| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.ScaledSmoothSeries |
| |
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory |
|
|
| def cascadePhase (D k : ℝ) (j : ℕ) (t : ℝ) : ℝ := |
| (2:ℝ)^j*(k*t-2*D)+2*D |
| |
| lemma cascadePhase_zero (D k t : ℝ) : cascadePhase D k 0 t = k*t := by |
| simp [cascadePhase] |
|
|
| lemma cascadePhase_succ (D k t : ℝ) (j : ℕ) : |
| cascadePhase D k (j+1) t = 2*(cascadePhase D k j t-D) := by |
| unfold cascadePhase |
| rw [pow_succ] |
| ring |
|
|
| lemma cascadePhase_add (D k t : ℝ) (j n : ℕ) : |
| cascadePhase D k (j+n) t = (2:ℝ)^n*(cascadePhase D k j t-2*D)+2*D := by |
| unfold cascadePhase |
| rw [pow_add] |
| ring |
|
|
| lemma cascadePhase_antitone {D k t : ℝ} (ht : k*t ≤ 2*D) : |
| Antitone (fun j => cascadePhase D k j t) := by |
| apply antitone_nat_of_succ_le |
| intro j |
| have hp : 0 ≤ (2:ℝ)^j := by positivity |
| rw [cascadePhase_succ] |
| have : cascadePhase D k j t ≤ 2*D := by |
| unfold cascadePhase |
| nlinarith [mul_nonpos_of_nonneg_of_nonpos hp (by linarith : k*t-2*D≤0)] |
| linarith |
|
|
| lemma cascadePhase_before {D k t : ℝ} (hD : 0 < D) {i j : ℕ} |
| (hij : i < j) (hj : 0 ≤ cascadePhase D k j t) : |
| D ≤ cascadePhase D k i t := by |
| by_cases ht : k*t ≤ 2*D |
| · have h := cascadePhase_antitone ht (show i+1≤j by omega) |
| dsimp only at h |
| rw [cascadePhase_succ] at h |
| linarith |
| · have hp : 0 ≤ (2:ℝ)^i := by positivity |
| have : 0 ≤ (2:ℝ)^i*(k*t-2*D) := mul_nonneg hp (by linarith) |
| unfold cascadePhase |
| linarith |
|
|
| lemma cascadePhase_after {D k t : ℝ} (hD : 0 < D) {i j : ℕ} |
| (hij : j+1 < i) (hj : cascadePhase D k j t ≤ D) : |
| cascadePhase D k i t ≤ -2*D := by |
| have ht : k*t ≤ 2*D := by |
| by_contra hh |
| have hp : 0 < (2:ℝ)^j := by positivity |
| have : 0 < (2:ℝ)^j*(k*t-2*D) := mul_pos hp (by linarith) |
| unfold cascadePhase at hj |
| linarith |
| have h := cascadePhase_antitone ht (show j + 2 ≤ i by omega) |
| dsimp only at h |
| rw [show j+2=(j+1)+1 by omega,cascadePhase_succ,cascadePhase_succ] at h |
| linarith |
|
|
| lemma cascadeProfile_only_two {L K k t : ℝ} (hL : 0 < L) (hK : 0 < K) |
| {j i : ℕ} (hj₁ : 0 ≤ cascadePhase (cascadeLength L K) k j t) |
| (hj₂ : cascadePhase (cascadeLength L K) k j t ≤ cascadeLength L K) |
| (hi : i ≠ j) (hi' : i ≠ j+1) : |
| cascadeProfile L K (cascadePhase (cascadeLength L K) k i t) = 0 := by |
| have hD := cascade_length_positive hL hK |
| by_cases hij : i < j |
| · apply cascadeProfile_zero_right hL |
| have hh := cascadePhase_before hD hij hj₁ |
| change K+2+2*L ≤ cascadePhase (cascadeLength L K) k i t at hh |
| linarith |
| · apply cascadeProfile_zero_left hL |
| have hh := cascadePhase_after hD (show j+1 < i by omega) hj₂ |
| change cascadePhase (cascadeLength L K) k i t ≤ -2*(K+2+2*L) at hh |
| linarith |
| |
| lemma cascadeProfile_terminal {L K k t : ℝ} (hL : 0 < L) (hK : 0 < K) |
| (ht : 2*cascadeLength L K ≤ k*t) (j : ℕ) : |
| cascadeProfile L K (cascadePhase (cascadeLength L K) k j t) = 0 := by |
| apply cascadeProfile_zero_right hL |
| have hp : 0 ≤ (2:ℝ)^j*(k*t-2*cascadeLength L K) := |
| mul_nonneg (by positivity) (by linarith) |
| unfold cascadePhase cascadeLength at * |
| linarith |
| |
| lemma exists_cascade_stage {D k t : ℝ} (_ : 0 < D) |
| (ht₀ : 0 ≤ k*t) (ht : k*t < 2*D) : |
| ∃ j : ℕ, 0 ≤ cascadePhase D k j t ∧ cascadePhase D k j t ≤ D := by |
| have hex : ∃ j : ℕ, cascadePhase D k j t ≤ D := by |
| obtain ⟨j,hj⟩ := pow_unbounded_of_one_lt (D/(2*D-k*t)) (by norm_num : (1:ℝ)<2) |
| refine ⟨j,?_⟩ |
| have hh : D ≤ (2:ℝ)^j*(2*D-k*t) := by |
| exact (div_le_iff₀ (by linarith : 0<2*D-k*t)).mp hj.le |
| unfold cascadePhase |
| nlinarith |
| let j := Nat.find hex |
| refine ⟨j,?_,Nat.find_spec hex⟩ |
| cases hj : j with |
| | zero => simpa [hj,cascadePhase_zero] using ht₀ |
| | succ n => |
| have hn : n < Nat.find hex := by change n < j; omega |
| have hh := Nat.find_min hex hn |
| rw [not_le] at hh |
| rw [cascadePhase_succ] |
| linarith |
| |
| def cascadeAxis (j : ℕ) : Fin 3 := if Even j then 1 else 2 |
| |
| lemma cascadeAxis_ne_zero (j : ℕ) : cascadeAxis j ≠ 0 := by |
| unfold cascadeAxis |
| split <;> decide |
| |
| lemma cascadeAxis_succ (j : ℕ) : cascadeAxis (j+1) = if Even j then 2 else 1 := by |
| simp only [cascadeAxis,Nat.even_add_one] |
| split <;> simp_all |
| |
| def cascadeCenter (D k : ℝ) (j : ℕ) : Coord3 := |
| Pi.single 0 ((2*D/k)*(1-1/(2:ℝ)^j)) |
| |
| lemma cascade_dilation_time {k : ℝ} (hk : k ≠ 0) (D : ℝ) (j : ℕ) (x : Coord3) : |
| ((k*2^j) • (x-cascadeCenter D k j)) 0 = cascadePhase D k j (x 0) := by |
| simp only [Pi.smul_apply,Pi.sub_apply,smul_eq_mul,cascadeCenter,Pi.single_eq_same,cascadePhase] |
| field_simp |
| ring |
| |
| lemma cascade_dilation_angular (D k : ℝ) (j : ℕ) (x : Coord3) : |
| ((k*2^j) • (x-cascadeCenter D k j)) (cascadeAxis j) = k*2^j*x (cascadeAxis j) := by |
| simp [cascadeCenter,Pi.single_eq_of_ne (cascadeAxis_ne_zero j)] |
| |
| def cascadeTerm (L K k A : ℝ) (j : ℕ) (x : Coord3) : ℝ := |
| (A*cascadeRatio L K^j)*cascadeProfile L K (cascadePhase (cascadeLength L K) k j (x 0))* |
| cos (k*2^j*x (cascadeAxis j)) |
| |
| def cascadeValue (L K k A : ℝ) (x : Coord3) : ℝ := ∑' j : ℕ, cascadeTerm L K k A j x |
|
|
| lemma cascadeTerm_scaled {k : ℝ} (hk : k ≠ 0) (L K A : ℝ) (j : ℕ) (x : Coord3) : |
| cascadeTerm L K k A j x = (A*cascadeRatio L K^j)* |
| cascadeMother L K (cascadeAxis j) ((k*2^j) • (x-cascadeCenter (cascadeLength L K) k j)) := by |
| rw [cascadeMother,cascade_dilation_time hk,cascade_dilation_angular] |
| unfold cascadeTerm |
| ring |
|
|
| theorem cascadeValue_C2 {L K k : ℝ} (hL : 1 ≤ L) (hK : 0 < K) (hk : k ≠ 0) (A : ℝ) : |
| ContDiff ℝ 2 (cascadeValue L K k A) := by |
| have hh := geometric_scaled_C2 |
| (f := fun j => cascadeMother L K (cascadeAxis j)) |
| (fun _ => cascadeMother_smooth _ _ _) |
| (fun n => by |
| obtain ⟨C,hC,hb⟩ := cascadeMother_iterated_bound (lt_of_lt_of_le (by norm_num) hL) n |
| exact ⟨C,hC,fun j x => hb (cascadeAxis j) x⟩) |
| (cascadeRatio_pos L K).le (cascadeRatio_small hL hK) A k |
| (cascadeCenter (cascadeLength L K) k) |
| convert hh using 1 |
| funext x |
| exact tsum_congr fun j => cascadeTerm_scaled hk L K A j x |
|
|
| lemma cascadeValue_terminal {L K k : ℝ} (hL : 0 < L) (hK : 0 < K) (A : ℝ) |
| (x : Coord3) (hx : 2*cascadeLength L K ≤ k*x 0) : cascadeValue L K k A x = 0 := by |
| change (∑' j : ℕ, cascadeTerm L K k A j x) = 0 |
| have h : ∀ j, cascadeTerm L K k A j x = 0 := by |
| intro j |
| rw [cascadeTerm,cascadeProfile_terminal hL hK hx,mul_zero,zero_mul] |
| simp only [h,tsum_zero] |
| |
| lemma cascadeValue_two_modes {L K k : ℝ} (hL : 0 < L) (hK : 0 < K) (A : ℝ) |
| (x : Coord3) (j : ℕ) (hj₁ : 0 ≤ cascadePhase (cascadeLength L K) k j (x 0)) |
| (hj₂ : cascadePhase (cascadeLength L K) k j (x 0) ≤ cascadeLength L K) : |
| cascadeValue L K k A x = cascadeTerm L K k A j x + cascadeTerm L K k A (j+1) x := by |
| have h : (∑' i : ℕ, cascadeTerm L K k A i x) = ∑ i ∈ ({j,j+1} : Finset ℕ), cascadeTerm L K k A i x := tsum_eq_sum (s := ({j,j+1} : Finset ℕ)) (f := fun i => cascadeTerm L K k A i x) (by |
| intro i hi |
| have hi₁ : i ≠ j := by simpa using fun h => hi (by simp [h]) |
| have hi₂ : i ≠ j+1 := by simpa using fun h => hi (by simp [h]) |
| rw [cascadeTerm,cascadeProfile_only_two hL hK hj₁ hj₂ hi₁ hi₂,mul_zero,zero_mul]) |
| simpa [cascadeValue,Finset.sum_pair (show j ≠ j+1 by omega)] using h |
|
|
| lemma cascadeValue_zero_phase {L K k : ℝ} (hL : 0 < L) (hK : 0 < K) (A : ℝ) |
| (x : Coord3) (hx : x 0 = 0) : cascadeValue L K k A x = A*cos (k*x 1) := by |
| have h₁ : 0 ≤ cascadePhase (cascadeLength L K) k 0 (x 0) := by simp [cascadePhase_zero,hx] |
| have h₂ : cascadePhase (cascadeLength L K) k 0 (x 0) ≤ cascadeLength L K := by |
| simpa [cascadePhase_zero,hx] using (cascade_length_positive hL hK).le |
| rw [cascadeValue_two_modes hL hK A x 0 h₁ h₂] |
| have hz : cascadeProfile L K (cascadePhase (cascadeLength L K) k 1 (x 0)) = 0 := by |
| apply cascadeProfile_zero_left hL |
| simp [cascadePhase,hx,cascadeLength] |
| linarith |
| rw [cascadeTerm,cascadeTerm,hz] |
| simp [cascadePhase_zero,hx,cascadeProfile_connector hL (le_refl 0) (show (0:ℝ)≤K+2 by linarith), |
| connectorProfile,cascadeAxis] |
|
|
| end ScalarConductivity |
| |
| end |
|
|
| end OAI |
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|
