| import Mathlib |
| import OAI.Analysis.Conductivity.Variational.CrossingHxx |
| import OAI.Analysis.Conductivity.Variational.CrossingEstimates |
| import OAI.Analysis.Conductivity.Branching.CascadeProfile |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ScalarConductivity |
| open Real Set Filter Topology MeasureTheory |
|
|
| def crossingValue (L z x y : ℝ) : ℝ := |
| crossingFirst L z*cos x + crossingSecond L z*cos (2*y) |
|
|
| def crossingXX (L z x y : ℝ) : ℝ := |
| 9 + crossingHxx (exp (-3*z)) (crossingSecond L z) (crossingResidualSecond L z) 1 2 x y |
|
|
| def crossingYY (L z x y : ℝ) : ℝ := |
| 1/4 + crossingHxx (exp (-z)) (crossingFirst L z) (crossingResidualFirst L z) 2 1 y x |
|
|
| def crossingXY (L z x y : ℝ) : ℝ := |
| crossingHxy (exp (-3*z)) (crossingResidualSecond L z) 1 2 x y + |
| crossingHxy (exp (-z)) (crossingResidualFirst L z) 2 1 y x |
|
|
| def crossingCurrentX (L z x y : ℝ) : ℝ := |
| -9*crossingFirst L z*sin x - crossingResidualSecond L z*cos x*sin x*cos (2*y) - |
| 2*crossingResidualFirst L z*(sin (2*y))^2*sin x |
|
|
| def crossingCurrentY (L z x y : ℝ) : ℝ := |
| -(crossingSecond L z/2)*sin (2*y) - crossingResidualSecond L z*(sin x)^2*sin (2*y) - |
| (crossingResidualFirst L z/2)*cos (2*y)*sin (2*y)*cos x |
|
|
| def crossingCurrentZ (L z x y : ℝ) : ℝ := |
| deriv (crossingFirst L) z*cos x + deriv (crossingSecond L) z*cos (2*y) |
|
|
| lemma crossing_first_on_left {L z : ℝ} (hL : 0 < L) (hz : z ≤ 0) : |
| crossingFirst L z = exp (-3*z) := by |
| rw [crossingFirst,crossingOff_eq_one hL (by linarith),one_mul] |
| lemma crossing_second_on_right {L z : ℝ} (hL : 0 < L) (hz : 0 ≤ z) : |
| crossingSecond L z = exp (-z) := by |
| rw [crossingSecond,crossingOn_eq_one hL (by linarith),one_mul] |
|
|
| lemma crossing_corrections_left {L z : ℝ} (hL : 0 < L) (hz : z ≤ 0) (x y : ℝ) : |
| crossingYY L z x y = 1/4 ∧ |
| crossingXY L z x y = crossingHxy (exp (-3*z)) (crossingResidualSecond L z) 1 2 x y := by |
| have hr := crossingResidualFirst_zero hL (show z < L/2 by linarith) |
| simp [crossingYY,crossingXY,hr,crossingHxx,crossingHxy] |
| lemma crossing_corrections_right {L z : ℝ} (hL : 0 < L) (hz : 0 ≤ z) (x y : ℝ) : |
| crossingXX L z x y = 9 ∧ |
| crossingXY L z x y = crossingHxy (exp (-z)) (crossingResidualFirst L z) 2 1 y x := by |
| have hr := crossingResidualSecond_zero hL (show -L/2 < z by linarith) |
| simp [crossingXX,crossingXY,hr,crossingHxx,crossingHxy] |
|
|
| lemma crossing_current_constitution {L : ℝ} (hL : 0 < L) (z x y : ℝ) : |
| crossingCurrentX L z x y = |
| crossingXX L z x y*(-crossingFirst L z*sin x) + |
| crossingXY L z x y*(-2*crossingSecond L z*sin (2*y)) ∧ |
| crossingCurrentY L z x y = |
| crossingXY L z x y*(-crossingFirst L z*sin x) + |
| crossingYY L z x y*(-2*crossingSecond L z*sin (2*y)) := by |
| have he₁ : exp (-3*z) ≠ 0 := (exp_pos _).ne' |
| have he₂ : exp (-z) ≠ 0 := (exp_pos _).ne' |
| by_cases hz : z ≤ 0 |
| · have hf := crossing_first_on_left hL hz |
| have hr := crossingResidualFirst_zero hL (show z < L/2 by linarith) |
| obtain ⟨hyy,hxy⟩ := crossing_corrections_left hL hz x y |
| rw [hyy,hxy] |
| constructor <;> dsimp only [crossingCurrentX,crossingCurrentY,crossingXX] <;> |
| rw [hr,hf] <;> simp only [crossingHxx,crossingHxy,one_mul,one_pow,mul_one] <;> |
| field_simp <;> ring |
| · have hg := crossing_second_on_right hL (le_of_not_ge hz) |
| have hr := crossingResidualSecond_zero hL (show -L/2 < z by linarith) |
| obtain ⟨hxx,hxy⟩ := crossing_corrections_right hL (le_of_not_ge hz) x y |
| rw [hxx,hxy] |
| constructor <;> dsimp only [crossingCurrentX,crossingCurrentY,crossingYY] <;> |
| rw [hr,hg] <;> simp only [crossingHxx,crossingHxy,one_mul,mul_one] <;> |
| field_simp <;> ring |
|
|
| lemma crossing_value_hasDerivAt_x (L z x y : ℝ) : |
| HasDerivAt (fun x => crossingValue L z x y) (-crossingFirst L z*sin x) x := by |
| exact (((hasDerivAt_cos x).const_mul (crossingFirst L z)).add_const |
| (crossingSecond L z*cos (2*y))).congr_deriv (g' := -crossingFirst L z*sin x) (by ring) |
| lemma crossing_value_hasDerivAt_y (L z x y : ℝ) : |
| HasDerivAt (fun y => crossingValue L z x y) (-2*crossingSecond L z*sin (2*y)) y := by |
| exact (((((hasDerivAt_id y).const_mul 2).cos).const_mul (crossingSecond L z)).const_add |
| (crossingFirst L z*cos x)).congr_deriv (g' := -2*crossingSecond L z*sin (2*y)) |
| (by simp only [id_eq,mul_one]; ring) |
| lemma crossing_value_hasDerivAt_z (L z x y : ℝ) : |
| HasDerivAt (fun z => crossingValue L z x y) (crossingCurrentZ L z x y) z := by |
| exact (((crossingFirst_smooth L).differentiable (by simp) z).hasDerivAt.mul_const (cos x)).add |
| (((crossingSecond_smooth L).differentiable (by simp) z).hasDerivAt.mul_const (cos (2*y))) |
|
|
| lemma crossing_current_divergence (L z x y : ℝ) : |
| deriv (fun z => crossingCurrentZ L z x y) z + |
| deriv (fun x => crossingCurrentX L z x y) x + |
| deriv (fun y => crossingCurrentY L z x y) y = 0 := by |
| have hsx := hasDerivAt_sin x |
| have hcx := hasDerivAt_cos x |
| have hy := (hasDerivAt_id y).const_mul 2 |
| have hsy := hy.sin |
| have hcy := hy.cos |
| have hx := ((hsx.const_mul (-9*crossingFirst L z)).sub |
| (((hcx.mul hsx).const_mul (crossingResidualSecond L z)).mul_const (cos (2*y)))).sub |
| (hsx.const_mul (2*crossingResidualFirst L z*(sin (2*y))^2)) |
| have hy' := ((hsy.const_mul (-(crossingSecond L z/2))).sub |
| (hsy.const_mul (crossingResidualSecond L z*(sin x)^2))).sub |
| (((hcy.mul hsy).const_mul (crossingResidualFirst L z/2)).mul_const (cos x)) |
| have hz := (((smooth_deriv_infty (crossingFirst_smooth L)).differentiable (by simp) z).hasDerivAt.mul_const |
| (cos x)).add (((smooth_deriv_infty (crossingSecond_smooth L)).differentiable (by simp) z).hasDerivAt.mul_const |
| (cos (2*y))) |
| simp only [Pi.mul_def,Pi.sub_def,Pi.add_def,id_eq,mul_one] at hx hy' hz |
| have hex : (fun x => crossingCurrentX L z x y) = |
| (fun x => -9*crossingFirst L z*sin x - crossingResidualSecond L z*(cos x*sin x)*cos (2*y) - |
| (2*crossingResidualFirst L z*(sin (2*y))^2)*sin x) := by |
| funext x; unfold crossingCurrentX; ring |
| have hey : (fun y => crossingCurrentY L z x y) = |
| (fun y => -(crossingSecond L z/2)*sin (2*y) - crossingResidualSecond L z*(sin x)^2*sin (2*y) - |
| (crossingResidualFirst L z/2)*(cos (2*y)*sin (2*y))*cos x) := by |
| funext y; unfold crossingCurrentY; ring |
| rw [hex,hey,hx.deriv,hy'.deriv] |
| simp only [crossingCurrentZ] |
| rw [hz.deriv] |
| have hr₁ : deriv (deriv (crossingFirst L)) z = crossingResidualFirst L z+9*crossingFirst L z := by |
| unfold crossingResidualFirst; ring |
| have hr₂ : deriv (deriv (crossingSecond L)) z = crossingResidualSecond L z+crossingSecond L z := by |
| unfold crossingResidualSecond; ring |
| rw [hr₁,hr₂] |
| linear_combination |
| -(crossingResidualSecond L z*cos (2*y)) * Real.sin_sq_add_cos_sq x + |
| -(crossingResidualFirst L z*cos x) * Real.sin_sq_add_cos_sq (2*y) |
| |
| lemma cascadeProfile_next_connector_zero {L K s : ℝ} (hL : 0 < L) |
| (hs : s ≤ K+2) : cascadeProfile L K (2*(s-cascadeLength L K)) = 0 := by |
| apply cascadeProfile_zero_left hL |
| unfold cascadeLength |
| linarith |
| |
| lemma cascadeProfile_crossing_current {L K s : ℝ} (hL : 0 < L) (hK : 0 < K) |
| (hs : K+2 ≤ s) : |
| cascadeProfile L K s = connectorProfile K (K+2)*exp (-3*L)* |
| crossingFirst L (s-(K+2+L)) := by |
| rw [cascadeProfile_outgoing hL hK hs,crossingFirst] |
| have he : exp (-3*(s-(K+2))) = exp (-3*L)*exp (-3*(s-(K+2+L))) := by |
| rw [←exp_add] |
| congr 1 |
| ring |
| rw [he] |
| ring |
| |
| lemma cascadeProfile_crossing_next {L K s : ℝ} (hL : 0 < L) (hK : 0 < K) |
| (hs : s ≤ cascadeLength L K) : |
| cascadeRatio L K*cascadeProfile L K (2*(s-cascadeLength L K)) = |
| connectorProfile K (K+2)*exp (-3*L)*crossingSecond L (s-(K+2+L)) := by |
| rw [cascadeProfile_incoming hL hK (by linarith),cascadeRatio,crossingSecond] |
| have ha : 2*(s-cascadeLength L K)/2+L = s-(K+2+L) := by |
| unfold cascadeLength |
| ring |
| rw [ha] |
| have he : exp (-4*L)*exp (-(2*(s-cascadeLength L K))/2) = |
| exp (-3*L)*exp (-(s-(K+2+L))) := by |
| rw [←exp_add,←exp_add] |
| congr 1 |
| unfold cascadeLength |
| ring |
| calc |
| _ = connectorProfile K (K+2)*crossingOn L (s-(K+2+L))* |
| (exp (-4*L)*exp (-(2*(s-cascadeLength L K))/2)) := by ring |
| _ = _ := by rw [he]; ring |
| |
| lemma cascade_overlap_value {L K s : ℝ} (hL : 0 < L) (hK : 0 < K) |
| (hs₁ : K+2 ≤ s) (hs₂ : s ≤ cascadeLength L K) (a x y : ℝ) : |
| a*cascadeProfile L K s*cos x + |
| (a*cascadeRatio L K)*cascadeProfile L K (2*(s-cascadeLength L K))*cos (2*y) = |
| (a*connectorProfile K (K+2)*exp (-3*L))*crossingValue L (s-(K+2+L)) x y := by |
| rw [cascadeProfile_crossing_current hL hK hs₁] |
| have hh := cascadeProfile_crossing_next hL hK hs₂ |
| dsimp only [crossingValue] |
| calc |
| _ = a*(connectorProfile K (K+2)*exp (-3*L)*crossingFirst L (s-(K+2+L)))*cos x + |
| a*(cascadeRatio L K*cascadeProfile L K (2*(s-cascadeLength L K)))*cos (2*y) := by ring |
| _ = _ := by rw [hh]; ring |
| |
| end ScalarConductivity |
| |
| end |
| |
| end OAI |
| |
