| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.CascadeGeometry |
| import OAI.Analysis.Conductivity.Branching.CascadeStreams |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory |
|
|
| lemma cascadePhase_lt_terminal {D k t : ℝ} {j : ℕ} |
| (hj : cascadePhase D k j t < 2*D) : k*t < 2*D := by |
| have hp : 0 < (2:ℝ)^j := by positivity |
| unfold cascadePhase at hj |
| nlinarith |
|
|
| lemma cascadePhase_before_wide {D k t : ℝ} {i j : ℕ} (ht : k*t ≤ 2*D) (hij : i < j) : |
| D+cascadePhase D k j t/2 ≤ cascadePhase D k i t := by |
| have hh := cascadePhase_antitone ht (show i+1 ≤ j by omega) |
| dsimp only at hh |
| rw [cascadePhase_succ] at hh |
| linarith |
|
|
| lemma cascadePhase_after_wide {D k t : ℝ} {i j : ℕ} (ht : k*t ≤ 2*D) (hij : j+1 < i) : |
| cascadePhase D k i t ≤ 4*cascadePhase D k j t-6*D := by |
| have hh := cascadePhase_antitone ht (show j+2 ≤ i by omega) |
| dsimp only at hh |
| rw [show j+2=(j+1)+1 by omega,cascadePhase_succ,cascadePhase_succ] at hh |
| linarith |
|
|
| lemma cascadePhase_inactive_wide {L K k t : ℝ} (hL : 0 < L) (hK : 0 < K) {j i : ℕ} |
| (hj₁ : -L/8 < cascadePhase (cascadeLength L K) k j t) |
| (hj₂ : cascadePhase (cascadeLength L K) k j t < cascadeLength L K+L/8) |
| (hi : i≠j) (hi' : i≠j+1) : |
| cascadePhase (cascadeLength L K) k i t < -7*L/2 ∨ |
| K+2+7*L/4 < cascadePhase (cascadeLength L K) k i t := by |
| have ht : k*t < 2*cascadeLength L K := cascadePhase_lt_terminal (hj₂.trans |
| (by unfold cascadeLength; linarith)) |
| by_cases hij : i < j |
| · right |
| have hh := cascadePhase_before_wide ht.le hij |
| unfold cascadeLength at * |
| linarith |
| · left |
| have hh := cascadePhase_after_wide ht.le (show j+1 < i by omega) |
| unfold cascadeLength at * |
| linarith |
|
|
| lemma cascadeProfile_deriv_zero_left {L K t : ℝ} (hL : 0 < L) (ht : t < -7*L/2) : |
| deriv (cascadeProfile L K) t = 0 := by |
| have he : cascadeProfile L K =ᶠ[𝓝 t] fun _ => (0:ℝ) := by |
| filter_upwards [isOpen_Iio.mem_nhds ht] with y hy |
| exact cascadeProfile_zero_left hL hy.le |
| simpa using he.deriv_eq |
| lemma cascadeProfile_deriv_zero_right {L K t : ℝ} (hL : 0 < L) (ht : K+2+7*L/4 < t) : |
| deriv (cascadeProfile L K) t = 0 := by |
| have he : cascadeProfile L K =ᶠ[𝓝 t] fun _ => (0:ℝ) := by |
| filter_upwards [isOpen_Ioi.mem_nhds ht] with y hy |
| exact cascadeProfile_zero_right hL hy.le |
| simpa using he.deriv_eq |
|
|
| lemma cascadeBaseStream_zero {L K : ℝ} (hL : 0 < L) (a : Fin 3) (x : Coord3) |
| (ht : x 0 < -7*L/2 ∨ K+2+7*L/4 < x 0) : cascadeBaseStream L K a x = 0 := by |
| have hd : deriv (cascadeProfile L K) (x 0) = 0 := |
| ht.elim (cascadeProfile_deriv_zero_left hL) (cascadeProfile_deriv_zero_right hL) |
| simp [cascadeBaseStream,hd] |
|
|
| lemma cascadeCrossStream_zero_left {L K : ℝ} (hL : 0 < L) (a b : Fin 3) (x : Coord3) |
| (ht : x 0 < K+2+L/4) : cascadeCrossStream L K a b x = 0 := by |
| have hf := crossingResidualFirst_zero hL (show x 0-(K+2+L) < L/2 by linarith) |
| have hg := crossingResidualSecond_zero_left hL (show x 0-(K+2+L) < -3*L/4 by linarith) |
| simp [cascadeCrossStream,hf,hg] |
| lemma cascadeCrossStream_zero_right {L K : ℝ} (hL : 0 < L) (a b : Fin 3) (x : Coord3) |
| (ht : K+2+7*L/4 < x 0) : cascadeCrossStream L K a b x = 0 := by |
| have hf := crossingResidualFirst_zero_right hL (show 3*L/4 < x 0-(K+2+L) by linarith) |
| have hg := crossingResidualSecond_zero hL (show -L/2 < x 0-(K+2+L) by linarith) |
| simp [cascadeCrossStream,hf,hg] |
|
|
| lemma cascadeMother_inactive_wide {L K k : ℝ} (hL : 0 < L) (hK : 0 < K) (hk : k≠0) |
| (x : Coord3) {j n : ℕ} |
| (hj₁ : -L/8 < cascadePhase (cascadeLength L K) k j (x 0)) |
| (hj₂ : cascadePhase (cascadeLength L K) k j (x 0) < cascadeLength L K+L/8) |
| (hn : n≠j) (hn' : n≠j+1) : |
| cascadeMother L K (cascadeAxis n) ((k*2^n) • (x-cascadeCenter (cascadeLength L K) k n)) = 0 ∧ |
| cascadeBaseStream L K (cascadeAxis n) ((k*2^n) • (x-cascadeCenter (cascadeLength L K) k n)) = 0 ∧ |
| cascadeCrossStream L K (cascadeAxis n) (cascadeAxis (n+1)) |
| ((k*2^n) • (x-cascadeCenter (cascadeLength L K) k n)) = 0 := by |
| let y := (k*2^n) • (x-cascadeCenter (cascadeLength L K) k n) |
| have ht : y 0 < -7*L/2 ∨ K+2+7*L/4 < y 0 := by |
| dsimp only [y] |
| rw [cascade_dilation_time hk] |
| exact cascadePhase_inactive_wide hL hK hj₁ hj₂ hn hn' |
| refine ⟨?_,cascadeBaseStream_zero hL _ y ht,?_⟩ |
| · have hp : cascadeProfile L K (y 0) = 0 := ht.elim |
| (fun h => cascadeProfile_zero_left hL h.le) (fun h => cascadeProfile_zero_right hL h.le) |
| exact show cascadeProfile L K (y 0)*cos (y (cascadeAxis n))=0 by rw [hp,zero_mul] |
| · rcases ht with ht | ht |
| · apply cascadeCrossStream_zero_left hL _ _ y |
| linarith |
| · exact cascadeCrossStream_zero_right hL _ _ y ht |
|
|
| lemma cascadeCrossStream_next_zero_wide {L K k : ℝ} (hL : 0 < L) (hK : 0 < K) (hk : k≠0) |
| (x : Coord3) {j : ℕ} |
| (hj₂ : cascadePhase (cascadeLength L K) k j (x 0) < cascadeLength L K+L/8) : |
| cascadeCrossStream L K (cascadeAxis (j+1)) (cascadeAxis (j+2)) |
| ((k*2^(j+1)) • (x-cascadeCenter (cascadeLength L K) k (j+1))) = 0 := by |
| apply cascadeCrossStream_zero_left hL |
| rw [cascade_dilation_time hk,cascadePhase_succ] |
| linarith |
|
|
| end ScalarConductivity |
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| end |
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| end OAI |
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