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

Goal state before:
pp: case inl
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 : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t = 0
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (s ∩ ({x} + b • t)) / μ ({x} + b • t) < ε

pp: case inr
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 : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (s ∩ ({x} + b • t)) / μ ({x} + b • t) < ε

Tactic:
· filter_upwards with r
    suffices H : μ (s ∩ ({x} + r • t)) = 0 by
      rw [H]; simpa only [ENNReal.zero_div] using εpos
    rw [← nonpos_iff_eq_zero]
    calc
      μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono inter_subset_right
      _ = 0 := by
        simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
          mul_zero]

Goal state after:
case inr
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 : Set E
ht : MeasurableSet t
h''t : μ t ≠ ∞
ε : ℝ≥0∞
εpos : 0 < ε
h't : μ t ≠ 0
⊢ ∀ᶠ (b : ℝ) in 𝓝[>] 0, μ (s ∩ ({x} + b • t)) / μ ({x} + b • t) < ε
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π₁ π₂ : Prepartition I
h₁ : π₁.IsPartition
h₂ : π₂.IsPartition
⊢ (π₁ ⊓ π₂).iUnion = ↑I

Tactic:
simp [h₁.iUnion_eq, h₂.iUnion_eq]

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

Goal state before:
pp: ι : Type u_1
I J : Box ι
i : ι
x : ℝ
y : ι → ℝ
h₁ : J ∈ split I i x
h₂ : y ∈ ↑I ∩ {y | y i ≤ x}
h₃ : x < y i
H : ↑J = ↑I ∩ {y | y i ≤ x}
⊢ False

Tactic:
exact h₃.not_ge h₂.2

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

Goal state before:
pp: K : Type u_1
inst✝⁹ : NormedField K
inst✝⁸ : LinearOrder K
inst✝⁷ : IsStrictOrderedRing K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
ι : Type u_3
hs : IsZLattice K L
b : Basis ι ℤ ↥L
i : ι
⊢ (ofZLatticeBasis K L b) i = ↑(b i)

Tactic:
by
  simp [Basis.ofZLatticeBasis]

Goal state after:
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 : EuclideanSpace 𝕜 ι
⊢ ∑ x_1, x.ofLp x_1 • b ((Equiv.refl ι) x_1) = ∑ x_1, x.ofLp x_1 • b x_1

Tactic:
Equiv.refl_apply

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'
⊢ orthonormalBasisOneI.repr.trans (v.repr.symm.trans f) = (orthonormalBasisOneI.repr.trans v.repr.symm).trans f

Tactic:
LinearIsometryEquiv.trans_assoc

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.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_4
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))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = 1 / (↑n ^ card ι).toReal * F x✝

Tactic:
ENNReal.toReal_pow,

Goal state after:
case refine_4
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))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = 1 / (↑n).toReal ^ card ι * F x✝
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
i j : ι
hij : i < j
b : Basis ι 𝕜 E
this✝¹ : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt 𝕜 ⇑b '' Set.Iio j)
this✝ : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (⇑b '' Set.Iio j)
this : ↑(b.repr (gramSchmidt 𝕜 (⇑b) i)).support ⊆ Set.Iio j
⊢ (b.repr (gramSchmidt 𝕜 (⇑b) i)) j = 0

Tactic:
exact (Finsupp.mem_supported' _ _).1 ((Finsupp.mem_supported 𝕜 _).2 this) j Set.self_notMem_Iio

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

Goal state before:
pp: ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (μ ↑I < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε

Tactic:
have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp [N]

Goal state after:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (μ ↑I < ∞)
hf : (μ.restrict ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x ↦ ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        l.MemBaseSet I c (r c) π → π.IsPartition → dist (integralSum f μ.toBoxAdditive.toSMul π) 0 ≤ ↑ε
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ
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))
b : OrthonormalBasis (Fin (n✝ + 1)) ℝ E
hb : b.toBasis.orientation = o
⊢ o.volumeForm = b.toBasis.det

Tactic:
· simp_rw [volumeForm]
    rw [same_orientation_iff_det_eq_det, hb]
    exact o.finOrthonormalBasis_orientation _ _

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

Goal state before:
pp: case succ.h0
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
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := ⋯
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ ∀ i ∈ Finset.univ, 0 ≤ |⟪b i, v i⟫|

Tactic:
intro i _

Goal state after:
case succ.h0
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
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
a✝ : i ∈ Finset.univ
⊢ 0 ≤ |⟪b i, v i⟫|
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case succ.h1
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
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := ⋯
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
⊢ ∀ i ∈ Finset.univ, |⟪b i, v i⟫| ≤ ‖v i‖

Tactic:
intro i _

Goal state after:
case succ.h1
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
this✝ : FiniteDimensional ℝ E
this : finrank ℝ E = Fintype.card (Fin n.succ)
b : OrthonormalBasis (Fin n.succ) ℝ E := gramSchmidtOrthonormalBasis this v
hb : b.toBasis.det v = ∏ i, ⟪b i, v i⟫
i : Fin n.succ
a✝ : i ∈ Finset.univ
⊢ |⟪b i, v i⟫| ≤ ‖v i‖
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: case neg.e_f
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⟫
h : ∀ (i : Fin (n + 1)), v i ≠ 0
i : Fin n.succ
⊢ |⟪b i, v i⟫| = ‖v i‖

Tactic:
have hb : b i = ‖v i‖⁻¹ • v i := gramSchmidtOrthonormalBasis_apply_of_orthogonal hdim hv (h i)

Goal state after:
case neg.e_f
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⟫
h : ∀ (i : Fin (n + 1)), v i ≠ 0
i : Fin n.succ
hb : b i = ‖v i‖⁻¹ • v i
⊢ |⟪b i, v i⟫| = ‖v i‖
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 ⋯
⊢ ‖f y‖ = ‖y‖

Tactic:
rw [coe_norm, ← norm_orthogonalProjectionOnto_apply Kᗮ (f y).2,
      orthogonalProjectionOnto_orthogonal, coe_norm, starProjection_minimal, eq_comm]

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 ⋯
⊢ ‖y‖ = ⨅ x, ‖↑(f y) - ↑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✝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
x y : E ⧸ K
r : 𝕜
⊢ ⟪r • K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal y⟫_𝕜 =
    (starRingEnd 𝕜) r * ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal y⟫_𝕜

Tactic:
inner_smul_left

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
x y : E ⧸ K
r : 𝕜
⊢ (starRingEnd 𝕜) r * ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal y⟫_𝕜 =
    (starRingEnd 𝕜) r * ⟪K.quotientEquivOrthogonal x, K.quotientEquivOrthogonal y⟫_𝕜
Module: Mathlib.Analysis.BoxIntegral.Basic

Goal state before:
pp: ι : Type u
E : Type v
F : Type w
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
I : Box ι
inst✝¹ : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst✝ : CompleteSpace F
h : Integrable I l f vol
π₁ π₂ : Prepartition I
hU : π₁.iUnion = π₂.iUnion
⊢ ∑ J ∈ π₁.boxes, integral J l f vol = ∑ J ∈ π₂.boxes, integral J l f vol

Tactic:
by
  refine tendsto_nhds_unique (h.tendsto_integralSum_sum_integral π₁) ?_
  rw [l.toFilteriUnion_congr _ hU]
  exact h.tendsto_integralSum_sum_integral π₂

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
⊢ covolume L₁ volume / covolume L₂ volume = ↑(L₁.toAddSubgroup.relIndex L₂.toAddSubgroup)

Tactic:
let b₁ := IsZLattice.basis L₁

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
⊢ covolume L₁ volume / covolume L₂ volume = ↑(L₁.toAddSubgroup.relIndex L₂.toAddSubgroup)
Module: Mathlib.Analysis.PSeries

Goal state before:
pp: α : Type u_1
inst✝² : Field α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
k n : ℕ
⊢ ((↑k + 1) ^ 2)⁻¹ + (↑k + 1)⁻¹ ≤ 1 / (↑k + 1) + 1 / (↑k + 1)

Tactic:
by
      have A : (1 : α) ≤ k + 1 := by simp only [le_add_iff_nonneg_left, Nat.cast_nonneg]
      simp_rw [← one_div]
      gcongr
      simpa using pow_right_mono₀ A one_le_two

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

Goal state before:
pp: 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
ι : Type u_3
b : Basis ι ℤ ↥L
x : ↥L
n : ℕ
hxn : ‖x‖ < normBound b * ↑n
i : ι
⊢ (normBound b)⁻¹ * ‖x‖ < ↑↑n

Tactic:
rwa [inv_mul_lt_iff₀ (normBound_pos b)]

Goal state after:
Module: Mathlib.Data.Finset.SMulAntidiagonal

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝³ : Preorder G
inst✝² : Preorder P
inst✝¹ : SMul G P
inst✝ : IsOrderedSMul G P
s : Set G
t : Set P
hs : s.IsPWO
ht : t.IsPWO
⊢ (s • t).IsPWO

Tactic:
← @image_smul_prod

Goal state after:
G : Type u_1
P : Type u_2
inst✝³ : Preorder G
inst✝² : Preorder P
inst✝¹ : SMul G P
inst✝ : IsOrderedSMul G P
s : Set G
t : Set P
hs : s.IsPWO
ht : t.IsPWO
⊢ ((fun x ↦ x.1 • x.2) '' s ×ˢ t).IsPWO
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: ι : Type u_1
inst✝³ : Fintype ι
L : Submodule ℤ (ι → ℝ)
inst✝² : DiscreteTopology ↥L
inst✝¹ : IsZLattice ℝ L
X : Set (ι → ℝ)
hX : ∀ ⦃x : ι → ℝ⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : (ι → ℝ) → ℝ
h₁ : ∀ (x : ι → ℝ) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x
h₂ : Bornology.IsBounded {x | x ∈ X ∧ F x ≤ 1}
h₃ : MeasurableSet {x | x ∈ X ∧ F x ≤ 1}
h₄ : volume (frontier {x | x ∈ X ∧ F x ≤ 1}) = 0
inst✝ : Nonempty ι
e : Free.ChooseBasisIndex ℤ ↥L ≃ ι := equivOfCardEq ⋯
b : Basis ι ℤ ↥L := (Free.chooseBasis ℤ ↥L).reindex e
⊢ volume (frontier (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0

Tactic:
rw [frontier_equivFun, volume_image_eq_volume_div_covolume, h₄, ENNReal.zero_div]

Goal state after:
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]
⊢ (f * g).toDirectSum = f.toDirectSum * g.toDirectSum

Tactic:
let to_hom : AddMonoidAlgebra M ι →+ ⨁ _ : ι, M :=
  { toFun := toDirectSum
    map_zero' := toDirectSum_zero
    map_add' := toDirectSum_add }

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' := ⋯ }
⊢ (f * g).toDirectSum = f.toDirectSum * g.toDirectSum
Module: Mathlib.Algebra.MvPolynomial.Comap

Goal state before:
pp: σ : Type u_1
τ : Type u_2
υ : Type u_3
R : Type u_4
inst✝ : CommSemiring R
f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R
g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R
x : υ → R
i : σ
⊢ (eval₂Hom (algebraMap R R) x).comp (algebraMap R (MvPolynomial υ R)) = algebraMap R R

Tactic:
ext r

Goal state after:
σ : Type u_1
τ : Type u_2
υ : Type u_3
R : Type u_4
inst✝ : CommSemiring R
f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R
g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R
x : υ → R
i : σ
r : R
⊢ ((eval₂Hom (algebraMap R R) x).comp (algebraMap R (MvPolynomial υ R))) r = (algebraMap R R) r
Module: Mathlib.Algebra.MvPolynomial.Nilpotent

Goal state before:
pp: case refine_1
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
P✝ : MvPolynomial σ R
inst✝¹ : Finite σ
this : Fintype σ
α β : Type u_1
inst✝ : Fintype β
e : α ≃ β
h₁ : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)
P : MvPolynomial β R
⊢ (∀ (i : α →₀ ℕ), IsNilpotent (coeff i ((rename ⇑e.symm) P))) ↔ ∀ (i : β →₀ ℕ), IsNilpotent (coeff i P)

Tactic:
(Finsupp.equivCongrLeft e).forall_congr_left

Goal state after:
case refine_1
σ : Type u_1
R : Type u_2
inst✝² : CommRing R
P✝ : MvPolynomial σ R
inst✝¹ : Finite σ
this : Fintype σ
α β : Type u_1
inst✝ : Fintype β
e : α ≃ β
h₁ : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)
P : MvPolynomial β R
⊢ (∀ (b : β →₀ ℕ), IsNilpotent (coeff ((Finsupp.equivCongrLeft e).symm b) ((rename ⇑e.symm) P))) ↔
    ∀ (i : β →₀ ℕ), IsNilpotent (coeff i P)
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: case pos
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
⊢ ∃ a',
    InvImage (GameAdd (DFinsupp.Lex r s) (DFinsupp.Lex r s)) snd a' (p, x₁, x₂) ∧
      (fun x ↦ x.2.1.piecewise x.2.2 x.1) a' = x

Tactic:
refine ⟨⟨{ j | r j i → j ∈ p }, piecewise x₁ x { j | r j i }, x₂⟩,
      .fst ⟨i, fun j hj ↦ ?_, ?_⟩, ?_⟩

Goal state after:
case pos.refine_1
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
j : ι
hj : r j i
⊢ (x₁.piecewise x {j | r j i}) j = x₁ j

case pos.refine_2
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
⊢ (fun {i} ↦ s i) ((x₁.piecewise x {j | r j i}) i) (x₁ i)

case pos.refine_3
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
⊢ (fun x ↦ x.2.1.piecewise x.2.2 x.1) ({j | r j i → j ∈ p}, x₁.piecewise x {j | r j i}, x₂) = x
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: case pos.refine_3
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
j : ι
⊢ (if r j i → j ∈ p then if r j i then x₁ j else x j else x₂ j) = x j

Tactic:
split_ifs with h₁ h₂

Goal state after:
case pos
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
j : ι
h₁ : r j i → j ∈ p
h₂ : r j i
⊢ x₁ j = x j

case neg
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
j : ι
h₁ : r j i → j ∈ p
h₂ : ¬r j i
⊢ x j = x j

case neg
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∈ p
hs : s i (x i) (x₁ i)
j : ι
h₁ : ¬(r j i → j ∈ p)
⊢ x₂ j = x j
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: case neg.refine_3
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
j : ι
⊢ (if r j i ∧ j ∈ p then x₁ j else if r j i then x₂ j else x j) = x j

Tactic:
split_ifs with h₁ h₂

Goal state after:
case pos
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
j : ι
h₁ : r j i ∧ j ∈ p
⊢ x₁ j = x j

case pos
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
j : ι
h₁ : ¬(r j i ∧ j ∈ p)
h₂ : r j i
⊢ x₂ j = x j

case neg
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
inst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)
p : Set ι
x₁ x₂ x : Π₀ (i : ι), α i
i : ι
hr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j
hp : i ∉ p
hs : s i (x i) (x₂ i)
j : ι
h₁ : ¬(r j i ∧ j ∈ p)
h₂ : ¬r j i
⊢ x j = x j
Module: Mathlib.Data.DFinsupp.WellFounded

Goal state before:
pp: ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)
b : ι
t : Finset ι
hb : b ∉ t
ih :
  ∀ (x : Π₀ (i : ι), α i), x.support = t → (∀ i ∈ t, Acc (DFinsupp.Lex r s) (single i (x i))) → Acc (DFinsupp.Lex r s) x
x : Π₀ (i : ι), α i
ht : x.support = insert b t
h : ∀ i ∈ insert b t, Acc (DFinsupp.Lex r s) (single i (x i))
⊢ (erase b x).support = t

Tactic:
support_erase,

Goal state after:
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
r : ι → ι → Prop
s : (i : ι) → α i → α i → Prop
hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)
b : ι
t : Finset ι
hb : b ∉ t
ih :
  ∀ (x : Π₀ (i : ι), α i), x.support = t → (∀ i ∈ t, Acc (DFinsupp.Lex r s) (single i (x i))) → Acc (DFinsupp.Lex r s) x
x : Π₀ (i : ι), α i
ht : x.support = insert b t
h : ∀ i ∈ insert b t, Acc (DFinsupp.Lex r s) (single i (x i))
⊢ x.support.erase b = t