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

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
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
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝¹ : SeparableSpace ↑s
this✝ : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
B :
  ∀ᵐ (ω : Ω) ∂μ,
    ∀ (k : ℕ),
      Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop
        (𝓝 (∫ (x : Ω), (fun ω ↦ ‖(X 0 - Y k 0) ω‖) x ∂μ))
ω : Ω
hω : ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
h'ω :
  ∀ (k : ℕ),
    Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop (𝓝 (∫ (x : Ω), ‖(X 0 - Y k 0) x‖ ∂μ))
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : δ + δ + δ < ε
k : ℕ
hk : ∫ (ω : Ω), ‖(X 0 - Y k 0) ω‖ ∂μ < δ
this : ‖∫ (x : Ω), Y k 0 x ∂μ - ∫ (x : Ω), X 0 x ∂μ‖ < δ
I : ∀ᶠ (n : ℕ) in atTop, (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n < δ
⊢ ∀ᶠ (n : ℕ) in atTop, ‖(↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω - ∫ (x : Ω), Y k 0 x ∂μ‖ < δ

Tactic:
specialize hω k

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁵ : IsProbabilityMeasure μ
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
h' : StronglyMeasurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
s : Set E := Set.range (X 0) ∪ {0}
zero_s : 0 ∈ s
this✝¹ : SeparableSpace ↑s
this✝ : Nonempty ↑s
φ : ℕ → SimpleFunc E E := SimpleFunc.nearestPt fun k ↦ Nat.casesOn k 0 (Subtype.val ∘ denseSeq ↑s)
Y : ℕ → ℕ → Ω → E := fun k i ↦ ⇑(φ k) ∘ X i
A : ∀ᵐ (ω : Ω) ∂μ, ∀ (k : ℕ), Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
B :
  ∀ᵐ (ω : Ω) ∂μ,
    ∀ (k : ℕ),
      Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop
        (𝓝 (∫ (x : Ω), (fun ω ↦ ‖(X 0 - Y k 0) ω‖) x ∂μ))
ω : Ω
h'ω :
  ∀ (k : ℕ),
    Tendsto (fun n ↦ (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n) atTop (𝓝 (∫ (x : Ω), ‖(X 0 - Y k 0) x‖ ∂μ))
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : δ + δ + δ < ε
k : ℕ
hω : Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω) atTop (𝓝 (∫ (x : Ω), Y k 0 x ∂μ))
hk : ∫ (ω : Ω), ‖(X 0 - Y k 0) ω‖ ∂μ < δ
this : ‖∫ (x : Ω), Y k 0 x ∂μ - ∫ (x : Ω), X 0 x ∂μ‖ < δ
I : ∀ᶠ (n : ℕ) in atTop, (∑ i ∈ Finset.range n, ‖(X i - Y k i) ω‖) / ↑n < δ
⊢ ∀ᶠ (n : ℕ) in atTop, ‖(↑n)⁻¹ • ∑ i ∈ Finset.range n, Y k i ω - ∫ (x : Ω), Y k 0 x ∂μ‖ < δ
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ : G
a : ↑A
s : ↥S
h : ¬(QuotientGroup.rightRel S) g₂ g₁
⊢ (A.indToCoindAux (↑s * g₁)) ((A.ρ s) a) g₂ = 0

Tactic:
indToCoindAux_of_not_rel

Goal state after:
case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ : G
a : ↑A
s : ↥S
h : ¬(QuotientGroup.rightRel S) g₂ g₁
⊢ 0 = 0

case inr.h
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ : G
a : ↑A
s : ↥S
h : ¬(QuotientGroup.rightRel S) g₂ g₁
⊢ ¬(QuotientGroup.rightRel S) g₂ (↑s * g₁)
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
h : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁
⊢ 0 = (A.indToCoindAux (g₁ * g₃)) a g₂

Tactic:
indToCoindAux_of_not_rel

Goal state after:
case inr
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
h : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁
⊢ 0 = 0

case inr.h
k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A : Rep.{w, u, v} k ↥S
g₁ g₂ g₃ : G
a : ↑A
h : ¬(QuotientGroup.rightRel S) (g₂ * g₃⁻¹) g₁
⊢ ¬(QuotientGroup.rightRel S) g₂ (g₁ * g₃)
Module: Mathlib.RepresentationTheory.Homological.FiniteCyclic

Goal state before:
pp: case succ
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
m : ℕ
a✝ : QuasiIsoAt (resolution.π k g) m
⊢ (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).f).range =
    (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).g).ker

Tactic:
by_cases hm : Odd (m + 1)

Goal state after:
case pos
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
m : ℕ
a✝ : QuasiIsoAt (resolution.π k g) m
hm : Odd (m + 1)
⊢ (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).f).range =
    (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).g).ker

case neg
k G : Type u
inst✝² : CommRing k
inst✝¹ : CommGroup G
inst✝ : Fintype G
g : G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
m : ℕ
a✝ : QuasiIsoAt (resolution.π k g) m
hm : ¬Odd (m + 1)
⊢ (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).f).range =
    (ModuleCat.Hom.hom
        ((HomologicalComplex.sc' ((chainComplexFunctor k g).obj (leftRegular k G)) (m + 2) (m + 1) m).map
            (forget₂ (Rep k G) (ModuleCat k))).g).ker
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.FiniteCyclic

Goal state before:
pp: 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
i : ℕ
x : leftRegular k G ⟶ A
⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =
    (ModuleCat.Hom.hom
        (((resolution k g hg).complex.linearYonedaObj k A).d i (i + 1) ≫ A.leftRegularHomEquiv.toModuleIso.hom))
      x

Tactic:
by_cases hi : Even i

Goal state after:
case pos
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
i : ℕ
x : leftRegular k G ⟶ A
hi : Even i
⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =
    (ModuleCat.Hom.hom
        (((resolution k g hg).complex.linearYonedaObj k A).d i (i + 1) ≫ A.leftRegularHomEquiv.toModuleIso.hom))
      x

case neg
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
i : ℕ
x : leftRegular k G ⟶ A
hi : ¬Even i
⊢ (ModuleCat.Hom.hom (A.leftRegularHomEquiv.toModuleIso.hom ≫ (moduleCatCochainComplex A g).d i (i + 1))) x =
    (ModuleCat.Hom.hom
        (((resolution k g hg).complex.linearYonedaObj k A).d i (i + 1) ≫ A.leftRegularHomEquiv.toModuleIso.hom))
      x
Module: Mathlib.RepresentationTheory.Homological.Resolution

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
x✝ : ↑((standardComplex k G).X 1)
⊢ (Hom.hom ((standardComplex k G).d 1 0 ≫ ε k G)).toLinearMap x✝ = (Hom.hom 0).toLinearMap x✝

Tactic:
have : (forget₂ToModuleCat k G).d 1 0
      ≫ (forget₂ (Rep k G) (ModuleCat.{u} k)).map (ε k G) = 0 := by
    rw [← forget₂ToModuleCatHomotopyEquiv_f_0_eq,
      ← (forget₂ToModuleCatHomotopyEquiv k G).1.2 1 0 rfl]
    exact comp_zero

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
x✝ : ↑((standardComplex k G).X 1)
this : (forget₂ToModuleCat k G).d 1 0 ≫ (forget₂ (Rep k G) (ModuleCat k)).map (ε k G) = 0
⊢ (Hom.hom ((standardComplex k G).d 1 0 ≫ ε k G)).toLinearMap x✝ = (Hom.hom 0).toLinearMap x✝
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree

Goal state before:
pp: G : Type u_1
M : Type u_2
inst✝² : Group G
inst✝¹ : CommGroup M
inst✝ : MulAction G M
f : G × G → M
hf : IsMulCocycle₂ f
g : G
⊢ g • f (g⁻¹, g) / f (g, g⁻¹) = f (1, 1) / f (g, 1)

Tactic:
have := hf g g⁻¹ g
  simp only [mul_inv_cancel, inv_mul_cancel, map_one_fst_of_isMulCocycle₂ hf g] at this
  exact div_eq_div_iff_mul_eq_mul.2 this.symm

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

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
m : MonomialOrder σ
B : Set (MvPolynomial σ R)
h : 0 ∈ B
⊢ span (m.leadingTerm '' insert 0 B) = span (m.leadingTerm '' B)

pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
m : MonomialOrder σ
B : Set (MvPolynomial σ R)
h : 0 ∉ B
⊢ span (m.leadingTerm '' insert 0 B) = span (m.leadingTerm '' B)

Tactic:
· rw [Set.insert_eq_of_mem h]

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
m : MonomialOrder σ
B : Set (MvPolynomial σ R)
h : 0 ∉ B
⊢ span (m.leadingTerm '' insert 0 B) = span (m.leadingTerm '' B)
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
f : σ → τ
inst✝² : TendstoCofinite f
inst✝¹ : DecidableEq σ
inst✝ : DecidableEq τ
x : τ →₀ ℕ
⊢ Finset.image
      (fun x ↦
        match x with
        | (fst, b) => (b.1 + b.2, b))
      ⋯.toFinset =
    ⋯.toFinset

Tactic:
by
  ext ⟨_, _, _⟩
  simp; grind [Finsupp.mapDomain_add]

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
⊢ Function.Surjective ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer A))

Tactic:
rintro ⟨x, hx⟩
  rcases range_d₁₀_eq_coinvariantsKer A ▸ hx with ⟨y, hy⟩
  use y, Subtype.ext hy

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
⊢ (ConcreteCategory.hom (d₂₁ A)) (single (g⁻¹, g * h) ((A.ρ g⁻¹) a) + single (g, h) a) =
    single g⁻¹ ((A.ρ g⁻¹) a) + single g a

Tactic:
simp only [map_add, d₂₁_single (G := G), inv_inv, self_inv_apply, inv_mul_cancel_left]
  abel

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
⊢ (shortComplexH0 A).Exact

Tactic:
rw [ShortComplex.moduleCat_exact_iff]
  intro x (hx : Coinvariants.mk _ _ = 0)
  rw [Coinvariants.mk_eq_zero, ← range_d₁₀_eq_coinvariantsKer] at hx
  rcases hx with ⟨x, hx, rfl⟩
  use x
  rfl

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
I : Ideal R
⊢ ∀ (i j : ℕ), ∃ k, I ^ k • ⊤ ≤ I ^ i • ⊤ ⊓ I ^ j • ⊤

Tactic:
suffices ∀ i j : ℕ, ∃ k, I ^ k ≤ I ^ i ∧ I ^ k ≤ I ^ j by
        simpa only [smul_eq_mul, mul_top, Algebra.algebraMap_self, map_id, le_inf_iff] using! this
      intro i j
      exact ⟨max i j, pow_le_pow_right (le_max_left i j), pow_le_pow_right (le_max_right i j)⟩

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

Goal state before:
pp: case neg
R : Type u_1
inst✝³ : CommRing R
I : Ideal R⟦X⟧
S : Set R
P : Ideal R⟦X⟧
inst✝² : P.IsPrime
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsDomain R
w✝¹ : R
w✝ : Set R⟦X⟧
h₁ : span w✝ ≠ ⊥
h₂ : (span w✝).IsPrime
hXP : X ∉ span w✝
h✝ : Ideal.map constantCoeff (span w✝) = R ∙ w✝¹
left✝ : w✝.Finite
h : w✝.ncard = {w✝¹}.ncard
⊢ ∃ x ∈ span w✝, Prime x

Tactic:
simp only [ncard_singleton, ncard_eq_one] at h

Goal state after:
case neg
R : Type u_1
inst✝³ : CommRing R
I : Ideal R⟦X⟧
S : Set R
P : Ideal R⟦X⟧
inst✝² : P.IsPrime
inst✝¹ : IsPrincipalIdealRing R
inst✝ : IsDomain R
w✝¹ : R
w✝ : Set R⟦X⟧
h₁ : span w✝ ≠ ⊥
h₂ : (span w✝).IsPrime
hXP : X ∉ span w✝
h✝ : Ideal.map constantCoeff (span w✝) = R ∙ w✝¹
left✝ : w✝.Finite
h : ∃ a, w✝ = {a}
⊢ ∃ x ∈ span w✝, Prime x
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case neg
σ✝ : Type u_1
R✝ : Type u_2
n✝ : ℕ
inst✝³ : CommRing R✝
inst✝² : Finite σ✝
σ : Type u_3
R : Type u_4
inst✝¹ : Finite σ
inst✝ : CommRing R
n : ℕ
h : n ≠ 0
⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1

Tactic:
rw [truncTotal_one h, map_one]

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

Goal state before:
pp: k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
⊢ ↑((v.sum fun a b ↦ single ((s ↑a)⁻¹ * a) ((A.ρ (s ↑a)⁻¹) b)) + v.sum fun a ↦ single (s ↑a)).support ⊆ ↑S

Tactic:
intro w hw

Goal state after:
k G : Type u
inst✝³ : CommRing k
inst✝² : Group G
A : Rep k G
S : Subgroup G
inst✝¹ : S.Normal
inst✝ : Representation.IsTrivial (MonoidHom.comp A.ρ S.subtype)
x : G →₀ ↑A
hxc : x ∈ cycles₁ A
hx :
  (ConcreteCategory.hom (H1π (A.ofQuotient S)))
      ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩) =
    0
y : ↑(ModuleCat.of k ((G ⧸ S) × G ⧸ S →₀ ↑(A.ofQuotient S)))
hy :
  (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y =
    ↑((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.resOfQuotientIso S).inv)) ⟨x, hxc⟩)
s : G ⧸ S → G
hs : ∀ (b : G ⧸ S), (QuotientGroup.mk' S) (s b) = b
hs₁ : QuotientGroup.mk ∘ s = id
z : G × G →₀ ↑A := (lmapDomain (↑A) k (Prod.map s s)) y
hz :
  (lmapDomain (↑A) k ⇑(QuotientGroup.mk' S)) ((ModuleCat.Hom.hom (d₂₁ A)) z) =
    (ModuleCat.Hom.hom (d₂₁ (A.ofQuotient S))) y
v : G →₀ ↑A := x - (ModuleCat.Hom.hom (d₂₁ A)) z
hv : mapDomain (s ∘ QuotientGroup.mk) v = 0
e : G → G × G := fun g ↦ (s ↑g, (s ↑g)⁻¹ * g)
he : Function.Injective e
ve : G × G →₀ ↑A := mapDomain e v
w : G
hw : w ∈ ↑((v.sum fun a b ↦ single ((s ↑a)⁻¹ * a) ((A.ρ (s ↑a)⁻¹) b)) + v.sum fun a ↦ single (s ↑a)).support
⊢ w ∈ ↑S
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂
a✝ : x ∈ ⊤
X : G →₀ ↥S →₀ ↑A
hX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x
Y : ↥S →₀ ↑A :=
  X.sum fun g f ↦ (mapRange.linearMap (A.ρ g⁻¹)) ((lmapDomain (↑A) k fun s ↦ (MulAut.conjNormal g⁻¹) s) f) - f
Z : G × G →₀ ↑A :=
  X.sum fun g f ↦ (lmapDomain (↑A) k fun s ↦ (g, g⁻¹ * ↑s * g)) f - (lmapDomain (↑A) k fun s ↦ (↑s, g)) f
⊢ (ModuleCat.Hom.hom (shortComplexH1 A).f) Z =
    (mapShortComplexH1 S.subtype (𝟙 (res S.subtype A))).τ₂.hom' Y -
      (mapShortComplexH1 (MonoidHom.id G) (A.coinvariantsShortComplex S).f).τ₂.hom' x

Tactic:
change d₂₁ A Z = mapRange id rfl (lmapDomain _ k Subtype.val Y) -
    mapRange.linearMap (Submodule.subtype _) (mapDomain id x)

Goal state after:
k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
A : Rep k G
S : Subgroup G
inst✝ : S.Normal
x : ↑(shortComplexH1 (A.coinvariantsShortComplex S).X₁).X₂
a✝ : x ∈ ⊤
X : G →₀ ↥S →₀ ↑A
hX : mapRange ⇑(ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) ⋯ X = x
Y : ↥S →₀ ↑A :=
  X.sum fun g f ↦ (mapRange.linearMap (A.ρ g⁻¹)) ((lmapDomain (↑A) k fun s ↦ (MulAut.conjNormal g⁻¹) s) f) - f
Z : G × G →₀ ↑A :=
  X.sum fun g f ↦ (lmapDomain (↑A) k fun s ↦ (g, g⁻¹ * ↑s * g)) f - (lmapDomain (↑A) k fun s ↦ (↑s, g)) f
⊢ (ConcreteCategory.hom (d₂₁ A)) Z =
    mapRange id ⋯ ((lmapDomain (↑A) k Subtype.val) Y) -
      (mapRange.linearMap (Coinvariants.ker (MonoidHom.comp A.ρ S.subtype)).subtype) (mapDomain id x)
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality

Goal state before:
pp: case h
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
x : ↥(cycles₁ (A.quotientToCoinvariants S))
y : ↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S)))
hy : (ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) ((A.toCoinvariants S).resOfQuotientIso S).inv)) y = x
Y : G →₀ ↑A
hY : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ Y = ↑y
this : (ConcreteCategory.hom (d₁₀ A)) Y ∈ Coinvariants.ker (MonoidHom.comp A.ρ S.subtype)
Z : ↥S →₀ ↑A
hZ : (ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) Z = ⟨(ConcreteCategory.hom (d₁₀ A)) Y, this⟩
⊢ ∃ a, (ConcreteCategory.hom (H1CoresCoinf A S).g) a = (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S))) x

Tactic:
have H : d₁₀ A (Y - mapDomain S.subtype Z) = 0 := by
    simpa [map_sub, sub_eq_zero, chains₁ToCoinvariantsKer, -LinearMap.sub_apply, d₁₀,
      sum_mapDomain_index_inj] using! Subtype.ext_iff.1 hZ.symm

Goal state after:
case h
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
x : ↥(cycles₁ (A.quotientToCoinvariants S))
y : ↑(ModuleCat.of k ↥(cycles₁ (A.toCoinvariants S)))
hy : (ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) ((A.toCoinvariants S).resOfQuotientIso S).inv)) y = x
Y : G →₀ ↑A
hY : mapRange ⇑(Coinvariants.mk (MonoidHom.comp A.ρ S.subtype)) ⋯ Y = ↑y
this : (ConcreteCategory.hom (d₁₀ A)) Y ∈ Coinvariants.ker (MonoidHom.comp A.ρ S.subtype)
Z : ↥S →₀ ↑A
hZ : (ConcreteCategory.hom (chains₁ToCoinvariantsKer (res S.subtype A))) Z = ⟨(ConcreteCategory.hom (d₁₀ A)) Y, this⟩
H : (ConcreteCategory.hom (d₁₀ A)) (Y - mapDomain (⇑S.subtype) Z) = 0
⊢ ∃ a, (ConcreteCategory.hom (H1CoresCoinf A S).g) a = (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S))) x
Module: Mathlib.RingTheory.DedekindDomain.GaussLemma

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
b : NNReal
hb : 1 < b
p : R[X]
hR : ¬IsField R
⊢ p.contentIdeal = ⊤ ↔ ∀ (v : HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1

Tactic:
convert_to _ ↔ ∀ (x : HeightOneSpectrum R), 1 ≤ gaussNorm (x.intAdicAbv hb) 1 p

Goal state after:
case e'_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
b : NNReal
hb : 1 < b
p : R[X]
hR : ¬IsField R
⊢ (∀ (v : HeightOneSpectrum R), gaussNorm (v.intAdicAbv hb) 1 p = 1) ↔
    ∀ (x : HeightOneSpectrum R), 1 ≤ gaussNorm (x.intAdicAbv hb) 1 p

case convert_2
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
b : NNReal
hb : 1 < b
p : R[X]
hR : ¬IsField R
⊢ p.contentIdeal = ⊤ ↔ ∀ (x : HeightOneSpectrum R), 1 ≤ gaussNorm (x.intAdicAbv hb) 1 p
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
a : R
⊢ a ∈ ↑T → a ∈ ↑(⊥.colon {↑(∏ a, g a) • m}).radical

Tactic:
ha

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
ha : a ∈ ↑T
⊢ a ∈ ↑(⊥.colon {↑(∏ a, g a) • m}).radical
Module: Mathlib.RingTheory.Depth.Rees

Goal state before:
pp: case succ
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
n : ℕ
ih :
  ∀ (M : ModuleCat R) [Module.Finite R ↑M],
    I • ⊤ < ⊤ →
      ∀ (rs : List R),
        (∀ r ∈ rs, r ∈ I) → RingTheory.Sequence.IsRegular (↑M) rs → rs.length = n → ∀ i < n, Subsingleton (Ext N M 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 = n + 1
⊢ ∀ i < n + 1, Subsingleton (Ext N M i)

Tactic:
rintro i hi

Goal state after:
case succ
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
n : ℕ
ih :
  ∀ (M : ModuleCat R) [Module.Finite R ↑M],
    I • ⊤ < ⊤ →
      ∀ (rs : List R),
        (∀ r ∈ rs, r ∈ I) → RingTheory.Sequence.IsRegular (↑M) rs → rs.length = n → ∀ i < n, Subsingleton (Ext N M 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 = n + 1
i : ℕ
hi : i < n + 1
⊢ Subsingleton (Ext N M i)
Module: Mathlib.RingTheory.DividedPowers.DPMorphism

Goal state before:
pp: case pos
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
hn : n = 0
a✝ : A
⊢ a✝ ∈ I → hJ.dpow n (f a✝) = f (hI.dpow n a✝)

Tactic:
ha

Goal state after:
case pos
A : Type u_1
B : Type u_2
inst✝¹ : CommSemiring A
inst✝ : CommSemiring B
I : Ideal A
J : Ideal B
hI : DividedPowers I
hJ : DividedPowers J
f : A →+* B
hIJ : map f I ≤ J
H : ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)
n : ℕ
hn : n = 0
a✝ : A
ha : a✝ ∈ I
⊢ hJ.dpow n (f a✝) = f (hI.dpow n a✝)