awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
i : ι
hi : i ∈ s
⊢ f i x • (g x • cov i σ x + ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)) =
    g x • f i x • cov i σ x +
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
        (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
have a : f i x • g x • cov i σ x = g x • f i x • cov i σ x
Tactic: simp [mvfderiv]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
⊢ ∑ i ∈ s,
      (g x • f i x • cov i σ x +
        HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
          (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
        ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
have rw : HAdd.hAdd (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
    (∑ x_1 ∈ s, g x • f x_1 x • cov x_1 σ x)
    (∑ x_1 ∈ s,
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f x_1 x)
        (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
  HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
      (∑ i ∈ s, f i x • cov i σ x) +
    HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
Tactic: rw [Finset.sum_add_distrib, Finset.smul_sum, Finset.sum_apply, Finset.sum_smul]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
rw :
  HAdd.hAdd (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      (∑ x_1 ∈ s, g x • f x_1 x • cov x_1 σ x)
      (∑ x_1 ∈ s,
        HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f x_1 x)
          (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
        ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
⊢ ∑ i ∈ s,
      (g x • f i x • cov i σ x +
        HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
          (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
        ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
have rw₁ : HAdd.hAdd (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
    (∑ x_1 ∈ s, g x • f x_1 x • cov x_1 σ x)
    (∑ x_1 ∈ s,
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f x_1 x)
        (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
  ∑ x_1 ∈ s, g x • f x_1 x • cov x_1 σ x +
    HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
Tactic: rw [Finset.sum_add_distrib, Finset.smul_sum, Finset.sum_apply, Finset.sum_smul]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
⊢ HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
        ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)) =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
have rw : HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
      (∑ i ∈ s, f i x • cov i σ x) +
    HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (1 x)
      (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)) =
  HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
      (∑ i ∈ s, f i x • cov i σ x) +
    ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
Tactic: rw [hf]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
⊢ ∑ i ∈ s, f i x • cov i (g • σ) x =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
have calc : Eq (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
  (∑ i ∈ s, f i x • cov i (g • σ) x)
  (∑ i ∈ s,
    (g x • f i x • cov i σ x +
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
        (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))))
Tactic: calc ∑ i ∈ s, f i x • cov i (g • σ) x
      _ = ∑ i ∈ s, (g x • f i x • cov i σ x + f i x • (d% g x).smulRight (σ x)) := by
          congr! 1 with i hi
          rw [(h i).leibniz hσ hg]
          simp [mvfderiv]
          module
      _ = g x • ∑ i ∈ s, f i x • cov i σ x + (∑ i ∈ s, f i) x • (d% g x).smulRight (σ x) := by
          rw [Finset.sum_add_distrib, Finset.smul_sum, Finset.sum_apply, Finset.sum_smul]
      _ = g x • ∑ i ∈ s, f i x • cov i σ x + (d% g x).smulRight (σ x) := by rw [hf]; simp
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
ι : Type u_7
s : Finset ι
u : Set M
cov : ι → ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
h : ∀ (i : ι), IsCovariantDerivativeOn F (V := V) (cov i) u
f : ι → M → 𝕜
hf : Eq (α := M → 𝕜) (∑ i ∈ s, f i) 1
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ u
calc :
  Eq (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
    (∑ i ∈ s, f i x • cov i (g • σ) x)
    (∑ i ∈ s,
      (g x • f i x • cov i σ x +
        HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
          (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))))
⊢ ∑ i ∈ s, f i x • cov i (g • σ) x =
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        (∑ i ∈ s, f i x • cov i σ x) +
      ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
have calc₁ : ∑ i ∈ s,
    (g x • f i x • cov i σ x +
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (f i x)
        (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))) =
  HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
      (∑ i ∈ s, f i x • cov i σ x) +
    HSMul.hSMul (α := 𝕜) (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      ((∑ i ∈ s, f i) x) (ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x))
Tactic: calc ∑ i ∈ s, f i x • cov i (g • σ) x
      _ = ∑ i ∈ s, (g x • f i x • cov i σ x + f i x • (d% g x).smulRight (σ x)) := by
          congr! 1 with i hi
          rw [(h i).leibniz hσ hg]
          simp [mvfderiv]
          module
      _ = g x • ∑ i ∈ s, f i x • cov i σ x + (∑ i ∈ s, f i) x • (d% g x).smulRight (σ x) := by
          rw [Finset.sum_add_distrib, Finset.smul_sum, Finset.sum_apply, Finset.sum_smul]
      _ = g x • ∑ i ∈ s, f i x • cov i σ x + (d% g x).smulRight (σ x) := by rw [hf]; simp
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
s : Set M
cov : ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
hcov : IsCovariantDerivativeOn F (V := V) cov s
A : (x : M) → V x →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
σ σ' : (x : M) → V x
x : M
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
hx : x ∈ s
⊢ HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov (σ + σ') x)
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) ((σ + σ') x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) =
    HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov σ x)
        ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
          ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) +
      HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov σ' x)
        ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ' x) :
          ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
have simp : cov σ x + cov σ' x +
    HAdd.hAdd (α := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (β :=
      ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ' x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) =
  HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov σ x)
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) +
    HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov σ' x)
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ' x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x))
Tactic: simp [hcov.add hσ hσ']
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
s : Set M
cov : ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
hcov : IsCovariantDerivativeOn F (V := V) cov s
A : (x : M) → V x →L[𝕜] ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
σ : (x : M) → V x
g : M → 𝕜
x : M
hσ : MDiffAt (T% σ) x
hg : MDiffAt g x
hx : x ∈ s
⊢ HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov (g • σ) x)
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) ((g • σ) x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) =
    g x •
        HAdd.hAdd (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (cov σ x)
          ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
            ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) +
      ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
have simp : g x • cov σ x + ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x) +
    HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
      ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
        ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) =
  g x • cov σ x +
      HSMul.hSMul (β := ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) (g x)
        ((A x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) :
          ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)) +
    ContinuousLinearMap.smulRight (S := 𝕜) (d% g x) (σ x)
Tactic: simp [hcov.leibniz hσ hg]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
cov cov' : ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
s : Set M
hcov : IsCovariantDerivativeOn F (V := V) cov s
hcov' : IsCovariantDerivativeOn F (V := V) cov' s
x : M
hx : x ∈ s
f : M → 𝕜
σ : (x : M) → V x
hf : MDiffAt f x
hσ : MDiffAt (T% σ) x
⊢ differenceAux (V := V) cov cov' (f • σ) x = f x • differenceAux (V := V) cov cov' σ x
have simp : f x • cov σ x - f x • cov' σ x = f x • (cov σ x - cov' σ x)
Tactic: simp [differenceAux, hcov.leibniz hσ hf, hcov'.leibniz hσ hf]
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
cov cov' : ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
s : Set M
hcov : IsCovariantDerivativeOn F (V := V) cov s
hcov' : IsCovariantDerivativeOn F (V := V) cov' s
x : M
hx : x ∈ s
σ σ' : (x : M) → V x
hσ : MDiffAt (T% σ) x
hσ' : MDiffAt (T% σ') x
⊢ differenceAux (V := V) cov cov' (σ + σ') x =
    differenceAux (V := V) cov cov' σ x + differenceAux (V := V) cov cov' σ' x
have simp : cov σ x + cov σ' x - (cov' σ x + cov' σ' x) = cov σ x - cov' σ x + (cov σ' x - cov' σ' x)
Tactic: simp [differenceAux, hcov.add hσ hσ', hcov'.add hσ hσ']
-- Lean 4 proof step from Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic (goal)
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
H : Type u_3
inst_3 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
F : Type u_5
inst_6 : NormedAddCommGroup F
inst_7 : NormedSpace 𝕜 F
V : M → Type u_6
inst_8 : TopologicalSpace (TotalSpace F V)
inst_9 : (x : M) → AddCommGroup (V x)
inst_10 : (x : M) → Module.{u_1, u_6} 𝕜 (V x)
inst_11 : (x : M) → TopologicalSpace (V x)
inst_12 : ∀ (x : M), IsTopologicalAddGroup.{u_6} (V x)
inst_13 : ∀ (x : M), ContinuousSMul.{u_1, u_6} 𝕜 (V x)
inst_14 : FiberBundle F V
cov cov' : ((x : M) → V x) → (x : M) → ContinuousLinearMap.{u_1, u_1, u_2, u_6} (RingHom.id 𝕜) (TangentSpace I x) (V x)
s : Set M
hcov : IsCovariantDerivativeOn F (V := V) cov s
hcov' : IsCovariantDerivativeOn F (V := V) cov' s
inst_15 : CompleteSpace 𝕜
inst_16 : FiniteDimensional 𝕜 F
inst_17 : VectorBundle 𝕜 F V
inst_18 : ContMDiffVectorBundle 1 F V I
x : M
hx : x ∈ s
σ : (x : M) → V x
hσ : MDiffAt (T% σ) x
⊢ (hcov.difference (V := V) (s := s) hcov' x : (a : V x) → TangentSpace I x →L[𝕜] V x) (σ x) = cov σ x - cov' σ x
have simp : (TensorialAt.mkHom (V := V) (fun (x_1 : (x : M) → V x) ↦ differenceAux (V := V) cov cov' x_1 x) x sorry :
      (a : V x) → TangentSpace I x →L[𝕜] V x)
    (σ x) =
  cov σ x - cov' σ x
Tactic: simp only [difference, hx, reduceDIte]