| import OAI.Geometry.IsometricImmersion.Pulses.ActualForcingCoefficient |
| import OAI.Geometry.IsometricImmersion.Energy.MultiplierIntegral |
| import Mathlib.Analysis.Convex.Topology |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set |
| open scoped ContDiff Topology Matrix |
|
|
| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
|
|
| theorem closedRectangle_eq_Icc (l r b t : ℝ) : |
| closedRectangle l r b t = Icc (![l,b]) (![r,t]) := by |
| ext p |
| change (p 0 ∈ Icc l r ∧ p 1 ∈ Icc b t) ↔ |
| (![l,b] ≤ p ∧ p ≤ ![r,t]) |
| constructor |
| · intro hp |
| constructor <;> intro i <;> fin_cases i |
| · exact hp.1.1 |
| · exact hp.2.1 |
| · exact hp.1.2 |
| · exact hp.2.2 |
| · intro hp |
| exact ⟨⟨hp.1 0,hp.2 0⟩,⟨hp.1 1,hp.2 1⟩⟩ |
|
|
| theorem closedRectangle_isPreconnected (l r b t : ℝ) : |
| IsPreconnected (closedRectangle l r b t) := by |
| rw [closedRectangle_eq_Icc] |
| exact (convex_Icc (![l,b] : Coord) (![r,t])).isPreconnected |
|
|
| def forcingFloor (e D C : ℝ) : ℝ := e/(2*max D 1*max C 1) |
|
|
| theorem forcingFloor_pos {e : ℝ} (he : 0 < e) (D C : ℝ) : 0 < forcingFloor e D C := by |
| unfold forcingFloor |
| have hD : 0 < max D 1 := zero_lt_one.trans_le (le_max_right _ _) |
| have hC : 0 < max C 1 := zero_lt_one.trans_le (le_max_right _ _) |
| positivity |
|
|
| theorem forcingCoefficient_floor_of_reference_margins |
| {g : MetricField} {z : Coord → ℝ} {U : Set Coord} |
| (hg : SmoothPositiveOn g U) {p : Coord} (hp : p ∈ U) |
| {e c D C : ℝ} (he : 0 < e) (hc : 0 < c) |
| (hE : e ≤ heightEnergy g z p) (hdet : (g p).det ≤ D) |
| (hxx : c ≤ |covHessian g z p 0 0|) (hxxup : |covHessian g z p 0 0| ≤ C) : |
| forcingFloor e D C ≤ |forcingCoefficient g z p| := by |
| exact (forcingCoefficient_bounds hg hp he (hg.2 p hp).det_pos hc hE le_rfl le_rfl |
| (hdet.trans (le_max_left _ _)) hxx (hxxup.trans (le_max_left _ _))).2.1 |
|
|
| theorem actual_forcingCoefficient_signed_floor |
| {g : MetricField} {z : Coord → ℝ} {U S : Set Coord} |
| (hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) |
| (hU : IsOpen U) (hSU : S ⊆ U) (hS : IsPreconnected S) |
| {p0 : Coord} (hp0 : p0 ∈ S) {e c D C : ℝ} (he : 0 < e) (hc : 0 < c) |
| (hE : ∀ p ∈ S, e ≤ heightEnergy g z p) |
| (hdet : ∀ p ∈ S, (g p).det ≤ D) |
| (hxx : ∀ p ∈ S, c ≤ |covHessian g z p 0 0|) |
| (hxxup : ∀ p ∈ S, |covHessian g z p 0 0| ≤ C) : |
| ∃ sigma : ℝ, |sigma| = 1 ∧ ∀ p ∈ S, |
| sigma*forcingCoefficient g z p = |forcingCoefficient g z p| ∧ |
| forcingFloor e D C ≤ sigma*forcingCoefficient g z p := by |
| have hxxne : ∀ p ∈ S, covHessian g z p 0 0 ≠ 0 := by |
| intro p hp hz0 |
| have hh := hxx p hp |
| rw [hz0,abs_zero] at hh |
| linarith |
| have hsign := forcingCoefficient_one_sign hg hz hU hSU hS |
| (fun p hp => he.trans_le (hE p hp)) hxxne |
| have hfloor (p : Coord) (hp : p ∈ S) := |
| forcingCoefficient_floor_of_reference_margins hg (hSU hp) he hc |
| (hE p hp) (hdet p hp) (hxx p hp) (hxxup p hp) |
| have hanchor : forcingCoefficient g z p0 ≠ 0 := |
| abs_pos.mp ((forcingFloor_pos he D C).trans_le (hfloor p0 hp0)) |
| rcases lt_or_gt_of_ne hanchor with hneg | hpos |
| · refine ⟨-1,by norm_num,?_⟩ |
| intro p hp |
| have hpneg : forcingCoefficient g z p < 0 := by |
| by_contra hn |
| have hn0 : 0 ≤ forcingCoefficient g z p := le_of_not_gt hn |
| have hnonpos := mul_nonpos_of_nonpos_of_nonneg hneg.le hn0 |
| linarith [hsign p0 hp0 p hp] |
| constructor |
| · rw [abs_of_neg hpneg] |
| ring |
| · simpa only [neg_one_mul,abs_of_neg hpneg] using hfloor p hp |
| · refine ⟨1,by norm_num,?_⟩ |
| intro p hp |
| have hppos : 0 < forcingCoefficient g z p := by |
| by_contra hn |
| have hn0 : forcingCoefficient g z p ≤ 0 := le_of_not_gt hn |
| have hnonpos := mul_nonpos_of_nonneg_of_nonpos hpos.le hn0 |
| linarith [hsign p0 hp0 p hp] |
| constructor |
| · rw [one_mul,abs_of_pos hppos] |
| · simpa only [one_mul,abs_of_pos hppos] using hfloor p hp |
|
|
| end SmoothLocal.Pulse |
|
|
| end |
|
|
| end OAI |
|
|
