| import Mathlib |
| import OAI.Analysis.Conductivity.Branching.CascadeCoefficient |
| |
| namespace OAI |
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| noncomputable section |
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory Matrix |
| open scoped Matrix.Norms.Elementwise |
| |
| lemma cascadeIndex_eq_of_strict_phase {D k : ℝ} (hD : 0<D) (x : Coord3) (j : ℕ) |
| (hj0 : 0<cascadePhase D k j (x 0)) (hjD : cascadePhase D k j (x 0)<D) : |
| cascadeIndex D k x=j := by |
| have ht : k*x 0<2*D := cascadePhase_lt_terminal (by linarith : cascadePhase _ k j (x 0)<2*_) |
| have ht0 : 0≤k*x 0 := by |
| have hh := cascadePhase_antitone ht.le (Nat.zero_le j) |
| dsimp only at hh |
| rw [cascadePhase_zero] at hh |
| exact le_trans hj0.le hh |
| have hs := cascadeIndex_spec x ht0 ht |
| rcases lt_trichotomy (cascadeIndex D k x) j with hij|he|hji |
| · have hh := cascadePhase_antitone ht.le (show cascadeIndex D k x+1≤j by omega) |
| dsimp only at hh |
| rw [cascadePhase_succ] at hh |
| linarith |
| · exact he |
| · have hh := cascadePhase_before hD hji hs.1 |
| linarith |
|
|
| lemma cascadeTensor_connector_eq {L K k : ℝ} (hL : 0<L) (hK : 0<K) |
| (x : Coord3) (j : ℕ) |
| (hj0 : 0<cascadePhase (cascadeLength L K) k j (x 0)) |
| (hjK : cascadePhase (cascadeLength L K) k j (x 0)<K+2) : |
| cascadeTensor L K k x = |
| pureModeTensor (connectorAngular K (cascadePhase (cascadeLength L K) k j (x 0))) (cascadeAxis j) := by |
| have hD : 0<cascadeLength L K := by unfold cascadeLength |
| have hjD : cascadePhase (cascadeLength L K) k j (x 0)<cascadeLength L K := by |
| have he : cascadeLength L K=K+2+2*L := rfl |
| linarith |
| have ht := cascadePhase_lt_terminal (by linarith : cascadePhase _ k j (x 0)<2*_) |
| have ht0 : 0≤k*x 0 := by |
| have hh := cascadePhase_antitone ht.le (Nat.zero_le j) |
| dsimp only at hh |
| rw [cascadePhase_zero] at hh |
| exact le_trans hj0.le hh |
| rw [cascadeTensor,ite_eq_left ⟨ht0,ht⟩,cascadeIndex_eq_of_strict_phase hD x j hj0 hjD, |
| cascadeStageTensor,ite_eq_left hjK] |
|
|
| lemma cascadeTensor_crossing_eq {L K k : ℝ} (hL : 0<L) (hK : 0<K) |
| (x : Coord3) (j : ℕ) |
| (hjK : K+2<cascadePhase (cascadeLength L K) k j (x 0)) |
| (hjD : cascadePhase (cascadeLength L K) k j (x 0)<cascadeLength L K) : |
| cascadeTensor L K k x = crossingTensor L (cascadeAxis j) (cascadeAxis (j+1)) |
| ((k*2^j) • (x-cascadeCenter (cascadeLength L K) k j)-Pi.single 0 (K+2+L)) := by |
| have hD : 0<cascadeLength L K := by unfold cascadeLength |
| have hj0 : 0<cascadePhase (cascadeLength L K) k j (x 0) := by linarith |
| have ht := cascadePhase_lt_terminal (by linarith : cascadePhase _ k j (x 0)<2*_) |
| have ht0 : 0≤k*x 0 := by |
| have hh := cascadePhase_antitone ht.le (Nat.zero_le j) |
| dsimp only at hh |
| rw [cascadePhase_zero] at hh |
| exact le_trans hj0.le hh |
| rw [cascadeTensor,ite_eq_left ⟨ht0,ht⟩,cascadeIndex_eq_of_strict_phase hD x j hj0 hjD, |
| cascadeStageTensor,ite_eq_right (not_lt.mpr hjK.le)] |
|
|
| lemma connectorAngular_smooth (K : ℝ) : ContDiff ℝ (↑(⊤:ℕ∞)) (connectorAngular K) := by |
| unfold connectorAngular |
| exact (smooth_deriv_infty (smooth_deriv_infty (connectorProfile_smooth K))).div |
| (connectorProfile_smooth K) (fun x => (connectorProfile_pos K x).ne') |
| |
| lemma crossingValue3_smooth (L : ℝ) (a b : Fin 3) : |
| ContDiff ℝ (↑(⊤:ℕ∞)) (crossingValue3 L a b) := by |
| have hf := crossingFirst_smooth L |
| have hg := crossingSecond_smooth L |
| unfold crossingValue3 crossingValue |
| fun_prop |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
