| 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 | |