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

Goal state before:
pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : A < 0
⊢ ∫ (y : ℝ), (Set.Ioc 0 A).indicator id y ^ n ∂Measure.map f μ = -∫ (x : ℝ) in Set.Ioc A 0, x ^ n ∂Measure.map f μ

pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : A < 0
⊢ AEStronglyMeasurable (fun z ↦ (Set.Ioc 0 A).indicator id z ^ n) (Measure.map f μ)

Tactic:
← integral_indicator M'

Goal state after:
case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : A < 0
⊢ ∫ (y : ℝ), (Set.Ioc 0 A).indicator id y ^ n ∂Measure.map f μ =
    -∫ (x : ℝ), (Set.Ioc A 0).indicator (fun x ↦ x ^ n) x ∂Measure.map f μ

case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : AEStronglyMeasurable f μ
A : ℝ
n : ℕ
hn : n ≠ 0
h'f : 0 ≤ f
M : MeasurableSet (Set.Ioc 0 A)
M' : MeasurableSet (Set.Ioc A 0)
hA : A < 0
⊢ AEStronglyMeasurable (fun z ↦ (Set.Ioc 0 A).indicator id z ^ n) (Measure.map f μ)
Module: Mathlib.Probability.Moments.SubGaussian

Goal state before:
pp: Ω : Type u_1
m mΩ : MeasurableSpace Ω
hm : m ≤ mΩ
inst✝¹ : StandardBorelSpace Ω
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
X : Ω → ℝ
c : ℝ≥0
h : HasCondSubgaussianMGF m hm X c μ
t : ℝ
⊢ ∀ᵐ (ω' : Ω) ∂μ.trim hm, μ[fun ω ↦ rexp (t * X ω) | m] ω' ≤ rexp (↑c * t ^ 2 / 2)

Tactic:
by
  have h_eq := condExp_ae_eq_trim_integral_condExpKernel hm (h.integrable_exp_mul t)
  simp_rw [condExpKernel_comp_trim] at h_eq
  filter_upwards [h.mgf_le, h_eq] with ω' h_mgf h_eq
  rw [h_eq]
  exact h_mgf t

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: 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 ∈ range K, ∫ (x : ℝ) in ↑k..↑(k + 1), 2 * x ∂ρ = 2 * ∫ (x : ℝ) in 0..↑K, x ∂ρ

Tactic:
rw [intervalIntegral.sum_integral_adjacent_intervals fun k _ => ?_]
      swap; · exact (continuous_const.mul continuous_id').intervalIntegrable _ _
      rw [intervalIntegral.integral_const_mul]
      norm_cast

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

Goal state before:
pp: case insert
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
ι : Type u_2
X : ι → Ω → ℝ
h_indep : iIndepFun X μ
c : ι → ℝ≥0
h_meas : ∀ (i : ι), AEMeasurable (X i) μ
this : IsProbabilityMeasure μ
i : ι
s : Finset ι
his : i ∉ s
h : (∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ) → HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ
h_subG : ∀ i_1 ∈ insert i s, HasSubgaussianMGF (X i_1) (c i_1) μ
⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ insert i s, X i ω) (∑ i ∈ insert i s, c i) μ

Tactic:
simp_rw [← Finset.sum_apply, Finset.sum_insert his, Pi.add_apply, Finset.sum_apply]
    have h_indep' := (h_indep.indepFun_finsetSum_of_notMem₀ h_meas his).symm
    refine add_of_indepFun (h_subG _ (Finset.mem_insert_self _ _)) (h ?_) ?_
    · exact fun i hi ↦ h_subG _ (Finset.mem_insert_of_mem hi)
    · convert! h_indep'
      rw [Finset.sum_apply]

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n ↦ truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n ↦ ∑ i ∈ range n, Y i
hS : S = fun n ↦ ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n ↦ ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
N : ℕ
⊢ ∑ i ∈ range N, ENNReal.ofReal (Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2) =
    ENNReal.ofReal (∑ i ∈ range N, Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2)

Tactic:
rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]
        exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝⁴ : IsProbabilityMeasure μ
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSpace E
X : ℕ → Ω → E
h' : Measurable (X 0)
hindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) μ μ
φ : SimpleFunc E E
⊢ ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, φ (X i ω)) atTop (𝓝 (∫ (x : Ω), (⇑φ ∘ X 0) x ∂μ))

Tactic:
by
  -- this follows from the one-dimensional version when `φ` takes a single value, and is then
  -- extended to the general case by linearity.
  refine SimpleFunc.induction (motive := fun ψ ↦ ∀ᵐ ω ∂μ,
    Tendsto (fun n : ℕ ↦ (n : ℝ)⁻¹ • (∑ i ∈ range n, ψ (X i ω))) atTop (𝓝 μ[ψ ∘ (X 0)])) ?_ ?_ φ
  · intro c s hs
    simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
      SimpleFunc.coe_zero, piecewise_eq_indicator, Function.comp_apply]
    let F : E → ℝ := indicator s 1
    have F_meas : Measurable F := (measurable_indicator_const_iff 1).2 hs
    let Y : ℕ → Ω → ℝ := fun n ↦ F ∘ (X n)
    have : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun (n : ℕ) ↦ (n : ℝ)⁻¹ • ∑ i ∈ Finset.range n, Y i ω)
        atTop (𝓝 μ[Y 0]) := by
      simp only [smul_eq_mul, ← div_eq_inv_mul]
      apply strong_law_ae_real
      · exact SimpleFunc.integrable_of_isFiniteMeasure
          ((SimpleFunc.piecewise s hs (SimpleFunc.const _ (1 : ℝ))
            (SimpleFunc.const _ (0 : ℝ))).comp (X 0) h')
      · exact fun i j hij ↦ IndepFun.comp (hindep hij) F_meas F_meas
      · exact fun i ↦ (hident i).comp F_meas
    filter_upwards [this] with ω hω
    have I : indicator s (Function.const E c) = (fun x ↦ (indicator s (1 : E → ℝ) x) • c) := by
      ext
      rw [← indicator_smul_const_apply]
      congr! 1
      ext
      simp
    simp only [I, integral_smul_const]
    convert! Tendsto.smul_const hω c using 1
    simp [F, Y, ← sum_smul, smul_smul]
  · rintro φ ψ - hφ hψ
    filter_upwards [hφ, hψ] with ω hωφ hωψ
    convert! hωφ.add hωψ using 1
    · simp [sum_add_distrib]
    · congr 1
      rw [← integral_add]
      · rfl
      · exact (φ.comp (X 0) h').integrable_of_isFiniteMeasure
      · exact (ψ.comp (X 0) h').integrable_of_isFiniteMeasure

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
⊢ ∀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 A : ∀ i, Integrable (X i) μ := fun i ↦ (hident i).integrable_iff.2 hint

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
⊢ ∀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.Semisimple

Goal state before:
pp: k : Type u_1
G : Type u_2
V : Type u_3
inst✝³ : Monoid G
inst✝² : Field k
inst✝¹ : AddCommGroup V
inst✝ : Module k V
ρ : Representation k G V
⊢ ρ.IsSemisimpleRepresentation ↔ ComplementedLattice (Submodule k[G] ρ.asModule)

Tactic:
exact OrderIso.complementedLattice_iff Subrepresentation.subrepresentationSubmoduleOrderIso

Goal state after:
Module: Mathlib.RepresentationTheory.FiniteIndex

Goal state before:
pp: k : Type u
G : Type v
inst✝² : CommRing k
inst✝¹ : Group G
S : Subgroup G
inst✝ : DecidableRel ⇑(QuotientGroup.rightRel S)
A B : Rep.{u_1, u, v} k ↥S
f : A ⟶ B
g₁ g₂ : G
a : ↑A
⊢ (B.indToCoindAux g₁) ((Hom.hom f) a) g₂ = (Hom.hom f) ((A.indToCoindAux g₁) a g₂)

Tactic:
rcases em ((QuotientGroup.rightRel S).r g₂ g₁) with ⟨s, rfl⟩ | h
  · simp [S.1.smul_def, hom_comm_apply]
  · simp [indToCoindAux_of_not_rel (h := h)]

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

Goal state before:
pp: 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
n : ℕ
⊢ cocyclesMap (φ.comp ψ) ((resFunctor ↑ψ).map f ≫ g) n = cocyclesMap φ f n ≫ cocyclesMap ψ g n

Tactic:
by
  simp [cocyclesMap, ← HomologicalComplex.cyclesMap_comp, ← cochainsMap_comp]

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

Goal state before:
pp: 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
σ : ↑(X.homogeneousCochains.X 0).toModuleCat
g : G
hσ :
  (((X.resolutionX (0 + 1)).ρ.coind₁ι ↑σ) 1) g⁻¹ = (((Hom.hom ((coind₁Functor k G).map (ofHom X.ρ.coind₁ι))) ↑σ) 1) g⁻¹
⊢ (X.ρ g) (↑σ 1) = ↑σ 1

Tactic:
simp only [Nat.reduceAdd, coind₁ι_toFun, ContinuousMap.const_apply, ConcreteCategory.hom_ofHom,
    coind₁Map_toFun, ContinuousMap.comp_apply, ContinuousMap.coe_mk] at hσ

Goal state after:
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
σ : ↑(X.homogeneousCochains.X 0).toModuleCat
g : G
hσ : ↑σ g⁻¹ = ↑σ 1
⊢ (X.ρ g) (↑σ 1) = ↑σ 1
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
f : Gal(L/K) → Lˣ
z : L
hz : aux f z ≠ 0
this : aux f z = ∑ h, ↑(f h) * h z
g : Gal(L/K)
hf : ∀ (g h : Gal(L/K)), ↑(f (g * h)) = g ↑(f h) * ↑(f g)
⊢ ∑ h, ↑(f h) * h z = (∑ x, g (↑(f x) * x z)) * ↑(f g)

Tactic:
map_mul,

Goal state after:
case h
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : FiniteDimensional K L
f : Gal(L/K) → Lˣ
z : L
hz : aux f z ≠ 0
this : aux f z = ∑ h, ↑(f h) * h z
g : Gal(L/K)
hf : ∀ (g h : Gal(L/K)), ↑(f (g * h)) = g ↑(f h) * ↑(f g)
⊢ ∑ h, ↑(f h) * h z = (∑ x, g ↑(f x) * g (x z)) * ↑(f g)
Module: Mathlib.RepresentationTheory.Homological.Resolution

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Monoid G
⊢ QuasiIso (((forget₂ (Rep k G) (ModuleCat k)).mapHomologicalComplex (ComplexShape.down ℕ)).map (εToSingle₀ k G))

Tactic:
by
  have h : QuasiIso (forget₂ToModuleCatHomotopyEquiv k G).hom := inferInstance
  rw [← εToSingle₀_comp_eq k G] at h
  exact quasiIso_of_comp_right (hφφ' := h)

Goal state after:
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90

Goal state before:
pp: case refine_2
K L : Type
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra K L
inst✝¹³ : FiniteDimensional K L
inst✝¹² : IsGalois K L
inst✝¹¹ : IsCyclic Gal(L/K)
g : Gal(L/K)
A : Type u_1
B : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra A L
inst✝⁶ : Algebra A K
inst✝⁵ : Algebra B L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : IsIntegralClosure B A L
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
η : B
hη : (Algebra.norm K) ((algebraMap B L) η) = 1
this✝¹ : Module.IsTorsionFree A L
this✝ : IsLocalization (Algebra.algebraMapSubmonoid B A⁰) L
η' : Lˣ := ⋯
ε : Lˣ
a : B
b : ↥(Algebra.algebraMapSubmonoid B A⁰)
h : IsLocalization.mk' L a b = ↑ε
t : A
ht : t ∈ ↑A⁰
ht' : (algebraMap A B) t = ↑b
this : t • IsLocalization.mk' L a b = (algebraMap B L) a
hε : (algebraMap B L) η * ↑g ((algebraMap B L) a) = (algebraMap B L) a
⊢ (algebraMap B L) (η * ((galRestrict A K L B) g) a) = (algebraMap B L) a

Tactic:
map_mul,

Goal state after:
case refine_2
K L : Type
inst✝¹⁶ : Field K
inst✝¹⁵ : Field L
inst✝¹⁴ : Algebra K L
inst✝¹³ : FiniteDimensional K L
inst✝¹² : IsGalois K L
inst✝¹¹ : IsCyclic Gal(L/K)
g : Gal(L/K)
A : Type u_1
B : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : CommRing B
inst✝⁸ : Algebra A B
inst✝⁷ : Algebra A L
inst✝⁶ : Algebra A K
inst✝⁵ : Algebra B L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : IsIntegralClosure B A L
hg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g
η : B
hη : (Algebra.norm K) ((algebraMap B L) η) = 1
this✝¹ : Module.IsTorsionFree A L
this✝ : IsLocalization (Algebra.algebraMapSubmonoid B A⁰) L
η' : Lˣ := Units.mk0 ((algebraMap B L) η) ⋯
ε : Lˣ
a : B
b : ↥(Algebra.algebraMapSubmonoid B A⁰)
h : IsLocalization.mk' L a b = ↑ε
t : A
ht : t ∈ ↑A⁰
ht' : (algebraMap A B) t = ↑b
this : t • IsLocalization.mk' L a b = (algebraMap B L) a
hε : (algebraMap B L) η * ↑g ((algebraMap B L) a) = (algebraMap B L) a
⊢ (algebraMap B L) η * (algebraMap B L) (((galRestrict A K L B) g) a) = (algebraMap B L) a
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.FiniteCyclic

Goal state before:
pp: k G : Type u
inst✝⁴ : CommRing k
inst✝³ : CommGroup G
inst✝² : Fintype G
A : Rep k G
g : G
inst✝¹ : DecidableEq G
hg : ∀ (x : G), x ∈ Subgroup.zpowers g
i : ℕ
inst✝ : NeZero i
hi : Even i
x y : ↥(LinearMap.ker A.ρ.norm)
⊢ (ConcreteCategory.hom (groupHomologyπEven A g hg i hi)) x =
      (ConcreteCategory.hom (groupHomologyπEven A g hg i hi)) y ↔
    ↑x - ↑y ∈ (Hom.hom (A.applyAsHom g - 𝟙 A)).range

Tactic:
by
  rw [← sub_eq_zero, ← map_sub, groupHomologyπEven_eq_zero_iff, AddSubgroupClass.coe_sub]

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

Goal state before:
pp: k : Type u
inst✝³ : CommRing k
G H K : Type u
inst✝² : Group G
inst✝¹ : Group H
inst✝ : Group K
A : Rep k K
B : Rep k H
C : Rep k G
f : H →* K
g : G →* H
φ : res f A ⟶ B
ψ : res g B ⟶ C
n : ℕ
⊢ cocyclesMap (f.comp g) ((resFunctor g).map φ ≫ ψ) n = cocyclesMap f φ n ≫ cocyclesMap g ψ n

Tactic:
by
  simp [cocyclesMap, ← HomologicalComplex.cyclesMap_comp, ← cochainsMap_comp]

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

Goal state before:
pp: k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : res f A ⟶ B
n✝ : ℕ
e : G ≃* H
e' : ↑B ≃ₗ[k] ↑A
he : ∀ (g : G), ↑e' ∘ₗ B.ρ g = A.ρ (e g) ∘ₗ ↑e'
n : ℕ
h : H
⊢ ↑e' ∘ₗ (MonoidHom.comp B.ρ ↑e.symm) h = A.ρ h ∘ₗ ↑e'

Tactic:
by simp [he]

Goal state after:
Module: Mathlib.RepresentationTheory.Tannaka

Goal state before:
pp: k G : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Finite G
η : Aut (forget k G)
α : (G → k) →ₗ[k] G → k := Hom.hom (η.hom.hom.app rightFDRep).hom
α_inv : (G → k) →ₗ[k] G → k := Hom.hom (η.inv.hom.app rightFDRep).hom
this h : α (α_inv 1) = 1
⊢ α 1 = 1

Tactic:
rwa [← one_mul (α_inv 1), map_mul_toRightFDRepComp, h, mul_one] at this

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

Goal state before:
pp: k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : res f A ⟶ B
n✝ n : ℕ
X✝ Y✝ Z✝ : Rep k G
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
⊢ map (MonoidHom.id G) (x✝¹ ≫ x✝) n = map (MonoidHom.id G) x✝¹ n ≫ map (MonoidHom.id G) x✝ n

Tactic:
simp only [← HomologicalComplex.homologyMap_comp]

Goal state after:
k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k H
B : Rep k G
f : G →* H
φ : res f A ⟶ B
n✝ n : ℕ
X✝ Y✝ Z✝ : Rep k G
x✝¹ : X✝ ⟶ Y✝
x✝ : Y✝ ⟶ Z✝
⊢ map (MonoidHom.id G) (x✝¹ ≫ x✝) n =
    HomologicalComplex.homologyMap (cochainsMap (MonoidHom.id G) x✝¹ ≫ cochainsMap (MonoidHom.id G) x✝) n
Module: Mathlib.RingTheory.MvPowerSeries.Rename

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
⊢ Function.Injective ⇑(rename ⇑e)

Tactic:
intro _ _ h; ext x
  simpa using MvPowerSeries.ext_iff.mp h (embDomain e x)

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

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
x : σ →₀ ℕ
r : R
⊢ killComplFun e ((monomial (embDomain e x)) r) = (monomial x) r

Tactic:
by
  classical
  ext; simp [coeff_killComplFun, coeff_monomial, embDomain_inj]

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

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
inst✝ : CommSemiring R
e : σ ↪ τ
i : σ
⊢ (killCompl e) (X (e i)) = X i

Tactic:
ext; simp [coeff_X, coeff_killCompl, ← embDomain_single]

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

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : DistribMulAction G A
x : G →₀ A
⊢ IsBoundary₁ x ↔ ∃ y, (y.sum fun g a ↦ single g.2 a - single (g.1 * g.2) (g.1 • a) + single g.1 (g.1 • a)) = x

Tactic:
by
  constructor
  · rintro ⟨y, hy⟩
    use y.sum (fun g a => single g (g.1⁻¹ • a))
    simp_all [sum_sum_index]
  · rintro ⟨x, hx⟩
    use x.sum (fun g a => single g (g.1 • a))
    simp_all [sum_sum_index]

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

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝² : Group G
inst✝¹ : AddCommGroup A
inst✝ : DistribMulAction G A
x : G × G →₀ A
⊢ IsBoundary₂ x ↔
    ∃ y,
      (y.sum fun g a ↦
          single (g.2.1, g.2.2) a - single (g.1 * g.2.1, g.2.2) (g.1 • a) + single (g.1, g.2.1 * g.2.2) (g.1 • a) -
            single (g.1, g.2.1) (g.1 • a)) =
        x

Tactic:
by
  constructor
  · rintro ⟨y, hy⟩
    use y.sum (fun g a => single g (g.1⁻¹ • a))
    simp_all [sum_sum_index]
  · rintro ⟨x, hx⟩
    use x.sum (fun g a => single g (g.1 • a))
    simp_all [sum_sum_index]

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

Goal state before:
pp: case mpr.h
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] ↑(J ^ n)
H₂ : ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
s : Set R
this : SubmodulesRingBasis fun n ↦ J ^ n • ⊤ := Ideal.adic_basis J
⊢ s ∈ 𝓝 0 ↔ ∃ i, True ∧ ↑(J ^ i) ⊆ s

Tactic:
constructor <;> intro H

Goal state after:
case mpr.h.mp
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] ↑(J ^ n)
H₂ : ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
s : Set R
this : SubmodulesRingBasis fun n ↦ J ^ n • ⊤ := Ideal.adic_basis J
H : s ∈ 𝓝 0
⊢ ∃ i, True ∧ ↑(J ^ i) ⊆ s

case mpr.h.mpr
R : Type u_1
inst✝¹ : CommRing R
top : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
H₁ : ∀ (n : ℕ), IsOpen[top] ↑(J ^ n)
H₂ : ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
s : Set R
this : SubmodulesRingBasis fun n ↦ J ^ n • ⊤ := Ideal.adic_basis J
H : ∃ i, True ∧ ↑(J ^ i) ⊆ s
⊢ s ∈ 𝓝 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
x : Option σ →₀ ℕ
r : R
n : ℕ
y : σ →₀ ℕ
h1 : ¬optionElim n y = x
h3 : ¬n = x none
⊢ 0 = (coeff y) 0

Tactic:
coeff_zero

Goal state after:
case neg
σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
x : Option σ →₀ ℕ
r : R
n : ℕ
y : σ →₀ ℕ
h1 : ¬optionElim n y = x
h3 : ¬n = x none
⊢ 0 = 0
Module: Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology

Goal state before:
pp: case right
R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
h : (∀ (n : ℕ), IsOpen[inst✝¹] ↑(J ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : V ∈ 𝓝 0
m : ℕ
hm : ↑(J ^ m) ⊆ V
⊢ ∃ n_1, ↑((J ^ n) ^ n_1) ⊆ V

Tactic:
use m

Goal state after:
case h
R : Type u_1
inst✝² : CommRing R
inst✝¹ : TopologicalSpace R
inst✝ : IsTopologicalRing R
J : Ideal R
h : (∀ (n : ℕ), IsOpen[inst✝¹] ↑(J ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(J ^ n) ⊆ s
n : ℕ
hn : 0 < n
V : Set R
hV : V ∈ 𝓝 0
m : ℕ
hm : ↑(J ^ m) ⊆ V
⊢ ↑((J ^ n) ^ m) ⊆ V
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case neg
R : Type u_2
inst✝ : CommSemiring R
n k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(x = 0 ∧ k = 1)
h3 : ¬k = 1
⊢ 0 = (coeff x) 0

Tactic:
coeff_zero

Goal state after:
case neg
R : Type u_2
inst✝ : CommSemiring R
n k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(x = 0 ∧ k = 1)
h3 : ¬k = 1
⊢ 0 = 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case neg
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
j : Fin n
k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(x = single j 1 ∧ k = 0)
h3 : ¬k = 0
⊢ 0 = (coeff x) 0

Tactic:
coeff_zero

Goal state after:
case neg
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
j : Fin n
k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(x = single j 1 ∧ k = 0)
h3 : ¬k = 0
⊢ 0 = 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: case neg
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
r : R
k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(k = 0 ∧ x = 0)
h3 : ¬k = 0
⊢ 0 = (coeff x) 0

Tactic:
coeff_zero

Goal state after:
case neg
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
r : R
k : ℕ
x : Fin n →₀ ℕ
h1 : ¬(k = 0 ∧ x = 0)
h3 : ¬k = 0
⊢ 0 = 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: 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
p : MvPolynomial σ R
x : σ →₀ ℕ
h : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) ↑p - p) = 0

Tactic:
rw [MvPolynomial.coeff_sub, sub_eq_zero, coeff_truncTotal _ h, MvPolynomial.coeff_coe]

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

Goal state before:
pp: k G H : Type u
inst✝² : CommRing k
inst✝¹ : Group G
inst✝ : Group H
A : Rep k G
B : Rep k H
f : G →* H
φ : A ⟶ res f B
n✝ : ℕ
e : G ≃* H
e' : ↑A ≃ₗ[k] ↑B
he : ∀ (g : G), ↑e' ∘ₗ A.ρ g = B.ρ (e g) ∘ₗ ↑e'
n : ℕ
⊢ ∀ (g : G), ↑e' ∘ₗ A.ρ g = (MonoidHom.comp B.ρ ↑e) g ∘ₗ ↑e'

Tactic:
by simp [he]

Goal state after: