mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜✝ : Type u_2
𝕜' : Type u_3
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup F
ε : Type u_6
inst✝⁵ : TopologicalSpace ε
inst✝⁴ : ContinuousENorm ε
β : Type u_7
inst✝³ : MeasurableSpace β
μb : Measure β
f✝ : α → β
𝕜 : Type u_8
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : α → β
hf : MeasurePreserving f μ μb
c : 𝕜
g : ↥(Lp E p μb)
⊢ (↑(compMeasurePreserving f hf)).toFun (c • g) = (RingHom.id 𝕜) c • (↑(compMeasurePreserving f hf)).toFun g

Tactic:
by rcases g with ⟨⟨_⟩, _⟩; rfl

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
have := coe_le_coe.2 <| NNReal.inner_le_weight_mul_Lp s hp.le (fun i ↦ ENNReal.toNNReal (w i))
    fun i ↦ ENNReal.toNNReal (f i)

Goal state after:
case neg
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0
H' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞
this :
  ↑(∑ i ∈ s, (w i).toNNReal * (f i).toNNReal) ≤
    ↑((∑ i ∈ s, (w i).toNNReal) ^ (1 - p⁻¹) * (∑ i ∈ s, (w i).toNNReal * (f i).toNNReal ^ p) ^ p⁻¹)
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_tsum_lt_top : (∑' (i : ℕ), B i) ^ p < ∞
⊢ 0 < p⁻¹

Tactic:
by simp [hp_pos]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
⊢ AEMeasurable (fun x ↦ ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) μ

Tactic:
exact AEMeasurable.tsum fun n => ((hf (n + 1)).sub (hf n)).enorm

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
hμs : μ s = 0
x : α
ε : ℝ≥0∞
hε : ε > 0
N k : ℕ
hk_lt_ε : 2 * 2⁻¹ ^ k < ε
n : ℕ
hNx : edist (f (ns n) x) (g x) < 2⁻¹ ^ n
hn_ge : n ≥ max N (k - 1)
⊢ 2 ^ k ≤ 2 ^ (n + 1)

Tactic:
gcongr
        · simp
        · omega

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
h : (∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i
hp_pos : 0 < p
h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ‖f (i + 1) a - f i a‖ₑ) ^ p ∂μ < ∞
rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ) ^ p < ∞
x : α
hx : ∑' (i : ℕ), ‖f (i + 1) x - f i x‖ₑ < ∞ ^ p⁻¹
⊢ 0 < p⁻¹

Tactic:
by simp [hp_pos]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁴ : SeminormedRing R
inst✝³ : NormedAlgebra 𝕜 R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : IsScalarTower 𝕜 R E
c : R
x : E
⊢ ‖((Algebra.lsmul 𝕜 𝕜 E).toLinearMap c) x‖ ≤ 1 * ‖c‖ * ‖x‖

Tactic:
simpa only [one_mul] using! norm_smul_le c x

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_5
Fₗ : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] Fₗ
h : ∀ z ∈ ball 0 r, ‖f z‖ ≤ c
k : 𝕜
hk : 1 < ‖k‖
⊢ ∃ C, ∀ (z : E), ‖f z‖ ≤ C * ‖z‖

Tactic:
use c * (‖k‖ / r)

Goal state after:
case h
𝕜 : Type u_1
E : Type u_5
Fₗ : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] Fₗ
h : ∀ z ∈ ball 0 r, ‖f z‖ ≤ c
k : 𝕜
hk : 1 < ‖k‖
⊢ ∀ (z : E), ‖f z‖ ≤ c * (‖k‖ / r) * ‖z‖
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case refine_1
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
aux₁ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
U : Set ((i : ι) → E i)
V : Set G
hU : IsVonNBounded 𝕜 (U, V).1
hV : (U, V).2 ∈ 𝓝 0
⊢ (U, ⇑g ⁻¹' V).2 ∈ 𝓝 0

pp: case refine_2
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
aux₁ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
U : Set ((i : ι) → E i)
V : Set G
hU : IsVonNBounded 𝕜 (U, V).1
hV : (U, V).2 ∈ 𝓝 0
⊢ ∀ x ∈ {f | MapsTo ⇑f (U, ⇑g ⁻¹' V).1 (U, ⇑g ⁻¹' V).2}, aux₁ x ∈ {f | MapsTo ⇑f (U, V).1 (U, V).2}

Tactic:
· exact (map_continuous g).tendsto 0 <| by simpa

Goal state after:
case refine_2
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
G : Type u_5
inst✝¹³ : NormedField 𝕜
inst✝¹² : (i : ι) → TopologicalSpace (E i)
inst✝¹¹ : (i : ι) → AddCommGroup (E i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E i)
inst✝⁹ : AddCommGroup F
inst✝⁸ : Module 𝕜 F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : IsTopologicalAddGroup F
inst✝⁵ : ContinuousConstSMul 𝕜 F
inst✝⁴ : AddCommGroup G
inst✝³ : Module 𝕜 G
inst✝² : TopologicalSpace G
inst✝¹ : IsTopologicalAddGroup G
inst✝ : ContinuousConstSMul 𝕜 G
g : F →L[𝕜] G
aux₁ : ContinuousMultilinearMap 𝕜 E F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E G :=
  { toFun := g.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯ }
U : Set ((i : ι) → E i)
V : Set G
hU : IsVonNBounded 𝕜 (U, V).1
hV : (U, V).2 ∈ 𝓝 0
⊢ ∀ x ∈ {f | MapsTo ⇑f (U, ⇑g ⁻¹' V).1 (U, ⇑g ⁻¹' V).2}, aux₁ x ∈ {f | MapsTo ⇑f (U, V).1 (U, V).2}
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
inst✝¹ : MeasurableSingletonClass α
inst✝ : SigmaFinite μ
s : Finset α
⊢ ∑ b ∈ s, μ.real {b} = μ.real ↑s

Tactic:
simp [measureReal_def, ← ENNReal.toReal_sum (fun _ _ ↦ ne_of_lt measure_singleton_lt_top)]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s : Set α
inst✝ : IsFiniteMeasure μ
h₁ : MeasurableSet s
⊢ μ.real (univ \ s) = μ.real univ - μ.real s

Tactic:
exact measureReal_sdiff (subset_univ s) h₁

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
s : Set α
hs : μ s ≤ ↑η
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) ≤ ‖c‖ₑ * ↑η ^ (1 / p.toReal)

Tactic:
hs

Goal state after:
case inr
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
hp : p ≠ ∞
c : E
ε : ℝ≥0∞
hε : ε ≠ 0
h'p : p ≠ 0
hp₀ : 0 < p
hp₀' : 0 ≤ 1 / p.toReal
hp₀'' : 0 < p.toReal
η : ℝ≥0
hη_pos : 0 < η
hη_le : ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ε
s : Set α
hs : μ s ≤ ↑η
⊢ ‖c‖ₑ * ↑η ^ (1 / p.toReal) ≤ ‖c‖ₑ * ↑η ^ (1 / p.toReal)
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
hm : ∃ i, ‖m i‖ = 0
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
rcases hm with ⟨i, hi⟩

Goal state after:
case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
i : ι
hi : ‖m i‖ = 0
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case refine_1
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this : {ξ₀}ᶜ ∈ 𝓝 0
⊢ Metric.closedBall 0 ε ∈ 𝓝 0

Tactic:
have : balancedCore 𝕜 {ξ₀}ᶜ ∈ @nhds 𝕜 t 0 := balancedCore_mem_nhds_zero this

Goal state after:
case refine_1
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
⊢ Metric.closedBall 0 ε ∈ 𝓝 0
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case inl
α : Type u_1
m : MeasurableSpace α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f 0
⊢ Integrable f ((0 univ)⁻¹ • 0)

pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
hne : μ ≠ 0
⊢ Integrable f ((μ univ)⁻¹ • μ)

Tactic:
· rwa [smul_zero]

Goal state after:
case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
hne : μ ≠ 0
⊢ Integrable f ((μ univ)⁻¹ • μ)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
δ : Type u_4
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α ≃ᵐ δ
g : δ → ε
⊢ MemLp g 1 (Measure.map (⇑f) μ) ↔ MemLp (g ∘ ⇑f) 1 μ

Tactic:
exact f.memLp_map_measure_iff

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝³ : NormedAddCommGroup β
inst✝² : Lattice β
inst✝¹ : HasSolidNorm β
inst✝ : IsOrderedAddMonoid β
f : α → β
hf : Integrable f μ
⊢ Integrable (fun a ↦ |f a|) μ

Tactic:
rw [← memLp_one_iff_integrable] at hf ⊢
  exact hf.abs

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case inr
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf : Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ (ε / ‖c‖) ^ #univ ≤ ∏ i, ‖m i‖

Tactic:
← prod_const

Goal state after:
case inr
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
hf : Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
h✝ : Nonempty ι
ε : ℝ
ε0 : 0 < ε
hε : ∀ (m : (i : ι) → E i), ‖m‖ < ε → ‖f m‖ < 1
c : 𝕜
hc : 1 < ‖c‖
this : 0 < (‖c‖ / ε) ^ Fintype.card ι
m : (i : ι) → E i
hcm : ∀ (i : ι), ε / ‖c‖ ≤ ‖m i‖
hm : ∀ (i : ι), ‖m i‖ < ε
⊢ ∏ _x, ε / ‖c‖ ≤ ∏ i, ‖m i‖
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
s_dim : finrank 𝕜 ↥s = n
this : IsUniformAddGroup ↥s := AddSubgroup.isUniformAddGroup s.toAddSubgroup
b : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 ↥s)) 𝕜 ↥s := Basis.ofVectorSpace 𝕜 ↥s
⊢ Fintype.card ↑(Basis.ofVectorSpaceIndex 𝕜 ↥s) = n

Tactic:
rw [← s_dim]

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
s_dim : finrank 𝕜 ↥s = n
this : IsUniformAddGroup ↥s := AddSubgroup.isUniformAddGroup s.toAddSubgroup
b : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 ↥s)) 𝕜 ↥s := Basis.ofVectorSpace 𝕜 ↥s
⊢ Fintype.card ↑(Basis.ofVectorSpaceIndex 𝕜 ↥s) = finrank 𝕜 ↥s
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
⊢ f x • ↑↑(u + v) x = ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x

Tactic:
hu,

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
⊢ f x • ↑↑(u + v) x = f x • ↑↑u x + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f : α → E
h' : ∀ s ∈ l, μ s = ∞
a : E
hf : Tendsto f l (𝓝 a)
H : ¬a = 0
ε : ℝ
εpos : 0 < ε
hε : ε < ‖a‖
u : Set α
ul : u ∈ l
hu : IntegrableOn f u μ
⊢ False

Tactic:
let v := u ∩ {b | ε < ‖f b‖}

Goal state after:
α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝ : NormedAddCommGroup E
μ : Measure α
l : Filter α
f : α → E
h' : ∀ s ∈ l, μ s = ∞
a : E
hf : Tendsto f l (𝓝 a)
H : ¬a = 0
ε : ℝ
εpos : 0 < ε
hε : ε < ‖a‖
u : Set α
ul : u ∈ l
hu : IntegrableOn f u μ
v : Set α := u ∩ {b | ε < ‖f b‖}
⊢ False
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
f : E →ₗ[𝕜] 𝕜
H : ¬finrank 𝕜 ↥f.range = 0
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f

Tactic:
have : finrank 𝕜 (LinearMap.ker f) = n := by
          have Z := f.finrank_range_add_finrank_ker
          rw [finrank_eq_card_basis ξ, hn] at Z
          have : finrank 𝕜 (LinearMap.range f) = 1 :=
            le_antisymm (finrank_self 𝕜 ▸ (LinearMap.range f).finrank_le) (zero_lt_iff.mpr H)
          rw [this, add_comm, Nat.add_one] at Z
          exact Nat.succ.inj Z

Goal state after:
case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
f : E →ₗ[𝕜] 𝕜
H : ¬finrank 𝕜 ↥f.range = 0
this : finrank 𝕜 ↥f.ker = n
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology β
⊢ IsSeparable (f '' s)

Tactic:
· exact .of_separableSpace _

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
inst✝ : RCLike 𝕜
f : α → ℝ
hf : Integrable f μ
⊢ Integrable (fun x ↦ ↑(f x)) μ

Tactic:
rw [← memLp_one_iff_integrable] at hf ⊢
  exact hf.ofReal

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case zero
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ max 1 ‖1‖

Tactic:
exact norm_mkPiAlgebraFin_zero.le.trans (le_max_right _ _)

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
x : Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 x
hU_tb : TotallyBounded U
⊢ FiniteDimensional 𝕜 Eᵤ

Tactic:
replace hU_nhds : x +ᵥ (-x) +ᵥ U ∈ 𝓝 x := by simpa
  rw [vadd_mem_nhds_self] at hU_nhds
  refine .of_totallyBounded_nhds_zero _ hU_nhds ?_
  have : -x +ᵥ U = (· - x) '' U := by simp [← Set.image_vadd, neg_add_eq_sub]
  exact this ▸ hU_tb.image (uniformContinuous_id.sub uniformContinuous_const)

Goal state after: