mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case pos
z₁ z₂ p : ℝ≥0∞
p_zero : p = 0
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)

pp: case neg
z₁ z₂ p : ℝ≥0∞
p_zero : ¬p = 0
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)

Tactic:
· simp [p_zero, LpAddConst_zero]

Goal state after:
case neg
z₁ z₂ p : ℝ≥0∞
p_zero : ¬p = 0
⊢ (z₁ + z₂) ^ p.toReal⁻¹ ≤ p.LpAddConst * (z₁ ^ p.toReal⁻¹ + z₂ ^ p.toReal⁻¹)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₂.inr.inl
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
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‖₊
⊢ essSup (fun x ↦ ‖‖f x‖ * ‖g x‖‖ₑ) μ ≤ essSup (fun x ↦ ‖‖f x‖‖ₑ) μ * essSup (fun x ↦ ‖‖g x‖‖ₑ) μ

Tactic:
enorm_mul,

Goal state after:
case h₂.inr.inl
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
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‖₊
⊢ essSup (fun x ↦ ‖‖f x‖‖ₑ * ‖‖g x‖‖ₑ) μ ≤ essSup (fun x ↦ ‖‖f x‖‖ₑ) μ * essSup (fun x ↦ ‖‖g x‖‖ₑ) μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₂.inr.inl
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
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‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) ∞ μ ≤ eLpNorm f ∞ μ * eLpNorm g ∞ μ

pp: case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ

Tactic:
· rw [← eLpNorm_norm f, ← eLpNorm_norm g]
    simp_rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm, enorm_mul, enorm_norm]
    exact ENNReal.essSup_mul_le (‖f ·‖ₑ) (‖g ·‖ₑ)

Goal state after:
case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ eLpNorm f ∞ μ * eLpNorm g p μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
hp₁ : 0 < p
hp₂ : p < ∞
⊢ ∫⁻ (x : α), ‖‖f x‖ * ‖g x‖‖ₑ ^ p.toReal ∂μ ≤
    (essSup (fun x ↦ ‖f x‖ₑ) μ * (∫⁻ (x : α), ‖g x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹) ^ p.toReal

Tactic:
enorm_mul,

Goal state after:
case h₂.inr.inr
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
p : ℝ≥0∞
f : α → E
g : α → F
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‖₊
hp : 0 < p.toReal
hp₁ : 0 < p
hp₂ : p < ∞
⊢ ∫⁻ (x : α), (‖‖f x‖‖ₑ * ‖‖g x‖‖ₑ) ^ p.toReal ∂μ ≤
    (essSup (fun x ↦ ‖f x‖ₑ) μ * (∫⁻ (x : α), ‖g x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹) ^ p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: 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
p : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
g : α → F
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
⊢ ∀ᵐ (x : α) ∂μ, ‖flip b (g x) (f x)‖₊ ≤ c * ‖g x‖₊ * ‖f x‖₊

Tactic:
convert! h using 3 with x

Goal state after:
case e'_4
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
p : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
g : α → F
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
x : α
⊢ c * ‖g x‖₊ * ‖f x‖₊ = c * ‖f x‖₊ * ‖g x‖₊
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case refine_2
ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
s : Finset ι
hp : 0 < p
hq : 0 < q
hr : 0 < r
⊢ (∑ i ∈ s, g i ^ q) ^ (r / q) ≤ (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
gcongr
      exact hg.sum_le_tsum _ (fun _ _ => zero_le)

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧
    ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
have H₁ : ∀ s : Finset ι,
      ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' i, f i ^ p) ^ (r / p) * (∑' i, g i ^ q) ^ (r / q) := by
    intro s
    obtain ⟨hp, hq, hr⟩ := hpqr.all_pos
    refine le_trans (Lr_rpow_le_Lp_mul_Lq s f g hpqr) (mul_le_mul ?_ ?_ bot_le bot_le)
    · gcongr
      exact hf.sum_le_tsum _ (fun _ _ => zero_le)
    · gcongr
      exact hg.sum_le_tsum _ (fun _ _ => zero_le)
  have bdd : BddAbove (Set.range fun s => ∑ i ∈ s, (f i * g i) ^ r) := by
    refine ⟨(∑' i, f i ^ p) ^ (r / p) * (∑' i, g i ^ q) ^ (r / q), ?_⟩
    rintro a ⟨s, rfl⟩
    exact H₁ s
  have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
  exact ⟨H₂, H₂.tsum_le_of_sum_le H₁⟩

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h_one_div_q : 1 / q = 1 - 1 / p
h :
  ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 - 1 / p)
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))

Tactic:
sub_eq_add_neg 1 (1 / p),

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
h_one_div_q : 1 / q = 1 - 1 / p
h :
  ∫⁻ (a : α), (f + g) a ^ p ∂μ ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 + -(1 / p))
⊢ 1 ≤
    (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
      ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p))
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : p = 1
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
h1,

Goal state after:
case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : p = 1
⊢ (∫⁻ (a : α), (f + g) a ^ 1 ∂μ) ^ (1 / 1) = (∫⁻ (a : α), f a ^ 1 ∂μ) ^ (1 / 1) + (∫⁻ (a : α), g a ^ 1 ∂μ) ^ (1 / 1)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
C : ℝ≥0∞
bound : ∀ᶠ (n : ι) in u, eLpNorm (f n) p μ ≤ C
hf : ∀ (n : ι), AEStronglyMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun x_1 ↦ f x_1 x) u (𝓝 (g x))
⊢ eLpNorm g p μ ≤ C

Tactic:
obtain ⟨v, hv⟩ := exists_seq_tendsto u
  have : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f (v n) x) atTop (𝓝 (g x)) := by
    filter_upwards [h_tendsto] with x hx
    exact hx.comp hv
  calc
  _ ≤ atTop.liminf (fun (n : ℕ) => eLpNorm (f (v n)) p μ) :=
    Lp.eLpNorm_lim_le_liminf_eLpNorm (fun n => hf (v n)) g this
  _ ≤ C := by
    refine liminf_le_of_le (by isBoundedDefault) (fun b hb => ?_)
    obtain ⟨n, hn⟩ := (hb.and (hv.eventually bound)).exists
    exact hn.1.trans hn.2

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
i : ι
h_ae_eq : f i =ᵐ[μ] f' i
x : α
hxf : f i x = f' i x
hxg : g x = g' x
⊢ ε ≤ edist (f' i x) (g' x) ↔ ε ≤ edist (f i x) (g x)

Tactic:
rw [hxg, hxf]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
hf_sum : Summable fun i ↦ f i ^ p
hg_sum : Summable fun i ↦ g i ^ q
hf' : 0 ≤ ∑' (i : ι), f i ^ p
hg' : 0 ≤ ∑' (i : ι), g i ^ q
hr : 0 < r
⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) =
    ((∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)) ^ (1 / r)

Tactic:
mul_rpow (rpow_nonneg hf' _) (rpow_nonneg hg' _),

Goal state after:
ι : Type u
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ (i : ι), 0 ≤ f i
hg : ∀ (i : ι), 0 ≤ g i
hf_sum : Summable fun i ↦ f i ^ p
hg_sum : Summable fun i ↦ g i ^ q
hf' : 0 ≤ ∑' (i : ι), f i ^ p
hg' : 0 ≤ ∑' (i : ι), g i ^ q
hr : 0 < r
⊢ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q) =
    ((∑' (i : ι), f i ^ p) ^ (r / p)) ^ (1 / r) * ((∑' (i : ι), g i ^ q) ^ (r / q)) ^ (1 / r)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → E
hf : MemLp f p μ
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ MemLp (fun x ↦ ‖f x‖ ^ p.toReal) 1 μ

Tactic:
by
  convert! hf.norm_rpow_div p
  rw [div_eq_mul_inv, ENNReal.mul_inv_cancel hp_ne_zero hp_ne_top]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ≥0∞
p : ℝ
hp : 1 ≤ p
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
by
  rcases eq_or_lt_of_le hp with hp | hp
  · simp [← hp]
  let q : ℝ := p / (p - 1)
  have hpq : p.HolderConjugate q := .conjExponent hp
  have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero
  have hq : 1 / q * p = p - 1 := by
    rw [← hpq.div_conj_eq_sub_one]
    ring
  simpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, ← ENNReal.rpow_mul, hp₁, hq, coe_one,
    one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_cast] using
    ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
f g₁ g₂ : ↥(Lp E p μ)
hg₁₂ : ↑↑g₁ ≤ᵐ[μ] ↑↑g₂
a✝ : α
h1 : ↑↑(f + g₁) a✝ = (↑↑f + ↑↑g₁) a✝
h2 : ↑↑(f + g₂) a✝ = (↑↑f + ↑↑g₂) a✝
h3 : ↑↑g₁ a✝ ≤ ↑↑g₂ a✝
⊢ ↑↑(f + g₁) a✝ ≤ ↑↑(f + g₂) a✝

Tactic:
h1,

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedAddCommGroup E
inst✝¹ : PartialOrder E
inst✝ : IsOrderedAddMonoid E
f g₁ g₂ : ↥(Lp E p μ)
hg₁₂ : ↑↑g₁ ≤ᵐ[μ] ↑↑g₂
a✝ : α
h1 : ↑↑(f + g₁) a✝ = (↑↑f + ↑↑g₁) a✝
h2 : ↑↑(f + g₂) a✝ = (↑↑f + ↑↑g₂) a✝
h3 : ↑↑g₁ a✝ ≤ ↑↑g₂ a✝
⊢ (↑↑f + ↑↑g₁) a✝ ≤ ↑↑(f + g₂) a✝
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
⊢ ∀ ⦃a : γ⦄, a ∈ Set.range ⇑(inl.trans f, inr.trans f).1 → ¬a ∈ Set.range ⇑(inl.trans f, inr.trans f).2

Tactic:
rintro _ ⟨a, h⟩ ⟨b, rfl⟩

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
f : α ⊕ β ↪ γ
a : α
b : β
h : (inl.trans f, inr.trans f).1 a = (inl.trans f, inr.trans f).2 b
⊢ False
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: case e_6
α : Type u_1
β : Type u_2
inst✝ : Unique α
x✝ : α ↪ β
x : α
⊢ default = x

Tactic:
exact Subsingleton.elim _ x

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

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
this : ∀ (i : ι), ∃ c, c ≠ 0 ∧ ∀ (c' : 𝕜), ‖c'‖ ≤ ‖c‖ → ∀ x ∈ s, c' • x i ∈ t i
⊢ Absorbs 𝕜 V (⇑f '' s)

Tactic:
choose c hc₀ hc using this

Goal state after:
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
⊢ Absorbs 𝕜 V (⇑f '' s)
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
x✝¹ x✝ : ContinuousMultilinearMap 𝕜 E₁ F
⊢ (x✝¹ + x✝).compContinuousLinearMap f = x✝¹.compContinuousLinearMap f + x✝.compContinuousLinearMap f

Tactic:
by ext; simp

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
x✝¹ : 𝕜
x✝ : ContinuousMultilinearMap 𝕜 E₁ F
⊢ (x✝¹ • x✝).compContinuousLinearMap f = (RingHom.id 𝕜) x✝¹ • x✝.compContinuousLinearMap f

Tactic:
by ext; simp

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

Goal state before:
pp: 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
x✝¹ x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ g.compContinuousMultilinearMap (x✝¹ + x✝) = g.compContinuousMultilinearMap x✝¹ + g.compContinuousMultilinearMap x✝

Tactic:
by ext; simp

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

Goal state before:
pp: 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
x✝¹ : 𝕜
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ g.compContinuousMultilinearMap (x✝¹ • x✝) = (RingHom.id 𝕜) x✝¹ • g.compContinuousMultilinearMap x✝

Tactic:
by ext; simp

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

Goal state before:
pp: 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
x✝¹ x✝ : F →L[𝕜] G
⊢ { toFun := (x✝¹ + x✝).compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } =
    { toFun := x✝¹.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } +
      { toFun := x✝.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

Tactic:
by ext; simp

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

Goal state before:
pp: 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
x✝¹ : 𝕜
x✝ : F →L[𝕜] G
⊢ { toFun := (x✝¹ • x✝).compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } =
    (RingHom.id 𝕜) x✝¹ • { toFun := x✝.compContinuousMultilinearMap, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }

Tactic:
by ext; simp

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
E₁ : ι → Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedField 𝕜
inst✝¹⁵ : (i : ι) → TopologicalSpace (E i)
inst✝¹⁴ : (i : ι) → AddCommGroup (E i)
inst✝¹³ : (i : ι) → Module 𝕜 (E i)
inst✝¹² : (i : ι) → TopologicalSpace (E₁ i)
inst✝¹¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E₁ 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
f : (i : ι) → E i ≃L[𝕜] E₁ i
g : ContinuousMultilinearMap 𝕜 E₁ F
⊢ (ContinuousMultilinearMap.compContinuousLinearMapL fun i ↦ ↑(f i).symm)
      ((↑(ContinuousMultilinearMap.compContinuousLinearMapL fun i ↦ ↑(f i))).toFun g) =
    g

Tactic:
by ext; simp

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
E₁ : ι → Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedField 𝕜
inst✝¹⁵ : (i : ι) → TopologicalSpace (E i)
inst✝¹⁴ : (i : ι) → AddCommGroup (E i)
inst✝¹³ : (i : ι) → Module 𝕜 (E i)
inst✝¹² : (i : ι) → TopologicalSpace (E₁ i)
inst✝¹¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E₁ 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
f : (i : ι) → E i ≃L[𝕜] E₁ i
g : ContinuousMultilinearMap 𝕜 E F
⊢ (↑(ContinuousMultilinearMap.compContinuousLinearMapL fun i ↦ ↑(f i))).toFun
      ((ContinuousMultilinearMap.compContinuousLinearMapL fun i ↦ ↑(f i).symm) g) =
    g

Tactic:
by ext; simp

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
E₁ : ι → Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedField 𝕜
inst✝¹⁵ : (i : ι) → TopologicalSpace (E i)
inst✝¹⁴ : (i : ι) → AddCommGroup (E i)
inst✝¹³ : (i : ι) → Module 𝕜 (E i)
inst✝¹² : (i : ι) → TopologicalSpace (E₁ i)
inst✝¹¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E₁ 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
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ((ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 E G F) ↑g.symm)
      ((↑((ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 E F G) ↑g)).toFun x✝) =
    x✝

Tactic:
by ext; simp

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
E₁ : ι → Type u_4
F : Type u_5
G : Type u_6
inst✝¹⁶ : NormedField 𝕜
inst✝¹⁵ : (i : ι) → TopologicalSpace (E i)
inst✝¹⁴ : (i : ι) → AddCommGroup (E i)
inst✝¹³ : (i : ι) → Module 𝕜 (E i)
inst✝¹² : (i : ι) → TopologicalSpace (E₁ i)
inst✝¹¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → Module 𝕜 (E₁ 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
x✝ : ContinuousMultilinearMap 𝕜 E G
⊢ (↑((ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 E F G) ↑g)).toFun
      (((ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 E G F) ↑g.symm) x✝) =
    x✝

Tactic:
by ext; simp

Goal state after: