| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.CrossingEstimates | |
| import OAI.Analysis.Conductivity.Variational.ScaledSmoothSeries | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Real Set Filter Topology MeasureTheory | |
| lemma crossingResidualFirst_smooth (L : ℝ) : | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (crossingResidualFirst L) := | |
| (smooth_deriv_infty (smooth_deriv_infty (crossingFirst_smooth L))).sub | |
| (contDiff_const.mul (crossingFirst_smooth L)) | |
| lemma crossingResidualSecond_smooth (L : ℝ) : | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (crossingResidualSecond L) := | |
| (smooth_deriv_infty (smooth_deriv_infty (crossingSecond_smooth L))).sub (crossingSecond_smooth L) | |
| lemma crossingResidualFirst_zero_right {L z : ℝ} (hL : 0 < L) (hz : 3*L/4 < z) : | |
| crossingResidualFirst L z = 0 := by | |
| have hu : 1 < 4*z/L-2 := by | |
| have : 3 < 4*z/L := (lt_div_iff₀ hL).mpr (by linarith) | |
| linarith | |
| rw [crossingResidualFirst_formula,crossingOff_second] | |
| simp only [deriv_crossingOff,(smoothTransition_flat_right hu).1,(smoothTransition_flat_right hu).2] | |
| ring | |
| lemma crossingResidualSecond_zero_left {L z : ℝ} (hL : 0 < L) (hz : z < -3*L/4) : | |
| crossingResidualSecond L z = 0 := by | |
| have hu : 4*z/L+3 < 0 := by | |
| have : 4*z/L < -3 := (div_lt_iff₀ hL).mpr (by linarith) | |
| linarith | |
| rw [crossingResidualSecond_formula,crossingOn_second] | |
| simp only [deriv_crossingOn,(smoothTransition_flat_left hu).1,(smoothTransition_flat_left hu).2] | |
| ring | |
| lemma crossingResidualFirst_compact {L : ℝ} (hL : 0 < L) : HasCompactSupport (crossingResidualFirst L) := by | |
| apply HasCompactSupport.intro (K := Icc (L/2) (3*L/4)) isCompact_Icc | |
| intro z hz | |
| simp only [mem_Icc,not_and_or,not_le] at hz | |
| exact hz.elim (crossingResidualFirst_zero hL) (crossingResidualFirst_zero_right hL) | |
| lemma crossingResidualSecond_compact {L : ℝ} (hL : 0 < L) : HasCompactSupport (crossingResidualSecond L) := by | |
| apply HasCompactSupport.intro (K := Icc (-3*L/4) (-L/2)) isCompact_Icc | |
| intro z hz | |
| simp only [mem_Icc,not_and_or,not_le] at hz | |
| exact hz.elim (crossingResidualSecond_zero_left hL) (crossingResidualSecond_zero hL) | |
| def cascadeBaseStream (L K : ℝ) (a : Fin 3) (x : Coord3) : ℝ := | |
| deriv (cascadeProfile L K) (x 0)*sin (x a) | |
| def cascadeCrossStream (L K : ℝ) (a b : Fin 3) (x : Coord3) : ℝ := | |
| connectorProfile K (K+2)*exp (-3*L)* | |
| (crossingResidualFirst L (x 0-(K+2+L))/4*sin (4*x b)*sin (x a) - | |
| crossingResidualSecond L (x 0-(K+2+L))/2*cos (x a)*sin (x a)*sin (2*x b)) | |
| lemma cascadeBaseStream_bounded {L K : ℝ} (hL : 0 < L) (a : Fin 3) : | |
| BoundedSmooth (cascadeBaseStream L K a) := by | |
| let P : Fin 3 → Coord3 →L[ℝ] ℝ := fun i => ContinuousLinearMap.proj i | |
| exact ((BoundedSmooth.of_compact (smooth_deriv_infty (cascadeProfile_smooth L K)) | |
| (cascadeProfile_compact hL).deriv).comp_linear (P 0)).mul | |
| (sin_boundedSmooth.comp_linear (P a)) | |
| lemma cascadeCrossStream_bounded {L K : ℝ} (hL : 0 < L) (a b : Fin 3) : | |
| BoundedSmooth (cascadeCrossStream L K a b) := by | |
| let P : Fin 3 → Coord3 →L[ℝ] ℝ := fun i => ContinuousLinearMap.proj i | |
| have hf := (((BoundedSmooth.of_compact (crossingResidualFirst_smooth L) | |
| (crossingResidualFirst_compact hL)).translate (K+2+L)).comp_linear (P 0)).smul (1/4) | |
| have hg := (((BoundedSmooth.of_compact (crossingResidualSecond_smooth L) | |
| (crossingResidualSecond_compact hL)).translate (K+2+L)).comp_linear (P 0)).smul (1/2) | |
| have hs₁ := sin_boundedSmooth.comp_linear (4 • P b) | |
| have hs₂ := sin_boundedSmooth.comp_linear (2 • P b) | |
| have hsa := sin_boundedSmooth.comp_linear (P a) | |
| have hca := cos_boundedSmooth.comp_linear (P a) | |
| convert (((hf.mul hs₁).mul hsa).sub (((hg.mul hca).mul hsa).mul hs₂)).smul | |
| (connectorProfile K (K+2)*exp (-3*L)) using 1 | |
| funext x | |
| simp only [cascadeCrossStream,Function.comp_def,smul_apply,P, | |
| ContinuousLinearMap.proj_apply] | |
| ring_nf | |
| end ScalarConductivity | |
| end | |
| end OAI | |