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

Goal state before:
pp: case refine_2
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → WithTop ι
f : ι → Ω → E
i : ι
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
inst✝ : SigmaFinite (μ.trim ⋯)
hτ_le : ∀ (x : Ω), τ x ≤ ↑i
h_int : Integrable ({ω | τ ω ≤ σ ω}.indicator (stoppedValue (fun n ↦ f n) τ)) μ
⊢ StronglyMeasurable ({ω | τ ω ≤ σ ω}.indicator (stoppedValue (fun n ↦ f n) τ))

Tactic:
refine StronglyMeasurable.indicator ?_ (hτ.measurableSet_le_stopping_time hσ)
      refine Measurable.stronglyMeasurable ?_
      exact measurable_stoppedValue h.stronglyAdapted.isStronglyProgressive_of_discrete hτ

Goal state after:
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
g : Y → ↑I := sigmoid ∘ embeddingReal Y
hg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)
hκg : IsMarkovKernel (κ.map g)
⊢ κ = κ.map (hg.invFun ∘ sigmoid ∘ embeddingReal Y)

Tactic:
LeftInverse.id hg.leftInverse_invFun,

Goal state after:
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
g : Y → ↑I := sigmoid ∘ embeddingReal Y
hg : MeasurableEmbedding (sigmoid ∘ embeddingReal Y)
hκg : IsMarkovKernel (κ.map g)
⊢ κ = κ.map id
Module: Mathlib.Probability.Moments.Tilted

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
μ : Measure Ω
X : Ω → ℝ
t : ℝ
s : Set Ω
hs : MeasurableSet s
ht : Integrable (fun ω ↦ rexp (t * X ω)) μ
⊢ (μ.tilted fun x ↦ t * X x) s = ∫⁻ (a : Ω) in s, ENNReal.ofReal (rexp (t * X a - cgf X μ t)) ∂μ

Tactic:
rcases eq_zero_or_neZero μ with rfl | hμ
  · simp
  · simp_rw [tilted_mul_apply_mgf' hs, exp_sub, exp_cgf ht]

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✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → WithTop ι
f : ι → Ω → E
inst✝ : SigmaFiniteFiltration μ ℱ
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
n : ι
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_sf_min : SigmaFinite (μ.trim ⋯)
this :
  ∀ᵐ (x : Ω) ∂μ,
    x ∈ {x | τ x ≤ σ x} → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
x : Ω
hx : τ x ≤ σ x → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
⊢ x ∈ {ω | σ ω ≤ τ ω}ᶜ → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x

Tactic:
hx_mem

Goal state after:
Ω : Type u_1
E : Type u_2
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : CompleteSpace E
ι : Type u_3
inst✝¹⁰ : LinearOrder ι
inst✝⁹ : LocallyFiniteOrder ι
inst✝⁸ : OrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : DiscreteTopology ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : BorelSpace ι
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : SecondCountableTopology E
ℱ : Filtration ι m
τ σ : Ω → WithTop ι
f : ι → Ω → E
inst✝ : SigmaFiniteFiltration μ ℱ
h : Martingale f ℱ μ
hτ : IsStoppingTime ℱ τ
hσ : IsStoppingTime ℱ σ
n : ι
hτ_le : ∀ (x : Ω), τ x ≤ ↑n
h_sf_min : SigmaFinite (μ.trim ⋯)
this :
  ∀ᵐ (x : Ω) ∂μ,
    x ∈ {x | τ x ≤ σ x} → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
x : Ω
hx : τ x ≤ σ x → μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
hx_mem : x ∈ {ω | σ ω ≤ τ ω}ᶜ
⊢ μ[stoppedValue f τ | ⋯.measurableSpace] x = μ[stoppedValue f τ | hσ.measurableSpace] x
Module: Mathlib.RepresentationTheory.Subrepresentation

Goal state before:
pp: A : Type u_1
G : Type u_2
W : Type u_3
M : Type u_4
inst✝⁵ : Semiring A
inst✝⁴ : Monoid G
inst✝³ : AddCommMonoid W
inst✝² : Module A W
ρ : Representation A G W
inst✝¹ : AddCommMonoid M
inst✝ : Module A[G] M
ρ₁ ρ₂ : Subrepresentation ρ
g : G
x : W
hx₁ : x ∈ ρ₁.toSubmodule
hx₂ : x ∈ ρ₂.toSubmodule
⊢ (ρ g) x ∈ ρ₁.toSubmodule ∧ (ρ g) x ∈ ρ₂.toSubmodule

Tactic:
exact ⟨ρ₁.apply_mem_toSubmodule g hx₁, ρ₂.apply_mem_toSubmodule g hx₂⟩

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 Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ rexp (↑cX * (t * ↑p) ^ 2 / 2) ^ (1 / ↑p) * rexp (↑cY * (t * ↑q) ^ 2 / 2) ^ (1 / ↑q) =
    rexp ((↑(NNReal.sqrt cX) + ↑(NNReal.sqrt cY)) ^ 2 * t ^ 2 / 2)

Tactic:
simp_rw [← exp_mul, ← exp_add]

Goal state after:
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
ν : Measure Ω'
κ : Kernel Ω' Ω
X Y : Ω → ℝ
cX cY : ℝ≥0
hX : HasSubgaussianMGF X cX κ ν
hY : HasSubgaussianMGF Y cY κ ν
hX0 : ¬cX = 0
hY0 : ¬cY = 0
p : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cX
q : ℝ≥0 := (NNReal.sqrt cX + NNReal.sqrt cY) / NNReal.sqrt cY
ω' : Ω'
hmX : ∀ (t : ℝ), mgf X (κ ω') t ≤ rexp (↑cX * t ^ 2 / 2)
hmY : ∀ (t : ℝ), mgf Y (κ ω') t ≤ rexp (↑cY * t ^ 2 / 2)
hlX : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * X ω)) (↑p) (κ ω')
hlY : ∀ (t : ℝ), MemLp (fun ω ↦ rexp (t * Y ω)) (↑q) (κ ω')
t : ℝ
⊢ rexp (↑cX * (t * ↑p) ^ 2 / 2 * (1 / ↑p) + ↑cY * (t * ↑q) ^ 2 / 2 * (1 / ↑q)) =
    rexp ((↑(NNReal.sqrt cX) + ↑(NNReal.sqrt cY)) ^ 2 * t ^ 2 / 2)
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✝⁸ : ConditionallyCompleteLinearOrderBot ι
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : OrderTopology ι
𝓕 : Filtration ι mΩ
X : ι → Ω → E
p : (ι → Ω → E) → Prop
inst✝⁵ : SecondCountableTopology ι
inst✝⁴ : IsFiniteMeasure P
inst✝³ : DenselyOrdered ι
inst✝² : NoMaxOrder ι
inst✝¹ : Zero E
inst✝ : 𝓕.IsRightContinuous
hp : IsStable 𝓕 p
⊢ Locally (fun Y ↦ Locally p 𝓕 Y P) 𝓕 X P ↔ Locally p 𝓕 X P

Tactic:
by
  refine ⟨fun hL ↦ ?_, fun hL ↦ ⟨hL.localSeq, hL.isLocalizingSequence_localSeq,
    fun n ↦ .of_prop <| hL.stoppedProcess_localSeq n⟩⟩
  choose τ hτ₁ hτ₂ using hL.stoppedProcess_localSeq
  obtain ⟨nk, hnk, hpre⟩ :=
    hL.isLocalizingSequence_localSeq.isPrelocalizingSequence_inf_extraction hτ₁
  refine locally_of_isPreLocalizingSequence hp hpre <| fun n ↦ ?_
  convert! hτ₂ n (nk n) using 1 with
  ext i ω
  rw [stoppedProcess_indicator_comm', stoppedProcess_indicator_comm',
    stoppedProcess_stoppedProcess, stoppedProcess_indicator_comm']
  simp only [lt_inf_iff, Set.indicator_indicator]
  congr 1
  · ext; grind
  · simp_rw [inf_comm]
    rfl

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, ℙ {ω | ↑(u i) * ε ≤ |S (u i) ω - ∫ (a : Ω), S (u i) a|} ≤ ENNReal.ofReal (ε⁻¹ ^ 2 * C)

Tactic:
calc
      ∑ i ∈ range N, ℙ {ω | (u i * ε : ℝ) ≤ |S (u i) ω - 𝔼[S (u i)]|} ≤
          ∑ i ∈ range N, ENNReal.ofReal (Var[S (u i)] / (u i * ε) ^ 2) := by
        gcongr with i _
        apply meas_ge_le_variance_div_sq
        · exact memLp_finsetSum' _ fun j _ => (hident j).aestronglyMeasurable_fst.memLp_truncation
        · apply mul_pos (Nat.cast_pos.2 _) εpos
          refine zero_lt_one.trans_le ?_
          apply Nat.le_floor
          rw [Nat.cast_one]
          apply one_le_pow₀ c_one.le
      _ = ENNReal.ofReal (∑ i ∈ range N, Var[S (u i)] / (u i * ε) ^ 2) := by
        rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]
        exact div_nonneg (variance_nonneg _ _) (sq_nonneg _)
      _ ≤ ENNReal.ofReal (ε⁻¹ ^ 2 * C) := by
        apply ENNReal.ofReal_le_ofReal
        simp_rw [div_eq_inv_mul, ← inv_pow, mul_inv, mul_comm _ (ε⁻¹), mul_pow, mul_assoc,
          ← mul_sum]
        gcongr
        simpa only [inv_pow] using I2 N

Goal state after:
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case hindep
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✝ : 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))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
⊢ Pairwise fun x y ↦ (fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) ((fun i ω ↦ ‖(X i - Y k i) ω‖) x) ((fun i ω ↦ ‖(X i - Y k i) ω‖) y)

Tactic:
simp_rw [I]

Goal state after:
case hindep
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✝ : 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))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
⊢ Pairwise fun x y ↦ (G k ∘ X x) ⟂ᵢ[μ] G k ∘ X y
Module: Mathlib.Probability.StrongLaw

Goal state before:
pp: case hident
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✝ : 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))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
i : ℕ
⊢ IdentDistrib (fun ω ↦ ‖(X i - Y k i) ω‖) (fun ω ↦ ‖(X 0 - Y k 0) ω‖) μ μ

Tactic:
simp_rw [I]

Goal state after:
case hident
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✝ : 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))
k : ℕ
G : ℕ → E → ℝ := fun k x ↦ ‖x - (φ k) x‖
G_meas : ∀ (k : ℕ), Measurable (G k)
I : ∀ (k i : ℕ), (fun ω ↦ ‖(X i - Y k i) ω‖) = G k ∘ X i
i : ℕ
⊢ IdentDistrib (G k ∘ X i) (G k ∘ X 0) μ μ
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
inst✝⁹ : Monoid G
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : TopologicalSpace V
inst✝⁵ : IsTopologicalAddGroup V
inst✝⁴ : Module R V
inst✝³ : AddCommGroup W
inst✝² : TopologicalSpace W
inst✝¹ : IsTopologicalAddGroup W
inst✝ : Module R W
ρ : ContRepresentation R G V
σ : ContRepresentation R G W
φ ψ : σ.Equiv ρ
h : φ.toContinuousLinearEquiv = ψ.toContinuousLinearEquiv
⊢ φ = ψ

Tactic:
by cases φ; cases ψ; simpa [ContIntertwiningMap.ext_iff] using h

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

Goal state before:
pp: k : Type u
inst✝² : Semiring k
G : Type v
inst✝¹ : Group G
inst✝ : Fintype G
A B : Rep k G
f : A ⟶ B
x✝ : ↑A
⊢ (Hom.hom (f ≫ B.norm)).toLinearMap x✝ = (Hom.hom (A.norm ≫ f)).toLinearMap x✝

Tactic:
simp [Representation.norm, hom_comm_apply]

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:
exact (NeZero.ne 0 rfl).elim

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.ρ.invariants
⊢ (ConcreteCategory.hom (iCocycles A 0)) (cocyclesMk ((ConcreteCategory.hom (cochainsIso₀ A).inv) ↑x) ⋯) =
    (ConcreteCategory.hom (iCocycles A 0)) ((ConcreteCategory.hom (cocyclesIso₀ A).inv) x)

Tactic:
iCocycles_mk

Goal state after:
k G : Type u
inst✝¹ : CommRing k
inst✝ : Group G
A : Rep k G
x : ↥A.ρ.invariants
⊢ (ConcreteCategory.hom (cochainsIso₀ A).inv) ↑x =
    (ConcreteCategory.hom (iCocycles A 0)) ((ConcreteCategory.hom (cocyclesIso₀ A).inv) 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:
exact (NeZero.ne 0 rfl).elim

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
⊢ idealOfVars σ R ^ n = restrictSupportIdeal R (⇑degree ⁻¹' Set.Ici n) ⋯

Tactic:
by
  rw [idealOfVars_eq_restrictSupportIdeal]
  apply Submodule.restrictScalars_injective R
  by_cases hn : n = 0
  · simp [hn, Set.Ici_zero_eq_univ]
  rw [Submodule.restrictScalars_pow hn]
  simp [← restrictSupport_nsmul, ← degree_preimage_nsmul, hn, Set.Ici_nsmul_eq]

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

Goal state before:
pp: σ : Type u_1
τ : Type u_2
R : Type u_4
f : σ → τ
inst✝¹ : TendstoCofinite f
inst✝ : Semiring R
x : σ →₀ ℕ
r : R
⊢ renameFun f ((monomial x) r) = (monomial (mapDomain f x)) r

Tactic:
by
  classical
  ext; simp [coeff_renameFun, coeff_monomial, eq_comm]

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

Goal state before:
pp: case pos
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
P : Ideal R⟦X⟧
h₁ : P ≠ ⊥
h₂ : P.IsPrime
hXP : X ∈ P
⊢ ∃ x ∈ P, Prime x

Tactic:
exact ⟨X, hXP, X_prime⟩

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

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Finite σ
l m n : ℕ
h : l ≤ m
h' : l ≤ n
p : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < l
⊢ MvPolynomial.coeff x ((truncTotal m) p - (truncTotal n) p) = 0

Tactic:
MvPolynomial.coeff_sub,

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : CommRing R
inst✝ : Finite σ
l m n : ℕ
h : l ≤ m
h' : l ≤ n
p : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < l
⊢ MvPolynomial.coeff x ((truncTotal m) p) - MvPolynomial.coeff x ((truncTotal n) p) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : CommRing R
inst✝ : Finite σ
p q : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) (p * q) - (truncTotal n) p * (truncTotal n) q) = 0

Tactic:
MvPolynomial.coeff_sub,

Goal state after:
σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : CommRing R
inst✝ : Finite σ
p q : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) (p * q)) - MvPolynomial.coeff x ((truncTotal n) p * (truncTotal n) q) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Equiv

Goal state before:
pp: σ : Type u_1
R : Type u_2
n : ℕ
inst✝¹ : CommRing R
inst✝ : Finite σ
p q : MvPowerSeries σ R
x : σ →₀ ℕ
hx : degree x < n
⊢ MvPolynomial.coeff x ((truncTotal n) (p * q) - (truncTotal n) p * (truncTotal n) q) = 0

Tactic:
rw [MvPolynomial.coeff_sub, sub_eq_zero, coeff_truncTotal _ hx,
    coeff_truncTotal_mul_truncTotal_eq_coeff_mul _ _ hx]

Goal state after:
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:
MvPolynomial.coeff_sub,

Goal state after:
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) - MvPolynomial.coeff x p = 0
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
x : Fin n → G
⊢ ModuleCat.ofHom (lsingle x) ≫ (chainsMap f φ).f n = ModuleCat.ofHom (lsingle (⇑f ∘ x) ∘ₗ (Hom.hom φ).toLinearMap)

Tactic:
by
  ext
  simp [chainsMap_f]

Goal state after:
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
hf : Function.Surjective ⇑f
inst✝ : Epi φ
i : ℕ
⊢ Epi ((chainsMap f φ).f i)

Tactic:
by
  simpa [ModuleCat.epi_iff_surjective] using!
    (mapRange_surjective φ.hom (map_zero _) ((Rep.epi_iff_surjective φ).1 inferInstance)).comp
    (mapDomain_surjective hf.comp_left)

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

Goal state before:
pp: case pos
R : Type u_1
inst✝⁷ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : IsNoetherianRing R
inst✝³ : Module.Finite R M
A : Type u_3
inst✝² : CommRing A
inst✝¹ : IsArtinianRing A
inst✝ : IsLocalRing A
f : ℕ → A
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]
n : ℕ
hn : maximalIdeal A ^ n = 0
m : ℕ
h : m ≤ n
⊢ f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]

pp: case neg
R : Type u_1
inst✝⁷ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : IsNoetherianRing R
inst✝³ : Module.Finite R M
A : Type u_3
inst✝² : CommRing A
inst✝¹ : IsArtinianRing A
inst✝ : IsLocalRing A
f : ℕ → A
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]
n : ℕ
hn : maximalIdeal A ^ n = 0
m : ℕ
h : ¬m ≤ n
⊢ f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]

Tactic:
· exact hf h

Goal state after:
case neg
R : Type u_1
inst✝⁷ : CommRing R
I : Ideal R
M : Type u_2
inst✝⁶ : AddCommGroup M
inst✝⁵ : Module R M
inst✝⁴ : IsNoetherianRing R
inst✝³ : Module.Finite R M
A : Type u_3
inst✝² : CommRing A
inst✝¹ : IsArtinianRing A
inst✝ : IsLocalRing A
f : ℕ → A
hf : ∀ {m n : ℕ}, m ≤ n → f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]
n : ℕ
hn : maximalIdeal A ^ n = 0
m : ℕ
h : ¬m ≤ n
⊢ f m ≡ f n [SMOD maximalIdeal A ^ m • ⊤]
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
φ : A ⟶ res 1 B
⊢ map 1 φ 1 = 0

Tactic:
simp only [← cancel_epi (H1π A), H1π_comp_map, Limits.comp_zero]
  ext x
  rw [ModuleCat.hom_comp]
  refine (H1π_eq_zero_iff _).2 ?_
  simpa [coe_mapCycles₁ _ φ x, mapDomain, map_finsuppSum] using
    (boundaries₁ B).finsuppSum_mem k x.1 _ fun _ _ => single_one_mem_boundaries₁ (A := B) _

Goal state after:
Module: Mathlib.RingTheory.Artinian.Algebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsArtinianRing R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : Algebra.IsIntegral R A
⊢ IsUnit.submonoid A = A⁰

Tactic:
by
  ext; simpa [IsUnit.mem_submonoid_iff] using isUnit_iff_nonZeroDivisor_of_isIntegral' (R := R)

Goal state after:
Module: Mathlib.RingTheory.Polynomial.ContentIdeal

Goal state before:
pp: R : Type u_3
inst✝¹ : CommRing R
inst✝ : NormalizedGCDMonoid R
p : R[X]
h_prin : Submodule.IsPrincipal p.contentIdeal
⊢ ∀ b ∈ p.support, Submodule.IsPrincipal.generator p.contentIdeal ∣ p.coeff b

Tactic:
exact fun n _ ↦ h_prin.contentIdeal_generator_dvd_coeff n

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.SInteger

Goal state before:
pp: R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x✝ : Kˣ
⊢ x✝ ∈ {x | ∀ v ∉ S, (HeightOneSpectrum.valuation K v) ↑x = 1} ↔
    x✝ ∈ ↑(⨅ v, ⨅ (_ : v ∉ S), (HeightOneSpectrum.valuation K v).valuationSubring.unitGroup)

Tactic:
by
      simp only [mem_ofPred, SetLike.mem_coe, Subgroup.mem_iInf, Valuation.mem_unitGroup_iff]

Goal state after: