openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Flow /CompactPeriodization.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Programs.CompactRapidEmbedding | |
| namespace OAI | |
| /-! Periodization in a chart strictly separated from every integer face. The | |
| fractional representative is only a definition: smoothness follows from local | |
| translation or local zero, so no differentiability of the fractional part at | |
| an integer is asserted. -/ | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| namespace RapidForcing.CompactEmbedding | |
| def latticeVector (k : Fin 3 → ℤ) : Space := WithLp.toLp 2 (fun i => (k i : ℝ)) | |
| def representative (x : Space) : Space := WithLp.toLp 2 (fun i => Int.fract (x i)) | |
| @[simp] theorem representative_apply (x : Space) (i : Fin 3) : | |
| representative x i = Int.fract (x i) := rfl | |
| @[simp] theorem latticeVector_apply (k : Fin 3 → ℤ) (i : Fin 3) : | |
| latticeVector k i = (k i : ℝ) := rfl | |
| theorem representative_add_lattice (x : Space) (k : Fin 3 → ℤ) : | |
| representative (x + latticeVector k) = representative x := by | |
| ext i | |
| exact Int.fract_add_intCast (x i) (k i) | |
| theorem representative_eq {x : Space} (hx : ∀ i, x i ∈ Ico (0 : ℝ) 1) : | |
| representative x = x := by | |
| ext i | |
| exact Int.fract_eq_self.mpr (hx i) | |
| def periodic {E : Type} (f : Field E) : Field E := fun t x => f t (representative x) | |
| theorem periodic_add_lattice {E : Type} (f : Field E) (t : ℝ) | |
| (x : Space) (k : Fin 3 → ℤ) : | |
| periodic f t (x + latticeVector k) = periodic f t x := by | |
| simp only [periodic, representative_add_lattice] | |
| theorem periodic_eq_on_chart {E : Type} (f : Field E) (t : ℝ) {x : Space} | |
| (hx : x ∈ chart) : periodic f t x = f t x := by | |
| rw [periodic, representative_eq] | |
| intro i | |
| exact ⟨(chart_inside_unit hx i).1.le, (chart_inside_unit hx i).2⟩ | |
| theorem representative_local_translation {x : Space} | |
| (hx : ∀ i, x i ≠ (⌊x i⌋ : ℤ)) : | |
| representative =ᶠ[𝓝 x] (fun y => y - latticeVector (fun i => ⌊x i⌋)) := by | |
| have hlocal : ∀ i : Fin 3, ∀ᶠ y in 𝓝 x, | |
| (⌊x i⌋ : ℝ) < y i ∧ y i < (⌊x i⌋ : ℝ) + 1 := by | |
| intro i | |
| exact ((EuclideanSpace.proj i).continuous.continuousAt).eventually | |
| (Ioo_mem_nhds ((Int.floor_le (x i)).lt_of_ne (Ne.symm (hx i))) | |
| (Int.lt_floor_add_one (x i))) | |
| filter_upwards [Filter.eventually_all.mpr hlocal] with y hy | |
| ext i | |
| have hk : ⌊y i⌋ = ⌊x i⌋ := Int.floor_eq_iff.mpr ⟨(hy i).1.le, (hy i).2⟩ | |
| simp only [representative_apply, PiLp.sub_apply, latticeVector_apply, Int.fract, hk] | |
| theorem fract_near_integer {x : ℝ} {k : ℤ} | |
| (hx : x ∈ Ioo ((k : ℝ) - 1 / 8) ((k : ℝ) + 1 / 8)) : | |
| Int.fract x < 1 / 4 ∨ 3 / 4 < Int.fract x := by | |
| have hfract : Int.fract (x - (k : ℝ)) = Int.fract x := Int.fract_sub_intCast x k | |
| rw [← hfract] | |
| by_cases hp : 0 ≤ x - (k : ℝ) | |
| · left | |
| rw [Int.fract_eq_self.mpr ⟨hp, by linarith [hx.2]⟩] | |
| linarith [hx.2] | |
| · right | |
| have hk : ⌊x - (k : ℝ)⌋ = (-1 : ℤ) := | |
| Int.floor_eq_iff.mpr ⟨by norm_num; linarith [hx.1], by norm_num; linarith⟩ | |
| rw [Int.fract, hk] | |
| norm_num | |
| linarith [hx.1] | |
| theorem representative_local_outside {x : Space} {i : Fin 3} | |
| (hx : x i = (⌊x i⌋ : ℤ)) : ∀ᶠ y in 𝓝 x, representative y ∉ chart := by | |
| have hl : (⌊x i⌋ : ℝ) - 1 / 8 < x i := by linarith [hx] | |
| have hr : x i < (⌊x i⌋ : ℝ) + 1 / 8 := by linarith [hx] | |
| filter_upwards [((EuclideanSpace.proj i).continuous.continuousAt).eventually | |
| (Ioo_mem_nhds hl hr)] with y hy | |
| intro hc | |
| rcases fract_near_integer hy with h | h | |
| · exact (not_lt_of_ge (hc.1 i)) h | |
| · exact (not_lt_of_ge (hc.2 i)) h | |
| theorem periodic_local {E : Type} [Zero E] (f : Field E) | |
| (hf : ∀ t x, x ∉ chart → f t x = 0) (x : Space) : | |
| (∃ k : Fin 3 → ℤ, ∀ᶠ y in 𝓝 x, ∀ t, periodic f t y = f t (y - latticeVector k)) ∨ | |
| (∀ᶠ y in 𝓝 x, ∀ t, periodic f t y = 0) := by | |
| by_cases hx : ∀ i, x i ≠ (⌊x i⌋ : ℤ) | |
| · left | |
| refine ⟨fun i => ⌊x i⌋, ?_⟩ | |
| filter_upwards [representative_local_translation hx] with y hy t | |
| simp only [periodic, hy] | |
| · right | |
| push Not at hx | |
| obtain ⟨i, hi⟩ := hx | |
| filter_upwards [representative_local_outside hi] with y hy t | |
| exact hf t (representative y) hy | |
| theorem periodic_smooth {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {f : Field E} (hs : ContDiff ℝ ∞ (Function.uncurry f)) | |
| (hf : ∀ t x, x ∉ chart → f t x = 0) : | |
| ContDiff ℝ ∞ (Function.uncurry (periodic f)) := by | |
| apply contDiff_iff_contDiffAt.mpr | |
| rintro ⟨t, x⟩ | |
| rcases periodic_local f hf x with ⟨k, hk⟩ | hz | |
| · have he : Function.uncurry (periodic f) =ᶠ[𝓝 (t, x)] | |
| (fun p : ℝ × Space => f p.1 (p.2 - latticeVector k)) := by | |
| filter_upwards [continuous_snd.continuousAt.eventually hk] with p hp | |
| exact hp p.1 | |
| exact (hs.comp (contDiff_fst.prodMk (contDiff_snd.sub contDiff_const))).contDiffAt.congr_of_eventuallyEq he | |
| · have he : Function.uncurry (periodic f) =ᶠ[𝓝 (t, x)] (fun _ => (0 : E)) := by | |
| filter_upwards [continuous_snd.continuousAt.eventually hz] with p hp | |
| exact hp p.1 | |
| exact contDiffAt_const.congr_of_eventuallyEq he | |
| end RapidForcing.CompactEmbedding | |
| end | |
| end OAI | |