l3lab/ntp-mathlib huggingface dataset:
/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => 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'
f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
⊢ uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂

Next tactic:
ext m
/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => by
        ext m
        

Proof state:
case H
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'
f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
m : (i : Fin (Nat.succ n)) → Ei i
⊢ (uncurryRight (f₁ + f₂)) m = (uncurryRight f₁ + uncurryRight f₂) m

Next tactic:
rfl
/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => by
        ext m
        rfl
      map_smul' := fun c f => 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'
c : 𝕜
f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
⊢ AddHom.toFun
      { toFun := uncurryRight,
        map_add' :=
          (_ :
            ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
              uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
      (c • f) =
    (RingHom.id 𝕜) c •
      AddHom.toFun
        { toFun := uncurryRight,
          map_add' :=
            (_ :
              ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
        f

Next tactic:
ext m
/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => by
        ext m
        rfl
      map_smul' := fun c f => by
        ext m
        

Proof state:
case H
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'
c : 𝕜
f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
m : (i : Fin (Nat.succ n)) → Ei i
⊢ (AddHom.toFun
        { toFun := uncurryRight,
          map_add' :=
            (_ :
              ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
        (c • f))
      m =
    ((RingHom.id 𝕜) c •
        AddHom.toFun
          { toFun := uncurryRight,
            map_add' :=
              (_ :
                ∀ (f₁ f₂ : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)),
                  uncurryRight (f₁ + f₂) = uncurryRight f₁ + uncurryRight f₂) }
          f)
      m

Next tactic:
rfl
/--
The space of continuous multilinear maps on `Π(i : Fin (n+1)), Ei i` is canonically isomorphic to
the space of continuous multilinear maps on `Π(i : Fin n), Ei <| castSucc i` with values in the
space of continuous linear maps on `Ei (last n)`, by separating the last variable. We register this
isomorphism as a continuous linear equiv in `continuousMultilinearCurryRightEquiv 𝕜 Ei G`.
The algebraic version (without topology) is given in `multilinearCurryRightEquiv 𝕜 Ei G`.

The direct and inverse maps are given by `f.uncurryRight` and `f.curryRight`. Use these
unless you need the full framework of linear isometric equivs.
-/
def continuousMultilinearCurryRightEquiv :
    ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei <| castSucc i) (Ei (last n) →L[𝕜] G) ≃ₗᵢ[𝕜]
      ContinuousMultilinearMap 𝕜 Ei G :=
  LinearIsometryEquiv.ofBounds
    { toFun := ContinuousMultilinearMap.uncurryRight
      map_add' := fun f₁ f₂ => by
        ext m
        rfl
      map_smul' := fun c f => by
        ext m
        rfl
      invFun := ContinuousMultilinearMap.curryRight
      left_inv := ContinuousMultilinearMap.curry_uncurryRight
      right_inv := ContinuousMultilinearMap.uncurry_curryRight } (fun f => by
        simp only [uncurryRight, LinearEquiv.coe_mk]
        

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'
f : ContinuousMultilinearMap 𝕜 (fun i => Ei (castSucc i)) (Ei (last n) →L[𝕜] G)
⊢ ‖MultilinearMap.mkContinuous
        (MultilinearMap.uncurryRight
          { toFun := fun m => ↑(f m),
            map_add' :=
              (_ :
                ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (x y : Ei (castSucc i)),
                  ↑(f (update m i (x + y))) = ↑(f (update m i x)) + ↑(f (update m i y))),
            map_smul' :=
              (_ :
                ∀ [inst : DecidableEq (Fin n)] (m : (i : Fin n) → Ei (castSucc i)) (i : Fin n) (c : 𝕜)
                  (x : Ei (castSucc i)), ↑(f (update m i (c • x))) = c • ↑(f (update m i x))) })
        ‖f‖
        (_ :
          ∀ (m : (i : Fin (Nat.succ n)) → Ei i), ‖(f (init m)) (m (last n))‖ ≤ ‖f‖ * ∏ i : Fin (Nat.succ n), ‖m i‖)‖ ≤
    ‖f‖

Next tactic:
exact MultilinearMap.mkContinuous_norm_le _ (norm_nonneg f) _
@[simp]
theorem ContinuousMultilinearMap.apply_zero_curry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ContinuousMultilinearMap.curry0 𝕜 G (f x) = f := 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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
x : Fin 0 → G
⊢ curry0 𝕜 G (f x) = f

Next tactic:
ext m
@[simp]
theorem ContinuousMultilinearMap.apply_zero_curry0 (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ContinuousMultilinearMap.curry0 𝕜 G (f x) = f := by
  ext m
  

Proof state:
case H
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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
x m : Fin 0 → G
⊢ (curry0 𝕜 G (f x)) m = f m

Next tactic:
simp [Subsingleton.elim x m]
theorem ContinuousMultilinearMap.uncurry0_curry0 (f : G[×0]→L[𝕜] G') :
    ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0 = f := 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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
⊢ curry0 𝕜 G (uncurry0 f) = f

Next tactic:
simp
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := 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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
x : Fin 0 → G
⊢ ‖f x‖ = ‖f‖

Next tactic:
obtain rfl : x = 0 := Subsingleton.elim _ _
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := by
  obtain rfl : x = 0 := Subsingleton.elim _ _
  

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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
⊢ ‖f 0‖ = ‖f‖

Next tactic:
refine' le_antisymm (by simpa using f.le_op_norm 0) _
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := by
  obtain rfl : x = 0 := Subsingleton.elim _ _
  refine' le_antisymm (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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
⊢ ‖f 0‖ ≤ ‖f‖

Next tactic:
simpa using f.le_op_norm 0
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := by
  obtain rfl : x = 0 := Subsingleton.elim _ _
  refine' le_antisymm (by simpa using f.le_op_norm 0) _
  

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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
⊢ ‖f‖ ≤ ‖f 0‖

Next tactic:
have : ‖ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0‖ ≤ ‖f.uncurry0‖ :=
    ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) fun m => by
      simp [-ContinuousMultilinearMap.apply_zero_curry0]
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := by
  obtain rfl : x = 0 := Subsingleton.elim _ _
  refine' le_antisymm (by simpa using f.le_op_norm 0) _
  have : ‖ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0‖ ≤ ‖f.uncurry0‖ :=
    ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) fun m => 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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
m : Fin 0 → G
⊢ ‖(curry0 𝕜 G (uncurry0 f)) m‖ ≤ ‖uncurry0 f‖ * ∏ i : Fin 0, ‖m i‖

Next tactic:
simp [-ContinuousMultilinearMap.apply_zero_curry0]
@[simp]
theorem ContinuousMultilinearMap.fin0_apply_norm (f : G[×0]→L[𝕜] G') {x : Fin 0 → G} :
    ‖f x‖ = ‖f‖ := by
  obtain rfl : x = 0 := Subsingleton.elim _ _
  refine' le_antisymm (by simpa using f.le_op_norm 0) _
  have : ‖ContinuousMultilinearMap.curry0 𝕜 G f.uncurry0‖ ≤ ‖f.uncurry0‖ :=
    ContinuousMultilinearMap.op_norm_le_bound _ (norm_nonneg _) fun m => by
      simp [-ContinuousMultilinearMap.apply_zero_curry0]
  

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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
this : ‖curry0 𝕜 G (uncurry0 f)‖ ≤ ‖uncurry0 f‖
⊢ ‖f‖ ≤ ‖f 0‖

Next tactic:
simpa [-Matrix.zero_empty] using this
theorem ContinuousMultilinearMap.uncurry0_norm (f : G[×0]→L[𝕜] G') : ‖f.uncurry0‖ = ‖f‖ := 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'
f : ContinuousMultilinearMap 𝕜 (fun i => G) G'
⊢ ‖uncurry0 f‖ = ‖f‖

Next tactic:
simp
@[simp]
theorem norm_domDomCongr (σ : ι ≃ ι') (f : ContinuousMultilinearMap 𝕜 (fun _ : ι => G) G') :
    ‖domDomCongr σ f‖ = ‖f‖ := 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'
σ : ι ≃ ι'
f : ContinuousMultilinearMap 𝕜 (fun x => G) G'
⊢ ‖domDomCongr σ f‖ = ‖f‖

Next tactic:
simp only [norm_def, LinearEquiv.coe_mk, ← σ.prod_comp,
    (σ.arrowCongr (Equiv.refl G)).surjective.forall, domDomCongr_apply, Equiv.arrowCongr_apply,
    Equiv.coe_refl, comp.left_id, comp_apply, Equiv.symm_apply_apply, id]