsAddHaarMeasure
e : F ≃L[ℝ] E
he : MeasurePreserving (⇑e) ν μ
i✝ : Free.ChooseBasisIndex ℤ ↥L
⊢ (Basis.ofZLatticeBasis ℝ (ZLattice.comap ℝ L ↑↑e) (Basis.ofZLatticeComap ℝ L (↑e) (Free.chooseBasis ℤ ↥L))) i✝ =
    ((Basis.ofZLatticeBasis ℝ L (Free.chooseBasis ℤ ↥L)).map ↑e.symm) i✝
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e_a.e_6
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : FiniteDimensional ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
b₀ : Basis ι ℝ E
⊢ ⇑(Basis.ofZLatticeBasis ℝ L b) = Subtype.val ∘ ⇑b

Tactic:
ext

Goal state after:
case e_a.e_6
E : Type u_1
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : FiniteDimensional ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
L : Submodule ℤ E
inst✝⁴ : DiscreteTopology ↥L
inst✝³ : IsZLattice ℝ L
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
b₀ : Basis ι ℝ E
x✝ : ι
⊢ (Basis.ofZLatticeBasis ℝ L b) x✝ = (Subtype.val ∘ ⇑b) x✝
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_2
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
⊢ Subtype.val ∘ ⇑b = ⇑(Basis.ofZLatticeBasis ℝ L b)

Tactic:
ext

Goal state after:
ι : Type u_2
inst✝² : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
b : Basis ι ℤ ↥L
x✝¹ x✝ : ι
⊢ (Subtype.val ∘ ⇑b) x✝¹ x✝ = (Basis.ofZLatticeBasis ℝ L b) x✝¹ x✝
Module: Mathlib.Analysis.SumOverResidueClass

Goal state before:
pp: m : ℕ
inst✝ : NeZero m
f : ℕ → ℝ
hf : Antitone f
k : ZMod m
H : Summable ({n | ↑n = k}.indicator f)
⊢ Summable f

Tactic:
rw [Finset.sum_indicator_mod m f]

Goal state after:
m : ℕ
inst✝ : NeZero m
f : ℕ → ℝ
hf : Antitone f
k : ZMod m
H : Summable ({n | ↑n = k}.indicator f)
⊢ Summable (∑ a, {n | ↑n = a}.indicator f)
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p q : ℕ
hpq : p < q
⊢ Tendsto (fun x ↦ x ^ (↑p - ↑q)) atTop (𝓝 0)

Tactic:
apply tendsto_zpow_atTop_zero

Goal state after:
𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
p q : ℕ
hpq : p < q
⊢ ↑p - ↑q < 0
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: α : Type u_1
u v : α → ℝ
l : Filter α
hv : 0 ≤ v
h : u ~[l] v
r : ℝ
φ : α → ℝ
hφ : Tendsto φ l (𝓝 1)
huφv : u =ᶠ[l] φ * v
hφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)
⊢ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u ^ r =ᶠ[l] φ * v ^ r

Tactic:
use (· ^ r) ∘ φ, hφr

Goal state after:
case h
α : Type u_1
u v : α → ℝ
l : Filter α
hv : 0 ≤ v
h : u ~[l] v
r : ℝ
φ : α → ℝ
hφ : Tendsto φ l (𝓝 1)
huφv : u =ᶠ[l] φ * v
hφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)
⊢ u ^ r =ᶠ[l] (fun x ↦ x ^ r) ∘ φ * v ^ r
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 := 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|

Tactic:
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.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e_a
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) = Matrix.of ⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)

Tactic:
ext

Goal state after:
case e_a
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
i✝ j✝ : ι
⊢ Matrix.of (Subtype.val ∘ ⇑b₁) i✝ j✝ = Matrix.of (⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)) i✝ j✝
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case e_a
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 ∘ ⇑b₂) = Matrix.of ⇑(Basis.ofZLatticeBasis ℝ L₂ b₂)

Tactic:
ext

Goal state after:
case e_a
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
i✝ j✝ : ι
⊢ Matrix.of (Subtype.val ∘ ⇑b₂) i✝ j✝ = Matrix.of (⇑(Basis.ofZLatticeBasis ℝ L₂ b₂)) i✝ j✝
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: α : Type u_1
inst✝ : NormedAddCommGroup α
f : ℕ → α
g : ℕ → ℝ
h : f =o[atTop] g
hg : 0 ≤ g
h'g : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, g i) atTop atTop
A : ∀ (i : ℕ), ‖g i‖ = g i
n : ℕ
⊢ ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i

Tactic:
rwa [Real.norm_eq_abs, abs_sum_of_nonneg']

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
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x
l : IntegrationParams
y : E
hy : HasIntegral I l f μ.toBoxAdditive.toSMul y
⊢ IntegrableOn f (↑I) μ

Tactic:
constructor

Goal state after:
case left
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x
l : IntegrationParams
y : E
hy : HasIntegral I l f μ.toBoxAdditive.toSMul y
⊢ AEStronglyMeasurable f (μ.restrict ↑I)

case right
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
hb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
hc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x
l : IntegrationParams
y : E
hy : HasIntegral I l f μ.toBoxAdditive.toSMul y
⊢ HasFiniteIntegral f (μ.restrict ↑I)
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
ι : Type u_3
inst✝ : Fintype ι
b : Basis ι ℤ ↥L
s : Set E
hs : NullMeasurableSet s volume
⊢ Fintype.card (Fin (finrank ℝ E)) = Fintype.card ι

Tactic:
by rw [Fintype.card_fin, finrank_eq_card_basis (b.ofZLatticeBasis ℝ)]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
⊢ Bornology.IsBounded (range f) ↔ f =O[cocompact D] 1

Tactic:
constructor

Goal state after:
case mp
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
⊢ Bornology.IsBounded (range f) → f =O[cocompact D] 1

case mpr
E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
D : Type u_2
inst✝ : TopologicalSpace D
f : D → E
hf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
⊢ f =O[cocompact D] 1 → Bornology.IsBounded (range f)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞ ↔ ∑' (b : ℕ), ↑(f b) = ∞

Tactic:
constructor

Goal state after:
case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
⊢ ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞ → ∑' (b : ℕ), ↑(f b) = ∞

case mpr
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
⊢ ∑' (b : ℕ), ↑(f b) = ∞ → ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
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]
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) ↔ x ∈ RHS

Tactic:
constructor

Goal state after:
case mp
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] := ⋯
⊢ x ∈ Ideal.span (⇑(of k G) '' s) → x ∈ RHS

case mpr
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] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (⇑(of k G) '' s)
Module: Mathlib.Algebra.MonoidAlgebra.Ideal

Goal state before:
pp: k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] :=
  { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ x ∈ Ideal.span (of' k A '' s) ↔ x ∈ RHS

Tactic:
constructor

Goal state after:
case mp
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ Ideal.span (of' k A '' s) → x ∈ RHS

case mpr
k : Type u_1
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : Semiring k
s : Set A
x : k[A]
RHS : Ideal k[A] := ⋯
⊢ x ∈ RHS → x ∈ Ideal.span (of' k A '' s)
Module: Mathlib.Data.Finset.SMulAntidiagonal

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt ↔ a = hs.min hns ∧ b = ht.min hnt

Tactic:
constructor

Goal state after:
case mp
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt → a = hs.min hns ∧ b = ht.min hnt

case mpr
G : Type u_1
P : Type u_2
inst✝³ : LinearOrder G
inst✝² : LinearOrder P
inst✝¹ : SMul G P
inst✝ : IsOrderedCancelSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hns : s.Nonempty
hnt : t.Nonempty
a : G
b : P
⊢ a = hs.min hns ∧ b = ht.min hnt → a ∈ s ∧ b ∈ t ∧ a • b = hs.min hns • ht.min hnt
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
⊢ s 0 = {0}

Tactic:
ext

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
a✝ : ι → ℤ
⊢ a✝ ∈ s 0 ↔ a✝ ∈ {0}
Module: Mathlib.Algebra.Module.ZLattice.Summable

Goal state before:
pp: case e_a.e_f
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
hs0 : s 0 = {0}
hs : ∀ {a b : ℕ}, a ≤ b → s a ⊆ s b
this : ∀ (k : ℕ), #(s (k + 1) \ s k) ≤ 2 * d * (2 * k + 3) ^ (d - 1)
ε : ℝ := normBound b
hε : 0 < ε
k : ℕ
⊢ (↑k + 1) ^ (d - 1) * (↑k + 1) ^ r = ↑(k + 1) ^ (↑d - 1 + r)

Tactic:
rw [← Real.rpow_natCast, ← Real.rpow_add (by positivity), Nat.cast_sub hd]

Goal state after:
case e_a.e_f
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝² : DiscreteTopology ↥L
ι : Type u_3
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℤ ↥L
n : ℕ
r : ℝ
s : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (-↑n) ↑n
d : ℕ := Fintype.card ι
hr : r < -↑d
hr' : r < 0
hd : 1 ≤ d
hs0 : s 0 = {0}
hs : ∀ {a b : ℕ}, a ≤ b → s a ⊆ s b
this : ∀ (k : ℕ), #(s (k + 1) \ s k) ≤ 2 * d * (2 * k + 3) ^ (d - 1)
ε : ℝ := normBound b
hε : 0 < ε
k : ℕ
⊢ (↑k + 1) ^ (↑d - ↑1 + r) = ↑(k + 1) ^ (↑d - 1 + r)