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Simplify

This commit is contained in:
Stefan Kebekus 2024-06-05 09:06:25 +02:00
parent 7741447426
commit 74d9636aa9
2 changed files with 24 additions and 64 deletions
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58
Nevanlinna/laplace.lean
View File

@ -141,55 +141,11 @@ theorem laplace_add_ContDiffAt
unfold Complex.laplace
simp
have hf₁ : ∀ z ∈ s, DifferentiableAt ℝ f₁ z := by
intro z hz
convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
apply ContDiffOn.differentiableOn h₁ one_le_two
have hf₂ : ∀ z ∈ s, DifferentiableAt ℝ f₂ z := by
intro z hz
convert DifferentiableOn.differentiableAt _ (IsOpen.mem_nhds hs hz)
apply ContDiffOn.differentiableOn h₂ one_le_two
have : partialDeriv ℝ 1 (f₁ + f₂) =ᶠ[nhds x] (partialDeriv ℝ 1 f₁) + (partialDeriv ℝ 1 f₂) := by
apply Filter.eventuallyEq_iff_exists_mem.2
use s
constructor
· exact IsOpen.mem_nhds hs hx
· intro z hz
apply partialDeriv_add₂_differentiableAt
exact hf₁ z hz
exact hf₂ z hz
rw [partialDeriv_eventuallyEq ℝ this]
have t₁ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₁) x := by
let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt ℝ B₀ 1
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
have t₂ : DifferentiableAt ℝ (partialDeriv ℝ 1 f₂) x := by
let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt ℝ B₀ 1
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
rw [partialDeriv_add₂_differentiableAt ℝ t₁ t₂]
have : partialDeriv ℝ Complex.I (f₁ + f₂) =ᶠ[nhds x] (partialDeriv ℝ Complex.I f₁) + (partialDeriv ℝ Complex.I f₂) := by
apply Filter.eventuallyEq_iff_exists_mem.2
use s
constructor
· exact IsOpen.mem_nhds hs hx
· intro z hz
apply partialDeriv_add₂_differentiableAt
exact hf₁ z hz
exact hf₂ z hz
rw [partialDeriv_eventuallyEq ℝ this]
have t₃ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₁) x := by
let B₀ := (h₁ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt ℝ B₀ Complex.I
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
have t₄ : DifferentiableAt ℝ (partialDeriv ℝ Complex.I f₂) x := by
let B₀ := (h₂ x hx).contDiffAt (IsOpen.mem_nhds hs hx)
let A₀ := partialDeriv_contDiffAt ℝ B₀ Complex.I
exact A₀.differentiableAt (Submonoid.oneLE.proof_2 ℕ∞)
rw [partialDeriv_add₂_differentiableAt ℝ t₃ t₄]
have h₁₁ : ContDiffAt ℝ 1 f₁ x := h₁.of_le one_le_two
have h₂₁ : ContDiffAt ℝ 1 f₂ x := h₂.of_le one_le_two
repeat
rw [partialDeriv_eventuallyEq ℝ (partialDeriv_add₂_contDiffAt ℝ h₁₁ h₂₁)]
rw [partialDeriv_add₂_differentiableAt]
-- I am super confused at this point because the tactic 'ring' does not work.
-- I do not understand why. So, I need to do things by hand.
@ -201,6 +157,10 @@ theorem laplace_add_ContDiffAt
rw [add_right_inj (partialDeriv ℝ Complex.I (partialDeriv ℝ Complex.I f₁) x)]
rw [add_comm]
repeat
apply fun v ↦ (partialDeriv_contDiffAt ℝ h₁ v).differentiableAt le_rfl
apply fun v ↦ (partialDeriv_contDiffAt ℝ h₂ v).differentiableAt le_rfl
theorem laplace_smul {f : ℂ → F} : ∀ v : ℝ, Δ (v • f) = v • (Δ f) := by
intro v

4
Nevanlinna/partialDeriv.lean
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@ -66,9 +66,9 @@ theorem partialDeriv_smul₂ {f : E → F} {a : 𝕜} {v : E} : partialDeriv
simp
theorem partialDeriv_add₂ {f₁ f₂ : E → F} {v : E} (h₁ : Differentiable 𝕜 f₁) (h₂ : Differentiable 𝕜 f₂) : partialDeriv 𝕜 v (f₁ + f₂) = (partialDeriv 𝕜 v f₁) + (partialDeriv 𝕜 v f₂) := by
theorem partialDeriv_add₂ {f₁ f₂ : E → F} (h₁ : Differentiable 𝕜 f₁) (h₂ : Differentiable 𝕜 f₂) : ∀ v : E, partialDeriv 𝕜 v (f₁ + f₂) = (partialDeriv 𝕜 v f₁) + (partialDeriv 𝕜 v f₂) := by
unfold partialDeriv
intro v
have : f₁ + f₂ = fun y ↦ f₁ y + f₂ y := by rfl
rw [this]
conv =>