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Added ederiv properties #15

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Added ederiv properties #15

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12 changes: 12 additions & 0 deletions DeRhamCohomology/DifferentialForm.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -55,6 +55,18 @@ def constOfIsEmpty (x : F) : Ω^0⟮E, F⟯ :=
def ederiv (ω : Ω^n⟮E, F⟯) : Ω^n + 1⟮E, F⟯ :=
fun x ↦ .uncurryFin (fderiv ℝ ω x)

theorem ederiv_add (ω₁ ω₂ : Ω^n⟮E, F⟯) {x : E} (hω₁ : DifferentiableAt ℝ ω₁ x)
(hω₂ : DifferentiableAt ℝ ω₂ x) : ederiv (ω₁ + ω₂) x = ederiv ω₁ x + ederiv ω₂ x := by
simp [ederiv, fderiv_add' hω₁ hω₂, uncurryFin_add]

theorem ederiv_smul (ω : Ω^n⟮E, F⟯) (c : ℝ) {x : E} (hω : DifferentiableAt ℝ ω x):
ederiv (c • ω) x = c • ederiv ω x := by
simp [ederiv, fderiv_const_smul' hω, uncurryFin_smul]

theorem ederiv_constOfIsEmpty (x : E) (y : F) :
ederiv (constOfIsEmpty y) x = .uncurryFin (fderiv ℝ (constOfIsEmpty y) x) :=
rfl

theorem Filter.EventuallyEq.ederiv_eq {ω₁ ω₂ : Ω^n⟮E, F⟯} {x : E}
(h : ω₁ =ᶠ[𝓝 x] ω₂) : ederiv ω₁ x = ederiv ω₂ x := by
ext v
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