l3lab/ntp-mathlib huggingface dataset:
/-- If the target space is complete, the space of continuous multilinear maps with its norm is also
complete. The proof is essentially the same as for the space of continuous linear maps (modulo the
addition of `Finset.prod` where needed. The duplication could be avoided by deducing the linear
case from the multilinear case via a currying isomorphism. However, this would mess up imports,
and it is more satisfactory to have the simplest case as a standalone proof. -/
instance completeSpace [CompleteSpace G] : CompleteSpace (ContinuousMultilinearMap 𝕜 E G) := by
  have nonneg : ∀ v : ∀ i, E i, 0 ≤ ∏ i, ‖v i‖ := fun v =>
    Finset.prod_nonneg fun i _ => norm_nonneg _
  -- We show that every Cauchy sequence converges.
  refine' Metric.complete_of_cauchySeq_tendsto fun f hf => _
  -- We now expand out the definition of a Cauchy sequence,
  rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩
  -- and establish that the evaluation at any point `v : Π i, E i` is Cauchy.
  have cau : ∀ v, CauchySeq fun n => f n v := by
    intro v
    apply cauchySeq_iff_le_tendsto_0.2 ⟨fun n => b n * ∏ i, ‖v i‖, _, _, _⟩
    · intro
      exact mul_nonneg (b0 _) (nonneg v)
    · intro n m N hn hm
      rw [dist_eq_norm]
      apply le_trans ((f n - f m).le_op_norm v) _
      exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v)
    · simpa using b_lim.mul tendsto_const_nhds
  -- We assemble the limits points of those Cauchy sequences
  -- (which exist as `G` is complete)
  -- into a function which we call `F`.
  

Proof state:
case intro.intro.intro
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁵ : Fintype ι
inst✝¹⁴ : Fintype ι'
inst✝¹³ : NontriviallyNormedField 𝕜
inst✝¹² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹⁰ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁸ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁷ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁶ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁵ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : NormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
c : 𝕜
f✝ g : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
inst✝ : CompleteSpace G
nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖
f : ℕ → ContinuousMultilinearMap 𝕜 E G
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => (f n) v
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Next tactic:
choose F hF using fun v => cauchySeq_tendsto_of_complete (cau v)
/-- If the target space is complete, the space of continuous multilinear maps with its norm is also
complete. The proof is essentially the same as for the space of continuous linear maps (modulo the
addition of `Finset.prod` where needed. The duplication could be avoided by deducing the linear
case from the multilinear case via a currying isomorphism. However, this would mess up imports,
and it is more satisfactory to have the simplest case as a standalone proof. -/
instance completeSpace [CompleteSpace G] : CompleteSpace (ContinuousMultilinearMap 𝕜 E G) := by
  have nonneg : ∀ v : ∀ i, E i, 0 ≤ ∏ i, ‖v i‖ := fun v =>
    Finset.prod_nonneg fun i _ => norm_nonneg _
  -- We show that every Cauchy sequence converges.
  refine' Metric.complete_of_cauchySeq_tendsto fun f hf => _
  -- We now expand out the definition of a Cauchy sequence,
  rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩
  -- and establish that the evaluation at any point `v : Π i, E i` is Cauchy.
  have cau : ∀ v, CauchySeq fun n => f n v := by
    intro v
    apply cauchySeq_iff_le_tendsto_0.2 ⟨fun n => b n * ∏ i, ‖v i‖, _, _, _⟩
    · intro
      exact mul_nonneg (b0 _) (nonneg v)
    · intro n m N hn hm
      rw [dist_eq_norm]
      apply le_trans ((f n - f m).le_op_norm v) _
      exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v)
    · simpa using b_lim.mul tendsto_const_nhds
  -- We assemble the limits points of those Cauchy sequences
  -- (which exist as `G` is complete)
  -- into a function which we call `F`.
  choose F hF using fun v => cauchySeq_tendsto_of_complete (cau v)
  -- Next, we show that this `F` is multilinear,
  let Fmult : MultilinearMap 𝕜 E G :=
    { toFun := F
      map_add' := fun v i x y => by
        

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁶ : Fintype ι
inst✝¹⁵ : Fintype ι'
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
c : 𝕜
f✝ g : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
inst✝¹ : CompleteSpace G
nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖
f : ℕ → ContinuousMultilinearMap 𝕜 E G
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => (f n) v
F : ((i : ι) → E i) → G
hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => (f n) v) atTop (𝓝 (F v))
inst✝ : DecidableEq ι
v : (i : ι) → E i
i : ι
x y : E i
⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)

Next tactic:
skip
/-- If the target space is complete, the space of continuous multilinear maps with its norm is also
complete. The proof is essentially the same as for the space of continuous linear maps (modulo the
addition of `Finset.prod` where needed. The duplication could be avoided by deducing the linear
case from the multilinear case via a currying isomorphism. However, this would mess up imports,
and it is more satisfactory to have the simplest case as a standalone proof. -/
instance completeSpace [CompleteSpace G] : CompleteSpace (ContinuousMultilinearMap 𝕜 E G) := by
  have nonneg : ∀ v : ∀ i, E i, 0 ≤ ∏ i, ‖v i‖ := fun v =>
    Finset.prod_nonneg fun i _ => norm_nonneg _
  -- We show that every Cauchy sequence converges.
  refine' Metric.complete_of_cauchySeq_tendsto fun f hf => _
  -- We now expand out the definition of a Cauchy sequence,
  rcases cauchySeq_iff_le_tendsto_0.1 hf with ⟨b, b0, b_bound, b_lim⟩
  -- and establish that the evaluation at any point `v : Π i, E i` is Cauchy.
  have cau : ∀ v, CauchySeq fun n => f n v := by
    intro v
    apply cauchySeq_iff_le_tendsto_0.2 ⟨fun n => b n * ∏ i, ‖v i‖, _, _, _⟩
    · intro
      exact mul_nonneg (b0 _) (nonneg v)
    · intro n m N hn hm
      rw [dist_eq_norm]
      apply le_trans ((f n - f m).le_op_norm v) _
      exact mul_le_mul_of_nonneg_right (b_bound n m N hn hm) (nonneg v)
    · simpa using b_lim.mul tendsto_const_nhds
  -- We assemble the limits points of those Cauchy sequences
  -- (which exist as `G` is complete)
  -- into a function which we call `F`.
  choose F hF using fun v => cauchySeq_tendsto_of_complete (cau v)
  -- Next, we show that this `F` is multilinear,
  let Fmult : MultilinearMap 𝕜 E G :=
    { toFun := F
      map_add' := fun v i x y => by
        skip
        

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁶ : Fintype ι
inst✝¹⁵ : Fintype ι'
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : NormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
c : 𝕜
f✝ g : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
inst✝¹ : CompleteSpace G
nonneg : ∀ (v : (i : ι) → E i), 0 ≤ ∏ i : ι, ‖v i‖
f : ℕ → ContinuousMultilinearMap 𝕜 E G
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
cau : ∀ (v : (i : ι) → E i), CauchySeq fun n => (f n) v
F : ((i : ι) → E i) → G
hF : ∀ (v : (i : ι) → E i), Tendsto (fun n => (f n) v) atTop (𝓝 (F v))
inst✝ : DecidableEq ι
v : (i : ι) → E i
i : ι
x y : E i
⊢ F (Function.update v i (x + y)) = F (Function.update v i x) + F (Function.update v i y)

Next tactic:
have A := hF (Function.update v i (x + y))
theorem norm_restr {k n : ℕ} (f : G[×n]→L[𝕜] G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) :
    ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) := by
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n✝ : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n✝) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
k n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
s : Finset (Fin n)
hk : card s = k
z : G
⊢ ‖restr f s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k)

Next tactic:
apply MultilinearMap.mkContinuous_norm_le
theorem norm_restr {k n : ℕ} (f : G[×n]→L[𝕜] G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) :
    ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k) := by
  apply MultilinearMap.mkContinuous_norm_le
  

Proof state:
case hC
𝕜 : Type u
ι : Type v
ι' : Type v'
n✝ : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n✝) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁵ : (i : Fin (Nat.succ n✝)) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin (Nat.succ n✝)) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
k n : ℕ
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
s : Finset (Fin n)
hk : card s = k
z : G
⊢ 0 ≤ ‖f‖ * ‖z‖ ^ (n - k)

Next tactic:
exact mul_nonneg (norm_nonneg _) (pow_nonneg (norm_nonneg _) _)
@[simp]
theorem norm_mkPiAlgebra_le [Nonempty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 := by
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : Nonempty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1

Next tactic:
refine op_norm_le_bound _ zero_le_one fun m => ?_
@[simp]
theorem norm_mkPiAlgebra_le [Nonempty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 := by
  refine op_norm_le_bound _ zero_le_one fun m => ?_
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : Nonempty ι
m : ι → A
⊢ ‖(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ 1 * ∏ i : ι, ‖m i‖

Next tactic:
simp only [ContinuousMultilinearMap.mkPiAlgebra_apply, one_mul]
@[simp]
theorem norm_mkPiAlgebra_le [Nonempty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1 := by
  refine op_norm_le_bound _ zero_le_one fun m => ?_
  simp only [ContinuousMultilinearMap.mkPiAlgebra_apply, one_mul]
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : Nonempty ι
m : ι → A
⊢ ‖∏ i : ι, m i‖ ≤ ∏ i : ι, ‖m i‖

Next tactic:
exact norm_prod_le' _ univ_nonempty _
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖1‖

Next tactic:
apply le_antisymm
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  apply le_antisymm
  · 

Proof state:
case a
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖

Next tactic:
apply op_norm_le_bound
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  apply le_antisymm
  · apply op_norm_le_bound <;> 

Proof state:
case a.hMp
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ 0 ≤ ‖1‖

Next tactic:
simp
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  apply le_antisymm
  · apply op_norm_le_bound <;> 

Proof state:
case a.hM
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ∀ (m : ι → A), ‖(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) m‖ ≤ ‖1‖ * ∏ i : ι, ‖m i‖

Next tactic:
simp
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  apply le_antisymm
  · apply op_norm_le_bound <;> simp
  · -- Porting note: have to annotate types to get mvars to unify
    

Proof state:
case a
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖

Next tactic:
convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => (1 : A)
theorem norm_mkPiAlgebra_of_empty [IsEmpty ι] :
    ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖(1 : A)‖ := by
  apply le_antisymm
  · apply op_norm_le_bound <;> simp
  · -- Porting note: have to annotate types to get mvars to unify
    convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun _ => (1 : A)
    

Proof state:
case h.e'_3
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : IsEmpty ι
⊢ ‖1‖ = ‖(ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A) fun x => 1‖ / ∏ i : ι, ‖1‖

Next tactic:
simp [eq_empty_of_isEmpty (univ : Finset ι)]
@[simp]
theorem norm_mkPiAlgebra [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 := by
  

Proof state:
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1

Next tactic:
cases isEmpty_or_nonempty ι
@[simp]
theorem norm_mkPiAlgebra [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1 := by
  cases isEmpty_or_nonempty ι
  · 

Proof state:
case inl
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁷ : Fintype ι
inst✝¹⁶ : Fintype ι'
inst✝¹⁵ : NontriviallyNormedField 𝕜
inst✝¹⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝¹² : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝¹⁰ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁸ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁷ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝⁶ : NormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : NormOneClass A
h✝ : IsEmpty ι
⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1

Next tactic:
simp [norm_mkPiAlgebra_of_empty]