mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: case refine_2
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
hs : IsZLattice K L
this✝³ : Module.Finite ℤ ↥L
this✝² : Module ℚ E
this✝¹ : IsAddTorsionFree E
b₀ : Basis (Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Free.chooseBasis ℤ ↥L
b : Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this✝ : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = L
h_spanE : span K (Set.range b) = ⊤
h_card : Fintype.card (Free.ChooseBasisIndex ℤ ↥L) = (Set.range b).toFinset.card
t : Set E
ht_inc : t ⊆ Set.range b
ht_span : span K t = span K (Set.range b)
ht_lin : LinearIndependent K Subtype.val
e : Basis (↑t) K E := Basis.mk ht_lin ⋯
this : Fintype ↑t
h : finrank K E < Fintype.card (Free.ChooseBasisIndex ℤ ↥L)
v : E
hv : v ∈ Set.range b \ Set.range ⇑e
a✝ : LinearIndepOn ℚ id (Set.range ⇑e)
x✝ : ℤ
⊢ ↑(floor e (x✝ • v)) ∈ span ℤ (Set.range b)

Tactic:
· exact span_mono (by simp [e, ht_inc]) (coe_mem _)

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

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
p : ι × ℝ
⊢ ((insert p s).inf fun p ↦ split I p.1 p.2) = splitMany I s ⊓ split I p.1 p.2

Tactic:
Finset.inf_insert,

Goal state after:
ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
p : ι × ℝ
⊢ (split I p.1 p.2 ⊓ s.inf fun p ↦ split I p.1 p.2) = splitMany I s ⊓ split I p.1 p.2
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
v : ι → E
inst✝ : DecidableEq E
v' : ι' → E
h : Orthonormal 𝕜 v'
s : Finset ι'
e₀' : Basis (↥s) 𝕜 ↥(span 𝕜 (range (v' ∘ Subtype.val))) := Basis.span ⋯
e₀ : OrthonormalBasis (↥s) 𝕜 ↥(span 𝕜 (range (v' ∘ Subtype.val))) := OrthonormalBasis.mk ⋯ ⋯
⊢ span 𝕜 ↑(Finset.image v' s) = span 𝕜 (range (v' ∘ Subtype.val))

Tactic:
by
        rw [Finset.coe_image, image_eq_range]
        rfl

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
x : ↥L
i✝ i : ι
⊢ (↑ℤ ↑(ofZLatticeBasis K L b).repr ∘ₗ L.subtype) (b i) =
    (Finsupp.mapRange.linearMap (Algebra.linearMap ℤ K) ∘ₗ ↑b.repr) (b i)

Tactic:
simp_rw [LinearMap.coe_comp, Function.comp_apply, LinearMap.coe_restrictScalars,
    LinearEquiv.coe_coe, coe_subtype, ← b.ofZLatticeBasis_apply K, repr_self,
    Finsupp.mapRange.linearMap_apply, Finsupp.mapRange_single, Algebra.linearMap_apply, map_one]

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

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

Tactic:
apply
      tendsto_addHaar_inter_smul_one_of_density_one_aux μ _ (measurableSet_toMeasurable _ _) _ _
        t ht h't h''t

Goal state after:
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
x : E
h : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
t : Set E
ht : MeasurableSet t
h't : μ t ≠ 0
h''t : μ t ≠ ∞
⊢ Tendsto (fun r ↦ μ (toMeasurable μ s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)
Module: Mathlib.Analysis.BoxIntegral.Partition.Filter

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ⊤.iUnion
⊢ (embedBox I J h) π ∈ {π | l.MemBaseSet J (max c π₀.compl.distortion) r π ∧ π.iUnion = π₀.iUnion}

Tactic:
Prepartition.iUnion_top

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
I J : Box ι
l : IntegrationParams
h : I ≤ J
c : ℝ≥0
π₀ : Prepartition J := Prepartition.single J I h
r : (ι → ℝ) → ↑(Set.Ioi 0)
hr : l.RCond r
π : TaggedPrepartition I
hπ : l.MemBaseSet I c r π ∧ π.iUnion = ↑I
⊢ (embedBox I J h) π ∈ {π | l.MemBaseSet J (max c π₀.compl.distortion) r π ∧ π.iUnion = π₀.iUnion}
Module: Mathlib.Analysis.Oscillation

Goal state before:
pp: E : Type u
F : Type v
inst✝¹ : PseudoEMetricSpace F
inst✝ : TopologicalSpace E
f : E → F
x : E
⊢ oscillation f x = 0 ↔ ContinuousAt f x

Tactic:
by
  rw [← oscillationWithin_univ_eq_oscillation, ← continuousWithinAt_univ f x]
  exact OscillationWithin.eq_zero_iff_continuousWithinAt f (mem_univ x)

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

Goal state before:
pp: case refine_1
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
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j : Fin (n + 1)
x : ℝ
hx : x ∈ Set.Ioo (J.lower i) (J.upper i)
⊢ Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower i x) +
      Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper i x) =
    f (J.upper i) (J.face i) - f (J.lower i) (J.face i)

pp: case refine_2
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
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j✝ : Fin (n + 1)
x : ℝ
j : Fin n
hx : x ∈ Set.Ioo (J.lower (i.succAbove j)) (J.upper (i.succAbove j))
⊢ Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower (i.succAbove j) x) +
      Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper (i.succAbove j) x) =
    f (J.upper i) (J.face i) - f (J.lower i) (J.face i)

Tactic:
· simp only [Box.splitLower_def hx, Box.splitUpper_def hx, update_self, ← WithBot.some_eq_coe,
          Option.elim', Box.face, Function.comp_def, update_of_ne (Fin.succAbove_ne _ _)]
        abel

Goal state after:
case refine_2
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
G : Type u
inst✝ : AddCommGroup G
I₀ : Box (Fin (n + 1))
i : Fin (n + 1)
f : ℝ → Box (Fin n) → G
fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → Fin n →ᵇᵃ[↑(I₀.face i)] G
hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : Box (Fin n)), f x J = (fb ⟨x, hx⟩) J
J : Box (Fin (n + 1))
hJ : J ≤ I₀
j✝ : Fin (n + 1)
x : ℝ
j : Fin n
hx : x ∈ Set.Ioo (J.lower (i.succAbove j)) (J.upper (i.succAbove j))
⊢ Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitLower (i.succAbove j) x) +
      Option.elim' 0 (fun J ↦ f (J.upper i) (J.face i) - f (J.lower i) (J.face i)) (J.splitUpper (i.succAbove j) x) =
    f (J.upper i) (J.face i) - f (J.lower i) (J.face i)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_1
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ∀ᵐ (x : ι → ℝ), ContinuousAt (s.indicator F) x
r : NNReal → (ι → ℝ) → ↑(Set.Ioi 0)
hr₁ : ∀ (c : NNReal), IntegrationParams.Riemann.RCond (r c)
hr₂ :
  ∀ (c : NNReal) (π : TaggedPrepartition B),
    IntegrationParams.Riemann.MemBaseSet B c (r c) π →
      π.IsPartition →
        dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul π)
            (∫ (x : ι → ℝ) in ↑B, s.indicator F x ∂volume) ≤
          ε / 2
n : ℕ
hn : n ≥ ⌈(↑(r 0 0))⁻¹⌉₊
this : NeZero n
⊢ 1 / ↑n ≤ ↑(r 0 0)

Tactic:
rw [one_div, inv_le_comm₀ (mod_cast (Nat.pos_of_neZero n)) (r 0 0).prop]

Goal state after:
case refine_1
ι : Type u_1
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
hF : Continuous F
hs₁ : Bornology.IsBounded s
hs₂ : MeasurableSet s
hs₃ : volume (frontier s) = 0
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
ε : ℝ
hε : ε > 0
h₁ : ∃ C, ∀ x ∈ Box.Icc B, ‖s.indicator F x‖ ≤ C
h₂ : ∀ᵐ (x : ι → ℝ), ContinuousAt (s.indicator F) x
r : NNReal → (ι → ℝ) → ↑(Set.Ioi 0)
hr₁ : ∀ (c : NNReal), IntegrationParams.Riemann.RCond (r c)
hr₂ :
  ∀ (c : NNReal) (π : TaggedPrepartition B),
    IntegrationParams.Riemann.MemBaseSet B c (r c) π →
      π.IsPartition →
        dist (integralSum (s.indicator F) volume.toBoxAdditive.toSMul π)
            (∫ (x : ι → ℝ) in ↑B, s.indicator F x ∂volume) ≤
          ε / 2
n : ℕ
hn : n ≥ ⌈(↑(r 0 0))⁻¹⌉₊
this : NeZero n
⊢ (↑(r 0 0))⁻¹ ≤ ↑n
Module: Mathlib.Analysis.InnerProductSpace.Orientation

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace ℝ E
ι : Type u_2
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
e : OrthonormalBasis ι ℝ E
x : Orientation ℝ E ι
inst✝ : Nonempty ι
v : ι → E
⊢ |(e.adjustToOrientation x).toBasis.det v| = |e.toBasis.det v|

Tactic:
by
  simp [toBasis_adjustToOrientation]

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

Goal state before:
pp: case refine_1
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)) ≤
    μ.real ↑I * ↑ε

Tactic:
← hπp.iUnion_eq,

Goal state after:
case refine_1
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
⊢ dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)) ≤
    μ.real π.iUnion * ↑ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_1
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
J : Box ι
x✝ : J ∈ π.boxes
⊢ ‖μ.real ↑J • g (π.tag J) - μ.real ↑J • (f (Nx (π.tag J))) (π.tag J)‖ ≤ μ.real ↑J * ↑ε

Tactic:
← smul_sub,

Goal state after:
case refine_1
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
J : Box ι
x✝ : J ∈ π.boxes
⊢ ‖μ.real ↑J • (g (π.tag J) - (f (Nx (π.tag J))) (π.tag J))‖ ≤ μ.real ↑J * ↑ε
Module: Mathlib.Analysis.BoxIntegral.Integrability

Goal state before:
pp: case refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi✝ : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi✝ : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ
hfgi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (fun x ↦ ‖(f n) x - g x‖) (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤
    ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ

Tactic:
← hπp.iUnion_eq,

Goal state after:
case refine_3
ι : Type u
E : Type v
inst✝⁴ : Fintype ι
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
I : Box ι
l : IntegrationParams
hl : l.bRiemann = false
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
g : (ι → ℝ) → E
hg : StronglyMeasurable g
this : SeparableSpace ↑(Set.range g ∪ {0})
hgi✝ : IntegrableOn g (↑I) μ
f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯
hfi✝ : ∀ (n : ℕ), IntegrableOn (⇑(f n)) (↑I) μ
hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (⇑(f n)) μ.toBoxAdditive.toSMul
hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖(f n) x - g x‖ ≤ ‖(f m) x - g x‖
ε : ℝ≥0
ε0 : 0 < ε
ε0' : 0 < ↑ε
N₀ : ℕ
hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖(f N₀) x - g x‖ ∂μ ≤ ↑ε
Nx : (ι → ℝ) → ℕ
hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x
hNxε : ∀ (x : ι → ℝ), dist ((f (Nx x)) x) (g x) ≤ ↑ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c✝ : ℝ≥0
hδc : HasSum δ c✝
hcε : c✝ < ε
r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x ↦ ⋯.convergenceR (↑(δ (Nx x))) c x
c : ℝ≥0
π : TaggedPrepartition I
hπ : l.MemBaseSet I c (r c) π
hπp : π.IsPartition
hfi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (⇑(f n)) (↑J) μ
hgi : ∀ J ∈ π, IntegrableOn g (↑J) μ
hfgi : ∀ (n : ℕ), ∀ J ∈ π, IntegrableOn (fun x ↦ ‖(f n) x - g x‖) (↑J) μ
⊢ dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, (f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in π.iUnion, g a ∂μ) ≤
    ∫ (x : ι → ℝ) in π.iUnion, ‖(f N₀) x - g x‖ ∂μ
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 (∑ a, {n | ↑n = a}.indicator f)

Tactic:
convert!
    summable_sum (s := Finset.univ) fun a _ ↦
      summable_indicator_mod_iff_summable_indicator_mod hf a H

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

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

Tactic:
obtain hC := tsum_schlomilch_le hf h_pos h_nonneg hu_strict.monotone h_succ_diff

Goal state after:
case mp
C : ℕ
u : ℕ → ℕ
f : ℕ → ℝ≥0
h_pos : ∀ (n : ℕ), 0 < u n
hu_strict : StrictMono u
hC_nonzero : C ≠ 0
h_succ_diff : SuccDiffBounded C u
h : ∑' (b : ℕ), ↑(↑(u (b + 1)) - ↑(u b)) * ↑(f (u b)) = ∞
hf : ∀ (m n : ℕ), 1 < m → m ≤ n → ↑(f n) ≤ ↑(f m)
h_nonneg : ∀ (n : ℕ), 0 ≤ ↑(f n)
hC : ∑' (k : ℕ), (↑(u (k + 1)) - ↑(u k)) * ↑(f (u k)) ≤ (↑(u 1) - ↑(u 0)) * ↑(f (u 0)) + ↑C * ∑' (k : ℕ), ↑(f k)
⊢ ∑' (b : ℕ), ↑(f b) = ∞
Module: Mathlib.Algebra.Module.ZLattice.Covolume

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃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 (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
⊢ ((fun x ↦
        ↑(Nat.card
              ↑(⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∩
                  x⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) /
          x ^ card ι) ∘
      fun x ↦ x ^ (↑(card ι))⁻¹) =ᶠ[atTop]
    fun c ↦ ↑(Nat.card ↑({x | x ∈ X ∧ F x ≤ c} ∩ ↑L)) / c

Tactic:
filter_upwards [eventually_gt_atTop 0] with c hc

Goal state after:
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
L : Submodule ℤ E
inst✝⁶ : DiscreteTopology ↥L
inst✝⁵ : IsZLattice ℝ L
ι : Type u_2
inst✝⁴ : Fintype ι
b : Basis ι ℤ ↥L
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : Nonempty ι
X : Set E
hX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X
F : E → ℝ
h₁ : ∀ (x : E) ⦃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 (⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1})) = 0
c : ℝ
hc : 0 < c
⊢ ((fun x ↦
          ↑(Nat.card
                ↑(⇑(Basis.ofZLatticeBasis ℝ L b).equivFun '' {x | x ∈ X ∧ F x ≤ 1} ∩
                    x⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))) /
            x ^ card ι) ∘
        fun x ↦ x ^ (↑(card ι))⁻¹)
      c =
    ↑(Nat.card ↑({x | x ∈ X ∧ F x ≤ c} ∩ ↑L)) / c
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✝ x y : k[G]
hy : y ∈ {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}
m : G
hm : m ∈ (x • y).coeff.support
⊢ ∃ m' ∈ s, ∃ d, m = d * m'

Tactic:
simp only [smul_eq_mul, mul_def, coeff_finsuppSum] at hm
        obtain ⟨xm, -, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
        obtain ⟨ym, hym, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
        obtain rfl := Finset.mem_singleton.mp (Finsupp.support_single_subset hm)
        refine (hy _ hym).imp fun sm p => And.imp_right ?_ p
        rintro ⟨d, rfl⟩
        exact ⟨xm * d, (mul_assoc _ _ _).symm⟩

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

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝ : SMul G P
s : Set G
t : Set P
hst : ∀ (a : P), (s.smulAntidiagonal t a).Finite
⊢ {a | (SMulAntidiagonal a ⋯).Nonempty} ⊆ s • t

Tactic:
grind [mem_smul, mem_smulAntidiagonal]

Goal state after: