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59d1c80 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
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