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785c727 | 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 | import OAI.Geometry.CylinderCovering.Physical.CapNormalizedDefinitions
import OAI.Geometry.CylinderCovering.Physical.CapFinite
import OAI.Geometry.CylinderCovering.Physical.Minimum
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace RadialPhysical
open CylinderCovering
lemma physicalHalfReference_eq (centered : Bool) (b : ℝ) :
physicalHalfReference centered b = ((if centered then 4 else 1)*b*height b)/2 := by
cases centered <;> simp only [physicalHalfReference, Bool.false_eq_true, ite_false, ite_true]
· ring
· ring
private lemma actualRelative_le (centered : Bool) {b : ℝ} (hb : 0 < b)
(eta : ℝ) (T : ℝ → ℝ) (N : ℕ) (tagU tagL : Fin N → ℝ)
(hU : ∀ i, ‖upperPhysicalAxis b eta (tagU i)‖ = 1)
(hL : ∀ i, ‖lowerPhysicalAxis b eta (tagL i)‖ = 1) :
physicalActualRelativeCost centered b eta T N tagU tagL ≤
physicalActualCost centered b eta T N tagU tagL /
((if centered then 4 else 1)*b*height b) := by
unfold physicalActualRelativeCost physicalActualCost
rw [add_div, Finset.sum_div, Finset.sum_div]
apply add_le_add
· apply Finset.sum_le_sum
intro i _
exact div_le_div_of_nonneg_left ENNReal.toReal_nonneg
(PhysicalMinimum.reference_pos centered hb)
(PhysicalMinimum.shadow_lower centered hb _ (hU i))
· apply Finset.sum_le_sum
intro i _
exact div_le_div_of_nonneg_left ENNReal.toReal_nonneg
(PhysicalMinimum.reference_pos centered hb)
(PhysicalMinimum.shadow_lower centered hb _ (hL i))
theorem PhysicalFamilyConclusion.eventually_both_half_bounds
{centered : Bool} {b eta : ℝ} (hb : 0 < b) {T : ℝ → ℝ}
{tagU tagL : (N : ℕ) → Fin N → ℝ}
{C : (N : ℕ) → Fin (2*N) → TriangularCovering.TriCylinder}
(hC : PhysicalFamilyConclusion centered b eta T tagU tagL C)
(hcost : scaledCost b eta T < physicalHalfReference centered b) :
∀ᶠ N in atTop, 0 < N ∧
PhysicalBothHalfBounds centered b eta T N (tagU N) (tagL N) (C N) := by
filter_upwards [hC.eventually_strict hcost] with N hN
have htriangle := PhysicalMinimum.both_half_bounds centered hb (C N)
(by simpa only [physicalHalfReference_eq] using hN.2.1)
have hactual : physicalActualCost centered b eta T N (tagU N) (tagL N) <
minimumShadowArea (modelBody centered b)/2 := by
change physicalActualCost centered b eta T N (tagU N) (tagL N) <
Similarity.minimumArea (modelBody centered b)/2
rw [PhysicalMinimum.minimum_modelBody centered hb]
simpa only [physicalHalfReference_eq] using hN.2.2
have hU : ∀ i, ‖upperPhysicalAxis b eta (tagU N i)‖ = 1 := by
intro i
rw [← (hC.local_properties N hN.1 i).1]
exact (C N (RadialPairedSweep.upperIndex i)).unit_axis
have hL : ∀ i, ‖lowerPhysicalAxis b eta (tagL N i)‖ = 1 := by
intro i
rw [← (hC.local_properties N hN.1 i).2.1]
exact (C N (RadialPairedSweep.lowerIndex i)).unit_axis
have hrelative := actualRelative_le centered hb eta T N (tagU N) (tagL N) hU hL
have hratio : physicalActualCost centered b eta T N (tagU N) (tagL N) /
((if centered then 4 else 1)*b*height b) < 1/2 := by
apply (div_lt_iff₀ (PhysicalMinimum.reference_pos centered hb)).2
have h := hN.2.2
rw [physicalHalfReference_eq] at h
linarith
exact ⟨hN.1, htriangle.1, htriangle.2, hactual, hrelative.trans_lt hratio⟩
theorem physical_cap_normalized : PhysicalCapNormalizedStatement := by
intro centered square b beta hb hbeta hbetamax
filter_upwards [eventually_physicalModelCap_covers centered square hb hbeta,
eventually_physicalModelCap_strict centered square hb hbeta hbetamax] with eta hcover hcost
intro tagU tagL htagU htagL
obtain ⟨C, hC⟩ := physical_family_of_sweep_cover centered b eta hb
(physicalModelCap centered square b eta beta)
(physicalModelCap_continuous centered square b eta beta).continuousOn
(fun s _ => physicalModelCap_nonneg centered square b eta beta s hbeta.le)
hcover tagU tagL htagU htagL
exact ⟨C, hC, hC.eventually_both_half_bounds hb hcost⟩
end RadialPhysical
end
end OAI
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