mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case hab
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := ⋯
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ IntervalIntegrable (fun x ↦ (G' x).toReal) volume a t

Tactic:
rw [intervalIntegrable_iff_integrableOn_Ioc_of_le ht.2.1]

Goal state after:
case hab
g' g φ : ℝ → ℝ
a b : ℝ
hab : a ≤ b
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ico a b, HasDerivWithinAt g (g' x) (Ioi x) x
φint : IntegrableOn φ (Icc a b) volume
hφg : ∀ x ∈ Ico a b, g' x ≤ φ x
ε : ℝ
εpos : 0 < ε
G' : ℝ → EReal
f_lt_G' : ∀ (x : ℝ), ↑(φ x) < G' x
G'cont : LowerSemicontinuous G'
G'int : Integrable (fun x ↦ (G' x).toReal) (volume.restrict (Icc a b))
G'lt_top : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), G' x < ⊤
hG' : ∫ (x : ℝ) in Icc a b, (G' x).toReal < (∫ (x : ℝ) in Icc a b, φ x) + ε
s : Set ℝ := {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Icc a b
s_closed : IsClosed s
t : ℝ
ht : t ∈ {t | g t - g a ≤ ∫ (u : ℝ) in a..t, (G' u).toReal} ∩ Ico a b
v : ℝ
t_lt_v : v ∈ Ioi t
y : ℝ
g'_lt_y' : ↑(g' t) < ↑y
y_lt_G' : ↑y < G' t
I1 : ∀ᶠ (u : ℝ) in 𝓝[>] t, (u - t) * y ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I2 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ (u - t) * y
I3 : ∀ᶠ (u : ℝ) in 𝓝[>] t, g u - g t ≤ ∫ (w : ℝ) in t..u, (G' w).toReal
I4 : ∀ᶠ (u : ℝ) in 𝓝[>] t, u ∈ Ioc t (min v b)
x : ℝ
hx : g x - g t ≤ ∫ (w : ℝ) in t..x, (G' w).toReal
h'x : x ∈ Ioc t (min v b)
⊢ IntegrableOn (fun x ↦ (G' x).toReal) (Ioc a t) volume
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
hf : IntegrableOn f (Ioi a) μ
hg : IntegrableOn f (Ioi b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Ioi a) μ →
      IntegrableOn f (Ioi b) μ →
        a ≤ b → ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in Ioi b, f x ∂μ = -∫ (x : ℝ) in b..a, f x ∂μ

Tactic:
← this hg hf hab.le,

Goal state after:
case inr
E : Type u_5
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
hf : IntegrableOn f (Ioi a) μ
hg : IntegrableOn f (Ioi b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Ioi a) μ →
      IntegrableOn f (Ioi b) μ →
        a ≤ b → ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in Ioi b, f x ∂μ = -(∫ (x : ℝ) in Ioi b, f x ∂μ - ∫ (x : ℝ) in Ioi a, f x ∂μ)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : IntegrableOn f (Iio b) μ
hg : IntegrableOn f (Iio a) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Iio b) μ →
      IntegrableOn f (Iio a) μ →
        a ≤ b → ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = -∫ (x : ℝ) in b..a, f x ∂μ

Tactic:
← this hg hf hab.le,

Goal state after:
case inr
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : IntegrableOn f (Iio b) μ
hg : IntegrableOn f (Iio a) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Iio b) μ →
      IntegrableOn f (Iio a) μ →
        a ≤ b → ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = -(∫ (x : ℝ) in Iio a, f x ∂μ - ∫ (x : ℝ) in Iio b, f x ∂μ)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: case inr
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : IntegrableOn f (Ici a) μ
hg : IntegrableOn f (Ici b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Ici a) μ →
      IntegrableOn f (Ici b) μ →
        a ≤ b → ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = -∫ (x : ℝ) in b..a, f x ∂μ

Tactic:
← this hg hf hab.le,

Goal state after:
case inr
E : Type u_5
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
a b : ℝ
f : ℝ → E
μ : Measure ℝ
inst✝ : NullSingletonClass μ
hf : IntegrableOn f (Ici a) μ
hg : IntegrableOn f (Ici b) μ
this :
  ∀ {a b : ℝ},
    IntegrableOn f (Ici a) μ →
      IntegrableOn f (Ici b) μ →
        a ≤ b → ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ
hab : b < a
⊢ ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = -(∫ (x : ℝ) in Ici b, f x ∂μ - ∫ (x : ℝ) in Ici a, f x ∂μ)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
hf : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑(f x)
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
⊢ False

Tactic:
A

Goal state after:
case pos
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
⊢ False
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hfi : IntervalIntegrable f volume a b
hpos : ∀ x ∈ Ioo a b, 0 < f x
hab : a < b
hsupp : Ioo a b ⊆ support f ∩ Ioc a b
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict (Ioo a b), 0 x ≤ f x

Tactic:
rw [ae_restrict_iff' measurableSet_Ioo]

Goal state after:
f : ℝ → ℝ
a b : ℝ
hfi : IntervalIntegrable f volume a b
hpos : ∀ x ∈ Ioo a b, 0 < f x
hab : a < b
hsupp : Ioo a b ⊆ support f ∩ Ioc a b
⊢ ∀ᵐ (x : ℝ), x ∈ Ioo a b → 0 x ≤ f x
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

Goal state before:
pp: case e'_4
g' g : ℝ → ℝ
a b : ℝ
hcont : ContinuousOn g (Icc a b)
hderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x
g'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x
hab : a < b
meas_g' : AEMeasurable g' (volume.restrict (Ioo a b))
H : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊
f : SimpleFunc ℝ ℝ≥0
fle : ∀ (x : ℝ), f x ≤ ‖g' x‖₊
fint : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) < ∞
F : ℝ → ℝ := NNReal.toReal ∘ ⇑f
hf : ENNReal.ofReal (g b - g a) < ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
intF : IntegrableOn F (Ioo a b) volume
A : ∫⁻ (x : ℝ) in Ioo a b, ↑(f x) = ENNReal.ofReal (∫ (x : ℝ) in Ioo a b, F x)
x : ℝ
hx : x ∈ Ioo a b
⊢ g' x = ↑‖g' x‖₊

Tactic:
simp only [Real.norm_of_nonneg (g'pos x hx), coe_nnnorm]

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.right.refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 s ×ˢ 𝓝[≥] 0 = 𝓝[univ ×ˢ Icc 0 1] (s, 0)

Tactic:
simp [nhdsWithin_prod_eq, nhdsWithin_univ]

Goal state after:
Module: Mathlib.Topology.Algebra.Order.Floor

Goal state before:
pp: case inl.right.refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
n : ℤ
⊢ 𝓝 (uncurry f (s, 0)) = 𝓝 ((uncurry f ∘ Prod.map id fract) (s, ↑n))

Tactic:
simp [nhdsWithin_prod_eq, nhdsWithin_univ]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case cpos.a
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := PositiveCompacts.map QuotientGroup.mk ⋯ ⋯ K
this : μ.IsMulLeftInvariant
s : Set G
finiteCovol : ν s ≠ ∞
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
h : ν (Quotient.mk (MulAction.orbitRel (↥Γ.op) G) ⁻¹' ↑K' ∩ s) = 0
i' : ν.IsOpenPosMeasure
⊢ ↑K ⊆ QuotientGroup.mk ⁻¹' QuotientGroup.mk '' ↑K

Tactic:
exact subset_preimage_image π K

Goal state after:
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map ((↑(IsScalarTower.toAlgHom F p.SplittingField E)).comp (algebraMap F p.SplittingField)) p).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y

Tactic:
AlgHom.comp_algebraMap

Goal state after:
case right
F : Type u_1
inst✝² : Field F
p : F[X]
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
h : Fact (map (algebraMap F E) p).Splits
y : ↑(p.rootSet E)
key :
  (map (algebraMap F E) p).roots =
    Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.SplittingField) p).roots
⊢ ∃ a, mapRoots p E a = y
Module: Mathlib.FieldTheory.PolynomialGaloisGroup

Goal state before:
pp: F : Type u_1
inst✝³ : Field F
p q : F[X]
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Fact (map (algebraMap F E) p).Splits
x✝² x✝¹ : p.Gal
x✝ : ↑(p.rootSet E)
⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝

Tactic:
simp only [smul_def, Equiv.symm_apply_apply, mul_smul]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: S : Type u_4
σ : Type u_5
R : Type u_6
inst✝³ : CommSemiring R
inst✝² : CommSemiring S
inst✝¹ : Fintype σ
inst✝ : Algebra R S
n : ℕ
f : σ → S
⊢ (aeval f) (esymm σ R n) = (Multiset.map f univ.val).esymm n

Tactic:
by
  simp_rw [esymm, aeval_sum, aeval_prod, aeval_X, esymm_map_val]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
⊢ (powersetCard n univ).sup val = univ.val

Tactic:
have : ((powersetCard n univ).sup (fun (x : Finset σ) => x)).val
        = sup (powersetCard n univ) val := by
      refine apply_sup_eq_sup_comp _ ?_ ?_ <;> simp

Goal state after:
σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : Nontrivial R
n : ℕ
hpos : 0 < n
hn : n ≤ Fintype.card σ
this✝ : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val
this : ((powersetCard n univ).sup fun x ↦ x).val = (powersetCard n univ).sup val
⊢ (powersetCard n univ).sup val = univ.val
Module: Mathlib.MeasureTheory.Integral.DivergenceTheorem

Goal state before:
pp: case refine_3
E : Type u
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
n : ℕ
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : Preorder F
inst✝¹ : MeasureSpace F
inst✝ : BorelSpace F
eL : F ≃L[ℝ] Fin (n + 1) → ℝ
he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y
he_vol : MeasurePreserving (⇑eL) volume volume
f : Fin (n + 1) → F → E
f' : Fin (n + 1) → F → F →L[ℝ] E
s : Set F
hs : s.Countable
a b : F
hle : a ≤ b
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)
Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x
DF : F → E
hDF : ∀ (x : F), DF x = ∑ i, (f' i x) (eL.symm (e i))
Hi : IntegrableOn DF (Set.Icc a b) volume
he_emb : MeasurableEmbedding ⇑eL
hIcc : ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b
hIcc' : Set.Icc (eL a) (eL b) = ⇑eL.symm ⁻¹' Set.Icc a b
⊢ IntegrableOn ((fun x ↦ ∑ i, (f' i (eL.symm x) ∘SL ↑eL.symm) (e i)) ∘ ⇑eL) (⇑eL ⁻¹' Set.Icc (eL a) (eL b)) volume

Tactic:
hIcc

Goal state after:
case refine_3
E : Type u
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
n : ℕ
F : Type u_1
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : Preorder F
inst✝¹ : MeasureSpace F
inst✝ : BorelSpace F
eL : F ≃L[ℝ] Fin (n + 1) → ℝ
he_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y
he_vol : MeasurePreserving (⇑eL) volume volume
f : Fin (n + 1) → F → E
f' : Fin (n + 1) → F → F →L[ℝ] E
s : Set F
hs : s.Countable
a b : F
hle : a ≤ b
Hc : ∀ (i : Fin (n + 1)), ContinuousOn (f i) (Set.Icc a b)
Hd : ∀ x ∈ interior (Set.Icc a b) \ s, ∀ (i : Fin (n + 1)), HasFDerivAt (f i) (f' i x) x
DF : F → E
hDF : ∀ (x : F), DF x = ∑ i, (f' i x) (eL.symm (e i))
Hi : IntegrableOn DF (Set.Icc a b) volume
he_emb : MeasurableEmbedding ⇑eL
hIcc : ⇑eL ⁻¹' Set.Icc (eL a) (eL b) = Set.Icc a b
hIcc' : Set.Icc (eL a) (eL b) = ⇑eL.symm ⁻¹' Set.Icc a b
⊢ IntegrableOn ((fun x ↦ ∑ i, (f' i (eL.symm x) ∘SL ↑eL.symm) (e i)) ∘ ⇑eL) (Set.Icc a b) volume
Module: Mathlib.RingTheory.Valuation.Archimedean

Goal state before:
pp: F : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : CommRing O
inst✝¹ : Algebra O F
v : Valuation F Γ₀
inst✝ : MulArchimedean ↥(MonoidHom.mrange v)
hv : v.Integers O
⊢ IsPrincipalIdealRing O ↔ ¬DenselyOrdered ↑(Set.range ⇑v)

Tactic:
refine ⟨fun _ ↦ not_denselyOrdered_of_isPrincipalIdealRing hv, fun H ↦ ?_⟩

Goal state after:
F : Type u_1
Γ₀ : Type u_2
O : Type u_3
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : CommRing O
inst✝¹ : Algebra O F
v : Valuation F Γ₀
inst✝ : MulArchimedean ↥(MonoidHom.mrange v)
hv : v.Integers O
H : ¬DenselyOrdered ↑(Set.range ⇑v)
⊢ IsPrincipalIdealRing O
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case mpr
R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
a✝ : MulArchimedean (ofClass v).ValueGroup₀
f : Additive (ofClass v).ValueGroup₀ˣ →+o ℝ
hf : Injective ⇑f
e : (ofClass v).ValueGroup₀ˣ →* Multiplicative ℝ := AddMonoidHom.toMultiplicativeRight ↑f
⊢ Nonempty v.RankOne

Tactic:
have he : StrictMono e := by
      simp only [AddMonoidHom.coe_toMultiplicativeRight, AddMonoidHom.coe_coe, e]
      -- toAdd_strictMono is already in an applied form, do defeq abuse instead
      exact StrictMono.comp strictMono_id (f.monotone'.strictMono_of_injective hf)

Goal state after:
case mpr
R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
a✝ : MulArchimedean (ofClass v).ValueGroup₀
f : Additive (ofClass v).ValueGroup₀ˣ →+o ℝ
hf : Injective ⇑f
e : (ofClass v).ValueGroup₀ˣ →* Multiplicative ℝ := AddMonoidHom.toMultiplicativeRight ↑f
he : StrictMono ⇑e
⊢ Nonempty v.RankOne
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
p_irr : Irreducible p
p_deg : Nat.Prime p.natDegree
p_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2
⊢ (map (algebraMap ℚ ℂ) p).Splits

Tactic:
exact IsAlgClosed.splits _

Goal state after:
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
hx : (hom v) x = 0
h_lt : 0 < x
⊢ False

Tactic:
have hs := strictMono v h_lt

Goal state after:
R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
hx : (hom v) x = 0
h_lt : 0 < x
hs : (hom v) 0 < (hom v) x
⊢ False
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y z : X
h : dist x y ≠ dist y z
⊢ max (dist x y) (dist y z) ≤ dist x z

Tactic:
rcases h.lt_or_gt with h | h

Goal state after:
case inl
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y z : X
h✝ : dist x y ≠ dist y z
h : dist x y < dist y z
⊢ max (dist x y) (dist y z) ≤ dist x z

case inr
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x y z : X
h✝ : dist x y ≠ dist y z
h : dist y z < dist x y
⊢ max (dist x y) (dist y z) ≤ dist x z
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: S : Type u_1
inst✝¹ : SeminormedGroup S
inst✝ : IsUltrametricDist S
x : S
n : ℕ
⊢ ‖x ^ n‖₊ ≤ ‖x‖₊

Tactic:
induction n with

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Ultra

Goal state before:
pp: M : Type u_1
ι : Type u_2
inst✝¹ : SeminormedCommGroup M
inst✝ : IsUltrametricDist M
f : ι → M
⊢ ‖∏' (i : ι), f i‖ ≤ ⨆ i, ‖f i‖

Tactic:
rcases isEmpty_or_nonempty ι with hι | hι
  · -- Silly case #1 : the index type is empty
    simp only [tprod_empty, norm_one', Real.iSup_of_isEmpty, le_refl]
  by_cases h : Multipliable f; swap
  · -- Silly case #2 : the product is divergent
    rw [tprod_eq_one_of_not_multipliable h, norm_one']
    by_cases h_bd : BddAbove (Set.range fun i ↦ ‖f i‖)
    · exact le_ciSup_of_le h_bd hι.some (norm_nonneg' _)
    · rw [Real.iSup_of_not_bddAbove h_bd]
  -- now the interesting case
  have h_bd : BddAbove (Set.range fun i ↦ ‖f i‖) :=
    h.tendsto_cofinite_one.norm'.bddAbove_range_of_cofinite
  refine le_of_tendsto' h.hasProd.norm' (fun s ↦ norm_prod_le_of_forall_le_of_nonneg ?_ ?_)
  · exact le_ciSup_of_le h_bd hι.some (norm_nonneg' _)
  · exact fun i _ ↦ le_ciSup h_bd i

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
P : Fin 3 → R
⊢ ![P x, P y, P z] = P

Tactic:
ext n

Goal state after:
R : Type r
P : Fin 3 → R
n : Fin (Nat.succ 0).succ.succ
⊢ ![P x, P y, P z] n = P n
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: case h
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
⊢ (Sieve.generate (Presieve.ofArrows 𝒰.X 𝒰.f)).arrows ≤ (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows

Tactic:
rw [← Sieve.generate_le_iff, Sieve.generate_sieve, Sieve.generate_le_iff,
      Presieve.ofArrows_le_iff]

Goal state after:
case h
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
⊢ ∀ (i : 𝒰.I₀), (Sieve.generate (Presieve.ofArrows 𝒱.X 𝒱.f)).arrows (𝒰.f i)
Module: Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes

Goal state before:
pp: case hS
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Limits.HasPullbacks C
K : Precoverage C
inst✝ : K.IsStableUnderBaseChange
S : C
F : Cᵒᵖ ⥤ Type u_2
𝒰 : K.ZeroHypercover S
𝒱 : K.ZeroHypercover S
f : Hom K 𝒰 𝒱
H₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)
H₂ : ∀ {X : C} (f : X ⟶ S), Presieve.IsSeparatedFor F (Presieve.ofArrows (pullback₂ f 𝒰).X (pullback₂ f 𝒰).f)
⊢ Presieve.IsSheafFor F (Sieve.generate (Presieve.ofArrows 𝒰.X 𝒰.f)).arrows

Tactic:
rwa [← Presieve.isSheafFor_iff_generate]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Preserves

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)
Y : Type w
⊢ Unique (Y ⟶ F.obj (op I))

Tactic:
choose t h using hF (by tauto) (by tauto)

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)
Y : Type w
t : F.obj (op I)
h :
  (fun t ↦ FamilyOfElements.IsAmalgamation (fun ⦃Y⦄ f a ↦ ⋯.elim) t) t ∧
    ∀ (y : F.obj (op I)), (fun t ↦ FamilyOfElements.IsAmalgamation (fun ⦃Y⦄ f a ↦ ⋯.elim) t) y → y = t
⊢ Unique (Y ⟶ F.obj (op I))
Module: Mathlib.Topology.Category.TopCat.ULift

Goal state before:
pp: J✝ : Type w
inst✝ : Category.{w', w} J✝
K : J✝ ⥤ TopCat
c : Cocone K
hc : IsColimit c
⊢ (uliftFunctor.mapCocone c).pt.str =
    ⨆ j,
      TopologicalSpace.coinduced (⇑(ConcreteCategory.hom ((uliftFunctor.mapCocone c).ι.app j)))
        ((K ⋙ uliftFunctor).obj j).str

Tactic:
ext s
    rw [Homeomorph.ulift.symm.isOpenEmbedding.isOpen_iff_preimage_isOpen (by simp),
      isOpen_iff_of_isColimit _ hc, isOpen_iSup_iff]
    congr!
    rw [Homeomorph.ulift.isOpenEmbedding.isOpen_iff_preimage_isOpen (by simp)]
    rfl

Goal state after:
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : EssFiniteType R A
p : PrimeSpectrum A
⊢ Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔
    Module.Free (Localization.AtPrime p.asIdeal) (LocalizedModule p.asIdeal.primeCompl Ω[A⁄R])

Tactic:
have := IsLocalizedModule.iso p.asIdeal.primeCompl
        (KaehlerDifferential.map R R A (Localization.AtPrime p.asIdeal))

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : EssFiniteType R A
p : PrimeSpectrum A
this : LocalizedModule p.asIdeal.primeCompl Ω[A⁄R] ≃ₗ[A] Ω[Localization.AtPrime p.asIdeal⁄R]
⊢ Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔
    Module.Free (Localization.AtPrime p.asIdeal) (LocalizedModule p.asIdeal.primeCompl Ω[A⁄R])