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namespace OAI
open scoped BigOperators Topology
open Set Filter
namespace RapidForcing
def EqBefore {E : Type} (R : ℝ) (v w : Field E) : Prop :=
∀ t, 0 ≤ t → t < R → ∀ x, v t x = w t x
namespace EqBefore
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
{R : ℝ} {v w : Field E}
lemma spatial (h : EqBefore R v w) (i : Fin 3) : EqBefore R (spatialD i v) (spatialD i w) := by
intro t ht hR x
have he : v t = w t := funext (h t ht hR)
simp only [spatialD, he]
lemma time (h : EqBefore R v w) : EqBefore R (timeD v) (timeD w) := by
intro t ht hR x
apply Filter.EventuallyEq.derivWithin_eq_of_mem _ ht
filter_upwards [self_mem_nhdsWithin, nhdsWithin_le_nhds (Iio_mem_nhds hR)] with s hs hsR
exact h s hs hsR x
lemma mixed (h : EqBefore R v w) (l : ℕ) (α : MultiIndex) :
EqBefore R (mixedD l α v) (mixedD l α w) := by
have hsp (i : Fin 3) (n : ℕ) {a b : Field E} (hab : EqBefore R a b) :
EqBefore R ((spatialD i)^[n] a) ((spatialD i)^[n] b) := by
induction n with
| zero => exact hab
| succ n ih => simpa only [Function.iterate_succ_apply'] using ih.spatial i
unfold mixedD spatialMulti
induction l with
| zero => exact hsp 0 _ (hsp 1 _ (hsp 2 _ h))
| succ l ih => simpa only [Function.iterate_succ_apply'] using ih.time
end EqBefore
namespace EffectiveProfile
namespace Formula
noncomputable def pack (ν t : ℝ) (x : Space) : Fin 5 → ℝ := ![t, x 0, x 1, x 2, ν]
def spaceIndex (i : Fin 3) : Fin 5 := ⟨i.val + 1, by omega⟩
noncomputable def scalarField (a : Formula 5) (ν : ℝ) : Field ℝ :=
fun t x => a.value (pack ν t x)
noncomputable def vectorField (a : Fin 3 → Formula 5) (ν : ℝ) : Field Space :=
fun t x => WithLp.toLp 2 (fun i => (a i).scalarField ν t x)
lemma pack_smooth (ν : ℝ) : ContDiff ℝ (⊤ : ℕ∞) (fun p : ℝ × Space => pack ν p.1 p.2) := by
apply contDiff_pi.mpr
intro i
fin_cases i <;> dsimp [pack] <;> fun_prop
lemma scalarField_smooth (a : Formula 5) (ν : ℝ) :
ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry (a.scalarField ν)) :=
a.contDiff.comp (pack_smooth ν)
lemma vectorField_smooth (a : Fin 3 → Formula 5) (ν : ℝ) :
ContDiff ℝ (⊤ : ℕ∞) (Function.uncurry (vectorField a ν)) := by
apply (contDiff_piLp 2).mpr
intro i
exact (a i).scalarField_smooth ν
lemma pack_space_line (ν t : ℝ) (x : Space) (i : Fin 3) (s : ℝ) :
pack ν t (x + s • basis i) = pack ν t x + s • Pi.single (spaceIndex i) 1 := by
ext j
fin_cases i <;> fin_cases j <;> simp [pack, basis, spaceIndex]
lemma pack_time_line (ν t : ℝ) (x : Space) (s : ℝ) :
pack ν (t + s) x = pack ν t x + s • Pi.single (0 : Fin 5) 1 := by
ext j
fin_cases j <;> simp [pack]
lemma spatial_scalarField (a : Formula 5) (ν : ℝ) (i : Fin 3) :
spatialD i (a.scalarField ν) = (a.diff (spaceIndex i)).scalarField ν := by
funext t x
have hline : HasDerivAt (fun s : ℝ => x + s • basis i) (basis i) 0 := by
simpa using ((hasDerivAt_id (0 : ℝ)).smul_const (basis i)).const_add x
have hd : DifferentiableAt ℝ (a.scalarField ν t) x :=
((a.scalarField_smooth ν).comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp) x
have hp := a.hasDerivAt_line (spaceIndex i) (pack ν t x) 0
have he : (fun s : ℝ => a.value (pack ν t x + s • Pi.single (spaceIndex i) (1 : ℝ))) =
fun s => a.scalarField ν t (x + s • basis i) := by
funext s
rw [← pack_space_line]
rfl
rw [he] at hp
have hd' : HasFDerivAt (a.scalarField ν t) (fderiv ℝ (a.scalarField ν t) x)
(x + (0 : ℝ) • basis i) := by simpa using hd.hasFDerivAt
have hc := hd'.comp_hasDerivAt 0 hline
simpa only [zero_smul, add_zero, spatialD, scalarField] using hc.unique hp
lemma time_scalarField (a : Formula 5) (ν : ℝ) (t : ℝ) (ht : 0 ≤ t) (x : Space) :
timeD (a.scalarField ν) t x = (a.diff 0).scalarField ν t x := by
have h := a.hasDerivAt_line 0 (pack ν 0 x) t
have he : (fun s => a.value (pack ν 0 x + s • Pi.single 0 1)) =
fun s => a.scalarField ν s x := by
funext s
rw [← pack_time_line, zero_add]
rfl
rw [he, ← pack_time_line, zero_add] at h
exact h.hasDerivWithinAt.derivWithin (uniqueDiffOn_Ici 0 t ht)
lemma spatial_vectorField (a : Fin 3 → Formula 5) (ν : ℝ) (i : Fin 3) :
spatialD i (vectorField a ν) = vectorField (fun j => (a j).diff (spaceIndex i)) ν := by
funext t x
ext j
have hd : DifferentiableAt ℝ (vectorField a ν t) x :=
((vectorField_smooth a ν).comp (contDiff_const.prodMk contDiff_id)).differentiable (by simp) x
change (fderiv ℝ (vectorField a ν t) x (basis i)) j = _
rw [← fderiv_coordinate hd]
exact congrFun (congrFun (spatial_scalarField (a j) ν i) t) x
lemma time_vectorField (a : Fin 3 → Formula 5) (ν : ℝ) (t : ℝ) (ht : 0 ≤ t) (x : Space) :
timeD (vectorField a ν) t x = vectorField (fun j => (a j).diff 0) ν t x := by
have hd : HasDerivAt (fun s => vectorField a ν s x)
(vectorField (fun j => (a j).diff 0) ν t x) t := by
apply (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).symm.toContinuousLinearMap.hasFDerivAt.comp_hasDerivAt t
apply hasDerivAt_pi.mpr
intro j
have h := (a j).hasDerivAt_line 0 (pack ν 0 x) t
have he : (fun s => (a j).value (pack ν 0 x + s • Pi.single 0 1)) =
fun s => (a j).scalarField ν s x := by
funext s
rw [← pack_time_line, zero_add]
rfl
rw [he, ← pack_time_line, zero_add] at h
exact h
exact hd.hasDerivWithinAt.derivWithin (uniqueDiffOn_Ici 0 t ht)
def mixed (l : ℕ) (α : MultiIndex) (a : Formula 5) : Formula 5 :=
(diff 0)^[l] ((diff (spaceIndex 0))^[α 0]
((diff (spaceIndex 1))^[α 1] ((diff (spaceIndex 2))^[α 2] a)))
lemma mixed_vectorField (a : Fin 3 → Formula 5) (ν : ℝ) (l : ℕ) (α : MultiIndex)
(t : ℝ) (ht : 0 ≤ t) (x : Space) :
mixedD l α (vectorField a ν) t x = vectorField (fun j => (a j).mixed l α) ν t x := by
have hsp (i : Fin 3) (n : ℕ) (a : Fin 3 → Formula 5) :
(spatialD i)^[n] (vectorField a ν) = vectorField (fun j => (diff (spaceIndex i))^[n] (a j)) ν := by
induction n with
| zero => rfl
| succ n ih => simp only [Function.iterate_succ_apply', ih, spatial_vectorField]
have htime (n : ℕ) (a : Fin 3 → Formula 5) :
∀ s, 0 ≤ s → ∀ y, (timeD)^[n] (vectorField a ν) s y =
vectorField (fun j => (diff 0)^[n] (a j)) ν s y := by
induction n with
| zero => intro s _ y; rfl
| succ n ih =>
intro s hs y
rw [Function.iterate_succ_apply']
have he : (fun r => (timeD)^[n] (vectorField a ν) r y) =ᶠ[𝓝[Ici 0] s]
fun r => vectorField (fun j => (diff 0)^[n] (a j)) ν r y := by
filter_upwards [self_mem_nhdsWithin] with r hr
exact ih r hr y
rw [timeD, he.derivWithin_eq_of_mem hs, ← timeD,
time_vectorField _ _ _ hs]
simp only [Function.iterate_succ_apply']
unfold mixedD spatialMulti mixed
rw [hsp, hsp, hsp]
exact htime l _ t ht x
end Formula
end EffectiveProfile
end RapidForcing
end OAI
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