mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h.refine_1
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ IntegrableOn (fun x ↦ 2 / (↑k + 1) * x ^ 2) (Set.Ioc ↑k ↑(k + 1)) ρ

pp: case h.refine_2
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ IntegrableOn (HMul.hMul 2) (Set.Ioc ↑k ↑(k + 1)) ρ

pp: case h.refine_3
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 / (↑k + 1) * x ^ 2 ≤ 2 * x

Tactic:
· apply Continuous.integrableOn_Ioc (by fun_prop)

Goal state after:
case h.refine_2
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ IntegrableOn (HMul.hMul 2) (Set.Ioc ↑k ↑(k + 1)) ρ

case h.refine_3
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 / (↑k + 1) * x ^ 2 ≤ 2 * x
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case h.refine_2
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
⊢ IntegrableOn (HMul.hMul 2) (Set.Ioc ↑k ↑(k + 1)) ρ

pp: case h.refine_3
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 / (↑k + 1) * x ^ 2 ≤ 2 * x

Tactic:
· apply Continuous.integrableOn_Ioc (by fun_prop)

Goal state after:
case h.refine_3
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
Y : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n
ρ : Measure ℝ := Measure.map X ℙ
Y2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ
k : ℕ
a✝ : k ∈ range K
Ik : ↑k ≤ ↑(k + 1)
x : ℝ
hx : x ∈ Set.Ioc ↑k ↑(k + 1)
⊢ 2 / (↑k + 1) * x ^ 2 ≤ 2 * x
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: case zero
A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f✝ : V →ₗ[A] W
f : ρ.IntertwiningMap σ
v : ρ.asModule
⊢ f.toLinearMap (0 • v) = (RingHom.id A[G]) 0 • f.toLinearMap v

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Intertwining

Goal state before:
pp: case zero
A : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝⁷ : CommSemiring A
inst✝⁶ : Monoid G
inst✝⁵ : AddCommMonoid V
inst✝⁴ : AddCommMonoid W
inst✝³ : AddCommMonoid U
inst✝² : Module A V
inst✝¹ : Module A W
inst✝ : Module A U
ρ : Representation A G V
σ : Representation A G W
τ : Representation A G U
f✝ : V →ₗ[A] W
f : ρ.IntertwiningMap ρ
⊢ (f ^ 0).toLinearMap = f.toLinearMap ^ 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case zero
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : StandardBorelSpace Ω
Y : ℕ → Ω → ℝ
cY : ℕ → ℝ≥0
ℱ : Filtration ℕ mΩ
inst✝ : IsZeroOrProbabilityMeasure μ
h_adapted : StronglyAdapted ℱ Y
h0 : HasSubgaussianMGF (Y 0) (cY 0) μ
h_subG : ∀ i < 0 - 1, HasCondSubgaussianMGF (↑ℱ i) ⋯ (Y (i + 1)) (cY (i + 1)) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ Finset.range 0, Y i ω) (∑ i ∈ Finset.range 0, cY i) μ

Tactic:
| zero =>

Goal state after:
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: case succ.zero
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : StandardBorelSpace Ω
Y : ℕ → Ω → ℝ
cY : ℕ → ℝ≥0
ℱ : Filtration ℕ mΩ
inst✝ : IsZeroOrProbabilityMeasure μ
h_adapted : StronglyAdapted ℱ Y
h0 : HasSubgaussianMGF (Y 0) (cY 0) μ
hn :
  (∀ i < 0 - 1, HasCondSubgaussianMGF (↑ℱ i) ⋯ (Y (i + 1)) (cY (i + 1)) μ) →
    HasSubgaussianMGF (fun ω ↦ ∑ i ∈ Finset.range 0, Y i ω) (∑ i ∈ Finset.range 0, cY i) μ
h_subG : ∀ i < 0 + 1 - 1, HasCondSubgaussianMGF (↑ℱ i) ⋯ (Y (i + 1)) (cY (i + 1)) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ Finset.range (0 + 1), Y i ω) (∑ i ∈ Finset.range (0 + 1), cY i) μ

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Invariants

Goal state before:
pp: case zero
k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
x : V
hx : (ρ g) x = x
⊢ (ρ ((fun x ↦ g ^ x) 0)) x = x

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
X : Type u_5
inst✝⁷ : CommRing k
inst✝⁶ : Monoid G
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
inst✝³ : AddCommGroup W
inst✝² : Module k W
inst✝¹ : AddCommGroup X
inst✝ : Module k X
ρ : Representation k G V
τ : Representation k G W
υ : Representation k G X
f : V →ₗ[k] W
h : ∀ (x : G), f ∘ₗ ρ x = f
x : V
x✝ : x ∈ Set.range fun gv ↦ (ρ gv.1) gv.2 - gv.2
g : G
y : V
hy : (fun gv ↦ (ρ gv.1) gv.2 - gv.2) (g, y) = x
⊢ x ∈ ↑f.ker

Tactic:
by
    simpa only [← hy, SetLike.mem_coe, LinearMap.mem_ker, map_sub, sub_eq_zero, LinearMap.coe_comp,
      Function.comp_apply] using LinearMap.ext_iff.1 (h g) y

Goal state after:
Module: Mathlib.RepresentationTheory.Coinvariants

Goal state before:
pp: k✝ : Type u_1
G✝ : Type u_2
V✝ : Type u_3
W : Type u_4
X : Type u_5
inst✝¹² : CommRing k✝
inst✝¹¹ : Monoid G✝
inst✝¹⁰ : AddCommGroup V✝
inst✝⁹ : Module k✝ V✝
inst✝⁸ : AddCommGroup W
inst✝⁷ : Module k✝ W
inst✝⁶ : AddCommGroup X
inst✝⁵ : Module k✝ X
ρ✝ : Representation k✝ G✝ V✝
τ : Representation k✝ G✝ W
υ : Representation k✝ G✝ X
k : Type u_6
G : Type u_7
V : Type u_8
inst✝⁴ : CommRing k
inst✝³ : Group G
inst✝² : AddCommGroup V
inst✝¹ : Module k V
ρ : Representation k G V
S : Subgroup G
inst✝ : S.Normal
g : ↥S
x : V
⊢ ((MonoidHom.comp (ρ.toCoinvariants S) S.subtype) g ∘ₗ mk (MonoidHom.comp ρ S.subtype)) x =
    (LinearMap.id ∘ₗ mk (MonoidHom.comp ρ S.subtype)) x

Tactic:
exact (Coinvariants.mk_eq_iff _).2 <| mem_ker_of_eq g x _ rfl

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
Y : ℕ → Ω → E := fun i ↦ AEStronglyMeasurable.mk (X i) ⋯
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))

Tactic:
have B : ∀ᵐ ω ∂μ, ∀ n, X n ω = Y n ω :=
    ae_all_iff.2 (fun i ↦ AEStronglyMeasurable.ae_eq_mk (A i).1)

Goal state after:
case neg
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
X : ℕ → Ω → E
hint : Integrable (X 0) μ
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
h : ¬∀ᵐ (ω : Ω) ∂μ, X 0 ω = 0
this : IsProbabilityMeasure μ
A : ∀ (i : ℕ), Integrable (X i) μ
Y : ℕ → Ω → E := fun i ↦ AEStronglyMeasurable.mk (X i) ⋯
B : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), X n ω = Y n ω
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, X i ω) atTop (𝓝 (∫ (x : Ω), X 0 x ∂μ))
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Basic

Goal state before:
pp: case zero
k : Type u_1
G : Type u_2
inst✝⁴ : Ring k
inst✝³ : Group G
inst✝² : TopologicalSpace k
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
⊢ X.d 0 ≫ X.d (0 + 1) = 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality

Goal state before:
pp: case zero
k : Type u
G : Type v
inst✝⁴ : Ring k
inst✝³ : TopologicalSpace k
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
X : TopRep k G
⊢ resolutionMap (ContinuousMonoidHom.id G) (𝟙 X) 0 = 𝟙 (X.resolutionX 0)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality

Goal state before:
pp: case zero
k : Type u
G H K : Type v
inst✝¹⁰ : Ring k
inst✝⁹ : TopologicalSpace k
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : Group H
inst✝⁴ : TopologicalSpace H
inst✝³ : IsTopologicalGroup H
inst✝² : Group K
inst✝¹ : TopologicalSpace K
inst✝ : IsTopologicalGroup K
X : TopRep k G
Y : TopRep k H
Z : TopRep k K
φ : H →ₜ* G
ψ : K →ₜ* H
f : res (↑φ) X ⟶ Y
g : res (↑ψ) Y ⟶ Z
⊢ resolutionMap (φ.comp ψ) ((resFunctor ↑ψ).map f ≫ g) 0 =
    (resFunctor ↑ψ).map (resolutionMap φ f 0) ≫ resolutionMap ψ g 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.ContCohomology.Functoriality

Goal state before:
pp: case zero
k : Type u
G H : Type v
inst✝⁷ : Ring k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalGroup G
inst✝² : Group H
inst✝¹ : TopologicalSpace H
inst✝ : IsTopologicalGroup H
X : TopRep k G
Y : TopRep k H
φ : H →ₜ* G
f : res (↑φ) X ⟶ Y
⊢ resolutionMap φ f 0 ≫ Y.d 0 = (resFunctor ↑φ).map (X.d 0) ≫ resolutionMap φ f (0 + 1)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.FiniteCyclic

Goal state before:
pp: case refine_1.zero
k : Type u_1
G : Type u_2
inst✝⁴ : CommRing k
inst✝³ : Group G
V : Type u_4
inst✝² : AddCommGroup V
inst✝¹ : Module k V
ρ : Representation k G V
g : G
inst✝ : Finite G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
α : V
⊢ (fun gv ↦ (ρ gv.1) gv.2 - gv.2) ((fun x ↦ g ^ x) 0, α) ∈ ↑(ρ g - LinearMap.id).range

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.FiniteCyclic

Goal state before:
pp: case zero
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
⊢ QuasiIsoAt (resolution.π k g) 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic

Goal state before:
pp: case zero
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
A : Rep k G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
h₀ : NeZero 0
hi : Even 0
⊢ (ComplexShape.up ℕ).Rel ((ComplexShape.up ℕ).prev 0) 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x y : G × G → ↑A
g : G × G × G
⊢ (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + ((A.ρ g.1) (y (g.2.1, g.2.2)) - y (g.1 * g.2.1, g.2.2)) +
      (x (g.1, g.2.1 * g.2.2) + y (g.1, g.2.1 * g.2.2) - (x (g.1, g.2.1) + y (g.1, g.2.1))) =
    (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + x (g.1, g.2.1 * g.2.2) - x (g.1, g.2.1) +
      ((A.ρ g.1) (y (g.2.1, g.2.2)) - y (g.1 * g.2.1, g.2.2) + y (g.1, g.2.1 * g.2.2) - y (g.1, g.2.1))

Tactic:
add_sub_add_comm,

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x y : G × G → ↑A
g : G × G × G
⊢ (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + ((A.ρ g.1) (y (g.2.1, g.2.2)) - y (g.1 * g.2.1, g.2.2)) +
      (x (g.1, g.2.1 * g.2.2) - x (g.1, g.2.1) + (y (g.1, g.2.1 * g.2.2) - y (g.1, g.2.1))) =
    (A.ρ g.1) (x (g.2.1, g.2.2)) - x (g.1 * g.2.1, g.2.2) + x (g.1, g.2.1 * g.2.2) - x (g.1, g.2.1) +
      ((A.ρ g.1) (y (g.2.1, g.2.2)) - y (g.1 * g.2.1, g.2.2) + y (g.1, g.2.1 * g.2.2) - y (g.1, g.2.1))
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : MulAction G A
f : G → A
hf : IsCocycle₁ f
g : G
⊢ g • f g⁻¹ = -f g

Tactic:
rw [← add_eq_zero_iff_eq_neg, ← map_one_of_isCocycle₁ hf, ← mul_inv_cancel g, hf g g⁻¹]

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
⊢ ∀ (x₂ : ↑(shortComplexH0 A).X₂),
    (ConcreteCategory.hom (shortComplexH0 A).g) x₂ = 0 → ∃ x₁, (ConcreteCategory.hom (shortComplexH0 A).f) x₁ = x₂

Tactic:
intro (x : A) (hx : d₀₁ _ x = 0)

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↑A
hx : (ConcreteCategory.hom (d₀₁ A)) x = 0
⊢ ∃ x₁, (ConcreteCategory.hom (shortComplexH0 A).f) x₁ = x
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic

Goal state before:
pp: case zero
k G : Type u
inst✝³ : CommRing k
inst✝² : CommGroup G
inst✝¹ : Fintype G
A : Rep k G
g : G
inst✝ : DecidableEq G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
h₀ : NeZero 0
hi : Even 0
⊢ (ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: case h₂.zero
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↑A
hy : (ModuleCat.Hom.hom (d₁₀ A)) 0 = x
⊢ x ∈ Submodule.span k (Set.range fun gv ↦ (A.ρ gv.1) gv.2 - gv.2)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
R : Type u_4
inst✝ : CommSemiring R
⊢ rename id = AlgHom.id R (MvPowerSeries σ R)

Tactic:
ext _ y
  simpa [coeff_rename] using Finset.sum_eq_single y (by simp) (by simp)

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
g h : G
a : ↑A
⊢ single (h, 1) ((A.ρ g⁻¹) a) - single (g * h, 1) a ∈ boundaries₂ A

Tactic:
by
  use single (g, h, 1) a
  simp [d₃₂]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Ideal

Goal state before:
pp: case zero
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R⟦X⟧
inst✝ : I.IsPrime
hXI : X ∉ I
n : ℕ
F : Fin n → R⟦X⟧
hJI : Submodule.span R⟦X⟧ (Set.range F) ≤ I
hJ : FG (Submodule.span R⟦X⟧ (Set.range F))
h' : Ideal.map constantCoeff I = span (Set.range (⇑constantCoeff ∘ F))
T : R⟦X⟧ → Fin n → R
hT : ∀ g ∈ I, ∑ i, T g i * constantCoeff (F i) = constantCoeff g
g : R⟦X⟧
hg : g ∈ I
G : ℕ → R⟦X⟧ := fun t ↦ Nat.rec g (fun x G' ↦ mk fun p ↦ (coeff (p + 1)) G' - ∑ i, T G' i * (coeff (p + 1)) (F i)) t
hG : ∀ (n : ℕ), G n ∈ I
hG' : ∀ (n_1 : ℕ), G n_1 - ∑ i, C (T (G n_1) i) * F i = X * G (n_1 + 1)
H : Fin n → R⟦X⟧ := fun i ↦ mk fun n ↦ T (G n) i
⊢ g - ∑ i, ↑((trunc 0) (H i)) * F i = X ^ 0 * G 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Topology

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
inst✝¹ : UniformSpace R
inst✝ : IsUniformAddGroup R
I : Ideal R
hI : IsAdic I
⊢ IsAdicComplete I R ↔ CompleteSpace R ∧ T2Space R

Tactic:
by
  rw [isAdicComplete_iff, hI.isHausdorff_iff, hI.isPrecomplete_iff, and_comm]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case zero
S : Type u_3
inst✝¹ : CommRing S
inst✝ : IsNoetherianRing S
⊢ IsNoetherianRing (MvPowerSeries (Fin 0) S)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.LocalRing

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
fg : (maximalIdeal R).FG
this : IsLocalRing (AdicCompletion (maximalIdeal R) R)
x : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)
y : AdicCompletion (maximalIdeal R) R
hy : (residue (AdicCompletion (maximalIdeal R) R)) y = x
z : R
hz : (Ideal.Quotient.mk (maximalIdeal R ^ 1 • ⊤)) z = ↑y 1
⊢ (algebraMap R (AdicCompletion (maximalIdeal R) R)) z - y ∈ maximalIdeal R ^ 1 • ⊤

Tactic:
rw [AdicCompletion.algebraMap_apply, pow_smul_top_eq_ker_eval fg]

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsLocalRing R
fg : (maximalIdeal R).FG
this : IsLocalRing (AdicCompletion (maximalIdeal R) R)
x : IsLocalRing.ResidueField (AdicCompletion (maximalIdeal R) R)
y : AdicCompletion (maximalIdeal R) R
hy : (residue (AdicCompletion (maximalIdeal R) R)) y = x
z : R
hz : (Ideal.Quotient.mk (maximalIdeal R ^ 1 • ⊤)) z = ↑y 1
⊢ (of (maximalIdeal R) R) ((algebraMap R R) z) - y ∈ (eval (maximalIdeal R) R 1).ker
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G H : Type u
inst✝⁴ : CommRing k
inst✝³ : Group G
inst✝² : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
⊢ Epi (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)

Tactic:
rw [ModuleCat.epi_iff_surjective]

Goal state after:
k G H : Type u
inst✝⁴ : CommRing k
inst✝³ : Group G
inst✝² : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
⊢ Function.Surjective ⇑(ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv))
Module: Mathlib.RingTheory.AdicCompletion.Completeness

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
s : Finset R
hs : Ideal.span ↑s = I ^ n
x : AdicCompletion I ↥(I ^ n • ⊤)
⊢ comap (I ^ n • ⊤).subtype ((lsum R) fun i ↦ ↑i • LinearMap.id).range = ⊤

Tactic:
← hs,

Goal state after:
R : Type u_1
inst✝² : CommRing R
I : Ideal R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
s : Finset R
hs : Ideal.span ↑s = I ^ n
x : AdicCompletion I ↥(I ^ n • ⊤)
⊢ comap (Ideal.span ↑s • ⊤).subtype ((lsum R) fun i ↦ ↑i • LinearMap.id).range = ⊤
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G H : Type u
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝ : S.Normal
⊢ Epi (H1CoresCoinf A S).g

Tactic:
rw [ModuleCat.epi_iff_surjective]

Goal state after:
k G H : Type u
inst✝³ : CommRing k
inst✝² : Group G
inst✝¹ : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n : ℕ
S : Subgroup G
inst✝ : S.Normal
⊢ Function.Surjective ⇑(ConcreteCategory.hom (H1CoresCoinf A S).g)
Module: Mathlib.RingTheory.Polynomial.ContentIdeal

Goal state before:
pp: R : Type u_3
inst✝ : CommSemiring R
p q : R[X]
h : (p * q).contentIdeal = ⊤
⊢ p.contentIdeal = ⊤

Tactic:
apply le_antisymm le_top

Goal state after:
R : Type u_3
inst✝ : CommSemiring R
p q : R[X]
h : (p * q).contentIdeal = ⊤
⊢ ⊤ ≤ p.contentIdeal
Module: Mathlib.RingTheory.Polynomial.ContentIdeal

Goal state before:
pp: R : Type u_3
inst✝ : CommRing R
p q : R[X]
hp : p.contentIdeal = ⊤
hq : q.contentIdeal = ⊤
⊢ (p * q).contentIdeal.radical = ⊤

Tactic:
apply le_antisymm le_top

Goal state after:
R : Type u_3
inst✝ : CommRing R
p q : R[X]
hp : p.contentIdeal = ⊤
hq : q.contentIdeal = ⊤
⊢ ⊤ ≤ (p * q).contentIdeal.radical
Module: Mathlib.RingTheory.Ideal.AssociatedPrime.Localization

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
R' : Type u_2
inst✝⁸ : CommRing R'
inst✝⁷ : Algebra R R'
hSR' : IsLocalization S R'
M : Type u_3
M' : Type u_4
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R M'
f : M →ₗ[R] M'
inst✝² : IsLocalizedModule S f
inst✝¹ : Module R' M'
inst✝ : IsScalarTower R R' M'
p : Ideal R'
hp : p.IsPrime
T : Finset R
hT : Ideal.span ↑T = Ideal.comap (algebraMap R R') p
m : M
s : ↥S
hx : p = (⊥.colon {IsLocalizedModule.mk' f m s}).radical
e : ↥T → ℕ
g : ↥T → ↥S
hg : ∀ (a : ↥T), g a • ↑a ^ e a • m = 0
⊢ ↑T ⊆ ↑(⊥.colon {↑(∏ a, g a) • m}).radical

Tactic:
intro a

Goal state after:
case refine_1
R : Type u_1
inst✝⁹ : CommRing R
S : Submonoid R
R' : Type u_2
inst✝⁸ : CommRing R'
inst✝⁷ : Algebra R R'
hSR' : IsLocalization S R'
M : Type u_3
M' : Type u_4
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : AddCommGroup M'
inst✝³ : Module R M'
f : M →ₗ[R] M'
inst✝² : IsLocalizedModule S f
inst✝¹ : Module R' M'
inst✝ : IsScalarTower R R' M'
p : Ideal R'
hp : p.IsPrime
T : Finset R
hT : Ideal.span ↑T = Ideal.comap (algebraMap R R') p
m : M
s : ↥S
hx : p = (⊥.colon {IsLocalizedModule.mk' f m s}).radical
e : ↥T → ℕ
g : ↥T → ↥S
hg : ∀ (a : ↥T), g a • ↑a ^ e a • m = 0
a : R
⊢ a ∈ ↑T → a ∈ ↑(⊥.colon {↑(∏ a, g a) • m}).radical
Module: Mathlib.RingTheory.Depth.Rees

Goal state before:
pp: case zero
R : Type u
inst✝⁴ : CommRing R
inst✝³ : Small.{v, u} R
inst✝² : IsNoetherianRing R
I : Ideal R
N : ModuleCat R
inst✝¹ : Module.Finite R ↑N
h_supp : Module.support R ↑N = PrimeSpectrum.zeroLocus ↑I
M : ModuleCat R
inst✝ : Module.Finite R ↑M
smul_lt : I • ⊤ < ⊤
h_ext : ∀ i < 0, Subsingleton (Ext N M i)
⊢ ∃ rs, rs.length = 0 ∧ (∀ r ∈ rs, r ∈ I) ∧ RingTheory.Sequence.IsRegular (↑M) rs

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.DividedPowers.Basic

Goal state before:
pp: case zero
A : Type u_1
inst✝ : CommSemiring A
I : Ideal A
a : A
hI : DividedPowers I
ha : a ∈ I
⊢ ↑0! * hI.dpow 0 a = a ^ 0

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.Depth.Rees

Goal state before:
pp: case zero
R : Type u
inst✝³ : CommRing R
inst✝² : Small.{v, u} R
inst✝¹ : IsNoetherianRing R
I : Ideal R
N : ModuleCat R
Nfin : Module.Finite R ↑N
Nsupp : Module.support R ↑N ⊆ PrimeSpectrum.zeroLocus ↑I
M : ModuleCat R
inst✝ : Module.Finite R ↑M
smul_lt : I • ⊤ < ⊤
rs : List R
mem : ∀ r ∈ rs, r ∈ I
reg : RingTheory.Sequence.IsRegular (↑M) rs
len : rs.length = 0
⊢ ∀ i < 0, Subsingleton (Ext N M i)

Tactic:
| zero =>

Goal state after:
Module: Mathlib.RingTheory.DividedPowerAlgebra.Init

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
ι : Type u_4
inst✝ : DecidableEq ι
s : Finset ι
q : ℕ
a : ι → R
x : ι → M
⊢ ∑ x_1 ∈ s.sym q,
      ∏ x_2 ∈ s,
        (algebraMap R (DividedPowerAlgebra R M)) (a x_2 ^ Multiset.count x_2 ↑x_1) *
          dp R (Multiset.count x_2 ↑x_1) (x x_2) =
    ∑ x_1 ∈ s.sym q,
      (∏ x ∈ s, (algebraMap R (DividedPowerAlgebra R M)) (a x ^ Multiset.count x ↑x_1)) *
        ∏ i ∈ s, dp R (Multiset.count i ↑x_1) (x i)

Tactic:
← Finset.prod_mul_distrib

Goal state after:
Module: Mathlib.RingTheory.DividedPowerAlgebra.Init

Goal state before:
pp: case zero
R : Type u_2
M : Type u_3
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
x : M
⊢ ↑0! * dp R 0 x = dp R 1 x ^ 0

Tactic:
| zero =>

Goal state after: