Simplify

Nevanlinna/laplace.lean

@@ -33,10 +33,10 @@ theorem laplace_eventuallyEq {f₁ f₂ : ℂ → F} {x : ℂ} (h : f₁ =ᶠ[nh
   rw [partialDeriv_eventuallyEq ℝ (partialDeriv_eventuallyEq' ℝ h Complex.I) Complex.I]
 
 
-theorem laplace_add 
+theorem laplace_add
   {f₁ f₂ : ℂ → F}
   (h₁ : ContDiff ℝ 2 f₁)
-  (h₂ : ContDiff ℝ 2 f₂) : 
+  (h₂ : ContDiff ℝ 2 f₂) :
   Δ (f₁ + f₂) = (Δ f₁) + (Δ f₂) := by
 
   unfold Complex.laplace
@@ -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
@@ -257,13 +217,13 @@ theorem laplace_compCLMAt {f : ℂ → F} {l : F →L[ℝ] G} {x : ℂ} (h : Con
   apply ContDiffAt.differentiableAt (partialDeriv_contDiffAt ℝ (ContDiffOn.contDiffAt hv₄ hv₁) 1) le_rfl
 
 
-theorem laplace_compCLM 
-  {f : ℂ → F} 
-  {l : F →L[ℝ] G} 
+theorem laplace_compCLM
+  {f : ℂ → F}
+  {l : F →L[ℝ] G}
   (h : ContDiff ℝ 2 f) :
   Δ (l ∘ f) = l ∘ (Δ f) := by
   funext z
-  exact laplace_compCLMAt h.contDiffAt 
+  exact laplace_compCLMAt h.contDiffAt
 
 
 theorem laplace_compCLE {f : ℂ → F} {l : F ≃L[ℝ] G} :

Nevanlinna/partialDeriv.lean

@@ -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 =>
@@ -110,16 +110,16 @@ theorem partialDeriv_add₂_contDiffAt
   · exact Filter.inter_mem hu₁ hu₂
   · intro x hx
     simp
-    apply partialDeriv_add₂_differentiableAt 𝕜 
+    apply partialDeriv_add₂_differentiableAt 𝕜
     exact (hf₁' x (Set.mem_of_mem_inter_left hx)).differentiableAt
     exact (hf₂' x (Set.mem_of_mem_inter_right hx)).differentiableAt
-  
 
-theorem partialDeriv_compContLin 
-  {f : E → F} 
-  {l : F →L[𝕜] G} 
-  {v : E} 
-  (h : Differentiable 𝕜 f) : 
+
+theorem partialDeriv_compContLin
+  {f : E → F}
+  {l : F →L[𝕜] G}
+  {v : E}
+  (h : Differentiable 𝕜 f) :
   partialDeriv 𝕜 v (l ∘ f) = l ∘ partialDeriv 𝕜 v f := by
   unfold partialDeriv