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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
e₁ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] (i : ι) → ↥(V i) := linearEquivFunOnFintype 𝕜 ι fun i ↦ ↥(V i)
e₂ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] E := LinearEquiv.ofBijective (coeLinearMap V) hV
v w : PiLp 2 fun i ↦ ↥(V i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
trans ⟪∑ i, (V i).subtypeₗᵢ (v i), ∑ i, (V i).subtypeₗᵢ (w i)⟫

Goal state after:
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
e₁ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] (i : ι) → ↥(V i) := ⋯
e₂ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] E := ⋯
v w : PiLp 2 fun i ↦ ↥(V i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁸ : RCLike 𝕜
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace ℝ F
F' : Type u_6
inst✝³ : NormedAddCommGroup F'
inst✝² : InnerProductSpace ℝ F'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : IsInternal V
hV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
e₁ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] (i : ι) → ↥(V i) := ⋯
e₂ : (⨁ (i : ι), ↥(V i)) ≃ₗ[𝕜] E := ⋯
v w : PiLp 2 fun i ↦ ↥(V i)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
⊢ Orthonormal 𝕜 fun i ↦ single i 1

Tactic:
by
  simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,
    PiLp.single_apply]
  intros
  trivial

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

Goal state before:
pp: case «0».«0»
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
⊢ inner ℝ (basisOneI ((fun i ↦ i) ⟨0, ⋯⟩)) (basisOneI ((fun i ↦ i) ⟨0, ⋯⟩)) =
    if (fun i ↦ i) ⟨0, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩ then 1 else 0

Tactic:
simp [real_inner_eq_re_inner]

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

Goal state before:
pp: case «0».«1»
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
⊢ inner ℝ (basisOneI ((fun i ↦ i) ⟨0, ⋯⟩)) (basisOneI ((fun i ↦ i) ⟨1, ⋯⟩)) =
    if (fun i ↦ i) ⟨0, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ then 1 else 0

Tactic:
simp [real_inner_eq_re_inner]

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

Goal state before:
pp: case «1».«0»
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
⊢ inner ℝ (basisOneI ((fun i ↦ i) ⟨1, ⋯⟩)) (basisOneI ((fun i ↦ i) ⟨0, ⋯⟩)) =
    if (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨0, ⋯⟩ then 1 else 0

Tactic:
simp [real_inner_eq_re_inner]

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

Goal state before:
pp: case «1».«1»
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁷ : RCLike 𝕜
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : InnerProductSpace 𝕜 E
F : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_6
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
⊢ inner ℝ (basisOneI ((fun i ↦ i) ⟨1, ⋯⟩)) (basisOneI ((fun i ↦ i) ⟨1, ⋯⟩)) =
    if (fun i ↦ i) ⟨1, ⋯⟩ = (fun i ↦ i) ⟨1, ⋯⟩ then 1 else 0

Tactic:
simp [real_inner_eq_re_inner]

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

Goal state before:
pp: case pos
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
a : ι × ℝ
s : Finset (ι × ℝ)
a✝ : a ∉ s
ihs : ∑ J ∈ (splitMany I s).boxes, f J = f I
J' : Box ι
hJ' : J' ∈ (splitMany I s).boxes
h : a.2 ∈ Set.Ioo (J'.lower a.1) (J'.upper a.1)
⊢ Option.elim' 0 f (J'.splitLower a.1 a.2) + Option.elim' 0 f (J'.splitUpper a.1 a.2) = f J'

Tactic:
exact hf _ ((WithTop.coe_le_coe.2 <| le_of_mem _ hJ').trans hI) h

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

Goal state before:
pp: 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 : ι
this : span 𝕜 (gramSchmidtNormed 𝕜 f '' Set.Iic j) ⟂ 𝕜 ∙ (gramSchmidtOrthonormalBasis h f) i
⊢ f j ∈ (𝕜 ∙ (gramSchmidtOrthonormalBasis h f) i)ᗮ

Tactic:
apply this

Goal state after:
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 : ι
this : span 𝕜 (gramSchmidtNormed 𝕜 f '' Set.Iic j) ⟂ 𝕜 ∙ (gramSchmidtOrthonormalBasis h f) i
⊢ f j ∈ span 𝕜 (gramSchmidtNormed 𝕜 f '' Set.Iic j)
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 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 j : ι
hij : i < j
⊢ ((gramSchmidtOrthonormalBasis h f).repr (f i)).ofLp j = 0

Tactic:
by
  simpa [OrthonormalBasis.repr_apply_apply] using gramSchmidtOrthonormalBasis_inv_triangular h f hij

Goal state after:
Module: Mathlib.LinearAlgebra.Orientation

Goal state before:
pp: case h
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm
H : LinearIndependent R ![f x, 1]
⊢ False

Tactic:
rw [linearIndependent_iff'] at H

Goal state after:
case h
R : Type u_1
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
M : Type u_2
inst✝² : AddCommGroup M
inst✝¹ : Module R M
ι : Type u_3
inst✝ : IsEmpty ι
x : M [⋀^ι]→ₗ[R] R
hx : x ≠ 0
h : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]
f : M [⋀^ι]→ₗ[R] R ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm
H :
  ∀ (s : Finset (Fin (Nat.succ 0).succ)) (g : Fin (Nat.succ 0).succ → R),
    ∑ i ∈ s, g i • ![f x, 1] i = 0 → ∀ i ∈ s, g i = 0
⊢ False
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
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
h : Integrable I l f vol
⊢ HasIntegral I l f vol (integral I l f vol)

Tactic:
rw [integral, dif_pos h]
  exact Classical.choose_spec h

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : b.toBasis.orientation ≠ o
⊢ positiveOrientation ≠ o

Tactic:
b.toBasis.orientation_isEmpty

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
_i : Fact (finrank ℝ E = 0)
o : Orientation ℝ E (Fin 0)
b : OrthonormalBasis (Fin 0) ℝ E
hb : positiveOrientation ≠ o
⊢ positiveOrientation ≠ o
Module: Mathlib.Analysis.Asymptotics.SpecificAsymptotics

Goal state before:
pp: case h
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
B : ∀ (n : ℕ), ‖∑ i ∈ Finset.range n, g i‖ = ∑ i ∈ Finset.range n, g i
ε : ℝ
εpos : 0 < ε
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b
this : (fun x ↦ ∑ i ∈ Finset.range N, f i) =o[atTop] fun n ↦ ∑ i ∈ Finset.range n, g i
n : ℕ
hn : ‖∑ i ∈ Finset.range N, f i‖ ≤ ε / 2 * ‖∑ i ∈ Finset.range n, g i‖
Nn : n ∈ Set.Ici N
⊢ Finset.Ico N n ⊆ Finset.range n

Tactic:
· rw [range_eq_Ico]
        exact Ico_subset_Ico zero_le le_rfl

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
ι : Type u_2
b : Basis ι ℤ ↥L
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : Submodule ℤ E}
    [inst_3 : DiscreteTopology ↥L] {ι : Type u_2} (b : Basis ι ℤ ↥L),
    IsZLattice ℝ L → ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
H : ¬IsZLattice ℝ L
E' : Submodule ℝ E := Submodule.span ℝ ↑L
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖

Tactic:
let L' : Submodule ℤ E' := ZLattice.comap ℝ L E'.subtype

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
ι : Type u_2
b : Basis ι ℤ ↥L
this :
  ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {L : Submodule ℤ E}
    [inst_3 : DiscreteTopology ↥L] {ι : Type u_2} (b : Basis ι ℤ ↥L),
    IsZLattice ℝ L → ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
H : ¬IsZLattice ℝ L
E' : Submodule ℝ E := Submodule.span ℝ ↑L
L' : Submodule ℤ ↥E' := ZLattice.comap ℝ L E'.subtype
⊢ ∃ ε, 0 < ε ∧ ∀ (x : ↥L) (i : ι), ε * ↑|(b.repr x) i| ≤ ‖x‖
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
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)
r : ℝ
hr : 0 < r
hr' : ↑u ⊆ Metric.closedBall 0 r
x : I → ℤ
hx : x ∈ u
hr'' : max ⌊r⌋ 0 = ⌊r⌋
this : ∀ (i : I), |x i| ≤ ⌊r⌋
⊢ x ∈ Fintype.piFinset fun x ↦ Finset.Icc (-↑⌊r⌋.toNat) ↑⌊r⌋.toNat

Tactic:
simpa only [Int.ofNat_toNat, Fintype.mem_piFinset, Finset.mem_Icc, ← abs_le, hr'']

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

Goal state before:
pp: E : Type u_1
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
inst✝ : Nontrivial E
X : Set E
F : E → ℝ
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
h₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ finrank ℝ E * 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
⊢ Tendsto (fun c ↦ ↑(Nat.card ↑({x | x ∈ X ∧ F x ≤ c} ∩ ↑L)) / c) atTop
    (𝓝 (volume.real {x | x ∈ X ∧ F x ≤ 1} / covolume L volume))

Tactic:
let b := Module.Free.chooseBasis ℤ L
  convert! tendsto_card_le_div'' b hX ?_ h₂ h₃ ?_
  · simp only [measureReal_def]
    rw [volume_image_eq_volume_div_covolume' L b h₃.nullMeasurableSet, ENNReal.toReal_div,
      ENNReal.toReal_ofReal (covolume_pos L volume).le]
  · have : Nontrivial L := nontrivial_of_finrank_pos <| (ZLattice.rank ℝ L).symm ▸ finrank_pos
    infer_instance
  · rwa [← finrank_eq_card_chooseBasisIndex, ZLattice.rank ℝ L]
  · rw [frontier_equivFun, volume_image_eq_volume_div_covolume', h₄, ENNReal.zero_div]
    exact NullMeasurableSet.of_null h₄

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

Goal state before:
pp: case hbc.H₁
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) := ⋯
μ' : Measure (ι → ℝ) := ⋯
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 := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
a✝ : J ∈ B'
⊢ ‖f (t₁ J)‖ ≤ C

Tactic:
exact hC _ (TaggedPrepartition.tag_mem_Icc _ J)

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

Goal state before:
pp: case H₂
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) := ⋯
μ' : Measure (ι → ℝ) := ⋯
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 := ⋯
t₂ : Box ι → ι → ℝ := ⋯
B : Finset (Box ι) := ⋯
B' : Finset (Box ι) := ⋯
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ∞
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
a✝ : J ∈ B'
⊢ ‖f (t₂ J)‖ ≤ C

Tactic:
exact hC _ (TaggedPrepartition.tag_mem_Icc _ J)

Goal state after:
Module: Mathlib.Data.Nat.Factorial.DoubleFactorial

Goal state before:
pp: n : ℕ
⊢ (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * (n.succ * n !)

Tactic:
succ_eq_add_one

Goal state after:
n : ℕ
⊢ (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * ((n + 1) * n !)
Module: Mathlib.Algebra.MvPolynomial.Comap

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
f : σ → τ
x : τ → R
⊢ comap (rename f) x = x ∘ f

Tactic:
by
  funext
  simp [rename_X, comap_apply, aeval_X]

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

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∈ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∉ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i

Tactic:
· simp [Finsupp.indicator_of_mem hi]

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
a : σ →₀ R
n : ℕ
x : σ → ℕ
hs : ((indicator a.support fun i x_1 ↦ x i).sum fun x m ↦ m) ≠ n
hx : a.support.sum x = n ∧ ∀ (i : σ), x i ≠ 0 → i ∈ a.support
i : σ
hi : i ∉ a.support
⊢ (indicator a.support fun i x_1 ↦ x i) i = x i
Module: Mathlib.Algebra.MvPolynomial.Division

Goal state before:
pp: case mp
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
⊢ (monomial i) r ∣ (monomial j) s → (s = 0 ∨ i ≤ j) ∧ r ∣ s

Tactic:
rintro ⟨x, hx⟩

Goal state after:
case mp
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
r s : R
i j : σ →₀ ℕ
x : MvPolynomial σ R
hx : (monomial j) s = (monomial i) r * x
⊢ (s = 0 ∨ i ≤ j) ∧ r ∣ s
Module: Mathlib.Algebra.MvPolynomial.Expand

Goal state before:
pp: case inl
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
p : ℕ
f : MvPolynomial σ R
hp : p = 0
⊢ ((expand p) f).totalDegree = f.totalDegree * p

pp: case inr
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
p : ℕ
f : MvPolynomial σ R
hp : p > 0
⊢ ((expand p) f).totalDegree = f.totalDegree * p

Tactic:
· simp [hp]

Goal state after:
case inr
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
p : ℕ
f : MvPolynomial σ R
hp : p > 0
⊢ ((expand p) f).totalDegree = f.totalDegree * p
Module: Mathlib.Algebra.MvPolynomial.Funext

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
σ : Type u_2
p q : MvPolynomial σ R
s : σ → Set R
hs : ∀ (i : σ), (s i).Infinite
h : ∀ x ∈ Set.univ.pi s, (eval x) p = (eval x) q
⊢ p = q

Tactic:
suffices ∀ p, (∀ x ∈ Set.pi .univ s, eval x p = 0) → p = 0 by
    rw [← sub_eq_zero, this (p - q)]
    intro x hx
    simp_rw [map_sub, h x hx, sub_self]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
σ : Type u_2
p q : MvPolynomial σ R
s : σ → Set R
hs : ∀ (i : σ), (s i).Infinite
h : ∀ x ∈ Set.univ.pi s, (eval x) p = (eval x) q
⊢ ∀ (p : MvPolynomial σ R), (∀ x ∈ Set.univ.pi s, (eval x) p = 0) → p = 0
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: α : Type u_1
r : α → α → Prop
inst✝ : Std.Trichotomous r
hr : WellFounded (Function.swap r)
s : ℕ → ℕ → Prop
hs : WellFounded s
hs0 : ∀ ⦃n : ℕ⦄, ¬s n 0
wft : WellFounded (Prod.Lex s (Finsupp.Lex r s))
⊢ WellFounded (Finsupp.DegLex r s)

Tactic:
rw [← Set.wellFoundedOn_univ] at wft

Goal state after:
α : Type u_1
r : α → α → Prop
inst✝ : Std.Trichotomous r
hr : WellFounded (Function.swap r)
s : ℕ → ℕ → Prop
hs : WellFounded s
hs0 : ∀ ⦃n : ℕ⦄, ¬s n 0
wft : Set.univ.WellFoundedOn (Prod.Lex s (Finsupp.Lex r s))
⊢ WellFounded (Finsupp.DegLex r s)
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

Goal state before:
pp: case inl
α : Type u_1
inst✝ : LinearOrder α
a b c : DegLex (α →₀ ℕ)
hbc :
  degree (ofDegLex b) < degree (ofDegLex c) ∨
    degree (ofDegLex b) = degree (ofDegLex c) ∧ toLex (ofDegLex b) < toLex (ofDegLex c)
hab : degree (ofDegLex a) < degree (ofDegLex b)
⊢ degree (ofDegLex a) < degree (ofDegLex c) ∨
    degree (ofDegLex a) = degree (ofDegLex c) ∧ toLex (ofDegLex a) < toLex (ofDegLex c)

pp: case inr
α : Type u_1
inst✝ : LinearOrder α
a b c : DegLex (α →₀ ℕ)
hbc :
  degree (ofDegLex b) < degree (ofDegLex c) ∨
    degree (ofDegLex b) = degree (ofDegLex c) ∧ toLex (ofDegLex b) < toLex (ofDegLex c)
hab : degree (ofDegLex a) = degree (ofDegLex b) ∧ toLex (ofDegLex a) < toLex (ofDegLex b)
⊢ degree (ofDegLex a) < degree (ofDegLex c) ∨
    degree (ofDegLex a) = degree (ofDegLex c) ∧ toLex (ofDegLex a) < toLex (ofDegLex c)

Tactic:
· rcases hbc with (hbc | hbc)
      · left; exact lt_trans hab hbc
      · left; exact lt_of_lt_of_eq hab hbc.1

Goal state after:
case inr
α : Type u_1
inst✝ : LinearOrder α
a b c : DegLex (α →₀ ℕ)
hbc :
  degree (ofDegLex b) < degree (ofDegLex c) ∨
    degree (ofDegLex b) = degree (ofDegLex c) ∧ toLex (ofDegLex b) < toLex (ofDegLex c)
hab : degree (ofDegLex a) = degree (ofDegLex b) ∧ toLex (ofDegLex a) < toLex (ofDegLex b)
⊢ degree (ofDegLex a) < degree (ofDegLex c) ∨
    degree (ofDegLex a) = degree (ofDegLex c) ∧ toLex (ofDegLex a) < toLex (ofDegLex c)
Module: Mathlib.Algebra.MvPolynomial.NoZeroDivisors

Goal state before:
pp: case pos
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : p = 0
⊢ degreeOf j (C c * p) = degreeOf j p

pp: case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ degreeOf j (C c * p) = degreeOf j p

Tactic:
· simp [hp]

Goal state after:
case neg
R : Type u_1
σ : Type u_2
inst✝ : CommSemiring R
p : MvPolynomial σ R
j : σ
c : R
hc : c ∈ R⁰
hp : ¬p = 0
⊢ degreeOf j (C c * p) = degreeOf j p
Module: Mathlib.Data.Finsupp.MonomialOrder.DegLex

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

Tactic:
simp [degLex_lt_iff, lt_iff]

Goal state after:
Module: Mathlib.Data.Fin.Tuple.Finset

Goal state before:
pp: n : ℕ
α : Fin (n + 1) → Type u_1
f : (i : Fin (n + 1)) → α i
s : (i : Fin (n + 1)) → Finset (α i)
⊢ f ∈ piFinset s ↔ f (last n) ∈ s (last n) ∧ init f ∈ piFinset (init s)

Tactic:
by
  simp only [Fintype.mem_piFinset, forall_fin_succ', init, and_comm]

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

Goal state before:
pp: case inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
hR : Subsingleton R
⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u

Tactic:
suffices ∃ m, m ≤ n by simpa [Subsingleton.elim _ p]

Goal state after:
case inl
R : Type u_1
σ : Type u_2
inst✝¹ : CommRing R
inst✝ : NoZeroDivisors R
p : MvPolynomial σ R
n : σ →₀ ℕ
hR : Subsingleton R
⊢ ∃ m, m ≤ n
Module: Mathlib.RingTheory.MvPolynomial.MonomialOrder

Goal state before:
pp: σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ m.toSyn (m.degree f) ≤ ⊥ ↔ f.totalDegree = 0

Tactic:
m.bot_eq_zero,

Goal state after:
σ : Type u_1
m : MonomialOrder σ
R : Type u_2
inst✝ : CommSemiring R
f : MvPolynomial σ R
⊢ m.toSyn (m.degree f) ≤ 0 ↔ f.totalDegree = 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
hfg : m.leadingCoeff f * m.leadingCoeff g ≠ 0
⊢ m.leadingCoeff f * m.leadingCoeff g ≠ 0

Tactic:
exact hfg

Goal state after: