mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.MeasurableSpace.Pi

Goal state before:
pp: case intro.a
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
val✝ : Encodable ι
i : ι
s : Set (α i)
hs : s ∈ C i
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
this : univ.pi (update (fun i' ↦ iUnion (t i')) i (⋃ x, s)) = univ.pi fun k ↦ ⋃ j, update (fun i' ↦ t i' j) i s k
⊢ MeasurableSet (univ.pi (update (fun x ↦ ⋃ n, t x n) i (⋃ x, s)))

Tactic:
rw [this, ← iUnion_univ_pi]

Goal state after:
case intro.a
ι : Type u_1
α : ι → Type u_2
inst✝ : Finite ι
C : (i : ι) → Set (Set (α i))
val✝ : Encodable ι
i : ι
s : Set (α i)
hs : s ∈ C i
t : (i : ι) → ℕ → Set (α i)
h1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i
h2t : ∀ (i : ι), ⋃ n, t i n = univ
this : univ.pi (update (fun i' ↦ iUnion (t i')) i (⋃ x, s)) = univ.pi fun k ↦ ⋃ j, update (fun i' ↦ t i' j) i s k
⊢ MeasurableSet (⋃ x, univ.pi fun a ↦ update (fun i' ↦ t i' (x a)) i s a)
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
hs : s.Finite
hs' : s.Nonempty
ht : s ⊆ t
⊢ (uniformOn s) t = 1

Tactic:
by
  have := isProbabilityMeasure_uniformOn hs hs'
  refine eq_of_le_of_not_lt prob_le_one ?_
  rw [not_lt, ← uniformOn_self hs hs']
  exact measure_mono ht

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

Goal state before:
pp: case insert
α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
f : ι → α →ₘ[μ] γ
a : ι
s : Finset ι
ha : a ∉ s
ih : ↑(∏ i ∈ s, f i) =ᵐ[μ] ∏ i ∈ s, ↑(f i)
⊢ ↑(∏ i ∈ insert a s, f i) =ᵐ[μ] ∏ i ∈ insert a s, ↑(f i)

Tactic:
simp only [ha, not_false_eq_true, Finset.prod_insert]

Goal state after:
case insert
α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
f : ι → α →ₘ[μ] γ
a : ι
s : Finset ι
ha : a ∉ s
ih : ↑(∏ i ∈ s, f i) =ᵐ[μ] ∏ i ∈ s, ↑(f i)
⊢ ↑(f a * ∏ i ∈ s, f i) =ᵐ[μ] ↑(f a) * ∏ i ∈ s, ↑(f i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
⊢ map (eval i) (Measure.pi μ) = (∏ j ∈ Finset.univ.erase i, (μ j) univ) • μ i

Tactic:
ext s hs

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : DecidableEq ι
i : ι
s : Set (α i)
hs : MeasurableSet s
⊢ (map (eval i) (Measure.pi μ)) s = ((∏ j ∈ Finset.univ.erase i, (μ j) univ) • μ i) s
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝⁴ : Fintype ι
inst✝³ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝² : ∀ (i : ι), SigmaFinite (μ i)
inst✝¹ : (i : ι) → PartialOrder (α i)
inst✝ : ∀ (i : ι), NullSingletonClass (μ i)
f g : (i : ι) → α i
⊢ (univ.pi fun i ↦ Ioo (f i) (g i)) =ᵐ[Measure.pi μ] univ.pi fun i ↦ Icc (f i) (g i)

Tactic:
exact pi_Ioo_ae_eq_pi_Icc

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
h0 : p ≠ 0
h_top : ¬p = ∞
⊢ eLpNorm 0 p μ = 0

Tactic:
simp [eLpNorm_eq_eLpNorm' h0 h_top, ENNReal.toReal_pos h0 h_top]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.SMul

Goal state before:
pp: α : Type u_1
F : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup F
f : α → F
𝕜 : Type u_3
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
c : 𝕜
x✝ : α
⊢ ‖(c • f) x✝‖₊ ≤ ‖c‖₊ * ‖f x✝‖₊

Tactic:
simp [nnnorm_smul_le]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.SMul

Goal state before:
pp: α : Type u_1
F : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝³ : NormedAddCommGroup F
f : α → F
𝕜 : Type u_3
inst✝² : NormedRing 𝕜
inst✝¹ : MulActionWithZero 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
c : 𝕜
x✝ : α
⊢ ‖(c • f) x✝‖₊ ≤ ‖c‖₊ * ‖f x✝‖₊

Tactic:
simp [nnnorm_smul_le]

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : ((-f) y - (-f) x) / (y - x) ≤ ((-f) z - (-f) y) / (z - y)
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)

Tactic:
simp only [Pi.neg_apply] at this

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : (-f y - -f x) / (y - x) ≤ (-f z - -f y) / (z - y)
⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : ((-f) y - (-f) x) / (y - x) < ((-f) z - (-f) y) / (z - y)
⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)

Tactic:
simp only [Pi.neg_apply] at this

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConcaveOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
this : (-f y - -f x) / (y - x) < (-f z - -f y) / (z - y)
⊢ (f z - f y) / (z - y) < (f y - f x) / (y - x)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
⊢ (log z - log y) / (z - y) < (log y - log x) / (y - x)

Tactic:
trans y⁻¹

Goal state after:
x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
⊢ (log z - log y) / (z - y) < y⁻¹

x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
⊢ y⁻¹ < (log y - log x) / (y - x)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
⊢ (log z - log y) / (z - y) < y⁻¹

Tactic:
have h : 0 < z - y := by linarith

Goal state after:
x y z : ℝ
hx : 0 < x
hz : 0 < z
hxy : x < y
hyz : y < z
hy : 0 < y
h : 0 < z - y
⊢ (log z - log y) / (z - y) < y⁻¹
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : AEStronglyMeasurable f μ
h : MemLp (fun x ↦ ‖f x‖) p μ
⊢ eLpNorm f p μ < ∞

Tactic:
by rw [← eLpNorm_norm]; exact h.2

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
hz : 0 ≤ z
hxy : x < y
hyz : y < z
hy : 0 < y
hy' : 0 < y ^ p
q : 0 < z - y
⊢ z / y - 1 ≠ 0

Tactic:
· rw [sub_ne_zero]
      exact ((one_lt_div hy).mpr hyz).ne'

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
p : ℝ
hp : 1 ≤ p
this : 0 < p
⊢ ∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)

Tactic:
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
p : ℝ
hp : 1 ≤ p
this : 0 < p
⊢ (∑ i ∈ s, w i * z i) ^ p ≤ ∑ i ∈ s, w i * z i ^ p

case hx
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
p : ℝ
hp : 1 ≤ p
this : 0 < p
⊢ 0 ≤ ∑ i ∈ s, w i * z i ^ p

ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
p : ℝ
hp : 1 ≤ p
this : 0 < p
⊢ 0 ≤ ∑ i ∈ s, w i * z i

ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
p : ℝ
hp : 1 ≤ p
this : 0 < p
⊢ 0 ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ ∑ i ∈ s, w i * z i = ∑ i ∈ s, w i * x

Tactic:
by
      refine sum_congr rfl fun i hi => ?_
      rcases eq_or_ne (w i) 0 with hwi | hwi
      · rw [hwi, zero_mul, zero_mul]
      · rw [hx i hi hwi]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p q : ℝ
a✝ b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
a : ℝ≥0
⊢ a ^ q = a ^ (p * (q / p))

Tactic:
div_eq_inv_mul,

Goal state after:
p q : ℝ
a✝ b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
a : ℝ≥0
⊢ a ^ q = a ^ (p * (p⁻¹ * q))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case neg
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
⊢ ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i ↔ ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i

Tactic:
· have hz' := fun i h => lt_of_le_of_ne (hz i h) (fun a => (ne_of_gt (hw i h)) (A i h a.symm))
    have := strictConvexOn_exp.map_sum_eq_iff hw hw' fun i _ => Set.mem_univ <| log (z i)
    simp only [exp_sum, smul_eq_mul, mul_comm (w _) (log _)] at this
    convert! this using 1
    · apply Eq.congr <;>
      [apply prod_congr rfl; apply sum_congr rfl] <;>
      intro i hi <;>
      simp only [exp_mul, exp_log (hz' i hi)]
    · constructor <;> intro h j hj
      · rw [← arith_mean_weighted_of_constant s w _ (log (z j)) hw' fun i _ => congrFun rfl]
        apply sum_congr rfl
        intro x hx
        simp only [mul_comm, h j hj, h x hx]
      · rw [← arith_mean_weighted_of_constant s w _ (z j) hw' fun i _ => congrFun rfl]
        apply sum_congr rfl
        intro x hx
        simp only [log_injOn_pos (hz' j hj) (hz' x hx), h j hj, h x hx]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case hf
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := ⋯
hq : ∑ i ∈ s, q i = 1
h2q : ∀ i ∈ s, 0 ≤ q i
⊢ AEMeasurable (f i₀) μ

Tactic:
exact hf i₀ <| mem_insert_self ..

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : ∑ i ∈ s, g i ^ q = 0
⊢ ∑ i ∈ s, f i * g i = ∑ i ∈ s, g i * f i

Tactic:
congr with i
        rw [mul_comm]

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₁
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hro_lt : 0 < r
hrp : r < p
hpqr : 1 / r = 1 / p + 1 / q
a : α
ha : ‖b (f a) (g a)‖₊ ≤ c * ‖f a‖₊ * ‖g a‖₊
⊢ ‖b (f a) (g a)‖₊ ≤ ‖↑c • ‖f a‖ * ‖g a‖‖₊

Tactic:
simpa using! ha

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
n : ι
a✝ : α
hx₁ : ↑↑(MemLp.toLp (f n - f_lim) ⋯) a✝ = (f n - f_lim) a✝
hx₂ :
  ↑↑(MemLp.toLp (f n) ⋯ - MemLp.toLp f_lim f_lim_ℒp) a✝ = (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝
⊢ (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝ = (f n - f_lim) a✝

Tactic:
rw [← hx₂]

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
ι : Type u_4
fi : Filter ι
inst✝ : Fact (1 ≤ p)
f : ι → α → E
f_ℒp : ∀ (n : ι), MemLp (f n) p μ
f_lim : α → E
f_lim_ℒp : MemLp f_lim p μ
n : ι
a✝ : α
hx₁ : ↑↑(MemLp.toLp (f n - f_lim) ⋯) a✝ = (f n - f_lim) a✝
hx₂ :
  ↑↑(MemLp.toLp (f n) ⋯ - MemLp.toLp f_lim f_lim_ℒp) a✝ = (↑↑(MemLp.toLp (f n) ⋯) - ↑↑(MemLp.toLp f_lim f_lim_ℒp)) a✝
⊢ ↑↑(MemLp.toLp (f n) ⋯ - MemLp.toLp f_lim f_lim_ℒp) a✝ = (f n - f_lim) a✝
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case e'_6
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
x : α
⊢ ‖f x‖ = |‖f x‖ ^ q.toReal| ^ q⁻¹.toReal

Tactic:
Real.abs_rpow_of_nonneg (norm_nonneg _),

Goal state after:
case e'_6
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
x : α
⊢ ‖f x‖ = (|‖f x‖| ^ q.toReal) ^ q⁻¹.toReal
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
g : E → F
c : ℝ≥0
hg : LipschitzWith c g
g0 : g 0 = 0
⊢ hg.compLp g0 0 = 0

Tactic:
by
  rw [Lp.eq_zero_iff_ae_eq_zero]
  apply (coeFn_compLp _ _ _).trans
  filter_upwards [Lp.coeFn_zero E p μ] with _ ha
  simp only [ha, g0, Function.comp_apply, Pi.zero_apply]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (L + L').compLp f = L.compLp f + L'.compLp f

Tactic:
ext1

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L L' : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ ↑↑((L + L').compLp f) =ᵐ[μ] ↑↑(L.compLp f + L'.compLp f)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ (c • L).compLp f = c • L.compLp f

Tactic:
ext1

Goal state after:
α : Type u_1
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹⁰ : NormedAddCommGroup E
inst✝⁹ : NormedAddCommGroup F
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜'
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝⁴ : RingHomIsometric σ
𝕜'' : Type u_8
inst✝³ : NormedRing 𝕜''
inst✝² : Module 𝕜'' F
inst✝¹ : IsBoundedSMul 𝕜'' F
inst✝ : SMulCommClass 𝕜' 𝕜'' F
c : 𝕜''
L : E →SL[σ] F
f : ↥(Lp E p μ)
⊢ ↑↑((c • L).compLp f) =ᵐ[μ] ↑↑(c • L.compLp f)
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g✝ : E → F
c : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
f g : ↥(Lp E p μ)
⊢ L.compLp (f + g) = L.compLp f + L.compLp g

Tactic:
ext1

Goal state after:
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g✝ : E → F
c : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
f g : ↥(Lp E p μ)
⊢ ↑↑(L.compLp (f + g)) =ᵐ[μ] ↑↑(L.compLp f + L.compLp g)
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g : E → F
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
c : 𝕜
f : ↥(Lp E p μ)
⊢ L.compLp (c • f) = σ c • L.compLp f

Tactic:
ext1

Goal state after:
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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
g : E → F
c✝ : ℝ≥0
𝕜 : Type u_6
𝕜' : Type u_7
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜'
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝ : RingHomIsometric σ
L : E →SL[σ] F
c : 𝕜
f : ↥(Lp E p μ)
⊢ ↑↑(L.compLp (c • f)) =ᵐ[μ] ↑↑(σ c • L.compLp f)
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NontriviallyNormedField 𝕜₂
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
h : RingHomIsometric σ₁₂
f : E →ₛₗ[σ₁₂] F
hf : Filter.comap (⇑f) (𝓝 0) ≤ 𝓝 0
ε : ℝ≥0
ε0 : 0 < ↑ε
hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1
c : 𝕜
hc : 1 < ‖σ₁₂ c‖
x : E
hx : ¬f x = 0
hc₀ : c ≠ 0
n : ℤ
hlt : ‖f (c ^ n • x)‖ < ↑ε
hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖
⊢ ‖x‖ = ‖c ^ n‖⁻¹ * ‖c ^ n • x‖

Tactic:
by
      rwa [← norm_inv, ← norm_smul, inv_smul_smul₀ (zpow_ne_zero _ _)]

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

Goal state before:
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 fun f hf ↦ hf

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
this : IsClosed[instTopologicalSpaceSubtype] {0}
⊢ T3Space ↥p

Tactic:
have : T1Space p := IsTopologicalAddGroup.t1Space _ this

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
this✝ : IsClosed[instTopologicalSpaceSubtype] {0}
this : T1Space ↥p
⊢ T3Space ↥p
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
ι : Type u_3
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : (μ univ).toReal < ∑ i ∈ s, (μ (t i)).toReal
⊢ ∑ i ∈ s, μ (t i) ≠ ∞

Tactic:
· exact (ENNReal.sum_lt_top.mpr (fun i hi ↦ measure_lt_top ..)).ne

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
c : E
⊢ indicatorConstLp p ⋯ ⋯ c = indicatorConstLp p hs hμs c + indicatorConstLp p ht hμt c

Tactic:
ext1

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
c : E
⊢ ↑↑(indicatorConstLp p ⋯ ⋯ c) =ᵐ[μ] ↑↑(indicatorConstLp p hs hμs c + indicatorConstLp p ht hμt c)
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
s : Set α
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ indicatorConstLp 2 hs hμs x = (ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1)

Tactic:
ext1

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
s : Set α
inst✝ : NormedSpace ℝ E
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ ↑↑(indicatorConstLp 2 hs hμs x) =ᵐ[μ]
    ↑↑((ContinuousLinearMap.toSpanSingleton ℝ x).compLp (indicatorConstLp 2 hs hμs 1))
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case inl
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✝ : IsEmpty ι
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖

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
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
· refine ⟨‖f 0‖ + 1, add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, fun m => ?_⟩
    obtain rfl : m = 0 := funext (IsEmpty.elim ‹_›)
    simp [univ_eq_empty, zero_le_one]

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
⊢ ∃ C, 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖