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Define partial over arbitrary fields

This commit is contained in:
Stefan Kebekus 2024-05-08 15:59:56 +02:00
parent 69a35228ee
commit 9579da6e39
3 changed files with 40 additions and 36 deletions
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4
Nevanlinna/cauchyRiemann.lean
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@ -45,9 +45,9 @@ theorem CauchyRiemann₃ : (DifferentiableAt ℂ f z)
theorem CauchyRiemann₄ {f : ℂ → ℂ} : (Differentiable ℂ f) theorem CauchyRiemann₄ {f : ℂ → ℂ} : (Differentiable ℂ f)
→ Real.partialDeriv Complex.I f = Complex.I • Real.partialDeriv 1 f := by → partialDeriv ℝ Complex.I f = Complex.I • partialDeriv ℝ 1 f := by
intro h intro h
unfold Real.partialDeriv unfold partialDeriv
conv => conv =>
left left

16
Nevanlinna/complexHarmonic.lean
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@ -17,10 +17,10 @@ import Nevanlinna.partialDeriv
noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by noncomputable def Complex.laplace : (ℂ → ℂ) → (ℂ → ℂ) := by
intro f intro f
let fx := Real.partialDeriv 1 f let fx := partialDeriv ℝ 1 f
let fxx := Real.partialDeriv 1 fx let fxx := partialDeriv ℝ 1 fx
let fy := Real.partialDeriv Complex.I f let fy := partialDeriv ℝ Complex.I f
let fyy := Real.partialDeriv Complex.I fy let fyy := partialDeriv ℝ Complex.I fy
exact fxx + fyy exact fxx + fyy
@ -35,8 +35,8 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
have f_is_real_C2 : ContDiff ℝ 2 f := have f_is_real_C2 : ContDiff ℝ 2 f :=
ContDiff.restrict_scalars ℝ (Differentiable.contDiff h) ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
have fI_is_real_differentiable : Differentiable ℝ (Real.partialDeriv 1 f) := by have fI_is_real_differentiable : Differentiable ℝ (partialDeriv ℝ 1 f) := by
exact (partialDeriv_contDiff f_is_real_C2 1).differentiable (Submonoid.oneLE.proof_2 ℕ∞) exact (partialDeriv_contDiff ℝ f_is_real_C2 1).differentiable (Submonoid.oneLE.proof_2 ℕ∞)
constructor constructor
· -- f is two times real continuously differentiable · -- f is two times real continuously differentiable
@ -49,7 +49,7 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
-- This lemma says that partial derivatives commute with complex scalar -- This lemma says that partial derivatives commute with complex scalar
-- multiplication. This is a consequence of partialDeriv_compContLin once we -- multiplication. This is a consequence of partialDeriv_compContLin once we
-- note that complex scalar multiplication is continuous ℝ-linear. -- note that complex scalar multiplication is continuous ℝ-linear.
have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → ℂ, Differentiable ℝ g → Real.partialDeriv v (s • g) = s • (Real.partialDeriv v g) := by have : ∀ v, ∀ s : ℂ, ∀ g : ℂ → ℂ, Differentiable ℝ g → partialDeriv ℝ v (s • g) = s • (partialDeriv ℝ v g) := by
intro v s g hg intro v s g hg
-- Present scalar multiplication as a continuous ℝ-linear map. This is -- Present scalar multiplication as a continuous ℝ-linear map. This is
@ -71,7 +71,7 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
have : s • g = sMuls ∘ g := by rfl have : s • g = sMuls ∘ g := by rfl
rw [this] rw [this]
rw [partialDeriv_compContLin hg] rw [partialDeriv_compContLin ℝ hg]
rfl rfl
rw [this] rw [this]

56
Nevanlinna/partialDeriv.lean
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@ -10,31 +10,33 @@ import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Symmetric import Mathlib.Analysis.Calculus.FDeriv.Symmetric
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
variable (𝕜)
noncomputable def Real.partialDeriv : E → (E → F) → (E → F) :=
fun v ↦ (fun f ↦ (fun w ↦ fderiv ℝ f w v))
theorem partialDeriv_smul₁ {f : E → F} {a : ℝ} {v : E} : Real.partialDeriv (a • v) f = a • Real.partialDeriv v f := by noncomputable def partialDeriv : E → (E → F) → (E → F) :=
unfold Real.partialDeriv fun v ↦ (fun f ↦ (fun w ↦ fderiv 𝕜 f w v))
theorem partialDeriv_smul₁ {f : E → F} {a : 𝕜} {v : E} : partialDeriv 𝕜 (a • v) f = a • partialDeriv 𝕜 v f := by
unfold partialDeriv
conv => conv =>
left left
intro w intro w
rw [map_smul] rw [map_smul]
theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : Real.partialDeriv (v₁ + v₂) f = (Real.partialDeriv v₁ f) + (Real.partialDeriv v₂ f) := by theorem partialDeriv_add₁ {f : E → F} {v₁ v₂ : E} : partialDeriv 𝕜 (v₁ + v₂) f = (partialDeriv 𝕜 v₁ f) + (partialDeriv 𝕜 v₂ f) := by
unfold Real.partialDeriv unfold partialDeriv
conv => conv =>
left left
intro w intro w
rw [map_add] rw [map_add]
theorem partialDeriv_smul₂ {f : E → F} {a : ℝ} {v : E} (h : Differentiable ℝ f) : Real.partialDeriv v (a • f) = a • Real.partialDeriv v f := by theorem partialDeriv_smul₂ {f : E → F} {a : 𝕜} {v : E} (h : Differentiable 𝕜 f) : partialDeriv 𝕜 v (a • f) = a • partialDeriv 𝕜 v f := by
unfold Real.partialDeriv unfold partialDeriv
have : a • f = fun y ↦ a • f y := by rfl have : a • f = fun y ↦ a • f y := by rfl
rw [this] rw [this]
@ -45,8 +47,8 @@ theorem partialDeriv_smul₂ {f : E → F} {a : ℝ} {v : E} (h : Differentiable
rw [fderiv_const_smul (h w)] rw [fderiv_const_smul (h w)]
theorem partialDeriv_add₂ {f₁ f₂ : E → F} {v : E} (h₁ : Differentiable ℝ f₁) (h₂ : Differentiable ℝ f₂) : Real.partialDeriv v (f₁ + f₂) = (Real.partialDeriv v f₁) + (Real.partialDeriv v f₂) := by 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
unfold Real.partialDeriv unfold partialDeriv
have : f₁ + f₂ = fun y ↦ f₁ y + f₂ y := by rfl have : f₁ + f₂ = fun y ↦ f₁ y + f₂ y := by rfl
rw [this] rw [this]
@ -57,8 +59,8 @@ theorem partialDeriv_add₂ {f₁ f₂ : E → F} {v : E} (h₁ : Differentiable
rw [fderiv_add (h₁ w) (h₂ w)] rw [fderiv_add (h₁ w) (h₂ w)]
theorem partialDeriv_compContLin {f : E → F} {l : F →L[ℝ] F} {v : E} (h : Differentiable ℝ f) : Real.partialDeriv v (l ∘ f) = l ∘ Real.partialDeriv v f := by theorem partialDeriv_compContLin {f : E → F} {l : F →L[𝕜] F} {v : E} (h : Differentiable 𝕜 f) : partialDeriv 𝕜 v (l ∘ f) = l ∘ partialDeriv 𝕜 v f := by
unfold Real.partialDeriv unfold partialDeriv
conv => conv =>
left left
@ -69,14 +71,14 @@ theorem partialDeriv_compContLin {f : E → F} {l : F →L[ℝ] F} {v : E} (h :
rfl rfl
theorem partialDeriv_contDiff {n : ℕ} {f : E → F} (h : ContDiff ℝ (n + 1) f) : ∀ v : E, ContDiff ℝ n (Real.partialDeriv v f) := by theorem partialDeriv_contDiff {n : ℕ} {f : E → F} (h : ContDiff 𝕜 (n + 1) f) : ∀ v : E, ContDiff 𝕜 n (partialDeriv 𝕜 v f) := by
unfold Real.partialDeriv unfold partialDeriv
intro v intro v
let A := (contDiff_succ_iff_fderiv.1 h).right let A := (contDiff_succ_iff_fderiv.1 h).right
simp at A simp at A
have : (fun w => (fderiv ℝ f w) v) = (fun f => f v) ∘ (fun w => (fderiv ℝ f w)) := by have : (fun w => (fderiv 𝕜 f w) v) = (fun f => f v) ∘ (fun w => (fderiv 𝕜 f w)) := by
rfl rfl
rw [this] rw [this]
@ -87,18 +89,21 @@ theorem partialDeriv_contDiff {n : ℕ} {f : E → F} (h : ContDiff ℝ (n + 1)
exact contDiff_const exact contDiff_const
lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff ℝ 2 f) (z a b : E) : lemma partialDeriv_fderiv {f : E → F} (hf : ContDiff 𝕜 2 f) (z a b : E) :
fderiv ℝ (fderiv ℝ f) z b a = Real.partialDeriv b (Real.partialDeriv a f) z := by fderiv 𝕜 (fderiv 𝕜 f) z b a = partialDeriv 𝕜 b (partialDeriv 𝕜 a f) z := by
unfold Real.partialDeriv unfold partialDeriv
rw [fderiv_clm_apply] rw [fderiv_clm_apply]
· simp · simp
· exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z · exact (contDiff_succ_iff_fderiv.1 hf).2.differentiable le_rfl z
· simp · simp
theorem partialDeriv_comm {f : E → F} (h : ContDiff ℝ 2 f) : theorem partialDeriv_comm
∀ v₁ v₂ : E, Real.partialDeriv v₁ (Real.partialDeriv v₂ f) = Real.partialDeriv v₂ (Real.partialDeriv v₁ f) := by {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
{f : E → F} (h : ContDiff ℝ 2 f) :
∀ v₁ v₂ : E, partialDeriv ℝ v₁ (partialDeriv ℝ v₂ f) = partialDeriv ℝ v₂ (partialDeriv ℝ v₁ f) := by
intro v₁ v₂ intro v₁ v₂
funext z funext z
@ -118,7 +123,6 @@ theorem partialDeriv_comm {f : E → F} (h : ContDiff ℝ 2 f) :
apply second_derivative_symmetric h₀ h₁ v₁ v₂ apply second_derivative_symmetric h₀ h₁ v₁ v₂
rw [← partialDeriv_fderiv ℝ h z v₂ v₁]
rw [← partialDeriv_fderiv h z v₂ v₁]
rw [derivSymm] rw [derivSymm]
rw [partialDeriv_fderiv h z v₁ v₂] rw [partialDeriv_fderiv ℝ h z v₁ v₂]