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a458723 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import Mathlib
import OAI.Analysis.Conductivity.Branching.CascadeCoefficient
namespace OAI
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; positivity
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; positivity
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
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