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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ

Tactic:
have e : E ≃ₗ[ℝ] ι → ℝ := LinearEquiv.ofFinrankEq E (ι → ℝ) this

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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
ι : Type := Fin (finrank ℝ E)
this✝ : FiniteDimensional ℝ (ι → ℝ)
this : finrank ℝ E = finrank ℝ (ι → ℝ)
e : E ≃ₗ[ℝ] ι → ℝ
⊢ map (⇑f) μ = ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ
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
r : ℝ
hr : r ≠ 0
⊢ QuasiMeasurePreserving (fun x ↦ r • x) μ μ

Tactic:
by
  refine ⟨measurable_const_smul r, ?_⟩
  rw [map_addHaar_smul μ hr]
  exact smul_absolutelyContinuous

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝¹ : ProperSpace E
inst✝ : Finite ι
s : Set E
hs : IsBounded s
this : DiscreteTopology ↥(span ℤ (Set.range ⇑b))
⊢ IsClosed ↑(span ℤ (Set.range ⇑b)).toAddSubgroup

Tactic:
exact AddSubgroup.isClosed_of_discrete

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

Goal state before:
pp: case inl
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
⊢ μ {x} = 0

Tactic:
measure_singleton

Goal state after:
case inl
E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : Nontrivial E
x : E
⊢ 0 = 0
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
⊢ ⇑((EuclideanSpace.basisFun ι 𝕜).equiv b (Equiv.refl ι)) = fun x ↦ ∑ x_1, x.ofLp x_1 • b x_1

Tactic:
← equiv_apply_euclideanSpace

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
F' : Type u_6
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
v : OrthonormalBasis (Fin 2) ℝ F
f : F ≃ₗᵢ[ℝ] F'
⊢ isometryOfOrthonormal (v.map f) = (isometryOfOrthonormal v).trans f

Tactic:
simp only [isometryOfOrthonormal, OrthonormalBasis.map, LinearIsometryEquiv.symm_trans,
    LinearIsometryEquiv.symm_symm]

Goal state after:
F : Type u_5
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
F' : Type u_6
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
v : OrthonormalBasis (Fin 2) ℝ F
f : F ≃ₗᵢ[ℝ] F'
⊢ orthonormalBasisOneI.repr.trans (v.repr.symm.trans f) = (orthonormalBasisOneI.repr.trans v.repr.symm).trans f
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_3
inst✝⁵ : RCLike 𝕜
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
m : Type u_7
n : Type u_8
inst✝² : Fintype n
inst✝¹ : DecidableEq n
inst✝ : Fintype m
b : OrthonormalBasis n 𝕜 E
b₂ : OrthonormalBasis m 𝕜 𝕜
x : E
⊢ (toMatrix b.toBasis b₂.toBasis) ((innerₛₗ 𝕜) x) = vecMulVec (star ⇑b₂) (star (b.repr x).ofLp)

Tactic:
by
  ext; simp [LinearMap.toMatrix_apply, vecMulVec_apply, OrthonormalBasis.repr_apply_apply, mul_comm]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: 𝕜 : Type u_7
E : Type u_8
F : Type u_9
ι : Type u_10
ι' : Type u_11
inst✝⁷ : RCLike 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 F
inst✝² : Finite ι
inst✝¹ : Fintype ι'
inst✝ : DecidableEq ι'
x : E
y : F
b : Basis ι 𝕜 E
b' : OrthonormalBasis ι' 𝕜 F
⊢ (toMatrix b'.toBasis b) ↑(((rankOne 𝕜) x) y) = Matrix.vecMulVec (⇑(b.repr x)) (star (b'.repr y).ofLp)

Tactic:
by
  have := Fintype.ofFinite ι
  rw [rankOne_def', ContinuousLinearMap.toLinearMap_comp, toLinearMap_toSpanSingleton,
    toMatrix_comp _ (OrthonormalBasis.singleton Unit 𝕜).toBasis, toMatrix_toSpanSingleton,
    toLinearMap_innerSL_apply, toMatrix_innerₛₗ_apply, OrthonormalBasis.toBasis_singleton,
    Basis.coe_singleton, Matrix.vecMulVec_one, OrthonormalBasis.coe_singleton, star_one,
    Matrix.one_vecMulVec, Matrix.vecMulVec_eq Unit]

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

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
h : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν'
⊢ tag n h.choose = tag n ν

Tactic:
(tag_injective n).eq_iff,

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
inst✝ : Fintype ι
ν : ι → ℤ
B : Box ι
hν : ν ∈ admissibleIndex n B
h : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν'
⊢ h.choose = ν
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
⊢ ∑ J ∈ (prepartition n B).boxes, (volume.toBoxAdditive.toSMul J) (s.indicator F ((prepartition n B).tag J)) =
    (∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i) / ↑n ^ card ι

Tactic:
Finset.sum_div,

Goal state after:
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
⊢ ∑ J ∈ (prepartition n B).boxes, (volume.toBoxAdditive.toSMul J) (s.indicator F ((prepartition n B).tag J)) =
    ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^ card ι
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
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
⊢ gramSchmidt 𝕜 f n ≠ 0

Tactic:
by_contra h

Goal state after:
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
n : ι
h₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)
h : gramSchmidt 𝕜 f n = 0
⊢ False
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁷ : RCLike 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝⁴ : LinearOrder ι
inst✝³ : LocallyFiniteOrderBot ι
inst✝² : WellFoundedLT ι
inst✝¹ : Fintype ι
inst✝ : FiniteDimensional 𝕜 E
h : finrank 𝕜 E = Fintype.card ι
f : ι → E
i : ι
hi : gramSchmidtNormed 𝕜 f i = 0
j k : ι
left✝ : k ∈ Set.Iic j
hk : ¬gramSchmidtNormed 𝕜 f k = 0
this : k ≠ i
⊢ ⟪(gramSchmidtOrthonormalBasis h f) k, (gramSchmidtOrthonormalBasis h f) i⟫_𝕜 = 0

Tactic:
exact (gramSchmidtOrthonormalBasis h f).orthonormal.2 this

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

Goal state before:
pp: 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
h₁ : π₁.IsPartition
h₂ : π₂.IsPartition
⊢ ∑ J ∈ (π₁.infPrepartition π₂.toPrepartition).boxes, (vol J) (f ((π₁.infPrepartition π₂.toPrepartition).tag J)) -
      ∑ J ∈ (π₂.infPrepartition π₁.toPrepartition).boxes, (vol J) (f ((π₂.infPrepartition π₁.toPrepartition).tag J)) =
    ∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, (vol x) (f ((π₁.infPrepartition π₂.toPrepartition).tag x)) -
      ∑ x ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes, (vol x) (f ((π₂.infPrepartition π₁.toPrepartition).tag x))

Tactic:
simp only [infPrepartition_toPrepartition, inf_comm]

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det ((∏ i, w i)⁻¹ • e.det)

Tactic:
e.det.eq_smul_basis_det (e.unitsSMul w),

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det ((∏ i, w i)⁻¹ • e.det ⇑(e.unitsSMul w) • (e.unitsSMul w).det)
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ (e.unitsSMul w).orientation = (∏ i, w i)⁻¹ • e.orientation

Tactic:
rw [Basis.orientation, Basis.orientation, smul_rayOfNeZero, ray_eq_iff,
    e.det.eq_smul_basis_det (e.unitsSMul w), det_unitsSMul_self, Units.smul_def, smul_smul]

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsStrictOrderedRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
e : Basis ι R M
w : ι → Rˣ
⊢ SameRay R (e.unitsSMul w).det ((↑(∏ i, w i)⁻¹ * ∏ i, ↑(w i)) • (e.unitsSMul w).det)
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: 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
h : e.toBasis.det = f.toBasis.det
⊢ e.toBasis.orientation = f.toBasis.orientation

Tactic:
dsimp [Basis.orientation]

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
h : e.toBasis.det = f.toBasis.det
⊢ rayOfNeZero ℝ e.toBasis.det ⋯ = rayOfNeZero ℝ f.toBasis.det ⋯
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
f : (E ⧸ K) ≃ₗ[𝕜] ↥Kᗮ := K.quotientEquivOfIsCompl Kᗮ ⋯
⊢ ‖Kᗮ.orthogonalProjectionOnto ↑(f y)‖ = ‖y‖

Tactic:
orthogonalProjectionOnto_orthogonal,

Goal state after:
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
f : (E ⧸ K) ≃ₗ[𝕜] ↥Kᗮ := K.quotientEquivOfIsCompl Kᗮ ⋯
⊢ ‖⟨↑(f y) - K.starProjection ↑(f y), ⋯⟩‖ = ‖y‖
Module: Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace

Goal state before:
pp: ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
⊢ volume = dirac 0

Tactic:
← (PiLp.volume_preserving_toLp ι).map_eq,

Goal state after:
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
⊢ map (toLp 2) volume = dirac 0
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := fun J ↦ (π₁.infPrepartition π₂.toPrepartition).tag J
⊢ ‖∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
        μ.toBoxAdditive J •
          (f ((π₁.infPrepartition π₂.toPrepartition).tag J) - f ((π₂.infPrepartition π₁.toPrepartition).tag J))‖ ≤
    ε

Tactic:
let t₂ (J : Box ι) : ℝⁿ := (π₂.infPrepartition π₁.toPrepartition).tag J

Goal state after:
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, ediam (f '' (eball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x_1 ↦ ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ∞
t₁ : Box ι → ι → ℝ := fun J ↦ (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J ↦ (π₂.infPrepartition π₁.toPrepartition).tag J
⊢ ‖∑ J ∈ (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes,
        μ.toBoxAdditive J •
          (f ((π₁.infPrepartition π₂.toPrepartition).tag J) - f ((π₂.infPrepartition π₁.toPrepartition).tag J))‖ ≤
    ε
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
⊢ |((Basis.ofZLatticeBasis ℝ L₂ b₂).toMatrix ⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)).det| =
    |↑(b₂.toMatrix fun i ↦ ⟨↑(b₁ i), ⋯⟩).det|

Tactic:
Int.cast_det

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
⊢ |((Basis.ofZLatticeBasis ℝ L₂ b₂).toMatrix ⇑(Basis.ofZLatticeBasis ℝ L₁ b₁)).det| =
    |((b₂.toMatrix fun i ↦ ⟨↑(b₁ i), ⋯⟩).map fun x ↦ ↑x).det|
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: f : ℕ → ℝ≥0∞
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
⊢ ∑' (k : ℕ), 2 ^ k * f (2 ^ k) ≤ f 1 + 2 * ∑' (k : ℕ), f k

Tactic:
ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id),

Goal state after:
f : ℕ → ℝ≥0∞
hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m
⊢ ⨆ i, ∑ a ∈ range i.succ, 2 ^ a * f (2 ^ a) ≤ f 1 + 2 * ∑' (k : ℕ), f k
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✝ : IsOrderedSMul G P
s : Set G
t : Set P
hs : s.IsWF
ht : t.IsWF
hsn : s.Nonempty
htn : t.Nonempty
⊢ ⋯.min ⋯ = hs.min hsn • ht.min htn

Tactic:
refine le_antisymm (IsWF.min_le _ _ (mem_smul.2 ⟨_, hs.min_mem _, _, ht.min_mem _, rfl⟩)) ?_
  rw [IsWF.le_min_iff]
  rintro _ ⟨x, hx, y, hy, rfl⟩
  exact IsOrderedSMul.smul_le_smul (hs.min_le _ hx) (ht.min_le _ hy)

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

Goal state before:
pp: a b : ℕ
⊢ multinomial univ ![a, b] = (a + b)! / (a ! * b !)

Tactic:
rw [multinomial, Fin.sum_univ_two, Fin.prod_univ_two]
  dsimp only [Matrix.cons_val]

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

Goal state before:
pp: ι : Type u_1
s : Finset ι
f : ι → ℕ
⊢ ((2 * ∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, (f i)! =
    ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, 2 ^ f i * (f i)!

Tactic:
by
      rw [mul_pow, ← prod_pow_eq_pow_sum, prod_mul_distrib]; ring

Goal state after:
Module: Mathlib.Data.Finsupp.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
a : WithBot (σ →₀ ℕ)
⊢ m.toWithBotSyn a = ⊥ ↔ a = ⊥

Tactic:
by
  simp [← m.toWithBotSyn.eq_symm_apply]

Goal state after:
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
⊢ x ≤ y → (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y

Tactic:
rw [le_iff]

Goal state after:
α : Type u_1
inst✝ : LinearOrder α
x y : DegLex (α →₀ ℕ)
⊢ degree (ofDegLex x) < degree (ofDegLex y) ∨
      degree (ofDegLex x) = degree (ofDegLex y) ∧ toLex (ofDegLex x) ≤ toLex (ofDegLex y) →
    (fun x ↦ degree (ofDegLex x)) x ≤ (fun x ↦ degree (ofDegLex x)) y
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
r : MvPolynomial σ R
hmn : m ≤ n
hr : r ∣ C a
h : p = (monomial m) 1 * r
⊢ ∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b

Tactic:
rw [dvd_C_iff_exists ha] at hr

Goal state after:
case mp
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
a : R
ha : a ≠ 0
m : σ →₀ ℕ
r : MvPolynomial σ R
hmn : m ≤ n
hr : ∃ b, b ∣ a ∧ r = C b
h : p = (monomial m) 1 * r
⊢ ∃ b, m ≤ n ∧ b ∣ a ∧ p = (monomial m) b
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: α : Type u_1
⊢ degLex.toSyn (single 0 1 + single 1 1) < degLex.toSyn (single 0 2)

Tactic:
rw [degLex_lt_iff, lt_iff, ofDegLex_toDegLex]
  simp only [Fin.isValue, map_add, degree_single, Nat.reduceAdd, ofDegLex_toDegLex,
    lt_self_iff_false, toLex_add, true_and, false_or]
  use 0
  simp

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
hf : m.degree f = 0
d : σ →₀ ℕ
hd : ¬d = 0
⊢ m.toSyn (m.degree f) < m.toSyn d

Tactic:
rw [hf, map_zero, lt_iff_le_and_ne, ne_eq, eq_comm, EmbeddingLike.map_eq_zero_iff]

Goal state after:
case neg
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
hf : m.degree f = 0
d : σ →₀ ℕ
hd : ¬d = 0
⊢ 0 ≤ m.toSyn d ∧ ¬d = 0
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
a b : σ →₀ ℕ
ha : m.toSyn (m.degree f) ≤ m.toSyn a
hb : m.toSyn (m.degree g) ≤ m.toSyn b
⊢ coeff (a + b) (f * g) = coeff a f * coeff b g

Tactic:
by
  classical
  rw [coeff_mul, Finset.sum_eq_single (a, b)]
  · rintro ⟨c, d⟩ hcd h
    simp only [Finset.mem_antidiagonal] at hcd
    by_cases hf : m.degree f ≺[m] c
    · rw [m.coeff_eq_zero_of_lt hf, zero_mul]
    · suffices m.degree g ≺[m] d by
        rw [coeff_eq_zero_of_lt this, mul_zero]
      rw [not_lt] at hf
      rw [← not_le]
      intro hf'
      apply h
      suffices c = a by
        simpa [Prod.mk.injEq, this] using hcd
      apply m.toSyn.injective
      apply le_antisymm (le_trans hf ha)
      apply le_of_add_le_add_right (a := m.toSyn b)
      rw [← map_add, ← hcd, map_add]
      simp only [add_le_add_iff_left]
      exact le_trans hf' hb
  · simp

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f g : MvPolynomial σ R
⊢ m.toWithBotSyn (m.withBotDegree f) ≤ m.toWithBotSyn (m.withBotDegree g) ↔
    m.toSyn (m.degree f) ≤ m.toSyn (m.degree g) ∧ (g = 0 → f = 0)

Tactic:
by_cases! +distrib h : f ≠ 0 ∧ g ≠ 0
  · simp [m.withBotDegree_eq, h, m.toWithBotSyn_apply]
  rcases h with h | _
  · simp [h]
  · aesop

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Zeta

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
f : ArithmeticFunction R
⊢ ↑ζ * f = f * ↑ζ

Tactic:
by
  ext n
  simp_rw [mul_apply, natCoe_apply, (cast_commute ..).eq]
  rw [sum_divisorsAntidiagonal fun x y ↦ f y * ζ x, sum_divisorsAntidiagonal' fun x y ↦ f x * ζ y]

Goal state after:
Module: Mathlib.Data.Nat.Factorization.PrimePow

Goal state before:
pp: n p k : ℕ
hp : Nat.Prime p
hk : 0 < k
hn : p ^ k = n
⊢ n.factorization = Finsupp.single p k

Tactic:
by rw [← hn, Nat.Prime.factorization_pow hp]

Goal state after:
Module: Mathlib.Data.Nat.Squarefree

Goal state before:
pp: n : ℕ
hn : Squarefree n
⊢ ∏ p ∈ n.primeFactors, p = n

Tactic:
by
  rw [← toFinset_factors, List.prod_toFinset _ hn.nodup_primeFactorsList,
    List.map_id', Nat.prod_primeFactorsList hn.ne_zero]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Misc

Goal state before:
pp: n : ℕ
⊢ 0 < Ω n ↔ 1 < n

Tactic:
by
  rw [cardFactors_apply, List.length_pos_iff, primeFactorsList_ne_nil]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Misc

Goal state before:
pp: ι : Type u_2
s : Finset ι
f : ι → ℕ
h : (↑s).Pairwise (Coprime on f)
⊢ ω (∏ i ∈ s, f i) = ∑ i ∈ s, ω (f i)

Tactic:
by
  induction s using cons_induction_on with
  | empty => simp
  | cons a s ha ih =>
    rw [prod_cons, sum_cons, cardDistinctFactors_mul, ih]
    · exact fun x hx y hy hxy => h (by simp [hx]) (by simp [hy]) hxy
    · exact Coprime.prod_right fun i hi =>
        h (by simp) (by simp [hi]) (ne_of_mem_of_not_mem hi ha).symm

Goal state after:
Module: Mathlib.Algebra.Order.Antidiag.Nat

Goal state before:
pp: n : ℕ
⊢ image (⇑(piFinTwoEquiv fun x ↦ ℕ)) (finMulAntidiag 2 n) = n.divisorsAntidiagonal

Tactic:
ext x
  simp [(piFinTwoEquiv <| fun _ => ℕ).symm.surjective.exists]

Goal state after:
Module: Mathlib.NumberTheory.ArithmeticFunction.Misc

Goal state before:
pp: N : ℕ
⊢ ∑ n ∈ Ioc 0 N, (sigma 0) n = ∑ n ∈ Ioc 0 N, N / n

Tactic:
rw [← zeta_mul_pow_eq_sigma, pow_zero_eq_zeta]
  convert! sum_Ioc_mul_zeta_eq_sum zeta N using 1
  simpa using sum_congr rfl (by grind)

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: R : Type u_2
inst✝ : CommRing R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.Monic (X i + -C r)

Tactic:
← map_neg

Goal state after:
R : Type u_2
inst✝ : CommRing R
ι : Type u_3
m : MonomialOrder ι
i : ι
r : R
⊢ m.Monic (X i + C (-r))