awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
⊢ ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)

Tactic:
have J0U : CMDiffAt[(extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s] (minSmoothness 𝕜 2)
      (T% U') (extChartAt I x x) :=
    contMDiffWithinAt_mpullbackWithin_extChartAt_symm hU hs hx A
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
⊢ ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)

Tactic:
have J0V : CMDiffAt[(extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s] (minSmoothness 𝕜 2)
      (T% V') (extChartAt I x x) :=
    contMDiffWithinAt_mpullbackWithin_extChartAt_symm hV hs hx A
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
⊢ ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)

Tactic:
have J0W : CMDiffAt[(extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s] (minSmoothness 𝕜 2)
      (T% W') (extChartAt I x x) :=
    contMDiffWithinAt_mpullbackWithin_extChartAt_symm hW hs hx A
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
⊢ minSmoothness 𝕜 2 ≠ ∞

Tactic:
simp
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
⊢ ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
      Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) y)

Tactic:
have J1U : ∀ᶠ y in 𝓝[s] x, CMDiffAt[(extChartAt I x).target ∩ (extChartAt I x).symm ⁻¹' s]
      (minSmoothness 𝕜 2) (T% U') (extChartAt I x y) :=
    eventually_contMDiffWithinAt_mpullbackWithin_extChartAt_symm hU hs hx A (by simp)
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
      Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) y)
⊢ minSmoothness 𝕜 2 ≠ ∞

Tactic:
simp
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
s : Set M
x : M
hs : UniqueMDiffOn I s
h's : x ∈ closure[inst_4] (interior s)
hx : x ∈ s
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) s x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) s x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) s x
A : minSmoothness 𝕜 2 + 1 ≤ minSmoothness 𝕜 3
s_inter_mem : s ∩ (extChartAt I x).source ∈ 𝓝[s] x
pre_mem : (↑(extChartAt I x) : M → E) ⁻¹' ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ∈ 𝓝[s] x
U' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) U (range (↑I : H → E))
V' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) V (range (↑I : H → E))
W' : (x : E) → TangentSpace 𝓘(𝕜, E) x :=
  mpullbackWithin 𝓘(𝕜, E) I (↑(extChartAt I x).symm : E → M) W (range (↑I : H → E))
J0U :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0V :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J0W :
  ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
    Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, W' x⟩)
    ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) x)
J1U :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
      Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, U' x⟩)
      ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) y)
J1V :
  ∀ᶠ (y : M) in 𝓝[s] x,
    ContMDiffWithinAt 𝓘(𝕜, E) (ModelWithCorners.prod 𝓘(𝕜, E) 𝓘(𝕜, E)) (M' :=
      Bundle.TotalSpace (B := E) E (TangentSpace 𝓘(𝕜, E))) (minSmoothness 𝕜 2) (fun (x : E) ↦ ⟨x, V' x⟩)
      ((extChartAt I x).target ∩ (↑(extChartAt I x).symm : E → M) ⁻¹' s) ((↑(extChartAt I x) : M → E) y)
⊢ minSmoothness 𝕜 2 ≠ ∞

Tactic:
simp
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
x : M
hU :
  ContMDiffAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) x
hV :
  ContMDiffAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) x
hW :
  ContMDiffAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) x
simp :
  ∀
    (hU :
      ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
        (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) univ x)
    (hV :
      ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
        (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) univ x)
    (hW :
      ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
        (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) univ x),
    mlieBracketWithin I U (mlieBracketWithin I V W univ) univ x =
      mlieBracketWithin I (mlieBracketWithin I U V univ) W univ x +
        mlieBracketWithin I V (mlieBracketWithin I U W univ) univ x
⊢ mlieBracket I U (mlieBracket I V W) x = mlieBracket I (mlieBracket I U V) W x + mlieBracket I V (mlieBracket I U W) x

Tactic:
simp only [← mlieBracketWithin_univ, ← contMDiffWithinAt_univ] at hU hV hW ⊢

Premises used: VectorField.mlieBracketWithin_univ, contMDiffWithinAt_univ
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
x : M
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) univ x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) univ x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) univ x
⊢ Membership.mem (γ := Set M) (closure[inst_4] (interior univ)) x

Tactic:
simp
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
x : M
hU :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, U x⟩) univ x
hV :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, V x⟩) univ x
hW :
  ContMDiffWithinAt I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I))
    (minSmoothness 𝕜 2) (fun (x : M) ↦ ⟨x, W x⟩) univ x
⊢ mlieBracketWithin I U (mlieBracketWithin I V W univ) univ x =
    mlieBracketWithin I (mlieBracketWithin I U V univ) W univ x +
      mlieBracketWithin I V (mlieBracketWithin I U W univ) univ x

Tactic:
exact leibniz_identity_mlieBracketWithin_apply uniqueMDiffOn_univ (by simp) (mem_univ _) hU hV hW

Premises used: uniqueMDiffOn_univ, VectorField.leibniz_identity_mlieBracketWithin_apply, Set.mem_univ
-- Module: Mathlib.Geometry.Manifold.VectorField.LieBracket
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
H : Type u_2
inst_1 : TopologicalSpace H
E : Type u_3
inst_2 : NormedAddCommGroup E
inst_3 : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace H M
inst_6 : IsManifold I (minSmoothness 𝕜 3) M
inst_7 : CompleteSpace E
U V W : (x : M) → TangentSpace I x
hU :
  ContMDiff I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I)) (minSmoothness 𝕜 2)
    fun (x : M) ↦ ⟨x, U x⟩
hV :
  ContMDiff I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I)) (minSmoothness 𝕜 2)
    fun (x : M) ↦ ⟨x, V x⟩
hW :
  ContMDiff I (I.prod 𝓘(𝕜, E)) (M' := Bundle.TotalSpace.{u_4, u_3, u_3} (B := M) E (TangentSpace I)) (minSmoothness 𝕜 2)
    fun (x : M) ↦ ⟨x, W x⟩
h :
  ∀ (x : M),
    mlieBracket I U (mlieBracket I V W) x =
      (mlieBracket I (mlieBracket I U V) W + mlieBracket I V (mlieBracket I U W)) x
⊢ mlieBracket I U (mlieBracket I V W) = mlieBracket I (mlieBracket I U V) W + mlieBracket I V (mlieBracket I U W)

Tactic:
ext x