awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
⊢ compContinuousLinearMapL (E := E) (E₁ := E₁)
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (f₁ + f₂)) =
    HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
      (compContinuousLinearMapL (E := E) (E₁ := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₁))
      (compContinuousLinearMapL (E := E) (E₁ := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₂))
have h : ∀ (g : ContinuousMultilinearMap 𝕜 E₁ G) (x : (i : ι) → E i),
  Eq.{wG + 1} (α := G)
    ((((compContinuousLinearMapL (E := E) (E₁ := E₁)
                  (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                    (f₁ + f₂)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
    ((((HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₁))
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₂)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
Tactic: ext g x
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ Eq.{wG + 1} (α := G)
    ((((compContinuousLinearMapL (E := E) (E₁ := E₁)
                  (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                    (f₁ + f₂)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
    ((((HAdd.hAdd (α := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₁))
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₂)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
have h : ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
    (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) j :
        (a : E j) → E₁ j)
      (x j)) =
  HAdd.hAdd (α := G) (β := G)
    ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
        (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ j :
            (a : E j) → E₁ j)
          (x j) :
      G)
    ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
        (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₂ j :
            (a : E j) → E₁ j)
          (x j) :
      G)
Tactic: change (g fun j ↦ update f i (f₁ + f₂) j <| x j) =
        (g fun j ↦ update f i f₁ j <| x j) + g fun j ↦ update f i f₂ j (x j)
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
      (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) j :
          (a : E j) → E₁ j)
        (x j)) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₂ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
have h : ∀ (x_1 : ι),
  (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) x_1 :
        (a : E x_1) → E₁ x_1)
      (x x_1) =
    update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
      (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i) ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
      x_1
Tactic: convert! g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i))
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
h :
  ∀ (x_1 : ι),
    (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) =
      update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
        (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i)
          ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
        x_1
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
      (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) j :
          (a : E j) → E₁ j)
        (x j)) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₂ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
have h₁ : ∀ (x_1 : ι),
  Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
Tactic: convert! g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i))
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
f₁ f₂ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
h :
  ∀ (x_1 : ι),
    (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) =
      update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
        (HAdd.hAdd (α := E₁ i) (β := E₁ i) ((f₁ : (a : E i) → E₁ i) (x i) : E₁ i)
          ((f₂ : (a : E i) → E₁ i) (x i) : E₁ i))
        x_1
h₁ :
  ∀ (x_1 : ι),
    Eq.{wE₁ + 1} (α := E₁ x_1)
      ((update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ x_1 :
            (a : E x_1) → E₁ x_1)
          (x x_1) :
        E₁ x_1)
      (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₁ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
      (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (f₁ + f₂) j :
          (a : E j) → E₁ j)
        (x j)) =
    HAdd.hAdd (α := G) (β := G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₁ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₂ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
have h₂ : ∀ (x_1 : ι),
  Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₂ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₂ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
Tactic: convert! g.map_update_add (fun j ↦ f j (x j)) i (f₁ (x i)) (f₂ (x i))
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
a : 𝕜
f₀ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
⊢ compContinuousLinearMapL (E := E) (E₁ := E₁)
      (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i (a • f₀)) =
    HSMul.hSMul (β := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G) a
      (compContinuousLinearMapL (E := E) (E₁ := E₁)
        (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i f₀))
have h : ∀ (g : ContinuousMultilinearMap 𝕜 E₁ G) (x : (i : ι) → E i),
  Eq.{wG + 1} (α := G)
    ((((compContinuousLinearMapL (E := E) (E₁ := E₁)
                  (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                    (a • f₀)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
    ((((HSMul.hSMul (β := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G) a
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₀)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
Tactic: ext g x
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
a : 𝕜
f₀ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ Eq.{wG + 1} (α := G)
    ((((compContinuousLinearMapL (E := E) (E₁ := E₁)
                  (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                    (a • f₀)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
    ((((HSMul.hSMul (β := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G) a
                  (compContinuousLinearMapL (E := E) (E₁ := E₁)
                    (update (β := fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) f i
                      f₀)) :
                (a : ContinuousMultilinearMap 𝕜 E₁ G) → ContinuousMultilinearMap 𝕜 E G)
              g :
            ContinuousMultilinearMap 𝕜 E G) :
          (a : (i : ι) → E i) → G)
        x :
      G)
have h : ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
    (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (a • f₀) j :
        (a : E j) → E₁ j)
      (x j)) =
  HSMul.hSMul (β := G) a
    ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
        (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₀ j :
            (a : E j) → E₁ j)
          (x j) :
      G)
Tactic: change (g fun j ↦ update f i (a • f₀) j <| x j) = a • g fun j ↦ update f i f₀ j (x j)
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
a : 𝕜
f₀ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
      (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (a • f₀) j :
          (a : E j) → E₁ j)
        (x j)) =
    HSMul.hSMul (β := G) a
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₀ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
have h : ∀ (x_1 : ι),
  (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (a • f₀) x_1 :
        (a : E x_1) → E₁ x_1)
      (x x_1) =
    update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
      (HSMul.hSMul (β := E₁ i) a ((f₀ : (a : E i) → E₁ i) (x i) : E₁ i)) x_1
Tactic: convert! g.map_update_smul (fun j ↦ f j (x j)) i a (f₀ (x i))
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
inst_11 : DecidableEq ι
f : (i : ι) → ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
i : ι
a : 𝕜
f₀ : ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)
g : ContinuousMultilinearMap 𝕜 E₁ G
x : (i : ι) → E i
h :
  ∀ (x_1 : ι),
    (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (a • f₀) x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) =
      update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i
        (HSMul.hSMul (β := E₁ i) a ((f₀ : (a : E i) → E₁ i) (x i) : E₁ i)) x_1
⊢ ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
      (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i (a • f₀) j :
          (a : E j) → E₁ j)
        (x j)) =
    HSMul.hSMul (β := G) a
      ((g : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
          (update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₀ j :
              (a : E j) → E₁ j)
            (x j) :
        G)
have h₁ : ∀ (x_1 : ι),
  Eq.{wE₁ + 1} (α := E₁ x_1)
    ((update (β := fun (a : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E a) (E₁ a)) f i f₀ x_1 :
          (a : E x_1) → E₁ x_1)
        (x x_1) :
      E₁ x_1)
    (update (β := E₁) (fun (j : ι) ↦ (f j : (a : E j) → E₁ j) (x j)) i ((f₀ : (a : E i) → E₁ i) (x i)) x_1 : E₁ x_1)
Tactic: convert! g.map_update_smul (fun j ↦ f j (x j)) i a (f₀ (x i))
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
f : (i : ι) → (fun (i : ι) ↦ ContinuousLinearMap.{u, u, wE, wE₁} (RingHom.id 𝕜) (E i) (E₁ i)) i
⊢ norm (E := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
      ((compContinuousLinearMapMultilinear 𝕜 E E₁ G :
            (a : (i : ι) → (fun (i : ι) ↦ E i →L[𝕜] E₁ i) i) →
              ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
          f :
        ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G) ≤
    HMul.hMul (α := ℝ) 1 (∏ i : ι, ‖f i‖)
have rw : norm (E := ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
    ((compContinuousLinearMapMultilinear 𝕜 E E₁ G :
          (a : (i : ι) → (fun (i : ι) ↦ E i →L[𝕜] E₁ i) i) →
            ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)
        f :
      ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G) ≤
  ∏ i : ι, ‖f i‖
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.Multilinear.Basic (goal)
Goal:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst : Fintype ι'
inst_1 : NontriviallyNormedField 𝕜
inst_2 : (i : ι) → SeminormedAddCommGroup (E i)
inst_3 : (i : ι) → NormedSpace.{u, wE} 𝕜 (E i)
inst_4 : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst_5 : (i : ι) → NormedSpace.{u, wE₁} 𝕜 (E₁ i)
inst_6 : SeminormedAddCommGroup G
inst_7 : NormedSpace 𝕜 G
inst_8 : SeminormedAddCommGroup G'
inst_9 : NormedSpace 𝕜 G'
inst_10 : Fintype ι
α : Type u_1
inst_11 : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : Equiv.{u_1 + 1, v + 1} α (↑s : Type v)
inst_12 : DecidablePred fun (x : ι) ↦ x ∈ s
x : (i : { a : ι // a ∉ s }) → E₁ (↑i : ι)
m : (i : α) → (i : ι) → E₁ i
⊢ norm.{wG} (E := G)
      ((((f.toMultilinearMap (M₁ := E₁).iteratedFDerivComponent (M₁ := E₁) (s := s) e :
                  (a : (i : { a : ι // a ∉ s }) → E₁ (↑i : ι)) → MultilinearMap 𝕜 (fun (x : α) ↦ (i : ι) → E₁ i) G)
                x :
              MultilinearMap 𝕜 (fun (x : α) ↦ (i : ι) → E₁ i) G) :
            (a : (i : α) → (i : ι) → E₁ i) → G)
          m :
        G) ≤
    (‖f‖ * ∏ i : { a : ι // a ∉ s }, ‖x i‖) * ∏ i : α, ‖m i‖
have simp : norm.{wG} (E := G)
    ((f : (a : (i : ι) → E₁ i) → G) fun (j : ι) ↦
        if h : j ∈ s then m ((e.symm : (a : (↑s : Type v)) → α) (Subtype.mk (p := fun (a : ι) ↦ a ∈ s) j sorry)) j
        else x (Subtype.mk (p := fun (a : ι) ↦ a ∉ s) j sorry) :
      G) ≤
  (‖f‖ * ∏ i : { a : ι // a ∉ s }, ‖x i‖) * ∏ i : α, ‖m i‖
Tactic: simp only [MultilinearMap.iteratedFDerivComponent, MultilinearMap.domDomRestrictₗ,
      MultilinearMap.coe_mk, MultilinearMap.domDomRestrict_apply, coe_coe]