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

Goal state before:
pp: E : Type u_1
inst✝ : MeasurableSpace E
μ : Measure E
Ω : Type u_2
x✝ : MeasurableSpace Ω
ℙ : Measure Ω
X : Ω → E
s : Set E
hns : μ s ≠ 0
hnt : μ s ≠ ∞
hu : IsUniform X s ℙ μ
⊢ HasPDF X ℙ μ

Tactic:
by
  let t := toMeasurable μ s
  apply hasPDF_of_map_eq_withDensity (hu.aemeasurable hns hnt) (t.indicator ((μ t)⁻¹ • 1)) <|
    (measurable_one.aemeasurable.const_smul (μ t)⁻¹).indicator (measurableSet_toMeasurable μ s)
  rw [hu, withDensity_indicator (measurableSet_toMeasurable μ s), withDensity_smul _ measurable_one,
    withDensity_one, restrict_toMeasurable hnt, measure_toMeasurable, ProbabilityTheory.cond]

Goal state after:
Module: Mathlib.Probability.Independence.InfinitePi

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
mΩ : MeasurableSpace Ω
P : Measure Ω
𝓧 : ι → Type u_3
m𝓧 : (i : ι) → MeasurableSpace (𝓧 i)
X : (i : ι) → Ω → 𝓧 i
inst✝ : IsProbabilityMeasure P
mX : AEMeasurable (fun ω i ↦ X i ω) P
h : map (fun ω i ↦ X i ω) P = infinitePi fun i ↦ map (X i) P
⊢ iIndepFun X P

Tactic:
have _ i := isProbabilityMeasure_map (mX.eval i)
    rw [iIndepFun_iff_finset]
    intro s
    rw [iIndepFun_iff_map_fun_eq_pi_map]
    · have : s.restrict ∘ (fun ω i ↦ X i ω) = fun ω i ↦ s.restrict X i ω := by ext; simp
      rw [← this, ← AEMeasurable.map_map_of_aemeasurable, h, infinitePi_map_restrict]
      · simp
      · fun_prop
      exact mX
    exact fun i ↦ mX.eval i

Goal state after:
Module: Mathlib.Probability.Kernel.Condexp

Goal state before:
pp: case pos
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : μ s = 0
⊢ μ[f | MeasurableSpace.generateFrom {s}] =ᵐ[μ.restrict s] fun x ↦ ∫ (x : Ω), f x ∂μ[|s]

pp: case neg
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
⊢ μ[f | MeasurableSpace.generateFrom {s}] =ᵐ[μ.restrict s] fun x ↦ ∫ (x : Ω), f x ∂μ[|s]

Tactic:
· rw [Measure.restrict_eq_zero.2 hμs]
    rfl

Goal state after:
case neg
Ω : Type u_1
F : Type u_2
mΩ : MeasurableSpace Ω
μ : Measure Ω
inst✝³ : IsFiniteMeasure μ
s : Set Ω
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
hs : MeasurableSet s
f : Ω → F
hf : Integrable f μ
hμs : ¬μ s = 0
⊢ μ[f | MeasurableSpace.generateFrom {s}] =ᵐ[μ.restrict s] fun x ↦ ∫ (x : Ω), f x ∂μ[|s]
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
𝓧 : Type u_4
mX : MeasurableSpace 𝓧
A : Set Ω
X : Ω → 𝓧
h : A.indicator 1 ⟂ᵢ[P] X
⊢ IndepSets {A} {s | MeasurableSet s} P

Tactic:
by
  rw [IndepSets_iff]
  rintro s - hs ⟨t, ht, rfl⟩
  rw [Set.mem_singleton_iff.1 hs]
  have hA' : A = A.indicator (1 : Ω → ℝ) ⁻¹' {1} := by ext; simp [Set.indicator]
  rw [hA']
  exact h.measure_inter_preimage_eq_mul _ _ (by simp) ht

Goal state after:
Module: Mathlib.Probability.Independence.BoundedContinuousFunction

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
G : Type u_6
H : Type u_7
inst✝⁸ : TopologicalSpace G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : HasOuterApproxClosed G
inst✝⁴ : TopologicalSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
inst✝¹ : HasOuterApproxClosed H
Z : Ω → G
U : Ω → H
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mU : AEMeasurable U P
h :
  ∀ (f : G →ᵇ ℝ) (g : H →ᵇ ℝ), ∫ (x : Ω), (⇑f ∘ Z * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
⊢ Z ⟂ᵢ[P] U

Tactic:
rw [indepFun_iff_map_prod_eq_prod_map_map mZ mU]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
G : Type u_6
H : Type u_7
inst✝⁸ : TopologicalSpace G
inst✝⁷ : MeasurableSpace G
inst✝⁶ : BorelSpace G
inst✝⁵ : HasOuterApproxClosed G
inst✝⁴ : TopologicalSpace H
inst✝³ : MeasurableSpace H
inst✝² : BorelSpace H
inst✝¹ : HasOuterApproxClosed H
Z : Ω → G
U : Ω → H
inst✝ : IsFiniteMeasure P
mZ : AEMeasurable Z P
mU : AEMeasurable U P
h :
  ∀ (f : G →ᵇ ℝ) (g : H →ᵇ ℝ), ∫ (x : Ω), (⇑f ∘ Z * ⇑g ∘ U) x ∂P = (∫ (x : Ω), (⇑f ∘ Z) x ∂P) * ∫ (x : Ω), (⇑g ∘ U) x ∂P
⊢ map (fun ω ↦ (Z ω, U ω)) P = (map Z P).prod (map U P)
Module: Mathlib.Probability.Independence.Conditional

Goal state before:
pp: case refine_1
Ω : Type u_1
β : Type u_3
β' : Type u_4
m' mΩ : MeasurableSpace Ω
inst✝¹ : StandardBorelSpace Ω
hm' : m' ≤ mΩ
μ : Measure Ω
inst✝ : IsFiniteMeasure μ
f : Ω → β
g : Ω → β'
mβ : MeasurableSpace β
mβ' : MeasurableSpace β'
hf : Measurable f
hg : Measurable g
h :
  ∀ (t1 t2 : Set Ω),
    MeasurableSet t1 →
      MeasurableSet t2 →
        μ[(t1 ∩ t2).indicator fun ω ↦ 1 | m'] =ᵐ[μ] μ[t1.indicator fun ω ↦ 1 | m'] * μ[t2.indicator fun ω ↦ 1 | m']
s : Set β
t : Set β'
hs : MeasurableSet s
ht : MeasurableSet t
⊢ μ[(f ⁻¹' s ∩ g ⁻¹' t).indicator fun ω ↦ 1 | m'] =ᵐ[μ] fun ω ↦
    μ[(f ⁻¹' s).indicator fun ω ↦ 1 | m'] ω * μ[(g ⁻¹' t).indicator fun ω ↦ 1 | m'] ω

Tactic:
exact h (f ⁻¹' s) (g ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩

Goal state after:
Module: Mathlib.Probability.Martingale.OptionalSampling

Goal state before:
pp: Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
ι : Type u_3
inst✝⁶ : LinearOrder ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝² : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
inst✝¹ : Nonempty ι
h✝ : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
i : WithTop ι
h : i = ⊤
hi : ∃ y, τ y = ⊤
⊢ False

Tactic:
obtain ⟨ω, hω⟩ := hi

Goal state after:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : CompleteSpace E
ι : Type u_3
inst✝⁶ : LinearOrder ι
inst✝⁵ : TopologicalSpace ι
inst✝⁴ : OrderTopology ι
inst✝³ : FirstCountableTopology ι
ℱ : Filtration ι m
inst✝² : SigmaFiniteFiltration μ ℱ
τ : Ω → WithTop ι
f : ι → Ω → E
n : ι
inst✝¹ : Nonempty ι
h✝ : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_countable_range : (Set.range τ).Countable
inst✝ : SigmaFinite (μ.trim ⋯)
this : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i}
i : WithTop ι
h : i = ⊤
ω : Ω
hω : τ ω = ⊤
⊢ False
Module: Mathlib.Probability.Kernel.Representation

Goal state before:
pp: X : Type u_1
Y : Type u_2
mX : MeasurableSpace X
inst✝² : Nonempty Y
mY : MeasurableSpace Y
inst✝¹ : StandardBorelSpace Y
κ : Kernel X Y
inst✝ : IsMarkovKernel κ
⊢ ∃ f, Measurable (uncurry f) ∧ ∀ (a : X), Measure.map (f a) volume = κ a

Tactic:
by
  let g := sigmoid ∘ embeddingReal Y
  have hg := measurableEmbedding_sigmoid_comp_embeddingReal Y
  have hκg : IsMarkovKernel (κ.map g) := IsMarkovKernel.map κ hg.measurable
  have hg'κ : κ = (κ.map g).map hg.invFun := by
    rw [← map_comp_right _ hg.measurable (by fun_prop), LeftInverse.id hg.leftInverse_invFun,
      map_id]
  obtain ⟨f', hf', hf'κ⟩ := (κ.map g).exists_measurable_map_eq_unitInterval_aux
  refine ⟨fun a u ↦ hg.invFun (f' a u), by fun_prop, fun a ↦ ?_⟩
  rw [hg'κ, map_apply _ (by fun_prop), ← hf'κ, Measure.map_map (by fun_prop) (by fun_prop)]
  rfl

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X : Ω → ℝ
c : ℝ≥0
h : HasSubgaussianMGF X c κ ν
⊢ ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), Integrable (fun ω ↦ rexp (t * X ω)) (κ ω')

Tactic:
have h_int (n : ℤ) : ∀ᵐ ω' ∂ν, Integrable (fun ω ↦ exp (n * X ω)) (κ ω') :=
    h.ae_integrable_exp_mul _
  rw [← ae_all_iff] at h_int
  filter_upwards [h_int] with ω' h_int t
  exact integrable_exp_mul_of_le_of_le (h_int _) (h_int _) (Int.floor_le t) (Int.le_ceil t)

Goal state after:
Module: Mathlib.Probability.Process.LocalProperty

Goal state before:
pp: ι : Type u_1
Ω : Type u_2
E : Type u_3
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝⁴ : LinearOrder ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p q : (ι → Ω → E) → Prop
inst✝³ : OrderBot ι
inst✝² : Zero E
inst✝¹ : TopologicalSpace ι
inst✝ : OrderTopology ι
hp : IsStable 𝓕 p
hq : IsStable 𝓕 q
x✝ : Locally p 𝓕 X P ∧ Locally q 𝓕 X P
hpX : Locally p 𝓕 X P
hqX : Locally q 𝓕 X P
n : ℕ
this :
  ∀ (p q : (ι → Ω → E) → Prop),
    IsStable 𝓕 p →
      IsStable 𝓕 q →
        ∀ (hpX : Locally p 𝓕 X P) (hqX : Locally q 𝓕 X P),
          p
            (stoppedProcess (fun i ↦ {ω | ⊥ < (hpX.localSeq ⊓ hqX.localSeq) n ω}.indicator (X i))
              ((hpX.localSeq ⊓ hqX.localSeq) n))
⊢ (fun Y ↦ p Y ∧ q Y)
    (stoppedProcess (fun i ↦ {ω | ⊥ < (hpX.localSeq ⊓ hqX.localSeq) n ω}.indicator (X i))
      ((hpX.localSeq ⊓ hqX.localSeq) n))

Tactic:
refine ⟨this p q hp hq hpX hqX, ?_⟩
    simp_rw [inf_comm hpX.localSeq]
    exact this q p hq hp hqX hpX

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

Goal state before:
pp: Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
Ω'' : Type u_3
mΩ'' : MeasurableSpace Ω''
Y : Ω'' → ℝ
cY : ℝ≥0
η : Kernel Ω Ω''
h : HasSubgaussianMGF Y cY η (⇑κ ∘ₘ ν)
⊢ HasSubgaussianMGF Y cY (prodMkLeft Ω' η) (ν ⊗ₘ κ)

Tactic:
by_cases hν : SFinite ν
  swap; · simp [hν]
  by_cases hκ : IsSFiniteKernel κ
  swap; · simp [hκ]
  constructor
  · simpa using h.integrable_exp_mul
  · have h2 := h.mgf_le
    rw [← Measure.snd_compProd, Measure.snd] at h2
    exact ae_of_ae_map (by fun_prop) h2

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
A : Tendsto (fun N ↦ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))
⊢ (fun N ↦ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) ≤ᶠ[atTop] fun x ↦ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)

Tactic:
filter_upwards [Ici_mem_atTop K] with N hN

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
A : Tendsto (fun N ↦ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))
N : ℕ
hN : N ∈ Set.Ici K
⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)
Module: Mathlib.RepresentationTheory.AlgebraRepresentation.Basic

Goal state before:
pp: A : Type u_1
V : Type u_2
k : Type u_3
inst✝¹⁰ : Field k
inst✝⁹ : Ring A
inst✝⁸ : Algebra k A
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : Module A V
inst✝⁴ : IsScalarTower k A V
inst✝³ : IsSimpleModule A V
inst✝² : FiniteDimensional k V
inst✝¹ : IsAlgClosed k
inst✝ : IsMulCommutative A
this : Nontrivial V
v : V
v_nz : v ≠ 0
a : A
w : V
hw : w ∈ (k ∙ v).carrier
⊢ a • w ∈ (k ∙ v).carrier

Tactic:
have ⟨t, ht⟩ := (IsSimpleModule.algebraMap_end_bijective_of_isAlgClosed k).2
        (Module.toModuleEnd A V a)
      simpa only [← show _ = a • w from congr($ht w)] using! (k ∙ v).smul_mem t hw

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

Goal state before:
pp: case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
c : ℝ≥0
hX : AEMeasurable X μ
h : HasSubgaussianMGF id c (Measure.map X μ)
⊢ HasSubgaussianMGF X c μ

Tactic:
rw [← Function.id_comp X]
    exact .of_map hX h

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Res

Goal state before:
pp: k : Type u
inst✝² : Semiring k
G : Type v1
H : Type v2
inst✝¹ : Monoid G
inst✝ : Monoid H
X Y : Rep k G
f : H →* G
p : X ⟶ Y
h : H
⊢ ↑(Hom.hom p) ∘ₗ (MonoidHom.comp X.ρ f) h = (MonoidHom.comp Y.ρ f) h ∘ₗ ↑(Hom.hom p)

Tactic:
by simpa using p.hom.2 (f h)

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
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ -↑n < X n ω

Tactic:
apply lt_of_lt_of_le _ (hnonneg n ω)

Goal state after:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
h : ¬(-↑n < X n ω ∧ X n ω ≤ ↑n)
⊢ -↑n < 0
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case neg
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : ∑' (j : ℕ), ℙ {ω | X j ω ∈ Set.Ioi ↑j} < ∞
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, X n ω ∉ Set.Ioi ↑n
n : ℕ
hn : X n ω ∉ Set.Ioi ↑n
npos : n ∈ Set.Ioi 0
this : -↑n < X n ω
h : ↑n < X n ω
⊢ 0 = 0 - X n ω

Tactic:
exact (hn h).elim

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.TopRep

Goal state before:
pp: k : Type u
G : Type v
X Y : Type w
inst✝¹² : TopologicalSpace k
inst✝¹¹ : Ring k
inst✝¹⁰ : Monoid G
inst✝⁹ : AddCommGroup X
inst✝⁸ : Module k X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : IsTopologicalAddGroup X
inst✝⁵ : ContinuousSMul k X
inst✝⁴ : AddCommGroup Y
inst✝³ : Module k Y
inst✝² : TopologicalSpace Y
inst✝¹ : IsTopologicalAddGroup Y
inst✝ : ContinuousSMul k Y
ρ : ContRepresentation k G X
σ : ContRepresentation k G Y
A B C : TopRep k G
X✝ Y✝ : TopRep k G
f : X✝ ⟶ Y✝
g : G
⊢ ConcreteCategory.hom
      ({ V := TopModuleCat.of k ↑X✝, ρ := (TopModuleCat.of k ↑X✝).endRingEquiv.symm.toMonoidHom.comp ↑X✝.ρ }.ρ g ≫
        Hom.toTopModuleCatHom f) =
    ConcreteCategory.hom
      (Hom.toTopModuleCatHom f ≫
        { V := TopModuleCat.of k ↑Y✝, ρ := (TopModuleCat.of k ↑Y✝).endRingEquiv.symm.toMonoidHom.comp ↑Y✝.ρ }.ρ g)

Tactic:
simp [TopModuleCat.endRingEquiv, f.hom.2 g]

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
τ : ContRepresentation R G U
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑ψ.toContinuousLinearEquiv ∘SL ↑φ.toContinuousLinearEquiv ∘SL ρ g =
    τ g ∘SL ↑ψ.toContinuousLinearEquiv ∘SL ↑φ.toContinuousLinearEquiv

Tactic:
φ.isIntertwining,

Goal state after:
R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
τ : ContRepresentation R G U
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑ψ.toContinuousLinearEquiv ∘SL σ g ∘SL ↑φ.toContinuousLinearEquiv =
    τ g ∘SL ↑ψ.toContinuousLinearEquiv ∘SL ↑φ.toContinuousLinearEquiv
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝¹³ : Monoid G
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : TopologicalSpace V
inst✝⁹ : IsTopologicalAddGroup V
inst✝⁸ : Module R V
inst✝⁷ : AddCommGroup W
inst✝⁶ : TopologicalSpace W
inst✝⁵ : IsTopologicalAddGroup W
inst✝⁴ : Module R W
inst✝³ : AddCommGroup U
inst✝² : Module R U
inst✝¹ : TopologicalSpace U
inst✝ : IsTopologicalAddGroup U
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
τ : ContRepresentation R G U
φ✝ φ : ρ.Equiv σ
ψ : σ.Equiv τ
g : G
⊢ ↑(φ.trans ψ.toContinuousLinearEquiv) ∘SL ρ g = τ g ∘SL ↑(φ.trans ψ.toContinuousLinearEquiv)

Tactic:
by
  rw [← ContinuousLinearEquiv.comp_coe, comp_assoc,
    φ.isIntertwining, ← comp_assoc, ψ.isIntertwining, comp_assoc]

Goal state after:
Module: Mathlib.RepresentationTheory.Continuous.Basic

Goal state before:
pp: R : Type u_1
G✝ : Type u_2
V : Type u_3
W : Type u_4
U : Type u_5
inst✝²³ : Monoid G✝
inst✝²² : Ring R
inst✝²¹ : AddCommGroup V
inst✝²⁰ : TopologicalSpace V
inst✝¹⁹ : IsTopologicalAddGroup V
inst✝¹⁸ : Module R V
inst✝¹⁷ : AddCommGroup W
inst✝¹⁶ : TopologicalSpace W
inst✝¹⁵ : IsTopologicalAddGroup W
inst✝¹⁴ : Module R W
inst✝¹³ : AddCommGroup U
inst✝¹² : Module R U
inst✝¹¹ : TopologicalSpace U
inst✝¹⁰ : IsTopologicalAddGroup U
H✝ : Type u_6
inst✝⁹ : Monoid H✝
π✝¹ : ContRepresentation R G✝ V
π' : ContRepresentation R H✝ W
G : Type u_7
H : Type u_8
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalSpace R
inst✝⁵ : ContinuousSMul R V
inst✝⁴ : ContinuousSMul R W
inst✝³ : Group H
inst✝² : TopologicalSpace H
φ : G →ₜ* H
π✝ : ContRepresentation R G V
inst✝¹ : IsTopologicalGroup G
inst✝ : IsTopologicalGroup H
π : ContRepresentation R G V
h : H
f : C(H, V)
hf : f ∈ coindV φ π
⊢ f.comp (ContinuousMap.mulRight h) ∈ coindV φ π

Tactic:
by simp [mul_assoc, hf _]

Goal state after:
Module: Mathlib.RepresentationTheory.Rep.Basic

Goal state before:
pp: k : Type u
G✝ : Type v
inst✝² : CommRing k
inst✝¹ : Monoid G✝
G : Type v
inst✝ : Group G
A B C : Rep k G
X Y : Rep k G
f : X ⟶ Y
g : G
⊢ (LinearMap.llcomp k ↑A ↑X ↑Y) (Hom.hom f).toLinearMap ∘ₗ (A.ρ.linHom X.ρ) g =
    (A.ρ.linHom Y.ρ) g ∘ₗ (LinearMap.llcomp k ↑A ↑X ↑Y) (Hom.hom f).toLinearMap

Tactic:
by
    ext; simp [Representation.IntertwiningMap.toLinearMap_apply, ← hom_comm_apply]

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

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
i : ℕ
σ : ↑(X.homogeneousCochains.X i).toModuleCat
⊢ ↑((TopModuleCat.Hom.hom (X.homogeneousCochains.d i (i + 1))) σ) = (Hom.hom (X.d (i + 1))) ↑σ

Tactic:
rw [homogeneousCochains.d_eq]
  dsimp [ContIntertwiningMap.mapInvariants_apply]

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

Goal state before:
pp: 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✝ : Module.IsTorsionFree A L
this : IsLocalization (Algebra.algebraMapSubmonoid B A⁰) L
h : (algebraMap B L) η = 0
⊢ False

Tactic:
simp [h] at hη

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

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
X : ShortComplex (Rep k G)
hX : X.ShortExact
y : ↑X.X₂
x : G → ↑X.X₁
hx : ⇑(Rep.Hom.hom X.f) ∘ x = (ConcreteCategory.hom (d₀₁ X.X₂)) y
⊢ x ∈ cocycles₁ X.X₁

Tactic:
apply Function.Injective.comp_left ((Rep.mono_iff_injective X.f).1 hX.2)

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
X : ShortComplex (Rep k G)
hX : X.ShortExact
y : ↑X.X₂
x : G → ↑X.X₁
hx : ⇑(Rep.Hom.hom X.f) ∘ x = (ConcreteCategory.hom (d₀₁ X.X₂)) y
⊢ (fun x ↦ ⇑(Rep.Hom.hom X.f) ∘ x) ((ModuleCat.Hom.hom (d₁₂ X.X₁)) x) = (fun x ↦ ⇑(Rep.Hom.hom X.f) ∘ x) 0
Module: Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence

Goal state before:
pp: k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
X : ShortComplex (Rep k G)
hX : X.ShortExact
y : G → ↑X.X₂
x : G × G → ↑X.X₁
hx : ⇑(Rep.Hom.hom X.f) ∘ x = (ConcreteCategory.hom (d₁₂ X.X₂)) y
⊢ x ∈ cocycles₂ X.X₁

Tactic:
apply Function.Injective.comp_left ((Rep.mono_iff_injective X.f).1 hX.2)

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
X : ShortComplex (Rep k G)
hX : X.ShortExact
y : G → ↑X.X₂
x : G × G → ↑X.X₁
hx : ⇑(Rep.Hom.hom X.f) ∘ x = (ConcreteCategory.hom (d₁₂ X.X₂)) y
⊢ (fun x ↦ ⇑(Rep.Hom.hom X.f) ∘ x) ((ModuleCat.Hom.hom (d₂₃ X.X₁)) x) = (fun x ↦ ⇑(Rep.Hom.hom X.f) ∘ x) 0
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)) + (A.ρ g.1) (y (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 * 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 (A.ρ _ _),

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) + 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))
Module: Mathlib.RepresentationTheory.Homological.GroupHomology.Basic

Goal state before:
pp: k✝ G✝ : Type u
inst✝³ : CommRing k✝
inst✝² : Group G✝
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
n✝ n : ℕ
⊢ inhomogeneousChains.d A (n + 1) ≫ inhomogeneousChains.d A n = 0

Tactic:
rw [inhomogeneousChains.d_eq, inhomogeneousChains.d_eq]
    slice_lhs 3 4 => rw [Iso.hom_inv_id]
    slice_lhs 2 4 => rw [Category.id_comp, ((barComplex k G).coinvariantsTensorObj A).d_comp_d]
    simp

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 : ↑A
hx : (ConcreteCategory.hom (d₀₁ A)) x = 0
g : G
⊢ (A.ρ g) x = x

Tactic:
rw [← sub_eq_zero]

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
g : G
⊢ (A.ρ g) x - x = 0
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
p : MvPowerSeries σ R
x : σ →₀ ℕ
⊢ x ∉ ⋯.toFinset → (coeff x) p = 0

Tactic:
by simp [← embDomain_eq_mapDomain]

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
x : ↥(cycles₁ A)
⊢ (ConcreteCategory.hom (iCycles A 1))
      (cyclesMk 1 0 cyclesMk₁_eq._proof_1 ((ConcreteCategory.hom (chainsIso₁ A).inv) ↑x) ⋯) =
    (ConcreteCategory.hom (iCycles A 1)) ((ConcreteCategory.hom (isoCycles₁ A).inv) x)

Tactic:
by
    rw [iCycles_mk]
    simp only [ChainComplex.of_X, isoCycles₁_inv_comp_iCycles_apply]
    rfl

Goal state after: